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PCE STAMP
Quantum Phase Transitions & the Spin Bath: DYNAMICS
PITP workshop – Q Info & the Many-Body Problem: Dec 1-3, 2007
Physics & Astronomy
UBC
Vancouver
Pacific Institute
for
Theoretical Physics
QUANTUM PHASE TRANSITIONS:
The BASIC IDEA
One imagines a system which has a phase
transition at T=0, which is not 1st order in the
T  0 limit (a condition not so easy to satisfy
in practise).
EXAMPLES: These can be classified into systems
with extended modes, and those with localised modes
Extended Modes
MnSi ; TiBe2 ; ZrZn2 : conductors with
FM/PM transition
Localised Modes
LiHoxY1-xF4 ; Mn-12 ; Fe-8 : insulators
with
order (FM; quantum spin glass) /
disorder (PM, spin liquid?) transition
Now – let’s suppose that we can
change the critical parameter g(t)
with time t, passing through gc
What happens?
A RELATED PROBLEM: Adiabatic Quantum Computation
Assume a system Hamiltonian:
H = (1- s) H1 + s H2
NB: Linear interpolation: s = vt
Assume also that the ground state of H1 is
easily accessible., and that the ground state
of H2 encodes the solution to a hard
computational problem.
E. Farhi et al., Science 292, 472 (2001)
One would like to analyse this in terms of the standard Landau-Zener formulae
for the transitions. However in reality we know the system is more complex
Landau  Zener
probabilit y :
2 / 2
PLZ  e
 ~ s ~ 1 / t f
SIMPLE EXAMPLE
Transverse field Q Ising model
(the ‘TFQIM’):
Simplify to 1-d nearest neighbour model:
Asymptotic ground states
Or else one of
The critical parameter is:
Energy gap is:
gc
SIMPLE EXAMPLE (cont.)
Assume that:
Then:
Define:
where
Then we can do a standard Landau-Zener calculation to get the probability
that there will be a non-adiabatic transition, corresponding to the nucleation
of a ‘kink’ during the passage through the QPT.
Then the result is that we get a density of defects, with a mean
distance between defects given by
One can argue that this resembles the Kibble-Zurek mechanism of
defect nucleation.
Zurek et al., PRL 95, 105701 (2005)
PROBLEM: Near a QPT we expect a huge number of other states at
low energy – this is particularly true when the interactions are
‘interesting’ (long-range, frustrating, etc.). What then?
Toy Model: LANDAU-ZENER plus ENVIRONMENT
Assume a model
Central Spin Hamiltonian:
with
Can also use:
COUPLINGS to ENVIRONMENT
Oscillator Bath:
Spin Bath:
Can also make the simplification:
(inert bath)
RESULTS: OSCILLATOR BATH ENVIRONMENT
If the coupling to the oscillator bath is diagonal, and we are at T=0, then
we get no change – we just get the original LZ formula.
If we have a non-diagonal coupling, or we are non-zero T, then the result
is more complicated. At finite T the results are controversial.
See, eg:
M Wubs et al., PRL 97, 200404 (2006)
RESULTS: SPIN BATH ENVIRONMENT
If the interaction between bath spins is zero (inert bath), and the
temperature T = 0, then we again find no change – we get the original
LZ formula.
ATS Wan et al., condmat /0703085
See, eg:
These restrictions make the results of
limited use – real systems are at finite T
(very much so!). Moreover the neglected
terms are usually there: and they are
Important even if they are small.
How much further can we go?
LANDAU-ZENER + SPIN BATH – INCLUDING BATH FLUCTUATIONS
Define
and assume
so that for no bath we have
With spin bath:
where
and
and the trace includes bath
fluctuations averaged according to:
with correlator
so that the average of
is just
For most experiments the interesting limit is the high T fast sweep limit,
with sweep rate s.t.
Then we find that
Candidates for Magnetic Qubits
One of the candidates discussed for quantum computations is magnetic system
that very large magnetic domain
walls have already shown macroscopic tunnelin
3+
ase
ofSQUID
Rare-earth
ions: Ho
in Y0.998
0.002LiF4on magnetic molecules and ions w
like
flux. Right
now interest
isHo
focussed
behave as 2-level systems- as ‘Qubits”.
Mn12 S = 10
Tetragonal symmetry
V15 S = 1/2
(Ho in S4)
J = L+S = 8;
gJ=5/4
Single-molecule magnets (SMM)
Giant spins
Ho ions in LiYF4 host
Ni12 S = 12
Fe8 S = 10
The DIPOLAR SPIN NET - REALISATION
The dipolar spin net is of great interest to solid-state theorists
because it represents the behaviour of a large class of systems
with “frustrating” interactions (spin glasses, ordinary dipolar
glasses). It is also a fascinating toy model for quantum
computation:
H = Sj (j tjx + ej tjz) + Sij Vijdip tiz tjz
+ HNN(Ik) + Hf(xq)
+ interactions
CERTAIN NANOMAGNETIC SYSTEMS ALLOW US
TO PROBE THIS SYSTEM IN GREAT DETAIL, &
TO VARY THE PARAMETERS OVER A VERY
WIDE RANGE
Almost all experiments so far are done in the
region where o is small- whether the dynamics
is dipolar-dominated or single molecule, it is
incoherent. However one can give a theory of this
regime.
The next great challenge is to understand the
dynamics in the quantum coherence regime, with
or without important inter-molecule interactions
Different Regimes for the
Spin Net System
The spin net offers a range of possibilities,
and our task is to find the behaviour in each
of the following regimes:
(i) DIPOLE INTERACTION-DOMINATED REGIME: If
one ignores the environment, this Quantum Ising
system simply localises into a glass if Vo>o.
However the environment has a profound effect even extremely small xo will delocalise the spins, &
give quantum relaxation. If we increase the quantum
parameter so that o>> xo (but still o< Vo) then very
complex multi-spin entangled dynamics ensues.
Vo
Dipole
Interaction
regime
xo
Decoherent
regime
Quantum
regime
(ii) DECOHERENT RELAXATION REGIME: Even with strong
o
decoherence/dissipation, the inter-spin correlations strongly affect
the relaxational dynamics. Again, the system is never frozen, even if o<< xo , at finite T; but
strong decoherence can freeze it at T=0
(iii) COHERENT QUANTUM REGIME: The most interesting but the most difficult to
understand – this is the full quantum computation problem, with N-spin entanglement.
The smallest environmental coupling eventually destroys coherent dynamics – higher spin
entanglement is the first to go. Many features of are not understood at all – this is a frontier
problem of great importance. It is commonly assumed in the quantum info literature that
for weak decoherence one can ignore all but uncorrelated errors (ie., single-spin
decoherence coming from interactions between individual qubits & the environment).
As we shall see below this is not in general correct.
The Fe-8 MOLECULE
Low-T Quantum regime- effective Hamiltonian
(T < 0.36 K):
Longitudinal bias:
Eigenstates:
Fe8 S = 10
Feynman Paths on the spin sphere for
a biaxial potential. Application of a
field pulls the paths towards the field
Which also defines orthonormal states:
HYPERFINE COUPLING to spin bath (NUCLEAR SPINS)
Hyperfine coupling:
Define the set of fields:
Static component is:
Component which flips is:
This gives a ‘central spin’ Hamiltonian:
Some of the couplings
in Fe-8 (at H=0)
Structure of NUCLEAR MULTIPLET
There are 215 nuclear sites in the molecu
Transitions between states of different total
polarisation (T1 process) are driven mainly by
molecular tunneling)
Total width of gaussian multiplet:
(NB: This decreases with increasing applied field)
For Fe-8 at H=0, Eo ~7 mK (depends on isotopic concentration
For large
, the precessional decoherence rate is just
gk ~  (Eo /)2
Coupling to PHONONSEffective coupling to qubit:
NV Prokof’ev PCE Stamp, J Low Temp Phys 104,
143 (1996)
PCE Stamp IS Tupitsyn Phys Rev B69,
014401 (2004)
A Morello, PCE Stamp, IS Tupitsn, PRL 97,
207206 (2006)
where
Giving a phonon decoherence rate:
Quantum Phase Transition or COHERENCE experiments in Fe-8
A. Morello, P.C.E. Stamp and I.S. Tupitsyn, Phys Rev Lett 97, 207206 (2006)
RESULTS for DECOHERENCE in the Fe-8 SPIN NET
A very startling result emerges when one looks at the low-T decoherence in a dipolar
spin net. Even for rather low T, the decoherence is dominated by correlated errors (ie.,
coming from pairs of qubits).
Here we see results
for the Fe-8 system.
Note that at low T
we can still get very
high coherence:
10
dipolar
0.05 K
0.1 K
0.2 K
0.4 K
1
0.1
0.01
1E-4
ar
cle
nu
gf
1E-3
optimal coherent
operation point
at T = 50 mK
1E-5
1E-6
1E-7
1E-9
1.5
Q  107
ph
on
on
1E-8
2.0
2.5
3.0
3.5
4.0
By (T)
A. Morello, P.C.E. Stamp, I.S. Tupitsyn, Phys Rev Lett 97, 207206 (2006)
QUANTUM RELAXATION REGIME: Derivation of Kinetic Eqtn.
In both the dipolar-dominated regime and the environment-dominated regime, the
dynamics is incoherent if o is small. The we can use a classical kinetic equation.
The kinetic eqtn for the magnetic qubit distribution Pa (x, r) is a BBGKY one, coupling it t
2-qubit distribution P2. Here r is the position of the qubit, a  +, is the polarisation of th
qubit along the z-axis, and x is the longitudinal field at r.
In this kinetic equation the interaction U(r-r’) is dipolar, and the relaxation rate t1N is
the inelastic, nuclear spin-mediated, single qubit tunneling flip rate, as a function of
the local bias field. As discussed before, this relaxation operates over a large bias
range xo where typically xo  Eo ( and Eo is the width
of the nuclear spin muliplet introduced before)
The BBGKY hierarchy can be truncated with the kinetic equation above if the initial 2-qubit
distribution factorizes. This happens if the system is either (i) initially polarized, or (ii) initia
strongly annealed. Then we have:
The kinetic equation can then be solved, and
gives the square root short-time behaviour:
M(x)
Quantum Relaxation in
a “Spin Net” of
Interacting MAGBITS
x
Vij >> Eo > 
At first glance the problem of a whole net of magbits, with
long-range “frustrating” dipole interactions between them,
looks insuperable. But actually the short-time dynamics
can be solved analytically, in the quantum relaxation regime!
This is because the dipole fields around the sample vary slowly
in time compared to the fluctuating hyperfine fields. This leads
IS Tupitsyn, PCE Stamp,
to universal analytic predictions:
NV Prokof’ev, Phys Rev B69,
NV Prokof’ev, PCE Stamp, PRL
80, 5794 (1998)
132406 (2004)
(1) Only magbits near resonance make incoherent flips
As tunneling occurs, the resonant surfaces move &
disintegrate- then, for ANY sample shape
dM(t) ~ [t/tQ]1/2
tQ ~ (2T2) Eo2N(xeH)/W
where W is the width of the dipolar field distribution,
and N(x) is the density of the distribution over bias.
(2) Tunneling digs a “hole” in this distribution, with initial
width Eo, and a characteristic spreading with time- so it
depends again on the nuclear hyperfine couplings.
HOLE DIGGING up close
We look at the time evolution of the INTERNAL DISTRIBUTION OF BIAS FIELDS M(x,t)
(recall that x is the longitudinal bias field. A key feature of the theory is ‘Hole-digging’
in this distribution; the tunneling spins deplete the distribution. Only spins in resonance
can tunnel, and this happens in a field range 2x0 (ie., controlled by the nuclear hyperfine
interactions). The time evolution is non-trivial because the dipolar interactions scatter
spins back into the hole (giving the square root time relaxation).
FAST SWEEPING: APPLICATION TO Fe-8 SYSTEM
Now we assume a full array of Fe-8 molecules with intermolecular dipole
coupling & a fluctuating nuclear bath. Assume the sample has been annealed
and then cooled to low T.
The we can assume
And the full kinetic
equation gives:
Near the nodes
where
and we can make the expansion
Then near the n-th node one
finds a decay rate given by:
For pictures see next page…..
For field along hard axis
Nodal regions
Field tilted away from nodes
by a 1 degree angle
FAST SWEEP EXPERIMENTS on Fe-8
The fast sweep of field gives a set of
hysteresis curves as a function of
sweep rate and temperature. For low T
(no thermal activation but well above
hyperfine energies) one gets the
graphs at right.
R Sessoli, W Wernsdorfer
Science 284, 133 (1999)
Interpretation of these
expts according to naïve
Landau-Zener gives the
curves at right.
The smoothing of the
curves is not due to
misalignment but instead
to the internal fields
NUCLEAR SPIN BATH in MAGNETIC
SYSTEMS: The LiHoxY1-xF4 system
The Ho ions interact with each other via dipolar interactions. This
system is usually treated as the
archetypal Quantum Ising system:
The single spin has
and
a 1-spin crystal-field Hamiltonian
with
and
In zero field there is a low-energy doublet, which we call
This is separated from a 3rd state
by a gap
The g-factor is extremely anisotropic (factor of 20) giving an
easy z-axis. Application of a transverse field gives a ‘tunneling
splitting’ between the 2 doublet states (with transitions through
the 3rd level) so that our low-E single spin effective Hamiltonian
is just that above, with
The dipolar interactions between the spins are just
However including crystal field effects strongly modifies this to the above form, with
the ‘zz’- interaction having strength
with
HYPERFINE COUPLING to the NUCLEAR SPIN BATH
We have a simple interaction
with
The hyperfine splitting between nuclear levels
is roughly 0.25K
This interaction has a profound effect on the
dynamics & on the effective Hamiltonian at low
energy – electronic spins cannot flip unless
multiple nuclear transitions also take place.
Consider first what happens in low transverse field; we
single out the 4 important states shown in the diagram.
This problem is easily solved without the transverse
hyperfine coupling; we get eigenstates
where
with mixing coefficient
At low transverse field this just produces a classical
Ising system:
with
and
& renormalised spin
The transverse hyperfine term
only becomes effective when
We then have a renormalised
Hamiltonian:
etc
&
&
QUANTUM
PHASE
TRANSITIONS:
HOW ARE
THEY AFFECTED
BY THE
SPIN BATH?