Download Thesis Truncated Free Fermion Space Approach

LOMONOSOV MOSCOW STATE UNIVERSITY
INSTITUTE FOR THEORETICAL AND MATHEMATICAL
PHYSICS
Analytical study of free energy in Ising field
theory in the presence of a magnetic field
MASTER THESIS
Stepanova Khristina
Stepanova Khristina
Under supervision by Alexey Litvinov
Moscow, 2024
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Table of Contents
1 Abstract
6
2 Introduction
6
3 Perturbation theory
6
4 Hamiltonian truncation
8
5 Beyond integrability: truncated space approach (TSA)
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5.1
Introduction to the approach . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2
Example of the truncated space approach: truncated free fermion space
approach (TFFSA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 TFFSA Results
18
6.1
Construction of the basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.2
TFFSA Data and Behavior of Ground State . . . . . . . . . . . . . . . . . 19
6.3
Behavior of Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
7 Conclusion
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1
Abstract
We study the 2D Ising model in the critical domain T Ñ Tc , H Ñ 0. The analysis of the
finite-size energy levels Ei pRq is based on numerical method called Truncated Free Fermion
Space Approach (TFFSA). We examined the behavior of these levels and introduced the
concept of the "false vacuum".
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Introduction
As is well known the Ising model free energy exhibits a singularity at the critical point H “
0, T “ Tc . This singularity is described in terms of the Euclidian quantum field theory
known as the Ising Field Theory (IFT). In this research work, we study the analytical
properties of the free energy in Ising Field Theory within the framework of a magnetic
field.
We consider a two-dimensional IFT in finite-size geometry, with one of the two Euclidian coordinates compactified on a circle of circumference R, x ` R „ x, and the other
y is treated as Euclidian time, which goes along the cylinder. To construct energy levels, the Truncated Free Fermion Space Approach (TFFSA) was used, which is one of the
Hamiltonian truncation methods family. In the process of research, we consider only those
eigenvalues of the total Hamiltonian that is less then chosen UV cutoff.
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Perturbation theory
Perturbation theory is a theory used when a problem cannot be solved exactly, but can
be divided into a part that is solved exactly and a part responsible for weak physical
deviations (perturbations). Usually the last part is series in finite powers of alpha, which
converge to exact values when summed. However, after a certain order n “ α1 , the results
become worse due to the divergence of the series. Before looking at the theory, let’s
mention cases where perturbation theory doesn’t work:
- Perturbation theory is not suitable for describing systems where the disturbance is
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not small (that is, the coupling constant must be small)
- Perturbation theory is not suitable for describing systems that are not generated
addiabatically from a free model, such as connected states and various collective phenomena.
Let us consider time-independent perturbation theory:
Perturbation theory in quantum mechanics is applied in the case when the Hamiltonian
of the system can be represented in the form:
(1)
H “ H0 ` λV
where H0 - is the unperturbed Hamiltonian (and the solution to the corresponding
Schrödinger equation is known exactly)
V - weak physical disturbance (Hermitian operator)
λ P r0, 1s - dimensionless parameter, where 0 means "no perturbation" and 1 means
"the full perturbation".
Unperturbed Hamiltonian is assumed to have no time dependence. It has known
energy levels and eigenstates:
H0 |np0q y “ Enp0q |np0q y
n “ 1, 2, 3, ...
(2)
Energy levels - are eigenstates of the perturbed Hamiltonian are given by the timeindependent Schrödinger equation:
pH0 ` λV q|ny “ En |ny
(3)
The aim is to express En and |ny in terms of the ebnergy levels and eigenstates of the
old Hamiltonian. If perturbation is sufficiently weak, they can be written as a (Maclaurin)
power series in λ,
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p2q
En “ Enp0q ` λEnp1q ` λ2 E2 ` ...
(4)
|ny “ |np0q y ` λ|np1q y ` λ2 |np2q y ` ...
(5)
where
Enpkq “
1 dk En ˇˇ
ˇ
k! dλk λ“0
(6)
|npkq y “
1 dk |ny ˇˇ
ˇ
k! dλk λ“0
(7)
and k “ 0 is for unperturbed values.
The energy levels and eigenstates should not deviate too much from unperturbed
values and the terms should rapidly become smaller as the order is increased.
Substituting the power series expansion into the Schrödinger equation:
pH0 ` λV qp|np0q y ` λ|np1q y ` ...q “ pEnp0q ` λEnp1q ` ...q
p|np0q y ` λ|np1q y ` ...q
4
Hamiltonian truncation
Hamiltonian truncation is a nonperturbative numerical method used to study quantum
field theories (QFTs) in d ě 2 spacetime dimensions of the form R ˆ M to compute the
spectrum of the Hamiltonian along R. It is an adaptation of the Rayleigh-Ritz method
from quantum mechanics.
One can define quantum field theory on any manifold, like Rd (flat space), R ˆ S d´1
(an infinite hollow cylinder), R ˆ T d´1 (space is taken to be a torus) or even Anti-de
Sitter in global coordinates. On such a manifold one can take time to run along R, such
8
that energies are conserved. The main idea is that for many QFTs the spectrum and
eigenstates of the Hamiltonian H cannot be find analytically or it’s difficult to do, but
the QFT Hamiltonian can be written as the sum of a "free" part H0 and an "interacting"
part that describes interactions:
H “ H0 ` gV
(8)
ş
where V “ M vdx, v - local operator of the theory over M .
Instead of a single interaction gV there may be multiple interaction terms g1 V1 `g2 V2 `
....
The recipe of the Hamiltonian truncation:
1. (a) Fix a UV cutoff Λ
(b) Find all eigenstates |iy of H0 with energy ei ď Λ:
ei |iy “ H0 |iy
(9)
(c) Normalize eigenstates such that xi|jy “ δij
(d) Let be N pΛq - number of low-energy states.
2. Compute the Hamiltonian restricted to these low-energy states:
HpΛqij “ ei δij ` gVij
(10)
3. Compute the energies and eigenstates of the finite matrix HpΛq, obeying HpΛq|ψα y “
Eα pΛq|ψα y
9
5
Beyond integrability: truncated space approach (TSA)
5.1
Introduction to the approach
The aim of this section is to present a comprehensive methodology, the truncated
space approach, that permits the study of perturbations of integrable and conformal models in one spatial dimension. The TSA methodology was first developed by V. Yurov and
Al. Zamolodchikov in two papers, one treating perturbations of the scaling Yang-Lee
model [1], and one treating the critical Ising model perturbed by a magnetic field [2].
These initial two papers sparked a sustained period of work on perturbed (both unitary
and non-unitary) conformal minimal models where the TSA was used to elucidate a wide
variety of the properties of these models: see, for example, [3]. Beyond this work on
conformal minimal models, the TSA has also been used to study variants of sine-Gordon
models. These papers all concern perturbed c “ 1 compact free bosons with the notable
exception of which considered a perturbation of the c “ 3{2 supersymmetric generalization of sine-Gordon and is the first paper to consider a model where the underlying
unperturbed theory had c ą 1. It also has been used extensively to study perturbations
of conformal theories with boundaries. The vast majority of the early works using TSA
studied perturbations of theories with central charge no greater than one. However, more
recently the TSA has been used to study more complicated cases, including multiboson
theories as well as perturbed WZNW theories.
In all of the cases, the basic problem the TSA treats is easy enough to state. The
TSA enables the study of a Hamiltonian of the following form:
H “ Hknown ` λVpert
(11)
Here Hknown is either an integrable or conformal theory, and Vpert is some perturbing
operator, which need not be of a form that renders the full Hamiltonian H integrable or
exactly solvable in any fashion. Hknown is a known" theory in the sense that we have a
complete understanding of its spectrum and matrix elements in finite volume.
We will see that the size of the system is a control parameter for the TSA, the varying
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of which allows us to explore different regimes of the theory, from the deep UV to far IR.
For the purpose of the TSA, it is important that we understand the spectrum in the
finite volume (as opposed to the infinite volume) as here the spectrum is discrete. We
will denote this spectrum by t|Ei yu8
i“1 , and portray it schematically in Fig. 1:
(a) A schematic depiction of the spectrum of Hknown in the infinite volume (left) and
the finite volume (right). In the infinite volume, there is a continuum of states, whilst in
the finite volume the spectrum is discrete (and possibly with finite degeneracy). (b) A
cartoon illustration of the TSA procedure; a cutoff energy Ec is introduced and states in
the spectrum of Hknown above this energy are discarded.
Figure 1: The process of the TSA
Having knowledge of the spectrum, the next ingredient that we require is an understanding of the matrix elements of the perturbing operator relative to unperturbed basis.
That is, we need to know
xEi |Vpert |Ej y
(12)
If the theory is a conformal theory, such matrix elements are readily computable. For
example, the states |Ei y will (at least) have a representation as a sum of products of the
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¯ i.e.
Virasoro generators,L´n , acting on some highest weight state |∆, ∆y,
ÿ
|Ei y “
j
cj
Mj
ź
L´nkj
kj “1
M̄j
ź
¯
L´nk¯ |∆, ∆y,
j
(13)
k̄j “1
with nkj , nk̄j ą 0. As we know how the Virasoro generators L´nj , L´nj̄ commute with
the perturbation Vpert , as well as how they commute with one another, we are able to
compute xEi |Vprt |Ej y in principle. In practice, we may need to compute these commutators numerically. For continuum relativistic integrable models, such matrix elements can
be computed in infinite volume via the form factor bootstrap, i.e. [4]. Under the bootstrap, they are computable by applying analyticity constraints based on the two-particle
S-matrix, crossing symmetry, and unitarity. For states in an integrable model with a relatively small number of particles, the matrix elements take on a tractable form. For matrix
elements involving states with many particles, the matrix elements can be formidable and,
while analytic expressions are available, they are typically not easily evaluated.
Supposing that we have full knowledge of both the unperturbed spectrum and the
matrix elements of the perturbing operator, we can represent the full Hamiltonian in
matrix form:
»
E1 ` λxE1 |Vpert |E1 y
—
—
—
—
—
—
H“—
—
—
—
—
—
–
λxE2 |Vpert |E1 y
λxE3 |Vpert |E1 y
.
.
.
fi
...
ffi
ffi
E2 ` λxE2 |Vpert |E2 y
λxE2 |Vpert |E3 y
...ffi
ffi
ffi
λxE3 |Vpert |E2 y
E3 ` λxE3 |Vpert |E3 y ...ffi
ffi
ffi
ffi
.
.
ffi
ffi
ffi
.
.
fl
.
.
λxE1 |Vpert |E2 y
λxE1 |Vpert |E3 y
(14)
As it stands H is an infinite dimensional matrix. So what to do? The most crude
thing we can imagine doing is simple truncating the space of states in energy. All states
whose unperturbed energy exceeds a cutff, Ec , we toss away, as pictured in Fig.1(b). This
leaves us with a infinite number of states (say N ) and a truncated Hamiltonian matrix,
HN that is infinite:
12
»
E ` λxE1 |Vpert |E1 y
λxE1 |Vpert |E2 y
λxE1 |Vpert |E3 y
— 1
—
— λxE2 |Vpert |E1 y
E2 ` λxE2 |Vpert |E2 y
λxE2 |Vpert |E3 y
—
—
— λxE3 |Vpert |E1 y
λxE3 |Vpert |E2 y
E3 ` λxE3 |Vpert |E3 y
—
—
HN “ —
.
.
.
—
—
—
.
.
.
—
—
—
.
.
.
–
λxEN |Vpert |E1 y
λxEN |Vpert |E2 y
λxEN |Vpert |E3 y
fi
...
λxE1 |Vpert |EN y
ffi
ffi
ffi
ffi
ffi
...
λxE3 |Vpert |EN y ffi
ffi
ffi
ffi
ffi
ffi
..
.
ffi
ffi
ffi
ffi
fl
... EN ` λxEN |Vpert |EN y
...
λxE2 |Vpert |EN y
(15)
This Hamiltonian we can easily diagonalize (e.g., numerically) and extract the spectrum. In this crude truncation scheme, we simply ignore the effects of the unperturbed
high energy Hilbert space; this works remarkably well for a surprisingly large number
of cases! The essential reason why the truncation may not strongly affect the results is
found in the relevancy (in the RG sense) of the perturbing operator. A strongly relevant
perturbing operator will not strongly mix the low and high energy Hilbert spaces of the
unperturbed theory and so the truncation goes unfelt in the low energy sector of the full
theory. This is not to say that the states in the low-energy sector are not mixed strongly
amongst themselves: indeed, they are. In this procedure we are not doing something akin
to perturbation theory!
5.2
Example of the truncated space approach: truncated free
fermion space approach (TFFSA)
The Ising model free energy exhibits a sibgularuty at the critical point H = 0 and T=Tc.
Th singularity is described in terms of the Euclidian quantum field theory known as the
Ising Field Theory (IFT).It can be defined as a perturbed conformal field theory through
the action
ż
AIF T “ Apc“1{2q ` τ
ż
2
ϵpxqd x ` h
σpxqd2 x
(16)
where Apc“1{2q stands fir the action of c “ 1{2 conformal field theory of free massless
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Majorana fermions, σpxq and ϵpxq are primary fields of conformal dimensions 1/16 and
1/2 respectively.
As is well known [5], at zero external filed the Ising model is equivalent to a free-fermion
theory, so (16) can be written as
ż
AIF T “ AF F ` h
1
AF F “
2π
σpxqd2 x
(17)
ż
rψ B̄ψ ` ψ̄B ψ̄ ` imψ̄ψsd2 x
(18)
Here B “ 21 pBx ´ iBy q, B̄ “ 12 pBx ` iBy q, where x “ px, yq are Cartesian coordinates, ψ, ψ̄
are chiral components of the Majorana fermi field, and σpxq is the "spin field" associated
with the fermion.
We start with the IFT (16) in finite-size geometry, with one of the two Euclidian
coordinates compactified on a circle of circumference R, x ` R „ x. If y is treated as
(Euclidian) time, the finite-size Hamiltonian associated with (17) can be written as
żR
HIF T “ HF F ` hV,
V “
σpxqdx,
(19)
0
where HF F is the Hamiltonian of the free-fermion theory (18). We are interested in
the eigenvalues of HIF T , particulary in its ground-state energy E0 pR, m, hq, because for
large R one expects to have
E0 pR, m, hq “ RF pm, hq ` Opexpp´M1 Rqq,
(20)
with M1 being the gap in the spectrum of HIF T at R ` 8 i.e. the mass of the lightest
particle of the field theory (16). We typically use the notation EpRq, with the arguments
m, h suppressed, for eigenvalues of the Hamiltonian (19) and E0 pRq will stand for the
ground-state eigenvalues of (19).
The free part HF F of the Hamiltonian (19) ris diagonal in the basis of N-particle
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states of free fermions of mass |m|. At finite R the space of states of (18) splits into
two sectors, the Neveu-Schwartz (NS) and Ramond (R) sector (with ψ, ψ̄ antiperiodic or
periodic as x Ñ x`R, respectively). In each sector teh particle momenta are quantized as
pn “ 2πn{R where n P Z ` 1{2 in NS sector, and n P Z in R sector. The N-particle states
can be obtained from the NS and R vacua |0yN S and |0yR be applying the corresponding
canonical fermionic creation operators,
NS sector:
|k1 , ...kN yN S “ a:k1 ...a:kN |0yN S
|n1 , ...nN yR “ a:n1 ...a:nN |0yR
R sector:
k1 , ...kN P Z ` 1{2
(21)
n1 , ...nN P Z
(22)
The normalizations of these states are fixed by conventional; anticommutators,
tan , a:n1 u “ δn,n1 .
tak , a:k1 u “ δk,k1 ,
(23)
In all cases the energies associated with the N-particle states, EN pRq, have the standart
form
EN pN Sq pRq “ E0pN Sq pRq `
N
ÿ
ωki pRq
(24)
ωni pRq
(25)
i“1
EN pRq pRq “ E0pRq pRq `
N
ÿ
i“1
where
d
ˆ
ωk pRq “
m2 `
2πk
R
d
˙2
ˆ
,
ωn pRq “
m2 `
2πn
R
˙2
(26)
(with positive branch of the square root taken, in particular, ω0 pRq “ |m|), and
ż8
E0pN Sq pRq “ RF pm, 0q ´ |m|
dθ
cosh θ logp1 ` e´|m|R cosh θ q
´8 2π
15
(27)
ż8
E0pRq pRq “ RF pm, 0q ´ |m|
dθ
cosh θ logp1 ´ e´|m|R cosh θ q
2π
´8
(28)
are the eigenvalues associated with |0yN S and |0yR , respectively. The term
F pm, 0q “
m2
log m2
8π
(29)
in (27) and (28) accounts for the famous Onsager’s singularity of the Ising free energy
at zero h [6].
In order to treat the IFT with nonzero h we can admit only the states which respect
the periodicity condition for the spit=n field σpx ` R, yq “ σpx, yq. This condition brings
a distinction between the cases m ą 0 (the "high-T regime") and m ă 0 (the "low-T
regime"). The admissible states are
m ą 0 : NS - states with N even, and R - states with N even
(30)
m ă 0 : NS - states with N even, and R - states with N odd
(31)
(at m “ 0 the odd-N states in the R sector can be viewed as even-N states, with one
a0 particle added).
The operator hV in the full Hamiltonian (19) generates transitions between the states
in NS and R sectors. Fortunately, all its matrix elements between the above states are
known exactly. They are related in a simple way to the finite-size formfactor of the field
σpxq, for which an explicit expression exists,
N S xk1 , k2 , ..., kK |σp0, 0q|n1 , n2 , ..., nN yR “
SpRq
K
ź
j“1
g̃pθkj q
N
ź
(32)
gpθni qFK,N pθk1 , ..., θkK |θn1 , ..., θnN q
i“1
where θn pθk q stand for the finite-size rapidities to the integer n (half-integer k) by the
equations
16
|m|R sinh θk “ 2πk,
|m|R sinh θn “ 2πn.
(33)
In (32) FK,N is the well-known spin-field formfactor in infinite-space [7]:
˙
θi ´ θj
FK,N pθk1 , ..., θkK |θn1 , ..., θnN q “ i
σ̄
tanh
2
0ăiăjďK
˜ 1
¸
ˆ
1
1 ˙
ź
ź
θp ´ θq
θs ´ θt
tanh
coth
.
2
2
0ăpăqďN
0ăsďK
ˆ
s
r K`N
2
ź
(34)
0ătďN
where
σ̄ “ s̄|m|1{8 ,
s̄ “ 21{12 e´1{8 A3{2 “ 1.35783834...
(35)
with A standing for the Glaisher’s constant, and the rest of the factors represent finite-size
effects. The overall factor SpRq is essentially the vacuum-vacuum matrix element
σ̄SpRq “
$
’
&N S x|σp0, 0q|0yR
for m ą 0
’
%N S x|µp0, 0q|0yR
for m ă 0
(36)
where µpxq is the dual spin field [8], and SpRq is given by
SpRq “ exp
$
8
& pmRq2 ij
dθ dθ
sinh θ1 sinh θ2
2
p2πq2 sinh mR cosh θ1 sinh mR cosh θ2
1
%
2
´8
(37)
ˇ
ˇ
ˇ
ˇ
θ
´
θ
1
2
ˇ
log ˇˇcoth
2 ˇ
which was obtained in [9]. The momentum-dependent leg factor g and g̃ are
eκpθq
gpθq “ a
,
|m|R cosh θ
where
17
e´κpθq
g̃pθq “ a
|m|R cosh θ
(38)
ż8
1
dθ
1
κpθq “
1 log
´8 2π cosh pθ ´ θ q
The phase factor ir
K`N
s
2
˜
1
1 ´ e´|m|R cosh θ
1
1 ` e´|m|R cosh θ
¸
(39)
(where r...s denotes the integer part of the number) appearing
in (34) can be removed by an appropriate phase rotation of the states, and thus play no
role in the computations.
And now we can note that the Hamiltonian (19) can be rewritten to make its scaling
form explicit,
HIF T “ E0pN Sq pRq ` |m|H0 prq ` |m|ξHσ prq
(40)
where
ξ“
h
|m|15{8
(41)
and the operators H0 prq and Hσ prq (corresponding to the terms HF F and hV in (19)
depend on a dimensionless parameter r “ |m|R only.
6
TFFSA Results
6.1
Construction of the basis
In the numerical analysis we used the technically simplest truncation scheme based on
the notion of level. A cutoff energy was defined as
Ec “
2πN
R
(42)
On the Fig. 2 one can find the plot of the size of basis as a function of cutoff-energy
Ec for m “ 1 and R “ 1 (blue line), R “ 2 (orange line), R “ 3 (green line), R “ 4 (red
line), R “ 5 (purple line) and R “ 6 (brown line) . As we can see the dimensionality of
the truncated space is rapidly increasing.
18
Figure 2: The size of the truncated space as a function of cutoff-energy Ec , where for left:
Ec P r0, 3s and for right: Ec P r0, 6s
6.2
TFFSA Data and Behavior of Ground State
In Fig. 3 we plot the lowest lying set of energies coming from the TFFSA (the parameters
chosen for the computation are given in the figure caption). We see that for sufficiently
large system size R all of these energies decrease linearly with increasing R (for smaller
values of R, the energies evolve into their unperturbed (h “ 0) forms). This linear decrease
reflects the negative energy density of the ground state, i.e.
Egs “ ´f pm, hqR
(43)
m2
logpm2 a2 q ` m2 Φpmh´8{15 q
8π
(44)
where
f pm, hq “
where Φpηq is a universal scaling function. In the presence of a magnetic field, h ąą
m15{8 , the dominant contribution to the ground state comes from a particular linear
combination of the near degenerate |0yR and |0yN S vacua,
|Egs y „ |0yN S ´ signphq|0yR ` ...,
(45)
where the ellipses denote states with finite fermion number. The ground state energy
19
Egs then reduces to
|Egs y „ 2σ̄m1{8 hR
(46)
That is, the ground state represents spins aligned in a direction anti-parallel to the
applied magnetic field.
However, what of the state with its spins parallel to the applied field? In the infinite
volume (R “ 8), this state would have infinite energy and hence would not exist in the
theory. In the finite volume (where we work when using the TFFSA), however, this state
indeed exists. It has finite positive energy, 2σ̄m1{8 hR “ ´Egs (at least for h ąą m15{8 )
and terms of the unperturbed basis is given roughly by
|Ef alsevac. y „ |0yN S ` signphq|0yR
(47)
The presence of this false vacuum state in the TFFSA data, Fig. 3, can be inferred
by regions where the energy of a particular state increases with system size R (say the
second excited state between R “ 1 and R “ 2). As we are typically interested in lowenergy excitations about the true vacuum, we have to be sure to note mistake a state
that is the false vacuum (or an excitation about the false vacuum) for one that is of direct
interest. This is always an issue in models where the unperturbed Hamiltonian has a
discrete spontaneous (near-) symmetry breaking where the perturbation explicitly breaks
this symmetry.
6.3
Behavior of Excited States
We now consider the behavior of the excited states. In Fig. 3 we plot the excited state
energies as well as ground state energy. We see clearly that there are regions in the finite
volume R where the lowest excited states are unchanging. This region in R is the region
in which we want to work within the TFFSA. We furthermore see that the energies can
be determined with relatively high precision (with small errors in the fourth significant
digit). We note that the data presented here is taken outside the region of validity of the
20
Figure 3: Raw TFFSA data for the lowest lying energy levels of the Hamiltonian (19) for
h “ p2mq15{8 , m “ 1 and η “ 0.5 plotted against the dimensionless system size, Rh8{15
The presented data focuses on the zero-momentum (ground state) sector. One sees that
the energy levels all roughly have a constant negative slope. The black line that has a
positive slope corresponds to a false vacuum state (equal to one of the linear combinations,
|0yN S ` signphq|0yR )
Bethe-Salpeter analysis (as h “ p2mq15{8 ).
One point to stress here is that different excitations have different "stability regions":
the first bound state, E1 becomes stable after the dimensionless system size Rh8{15 exceeds
2.4, while the third meson state E3 stabilizes when Rh8{15 ą 4.1. For the particular choice
of m and h presented, only the first three mesons are stable. States higher in energy
coming from the TFFSA represent multi-meson states.
7
Conclusion
In the course of the work, the basis of the states was build and the graphs of energy levels
were obtained depending on the circumference R of the base of the cylinder for certain
value of the parameter (the scaling parameter) connecting the deviation from the critical
value of the temperature with the deviation from the critical value of the magnetic field.
The analysis of the obtained graphs of finite-size energy levels Ei pRq is presented. The
21
concept of the "false vacuum" was discussed.
References
[1] V. P. Yurov and A. B. Zamolodchikov, “Truncated conformal space approach to scaling
Lee-Yang model,” Int. J. Mod. Phys., pp. 3221–3245, 1990.
[2] ——, “Truncated fermionic - space approach to the critical 2D Ising model with magnetic field,” Int. J. Mod. Phys., pp. 4557–4578, 1991.
[3] R. S. K. Mong, D. J. Clarke, J. Alicea, N. H. Lindner, P. Fendley, C. Nayak, Y. Oreg,
A. Stern, E. Berg, K. Shtengel, and M. P. A. Fisher, “Universal topological quantum
computation from a superconductor-Abelian quantum Hall heterostructure,” Phys.
Rev., pp. 2160–3308, 2014.
[4] F. A. Smirnov, “Form factors in completely integrable models of quantum field theory,”
World Scientific, vol. 14, 1992.
[5] T. T.Wu and B. M. McCoy, “The two-dimensional Ising model,” Harvard University
Press, 1973.
[6] L. Onsager, “Crystal statistics. I. a two-dimensional model with an order-disorder
transition,” Phys. Rev., p. 117–149, 1944.
[7] B. Berg, M. Karowski, and P. Weisz, “Construction of Green’s functions from an exact
S matrix,” Phys. Rev., pp. 2477–2479, 1979.
[8] L. P. Kadanoff and H. Ceva, “Determination of an operator algebra for the twodimensional Ising model,” Phys. Rev., p. 3918–3939, 1971.
[9] S. Sachdev, “Universal, finite temperature, crossover functions of the quantum transition in the Ising chain in a transverse field,” Nucl. Phys., pp. 576–595, 1996.
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