Download Eigenstate Phase Transitions

Eigenstate Phase Transitions
Bo Zhao
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Adviser: Professor David A. Huse
September 2015
c Copyright by Bo Zhao, 2015.
All rights reserved.
Abstract
Phase transitions are one of the most exciting physical phenomena ever discovered.
The understanding of phase transitions has long been of interest. Recently eigenstate
phase transitions have been discovered and studied; they are drastically different
from traditional thermal phase transitions. In eigenstate phase transitions, a sharp
change is exhibited in properties of the many-body eigenstates of the Hamiltonian of
a quantum system, but not the thermal equilibrium properties of the same system.
In this thesis, we study two different types of eigenstate phase transitions. The first
is the eigenstate phase transition within the ferromagnetic phase of an infinite-range
spin model. By studying the interplay of the eigenstate thermalization hypothesis
and Ising symmetry breaking, we find two eigenstate phase transitions within the
ferromagnetic phase: In the lowest-temperature phase the magnetization can macroscopically oscillate by quantum tunneling between up and down. The relaxation of
the magnetization is always overdamped in the remainder of the ferromagnetic phase,
which is further divided into phases where the system thermally activates itself over
the barrier between the up and down states, and where it quantum tunnels. The
second is the many-body localization phase transition. The eigenstates on one side
of the transition obey the eigenstate thermalization hypothesis; the eigenstates on
the other side are many-body localized, and thus thermal equilibrium need not be
achieved for an initial state even after evolving for an arbitrary long time. We study
this many-body localization phase transition in the strong disorder renormalization
group framework. After setting up a set of coarse-graining rules for a general one
dimensional chain, we get a simple “toy model” and obtain an almost purely analytical solution to the infinite-randomness critical fixed point renormalization group
equation. We also get an estimate of the correlation length critical exponent ν ≈ 2.5.
iii
Acknowledgements
No one can achieve a PhD without others’ help. I am very thankful to have this
opportunity of pursuing the PhD in physics department in Princeton. First of all,
the honor should be given to my advisor, Professor David Huse. It is he who led me
to the palace of physics. Before entering graduate school in Princeton, I was just a
student who had some ability in learning in courses and doing homework. One of the
most important lessons David has shown me is how to tackle a research topic where
there might be no preexisting theory. By doing research with David during these PhD
years , I have experienced both exciting and painful moments. But David has taught
me, through his own attitude, that when doing research, one should keep cautious,
positive and energetic. What typically happens during research is that one keeps
trying and failing until stages where a new method, model or direction are found. I
am very grateful to learn from David, in terms of both his strong physical intuition
and his encouraging personality.
Secondly, I would like to thank my colleagues who have helped me in my academic
life in my PhD years. In David’s group, I have had many meaningful discussions about
physics with Hyungwon Kim and Liangsheng Zhang. Both of them have helped me a
lot in digesting the background physics in our research projects. I would also like to
thank my colleagues, Aris Alexandradinata and Vedika Khemani who have also given
me lots of advice about many academic aspects.
Thirdly, PhD’s life is not all about research. I enjoyed these years in Jadwin with
all of my friends, including, but not limited to, Chaney Lin, Akshay Kumar, Bin Xu,
Yu Shen and many more graduate students who often appeared on the 4th floor in
Jadwin. Without them, my life in Jadwin would be much more lonely and much less
interesting. There are countless many friends who I enjoyed staying with, so I also
would like to thank all of them.
iv
Last, but definitely not the least, I must give special thanks to my family. The
most selfless love and support were consistently given by my parents. Without them,
this thesis would have never been completed.
v
To my parents.
vi
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
1 Introduction
1
1.1
Quantum Thermalization . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Eigenstate Thermalization Hypothesis . . . . . . . . . . . . . . . . .
5
1.3
Many-body Localization . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.4
Eigenstate Phase Transition . . . . . . . . . . . . . . . . . . . . . . .
10
1.5
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2 Three ‘Species’ of Schrödinger Cat States in an Infinite-range Spin
Model
12
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2
Infinite Range Transverse Field Ising Model . . . . . . . . . . . . . .
16
2.2.1
Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2.2
Energy Level Degeneracy . . . . . . . . . . . . . . . . . . . . .
17
2.2.3
Discrete WKB Method . . . . . . . . . . . . . . . . . . . . . .
18
2.3
Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.4
Eigenstate Thermalization . . . . . . . . . . . . . . . . . . . . . . . .
24
vii
2.5
2.6
Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.5.1
Thermal Activation and Quantum Tunneling . . . . . . . . . .
26
2.5.2
Paired States and Unpaired States . . . . . . . . . . . . . . .
31
Analytical Calculation of the Thermal Activation and Quantum Tunneling Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7
32
Asymptotic Behaviors of Both Transitions Near Both QCP and
u = 0 , Γ = 0 Point . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.7.1
Near QCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.7.2
Near u = 0 , Γ = 0 Point . . . . . . . . . . . . . . . . . . . . .
43
2.8
Numerical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.9
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
3 Strong Disorder Renormalization Group Approach to Many-body
Localization Transition
54
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
3.2
RG Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.2.1
RG Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.2.2
RG Flow of Probability Distributions . . . . . . . . . . . . . .
67
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.3.1
Fixed Point Distribution . . . . . . . . . . . . . . . . . . . . .
68
3.3.2
Critical Exponent . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.3.3
Numerical Evidence . . . . . . . . . . . . . . . . . . . . . . . .
91
Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . .
95
3.3
3.4
A Mathematical Proof of the Degeneracy Formula
B Estimation of the Magnitude of Disorder
97
103
B.1 Lower Bound of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . 104
B.2 Upper Bound of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . 107
viii
C Divergence of First Order Derivative of γ(s̄) at s̄ = s̄min
109
D Discussion on Convexity of Function γ(s̄)
112
Bibliography
116
ix
List of Tables
1.1
A list of some properties of the many-body localized phase contrasted
with properties of the thermal phases. Table from [19]. . . . . . . . .
x
9
List of Figures
2.1
The phase diagram of our model. u is the energy per spin and Γ is the
transverse field. The ground state (zero temperature) is indicated by
the black (solid) line, with the quantum critical point (QCP) indicated. The green (dot-dashed) line is the thermodynamic phase transition
(PT) between the paramagnetic and ferromagnetic (F) phases. There
are two dynamical (eigenstate) phase transitions within the ferromagnetic phases, indicated by the blue (dashed) lines. See the text for
discussions of the sharp distinctions between these three phases F1, F2
and F3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2
Two “potential-energy curves” in our model. . . . . . . . . . . . . . .
20
2.3
Constant energy (E) line described by equation (2.25) by setting s̄y = 0
or just by equation (2.22). It determines the WKB turning point xt
when the total spin density s̄ is fixed. . . . . . . . . . . . . . . . . . .
27
2.4
A sketch of the entropy Σ(s̄) and the tunneling rate γ(s̄).
29
2.5
A sketch of the difference between thermal activation and quantum
. . . . . .
tunneling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
30
2.6
The level-spacing statistics using 100 realizations of H at N = 15 in
phase F1 within the even sector. δ< /δ> is the ratio between the smaller
level spacing δ< to the larger level spacing δ> for three consecutive
eigenenergies in the even sector. f is the relative frequency for each
bin in this histogram. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7
46
Averages of log (D) in phases F1 and F3, respectively, where D is the
‘eigenstate distance’ defined in the text.. The energy density range we
used in F1 is from the first excited state in each sector up to uc − 0.02
where uc is the energy density at the phase boundary between F1 and
F2, whereas in F3 we used the phase’s full energy density range. N
is the total number of spins varying from 8 to 15. The exponential
decrease of D with increasing N indicates thermalization. The error
bars come from averaging over 100 realizations. . . . . . . . . . . . .
2.8
48
The mean ᾱn and the standard deviation ∆αn of the quantity αn defined in Eq. (2.81). The number of realizations is 1600 for N = 11
(blue dash-dotted lines) and 100 for N = 15 (red dashed lines). The
green (solid) line gives the theoretical quantity α(u, Γ) defined in Eq.
(2.32) for the system size N → ∞. . . . . . . . . . . . . . . . . . . .
3.1
49
A sketch of typical RG moves. For example, (a) is to fuse two adjacent
thermal (T) blocks into one thermal block, called “TT move”; all others
are similar.
3.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
Q∗ (η) up to η = 20 using composite Trapezoidal rule. (a) shows the
curve and (b) demonstrates it in a semi-log plot with a linear regression
fit. (c) plots the cumulative area under the curve. . . . . . . . . . . .
3.3
80
The eigenfunction f− (η) corresponding to eigenvalue 0.3995: (a) Linear
scale (b) Absolute value on log scale. . . . . . . . . . . . . . . . . . .
xii
90
3.4
The eigenfunction f− (η) using either the numerical integration directly
or the diagonalization. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5
91
The cumulative distribution function (Cdf) for both the T-blocks and
I-blocks as the total number of blocks N decreases from 107 to 103 and
the cutoff Λ grows. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
93
The difference of cumulative distribution functions (Cdf) between the
T-blocks and I-blocks as the total number of blocks N decreases from
107 to 103 and the cutoff Λ grows. . . . . . . . . . . . . . . . . . . . .
94
D.1 Comparison between numerical and analytical result of the thermal activation and quantum tunneling transition line; the dots are numerical
points and the line underneath the dots is the analytical result given
by equation (2.47). They fit perfectly well. . . . . . . . . . . . . . . . 113
D.2 Numerical results of
∂γ
∂s̄
as a function of s̄ by sowing different pairs of
Γ and u randomly within the entire ferromagnetic phase region. . . . 114
xiii
Chapter 1
Introduction
Our understanding of the laws of nature has taken giant leaps in the past several
centuries, beginning from the time of Galileo. For mechanics, Newton first postulated
the classical equations of motion that every macroscopic object should obey, these
equations, we now call “Newton’s Laws”. These laws solve nearly every mechanical phenomenon we observe in daily life! However, when we would like to further
understand the microscopic structures of thermal behaviors, we must consider many
interacting degrees of freedom, too many to explicitly solve for the dynamics, say
from Newton’s Laws or from Schrödinger equation for a quantum system. This gave
birth to the area of statistical mechanics.
Classical statistical mechanics is very successful at explaining equilibrium thermal behavior in daily life, for example the concept of temperature. Thus, at least
phenomenologically speaking, we know the solution based on the essence of classical
equilibrium statistical mechanics: the ensemble theory. But the foundation of the
ensemble theory is to assume a probabilistic nature of so-called microstates. If we
want to get the equilibrium probability distributions in ensemble theory purely from
Newton’s Laws, either we assume that the initial state is itself an ensemble, or we get
the ensemble by including all long time states of the system (ergodicity). Fortunately,
1
people have successfully discovered a more fundamental theory, quantum mechanics,
which is itself a probabilistic theory. Quantum statistical mechanics is the product
of combining quantum mechanics and statistical mechanics. In quantum statistical
mechanics, ensemble theory naturally arises from the probabilistic nature of quantum
mechanics, so the equilibrium ensemble can emerge at any specific long time even if
the initial state is a pure state.
In quantum statistical mechanics, one of the most fundamental questions is
whether a closed quantum system equilibrates. This question is by no means as easy
as it appears. One of the difficulties is the apparent time reversal symmetry breaking
which arises from low entropy initial states. However, the quantum mechanical
formalism obeys the time reversal symmetry, through the unitary time evolution of
a quantum state. Thus, at least, we need to formulate a more rigorous definition of
equilibration called quantum thermalization.
In the remaining sections of this chapter, I will briefly review the concepts of
quantum statistical mechanics, including, but not limited to, quantum thermalization,
eigenstate thermalization hypothesis and many-body localization.
1.1
Quantum Thermalization
Consider a closed quantum system with a time independent Hamiltonian H. In
quantum mechanics, the system is described by a ket vector |ψi in the Hilbert space.
But in quantum statistical mechanics, |ψi is not enough. It only covers the pure
states, a subset of the quantum states described by the probability operator (also
known as density matrix) ρ. As a probability operator, ρ needs to satisfy the following
constraints:
ρ† = ρ ,
tr ρ = 1 ,
2
(1.1)
and also, the spectrum of ρ is in the interval [0, 1].
After describing the quantum state of the system, what we need next is the law
of time evolution. If we choose the Schrödinger picture, the time evolution of ρ(t) as
a function of time t is given by the unitary operator e−iHt/~ :
ρ(t) = e−iHt/~ ρ(0)eiHt/~ ,
(1.2)
or the differential equation form:
i~
d
ρ(t) = [H, ρ(t)] .
dt
(1.3)
By establishing both the quantum state and the rule of time evolution, we are
able to formulate a definition for quantum thermalization. Roughly speaking, as
mentioned earlier, thermalization means that the system goes to equilibrium in the
limit of a long time evolution. But if we consider the full probability operator of an
entire closed quantum system, the time evolution is unitary; it will never reach equilibrium, because such a state is, by definition, fully determined by initial conditions.
In other words, the unitary time evolution “remembers” all information about the
initial state. Instead, we consider thermalization in terms of subsystems. And the
idea is to use the rest of the full system to serve as a heat bath to thermalize the
selected subsystem.
When we are talking about thermalization, we need to send our system to the
thermodynamic limit where the total degrees of freedom N goes to infinity. In general, the full system can have several extensive conserved quantities, for example, total
energy, particle number, total spin, etc. But to keep the discussion simple, we assume
the system only has one extensive conserved quantity, the total energy. So if the system goes to thermal equilibrium, the equilibrium state has only one thermodynamic
parameter, the temperature T . In order to take the thermodynamic limit, we need
3
a sequence of both the initial states ρN (t = 0) and the Hamiltonian HN labeled by
N . In addition, we need to restrict to initial states ρN (0) where the uncertainty in
total energy is subextensive, so T has zero uncertainty for N → ∞. Now we consider
a subsystem S. The degrees of freedom in the full system F can be decomposed
into the degrees of freedom in the subsystem S and the remaining degrees of freedom
—called the “bath”— B:
F =S⊕B,
(1.4)
for the sets of degrees of freedom. (It is a tensor product in Hilbert space.) Then
the thermodynamic limit corresponds to taking both F and B to infinity but keeping
S finite. We consider the probability operator ρS (t) on S (known as the “reduced”
density matrix)
ρS (t) = trB ρN (t) .
(1.5)
Thermalization on the subsystem S means that by sending the system to the thermodynamic limit, the probability operator of the subsystem S goes to its thermal
equilibrium value indicated by the temperature T :
lim ρS (t) = ρeq
S (T ) ,
N →∞
t→∞
(1.6)
where
eq
ρeq
S (T ) = lim trB ρN (T )
N →∞
4
(1.7)
and
ρeq
N (T ) =
1 −βH
e
.
Z
(1.8)
The last expression is the standard quantum statistical mechanical formula describing
the equilibrium state in the canonical ensemble where Z is the partition function
Z = trF e−βH and β is the temperature parameter β = 1/kB T .
In other words, quantum thermalization of a subsystem S means that the subsystem behaves as if the full system is exactly at the thermal equilibrium state. And
the rest of the full system serves as a heat bath. And if for all subsystems S, at long
time t the above thermalization condition is true for the same temperature T , and
for all initial states corresponding to that T , we say the system thermalizes for this
temperature. [10, 26, 27, 22, 19]
1.2
Eigenstate Thermalization Hypothesis
By defining quantum thermalization, we can further study the destiny of a given
closed quantum system H. From everyday thermal phenomena, one would expect
that if a system can equilibrate, then the system must thermalize from any initial
conditions. Then, it follows that all the exact many-body eigenstates of the full
system must be thermal because the probability operators, ρeigen = |ψihψ|, do not
change with time, where |ψi satisfies H|ψi = E|ψi. This motivates the following
hypothesis called the Eigenstate Thermalization Hypothesis (ETH):[10, 26, 27, 22]
Every single eigenstate thermalizes.
By saying eigenstate, we are making no approximations. We mean the exact manybody eigenstate of the full system. By saying thermalize, we mean all subsystems
thermalize for the temperature determined by the eigenenergy of the eigenstate.
5
ETH is a hypothesis. It is not true for systems that are many-body localized
(MBL) as I will discuss in the next section. Also, it is not true for integrable systems
which contain infinite number of local conservation laws. By saying local, we do
not necessarily mean local in real space, it can be local in for example momentum
space. But it should not be global because it is trivial that for any given full system
eigenstate |ni, the projection operator |nihn| is a conserved operator. And so is any
weighted sum over them. Thus there are infinite number of “global” conservation
laws for any given system. But these projection operators are global and presently
inaccessible to measurement.
ETH is a hypothesis in another sense, that even for those systems where ETH
appears to be true, for example everyday thermal phenomena, it is extremely hard to
prove, even numerically, because one needs to test the exact many-body eigenstates
which requires the exact diagonalization of the full system. In practice, this approach
is limited to N only up to about 20, due to the exponential growth of the dimension
of H. But that has not deferred people from finding numerical evidence for ETH.
Indeed, there has been plenty of research tackling the numerical test of ETH and
there is strong support that ETH is true for a large number of systems, for example
Rigol, Dunjko and Olshanii [22]; Pal and Huse [20]; Kim and Huse [16]; Beugeling,
Moessner, and Haque [6] and many more.
Thus it is worthwhile to mention some consequences based on ETH. One of them
is for the diagonal ensemble. We write the probability operator in the basis of the
exact many-body eigenstates of the full Hamiltonian:
ρ=
X
|miρmn hn| ,
(1.9)
m,n
then based on the time evolution of the probability operator, we have that the diagonal
matrix element ρnn stays constant and the off-diagonal matrix element ρmn , m 6= n,
6
evolves by multiplying a “simple” phase factor:
ρmn (t) = e−i(Em −En )t/~ ρmn (0) ,
m 6= n ,
(1.10)
where Em and En are the eigenenergies of the corresponding eigenstates |mi and |ni:
H|ni = En |ni .
(1.11)
When t → ∞, the phase factors in the off-diagonal terms become essentially random
and when they contribute to the probability operator of any local subsystem, they
effectively go to their mean value 0. This procedure is called dephasing which causes
the full system to thermalize when ETH is true. After dephasing, the probability
operator becomes diagonal:
ρD =
X
|niρnn hn| ,
(1.12)
n
which is called the diagonal ensemble. It neglects all the off-diagonal terms, and the
diagonal terms are set by the initial condition. As being two special cases of the
diagonal ensemble, we also have the canonical ensemble as appears in the standard
equilibrium statistical mechanics
ρeq =
1 −βH
1X
e
=
|nie−βEn hn|
Z
Z n
(1.13)
and the single many-body eigenstate “ensemble”
ρ(n) = |nihn|
(1.14)
as a limit of the standard microcanonical ensemble by sending the available energy
window ∆E → 0. When ETH is true, all different forms of the diagonal ensembles
7
including the two special cases above are equivalent for small subsystems, provided
the uncertainty in the total energy remains less than extensive.
1.3
Many-body Localization
As mentioned in the previous section, there exists a large class of systems, localized
systems, which do not obey ETH. The idea of localization first came from Anderson [1]
in 1958. In this section, I will focus on one subset of the localized systems: interacting
many-body localized (MBL) systems.
To briefly demonstrate the idea of many-body localization, I use the model of Pal
and Huse [20]. It is a one dimensional spin chain with spin-1/2 with Hamiltonian:
H=
X
hi σiz +
i
X
J~σi · ~σi+1 ,
(1.15)
i
where σi are the Pauli operators for the spin-1/2 at ‘site’ i. The onsite magnetic
fields hi are static random variables with a probability distribution that is uniform in
[−h, h].
At J = 0, the many-body eigenstates are just the product states which are the
basis of the σ z representation and the system is fully localized. For nonzero J, if
we assume J h, we can apply perturbation theory with zeroth order J = 0 to
construct the many-body eigenstates [5]. Because the local level spacing produced
by the J = 0 product states are comparable to h, it is in general much larger than
the interaction J. It implies that the eigenstates are very weakly hybridized. This
argument further implies that there is no DC spin transport or energy transport
and so the quantum thermalization is violated [5]. In addition to this perturbative
argument, Pal and Huse [20] demonstrated numerical evidence for the generic nonperturbative case by doing the exact diagonalization up to 16 spins. They showed
8
that when h/J > hc /J ≈ 3.5, the system fails to behave thermal and goes into the
localized phase.
There are several properties that differ between thermalization and localization.
Some are summarized in Table 1.1 cited from [19]. Note that these differences are all
in dynamical properties or properties of the exact eigenstates. In fact, the two phases
do not differ at all in their static thermal equilibrium properties.
Thermal Phase
Many-body Localized Phase
Memory of initial conditions ‘hid- Some memory of local initial conden’ in global operators at long ditions preserved in local observtimes
ables at long times
ETH true
ETH false
May have non-zero DC conduc- Zero DC conductivity
tivity
Continuous local spectrum
Discrete local spectrum
Eigenstates with volume-law entanglement
Eigenstates with area-law entanglement
Power-law in time spreading of Logarithmic in time spreading of
entanglement from non-entangled entanglement from non-entangled
initial condition
initial condition
Dephasing and dissipation
Dephasing but no dissipation
Table 1.1: A list of some properties of the many-body localized phase contrasted with
properties of the thermal phases. Table from [19].
I close this section by mentioning the definition of temperature in the MBL phase.
As we know in the standard statistical mechanics, temperature is a crucial concept
in the equilibrium state. But in the MBL phase, the system fails to obey the ETH
hence the system cannot reach the equilibrium state. So temperature is ill-defined in
the MBL phase. One way to define temperature is to say that it is the corresponding
temperature if the system could thermalize. This implies a brand new type of phase
transition called the eigenstate phase transition and it will be discussed in the next
section.
9
1.4
Eigenstate Phase Transition
The eigenstate phase transition, as formulated e.g. in [13], is a brand new type
of phase transition compared with traditional thermal phase transition. In thermal
phase transition, we see a sharp change in terms of the thermal equilibrium when we
go from one phase to another. For example, from water to ice, the sharp change can
be observed directly by solving the minimal value of the free energy in the standard
statistical mechanical way and the minimal point of the free energy just corresponds to
the thermal equilibrium. However, in eigenstate phase transitions, the sharp change is
in properties of the many-body eigenstates but not the thermal equilibrium because
thermal equilibrium is an average over lots of eigenstates and this average washes
out the sharp change in terms of single many-body eigenstates. Thus, eigenstate
phase transition is invisible to the equilibrium statistical mechanics. For example,
as I mentioned in the previous section, the phase transition between the thermal
phase and the many-body localized phase is an eigenstate phase transition. In the
MBL phase, the system cannot even evolve to its thermal equilibrium value and some
information is still stored locally for a long time. In fact, if one would use the standard
equilibrium statistical mechanics to solve the MBL system, one could still get some
result and there would be no sharp change between the two phases in this sense.
The traditional equilibrium statistical mechanics however, does not correctly give the
long-time behavior in the MBL phase.
Because the eigenstate phase transition is a brand new type of phase transition,
it by itself has become a topic of research. Also, the usual equilibrium statistical
mechanics may break down when applied to the eigenstate phase transition, so the
properties of the critical point in the eigenstate phase transition, for example the
MBL transition, need to be reestablished and many questions about it are still open
[19]. In addition, eigenstate phase transition is not limited to the MBL transition.
Even in the regime where ETH is true on both sides, there remain eigenstate phase
10
transitions, at least in the limit of long-ranged interaction [32]. These are one of the
main topics in this thesis.
1.5
Thesis Outline
In this thesis, I will discuss two examples of eigenstate phase transitions as mentioned in the previous section. In Chapter 2, we explore a transverse-field Ising
model that exhibits both spontaneous symmetry-breaking and eigenstate thermalization. Within its ferromagnetic phase, the exact eigenstates of the Hamiltonian of any
large but finite-sized system are all Schrödinger cat states: superpositions of states
with ‘up’ and ‘down’ spontaneous magnetization. This model exhibits two eigenstate
phase transitions within its ferromagnetic phase: In the lowest-temperature phase the
magnetization can macroscopically oscillate by quantum tunneling between up and
down. The relaxation of the magnetization is always overdamped in the remainder
of the ferromagnetic phase, which is further divided into phases where the system
thermally activates itself over the barrier between the up and down states, and where
it quantum tunnels.
In Chapter 3, we study the many-body localization transition in one dimensional
systems via the strong disorder renormalization group approach. In this framework,
we impose a set of rules for coarse-graining the system. The result from this set
of rules turns out to be a beautifully simple “toy” renormalization group. We can
almost solve for the critical fixed point distribution analytically. In addition, we also
get an estimate of the correlation length critical exponent ν ≈ 2.5 by both solving the analytical equations numerically and directly simulating the coarse-graining
procedure.
11
Chapter 2
Three ‘Species’ of Schrödinger Cat
States in an Infinite-range Spin
Model
2.1
Introduction
The dynamical properties of isolated many-body quantum systems have long been
of interest, due to their role in the fundamentals of quantum statistical mechanics.
More recently, experiments approximating this ideal of isolated many-body quantum
systems have become feasible in systems of trapped atoms [17, 8] and ions [7], and
as a consequence this topic has received renewed attention. It appears that a broad
class of such systems obey the Eigenstate Thermalization Hypothesis (ETH) [10, 26,
27, 22, 19]. The ETH asserts that each exact many-body eigenstate of a system’s
Hamiltonian is, all by itself, a proper microcanonical ensemble in the thermodynamic
limit, in which any small subsystem is thermally equilibrated, with the remainder of
the system acting as a reservoir. In the present chapter we present some interesting
12
results for an infinite-range transverse-field Ising model that obeys the ETH and also
has spontaneous symmetry-breaking.
Quantum many-body systems with static randomness may fail to obey the ETH
due to many-body Anderson localization stopping thermalization [1, 5, 20]. The
interesting interplay of many-body localization and discrete symmetry-breaking was
recently explored in Refs. [13, 21, 29, 18]; the present chapter instead explores an
example of the interplay of the ETH and Ising symmetry breaking.
We start with the infinite-range transverse-field Ising model:
N
N
X
1 X z z
H0 = −
si sj − Γ
sxi ,
N 1=i<j
i=1
(2.1)
where ~si = (sxi , syi , szi ) = ~σi /2, and ~σi are the Pauli operators for the spin-1/2 at ‘site’
i. We choose to set ~ = kB = 1. This model has been extensively studied recently,
particularly as an example for exploring quantum information issues where it is known
as the ‘Lipkin-Meshkov-Glick model’; see, e.g., [31] and references therein. One can
determine and use many properties of the exact eigenstates of this Hamiltonian,
but there are extensive degeneracies in its spectrum due to its symmetry under any
permutation of the N spins, which give it eigenstates that do not obey the ETH. Thus
we add static random Ising interactions to the Hamiltonian to break the permutation
symmetry and lift all the degeneracies (with probability one), so the full Hamiltonian
of the system we consider is
H = H0 + H1 = H0 +
N
λ X
εij szi szj ,
N p 1=i<j
(2.2)
where 1 ≥ λ > 0, the εij are independent Gaussian random numbers of mean zero
and variance one, and the power p satisfies 1/2 < p < 1. As we argue below,
the eigenstates of this Hamiltonian should obey the ETH, but the randomness is
weak enough so that many of the properties of H0 , such as the thermodynamics, are
13
unchanged and can be used in our analysis. This randomness is too weak to produce
any localization. We have chosen to put the randomness on the interactions, since we
have found in exact diagonalizations that this produces much better thermalization
at numerically accessible system sizes as compared to, e.g., only making the local
transverse fields random.
0
Para
−0.05
−0.1
Ground State
Thermodynamic PT
Eigenstate PT
QCP
F1
u−0.15
F3
F2
−0.2
−0.25
−0.3
−0.35
0
0.1
0.2
0.3
Γ
0.4
0.5
0.6
0.7
Figure 2.1: The phase diagram of our model. u is the energy per spin and Γ is the
transverse field. The ground state (zero temperature) is indicated by the black (solid)
line, with the quantum critical point (QCP) indicated. The green (dot-dashed) line is
the thermodynamic phase transition (PT) between the paramagnetic and ferromagnetic (F) phases. There are two dynamical (eigenstate) phase transitions within the
ferromagnetic phases, indicated by the blue (dashed) lines. See the text for discussions
of the sharp distinctions between these three phases F1, F2 and F3.
We now briefly summarize our results, before deriving and discussing them in
more detail below. The phase diagram of this spin model as a function of the energy
14
per spin u = hHi/N and the transverse field Γ is shown in Fig 2.1. There are the
usual two thermodynamic phases of a ferromagnet: the paramagnetic phase (Para)
at high energy and/or high |Γ|, and the ferromagnetic phase (F) when both |Γ| and
the energy are low enough. In the ferromagnetic phase, for any finite N , we can
ask about the dynamics of the system’s order parameter. There are three regimes
of behavior that are sharply distinguished from one another in the thermodynamic
limit N → ∞: At the highest energies within region F3 of the ferromagnetic phase
the system is a sufficiently large thermal reservoir for itself so that the most probable
path by which it flips from ‘up’ to ‘down’ magnetization under unitary time evolution
is by thermally activating itself over the free energy barrier between the two ordered
states. At lower energies (F1 and F2) the barrier is higher and wider and as a
result the reservoir is inadequate, so the system quantum tunnels through the barrier
when it flips the Ising order parameter. At the lowest energies in region F1 one
can in principle prepare a state that is a linear combination of two Schrödinger cat
eigenstates of H that will coherently oscillate via macroscopic quantum tunneling
between up and down magnetizations. In the intermediate energy regime (F2) the
magnetization dynamics due to quantum tunneling is always overdamped.
Throughout the ferromagnetic phase, the exact eigenstates of H for any finite N
are Schrödinger cat states that are superpositions of up and down magnetized states,
and the properties of these cats differ in the three regimes of the ferromagnetic phase
that are indicated in Fig 2.1. Thus the two phase transitions between these three
dynamically distinct ferromagnetic phases are not only dynamical phase transitions
but also ‘eigenstate phase transitions’ [13], while the equilibrium thermodynamic
properties are perfectly analytic through these two phase transitions.
We consider the infinite-range model not only because this allows a controlled
calculation of this novel physics within the ferromagnetic phases, but also because
finite-range, finite-dimensional models obeying ETH do not show these features. In
15
the latter models the free energy needed to flip the magnetization by making a domain
wall and sweeping it across the system is sub-extensive, while at any nonzero temperature the system is a reservoir of extensive size, so a large system will always flip
via the thermal process without macroscopic quantum tunneling through the energy
barrier; phases F1 and F2 thus do not exist for such models. One can also consider
intermediate cases of transverse-field Ising models with interactions that fall off as a
power of the distance between spins. When this power is small enough, the resulting
free energy barrier to flip the magnetization in the ferromagnetic phase is extensive,
and we thus expect phases F1 and F2 to also occur in those models, although we do
not see a way to simply calculate the locations of the phase boundaries as we can for
the infinite-range model.
2.2
2.2.1
Infinite Range Transverse Field Ising Model
Integrability
First we examine the the unperturbed Hamiltonian H0 , which has the same thermodynamics as our full model H. The Hamiltonian H0 commutes with all permutations of the spins, as does the total spin operator:
~≡
S
N
X
s~i .
(2.3)
i=1
The magnitude S 2 of the total spin squared also commutes with H0 , so we can choose a
set of eigenstates of H0 that are also eigenstates of S 2 . This unperturbed Hamiltonian
16
only depends on the total spin and thus can be written as
N
N
X
1 X z z
H0 = −
s s −Γ
sxi
N 1=i<j i j
i=1
=−
N
N
X
1 X z z
si sj − Γ
sxi
2N i6=j=1
i=1
N
N
N
X
1 X z z X z2 si sj −
si − Γ
sxi
=−
2N i,j=1
i=1
i=1
N
N
X
1 X z 2 1
=−
si + − Γ
sxi
2N i=1
8
i=1
=−
1
1 2
Sz − ΓSx + .
2N
8
(2.4)
The magnitude S of the total spin is of order N and ranges from zero up to S = N/2.
2.2.2
Energy Level Degeneracy
~ 2 = 0 and also H0 , S
~ 2 = H0 , S
~ 2 = H0 , S
~ 2 = · · · = 0,
Manifestly, H0 , S
1
12
123
~123···n ≡ Pn ~si . It means if we couple the angular momentum one by one
where S
i=1
from the very beginning site, all “partial summations” are good quantum numbers.
So the Hamiltonian is block diagonal with each specific angular momentum coupling.
~tot , it means all
Moreover, since H0 only contains the information of total spin S
different ways of the angular momentum coupling have exactly the same spectrum if
the total spin is unchanged. It implies a large amount of degeneracy of each energy
level. The degeneracy is then determined by counting how many different ways of
constructing an angular momentum coupling to have a specific value of total spin S.
Let us denote the degeneracy by fN (S) where N is the total number of sites and S
17
is the total spin value, then it is shown1 that
N
fN (S) = CN2
−S
N
− CN2
−S−1
.
(2.5)
Then the entropy per spin Σ(S/N ) defined by
S
1
)≡
ln fN (S)
N
N
(2.6)
1
S
1
S
1
S
1
S S
) = − ( − ) ln( − ) + ( + ) ln( + ) ,
N
2 N
2 N
2 N
2 N
(2.7)
Σ(
is given by2
Σ(
due to all the different ways one can add together N spin-1/2’s to get total spin S
depends only on the ratio S/N .
For each value of S the spectrum of H0 has (2S + 1) eigenenergies. These eigenenergies and the corresponding eigenstates can be approximated for large S using a
discrete version of the WKB method [9], as we discuss below.
2.2.3
Discrete WKB Method
After solving the degeneracy of the energy level, the remaining task we need to
deal with is one copy of the block diagonal Hamiltonian with the total spin S. The
subspace is 2S + 1 dimensional and the Hamiltonian in the subspace is
1 2
S − ΓSx
2N z
1 2 Γ
=−
S − (S+ + S− ) ,
2N z
2
H0 = −
1
2
Please see Appendix A for the mathematical proof.
Please see the end of Appendix A for the proof.
18
(2.8)
where S+ , S− are raising and lowering operators. In (S 2 , Sz ) representation, we have



Sz2 |S, mi = m2 |S, mi




p
S+ |S, mi = (S − m)(S + m + 1)|S, m + 1i



p


S− |S, mi = (S + m)(S − m + 1)|S, m − 1i .
(2.9)
If we set the eigenstate wave function as
|ψi =
S
X
Cm |S, mi ,
(2.10)
m=−S
we arrive at the discrete version of the Schrödinger equation:
−
1 2
Γp
Γp
m Cm −
(S + m)(S − m + 1)Cm−1 −
(S − m)(S + m + 1)Cm+1 = E · Cm .
2N
2
2
(2.11)
By applying the discrete WKB method reviewed by P.A.Braun[9], we define
1 2
m
2N
Γp
pm =
(S + m)(S − m + 1) ,
2
wm =
(2.12)
then the Schrödinger equation reduces to the standard form in [9]:3
pm Cm−1 + (wm + E)Cm + pm+1 Cm+1 = 0 .
(2.13)
∂
By following the steps mentioned in [9], we introduce “momentum” ϕ = −i ∂m
, then
H0 = −(wm + pm e−iϕ + pm+1 eiϕ )
' −(wm + 2pm+ 1 cos ϕ) .
2
3
(2.14)
In the original paper, the coefficient before Cm is “wm − E”, what we only need to do is to
replace all E’s into −E.
19
From now on, we will simplify the notation by denoting w ≡ wm and p ≡ pm+ 1 =
2
q
Γ
(S + 12 )2 − m2 . Then we introduce the “potential-energy curve”:4
2


U + (m) ≡ w + 2|p|
(2.15)

U − (m) ≡ w − 2|p| .
The result given by [9] about U + and U − is that the classical accessible energy region
is confined by U + and U − :
−U + (m) ≤ E ≤ − U − (m) .
(2.16)
In our model, the relation among U + , U − and E is demonstrated by Fig. 2.2. From
U
-U -
m_t
m
E
-U +
Figure 2.2: Two “potential-energy curves” in our model.
Fig. 2.2, the classical accessible region is between the two curves, so if the energy is
specified as shown in Fig. 2.2, it would generically have the quantum tunneling effect
between the left and right well.
In the original paper, the definition was U ± ≡ w ± 2p by assuming p > 0. Actually when p < 0,
the situation is the same as p > 0 if we define U ± ≡ w ± 2|p|.
4
20
Based on the connection relation associated by U + turning point mt , we have the
WKB wave function[9]:

Z m
A
π


C
=
cos
arccos
B
dm
−

√
m>m
t

vm
4
mt
Z mt
A


p

C
=
arccosh
B
dm
,
exp
−
 m<mt
2 |vm |
m
p
where A is the normalization factor; vm ≡
ity; B ≡
−E−w
2|p|
(2.17)
(U + + E)(U − + E) is the classical veloc-
has the property when m is reaching the edge of the classical region or
the turning point mt , B(m = mt ) = 1. Thus, the leading effect of the level splitting
due to double-well quantum tunneling is given by:
Z
mt
(2.18)
arccosh B dm
(2.19)
arccosh B dm ≡ e−γN ,
∆E ∼ exp −
−mt
where
1
γ=
N
Z
mt
−mt
is the scaled logarithm of the tunneling rate and mt satisfies the turning point equation
B(mt ) = 1 which is
r
1
1 2
mt + |Γ| (S + )2 − m2t + E = 0 .
2N
2
(2.20)
When we send the number of sites N → ∞, we would notice that the total energy
E, total spin S and the z-component of the total spin m are all proportional to N .
Therefore it is better to define the number density of all the above quantities in order
to explicitly demonstrate the N dependence:
u≡
E
,
N
s̄ ≡
S
,
N
21
x≡
m
.
N
(2.21)
Then the turning point equation (2.20) reduces to:
p
1 2
xt + |Γ| s̄2 − x2t + u = 0 ;
2
⇒
xt =
√
q
p
2 · −u − Γ2 − Γ2 (2u + s̄2 + Γ2 ) .
(2.22)
(2.23)
Meanwhile, the scaled logarithm of the tunneling rate γ reduces to:
1
−E − 2N
m2
arccosh √
dm
|Γ| S 2 − m2
−mt
xt
−u − 21 x2
arccosh √
dx .
|Γ| s̄2 − x2
0
1
γ=
N
Z
=2
2.3
Z
mt
(2.24)
Thermodynamics
For the thermodynamics (but not the dynamics) of this system in the limit of large
~ classically and ignore their nonzero commutators
N we can treat the components of S
when obtaining the extensive thermodynamic properties (energies, entropies, magnetizations). Then we obtain the energy density
u=
E
H0
1
=
= − s̄2z − Γs̄x ,
N
N
2
(2.25)
where
q
s̄x = ± s̄2 − s̄2y − s̄2z .
(2.26)
Note that this equation is the same as the turning point equation (2.22) by setting
s̄y = 0 and s̄z = xt and choosing the sign of s̄x as the same as the sign of Γ. It
means that we can also use the semiclassical point of view to determine the WKB
22
turning point xt and further obtain the tunneling rate γ(s̄). The ground state is given
by minimizing the energy density u as a function of s̄, s̄y and s̄z . Here we choose
the sign of s̄x as the same as the sign of Γ so that the term −Γs̄x gives a negative
contribution to u. Since s̄ ≤ 1/2, it is easy to see that when s̄ = 1/2 and s̄y = 0,
u will be minimized in terms of them. By plugging these two conditions into u, we
have
r
1 2
1
u = − s̄z − |Γ|
− s̄2z .
2
4
(2.27)
By setting the first order derivative of s̄z equals zero, we have s̄z = 0 or s̄z =
q
1
4
− Γ2 .
After evaluating the second order derivative, it becomes clear that when 41 − Γ2 < 0,
q
1
2
s̄z = 0 is the minimal point; when 4 − Γ > 0, s̄z = 14 − Γ2 minimizes u:

q

 when |Γ| < 1 , s̄z = 1 − Γ2 , umin = − 1 − 1 Γ2
2
4
8
2

 when |Γ| > 1 , s̄z = 0,
2
.
(2.28)
umin = − 21 |Γ|
Thus, the ground state of H0 always has the maximum value of S = N/2. For
|Γ| ≥ 1/2, the ground state is paramagnetic with the spins polarized along the xdirection: s̄y = s̄z = 0, s̄x = 21 sign(Γ) and u = hH0 i/N = −|Γ|/2. For |Γ| < 1/2, the
√
two nearly-degenerate ground states are ferromagnetic, with s̄z = ±(1/2) 1 − 4Γ2 ,
s̄x = Γ, s̄y = 0 and u = −(1 + 4Γ2 )/8. So the system has ground state quantum
phase transitions at Γ = ±1/2 between ferromagnetic phase (s̄z 6= 0, |Γ| < 1/2) and
paramagnetic phase (s̄z = 0, |Γ| > 1/2). The corresponding quantum critical points
are at Γc = ±1/2 , uc = −1/4.
For the excited states, the phase transition between ferromagnetic and paramagnetic phase also exists. Since we are interested here in eigenstates, which are at a
given energy E = N u, we will do the statistical mechanics in the microcanonical ensemble. For a given transverse field Γ and energy E, the equilibrium (most probable)
23
state of the system is the one that maximizes the entropy, which means minimizing
the total spin S. To minimize S for a given E, clearly we set Sy = 0, since Sy does not
appear in the Hamiltonian. In the paramagnetic phase the total spin points along
the x-direction and the equilibrium value of the total spin is thus s̄eq = |u/Γ|. In
the ferromagnetic phase the system can go to higher entropy (lower total spin) for
a given u by making s̄z 6= 0. Some algebra shows that in the ferromagnetic phase,
which is |Γ| < 1/2 and −(1 + 4Γ2 )/8 ≤ u < −Γ2 , the equilibrium is at s̄x = Γ,
p
√
s̄z = ± 2(−u − Γ2 ) and s̄eq = −2u − Γ2 . The line of critical points separating the
para- and ferromagnetic phases is u = −Γ2 , for |Γ| < 1/2, as indicated in Fig 2.1; this
critical line ends at the quantum critical points at |Γ| = 1/2, u = −1/4. Note that
in this whole ferromagnetic phase regime, the sign of s̄x is the same as the sign of Γ.
Since all the quantities related to Γ, except s̄x , are even functions of Γ (or functions
of |Γ|), from now on we assume Γ > 0 and use Γ and |Γ| interchangeably. So it also
implies that s̄x ≥ 0.
2.4
Eigenstate Thermalization
Due to its full symmetry under all permutations of the N spins, the Hamiltonian
H0 is integrable, with all the good quantum numbers associated with this permutation
symmetry, including the magnitude S of the total spin. We want to study a more
generic system, so we add to the Hamiltonian the small term H1 (see Eq. (2.2))
to break the permutation symmetry, lift all degeneracies, and make the eigenstates
thermal. The only symmetry that remains in our full H is the Ising (Z2 ) symmetry
under a global rotation of all spins by angle π about their x axes.
For a given E and Γ, the eigenstates of H0 have total spin ranging from the
minimum and equilibrium value Seq up to the maximum value of S = N/2. To make
these in to thermal eigenstates we need H1 to perturb the system enough so that
24
the eigenstates of the full H are linear combinations of all these total spin values,
weighted as at thermal equilibrium. At first order, the perturbation we are adding,
H1 , flips at most two spins, so it can change the total spin by at most ±2. The
spectrum of H0 at each value of total spin S contains (2S + 1) eigenenergies spread
over a range of energy that is of order S. Thus the level-spacing in the spectrum of
H0 at a given S remains of order one in the limit of large N . This is reflected in the
dynamics under H0 , which is spin precession about the mean field, and the mean field
is of order one, so the rate of precession is also of order one.
For the eigenstates of H to strongly and thermally mix the different values of S
we thus need the matrix elements of H1 between states at different S to be large
compared to the (order-one) level spacing of H0 in the large N limit. This is why we
require that the exponent p in the definition of H1 satisfies p < 1, since this is the
condition for these matrix elements to diverge in the large N limit.5 This should be
sufficient to make all the eigenstates of H thermal in that limit. We have not yet
found a way to actually prove that this is sufficient to make all the eigenstates of H
satisfy the ETH, but below we provide some numerical evidence for this from exact
diagonalization of finite-size systems.
2.5
Dynamics
In addition to making sure that H1 is strong enough to thermalize the system, we
also want it to be weak enough so that we can use the well-understood dynamics and
thermodynamics of H0 in our analysis. By restricting the exponent p to be greater
than 1/2, in the large N limit the effective field that each spin is precessing about
is the mean field from H0 , with only a small correction from H1 that vanishes as
N → ∞.6 This small correction is enough to thermalize the system for 1/2 < p < 1,
5
6
For the details of the order analysis of the lower bound of disorder, see Appendix B.1.
For the details of the order analysis of the upper bound of disorder, see Appendix B.2.
25
which is the range where the perturbation due to H1 on a single spin’s dynamics
vanishes for N → ∞, while the perturbation to the dynamics of the full many-body
system diverges. In this regime, the system’s primary dynamics is the S-conserving
dynamics due to H0 , which is spin precession at a rate of order one, and, assuming
we are in the ferromagnetic phase, ‘attempts’ at rate of order one to tunnel through
the energy barrier between total Sz up and down. At a rate that is slower by a
power of N , the dynamics due to H1 allow ‘hopping’ between different values of the
total spin S, and thus thermalization to the equilibrium probability distribution of
S dictated by the entropy Σ. And at a rate that is slower still, exponentially slow
in N , the system succeeds in crossing the free energy barrier between up and down
magnetizations. It is the separation between these three time scales that allows us to
systematically understand the dynamics of this system for large N .
Since our full Hamiltonian H has Ising symmetry under a global spin flip, and
the randomness in H1 means there are no exact degeneracies in the spectrum of H,
for finite N any eigenstate of H is either even or odd under this Ising symmetry
(with probability one). In the ferromagnetic phase, this means the exact eigenstates
of H are all Schrödinger cat states that are either even or odd linear combinations
of states with total Sz up and down. These two equal (in magnitude) and opposite
(in sign) values of Sz are extensive, thus ‘macroscopically’ different, which is why it
is appropriate to call these eigenstates ‘Schrödinger cats’.
2.5.1
Thermal Activation and Quantum Tunneling
Next we examine the rate at which this system, in its ferromagnetic phase, will flip
from the up state with positive total Sz to the down state with negative Sz under the
unitary time evolution due to its Hamiltonian H. In Section 2.2.3, we noticed that
the system has an energy level splitting due to double-well quantum tunneling which
is given by ∆E ∼ e−γ(s̄)N where γ(s̄) is the scaled logarithm of the tunneling rate
26
as a function of the total spin density s̄ = S/N . By drawing the classically constant
energy line shown in Fig. 2.3 described by equation (2.25) by setting s̄y = 0 or (2.22)
s
E
s_0
s_min
x_t
s_z
Figure 2.3: Constant energy (E) line described by equation (2.25) by setting s̄y = 0
or just by equation (2.22). It determines the WKB turning point xt when the total
spin density s̄ is fixed.
in the ferromagnetic phase region, we obtain a double-well shaped curve which has
a local maximum at s̄z = 0 where s̄ = s̄0 = −u/Γ and two local minima at |s̄z | 6= 0
where s̄ = s̄min = Seq /N .
There are two steps to this process: First the system gets ‘excited’ from its usual
(high entropy) total spin Seq ‘up’ to a larger total spin S with lower entropy, with a
probability ∼ exp {−N (Σ(Seq /N ) − Σ(S/N ))} given by the resulting decrease of the
entropy. As S is increased, the energy barrier, whose top is at energy E = −ΓS,
decreases. For E ≥ −ΓN/2, one way the system can flip is to increase S enough
so that E ≥ −ΓS and then it will simply cross over the top of the barrier without
quantum tunneling. In the higher-energy part (F3) of the ferromagnetic phase, in the
limit of large N this is the dominant process that flips the magnetization: the system
‘thermally activates’ itself (via its unitary time-evolution) to a low-entropy, hightotal-spin state where the energy barrier can be crossed without quantum tunneling.
27
The ‘height’ of the entropy barrier it must cross to do this is extensive:
√
N ∆Σ = N (Σ( −2u − Γ2 ) − Σ(−u/Γ)) ,
(2.29)
where ∆Σ is the reduction in entropy per site needed to go over the barrier.
If the system does not or can not go over the energy barrier by increasing the total
spin S, then in order to flip the magnetization it must quantum tunnel through the
barrier. For large N , this tunneling probability can be estimated using a version of the
WKB method [9] I have explained in Section 2.2.3. As a small summary of Section
2.2.3, the total Sz serves as the ‘position’, while the operator −ΓSx serves as the
‘kinetic energy’. What we need to calculate is the probability of the system tunneling
between positive and negative total Sz for a given total spin S = N s̄ satisfying
Seq ≤ S < −E/Γ. This probability behaves as ∼ exp (−N γ), with γ(u, s̄, Γ) being
an intensive quantity. If we define a scaled ‘position’ x = Sz /N , the WKB ‘turning
points’ adjacent to the barrier are at x = ±xt with
xt =
√
q
p
2 · −u − Γ2 − Γ2 (2u + s̄2 + Γ2 ) .
(2.30)
Then the intensive factor in the exponent of the tunneling probability is given by the
WKB tunneling integral
Z
γ=2
0
xt
−u − 1 x2
arcosh √ 2
dx .
Γ s̄2 − x2
(2.31)
Since the probabilities of being ‘excited’ to total spin S and of quantum tunneling
through the barrier with total spin S are both exponentially small in N , in the limit of
large N the dominant process by which the magnetization flips is given by a standard
‘saddle point’ condition. The total spin S = N s̄ at which the system tunnels is the
value that maximizes the product of these two probabilities and thus minimizes the
28
quantity
α(u, Γ) = mins̄ {Σ(s̄min =
√
−2u − Γ2 ) − Σ(s̄) + γ(u, s̄, Γ)} .
(2.32)
This means in order to tunnel through the barrier, the system will finally choose one
particular value of the total spin s̄ to maximize {Σ(s̄) − Σ(s̄min ) − γ(u, s̄, Γ)}. Then
the distinction between thermal and quantum tunneling becomes clear: if the total
spin is chosen to be in the interval (s̄min , s̄0 ) which is below the top of the barrier, it is
quantum tunneling; if the total spin is chosen to be bigger than or equal to s̄0 = −u/Γ
which is at the top of the barrier, it is thermal activation over the barrier.
The sketch of Σ(s̄) and γ(s̄) is shown in Fig. 2.4, where
dγ(s̄)
ds̄
diverges7 at s̄ = s̄min ;
S, Γ
S(s)
Γ(s)
0
s_min
s_0
1/2
s
Figure 2.4: A sketch of the entropy Σ(s̄) and the tunneling rate γ(s̄).
γ(s̄) becomes zero at s̄ = s̄0 since it is at the top of the barrier. From Fig. 2.4, if
we assume γ(s̄) is a convex8 function of s̄, then the distinction between the thermal
activation and quantum tunneling would be shown straightforward in Fig. 2.5 where
in thermal activation, Σ − γ reaches its maximum at s̄ = s̄0 ; in quantum tunneling,
Σ − γ reaches its maximum in the middle between s̄min and s̄0 . Then, the condition
7
8
Please see Appendix C for proof of the divergence.
For detailed discussion of this issue, see Appendix D.
29
S-Γ
S-Γ
s_min
s_0
s_min
s
(a) Thermal activation
s_0
s
(b) Quantum tunneling
Figure 2.5: A sketch of the difference between thermal activation and quantum tunneling.
reduces to:


 if
− γ)
> 0 ⇒ Thermal Activation;
s̄=s̄0 −0

 if ∂ (Σ − γ)
< 0 ⇒ Quantum Tunneling.
∂s̄
∂
(Σ
∂s̄
(2.33)
s̄=s̄0 −0
We have located the saddle point numerically at many points within the ferromagnetic phase and it appears to always be unique, without any discontinuities as
the parameters u and Γ are varied. Some straightforward analysis as I will discuss in
Section 2.6 in details shows that in the higher-energy part (F3) of the ferromagnetic
phase where
√
πΓ
Γ − 2u
≥ ln
Γ + 2u
−u − Γ2
(2.34)
the saddle point is ‘thermal’: the ‘entropy cost’ of going to higher S is less than the
‘tunneling cost’, and the system goes over the barrier without any quantum tunneling.
We call the eigenstates in this regime ‘thermal cats’, since these Schrödinger cat states
flip by thermally activating themselves over the barrier. In regions F1 and F2 this
inequality is instead false, and the system quantum tunnels through the barrier at a
value of S satisfying Seq ≤ S < −E/Γ, so the eigenstates are instead ‘quantum cats’.
30
The location of the dynamical phase transition between phases F2 and F3 is given
by converting the above inequality (2.88) to an equality.
The asymptotic behavior of this transition curve will be discussed in details in
Section 2.7 and the result is as follows. Near the quantum critical points (∆Γ =
1/2 − |Γ| 1), this transition line becomes exponentially adjacent to (and above)
√
the straight line u = −|Γ|/2: u = −1/4 + (∆Γ/2) + O(exp(−1/ ∆Γ)); near Γ = 0,
p
it behaves as a power law: −u ≈ (|Γ| π/4)4/3 .
2.5.2
Paired States and Unpaired States
Within the lower-energy phases (F1 and F2) of ‘quantum cats’, there is a second
dynamical phase transition within the ferromagnetic phase. This is also a ‘spectral
phase transition’ [13] in the level-spacing statistics of the eigenenergies. If one starts
with a state (not an eigenstate) that is magnetized up, the rate at which the system
crosses the barrier to down is ∼ exp (−N α(u, Γ)), and as a result the uncertainty of
the energy of this initial ‘up’ state must be at least this large, by the time-energy
uncertainty relation. Compare this minimum energy uncertainty to the typical many√
body level spacing ∼ exp (−N Σ( −2u − Γ2 )) of the eigenstates of H at that energy.
There is clearly a sharp change at the phase transition line, which is the line where
√
α(u, Γ) = Σ( −2u − Γ2 ). In another word,


 if max(Σ − γ) < 0 ⇒ Paired States;
s̄
(2.35)

 if max(Σ − γ) > 0 ⇒ Unpaired States.
s̄
This transition line between phases F1 and F2 is shown in Fig 2.1. The location of
this transition was obtained numerically, since we do not have a simple closed-form
expression for α(u, Γ).
The asymptotic behavior of this transition curve will also be discussed in details
in Section 2.7 and the result is as follows. Near the quantum critical points (∆Γ =
31
1/2 − |Γ| 1), this transition line also becomes exponentially adjacent to (but now
√
below) the straight line u = −|Γ|/2: u = −1/4 + (∆Γ/2) − O(exp(−1/ ∆Γ)); near
Γ = 0, it is logarithmically tangent to the u-axis: −u ∼ 1/(ln |Γ|)2 .
2.6
Analytical Calculation of the Thermal Activation and Quantum Tunneling Transition
The transition between thermal activation and quantum tunneling is illustrated
by condition (2.33). Thus, in order to find the transition, we need to calculate the
derivatives
∂Σ
∂s̄
and
∂γ
∂s̄
at s̄ = s̄0 − 0. The first one is straightforward from equation
(2.7):
∂Σ
1 + 2s̄
= − ln
.
∂s̄
1 − 2s̄
Therefore we need to focus on solving
∂γ
∂s̄
(2.36)
at s̄ = s̄0 − 0.
When s̄ = s̄0 , by definition we have the turning point xt = 0 (It implies γ(s̄0 ) = 0.).
Plugging xt = 0 into the turning point equation (2.22) gives us:
s̄0 =
−u
.
Γ
Now we perturb xt a little away from 0 by setting xt = ε 1. Then we have:
√
1 2
ε + Γ s̄2 − ε2 + u = 0
2
√
1
Γ s̄2 − ε2 = −u − ε2
2
1
1
s̄2 − ε2 = 2 (−u − ε2 )2
Γ
2
2
2
u
uε
s̄2 = 2 + ε2 + 2 + O(ε4 ) .
Γ
Γ
32
(2.37)
Then,
r
u2
uε2
2+
+
ε
+ O(ε4 )
2
Γr
Γ2
−u
u + Γ2 2
1+
ε + O(ε4 )
=
Γ
u2
i
u + Γ2 2
−u h
4
1+
=
ε + O(ε )
Γ
2u2
u u + Γ2 2
=− −
ε + O(ε4 ) .
Γ
2uΓ
⇒
s̄ =
Thus the infinitesimal change of s̄ is:
∆s̄ = −
u + Γ2 2
ε + O(ε4 ) < 0 .
2uΓ
(2.38)
Now it is time to see ∆γ:
ε
Z
∆γ = γ(xt = ε) = 2
0
−u − 12 x2
arccosh √
dx .
Γ s̄2 − x2
(2.39)
Since in this integral x runs from 0 to ε, we may do the rescaling to let x ≡ εb where
b runs from 0 to 1. After that, we have:
−u − 21 ε2 b2
B= √
Γ s̄2 − ε2 b2
−u − 12 ε2 b2
= q
2
Γ Γu2 + (1 + Γu2 )ε2 − ε2 b2 + O(ε4 )
−u − 21 ε2 b2
=p
u2 + (u + Γ2 − b2 Γ2 )ε2 + O(ε4 )
−u − 21 ε2 b2
q
=
2
2 Γ2
(−u) 1 + u+Γ u−b
ε2 + O(ε4 )
2
=
b2 2
ε
2u
u+Γ2 −b2 Γ2 2
+
ε +
2u2
2
2
1+
O(ε4 )
b
u + Γ − b2 Γ 2 2
=1+
−
ε + O(ε4 ) .
2u
2u2
1
33
(2.40)
Since arccosh B = ln(B +
√
B 2 − 1), then if B = 1 + ∆ where ∆ 1, we have
√
2∆ + ∆2 )
√
√
= ln[1 + ∆ + 2∆ + O(∆ ∆)]
√
√
√
√
√
1
= ∆ + 2∆ + O(∆ ∆) − [∆ + 2∆ + O(∆ ∆)]2 + O(∆ ∆)
2
√
√
1
= ∆ + 2∆ − · 2∆ + O(∆ ∆)
2
√
√
= 2∆ + O(∆ ∆) .
(2.41)
arccosh(1 + ∆) = ln(1 + ∆ +
So if ∆ =
b2
2u
−
u+Γ2 −b2 Γ2
2u2
ε2 + O(ε4 ), then
r
arccosh(1 + ∆) =
1
r
Z
1
Z
⇒
γ(xt = ε) = 2
0
= 2ε
2
0
= 2ε
2
Z
0
1
b2 u + Γ2 − b2 Γ2
−
ε + O(ε3 ) .
2
u
u
(2.42)
b2 u + Γ2 − b2 Γ2
−
ε · ε db + O(ε4 )
u
u2
1 p 2
ub − (Γ2 + u) + b2 Γ2 db + O(ε4 )
−u
√
−u − Γ2 √
1 − b2 db + O(ε4 ) .
−u
If we change the variable b = sin θ, then
R1√
0
1 − b2 db =
Rπ
2
0
cos θ ·cos θ dθ =
(2.43)
π
4
. Thus
we arrive at:
√
−u − Γ2 π 2
γ(xt = ε) = 2 ·
· · ε + O(ε4 ) ,
−u
4
34
(2.44)
∂γ γ(xt = ε) − 0
= lim
∂s̄ s̄=s̄0 −0 ε→0√
∆s̄
−u−Γ2 π
2 · −u · 4
=
2
− u+Γ
2uΓ
πΓ
.
= −√
−u − Γ2
(2.45)
In addition,
∂Σ 1 + 2s̄0
= − ln
∂s̄ s̄=s̄0
1 − 2s̄0
Γ − 2u
.
= − ln
Γ + 2u
(2.46)
Therefore the boundary between thermal activation and quantum tunneling is given
by:
⇒
∂γ ∂Σ =
∂s̄ s̄=s̄0 −0
∂s̄ s̄=s̄0
πΓ
Γ − 2u
√
.
= ln
Γ + 2u
−u − Γ2
(2.47)
To be more specific, we arrive at the result:


 if
√ πΓ
−u−Γ2
⇒
> ln Γ−2u
Γ+2u

 if
√ πΓ
−u−Γ2
< ln Γ−2u
⇒
Γ+2u
− γ)
>0 ⇒
s̄=s̄0 −0
∂
(Σ − γ)
<0 ⇒
∂s̄
∂
(Σ
∂s̄
s̄=s̄0 −0
Thermal Activation;
Quantum Tunneling.
(2.48)
2.7
Asymptotic Behaviors of Both Transitions Near Both QCP and u = 0 , Γ = 0 Point
Up to now, we have already obtained the mathematical condition of both transition lines. To further explore the properties of both lines, it is better to also have
the asymptotic behaviors near the edges of the ferromagnetic regime demonstrated
35
by Fig. 2.1. One edge is the Quantum Critical Point(QCP); the other is the origin
of the phase diagram: u = 0 , Γ = 0 point. We shall study the asymptotic behaviors
one by one.
2.7.1
Near QCP
The QCP is at Γc = 1/2 , uc = −1/4. We perturb the system a little away from
the QCP into the ferromagnetic region to let Γ =
1
2
− ∆Γ where ∆Γ 1. Then the
goal is to see what asymptotic behavior the energy density u should have in terms of
∆Γ. Let us discuss these two transitions one by one.
Thermal Activation and Quantum Tunneling Transition
In the thermal activation and quantum tunneling transition case, the transition
line must be placed above the straight line u = −Γ/2. If not, the top of the barrier
s̄0 =
−u
Γ
in Fig. 2.3 would be greater than 1/2 which is the upper bound of the
available total spin density s̄. Then there would be always a quantum tunneling
effect and no thermal activation can be achieved. So the transition line must be
placed in the regime where s̄0 =
−u
Γ
≤
1
2
or u ≥ −Γ/2. Also we know that u ≤ −Γ2
since it is in ferromagnetic region. Note that u = −Γ2 has a first order derivative −1
at Γc = 1/2. Therefore we set9
1
1
u = − + ( + δ) · ∆Γ ,
4
2
(2.49)
where 0 ≤ δ ≤ 1/2. Plugging this definition into the transition line equation:
Γ − 2u
πΓ
√
= ln
,
2
Γ + 2u
−u − Γ
9
(2.50)
This is only a redefinition of a variable but not a Taylor expansion since the essential singularity
occurs in general. And the same concern applies to all cases afterwards.
36
we then arrive at:
π( 21 − ∆Γ)
1 L.H.S. = q
;
&O √
∆Γ
( 12 − δ)∆Γ − ∆Γ2
1 − 2(1 + δ)∆Γ
R.H.S. = ln
∼ O − ln(δ∆Γ) .
2δ∆Γ
(2.51)
Thus, if δ ∼ O(1), then the right hand side would be O − ln(∆Γ) which is much
1
smaller than O √∆Γ
. So in order to let the right hand side big enough, we need
1
. By setting the
δ 1. Under this condition, the left hand side is exactly O √∆Γ
left hand side to be equal to the right hand side, we finally get:
1
∼ − ln(δ∆Γ)
∆Γ
1 − √1
⇒ δ∼
e ∆Γ .
∆Γ
√
Note that the form e
− √1
∆Γ
(2.52)
behaves essentially singular at ∆Γ = 0 point and does
not have a proper Taylor expansion in terms of ∆Γ; in fact, it is much smaller than
any finite power of ∆Γ, so all the Taylor coefficients would be zero. It implies the
transition line is extremely near (and above) the straight line u = −Γ/2. So finally,
the asymptotic behavior of the thermal activation and quantum tunneling transition
is:

1

Γ = − ∆Γ
2
1

 u = − 1 + 1 ∆Γ + O e− √∆Γ
,
4 2
where ∆Γ 1.
37
(2.53)
Paired and Unpaired States Transition
In the paired and unpaired states transition case, on the contrary, the transition
line must be placed below the straight line u = −Γ/2. If not, then s̄0 =
−u
Γ
would be
an available candidate for taking the maximal value of Σ(s̄) − γ(s̄). By noting that
γ(s̄0 ) = 0, we would have
max(Σ − γ) ≥ Σ(s̄0 ) − γ(s̄0 )
s̄
= Σ(s̄0 ) > 0 ,
(2.54)
then it would be always in unpaired states region. So the transition line must be
placed in the regime where u ≤ − Γ/2. Also we know that u ≥ − 81 − 12 Γ2 since the
right hand side is the ground state energy. Note that at Γc = 1/2, u = − 81 − 12 Γ2 has
a first order derivative −1/2 which is the same as that of the straight line u = −Γ/2.
Since the transition line must be placed between these two bounds, we obtain the
first order derivative of the transition line must also be −1/2. Therefore we set
1 1
u = − + ∆Γ − λ1 ∆Γ2 ,
4 2
(2.55)
where 0 ≤ λ1 ≤ 1/2.
In order to solve max(Σ − γ), we need to first solve the upper and lower bound
s̄
of the available s̄. The upper bound is 1/2 since in this case s̄0 ≥ 1/2. The lower
bound is exactly s̄min indicated in Fig. 2.3. By solving s̄ in the turning point equation
(2.22), we arrive at:
s̄2 =
u
1
u2
+ (1 + 2 )x2t + 2 x4t ,
2
Γ
Γ
4Γ
38
(2.56)
then the minimal available s̄2 is exactly
s̄2min = −2u − Γ2
(2.57)
at (xt )2min = −2u − 2Γ2 . By plugging the definition of u and Γ in terms of ∆Γ and
λ1 into s̄2min , we have:
s̄2min = −2u − Γ2 =
1
− (1 − 2λ1 )∆Γ2 .
4
(2.58)
Then we define
s̄2 ≡
1
− (1 − 2λ1 ) · α1 · ∆Γ2 ,
4
(2.59)
where 0 ≤ α1 ≤ 1. Then the task converts into finding the maximal value of Σ − γ
in terms of α1 .
Since ∆Γ 1 and λ1 , α1 . O(1), we have
s̄ =
⇒
2 1
− (1 − 2λ1 )α1 ∆Γ2 + O (1 − 2λ1 )α1 ∆Γ2
2
(2.60)
1
1
1
1
Σ(s̄) = − ( − s̄) ln( − s̄) + ( − s̄) ln( − s̄)
2
2
2
2
h
i
2
2
2
= − (1 − 2λ1 )α1 ∆Γ ln (1 − 2λ1 )α1 ∆Γ + O (1 − 2λ1 )α1 ∆Γ
. − ∆Γ2 ln(∆Γ2 ) .
(2.61)
Then for
−u − 21 x2
γ(s̄) = 2
arccosh √
dx
Γ s̄2 − x2
0
Z 1
−u − 1 x2 b2 =2
arccosh p 2 t
xt db ,
Γ s̄2 − x2t b2
0
Z
xt
39
(2.62)
we have:
x2t
=2·
√
2
2
− u − Γ − Γ 2u + s̄ + Γ
2
1
1
− ∆Γ + λ1 ∆Γ2 − ( − ∆Γ)2
4 2
2
r
i
1
1
1
2
2
− ( − ∆Γ)
− (1 − 2λ1 )α1 ∆Γ − + (1 − 2λ1 )∆Γ
2
4
4
h 1 1 p
i
p
=2·
−
(1 − 2λ1 )(1 − α1 ) ∆Γ + λ1 − 1 + (1 − 2λ1 )(1 − α1 ) ∆Γ2
2 2
=2·
h1
≡ 2 · (a1 ∆Γ + a2 ∆Γ2 ) ,
where a1 ≡
1
2
−
1
2
(2.63)
p
p
(1 − 2λ1 )(1 − α1 ) , a2 ≡ λ1 − 1 + (1 − 2λ1 )(1 − α1 ) . Then, we
have:
−u − 12 x2t b2
p
Γ s̄2 − x2t b2
1
− 21 ∆Γ + λ1 ∆Γ2 − (a1 ∆Γ + a2 ∆Γ2 )b2
4
q
=
1
( 2 − ∆Γ) 14 − (1 − 2λ1 )α1 ∆Γ2 − 2(a1 ∆Γ + a2 ∆Γ2 )b2
i
h 1 λ ∆Γ2 − a b2 ∆Γ − a b2 ∆Γ2 i h
1
1
2
2
2
2
2
2
+
·
2
1
+
2(1
−
2λ
)α
∆Γ
+
4a
b
∆Γ
+
4a
b
∆Γ
≈
2
1
1
1
1
2
− ∆Γ
2
2
2(λ1 ∆Γ − a1 b2 ∆Γ − a2 b2 ∆Γ2 )
+ 2(1 − 2λ1 )α1 ∆Γ2 + 4a1 b2 ∆Γ + 4a2 b2 ∆Γ2
=1+
1
− ∆Γ
2
h
1
=1+ 1
2λ1 ∆Γ2 − 2a1 b2 ∆Γ − 2a2 b2 ∆Γ2 + (1 − 2∆Γ)(1 − 2λ1 )α1 ∆Γ2
−
∆Γ
2
i
+ 2a1 b2 ∆Γ − 4a1 b2 ∆Γ2 + 2a2 b2 ∆Γ2 − 4a2 b2 ∆Γ3
h
i
≈ 1 + 2∆Γ2 2λ1 + (1 − 2λ1 )α1 − 4a1 b2 − 4a2 b2 ∆Γ ,
(2.64)
where all the terms I have omitted are definitely much smaller than at least one of
the present terms in the last line given ∆Γ 1 , λ1 . O(1) , α1 . O(1) .
Next comes the key point: in order to have Σ and γ in same order at some
value of α1 , both α1 and λ1 have to be much smaller than 1. Let me prove it by
contradiction. If the conclusion is not true, then at least one of α1 or λ1 would be
40
√
O(1). Then, we would have both a1 ∼ O(1) and a2 ∼ O(1). So xt ∼ ∆Γ and
√
√
−u− 12 x2t b2
√ 2 2 2 ∼ 1 + O(∆Γ2 ). Since we know arccosh(1 + ∆) = 2∆ + O(∆ ∆) when
Γ
s̄ −xt b
∆ 1, it means that:
−u − 21 x2t b2 arccosh p
γ(s̄) = 2
xt db
Γ s̄2 − x2t b2
0
√
√
∼ ∆Γ2 · ∆Γ
Z
1
3
= ∆Γ 2 .
(2.65)
However, from equation (2.61), we have Σ . ∆Γ2 ln(∆Γ2 ). So we would arrive at
Σ(s̄) γ(s̄) which is not the condition of the transition line. Thus, by contradiction,
we conclude both α1 1 and λ1 1 have to be true in order to satisfy the transition
condition.
By knowing α1 1 and λ1 1, all quantities are further simplified. First, we
have:
1
1
a1 ≈ λ1 + α1 ,
2
4
1
a2 = λ1 − 2a1 ≈ − α1 ,
2
(2.66)
which are both linear in λ1 and α1 . Then, we get:
r
p
xt ≈ 2a1 ∆Γ ≈
1
(λ1 + α1 )∆Γ ,
2
(2.67)
and
−u − 21 x2t b2
p
≈ 1 + 2(2λ1 + α1 − 4a1 b2 )∆Γ2
2 2
2
Γ s̄ − xt b
≈ 1 + 2(2λ1 + α1 )(1 − b2 )∆Γ2 .
41
(2.68)
r
p
1
γ(s̄) ≈ 2
2 · 2(2λ1 + α1 )(1 − b2 )∆Γ2 (λ1 + α1 )∆Γ db
2
0
Z 1√
√
= 2 2(2λ1 + α1 )∆Γ3/2
1 − b2 db
0
√
2
=
π(2λ1 + α1 )∆Γ3/2 .
2
Z
⇒
1
(2.69)
In addition, from equation (2.61) we have:
Σ(s̄) ≈ − (1 − 2λ1 )α1 ∆Γ2 ln (1 − 2λ1 )α1 ∆Γ2
≈ α1 ∆Γ2 ln α1 ∆Γ2 .
(2.70)
In order to let Σ and γ be in same order, we finally arrive at:
(2λ1 + α1 )∆Γ3/2 ∼ α1 ∆Γ2 ln α1 ∆Γ2 .
(2.71)
Then it is better to let O(λ1 ) ≤ O(α1 ), since if not, both λ1 and α1 would have an
extra factor λ1 /α1 on the exponent which would make the value far smaller than the
case O(λ1 ) ≤ O(α1 ). Therefore we finally get the result:
α1 ∼
1 − √1
e ∆Γ ,
∆Γ2
λ1 ∼
1 − √1
e ∆Γ .
∆Γ2
(2.72)
They also behave essentially singular at ∆Γ = 0 point so the transition line is also extremely near (but this time below) the straight line u = −Γ/2. Finally, the
asymptotic behavior of the paired and unpaired transition is also:

1

Γ = − ∆Γ
2
1

 u = − 1 + 1 ∆Γ + O e− √∆Γ
,
4 2
where ∆Γ 1.
42
(2.73)
2.7.2
Near u = 0 , Γ = 0 Point
The other interested point is the origin of the phase diagram: u = 0 , Γ = 0 point.
Also, we will perturb the system a little away from Γ = 0 or equivalently let Γ 1.
Then the goal is to see what u behaves in terms of Γ asymptotically when Γ → 0.
Thermal Activation and Quantum Tunneling Transition
In this case, we know that Γ2 ≤ − u ≤ Γ/2. First consider two extreme cases:
√
πΓ
Γ−2u
& O(1). So we have
1. If −u ∼ O(Γ), then √−u−Γ
∼
O(
Γ)
,
ln
2
Γ+2u
√ πΓ
ln Γ−2u
. It is in thermal activation region.
Γ+2u
−u−Γ2
πΓ
Γ−2u
2. If −u ∼ O(Γ2 ), then √−u−Γ
∼
O(1)
,
ln
∼ O(Γ). Then we have
2
Γ+2u
√ πΓ
. It is in quantum tunneling region.
ln Γ−2u
Γ+2u
−u−Γ2
Thus, the transition line must be placed in between where Γ2 − u Γ. When
this is true,
πΓ
≈
−u − Γ2
Γ − 2u ln
≈
Γ + 2u
√
⇒
πΓ
√
,
−u
−4u
.
Γ
πΓ
−4u
√
=
,
Γ
−u
3
π 2
Γ = (−u) 2 ,
4
r π 34
4
−u =
Γ ∼ Γ3 .
4
(2.74)
(2.75)
This is a power law behavior when Γ → 0.
Paired and Unpaired States Transition
In this case, we have Γ/2 ≤ − u ≤ 1/8. Also consider the two extreme cases first:
43
1. If −u ∼ O(1), then xt =
−u− 21 x2t b2
√
√ p
√
√
2 −u − Γ2 − Γ 2u + s̄2 + Γ2 ≈ −2u ∼ O(1).
) → ∞.
∼ O( −u
Γ
In addition, arccosh B = ln(B +
Z 1
√
−u − 1 x2 b2 2
B − 1) → ln 2B when B → ∞. So γ = 2
arccosh p 2 t
xt db ∼
Γ s̄2 − x2t b2
0
− ln Γ → ∞. But Σ . O(1), so it means max(Σ − γ) < 0. It is in paired states
But B =
Γ
s̄2 −x2t b2
s̄
region. Thus the transition line tends to reach u = 0 point when Γ → 0.
2. If −u ∼ O(Γ), then we define −u = α2 Γ, where α2 ∼ O(1) and α2 ≥ 1/2 since
we have −u ≥ Γ/2. Under this assumption, we have:
q
√
√
xt = 2 −u − Γ2 − Γ 2u + s̄2 + Γ2
p
√ q
= 2 α2 Γ − Γ2 − Γ s̄2 − 2α2 Γ + Γ2
p
√ q
≈ 2 α2 Γ − Γ s̄2 − 2α2 Γ
p
√ √ q
= 2 · Γ · α2 − s̄2 − 2α2 Γ
√
. O( Γ) ,
(2.76)
whereas,
−u − 1 x2 b2
B= p 2 t
Γ s̄2 − x2t b2
√
α2 Γ − b2 Γ(α2 − s̄2 − 2α2 Γ)
p
≈
Γ s̄2 − x2t b2
√
α2 − b2 (α2 − s̄2 − 2α2 Γ)
p
.
=
s̄2 − x2t b2
(2.77)
Z
1
So for a given finite s̄, we have B ∼ O(1). Thus, γ = 2
(arccosh B)xt db .
0
√
O( Γ) → 0, but Σ ∼ O(1), so it means max(Σ − γ) > 0. It is in unpaired
s̄
states region.
Thus, the transition line must also be placed in between where Γ − u 1.
√ p
√
√
−u− 1 x2 b2
Then we get xt = 2 −u − Γ2 − Γ 2u + s̄2 + Γ2 ∼ O( −u) and B = √ 22 t2 2 ∼
Γ
44
s̄ −xt b
O( −u
) 1.
Γ
Z
⇒
γ=2
0
1
−u √ i
(arccosh B)xt db ∼ O ln
· −u .
Γ
h
(2.78)
In order to let γ comparable to Σ which is typically O(1), we should have:
⇒
h −u √ i
O ln
· −u ∼ O(1)
Γ
√
ln(−u) − ln Γ −u ∼ 1
⇒
√
− ln Γ −u ∼ 1
⇒
This is equivalent to Γ ∼ e
1
− √−u
−u∼
1
.
(ln Γ)2
(2.79)
which is also an essential singularity at u = 0 , Γ = 0
point. It means that the transition line is extremely near the u-axis but not the
u = −Γ/2 line.
2.8
Numerical Evidence
We now present some numerical results about the thermalization properties and
the distinction between phases F1 and F3 (phase F2 is too narrow to clearly see in
the size systems we can diagonalize). We exactly diagonalize Hamiltonians with up
to N = 15 spins. We first put the Hamiltonian into a block diagonal form using basis
states that are even and odd with respect to the system’s Z2 Ising symmetry, and then
diagonalize each sector numerically to obtain all of the eigenstates within that sector.
For finite N , the coefficient λ in the disorder term H1 in Eq. (2.2) matters. If λ is too
large the system will become a spin glass rather than a ferromagnet. Thus λ needs to
be carefully chosen to be large enough for our finite systems to show thermalization,
45
but small enough to avoid the spin glass regime. After some exploration, we chose to
use the parameters λ = 0.7, p = 3/4 and Γ = 1/8 for our exact diagonalizations.
0.05
0.04
0.03
f
0.02
0.01
0
0
0.2
0.4
0.6
0.8
1
δ< /δ>
Figure 2.6: The level-spacing statistics using 100 realizations of H at N = 15 in phase
F1 within the even sector. δ< /δ> is the ratio between the smaller level spacing δ< to
the larger level spacing δ> for three consecutive eigenenergies in the even sector. f is
the relative frequency for each bin in this histogram.
First, we show the level-spacing statistics, which should be GOE if the eigenstates
are thermal. This must be done within one Z2 symmetry sector, since there is no levelrepulsion between states in different sectors. We look at each set of three consecutive
levels in one sector and denote δ< as the smaller level spacing and δ> as the larger
level spacing. Then the histogram of the ratio δ< /δ> can be compared to GOE level
statistics [2]. The even sector results in the F1 phase for 100 realizations at N = 15
are shown in Fig. 2.6. We see the expected strong level repulsion, consistent with the
46
thermalization. All other phases and symmetry sectors were also examined and the
results are also thermal, since phase F1 is the lowest-energy phase and thus the most
difficult to thermalize.
Next, we examine a ‘distance’ between two eigenstates that are adjacent in the
energy spectrum by comparing their probability distributions for the total spin Sz .
We define this distance between eigenstate 1 and eigenstate 2 as
D12 =
N/2
X
P1 (Sz ) − P2 (Sz ) ,
(2.80)
Sz =−N/2
where P1 (Sz ) and P2 (Sz ) are the probability distributions of Sz in eigenstates 1 and
2, respectively. We tested three different distances: Deo the distance between an even
parity state (eigenstate 1) and the nearest-energy odd parity state (eigenstate 2), and
similarly for Dee and Doo . If a system thermalizes, each eigenstate is equivalent to
a microcanonical ensemble characterized by its energy. For two eigenstates that are
adjacent in energy, the energy difference ∆E ∼ 2−N , therefore we expect the eigenstate distances Deo , Dee and Doo should decrease exponentially with N . In phase F1,
since the spectrum consists of nearly-degenerate pairs of states, the energy differences
satisfy ∆Eeo ∆Eee , ∆Eoo . Thus we expect in phase F1, Deo Dee , Doo . In addition, if we choose the upper bound of the energy window we average over to be well
within the F1 phase, we would also expect that Eq. (2.84) holds, so the exponential
decay rate of Deo would be greater than those of Dee and Doo . Meanwhile, in phases
other than F1, we expect all three D’s are well coincident. The numerical results are
shown in Fig. 2.7. As we expected, all these eigenstate distances decay exponentially
with N . In addition, Deo in phase F1 is much smaller and decreasing much faster
than the other two distances, and in phase F3 all three D’s are well coincident. This
demonstrates the clear distinction between phases F1 and F3, and further tests the
thermalization in phase F3.
47
−1
log(D)
−2
−3
e−o
e−e
o−o
−4
−5
−6
8
9
10
11
12
13
14
15
N
(a) Eigenstate distance in phase F1
−1.5
e−o
e−e
o−o
log(D)
−2
−2.5
−3
−3.5
8
9
10
11
12
13
14
15
N
(b) Eigenstate distance in phase F3
Figure 2.7: Averages of log (D) in phases F1 and F3, respectively, where D is the
‘eigenstate distance’ defined in the text.. The energy density range we used in F1
is from the first excited state in each sector up to uc − 0.02 where uc is the energy
density at the phase boundary between F1 and F2, whereas in F3 we used the phase’s
full energy density range. N is the total number of spins varying from 8 to 15. The
exponential decrease of D with increasing N indicates thermalization. The error bars
come from averaging over 100 realizations.
48
1.2
ᾱn at N = 11
ᾱn at N = 15
∆αn at N = 11
∆αn at N = 15
N →∞
1
F1
0.8
αn
0.6
F2
0.4
F3
0.2
0
−0.14
−0.12
−0.1
−0.08
−0.06
u
−0.04
−0.02
0
Figure 2.8: The mean ᾱn and the standard deviation ∆αn of the quantity αn defined
in Eq. (2.81). The number of realizations is 1600 for N = 11 (blue dash-dotted lines)
and 100 for N = 15 (red dashed lines). The green (solid) line gives the theoretical
quantity α(u, Γ) defined in Eq. (2.32) for the system size N → ∞.
In the ferromagnetic phases, the system spontaneously flips between magnetization up and down at a rate that behaves as ∼ exp(−N α(u, Γ)), where the quantity
α(u, Γ) is defined in Eq. (2.32). The probability of the system having total magnetization zero (or 1/2 for systems with odd N ) also behaves as ∼ exp(−N α(u, Γ)). Thus
from a single many-body eigenstate |ni we can obtain an estimate of the quantity α
as
αn =
1
maxSz {Pn (Sz )}
ln
,
N Pn (Sz = 0 or 1/2)
49
(2.81)
where Pn (Sz ) is the probability distribution of the total magnetization Sz in this
eigenstate. If the ETH is true, the magnetization thermalizes and these estimates αn
will converge to α(u, Γ) in the limit of large N .
In Fig. 2.8 we show the mean (ᾱn ) and the standard deviation (∆αn ) of αn within
energy bins for systems of size N = 11 and N = 15. The standard deviation decreases
with increasing N , as expected for these Schrödinger cat states that obey the ETH.
In the limit of large N every eigenstate at a given energy density will have the same
probability distribution of the total magnetization. That distribution is a thermal
distribution for the magnetizations that are accessed by thermal excitation and ‘tails’
due to quantum tunneling in the remainder of the distribution.
The model we are studying here is a ferromagnet (H0 ) with a small added spinglass term (H1 ) in its Hamiltonian. In terms of its affect on the system’s thermodynamics, the relative strength of the spin-glass term scales as ∼ N −1/4 for the case
p = 3/4 that we have here. Thus it is perhaps reasonable to expect the mean value
of αn to exhibit a finite-size correction that vanishes for large N as ∼ N −1/4 . The
spin-glass term weakly frustrates the ferromagnetism, so will cause a reduction in the
apparent α. Given the very modest range of N for which we can do exact diagonalizations, this ∼ N −1/4 is a very slow convergence towards the thermodynamic limit.
In Fig. 2.8 we also show the expected value of α in the limit of large N . As N is
increased from 11 to 15, ᾱn does indeed increase slowly toward α(u, Γ), as expected.
Thus the results of these exact diagonalizations provide some numerical support
for the theoretical results derived above assuming this system obeys the ETH. Of
course, given the very small size systems that can be diagonalized, there are strong
finite-size effects that make a detailed demonstration of thermalization in the large
system limit not possible. This is the situation with all numerical tests of thermalization.
50
In addition to the model discussed above, we tested two other ways of adding
disorder to H0 . The other Hamiltonians that we diagonalized are
N
1 X x
H = H0 + p0
εi s i ,
N i=1
(2.82)
N
N
X
1 X
z z
H =−
(ηi + 1)(ηj + 1)si sj − Γ
sxi ,
N 1=i<j
i=1
(2.83)
0
and
00
where in H 0 , the εi are also independent Gaussian random variables of mean zero
and variance one, and the power p0 here satisfies 0 < p0 < 1/2; specifically we looked
at p0 = 1/4. In H 00 , the ηi are independent random variables uniformly distributed
in the interval [−1, 1]. These latter two models only have N random parameters,
one per spin, unlike Eq. (2.2) which has one per pair of spins: N (N − 1)/2 random
couplings. This difference appears to be quantitatively important, as neither of these
latter two models showed good evidence of thermalization under the tests illustrated
above for the sizes that can be exactly diagonalized. But we expect that in the limit
of large N this difference should go away, with all of these models thermalizing.
2.9
Conclusion
In this chapter, we confirm that there are indeed three different eigenstate phases
(F1, F2, F3) in the ferromagnetic phase of our infinite-range spin model by demonstrating both analytical and numerical evidence. Let me conclude this chapter by
indicating the physical differences among these phases.
In the lowest-energy phase (F1) where
√
α(u, Γ) > Σ( −2u − Γ2 ) ,
51
(2.84)
the many-body eigenstates come in almost-degenerate pairs that are well separated in energy from other pairs of eigenstates. These eigenstates are still thermal in
their fluctuations near equilibrium, so the inter-pair eigenenergy spacings have the
level statistics of the Gaussian Orthogonal Ensemble (GOE). Each pair of eigenstates
within phase F1 is well-approximated by the two states
1
|n, ±i = √ (| ↑in ± | ↓in ) ,
2
(2.85)
where | ↑in is a thermal state that is magnetized up, while | ↓in is its opposite under
the global spin flip symmetry; here the label n refers to such a pair of eigenstates.
In this regime (F1), one can make an up-magnetized initial state using a simple
linear combination of only these two eigenstates, and this linear combination will then
oscillate with a frequency ∼ exp (−N α(u, Γ)). Thus in this phase the Schrödinger
cats can in principle be made to oscillate between ‘alive’ (up) and ‘dead’ (down).
In the higher-energy parts of the ferromagnetic phase (F2 and F3 in Fig 2.1) where
√
α(u, Γ) < Σ( −2u − Γ2 ) ,
(2.86)
the energy associated with the tunneling is large compared to the many-body level
spacing, so there are no closely degenerate pairs of states from which we can make a
coherently oscillating cat. Instead, the eigenstates are of the form
1
|ni = √ (| ↑in + (−1)zn | ↓in ) ,
2
(2.87)
with each eigenstate being either even (zn = 0) or odd (zn = 1) under the global
spin flip; here the label n refers to just one eigenstate. States of opposite symmetry
(zn 6= zm ) that are nearly degenerate are not made out of the same thermal states in
each well, so n h↑ | ↑im → δnm for N → ∞. To make a state that is initially magnetized
52
up requires coherently adding together exponentially many eigenstates in order to
destructively cancel all the amplitudes for down magnetizations. Under the unitary
time evolution this special initial linear combination will dephase and the probability
of down magnetization will increase, but presumably in an overdamped fashion as
the state relaxes to equal probability of up and down magnetization. Thus we expect
the dynamics of the average magnetization to always be an overdamped relaxation
in phases F2 and F3. In phase F1, on the other hand, one can in principle prepare
an initial state whose macroscopic magnetization will oscillate, as discussed above.
However, even in phase F1, if one starts in a generic state that has a given energy
density and is magnetized up, this will also be a linear combination of exponentially
many eigenstates and also presumable show an overdamped relaxation of the average
magnetization.
In the highest-energy phase (F3) where
√
Γ − 2u
πΓ
≥ ln
Γ + 2u
−u − Γ2
(2.88)
the saddle point is ‘thermal’: the ‘entropy cost’ of going to higher S is less than the
‘tunneling cost’, and the system goes over the barrier without any quantum tunneling.
We call the eigenstates in this regime ‘thermal cats’, since these Schrödinger cat states
flip by thermally activating themselves over the barrier. In regions F1 and F2 this
inequality is instead false, and the system quantum tunnels through the barrier at a
value of S satisfying Seq ≤ S < −E/Γ, so the eigenstates are instead ‘quantum cats’.
53
Chapter 3
Strong Disorder Renormalization
Group Approach to Many-body
Localization Transition
3.1
Introduction
As mentioned in Chapter 1, the many-body localization transition is an eigenstate
phase transition. But it is not as standard as the eigenstate phase transition in Chapter 2 where the eigenstates on both sides of the transition and at the transition are
all satisfying the ETH. Instead, the MBL phase is the only known generic exception
to thermalization [19]. We know that there are many properties of the MBL phase
which are not shared by the thermal phase, which does obey the ETH. For example,
the area law of the eigenstate entanglement entropy for the MBL phase versus the
volume law of the entanglement entropy for the thermal phase. These distinctions
between the two phases drive us to study this special phase transition. Although
there has been much recent progress in understanding the MBL phase by itself, for
example [28, 25, 14, 12, 15, 3, 23, 33, 4, 24], very little is known about the eigenstate
54
phase transition between the MBL phase and thermal phases. In 2014, Vosk, Huse
and Altman [30] formulated a new theoretical approach, a strong disorder renormalization group (RG) framework, to explore the physics of this critical point. However,
in their framework, parts of the RG rules are ad hoc and, in addition, their approach
seems to be more complicated than is necessary. This motivates us to think about
variations of the rules.
Ultimately, the set of RG rules we want is different from that found in [30], simpler
and less ad hoc. But it is not easy, and may not even be possible, to simultaneously
achieve all of these goals. I will not explain the RG scheme in detail until the next
section, but in order to get a rough idea of how we arrived at a certain set of RG
rules, here I would like to briefly mention the rules we initially tried. The RG scheme
is for a one dimensional chain containing many blocks. Each block is a section of the
system that all by itself looks either thermal or insulating and has a dimensionless
quantity g to measure how thermal or how insulating the block is. The RG rules are
just the rules for coarse-graining: we fuse several adjacent blocks into a new longer
block and determine the parameters of the new block. As the RG flows, the number
of blocks reduces steadily. The first two simple rules are the case of combining two
adjacent thermal blocks and the case of combining two adjacent insulating blocks as
shown in Fig. 3.1(a) and 3.1(b). As expected, after the combination, the new block
becomes more thermal, or more insulating, respectively. But these two rules are of
course not sufficient since they do not address the situation of a thermal block next
to an insulating block. Thus, we need to construct rules for the cases where a thermal
block meets with an insulating block. But not all such cases are simple, and it is for
such cases that Ref. [30] introduced somewhat arbitrary rules, and in the end, so
do we here. Also, it is in these cases where the competition between thermalization
and localization lies. A configuration for which there exists a natural expectation for
the RG flow is when two blocks of the same type surround a block of the opposite
55
type: that is, either a sequence of thermal, insulating and thermal, or a sequence of
insulating, thermal and insulating blocks as shown in Fig. 3.1(c) and 3.1(d). If the
Thermal
Block
Thermal
Block
Insulating Insulating
Block
Block
Thermal Block
Insulating Block
(a) TT move
Thermal
Block
Insulating
Block
(b) II move
Thermal
Block
Insulating
Block
Thermal Block
Thermal
Block
Insulating
Block
Insulating Block
(c) TIT move
(d) ITI move
Figure 3.1: A sketch of typical RG moves. For example, (a) is to fuse two adjacent
thermal (T) blocks into one thermal block, called “TT move”; all others are similar.
blocks are consecutively thermal, insulating and thermal, we are able to set a simple
rule that combines them into a new effective block that ensures, as expected, that
this new block is thermal. This RG rule applies when the two thermal blocks are
large enough to thermalize the insulating block in between them.
However, difficulties arise in the other scenario, a sequence of insulating, thermal
and insulating blocks. Our first trial at setting this rule in this case was an attempt
to stay close to the microscopic physics and avoid an ad hoc RG rule. But the natural
formula for such an RG step does not ensure the new block is insulating. In this first
56
trial, we simply forbid this RG step from happening if the new block turns out not
to be insulating. But the numerical results show that this version of our RG gets
stuck when the system is near the phase transition. This means that the system flows
under this RG to configurations from which no moves are possible. This means that
our basic goal of studying the RG fixed point governing the phase transition is not
possible since no such fixed point exists in this version of the RG. In order to resolve
this, we apparently need to modify the rules so the system never gets stuck. This
motivates our second trial: allowing the move (combining insulating (I), thermal (T)
and insulating (I) blocks, called “ITI move”) any time. And in this scenario, when
the new block happens to not be insulating, we set it as a thermal block, and at
the same time combine it with the two other thermal blocks which are adjacent to
this new thermal block. After this treatment, the final new block is thermal. This
successfully solves the previous issue of the RG getting stuck, however, this introduces
new difficulties. When we tried this set of RG rules, we found that the RG becomes
very unstable and oscillates, such that we cannot even see an RG fixed point and the
RG flow appears to go towards a limit cycle at the phase transition, again preventing
our goal of studying the fixed point. After these two failed attempts at developing a
new RG, we decided to be less ambitious and to give up the goal of avoiding ad hoc
rules. The hope is to develop a simple “toy” RG for this phase transition that can
then be improved and built upon.
One solution to the difficulties with the first two RG schemes is to impose a
constraint that the new block of the basic ITI move must be an insulating block.
We include a rule for our ITI move that satisfies this constraint. This gives us our
“final” version of the RG rules. Although this version is still ad hoc, the RG equation
is largely simplified and we can nearly get an analytical solution to the fixed point
distribution and also the critical exponent. Thus, this version becomes a “toy model”
and we can explore this model in great detail. It also gives us hope for studying the
57
real physical models by perturbing this toy model to be less ad hoc and to use this
toy model as a reference point. Here in this chapter, I will discuss this toy model in
detail after setting the RG scheme.
3.2
RG Scheme
Let us consider a generic one dimensional chain of “blocks” with different length.
The microscopic details of the chain are coarse-grained by doing the renormalization
group flow and only a minimal set of parameters remains in the coarse-grained system.
This suggests that this approach can describe several microscopically distinct systems,
including, but not limited to, spin chains and lattice particles. For now, let us assume
for specificity that it is a spin-1/2 chain.
Assume that there are, in total, N blocks numbered from 1 to N . We impose
periodic boundary conditions such that the N th block is adjacent to the 1st block.
This is convenient because the final goal is to take the thermodynamic limit N → ∞
and the periodic boundary conditions eliminate boundary effects when we coarsegrain the system with finite N . We characterize each block i by only two parameters:
Γi and ∆i . Γi is the end-to-end entanglement rate. The physical meaning of Γi can
be obtained in the following setup: One connects the block i at one end to a much
longer thermalizing block j where j can serve as a heat bath to thermalize i, and one
isolates them from the environment. Then the two block system starts with a pure
product state (no entanglement between blocks i and j) and one evolves the state
under the unitary time evolution due to the Hamiltonian of these two blocks. The
entanglement entropy will grow from 0 and saturate at the time scale τi = 1/Γi . ∆i is
the typical many-body level spacing of block i which is set by the length of the block.
If we denote the length of block i as li , then for a spin-1/2 chain the dimension of
the Hilbert space is 2li . So we have the typical level spacing ∆i ∼ 2−li , up to some
58
prefactors that are a power of li . Note that we are considering the strong disorder RG
and assuming the critical fixed point is at infinite randomness, which means that at
the fixed point we study, the logarithm of the rates Γi and ∆i have a broad probability
distribution [11]. Thus, these power law prefactors can be neglected in this strong
disorder RG scheme.
Next, we divide these N blocks into two groups, depending on whether a block
locally behaves more like an insulator (called “I-block”) or more like a thermalizing
system (called “T-block”). If a block only exhibits short range entanglement in its
eigenstates, it is considered an insulator. On the other hand, in a thermal block, the
entanglement is spreading fast and the whole block is entangled. In terms of the two
parameters Γi and ∆i , block i is considered as an I-block when Γi ∆i and it is a
T-block when Γi ∆i . If we define a dimensionless parameter gi as follows:
gi ≡
Γi
,
∆i
(3.1)
we compare whether gi is much smaller or larger than 1 to decide whether block i
is insulating or thermal, respectively. Still, note that we are considering the strong
disorder RG, so when we are approaching the critical point, the probability of having
g ∼ 1 vanishes. In practice, we treat any g > 1 as g 1, and any g < 1 as g 1.
Before proceeding, let me mention several issues about Γ and ∆. The first issue
is the dimension. Although both Γ and ∆ are dimensionful quantities with the same
dimensions (energy), we can redefine them as dimensionless quantities, relative to a
typical microscopic energy scale in the system: Γ0i ≡ Γi /E , ∆0i ≡ ∆i /E where E is a
typical energy scale at the microscopic scale. The second issue is the regime. We are
interested in the low energy regime where both Γi 1 and ∆i 1 for any block i.
For ∆i , because ∆i ∼ 2−li for block i where li is the length of the block, it is reasonable
to assume ∆i 1, since we are considering long, coarse-grained blocks. For Γi , we
59
split the discussion into two cases: one for insulating blocks, one for thermal blocks.
First, if the block is insulating, the many-body level spacing for that single isolated
block is large and the eigenstates are localized. This means that the entanglement
spreading is basically due to quantum tunneling and is exponentially slow in the
length. In the second case, for a thermal block, if the block were deeply thermal,
the entanglement would spread fast. But the system is now near the critical point
and the Griffiths effect takes place [30]. The Griffiths effect says that even within a
thermal block there will be shorter segments that are localized and act as a bottleneck
for the entanglement spreading across the thermal block. The entanglement needs to
tunnel through these bottlenecks. Hence the entanglement spreading rate Γi is also
slow even in thermal blocks, when we are near the phase transition. Thus, we have
Γi 1 for all blocks.
After parametrizing each block, we can discuss the coarse-graining procedure, the
essence of the RG flow. By coarse-graining the chain, we are combining several blocks
into one single “new” block with only two new parameters Γnew and ∆new remaining
as a function of the previous parameters. In this way, we reduce the amount of microscopically detailed information. In this framework, we focus more and more on the
physics on longer length scales, which correspond to lower energy scales. In the theory
of phase transition, it is this low-energy, long-length-scale limit where the behavior
is universal, and the RG is set up to capture that universal behavior by focusing on
these limits. Here we are assuming that this general framework will also apply to this
new type of eigenstate phase transition between MBL and thermalization.
3.2.1
RG Rules
After briefly describing the RG framework, we are now ready to discuss the RG
rules, which govern how the new parameters relate to the old ones.
60
Building Up from Simple Cases
First comes the simple rules for combining two adjacent I-blocks or two adjacent
T-blocks. Consider the two adjacent I-blocks case, and call the blocks block 1 and
block 2. Let the parameters for block i be Γi and ∆i as before, where i = 1, 2. For the
new Γ, we multiply the two old Γ’s. This is because in insulators, the entanglement
spreads basically by quantum tunneling and the spreading time is exponentially slow.
For the new ∆, we similarly multiply the two old ∆’s. This is because ∆ is roughly
the inverse of the dimension of the Hilbert space of the corresponding block and when
we combine the blocks, the dimension of the Hilbert space multiplies. And this is
true in general, so for all the rules for ∆, we just get the new ∆ by multiplying all
the original ∆’s involved in the combination. So for the two I-blocks case, we get the
following RG rule:


 Γnew = Γ1 Γ2 ,
(3.2)

 ∆new = ∆1 ∆2 .
Consider now two adjacent T-blocks. For the new Γ, because both blocks are thermal
blocks, they are both well entangled and the entanglement is spreading fast and the
spreading time basically adds. This means that the inverse of the rates add:
Γnew =
1
.
1
1
+
Γ1 Γ2
(3.3)
But we are dealing with the strong disorder RG and we will assume that we are
working near an infinite-randomness fixed point, which means that Γ1 will be either
much larger than or much smaller than Γ2 . In both cases, the right hand side of the
above formula is approximately equal to the minimum of Γ1 and Γ2 (the slower of the
61
two rates is the “bottleneck”):
Γnew ≈ min{Γ1 , Γ2 } .
(3.4)
And the formula for ∆new is the same as in the I-blocks case, for the same reason. So
for the two T-blocks case, we arrive at the following rules:


 Γnew = min{Γ1 , Γ2 } ,
(3.5)

 ∆new = ∆1 ∆2 .
By combining neighboring blocks of the same type, one would naturally expect
that the resulting new block must share the same type. This can be seen by comparing
the new Γ and new ∆. For the insulating case, we have Γ1 ∆1 and Γ2 ∆2 by
definition. So we have
Γnew = Γ1 Γ2 ∆1 ∆2 = ∆new .
(3.6)
Thus, the new block is an insulating block, as expected. And in fact, the new block
is more insulating than the original ones in the sense that the ratio g ≡ Γ/∆ in the
new block is smaller than the original g’s. Note that g 1 is the condition for a
block being insulating, and the smaller value a block has for g, the more insulating it
is. For the thermal case, we have Γ1 ∆1 and Γ2 ∆2 by definition. As mentioned
before, we are interested in the regime where Γi , ∆i 1 for any block i. So we have
Γ1 ∆1 ∆2
Γ2 ∆1 ∆2 .
(3.7)
Γnew = min{Γ1 , Γ2 } ∆1 ∆2 = ∆new .
(3.8)
and
Thus,
62
Therefore, the new block is a thermal block, as expected. Similarly, we can also show
as follows that the new ratio gnew ≡ Γnew /∆new is greater than both g1 and g2 which
indicates that the new block is more thermal. By the exchange symmetry between
subscripts 1 and 2, we only need to show gnew > g1 which is
gnew =
min{Γ1 , Γ2 }
Γ1
> g1 =
.
∆1 ∆2
∆1
(3.9)
We know that
Γ1
Γ1
>
∆1 ∆2
∆1
(3.10)
Γ2
1
Γ1
>
>
∆1 ∆2
∆1
∆1
(3.11)
because ∆2 < 1;
because g2 > 1 and Γ1 < 1. Combining the above two expressions, we get the desired
statement gnew > g1 .
In these two simple cases, we showed that any two adjacent blocks which are both
insulating or both thermal can combine into one block which is even more insulating
or more thermal, respectively. This means that as a first step in the RG, we can
combine sequences of adjacent I-blocks or T-blocks together, which then result in a
sequence of the blocks alternating between I-blocks and T-blocks:
···
I T I T I T
··· .
Next we must set up a series of RG rules to do the coarse-graining starting from this
new pattern.
63
Order of Flow: the Cutoff
In our “toy” model, we have two RG rules. The first rule is to combine 3 adjacent
T-I-T blocks into a new T-block (called “TIT rule”) and the second rule is to combine
3 adjacent I-T-I blocks into a new I-block (called “ITI rule”). Then the RG will run
by “integrating out” the fastest block. To determine how fast a block is, we define
the rate Ωi for each block i as follows. For thermal blocks we use ∆ and for insulating
blocks we use Γ:


 ∆T ,
Ωi =

 Γ ,
I
if the block is thermal;
(3.12)
if the block is insulating.
In this way, we can make a list ordered by Ωi . Then the maximal value in this list is
defined as the cutoff Ω:
Ω = max{Ωi } .
i
(3.13)
At every iteration of the RG, we select the move in which the central block i has the
Ωi value equal to the cutoff Ω. In this way, we first ensure that the flow will never
get stuck because the maximal value of Ωi always exists. Second, by selecting the
maximal energy Ω every time, we lower the cutoff steadily, and focus more and more
on the lower energy behavior, which is the spirit of the RG framework. After setting
the cutoff Ω, let me discuss both rules one by one in detail.
TIT Rule
In order to address both rules, we need to understand what happens physically
when we connect a T-block with parameters ΓT , ∆T to an I-block with parameters
ΓI , ∆I . If we treat the combined two blocks as an isolated quantum system, it is
natural to expect the final result depends on whether the thermal block can serve
64
as a heat bath that can thermalize the I-block. We argue that the condition for a
T-block to thermalize an I-block is ΓI ∆T . Note that, by definition, ΓI is the
end-to-end entanglement rate of the I-block when it combines with a large enough
thermal block. So the rate will remain ΓI as long as the T-block considered here
has the level spacing ∆T ΓI . In this way, the T-block has a continuum spectrum
relative to ΓI . So ΓI ∆T is the condition that the T-block serves as a heat bath
and thermalizes the I-block.
After getting the condition for a T-block thermalizing an adjacent I-block, we then
set our TIT rule. We denote these three blocks as 1, 2 and 3 with the type T, I and
T, respectively. In order for these three blocks to combine to one new thermal block,
we need at least one of the T-blocks to be able to thermalize the middle I-block. The
RG only does this move if the I-block is faster than both T-blocks, since it “integrates
out” the fastest blocks first. So we have
Γ2 ∆1
and
Γ2 ∆3 .
(3.14)
Under this condition, we can derive the parameters of the new block Γnew and ∆new .
For Γnew , the argument is similar as the two T-blocks (TT) simple case. Since the
middle block is successfully thermalized, it is well entangled with the thermal blocks
and the entanglement spreading is fast. Thus the entanglement time still adds as in
the TT case. So Γnew becomes the minimal value of all 3 original Γ’s as it is in the
TT case:
Γnew = min{Γ1 , Γ2 , Γ3 } .
(3.15)
And the rule for ∆new is still as simple as before:
∆new = ∆1 ∆2 ∆3 .
65
(3.16)
Note that these two formulae ensure that the new block is indeed a thermal block.
We can check it by evaluating the parameter gnew = Γnew /∆new . Because
Γ1
Γ1
1
∆1 ∆2 ∆3
∆1
(3.17)
and similarly for Γ2 or Γ3 as being the numerator. Then we have
gnew =
Γnew
min{Γ1 , Γ2 , Γ3 }
=
1
∆new
∆1 ∆2 ∆3
(3.18)
which indicates the new block is automatically thermal.
ITI Rule
Next comes the ITI rule, which is the RG rule that is difficult to determine in
any controlled way, and was the source of the difficulties in our attempts to avoid
making ad hoc assumptions. Here we also denote the 3 consecutive blocks as 1, 2 and
3; this time block 2 is the middle T-block, which is the fastest of these three blocks.
Similar to the TIT rule, this time we need the middle T-block to not be able to by
itself thermalize either block 1 or block 3. So one condition for the ITI rule is that
∆2 Γ1
and
∆2 Γ3 .
(3.19)
The ∆new is still the product of the original ∆’s; this follows simply from the dimension
of the product Hilbert space:
∆new = ∆1 ∆2 ∆3 .
(3.20)
For Γnew in this case, things become a little arbitrary, as mentioned before, and I
will explain this later on. Because neither block 1 nor block 3 is thermalized by the
middle T-block, both of them are insulating. The entanglement spreads very slowly
66
in both block 1 and block 3. As in the two I-blocks case, Γnew must at least contain
the factor Γ1 Γ3 . But if we just set Γ1 Γ3 as the new Γ, it would not guarantee that the
new block is insulating or gnew 1 as we would expect. In order to fix this issue, we
add a multiplier which is smaller than or equal to ∆2 to ensure that the new block is
insulating. To minimize the modification, we set
Γnew = Γ1 ∆2 Γ3 .
(3.21)
Although this ITI rule for Γnew is a bit arbitrary, it turns out to give us a beautifully
simple solution to the RG fixed point. Moreover, we can almost tackle it analytically
a la Fisher [11]! I will present the results in the next section.
3.2.2
RG Flow of Probability Distributions
We can run this RG flow from any given chain of alternating T and I blocks. For
example, we can start from drawing Γ’s and ∆’s from a given probability distribution
and form the chain. Then our RG rules give us an evolution of the probability
distribution. In general, the distribution is 2n dimensional if the total number of
blocks is n:
P ({ln Γi }, {ln ∆i }) ,
(3.22)
where we take the natural logarithm for convenience. Here we assume the parameters
in different blocks are independent and one can show that this remains true under the
RG. So the distribution simplifies to the following two joint probability distributions:
PT (ln ΓT , ln ∆T )
and
67
PI (ln ΓI , ln ∆I ) ,
(3.23)
where the subscripts T and I indicate the T-block distribution and I-block distribution
because these two distributions should in general differ from each other. One further
simplification can be made based on the following observation. If we focus on the
parameter ΓT , by checking all the formulae, we see that the ΓT ’s do not influence any
of the other quantities, only themselves. So we can ignore the parameter ΓT in the
distribution PT . The distributions then become
PT (ln ∆T )
and
PI (ln ΓI , ln ∆I ) .
(3.24)
Our aim is to find out the scaled distributions for Γ’s and ∆’s at the critical point
which, by assumption, should not change because the critical point typically behaves
as a fixed point in the RG flow. And note that the parameters of the distributions
should also be rescaled by the cutoff, so in the end, we are interested in the evolution
of the following distribution functions:
PT
3.3
ln ∆ T
ln Ω
and
PI
ln ∆I .
ln Ω ln Ω
ln Γ
I
,
(3.25)
Results
As mentioned previously, the solution to the fixed point distribution can be found
nearly analytically. In this section, I will present both analytical and numerical results
regarding the fixed point distribution and the critical exponent.
3.3.1
Fixed Point Distribution
First, let us try to simplify the form of the fixed point distribution. We have
3 variables: ln ∆T , ln ΓI and ln ∆I . All of them are proportional (or extensive) to
ln Ω when we send the cutoff ln Ω → −∞. One nice feature is that when the RG
flows, ln gI becomes sub-extensive to ln Ω because when we apply the ITI rule to three
68
consecutive blocks 1,2 and 3,
ln gInew
(ln Γ1 + ln ∆2 + ln Γ3 ) − (ln ∆1 + ln ∆2 + ln ∆3 )
=
new
ln ΓI
ln Γ1 + ln ∆2 + ln Γ3
ln g1 + ln g3
=
.
ln Γ1 + ln Γ3 + ln ∆2
(3.26)
If ln ∆2 were zero, then the ratio ln gI / ln ΓI would become fixed when the distribution
goes towards the fixed point. But here ln ∆2 is extensive, which means that when all
the other quantities approach the fixed point, ln gI / ln ΓI are still steadily decaying.
Thus,
ln gI
ln gI
∼
→ 0.
ln Ω
ln ΓI
(3.27)
This is a very nice feature because we can then treat the following two intensive
quantities as equal at the fixed point:
ln ΓI
ln ∆I
=
.
ln Ω
ln Ω
(3.28)
Note that this does not affect the strong disorder of the fixed point where ln gT,I are
far away from 0, because we have
ln gI = ln g1 + ln g3 ,
(3.29)
the absolute value of ln gI becomes larger and larger. The numerical simulation of
the RG flow shows that | ln gI | ∼ | ln Ω|p where the power p ≈ 0.8. It verifies while
ln gI is growing by itself, ln gI / ln Ω is still an irrelevant operator. Then we only have
two variables: ∆T and ∆I . Because the rules for ∆T and ∆I are symmetric between
T and I, their distributions will be equal at the fixed point. This further reduces the
number of variables to one when we consider only the fixed point; we denote this
69
variable by ln ∆. And the RG rule for it is simply
ln ∆new = ln ∆1 + ln ∆2 + ln ∆3
(3.30)
when we merge blocks 1, 2 and 3, which we do only when ∆2 = Ω, so ∆2 ∆1 and
∆2 ∆3 .
After reducing the problem to a single variable problem, we may then construct
the analytical equation for the probability distribution, following Fisher [11]. We
define two quantities as Fisher did:
Λ ≡ − ln Ω
(3.31)
and
ζ ≡ ln
Ω
∆
.
(3.32)
Then we have the constraint ζ ≥ 0 and large ζ corresponds to small ∆. In terms of
Λ and ζ, the RG rule Eq. (3.30) becomes
ζnew = ζ1 + ζ2 + ζ3 + 2Λ = ζ1 + ζ3 + 2Λ ,
(3.33)
where ζ2 = 0 because the second block is at the cutoff: ∆2 = Ω.
Let the distribution density of ζ at cutoff Λ be ρ(ζ, Λ), such that the probability
of ζ in the interval [ζ, ζ + dζ] is ρ(ζ, Λ)dζ. As the cutoff Λ gets large, the distribution
of ζ becomes broad. This leads us to consider the following transformation: let
η≡
ζ
,
Λ
ρ(ζ, Λ) ≡
1
1
ζ
Q(η, Λ) = Q( , Λ) .
Λ
Λ Λ
70
(3.34)
Thus, Q(η, Λ) is just the probability distribution for η when the cutoff is Λ, and the
RG rule for η is
ηnew = η1 + η3 + 2 ,
(3.35)
which follows from Eq. (3.33).
What we want to find out is the evolution of the distribution Q(η, Λ) when the
cutoff Λ is increasing, namely
distribution ρ(ζ, Λ), namely
∂Q(η,Λ)
.
∂Λ
∂ρ(ζ,Λ)
.
∂Λ
But it is easier to find out the evolution of the
Thus we analyze the process for the change of ρ
when the cutoff Λ changes by dΛ. In order to successfully change the cutoff, we need
to first eliminate all the blocks which have the ln ∆ in the interval [ln Ω + d ln Ω, ln Ω],
or [ln Ω − dΛ, ln Ω] in terms of dΛ. In terms of ζ, the points (blocks) we need to
remove have the value ζ in the interval [0, dΛ] which has a fraction of points equal
to ρ(0, Λ)dΛ. After removing all the points in this interval, we then re-normalize the
distribution so that it integrates to 1. Up to now, the available ζ is in the interval
[dΛ, +∞). Thus, the next step after re-normalization is to shift the lower edge of the
the distribution horizontally by dΛ so that the domain of ζ returns to be [0, +∞).
By finishing these steps we get the evolution of the distribution ρ(ζ, Λ), and the final
step is to rescale the variable ζ into η to get the evolution of the distribution Q(η, Λ).
Let me analyze these steps one by one.
The first step is to remove all ζ in the interval [0, dΛ]. These are the fast blocks that
are being integrated out. Each time we remove one ζ in this interval, we randomly pick
up two other points ζa and ζb with probabilities ρ(ζa , Λ)dζa and ρ(ζb , Λ)dζb separately.
These are the two neighboring blocks. These three points form the three original
blocks in the TIT or ITI move. Based on the RG rule Eq. (3.33), the new block we
generate has ζc = ζa + ζb + 2Λ. But the increase at ζc is not only from the pair ζa and
ζb , but also from all other pairs, so long as the sum of the pair is equal to ζa + ζb , or
71
ζc − 2Λ. Thus, the total probability of selecting ζc in dζc is
∞
Z
dζc
∞
Z
dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζc − ζa − ζb − 2Λ) .
dζa
0
(3.36)
0
If we consider the change of ρ at ζ, we have three cases: ζ = ζa , ζ = ζb or ζ = ζc . In
other words, for the selection of ζa (or ζb , ζc ), it only affects the distribution at point
ζa (or ζb , ζc ). And in the first step, we in total need to do a fraction ρ(0, Λ)dΛ of RG
moves. Therefore, in this step the total change in the probability density function at
ζ is
h
dρ(ζ, Λ) = ρ(0, Λ)dΛ − ρ(ζ, Λ) − ρ(ζ, Λ)
Z ∞
Z ∞
i
dζa
dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ) .
+
0
0
(3.37)
The second step is to re-normalize ρ such that the integral of ζ is 1. In the first
step, the integral of ρ(ζ, Λ) is decreased: for the selection of ζa , we lose
∞
Z
ρ(0, Λ)dΛ
dζa ρ(ζa , Λ) = ρ(0, Λ)dΛ ;
(3.38)
dζb ρ(ζb , Λ) = ρ(0, Λ)dΛ ;
(3.39)
0
for the selection of ζb , we lose
Z
ρ(0, Λ)dΛ
∞
0
for the selection of ζc , we gain
Z
ρ(0, Λ)dΛ
∞
Z
dζc
0
∞
Z
0
∞
dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζc − ζa − ζb − 2Λ) = ρ(0, Λ)dΛ ;
dζa
0
(3.40)
72
for the selection at the cutoff, ζ = 0, we lose ρ(0, Λ)dΛ. Thus, we in total lose
2ρ(0, Λ)dΛ. In order to re-normalize ρ, we need to amplify it everywhere by a factor
of 1 + 2ρ(0, Λ)dΛ. The resulting ρ after re-normalization is
n
h
ρ(ζ, Λ) + ρ(0, Λ)dΛ − 2ρ(ζ, Λ)
Z ∞
Z ∞
io
+
dζa
dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ) · (1 + 2ρ(0, Λ)dΛ)
0
0
h
= ρ(ζ, Λ) + ρ(0, Λ)dΛ − 2ρ(ζ, Λ)
Z ∞
Z ∞
i
dζa
dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ) + ρ(ζ, Λ) · 2ρ(0, Λ)dΛ
+
0
0
Z ∞
Z ∞
= ρ(ζ, Λ) + ρ(0, Λ)dΛ
dζa
dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ) .
(3.41)
0
0
The third step is to shift the whole function horizontally to start from 0. The
amount of this shift is just dΛ. So the change ρ in this step is
∂ρ
dΛ.
∂ζ
After this, we
complete one infinitesimal process of changing the distribution. The total amount of
change is
Z
∞
Z
∞
dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ) +
dζa
dρ = ρ(0, Λ)dΛ
0
0
∂ρ
dΛ
∂ζ
∂ρ
≡
dΛ .
∂Λ
(3.42)
Thus, we obtain the equation for the RG flow of the probability distribution ρ(ζ, Λ):
∂ρ
∂ρ
=
+ ρ(0, Λ)
∂Λ
∂ζ
Z
∞
Z
0
∞
dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ) .
dζa
(3.43)
0
The last step is to rescale ζ into η, and we have
i
∂ρ
∂ h1
ζ
=
Q( , Λ)
∂Λ
∂Λ Λ Λ
1
η ∂Q
1 ∂Q
+
;
= − 2Q − 2
Λ
Λ ∂η
Λ ∂Λ
73
(3.44)
and
∂ρ
1 ∂Q
= 2
.
∂ζ
Λ ∂η
(3.45)
The integral becomes
Z
∞
Z
∞
dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ)
dζa
0
0
ζ−2Λ
Z
ρ(ζa , Λ)ρ(ζ − ζa − 2Λ, Λ)dζa
=Θ(ζ − 2Λ)
0
1
= Θ(η − 2)
Λ
Z
η−2
Q(ηa , Λ)Q(η − ηa − 2, Λ)dηa ,
(3.46)
0
where Θ(x) is the step function. By plugging these relations into the equation of ρ,
we arrive at
Z η−2
i
∂Q
∂ h
=
(1 + η)Q + Q(0, Λ)Θ(η − 2)
Q(ηa , Λ)Q(η − ηa − 2, Λ)dηa .
Λ
∂Λ
∂η
0
(3.47)
Given the “toy” RG rules we are using, this (3.47) is the RG equation for the probability distribution Q flowing as a function of the cutoff Λ at the critical point, where
the T and I blocks have the same distributions of ∆. We will return to the RG flow
away from this fixed point later. When the left hand side of this equation is equal
to zero, the solution, denoted by Q∗ (η), becomes the critical fixed point distribution
which is independent of Λ. The fixed point distribution Q∗ (η) satisfies
i
dh
(1 + η)Q∗ + Q∗ (0)Θ(η − 2)
dη
Z
η−2
Q∗ (ηa )Q∗ (η − ηa − 2)dηa = 0 .
0
74
(3.48)
After getting the equation Eq. (3.48) for the fixed point distribution Q∗ (η), we
can then try to solve it. First is the value at zero, Q∗ (0). We can show that
Q∗ (0) =
1
2
(3.49)
based on the following argument: Consider the sum of ln ∆i over all the blocks indicated by i when the system is at the fixed point. By the rule of ln ∆, Eq. (3.30)
conserves this summation, so we know that as the RG flows, this quantity stays
constant. And we have
X
ln ∆i = −Λ
X
(1 + ηi ) = −ΛN (Λ)hηif.p.
i
(3.50)
i
where N (Λ) is the total number of blocks when the cutoff is Λ and hηif.p. is the
average value of η at the fixed point which is independent of Λ. This means the
product ΛN is a constant, so
ΛdN + N dΛ = 0 .
(3.51)
In addition, based on the analysis of obtaining ρ(ζ, Λ), when the cutoff is moving by
an infinitesimal dΛ, we lose a fraction of 2ρ(0, Λ)dΛ points (blocks). This means
dN
= −2ρ(0, Λ)dΛ .
N
(3.52)
Combining the two equations we get
ρ(0, Λ) = −
1 dN
1
=
,
2 N dΛ
2Λ
75
(3.53)
thus
Q∗ (0) ≡ Λρ(0, Λ) =
1
.
2
(3.54)
It is very helpful to obtain this Q∗ (0) value because it serves as an initial condition for determining the rest of the Q∗ (η), and in fact, we can integrate Eq. (3.48)
iteratively as follows. First consider the interval 0 < η < 2. In this interval, the step
function Θ(η − 2) kills the contribution from the integral in Eq. (3.48). And the
equation reduces to
i
dh
(1 + η)Q∗ = 0 ,
dη
for 0 < η < 2 .
(3.55)
Then the solution in this interval is simply given Q∗ (0) = 1/2:
Q∗ (η) =
1
,
2(1 + η)
for 0 < η < 2 .
(3.56)
Second is for the interval 2 < η < 4. When η is in this interval, the integral in Eq.
(3.48) starts contributing. But it contributes in a simple way: the integrand only
depends on values of Q∗ (η) up to η = 2, which I have just presented. So we have
Z
i
1 η−2
1
1
dh
∗
(1 + η)Q = −
·
dηa
dη
2 0
2(1 + ηa ) 2[1 + (η − ηa − 2)]
ln(η − 1)
=−
, for 2 < η < 4 .
4η
(3.57)
Thus, we obtain a closed form for Q∗ (η) in 2 < η < 4:
1 h1
Q (η) =
−
1+η 2
∗
Z
2
η
ln(η 0 − 1) 0 i
dη ,
4η 0
76
for 2 < η < 4 ,
(3.58)
where [(1 + η)Q∗ (η)]|η=2 = 1/2 serves as the initial condition. In principle, we can
get the analytical form of Q∗ (η) for any η ≥ 0 by using exactly the same iterating
method and treating Q∗ (η) piecewise. This shows that the physical solution to Eq.
(3.48) is unique. In practice, we obtain Q∗ (η) to numerical precision by doing this
iteration numerically, as is discussed below.
After getting the solution Q∗ (η), let me study the behavior of this function to get
an idea of what this function looks like. First, one can show that although Q∗ (η) is
piecewise, it is continuous up to (2l − 1)th order derivative at the connecting point
η = 2l, where l is a positive integer. This can be proven using mathematical induction
as follows. At η = 2, from the analytical solution from both sides, Eq. (3.56) and
(3.58), we see that Q∗ is continuous in its first order derivative but the second order
derivative has a finite jump: ∆Q∗(2) (2) = Q∗(2) (2+ ) − Q∗(2) (2− ) 6= 0. Then by taking
nth order derivatives on Eq. (3.48) at the argument η > 2, we have
n−2
∗(n)
(1 + η)Q
∗(n−1)
+ nQ
1 X ∗(k)
Q (η − 2)Q∗(n−2−k) (0+ )
+
2 k=0
Z
1 η−2 ∗
+
Q (ηa )Q∗(n−1) (η − ηa − 2)dηa = 0 ,
2 0
(3.59)
where we have already plugged in Q∗ (0) = 1/2. Then, we evaluate this equation at
η = 2l+ and η = 2l− and take the difference between them to get
h
i
h
i
(1 + 2l) Q∗(n) (2l+ ) − Q∗(n) (2l− ) + n Q∗(n−1) (2l+ ) − Q∗(n−1) (2l− )
n−2
i
1 X ∗(n−2−k) + h ∗(k)
+
Q
(0 ) Q (2(l − 1)+ ) − Q∗(k) (2(l − 1)− )
2 k=0
Z
+
1 2(l−1) ∗
+
Q (ηa )Q∗(n−1) (2l − ηa − 2)dηa = 0 .
2 2(l−1)−
(3.60)
Assuming Q∗ (η) is continuous at 2l − 2 up to (2l − 3)th order derivative and with a
finite jump at (2l − 2)th order derivative, and since by definition Q∗ (η) is everywhere
77
continuous, we can easily see that Q∗ (η) is continuous up to (2l−1)th order derivative.
In addition, we have
∆Q∗(2l) (2l) =
1
∆Q∗(2l−2) (2l − 2) ,
2(1 + 2l)
(3.61)
so the jump size decreases exponentially fast.
Second, one can also study the asymptotic behavior as η → ∞. In Fisher’s paper
[11], he got the simple exponential distribution for the fixed point distribution from his
RG equation. So we expect that in our Q∗ (η), it will also have an exponential decay
when η → ∞. I can verify1 here that the Q∗ (η) in our case also decays exponentially.
Assume
Q∗ (η) ∼ CQ exp(−λQ η)
(3.62)
when η 1. Then we have
Z
η−2
∗
∗
Z
η−2−A
Q (ηa )Q (η − ηa − 2)dηa ≈
0
CQ2 e−λQ ηa e−λQ (η−ηa −2) dηa
A
≈ ηCQ2 e−λQ η+2λQ ,
(3.63)
where A is a large constant. Thus, Eq. (3.48) becomes
1
CQ e−λQ η − λQ CQ (1 + η)e−λQ η + ηCQ2 e−λQ η+2λQ ≈ 0 .
2
(3.64)
So the leading order is O(ηe−λQ η ) when η 1. By setting this leading order equal
to zero, we get
1
−λQ CQ ηe−λQ η + ηCQ2 e−λQ η+2λQ = 0 .
2
1
(3.65)
Although it is not a proof, the numerical integration result I will present later shows that it is
indeed the case.
78
Then we have an estimation of the exponent λQ which satisfies
1
λQ = CQ e2λQ .
2
(3.66)
This verifies the exponential decaying behavior when η 1.
After getting the analytical solution of Q∗ (η) for small η and some other analytical
properties, I would like to move on to the numerical solution of Q∗ (η) by doing the
numerical integral iteratively. In particular, 20000 points are evenly sampled in each
interval [2l, 2l + 2) for l = 1, 2, 3, · · · and the double integration for solving (1 +
η)Q∗ (η) is numerically performed using composite trapezoidal rule. The integration
is performed up to η = 20. The results are shown in Fig. 3.2. Fig. 3.2(a) gives
the numerical curve and Fig. 3.2(b) shows the curve in a semi-log plot as well as
a straight line from a linear regression for η > 5. It can be seen that the plot
already looks exponential for small η ∼ 2 and the linear fit for relative large η is
almost perfect. The fitting generates λQ ≈ 0.3725 and a normalization CQ ≈ 0.3537.
Since 2λQ exp(−2λQ ) ≈ 0.3537, we see 2λQ exp(−2λQ ) = CQ is satisfied. Fig. 3.2(c)
reports the cumulative area under the curve Q∗ (η) up to η = 20, and we see it seems to
converge to 1, confirming the normalization of Q∗ (η) to 1. In particular, until η = 20
the total area under the curve is 0.9995, and if we assume Q∗ (η) = CQ exp(−λQ η)
after η > 20, we can get an approximate asymptotic area 1.0000, which again confirms
the normalization.
3.3.2
Critical Exponent
After getting the theoretical solution for the critical fixed point distribution Q∗ (η)
for this “toy” RG, we may perturb the distribution a bit away from Q∗ (η) to study
the critical exponent given by the RG flow. In order to move off the critical point, we
need to allow T and I blocks to have different distributions of η when the cutoff is Λ:
79
100
0.5
10-1
0.3
Q ∗ ( η)
Q ∗ ( η)
0.4
0.2
10-2
10-3
0.1
0.00
RG solution
Fitted Curve
5
10
η
15
10-4 0
20
(a) Q∗ (η)
5
10
η
15
20
(b) Q∗ (η) in log scale
1.0
Cumulative Q ∗
0.8
0.6
0.4
0.2
0.00
5
10
η
15
20
(c) Cumulative Q∗ (η)
Figure 3.2: Q∗ (η) up to η = 20 using composite Trapezoidal rule. (a) shows the curve
and (b) demonstrates it in a semi-log plot with a linear regression fit. (c) plots the
cumulative area under the curve.
QT (η, Λ) and QI (η, Λ). In this section, we focus on perturbations that moves away
from the critical point, so QT (η, Λ) 6= QI (η, Λ). In general, this difference between
the distributions of ∆T and ∆I will drive the system away from the critical point and
move to either a thermal phase or a localized phase as the RG flows. But for the
same reason in the critical fixed point case, here we still set ln gI as sub-extensive, i.e.
ln gI / ln Ω → 0. This gives us ΓI = ∆I . So what we obtain is a set of two-variable
RG equations, and the cutoff Ω becomes
Ω = max{∆T , ∆I } .
80
(3.67)
Here we still use the parameters Λ, ζ and η defined in Eq. (3.31), (3.32) and (3.34).
We set ρT (ζ, Λ) and ρI (ζ, Λ) as the distributions of ζ for thermal blocks and insulating
blocks when the cutoff is Λ. Thus the relations between Q’s and ρ’s are still similar:
QT (η, Λ) ≡ ΛρT (ζ, Λ) and QI (η, Λ) ≡ ΛρI (ζ, Λ) .
(3.68)
Similar to the derivation of the critical fixed point distribution, here we also consider
the infinitesimal process when the cutoff is moving by dΛ. But now the contributions
from TIT move and ITI move are different. For TIT move, we use points in the regime
ρI (0, Λ)dΛ, because here the middle I-block will be at the cutoff. Then we randomly
select ζa , ζb from the distribution ρT (ζ, Λ) and merge these three points into a new
point with ζnew = ζa + ζb + 2Λ that adds to the T distribution. And similarly for ITI
move because of the exchange symmetry T ↔ I. Only after we use all the points in
ρT (0, Λ)dΛ and ρI (0, Λ)dΛ can we move the cutoff by dΛ.
Let us focus on the change in ρT before re-normalization. The change contains
two parts: first is to get rid of all points in ρT (0, Λ)dΛ and second is to get rid of
ζa , ζb and add a new point at ζa + ζb + 2Λ. For the second part, the change of ρT is
(similar to the analysis for the fixed point distribution)
Z
ρI (0, Λ)dΛ
∞
Z
dζa
0
∞
dζb ρT (ζa , Λ)ρT (ζb , Λ)·
0
h
i
· − δ(ζ − ζa ) − δ(ζ − ζb ) + δ(ζ − ζa − ζb − 2Λ) .
(3.69)
The prefactor ρI (0, Λ)dΛ is because this whole second part is from TIT move where
the middle I-block is at the cutoff. And the next step is the re-normalization. The
area decreases by
ρT (0, Λ)dΛ + ρI (0, Λ)dΛ ,
81
(3.70)
where the first term comes from ITI move and the second term comes from TIT
move. Thus, we need to multiply the whole distribution by a factor (1 + ρT (0, Λ)dΛ +
ρI (0, Λ)dΛ). In this way, ρT becomes
Z
n
ρT (ζ, Λ) + ρI (0, Λ)dΛ
∞
Z
∞
h
dζb ρT (ζa , Λ)ρT (ζb , Λ) · − δ(ζ − ζa )
dζa
0
0
io
− δ(ζ − ζb ) + δ(ζ − ζa − ζb − 2Λ) (1 + ρT (0, Λ)dΛ + ρI (0, Λ)dΛ)
h
i
= ρT (ζ, Λ) + ρT (ζ, Λ) ρT (0, Λ)dΛ − ρI (0, Λ)dΛ
Z ∞
Z ∞
+ ρI (0, Λ)dΛ
dζa
dζb ρT (ζa , Λ)ρT (ζb , Λ)δ(ζ − ζa − ζb − 2Λ)
0
0
(3.71)
The remaining procedure of shifting the function horizontally is exactly the same as
that in the fixed point distribution. Thus, we get the equation for ρT (ζ, Λ):
h
i
∂ρT
∂ρT
=
+ ρT (ζ, Λ) ρT (0, Λ) − ρI (0, Λ)
∂Λ
∂ζ
Z ∞
Z ∞
+ ρI (0, Λ)
dζa
dζb ρT (ζa , Λ)ρT (ζb , Λ)δ(ζ − ζa − ζb − 2Λ) . (3.72)
0
0
For ρI , all things are the same except for change in the subscript T ↔ I:
h
i
∂ρI
∂ρI
=
+ ρI (ζ, Λ) ρI (0, Λ) − ρT (0, Λ)
∂Λ
∂ζ
Z ∞
Z ∞
+ ρT (0, Λ)
dζa
dζb ρI (ζa , Λ)ρI (ζb , Λ)δ(ζ − ζa − ζb − 2Λ) . (3.73)
0
0
These two equations together form the set of RG equations for ρT (ζ, Λ) and ρI (ζ, Λ).
Then we still would like to transform the argument from ζ to η ≡ ζ/Λ. Similar to
82
the single variable case, after change of variables, we arrive at

h
i
∂QT
∂QT


Λ
= QT + (1 + η)
+ QT (η, Λ) QT (0, Λ) − QI (0, Λ)


∂Λ
∂η


Z η−2





dηa QT (ηa , Λ)QT (η − ηa − 2, Λ) ;
+ QI (0, Λ)Θ(η − 2)

0
h
i
∂QI
∂QI



Λ
= QI + (1 + η)
+ QI (η, Λ) QI (0, Λ) − QT (0, Λ)


∂Λ
∂η


Z η−2





dηa QI (ηa , Λ)QI (η − ηa − 2, Λ) .
+ QT (0, Λ)Θ(η − 2)
(3.74)
0
After getting the general set of RG equations for QT (η, Λ) and QI (η, Λ), we set these
two functions close to the fixed point distribution Q∗ (η):


QT (η, Λ) ≡ Q∗ (η) + δT (η, Λ) ,
(3.75)

 QI (η, Λ) ≡ Q∗ (η) + δI (η, Λ) ,
where δT,I are small compared to Q∗ so that we can omit the second order terms in
δT,I . And also, δT,I satisfy
Z
∞
δT,I (η, Λ)dη = 0
(3.76)
0
so that the QT,I are normalized. By plugging these into the equations for QT and QI
and omitting the second order terms in δT,I , we obtain
h
i
∂δT
∂δT
∗
Λ
= δT + (1 + η)
+ Q (η) δT (0, Λ) − δI (0, Λ)
∂Λ
∂η
Z η−2
+ δI (0, Λ)Θ(η − 2)
dηa Q∗ (ηa )Q∗ (η − ηa − 2)
Z η−2 0
+ Θ(η − 2)
dηa δT (ηa , Λ)Q∗ (η − ηa − 2)
0
83
(3.77)
and
h
i
∂δI
∂δI
∗
Λ
= δI + (1 + η)
+ Q (η) δI (0, Λ) − δT (0, Λ)
∂Λ
∂η
Z η−2
dηa Q∗ (ηa )Q∗ (η − ηa − 2)
+ δT (0, Λ)Θ(η − 2)
0
Z η−2
dηa δI (ηa , Λ)Q∗ (η − ηa − 2) .
+ Θ(η − 2)
(3.78)
0
In order to decouple δT and δI , we set their sum and difference as new variables:


δ+ (η, Λ) ≡ δT (η, Λ) + δI (η, Λ) ,
(3.79)

δ− (η, Λ) ≡ δT (η, Λ) − δI (η, Λ) .
By adding and subtracting Eq. (3.77) and (3.78) separately, we get the decoupled
equations for δ+ and δ− :
Z η−2
∂δ+
∂δ+
Λ
= δ+ + (1 + η)
+ δ+ (0, Λ)Θ(η − 2)
dηa Q∗ (ηa )Q∗ (η − ηa − 2)
∂Λ
∂η
0
Z η−2
+ Θ(η − 2)
dηa δ+ (ηa , Λ)Q∗ (η − ηa − 2)
(3.80)
0
and
Λ
∂δ−
∂δ−
= δ− + (1 + η)
+ 2Q∗ (η)δ− (0, Λ)
∂Λ
∂η
Z η−2
− δ− (0, Λ)Θ(η − 2)
dηa Q∗ (ηa )Q∗ (η − ηa − 2)
0
Z η−2
+ Θ(η − 2)
dηa δ− (ηa , Λ)Q∗ (η − ηa − 2) .
0
84
(3.81)
In general, they are both linear integro-differential equations, so the method of separating of variables works here. If we set


δ+ (η, Λ) ≡ f+ (η)g+ (Λ) ,
(3.82)

δ− (η, Λ) ≡ f− (η)g− (Λ) ,
the equation for δ+ reduces to
Z η−2
d ln g+
1 + η df+ f+ (0)
dηa Q∗ (ηa )Q∗ (η − ηa − 2)
=1+
+
Θ(η − 2)
d ln Λ
f+ (η) dη
f+ (η)
0
Z η−2
f+ (ηa ) ∗
dηa
+ Θ(η − 2)
Q (η − ηa − 2) .
(3.83)
f+ (η)
0
The left hand side of the above equation is a function of Λ while the right hand side
is a function of η. So they must be constant. We set this constant to 1/ν+ and we
get
1
d ln g+
=
,
d ln Λ
ν+
(3.84)
g+ ∝ Λ1/ν+ .
(3.85)
so
This gives us a general power law relation to the cutoff Λ which is the typical behavior
near the critical point of a phase transition. Then the remaining equation of f+ (η)
becomes an eigenvalue equation:
Z η−2
1
df+
f+ (η) = f+ (η) + (1 + η)
+ f+ (0)Θ(η − 2)
dηa Q∗ (ηa )Q∗ (η − ηa − 2)
ν+
dη
0
Z η−2
+ Θ(η − 2)
dηa f+ (ηa )Q∗ (η − ηa − 2) .
(3.86)
0
85
Similar analysis applies to the equation for δ− , by setting the form
g− ∝ Λ1/ν− ,
(3.87)
1
df−
f− (η) = f− (η) + (1 + η)
+ 2Q∗ (η)f− (0)
ν−
dη
Z η−2
dηa Q∗ (ηa )Q∗ (η − ηa − 2)
− f− (0)Θ(η − 2)
0
Z η−2
+ Θ(η − 2)
dηa f− (ηa )Q∗ (η − ηa − 2) .
(3.88)
we arrive at
0
These two eigenvalue equations can be solved numerically by discretizing them
into a matrix form and numerically diagonalizing the matrix. Before directly doing
that, let me discuss one property of the eigenfunctions f± (η), the constraint
Z
∞
f± (η)dη = 0
(3.89)
0
which comes from the similar constraint of δT,I , Eq. (3.76), and hence δ± . By
integrating η from 0 to ∞ in both sides of Eq. (3.86), we have
Z
0
∞
1
dη f+ (η) =
ν+
Z
∞
i
dh
dη
(1 + η)f+
dη
0
Z ∞ Z η−2
+ f+ (0)
dη
dηa Q∗ (ηa )Q∗ (η − ηa − 2)
Z ∞ 2Z η−2 0
+
dη
dηa f+ (ηa )Q∗ (η − ηa − 2) .
2
(3.90)
0
The first term in the right hand side of the above equation gives us −f+ (0) if we
R∞
assume that f+ (η) decays faster than 1/η when η → ∞ or the integral 0 f+ (η)dη is
finite. For the second and the third terms, we may apply the change of variables to
86
get the result:
∞
Z
Z
dη
2
η−2
dηa Q∗ (ηa )Q∗ (η − ηa − 2) = 1
(3.91)
0
and
Z
∞
Z
because
R∞
0
∗
Z
dηa f+ (ηa )Q (η − ηa − 2) =
dη
2
η−2
0
∞
f+ (η)dη
Q∗ (η)dη = 1. These together imply
1
Z ∞
−1
f+ (η)dη = 0
ν+
0
if
R∞
0
(3.92)
0
f+ (η)dη is finite. This gives us a trivial eigenvalue
(3.93)
1
ν+
= 1, but we need to
give up this solution because the corresponding eigenfunction does not satisfy the
R∞
constraint 0 f+ (η)dη = 0, and hence it is not physical. But this also means that
if we find an eigenvalue which is not 1 and also the eigenfunction is integrable, the
integral must be zero. So when we are doing the numerical diagonalization, we can
just rule out the solution with eigenvalue 1 if the integral of the eigenfunction is not
zero. The same happens for f− (η). By doing the same with Eq. (3.88), we also arrive
at
1
Z ∞
−1
f− (η)dη = 0
ν−
0
(3.94)
The same argument applies: we can just rule out the solution with eigenvalue 1 if the
integral of the eigenfunction is not zero.
Next comes the numerical diagonalization. We evenly discretize the η axis from
0 up to ηu and set the minimal length δη so that ηu is a large multiple of δη. In this
87
way, we can have a matrix form of the eigenvalue equation.
M± f± =
1
f± ,
ν±
(3.95)
where M± is the matrix we need to diagonalize, f± is the discretized version of f± (η)
and 1/ν± is still the eigenvalue. When constructing the matrix M± , we make the
following considerations. The first consideration is the derivative: Because
f± (η) + (1 + η)
i
df±
dh
=
(1 + η)f±
dη
dη
(3.96)
and this whole term helps us to determine the constraint equation Eq. (3.93) and
Eq. (3.94), we would like to discretize the full derivative as a whole:
i
i
dh
1h
(1 + η)f± (η) →
1 + (i + 1)δη f± (i + 1)δη − (1 + iδη)f± (iδη)
dη
δη
(3.97)
for the place η = iδη where i is a non negative integer. Note that here we also set the
derivative as the right derivative, because in the last step of the derivation of ρ(ζ, Λ),
when we shift the whole function horizontally by an amount −dΛ, this actually corresponds in the discrete case to the right derivative. The second consideration is the
boundary condition at η = ηu . In order to match the exponential decaying behavior
of Q∗ (η) at large η, it turns out that the f± (η) must also be exponentially decaying
with the same decay rate λQ . In other words, f± (η) ∼ exp(−λQ η) up to a prefactor
for large η. This gives us the boundary condition, which is to match the first order
derivative at the point ηu . Roughly speaking,
f± (ηu + δη) ≈ f± (ηu ) exp(−λQ δη)
88
(3.98)
so we have the discretized version of the derivative at the boundary:
i
i
dh
1h
(1 + η)f± (η) (1 + ηu + δη)f± (ηu + δη) − (1 + ηu )f± (ηu )
→
dη
δη
η=ηu
i
1h
≈
(1 + ηu + δη)e−λQ δη − (1 + ηu ) f± (ηu ) .
(3.99)
δη
The third consideration is the integral, where we just use the standard trapezoidal
rule to discretize the integral.
After constructing the matrices, we can numerically diagonalize them. For δη =
0.01 and ηu = 39.99, we have two 4000 × 4000 matrices to diagonalize. Note that
previously we only have the Q∗ (η) values up to η = 20; here for η > 20, we use the
approximation Q∗ (η) ≈ CQ exp(−λQ η) directly and the values CQ and λQ from the
linear fit in Fig. 3.2(b). The result shows that for M+ , if we sort the eigenvalues
based on their real parts, then the largest eigenvalue is equal to 0.9993 which corresponds to the trivial
1
ν+
= 1 solution and the eigenfunction always has the same sign
so the integral is nonzero. This confirms our analysis for the constraint about the
integral for f± (η). The second largest has real part −3.4. It means that this is an
irrelevant solution because it decays as Λ−3.4 when the RG is flowing. And all the
other eigenvalues have real parts even smaller than or equal to −3.4, which means
that for M+ or f+ (η), no relevant solution is found. This is as expected because this
means that if we perturb the QT and QI in the same direction, this perturbation
would in general decay back to zero as the cutoff Λ grows. Thus, in this direction,
the fixed point distribution Q∗ (η) is stable because we actually do not put the system
away from the critical point by setting QT and QI equal.
Next comes the result for M− , which gives us the critical exponent. If we as before
sort the eigenvalues based on their real parts, we again find the largest eigenvalue
equal to 1.0000 corresponding to the trivial
1
ν−
= 1 solution, and the integral of the
eigenfunction is also nonzero. So we just rule out this solution as before. The next
89
largest eigenvalue is the most important one. It is real and equal to 0.3995. The next
eigenvalue in the sorted list has real part −1.8, which is negative and hence irrelevant.
This means that the only relevant solution is the one with eigenvalue 0.3995. We will
focus on this solution from now on.
1.0
100
0.8
10-1
0.6
10-2
|f−(η)|
f−(η)
First we plot the eigenfunction as shown in Fig. 3.3. Fig. 3.3(a) gives us the eigen-
10-3
0.4
0.2
10-4
0.0
10-5
0.20
5
10
15
20
η
25
30
35
10-6 0
40
5
10
15
20
η
25
30
35
40
(b) |f− (η)| in log scale
(a) f− (η)
Figure 3.3: The eigenfunction f− (η) corresponding to eigenvalue 0.3995: (a) Linear
scale (b) Absolute value on log scale.
function on a linear scale. We choose f− (0) = 1 as a normalization factor because the
eigenfunction has an arbitrary multiplier, meaning if we multiply an arbitrary constant everywhere in an eigenfunction f− (η), the new function still solves the eigenvalue
equation with the same eigenvalue. One nice thing about this eigenfunction is that it
has both positive and negative values so that the integral can be close to 0. Actually,
R∞
in this case the area is −0.005 which confirms the constraint 0 f− (η) = 0. Fig.
3.3(b) gives us the absolute value of the eigenfunction on a log scale. The tip in the
regime 0 < η < 5 is due to the changing sign of f− (η) at that point. Fig. 3.3(b) shows
clearly that the eigenfunction f− (η) also decays exponentially at large η as expected.
We can also do the numerical integration to directly obtain the eigenfunction by
setting 1/ν− = 0.4 and f− (0) = 1. The result is shown in Fig. 3.4. Fig. 3.4 shows
the two resulting curves of f− (η) coincide perfectly. This result further confirms that
90
1.0
Direct Integration
Diagonalization
0.8
f−(η)
0.6
0.4
0.2
0.0
0.20
5
10
15
η
20
Figure 3.4: The eigenfunction f− (η) using either the numerical integration directly
or the diagonalization.
f− (η) is the correct solution to the eigenvalue equation. After solving this eigenvalue
equation, we get an estimate of the critical exponent:
1
≈ 0.4 or ν− ≈ 2.5 .
ν−
3.3.3
(3.100)
Numerical Evidence
After deducing the analytical RG equations for the distributions and discussing
the solutions to them, we now move on the numerical test by initializing a long one
dimensional chain and running the RG numerically. The initial number of blocks is
107 and the final number of blocks we choose is 103 .
The first step is to test the fixed point distribution Q∗ (η). We set the initial cutoff
Λ equal to 1 and use the theoretical Q∗ (η) distribution, shown in Fig. 3.2, as the
91
initial distribution for both ηT ≡
ln ∆T
ln Ω
−1 and ηI ≡
ln ∆I
ln Ω
−1. And we set ln gI = −10−9
as a number close to zero. The resulting cumulative distribution function (Cdf) for
the RG distribution flow is shown in Fig. 3.5. Fig. 3.5 clearly shows that for both
the distributions in T-block and I-block, they do not change when the total number
of blocks is decreasing from 107 all the way to 103 and simultaneously when the cutoff
Λ is increasing. This confirms that our Q∗ (η) is indeed a fixed point distribution. In
addition, we can also see the pairs of N and Λ in the legend in Fig. 3.5, and their
product also keeps the same as expected in Eq. (3.51). Note that Eq. (3.51) is only
true when the system is at the fixed point. This is further evidence that the system
is at the fixed point.
After verifying the fixed point distribution Q∗ (η), we can then perturb the initial
distribution away from Q∗ (η) by setting


QT (η) = Q∗ (η) + a · f− (η) ,
(3.101)

 QI (η) = Q∗ (η) − a · f− (η) ,
where f− (η) is the eigenfunction with eigenvalue ≈ 0.4 and a is a parameter we can
tune. Note that we have an arbitrary multiplier in the eigenfunction f− (η) and it
actually can be absorbed in the factor a in our numerical test. So we do not need to
normalize the f− (0) value to 1 and we just use the numerical value directly outputted
by diagonalization: f− (0) = 0.1119. In addition, we need to keep the a value neither
too large nor too small. If a were too large, the perturbation approximation by
omitting second order terms in δT,I would no longer be true. If a were too small, it
would amplify the error coming from the random sampling and we can hardly tell
the difference between QT (η) and QI (η). Based on these concerns, we choose a = 0.1
together with the normalization factor f− (0) = 0.1119. The result of the difference
between the Cdfs is shown in Fig. 3.6. Fig. 3.6(a) shows that when the cutoff Λ is
increasing, the difference is indeed increasing. In order to see whether this increasing
92
1
N = 10000000; Λ = 1
N = 2154434; Λ = 4.644
N = 464158; Λ = 21.5743
N = 99998; Λ = 99.9655
N = 21544; Λ = 465.511
N = 4640; Λ = 2152.89
N = 1000; Λ = 10001
0.9
0.8
0.7
Cdf(ηT)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
η
25
30
35
40
T
(a) Cdf flow of T-blocks
1
N = 10000000; Λ = 1
N = 2154434; Λ = 4.644
N = 464158; Λ = 21.5743
N = 99998; Λ = 99.9655
N = 21544; Λ = 465.511
N = 4640; Λ = 2152.89
N = 1000; Λ = 10001
0.9
0.8
0.7
Cdf(ηI)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
ηI
25
30
35
40
(b) Cdf flow of I-blocks
Figure 3.5: The cumulative distribution function (Cdf) for both the T-blocks and
I-blocks as the total number of blocks N decreases from 107 to 103 and the cutoff Λ
grows.
93
0.6
N = 10000000; Λ = 1
N = 2154434; Λ = 4.6415
N = 464158; Λ = 21.5246
N = 99998; Λ = 98.7657
N = 21544; Λ = 453.086
N = 4640; Λ = 1966.02
N = 1000; Λ = 7493.21
0.5
∆ Cdf(η)
0.4
0.3
0.2
0.1
0
−0.1
0
5
10
15
20
η
25
30
35
40
(a) Difference of Cdf flow between T-blocks and I-blocks
−3
20
x 10
N = 10000000; Λ = 1
N = 2154434; Λ = 4.6415
N = 464158; Λ = 21.5246
N = 99998; Λ = 98.7657
N = 21544; Λ = 453.086
N = 4640; Λ = 1966.02
N = 1000; Λ = 7493.21
15
∆ Cdf(η)
10
5
0
−5
0
5
10
15
20
η
25
30
35
40
(b) Collapsing of difference of Cdf by multiplying Λ−0.4
Figure 3.6: The difference of cumulative distribution functions (Cdf) between the
T-blocks and I-blocks as the total number of blocks N decreases from 107 to 103 and
the cutoff Λ grows.
94
is just due to the power law formula
δ− (η, Λ) ∝ Λ1/ν− f− (η) ,
(3.102)
we multiply every single function by a factor Λ−0.4 . The result is in Fig. 3.6(b) and
all the resulting functions collapse pretty well. This confirms our estimation of the
critical exponent
1
≈ 0.4 .
ν−
(3.103)
Also note that products of the pairs N and Λ here are not constant. This also
indicates that the RG in this case is indeed flowing and is not at the fixed point.
3.4
Conclusion and Outlook
In this chapter, we studied a “toy model” in the strong disorder renormalization
group framework to explore the eigenstate phase transition between MBL and thermalization. This model turns out to be quite simple. We solved the critical fixed
point distribution Q∗ (η) almost analytically from the fixed point RG equation. It is
an infinite-randomness fixed point with the energy scale Ω related to the length scale
L by ln Ω ∼ −L, as in [30]. We also analytically derived the RG equations when the
distributions in both T-blocks and I-blocks deviate from the fixed point distribution,
and solved for the correlation length critical exponent
ν = ν− ≈ 2.5
by both solving the eigenvalue equation and directly simulating the RG flow.
95
(3.104)
One apparent issue of this model is that we do not avoid ad hoc rules. In the
ITI move, we imposed that the move always flows to an insulating block to avoid
the RG getting stuck and unstable as in the case of our initial trials. The solution
to the issue of this “toy” model is to modify the RG rules such that the RG flow
becomes closer to the microscopic physics and still shares the features of not getting
stuck and unstable. The new model may not be as simple as our “toy model”, but
we hope that our “toy model” can serve as a useful reference point for studying the
more physical model, and thus for understanding the physics of the eigenstate phase
transition between MBL and thermalization.
96
Appendix A
Mathematical Proof of the
Degeneracy Formula
In Section 2.2.2, we stated that the energy level degeneracy denoted by fN (S) has
the form:
N
fN (S) = CN2
−S
N
− CN2
−S−1
,
(A.1)
where N is the total number of sites and S is the total spin. Let me briefly prove this
formula here.
The method is basically induction by counting N from 1 to ∞. Let me list the
first few cases:
1. N = 1: S = 1/2 is single degenerate. f1 ( 21 ) = 1.
2. N = 2: S = 0 or 1. Both are single degenerate. f2 (0) = f2 (1) = 1.
3. N = 3: Based on angular momentum coupling procedure, if the first two spins
are coupled to have S12 = 0, then the total spin S would be 1/2; if the first
two are coupled to have S12 = 1, then the total spin S would be 1/2 or 3/2.
97
So S = 1/2 is double degenerate; S = 3/2 is single degenerate. f3 ( 12 ) = 2,
f3 ( 32 ) = 1.
4. N = 4: If S123 = 1/2, we would have either S = 0 or S = 1; if S123 = 3/2, we
would have either S = 1 or S = 2. So f4 (0) = 2, f4 (1) = 3, f4 (2) = 1. As a
double check, every single S expands a 2S + 1 dimensional Hilbert space, so the
total dimension is (2 × 0 + 1) · 2 + (2 × 1 + 1) · 3 + (2 × 2 + 1) · 1 = 16 = 24 as
expected.
5. N = 5: If S1234 = 0, then S = 1/2; if S1234 = 1, then S = 1/2 or S = 3/2; if
S1234 = 2, then S = 3/2 or S = 5/2. So f5 ( 21 ) = 2 + 3 = 5, f5 ( 32 ) = 3 + 1 = 4,
f5 ( 52 ) = 1. Also, as a double check, the total dimension of the Hilbert space is
(2 × 12 + 1) · 5 + (2 × 23 + 1) · 4 + (2 × 52 + 1) · 1 = 32 = 25 as expected.
From these examples, we conclude a general recursion relation of fN (S):















1
f2N +1 ( ) = f2N (0) + f2N (1)
2
3
f2N +1 ( ) = f2N (1) + f2N (2)
2
..
.
(A.2)




1


f2N +1 (N − ) = f2N (N − 1) + f2N (N )



2




f2N +1 (N + 1 ) = f2N (N )
2
and





























1
f2N +2 (0) = f2N +1 ( )
2
1
3
f2N +2 (1) = f2N +1 ( ) + f2N +1 ( )
2
2
..
.
1
1
f2N +2 (N ) = f2N +1 (N − ) + f2N +1 (N + )
2
2
1
f2N +2 (N + 1) = f2N +1 (N + ) .
2
98
(A.3)
First, I will show that the formula (A.1) is true for any positive even N . By
induction, when N = 2, the conclusion is true from the above example. We assume
for 2N case, the formula is true:
N −S
N −S−1
f2N (S) = C2N
− C2N
.
(A.4)
Then, for 2N + 2 case, by recursion relation, we have:


 f2N +2 (0) = f2N (0) + f2N (1)

f2N +2 (S) = f2N (S − 1) + 2f2N (S) + f2N (S + 1) ,
(A.5)
∀S ≥ 1.
n
= 0 if n < 0.
We do not need to worry about the case S ≥ N . Since by definition Cm
So the above equation is also valid for S ≥ N by setting f2N (S ≥ N + 1) = 0 ≡
N −S
N −S−1
C2N
− C2N
where N − S < 0 and N − S − 1 < 0.
So for S = 0 case, we have:
f2N +2 (0) = f2N (0) + f2N (1)
N −1
N −1
N −2
N
= (C2N
− C2N
) + (C2N
− C2N
)
N −1
N −1
N −2
N
= (C2N
+ C2N
) − (C2N
+ C2N
)
N −1
N
= C2N
+1 − C2N +1
N +1
N −1
N
N
= (C2N
+1 + C2N +1 ) − (C2N +1 + C2N +1 )
N +1
N
= C2N
+2 − C2N +2 ,
which is the desired equation.
99
(A.6)
For S ≥ 1 case, we have:
f2N +2 (S) = f2N (S − 1) + 2f2N (S) + f2N (S + 1)
= [f2N (S − 1) + f2N (S)] + [f2N (S) + f2N (S + 1)]
h
i
N −S+1
N −S
N −S
N −S−1
= (C2N
− C2N
) + (C2N
− C2N
)
h
i
N −S
N −S−1
N −S−1
N −S−2
+ (C2N
− C2N
) + (C2N
− C2N
)
h
i
N −S+1
N −S
N −S
N −S−1
= (C2N
+ C2N
) − (C2N
+ C2N
)
h
i
N −S
N −S−1
N −S−1
N −S−2
+ (C2N
+ C2N
) − (C2N
+ C2N
)
N −S+1
N −S
N −S
N −S−1
= C2N
− C2N
+1
+1 + C2N +1 − C2N +1
N −S+1
N −S
N −S
N −S−1
= (C2N
+ C2N
+1
+1 ) − (C2N +1 + C2N +1 )
N −S+1
N −S
= C2N
− C2N
+2
+2
(N +1)−S
= C2N +2
(N +1)−S−1
− C2N +2
,
(A.7)
which is also the desired equation. Thus, by induction, we conclude that for any
positive even N , the formula is valid.
The next step is to prove the formula is also valid for positive odd N . Since now
N −S
N −S−1
f2N (S) = C2N
− C2N
is true, then for 2N + 1 case, we have:
1
1
f2N +1 (S) = f2N (S − ) + f2N (S + )
2
2
N −S+ 21
= C2N
N −S− 21
− C2N
N −S+ 12
= (C2N
N −S+ 12
= C2N +1
2N +1
−S
= C2N2+1
N −S− 21
+ C2N
N −S− 12
+ C2N
N −S− 12
) − (C2N
N −S− 32
− C2N
N −S− 32
+ C2N
)
N −S− 21
− C2N +1
2N +1
−S−1
− C2N2+1
which is the desired equation for the odd case.
100
,
(A.8)
After these two steps, we conclude that the degeneracy formula:
N
fN (S) = CN2
−S
N
− CN2
−S−1
(A.9)
is valid for ∀N ∈ Z+ .
Also, the total dimension of the Hilbert space is in general given by:
N/2
X
(2S + 1)fN (S) = CN0 + CN1 + · · · + CNN = 2N
(A.10)
S=0 or 1/2
as expected.
Then, by using Stirling’s formula ln N ! = N ln N − N + O(ln N ), we may have the
expression of the entropy defined by Σ(S) ≡ ln fN (S):
N
fN (S) = CN2
−S
N
− CN2
−S−1
N!
N!
− N
N
− S)!( 2 + S)! ( 2 − S − 1)!( N2 + S + 1)!
h N
i
N!
N
· ( + S + 1) − ( − S)
= N
2
2
( 2 − S)!( N2 + S + 1)!
N!
= (2S + 1) · N
;
( 2 − S)!( N2 + S + 1)!
=
( N2
101
(A.11)
then
N
N
− S)! − ln( + S + 1)!
2
2
N
N
N
= N ln N − N − ( − S) ln( − S) + ( − S)
2
2
2
N
N
N
− ( + S + 1) ln( + S + 1) + ( + S + 1) + O(ln N )
2
2
2
N
N
N
N
= N ln N − ( − S) ln( − S) − ( + S + 1) ln( + S + 1) + O(ln N )
2
2
2
2
N
N
= N ln N − ( − S) ln N − ( + S + 1) ln N
2
2
N
1
S
N
1 S+1
− ( − S) ln( − ) − ( + S + 1) ln( +
) + O(ln N )
2
2 N
2
2
N
h 1
S
1
S
1 S+1
1 S+1 i
= −N ( − ) ln( − ) + ( +
) ln( +
) + O(ln N )
2 N
2 N
2
N
2
N
h 1
S
1
S
1
S
1
S i
= −N ( − ) ln( − ) + ( + ) ln( + ) + O(ln N ) ,
(A.12)
2 N
2 N
2 N
2 N
ln fN (S) = ln(2S + 1) + ln N ! − ln(
which is exactly the entropy equation (2.7) in Sec 2.2.2.
102
Appendix B
Estimation of the Magnitude of
Disorder
In Section 2.4 and 2.5, we added a disorder term in the original Hamiltonian to
get:
H=−
N
N
N
X
1 X z z
λ X
si sj − Γ
sxi + p
εij szi szj ,
N 1=i<j
N
i=1
1=i<j
(B.1)
where 1 ≥ λ > 0, εij are independent and identically distributed random variables
for different site i with expectation value zero and standard deviation one. This task
is to estimate the upper and lower bound for the power p. For further convenience,
we denote
N
N
X
1 X z z
si sj − Γ
sxi ,
H0 ≡ −
N 1=i<j
i=1
N
λ X
Hr ≡ p
εij szi szj .
N 1=i<j
(B.2)
So H = H0 + Hr . Then the constrains acting on the magnitude of the disorder term
Hr are:
1. Lower Bound: Hr should be big enough for the system to look thermal.
103
2. Upper Bound: Hr should be small enough not to vary macroscopic thermal
property such as free energy.
Let us discuss them one by one.
B.1
Lower Bound of Disorder
First is for the lower bound of disorder. When adding Hr , all degenerated energy
levels are split due to Hr . By letting the system look thermal, we mean the typical
value of the level splitting due to Hr should be big enough to compare with the
typical value of the original level splitting of H0 which is typically of order 1. Using
degenerate perturbation theory treating Hr perturbatively, we need to diagonalize Hr
in the degenerated H0 subspace:

 Hr11 Hr12 · · · Hr1n

 Hr
Hr22 · · · Hr2n
21

 .
..
..
...
 ..
.
.


Hrn1 Hrn2 · · · Hrnn


0
···
0

 ∆E1




∆E2 · · ·
0
 diagonalize  0
 −−−−−−→  .
..
..
...

 ..
.
.




0
0
· · · ∆En





,



(B.3)
where Hrmn ≡ hm|Hr |ni is the matrix element of Hr in H0 representation; n = eΣ(S)
is the degeneracy. Since the diagonalization process keeps the trace invariant, we
would have an estimation by taking the trace:
X
hn|Hr |ni =
n
n
X
∆Ei .
(B.4)
i=1
Then, the left hand side is equal to:
X
n
hn|Hr |ni =
X
n
N
N
X
λ X
λ X
z z
hn| p
εij si sj |ni = p
εij
hn|szi szj |ni .
N 1=i<j
N 1=i<j
n
104
(B.5)
Since the Hamiltonian H0 is invariant under permutation Pjk for any two sites j and k,
we have [H0 , Pjk ] = 0, so H0 Pjk |ni = En Pjk |ni. It means Pjk is a block diagonalized
form in terms of the same energy subspace. In another word, Pjk is also an operator in
the same energy subspace. So we can deal with the operator totally in the subspace:
X
hn|szi szj |ni = trsubspace (szi szj )
n
−1
= trsubspace (Pjk szi szj Pjk
)
= trsubspace (szi szk )
X
=
hn|szi szk |ni .
(B.6)
n
It implies that
x x
n hn|si sj |ni
P
is independent of site indices i and j as long as i 6= j.
Because
X
X
X
hn|Sz · Sz |ni = (N 2 − N )
hn|szi szj |ni + N
hn|sz2
i |ni
n
n
= (N 2 − N )
X
n
n
1
hn|szi szj |ni + N · n ,
4
(B.7)
we have:
X
hn|szi szj |ni =
n
X
1
1
2
hn|Sz |ni − N · n .
N2 − N n
4
(B.8)
Every |ni in the subspace has the same quantum number of total S and m. Therefore,
hn|Sz2 |ni ≡ hS, m|Sz2 |S, mi = m2 .
(B.9)
X
m2 − 41 N
z z
hn|si sj |ni = n · 2
.
N −N
n
(B.10)
Then we arrive at:
105
Plugging this back we have:
λ X m2 − 14 N
tr(Hr ) = n p
εij · 2
N
N −N
i6=j
=
n
X
∆Ei ,
(B.11)
i=1
n
1X
λ X m2 − 14 N
∆Ei = p
.
εij · 2
n i=1
N
N −N
i6=j
(B.12)
We need the typical value of the energy shift ∆Ei to be O(1) in order to fully mix the
energy spectrum up. Since ∆Ei s come from random variables εij s, they themselves
can also be treated as random variables. From the law of large number in probability
theory,
n
1X
∆Ei = E(∆Ei ) & O(1) ,
n i=1
(B.13)
where E(X) is the expectation value of a random variable X. Since εij has a variance
1 as mentioned at the beginning, we apply to the central limit theorem and have:
X
εij ∼
√
N 2 − N = O(N ) .
(B.14)
i6=j
In addition, m2 = O(N 2 ), so
m2 − 41 N
N 2 −N
∼ O(1). Thus the requirement finally reduces
to:
N
O(1)
Np
⇒
106
p < 1.
(B.15)
B.2
Upper Bound of Disorder
For the upper bound of disorder, we require free energy should not change much
by adding disorder. Let f0 be the free energy of the system H0 and f˜ be the free
energy of the full system H0 + Hr . The requirement becomes:
f˜ − f0
= 0.
N →∞
f0
lim
(B.16)
Obviously, f0 ∼ O(N ), so the only thing we need to solve is the difference: f˜ − f0 .
By definition,
h
i
−β f˜ = ln Z̃ = ln tr e−β(H0 +Hr ) .
(B.17)
To take the trace, it is equivalent to use the saddle point approximation: to take the
minimal eigenvalue of the following expression1
Σ(S) − β(H0 + Hr ) .
(B.18)
We know that
Σ(S) ∼ O(N )
and
βH0 ∼ O(N ) .
(B.19)
Then the task is to estimate the ground state energy of Hr . If the system only contains
Hr , the system is a typical “spin glass”. And the ground state of this spin glass can
be argued this way: for one spin, there are N adjacent spins exerting “random” fields
and each one of the fields has a magnitude O( N1p ). So by the central limit theorem,
√
the total effective field for one spin is O( N · N1p ). If we choose every spin to be
The approximation exp Σ(S) exp − β(H0 + Hr ) ≈ exp Σ(S) − β(H0 + Hr ) is used since
the commutator has a subleading order of N .
1
107
aligned with this effective field, the energy of Hr can be as low as O(N ·
√
N·
1
)
Np
which is the order of magnitude of the ground state energy of a spin glass system.
√
√
Thus if N · N · N1p O(N ) or N · N · N1p ∼ O(N ), the disorder term would have
a non-eligible or even a leading effect on free energy. Therefore in order to maintain
the same free energy as N → ∞, we need
Hr ∼ N ·
√
N·
1
O(N )
Np
⇒
p>
1
.
2
(B.20)
As a conclusion of the discussion in both Section B.1 and Section B.2, we finally
get that the magnitude of the disorder should be within the following interval:
1
< p < 1.
2
108
(B.21)
Appendix C
Divergence of First Order
Derivative of γ(s̄) at s̄ = s̄min
In Section 2.5.1, we stated that
dγ(s̄)
ds̄
diverges at s̄ = s̄min . Let me show this
divergence briefly here.
The original expression of γ(s̄) is given by:
Z
γ(s̄) = 2
0
xt
−u − 1 x2
arccosh √ 2
dx ,
Γ s̄2 − x2
(C.1)
−u− 1 x2
2
where xt is the turning point. We denote B(x) ≡ arccosh Γ√s̄2 −x
2 , then xt satisfies
√ p
√
the turning point equation B(xt ) = 1 or xt = 2 −u − Γ2 − Γ 2u + s̄2 + Γ2 . Since
the upper bound of the integral in γ(s̄) is also a function of s̄, then the derivative
contains generically two terms:
−u − 1 x2 ∂x
∂γ
t
= 2 arccosh p 2 t ·
+2
2
2
∂s̄
∂s̄
Γ s̄ − xt
Z
0
xt
∂
arccosh B dx ,
∂s̄
(C.2)
1 2
−u− x
since B(xt ) = √ 22 t2 = 1, arccosh B(xt ) = 0, the first term would vanish. But there
Γ
are points where
s̄ −xt
∂xt
∂s̄
= ∞ and it happens when xt = 0 and s̄ = s̄min . However, we
do not need to take a special care even at these points since what we really care is
109
the asymptotic behavior near these points and every individual point does not matter
that much. So the derivative reduces to:
Z
∂γ
=2
∂s̄
xt
Z0 xt
=2
0
∂
arccosh B dx
∂s̄
B
−s̄
√
· 2
dx .
B 2 − 1 s̄ − x2
(C.3)
If we study the above equation at the point xt = 0 or s̄ = s̄0 , we would recover:
πΓ
∂γ ,
= −√
∂s̄ s̄=s̄0 −0
−u − Γ2
(C.4)
but here the meaning is slightly different from that in Section 2.6. In Section 2.6, we
calculated the derivative exactly at s̄ = s̄0 ; here, on the other hand, the procedure is
that we first take the derivative at some other point and then send the point to s̄0 .
The consistency of the two result shows the left continuity of the function
∂γ
∂s̄
at s̄0 .
Now it is time to see the asymptotic behavior of ∂γ
when s̄ → s̄min . When
∂s̄
√
√
s̄ → s̄min = −2u − Γ2 , we have xt → −2u − 2Γ2 ≡ xt0 . Thus,
∂γ =2
∂s̄ s̄=s̄min
Z
0
xt0
B(s̄min )
p
B(s̄min )2 − 1
·
−s̄min
dx ,
− x2
s̄2min
(C.5)
when x → xt0 , the integrand behaves as: B(s̄min ) → 1, s̄2min − x2 → Γ2 , but
√
1
B(s̄min )2 −1
→ ∞. We need to study how fast it goes to ∞.
110
If we perturb x a little away from xt0 by setting x = xt0 − ∆ and expand B(s̄min )
in terms of ∆, we have:
B(s̄min )
−u − 12 x2
= √
Γ −2u − Γ2 − x2
−u − 12 (xt0 − ∆)2
= p
Γ −2u − Γ2 − (xt0 − ∆)2
−u − 12 x2t0 + xt0 ∆ − 12 ∆2
√
=
Γ Γ2 + 2xt0 ∆ − ∆2
Γ2 + xt0 ∆ − 12 ∆2
= q
Γ2 1 + 2xΓ2t0 ∆ − Γ12 ∆2
2x
xt0
1 2 h
1 2 i− 12
t0
= 1 + 2 ∆ − 2∆ 1 +
∆ − 2∆
Γ
2Γ
Γ2
Γ
h
i
xt0
1 2
1 2xt0
1 2 3 2xt0
1 2 2
3
= 1 + 2 ∆ − 2∆ 1 −
∆ − 2∆ +
∆ − 2∆
+ O(∆ )
Γ
2Γ
2 Γ2
Γ
8 Γ2
Γ
xt0
1
1 2xt0
xt0 1 2xt0
1
3 2xt0 2
= 1 + 2 ∆ − 2 ∆2 − · 2 ∆ − 2 ∆ · 2 ∆ + 2 ∆2 +
∆ + O(∆3 )
Γ
2Γ
2 Γ
Γ
2 Γ
2Γ
8 Γ2
x2
= 1 + t04 ∆2 + O(∆3 ) ,
(C.6)
2Γ
then,
r
xt0
x2t0 2
∆ + O(∆3 ) =
∆[1 + O(∆)] ,
4
Γ
Γ
(C.7)
1
Γ
Γ
p
=
[1 + O(∆)] =
+ O(1) .
2
xt0 ∆
xt0 ∆
B(s̄min ) − 1
(C.8)
p
B(s̄min )2 − 1 =
⇒
Hence,
∂γ ∼ −
∂s̄ s̄=s̄min
Z
xt0
0
which diverges logarithmically.
111
1
d∆ → − ∞ ,
∆
(C.9)
Appendix D
Discussion on Convexity of
Function γ(s̄)
In Section 2.5.1, we assumed that γ(s̄) is a convex function of s̄ in order to get
the analytical result of the thermal activation and quantum tunneling transition line.
By saying convexity, we mean
∂ 2γ
> 0.
∂s̄2
(D.1)
In fact, we can do the numerics following directly from the thermal activation and
quantum tunneling transition criteria which is to minimize the quantity α(u, Γ) in
Eq. 2.32 without assuming the convexity of γ(s̄). The result is shown in Fig. D.1.
The dots in Fig. D.1 are the numerical results given by actually doing the numerical
integral for γ(s̄) without assuming the convexity of function γ(s̄). But this numerical
result still fits the analytical one which is the line underneath the dots in Fig. D.1.
This convinced us that the analytical result must be the right answer. While we
already have the numerical evidence, we would also like to discuss the convexity of
function γ(s̄) directly and see how much we could get.
112
0.00
-0.05
-0.10
u
-0.15
-0.20
-0.25
G
-0.30
0.0
0.1
0.2
0.3
0.4
0.5
Figure D.1: Comparison between numerical and analytical result of the thermal activation and quantum tunneling transition line; the dots are numerical points and the
line underneath the dots is the analytical result given by equation (2.47). They fit
perfectly well.
One attempt is to directly calculate the integral:
∂γ
=2
∂s̄
Z
0
xt
B
−s̄
√
dx
· 2
B 2 − 1 s̄ − x2
(D.2)
numerically by sowing different pairs of Γ and u randomly. Some of the results are
shown in Fig. D.2. In Fig. D.2, it is clear that for all pairs of Γ and u which have been
randomly chosen,
∂γ
∂s̄
is a monotonic increasing function of s̄. This adds our belief
that γ(s̄) is a convex function although we do not have a mathematical proof.
113
dΓ
dΓ
dΓ
ds
0.356
0.357
0.358
0.359
0.360
ds
ds
s
0.361
0.5
1.0
1.5
2.0
s
2.5
-1
0.358
0.360
0.362
s
0.366
0.364
-1
-0.2
-2
-2
-0.4
-3
-3
-0.6
-4
-4
-5
-0.8
-6
-1.0
-5
-6
(a) Γ
=
−0.106658
0.295502 , u
= (b) Γ
=
−0.0937125
dΓ
0.031665 , u
= (c) Γ
=
−0.104641
dΓ
ds
0.3856
0.3858
0.3860
0.3862
0.3864
0.3866
ds
s
0.45
-2
-0.5
-4
-1.0
=
dΓ
ds
0.3854
0.285948 , u
0.50
0.55
0.60
0.65
0.70
s
0.49865
0.49870
s
0.49880
0.49875
-5
-10
-1.5
-6
-15
-2.0
-8
-2.5
-20
-10
(d) Γ
=
−0.136793
0.353770 , u
= (e) Γ
=
−0.106585
0.145284 , u
= (f) Γ
=
−0.242333
0.485834 , u
=
as a function of s̄ by sowing different pairs of Γ
Figure D.2: Numerical results of ∂γ
∂s̄
and u randomly within the entire ferromagnetic phase region.
Another attempt is based on trying to calculate it analytically by doing another
order derivative from the expression of
Z
∂γ
=2
∂s̄
xt
0
Z
=2
0
1
∂γ
.
∂s̄
Since
−s̄
B
√
· 2
dx
B 2 − 1 s̄ − x2
B
−s̄
√
· 2
xt db ,
B 2 − 1 s̄ − x2t b2
(D.3)
we have:
∂ 2γ
= −2
∂s̄2
Z
0
1
∂
B
−s̄
√
· 2
x
t db .
∂s̄
B 2 − 1 s̄ − x2t b2
(D.4)
−s̄
· s̄2 −x
2 b2 xt where in general both B and xt are functions of s̄: B =
t
√ p
√
≡ B(s̄, xt ) = B(s̄, xt (s̄)) and xt = 2 −u − Γ2 − Γ 2u + s̄2 + Γ2 ≡
√ B
B 2 −1
−u− 12 x2t b2
Let g(s̄, b) ≡
arccosh √
s̄2 −x2t b2
Thus if ∂g
∂s̄
Γ
xt (s̄).
< 0 for all 0 < b < 1, we would have
do is to test whether
∂g
∂s̄
∂2γ
∂s̄2
> 0. Then all we need to
< 0 or not. After a messy calculation, we successfully turn
114
this condition into whether a big polynomial containing about 50 terms is smaller
than zero. Then the method of sowing points randomly also applies. After sowing
more than 106 points and selecting the maximal value for that big polynomial, we
arrive at a negative number with a tiny absolute value. It implies that
∂g
∂s̄
should
be smaller than zero in the entire region we care about. This will provide another
evidence of the statement of the convexity of function γ(s̄).
115
Bibliography
[1] P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[2] Y. Y. Atas, E. Bogomolny, O. Giraud and G. Roux, Phys. Rev. Lett. 110, 084101
(2013).
[3] Y. Bahri, R. Vosk, E. Altman, and A. Vishwanath, arXiv:1307.4092 (2013).
[4] J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys. Rev. Lett. 109, 017202
(2012).
[5] D. M. Basko, I. L. Aleiner and B. L. Altshuler, Annals of Physics 321, 1126
(2006); cond-mat/0602510.
[6] W. Beugeling, R. Moessner, and M. Haque, Phys. Rev. E 89, 042112 (2014).
[7] R. Blatt and C. F. Roos, Nature Physics 8, 277 (2012).
[8] I. Bloch, J. Dalibard and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).
[9] P. A. Braun, Rev. Mod. Phys. 65, 115 (1993).
[10] J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).
[11] D. S. Fisher, Phys. Rev. B 50, 3799 (1994).
[12] D. A. Huse, R. Nandkishore, and V. Oganesyan, Phys. Rev. B 90, 174202 (2014).
[13] D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal and S. L. Sondhi, Phys. Rev.
B 88, 014206 (2013).
[14] D. A. Huse and V. Oganesyan, arXiv:1305.4915 (2013).
[15] J. Z. Imbrie, arXiv:1403.7837 (2014).
[16] H. Kim and D. A. Huse, Phys. Rev. Lett. 111, 127205 (2013).
[17] T. Kinoshita, T. Wenger and D. S. Weiss, Nature 440, 900 (2006).
[18] J. A. Kjäll, J. H. Bardarson and F. Pollmann, Phys. Rev. Lett. 113, 107204
(2014).
116
[19] R. Nandkishore and D. A. Huse, Ann. Rev. C. M. P. 6, 15-38 (2015).
[20] A. Pal and D. A. Huse, Phys. Rev. B 82, 174411 (2010).
[21] D. Pekker, G. Refael, E. Altman, E. Demler and V. Oganesyan, Phys. Rev. X 4,
011052 (2014).
[22] M. Rigol, V. Dunjko and M. Olshanii, Nature 452, 854-858 (2008).
[23] M. Serbyn, M. Knap, S. Gopalakrishnan, Z. Papić, N. Y. Yao, C. R. Laumann,
D. A. Abanin, M. D. Lukin, and E. A. Demler, Phys. Rev. Lett. 113, 147204
(2014).
[24] M. Serbyn, Z. Papić, and D. A. Abanin, Phys. Rev. Lett. 110, 260601 (2013).
[25] M. Serbyn, Z. Papić, and D. A. Abanin, Phys. Rev. Lett. 111, 127201 (2013).
[26] M. Srednicki, Phys. Rev. E 50, 888 (1994).
[27] H. Tasaki, Phys. Rev. Lett. 80, 1373 (1998).
[28] R. Vosk and E. Altman, Phys. Rev. Lett. 110, 067204 (2013).
[29] R. Vosk and E. Altman, Phys. Rev. Lett. 112, 217204 (2014).
[30] R. Vosk, D. A. Huse and E. Altman, arXiv: 1412.3117 (2014).
[31] J. Wilms, J. Vidal, F. Verstraete and S. Dusuel, J. Stat. Mech., P01023 (2012).
[32] B. Zhao, M. C. Kerridge and D. A. Huse, Phys. Rev. E 90, 022104 (2014).
[33] M. Žnidarič, T. Prosen, and P. Prelovšek, Phys. Rev. B 77, 064426 (2008).
117