PCE STAMP

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PCE STAMP
PITP/BIRS Workshop SPIN, CHARGE, TOPOLOGY
Physics Astronomy UBC Vancouver
Pacific Institute for Theoretical
Physics
http//pitp.physics.ubc.ca/index.html
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REDUCTION to a LOW-ENERGY OSCILLATOR BATH FORM
Classical Dynamics
Quantum Dynamics
Heff
Suppose we want to describe the dynamics of
some quantum system in the presence of
decoherence. As pointed out by Feynman and
Vernon, if the coupling to all the enevironmental
modes is WEAK, we can map the environment to an
oscillator bath, giving an effective Hamiltonian
like
A much more radical argument was given by
Caldeira and Leggett- that for the purposes of
TESTING the predictions of QM, one can pass
between the classical and quantum dynamics of a
quantum system in contact with the environment
via Heff. Then, it is argued, one can connect the
classical dissipative dynamics directly to the
low-energy quantum dynamics, even in the regime
where the quantum system is showing phenomena
like tunneling, interference, coherence, or
entanglement and even where it is MACROSCOPIC.
This is a remarkable claim because it is very
well-known that the QM wave- function is far
richer than the classical state- and contains far
more information.
Feynman Vernon, Ann. Phys. 24, 118
(1963) Caldeira Leggett, Ann. Phys. 149, 374
(1983) AJ Leggett et al, Rev Mod Phys 59, 1 (1987
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CONDITIONS for DERIVATION of OSCILLATOR BATH
MODELS
Starting from some system interacting with an
environment, we want an effective low-energy
Hamiltonian of form
(1) PERTURBATION THEORY
Assume environmental states
and energies
The system-environment coupling is
Weak coupling
where
In this weak coupling limit we get oscillator
bath with
and couplings
(2) BORN-OPPENHEIMER (Adiabatic) APPROXIMATION
Suppose now the couplings are not weak, but the
system dynamics is SLOW, ie., Q changes with a
characteristic low frequency scale Eo . We define
slowly-varying environmental functions as
follows
Quasi-adiabatic eigenstates
Quasi-adiabatic energies
Slow means
Then define a gauge potential
We can now map to an oscillator bath if
The oscillator bath models are good for
describing delocalised modes then Fq(Q)
O(1/N1/2) (normalisation factor)
Here the bath oscillators have energies
and couplings
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WHAT ARE THE LOW- ENERGY EXCITATIONS IN A SOLID
?
DELOCALISED Phonons, photons, magnons, electrons,
These always dominate at high energy/high T
LOCALISED Defects, Dislocations, Paramagnetic
impurities, Nuclear Spins, . These
always dominate at low T
.
.
..
.
.,.,
..
.
.

.

.
/

.
At right- artists view of energy
distribution at low T in a solid- at low T
most energy is in localised states. INSET
heat relaxation in bulk Cu at low T
.
,
.
..
.,
.,
.
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How do REAL Solids (99.9999) behave at low
Energy?
In almost all real solids, a combination of
frustrating interactions, residual long-range
interactions, and boundaries leads to a very
complex hierarchy of states. These often have
great difficulty communicating with each other,
so that the long-time relaxation properties and
memory/aging effects are quite interesting- for
the system to relax, a large number of
objects (atoms, spins, etc.) must
simultaneously reorganise themselves . This
happens even in pure systems
Results for Capacitance (Above) Sound velocity
and dielectric absorption (Below) for pure SiO2 ,
at very low T
A model commonly used to describe the low-energy
excitations (which is certainly appropriate for
many of them) is the interacting TLS model,
with effective Hamiltonian
ABOVE structure of low-energy eigenstates for
interacting TLS model, before relaxation
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QUANTUM ENVIRONMENTS of LOCALISED MODES
Consider now the set of localised modes that
exist in all solids (and all condensed matter
systems except the He liquids). As we saw
before, a simple description of these on their
own is given by the bare spin bath Hamiltonian
where the spins represent a set of discrete
modes (ie., having a restricted Hilbert space).
These must couple to the central system with a
coupling of general form
We are thus led to a general description of a
quantum system coupled to a spin bath, of the
form shown at right. This is not the most
general possible Hamiltonian, because the bath
modes may have more than 2 relevant levels.
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ENERGY SCALES in SUPERCONDUCTORS
Again one has a broad hierarchy of energy scales
(here shown for conventional s/c) electronic
energy scales U, eF, (or t) phonon
energies qD
Gap/condensation energy DBCS p/m
impurities (not shown) J, TK Coupling of
y(r) to spins wk Total coupling to spin
bath E0 N1/2wk A superconducting
device has other energy scales eg., in a
SQUID Josephson plasma energy W0J
Tunneling splitting D0 These
are of course not all the energies that can be
relevant in a superconductor. However we note
that in general magnetic systems have a more
complex hierarchy of interactions than a
superconductor.
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MICROSCOPIC ENERGY SCALES in
MAGNETS
The standard electronic coupling energies are
(shown here for Transition metals) Band
kinetic interactions t, U Crystal
field DCF
Exchange, superexchange J Spin-orbit
lso
Magnetic anisotropy KZ
inter-spin dipole coupling VD
p/m impurities (not shown) J, TK which
for large spin systems lead to
Anisotropy barriers EB EKZ
small oscillation energies EG KZ
Spin tunneling amplitude D0
Also have couplings to various thermal baths,
with energy scales Debye frequency
qD Hyperfine couplings
Aik Total spin bath energy
E0 N1/2wk Inter-nuclear couplings
Vkk
NOTE all of these are parameters in effective
Hamiltonians for magnets at low T.
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1 DECOHERENCE in QUANTUM WALKS
Quantum walks refer to the dynamics of a particle
on some arbitrary mathematical graph. Their
importance is twofold. First, they can
be mapped to a very large class of quantum
information processing systems.
Second, it is hoped that they will be used to
generate new kinds of quantum information
processing algorithm. The whole field of
quantum walks is rather new, and there is still
elementary basic work to be done. One of the
most interesting things is the investigation of
decoherence on quantum walks. Here I briefly
describe some recent results in this field (see
also talk of A. Hines at this workshop)
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Remarks on NETWORKS- the QUANTUM WALK
Computer scientists have been interested in
RANDOM WALKS on various mathematical GRAPHS, for
many years. These allow a general analysis of
decision trees, search algorithms, and indeed
general computer programmes (a Turing machine
can be viewed as a walk). One of the most
important applications of this has been to error
correction- which is central to modern software.
Starting with papers by Aharonov et al (1994),
Farhi Gutmann (1998), the same kind of
analysis has been applied to QUANTUM COMPUTATION.
It is easy to show that many quantum
computations can be modeled as QUANTUM WALKs on
some graph. The problem then becomes one of
QUANTUM DIFFUSION on this graph, and one easily
finds either power-law or exponential speed-up,
depending on the graph. Great hopes have been
pinned on this new development- it allows very
general analyses, and offers hope of new kinds
of algorithm, and new kinds of quantum error
correction- and new circuit designs.
Thus we are interested in simple walks
described by Hamiltonians like
which can be mapped to a variety of gate
Hamiltonians, spin Hamiltonians, and interacting
qubit networks (see talk of A Hines).
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A VARIETY of MAPPINGS
A Hines PCE Stamp A Hines G Milburn PCE Stamp
One can make a lot of useful mappings between
qubit Hamiltonians, Hamiltonians for spin chains
and other spin networks, quantum gate systems,
and quantum walk Hamiltonians. This is very
useful in the exploration of different quantum
algorithms and quantum information processing
hierarchies.
One of the most important goals in this field is
to try and produce new kinds of quantum
algorithm. So far the 2 most important ones are
the Shor and Grover algorithms. The hope is that
new representations, like the quantum walk, will
allow us to do this.
Another important use of quantum walks is the
possibility of more easily
studying decoherence in different quantum
information processing systems.
The mappings above allow us to easily move
between different
representations of this, and to easily study the
dynamics of quantum
information processing.
QUESTION how does decoherence affect the quantum
walk dynamics?
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ANALYSIS of DECOHERENCE for QUANTUM WALKS
Adding couplings of a quantum walker to an
oscillator bath gives both diagonal and
non-diagonal terms
The same happens when we couple the walker
to a spin bath (here written for a bath of
2-level systems)
In what follows we will consider coupling to a
spin bath, and look at the role of the
non-diagonal terms. We will drop the interactions
Vkk , and look at 2 limiting cases
(1) Weak field (hk small)
(2) Strong field (hk large)
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DECOHERENCE QUANTUM WALKS a MODEL EXAMPLE

We look at a simple but interesting example -
a
d-dimensional hyperlattice, with a
non-diagonal coupling
to a spin bath
NV Prokofev, PCE Stamp condmat/0605097 (PRB in
press) to be published
We assume a hypercubic lattice
FREE QUANTUM BEHAVIOUR
Suppose initial state is at origin
So that
Then, since
One gets
More generally, we can start with a wave-packet
which gives
Thus, quite generally one has
and that
Now this is to contrasted with diffusive
behaviour
and
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DECOHERENCE DYNAMICS
For the decoherent Quantum Walk Hamiltonian
Or, for an initial wave-packet
Now this produces a very surprising result
BUT.
In other words, the particle spends more time
near the origin than classical diffusion would
predict, BUT it also has a BALLISTIC part (in
the long-time limit)!!
More detailed evaluation of the integrals
fills this picture out
Density matrix after time t such that z2Dt gtgtR2,
with z 2000 and R10. Long-range part is
ballistic, short-range part is sub-diffusive.
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2. The DISSIPATIVE W.A.H. MODEL
We are interested in topological field theories
because they possess hidden topological
quantum numbers which are conserved even when the
system is subject to quite severe perturbations.
A model of central interest is the dissipative
W.A.H. model (named after Wannier, Azbel,
Hofstadter). This is produced by synthesizing
elements from 2 simpler models which are very
interesting on their own the W.A.H. model
(non-interacting charged particles moving on a
2-d lattice in a uniform magnetic field), and the
Schmid model (a particle moving in a periodic
potential, coupled to an oscillator bath).
Combining these gives a model in which W.A.H.
particles couple dissipatively to an oscillator
bath. This model is believed to have an SL(2,Z)
symmetry, in common with some other field
theories which attempt to describe the
Fractional Quantum Hall liquid, certain systems
of interacting quantum wires, and possibly
Josephson junction arrays. However the model was
actually originally introduced by string
theorists (Callan et al.) to deal with a class of
open string theories, and it is still of central
interest in string theory. It is of potential
interest for topological quantum computation.
In what follows I first describe key results
for the W.A.H. and Schmid models on their own,
and then go on to discuss results for the
dynamics of the dissipative WAH model. It is
found that there are some important outstanding
problems here in particular, the older results
of the string theorists seem to conflict with
more recent results (see also talk of Taejin Lee
in this workshop)
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The W.A.H. MODEL
The Hamiltonian involves a set of charged
fermions moving on a periodic lattice-
interactions between the fermions are ignored.
The charges couple to a uniform flux through
the lattice plaquettes.
Often one looks at a square lattice, although it
turns out much depends on the lattice symmetry.
One key dimensionless parameter in the problem is
the FLUX per plaquette, in units of the flux
quantum
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The crucial effect of the applied field is
in the extra phase accumulated around each
lattice plaquette- these phases of course
interfere with each other. To see this we choose
the Landau gauge
Then , wrting the Schrodinger eqtn as a
difference eqtn. around a plaquette, we have
In terms of reduced variables (where Eo is just
the bandwidth) we can then write the solution in
the form
The Schrodinger eqtn. takes the iterative form
Solutions exist provided
This is just a condition on the flux- it must be
rational
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The HOFSTADTER BUTTERFLY
The graph shows the support of the density of
states- provided a is rational
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The recursive nature of the Schrodinger eqtn
is then directly responsible for the recursive
form of the density of states. One has a nested
pattern in which the entire form is repeated in
any subset (reflecting of course the structure of
the rational numbers). The shape of the
nesting pattern depends on the lattice
structure. For a finite lattice the adjustment
of levels between very close values of flux is
effected by level crossing between band edges.
For infinite lattices this happens infinitely
fast. In finite lattices, EDGE states are crucial.
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Another way of looking at W.A.H. (Chern ) is
shown at right
For various aspects of the WAH model, see the
references below
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II Schmid model- particle coupled to oscillator
bath
In the Schmid model a particle moves on a 1-d
periodic lattice, but is now coupled to an
oscillator bath. It is then interesting to
apply a weak field. The quantity of crucial
interest is then the particle mobility
(hopefully well-defined!).
The particle-bath interaction is bilinear in the
coordiantes of the two. The individual couplings
are weak (delocalised modes), but their
cumulative effect on the particle depends on the
form of the Feynman-Vernon/Caldeira-Leggett
spectral density, defined as follows
In this study we choose an Ohmic spectral form
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The Schmid model is a very rich field theory.
We first separate the action into 2 terms The
bare action contains the interactions with the
bath- this is the Caldeira-Leggett action
The interaction term is the periodic potential
The reasons for making this choice will become
clear. The bare action is a simple quadratic
form The propagator describes quantum Brownian
motion
A crucial feature of the Ohmic form is that we
get a logarithmic interaction generated between
states of the particle separated by long time
intervals- leading to IR divergence at low
energy in the particle dynamics.
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DYNAMICS of the SCHMID MODEL (traditional
approach)
The partition function is written as a path
integral over trajectories
(1) EXPANSION in POTENTIAL
Let us assume that we can expand in g
In the action we easily get
We can consider the ej to be classical
charges located at tj. We deal with the
standard Coulomb gas the partition function is
only well-defined if the system is globally
neutral. We give it a charge density
We then have
with the usual normalisation
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(2) DUAL INSTANTON EXPANSION
The WKB/instanton expansion is valid in the
regime where
The particle then tunnels between wells
through large barriers- this is the large g
limit. We can write the trajectory in the form
at right
We then have an action with interactions between
local instanton charges
Again, we require global charge neutrality
Again, we have long-range log interactions
(here, wo is the bounce frequency, and
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We now see that
the duality can be written provided we make the
following change
DUALITY

This system is governed by the
extremely
well-known Kosterlitz-Thouless
scaling. The 2 phases differ in the mobility
of the particle, defined in terms of the
partition function by
PHASE DIAGRAM
The KT scaling theory then shows that one has a
localised phase at T 0 when a gt 1, and a
delocalised phase when a lt 1. This general
conclusion can also be arrived at by direct
calculation. The duality appears
in the mobility in the following form
REFERENCES
One can give a quite different approach to this
model, not relying on these expansions M
Hasselfield, T Lee, GW Semenoff, PCE Stamp
hep-th/0512219 (Ann Phys, in press)
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III W.A.H. Boson Field/ Oscillator bath/gauge
fluctns
So now we arrive at the model we really want
to study. This problem is produced by combining
the 2 previous problems- we have a 2-d WAH
lattice with particles coupled to an oscillator
bath
There are now TWO dimensionless couplings in the
problem- to the external field, and to the bath
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The effective Hamiltonian is also written as H
- t Sij ci cj exp iAij H.c.
. WAH lattice
SnSq lq Rn . xq Hosc (xq)
coupled to

oscillators
(i) the the WAH (Wannier-Azbel-Hofstadter)
Hamiltonian describes the motion of
spinless fermions on a 2-d square lattice, with a
flux f per plaquette (coming from the
gauge term Aij). (ii) The particles at
positions Rn couple to a set of oscillators.
This can be related to many systems- from 2-d J.
Junction arrays in an external field to flux
phases in HTc systems, to one kind of open
string theory. It is also a model for the
dynamics of information propagation in a QUIP
array, with simple flux carrying the info.
There are also many connections with other models
of interest in mathematical physics and
statistical physics.
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EXAMPLE Superconducting arrays
The bare action is
Plus coupling to Qparticles, photons, etc
Interaction kernel (shunt resistance is RN)
Another EXAMPLE 3-wire junctions
C. Chamon, M. Oshikawa, I Affleck, PRL 91,
206403 (2004)
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Another EXAMPLE FQHE
Resulting Phase diagram (Lutken Ross (1993)
RG flow (Laughlin (1984) Lutken Ross
(1992-4))
Another phase diagram (Zhang, Lee, Kivelson (1994)
Expt (Kravchenko, Coleridge,..)
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The TOPOLOGICAL QUANTUM COMPUTER
Kitaev, 1997 Freedman et al (2003, 2004)
Basic idea is to try and construct some lattice
realisation of an anyon system, use the anyons
to do quantum computation. The preliminary theory
indicates almost no decoherence
Problem is that so far, the only realisations of
this involve very strange spin Hamiltonians-
which can only be analysed using topological
methods.
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ACTION for the DISSIPATIVE WAH MODEL
The action is an obvious generalisation
The propagator now has a typical Quantum Hall
form
Mij satisfies the following relations
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PHASE DIAGRAM ?
Arguments leading to this phase diagram based
mainly on duality, assumption of localisation
for strong coupling to bosonic bath. The
duality is now that of the generalised vector
Coulomb gas, in the complex z- plane.
Mapping of the line a1 under z ? 1/(1 inz)
Proposed phase diagram (Callan Freed, 1992)
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DIRECT CALCULATION of m (Chen Stamp)
We add a finite external field
We also make a change in notation the
dimensionless parameters will now be called
Dimensionless coupling To bath
Dimensionless coupling to field
We wish to calculate directly the time
evolution of the reduced density matrix of the
particle. It is convenient to write this in
Wigner form
Time evolution of density matrix
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RESULTS of DIRECT CALCULATION
We get exact results on a particular circle in
the phase plane- the one for which K 1/2
The reason is that on this circle, one finds
that both the long- and short-range parts of the
interaction permit a dipole phase, in which
the system form closes dipoles, with the dipoles
widely separated. This happens nowhere else.
One then may immediately evaluate the dynamics,
which is well-defined. If we write this in terms
of a mobility we have the simple results shown
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RESULTS on CIRCLE K 1/2
The results can be summarized as shown in
the figure. For a set of points on the circle
the system is localised. At all other points on
the circle, it is delocalised.
This behaviour is quite different from the
previous results! The explanation is
almost certainly the existence of hidden fixed
points
.
The behaviour on this circle should be testable
in experiments.
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SOME DETAILS of the CALCULATION
In the next 6 pages some details of the
calculations are given, for specialists. The
dynamics of the density matrix is calculated
using path integral methods. We define the
propagator for the density matrix as follows
This propagator is written as a path integral
along a Keldysh contour
All effects of the bath are contained in
Feynmans influence functional, which averages
over the bath dynamics, entangled with that of
the particle
The reactive part the decoherence
part of the influence functional depend on
the spectral function
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Influence of the periodic potential
We do a weak potential expansion, using the
standard trick
Without the lattice potential, the path integral
contains paths obeying the simple Q Langevin
eqtn
The potential then adds a set of delta-fn.
kicks
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One can calculate the dynamics now in a quite
direct way, not by calculating an
autocorrelation function but rather by evaluating
the long-time behaviour of the density matrix.
If one evaluates the long-time behaviour
of the Wigner function one then finds the
following, after expanding in the potential
We now go to some rather detailed exact results
for this velocity, in the next 3 slides .
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LONGITUDINAL COMPONENT
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TRANSVERSE COMPONENT
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DIAGONAL CROSS-CORRELATORS
It turns out from these exact results that not
all of the conclusions which come from a simple
analysis of the long-time scaling are confirmed.
In particular we do not get the same phase
diagram as Callan et al., but instead the results
that were summarized a few pages back, for the
circle K ½.
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SUMMARY CONCLUSIONS

• The effect of decoherence on rather important
  problems is often
• interesting and counter-intuitive.

• It is often useful to study quantum walks in
  quantum information
• processing systems, but such studies must
  look at realistic models.
• In particular, non-diagonal processes cause
  counter-intuitive results.

• In the case of the very important dissipative WAH
  model, the
• dynamics is highly non-trivial. The nature
  of the pase diagram is
• crucial for the understanding of a wide
  variety of quantum systems,
• and it apparently has a different structure
  from that previously
• thought. The complete phase structure is not
  yet known.

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