SPINS, CHARGES, LATTICES,

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SPINS, CHARGES, LATTICES, TOPOLOGY IN LOW d
Meeting of QUANTUM CONDENSED MATTER network of
PITP (Fri., Jan 30- Sunday, Feb 1,
2004 Vancouver, Canada)
For all current information on this workshop go to
http//pitp.physics.ubc.ca/Conferences/20030131/in
dex.html
All presentations will go online in the next week
on PITP archive page
http//pitp.physics.ubc.ca/CWSSArchives/CWSSArchiv
es.html
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DECOHERENCE in SPIN NETS RELATED LATTICE MODELS
PCE Stamp (UBC) YC Chen
(Australia?)
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PROBLEM 1
The theoretical problem is to calculate the
dynamics of the M-qubit reduced density matrix
for the following Hamiltonian, describing a set
of N interacting qubits (with N gt M
typically) H Sj ( Dj tjx ej tjz )
Sij Vij tiz tjz
Hspin(sk) Hosc(xq) int.
The problem is to integrate out the 2
different environments coupling to the qubit
system- this gives the N-qubit reduced density
matrix. We may then average over other qubits if
necessary to get the M-qubit density matrix
operator
rNM(tj
t) The N-qubit density matrix contains
all information about the dynamics of this QUIP
(QUantum Information Processing) system- all
the quantum information is encoded in it.
A question of some theoretical interest is- how
do decoherence rates in this quantity vary with
N and M ?
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Feynman Vernon, Ann. Phys. 24, 118 (1963) PW
Anderson et al, PR B1, 1522, 4464 (1970) Caldeira
Leggett, Ann. Phys. 149, 374 (1983) AJ Leggett
et al, Rev Mod Phys 59, 1 (1987) U. Weiss,
Quantum Dissipative Systems (World Scientific,
1999)
A qubit coupled to a bath of delocalised
excitations the SPIN-BOSON Model
Suppose we have a system whose low-energy
dynamics truncates to that of a 2-level system t.
In general it will also couple to DELOCALISED
modes around (or even in) it. A central feature
of many-body theory (and indeed quantum field
theory in general) is that (i) under
normal circumstances the coupling to each mode is
WEAK (in fact O (1/N1/2)), where N is the
number of relevant modes, just BECAUSE the modes
are delocalised and (ii) that then we map
these low energy environmental modes to a set
of non-interacting Oscillators, with canonical
coordinates xq,pq and frequencies wq.
It then follows that we can write the
effective Hamiltonian for this coupled system in
the SPIN-BOSON form H (Wo) Dotx eotz

qubit 1/2 Sq
(pq2/mq mqwq2xq2)
oscillator Sq
cqtz (lqt H.c.) xq
interaction
Where Wo is a UV cutoff, and the cq, lq N-1/2.
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UCL 16
A qubit coupled to a bath of localised
excitations the CENTRAL SPIN Model
P.C.E. Stamp, PRL 61, 2905
(1988) AO Caldeira et al., PR B48,
13974 (1993) NV Prokofev, PCE Stamp, J
Phys CM5, L663 (1993) NV Prokofev, PCE
Stamp, Rep Prog Phys 63, 669 (2000)
Now consider the coupling of our 2-level system
to LOCALIZED modes. These have a Hilbert space of
finite dimension, in the energy range of
interest- in fact, often each localised
excitation has a Hilbert space dimension 2. From
this we see that our central Qubit is coupling
to a set of effective spins ie., to a SPIN
BATH. Unlike the case of the oscillators, we
cannot assume these couplings are weak.
For simplicity assume here that the bath spins
are a set sk of 2-level systems. Now actually
these interact with each other very weakly
(because they are localised), but we cannot drop
these interactions. What we then get is the
following low-energy effective Hamiltonian
(recall previous slide)
H (Wo) Dt exp(-i Sk
ak.sk) H.c. eotz
(qubit)
tzwk.sk hk.sk
(bath spins)

inter-spin interactions
The crucial
thing here is that now the couplings

wk , hk to the bath spins- the first
between bath
spin and qubit,
the second to external fields- are

often very strong (much larger than either
the
inter-spin interactions
or even than D).
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Dynamics of Spin-Boson System

The easiest way to solve for the dynamics of the
spin-boson model is in a path integral
formulation. The qubit density matrix
propagator is written as an integral over an
influence functional
The influence functional is defined as
For an oscillator bath
with bath propagator
For a qubit the path reduces to
Thence
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Dynamics of Central Spin model (Qubit
coupled to spin bath)
Consider following averages
Topological phase average
Orthogonality average
Bias average
The reduced density matrix after a spin bath is
integrated out is quite generally given by
Eg., for a single qubit, we get the return
probability
NB can also deal with external noise
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UCL 28
DYNAMICS of DECOHERENCE
At first glance a solution of this seems very
forbidding. However it turns out that one can
solve for the reduced density matrix of the
central spin exactly, in the interesting
parameter regimes. From this soltn the
decoherence mechanisms are easy to identify
(i) Noise decoherence Random phases added to
different Feynman paths by the noise field.

(ii) Precessional decoherence
the
phase accumulated by environmental
spins between
qubit flips.
(iii) Topological Decoherence The

phase induced in the environmental
spin dynamics by
the qubit flip itself
USUALLY THE 2ND
MECHANISM
(PRECESSIONAL DECOHERENCE)

is DOMINANT
Precessional decoherence
Noise decoherence source
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Pey 1.34

• The oscillator bath decoherence rate goes like
• tf-1 Do g(D,T) coth (D/2kT)
• with the spectral function g(w,T) shown below
  for an Al
• SQUID (contribution from electrons phonons).
  All of this is
• well known and leads to a decoherence rate tf-1
  paDo once
• kT lt Do. By reducing the flux change df (f -
  f- ) 10-3 , it
• has been possible to make a 10-7 (Delft expts),
  ie., a
• decoherence rate for electrons O(100 Hz). This
  is v small!

Decoherence in SQUIDs
A.J. Leggett et al., Rev. Mod Phys. 59, 1
(1987) AND PCE Stamp, PRL 61, 2905
(1988) Prokofev and Stamp Rep Prog Phys 63, 669
(2000)
On the other hand paramagnetic spin
impurities (particularly in the junctions),
nuclear spins have a Zeeman coupling to the
SQUID flux peaking at low energies- at energies
below Eo, this will cause complete incoherence.
Coupling to charge fluctuations (also a spin
bath of 2-level systems) is not shown here, but
also peaks at very low frequencies.
However when Do gtgt Eo, the spin bath decoherence
rate is 1/tf Do (Eo/8D0)2
as before
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WRITE on PAPER SHEETS
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PROBLEM 2 The DISSIPATIVE HOFSTADTER Model
This problem describes a set of fermions on a
periodic potential, with uniform flux threading
the plaquettes. The fermions are then coupled to
a background oscillator bath
We will assume a square lattice, and a simple
cosine potential
There are TWO dimensionless couplings in the
problem- to the external field, and to the
bath
The coupling to the oscillator bath is assumed
Ohmic
where
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The W.A.H. MODEL
This famous model was first investigated in
preliminary way by Peierls, Harper,, Kohn, and
Wannier in the 1950s. The fractal structure was
shown by Azbel in 1964. This structure was first
displayed on a computer by Hofstadter in 1976,
working with Wannier.
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 HOFSTADTER BUTTERFLY
The graph shows the support of the density of
states- provided a is rational
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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 S/cond arrays
The bare action is
Plus coupling to Qparticles, photons, etc
Interaction kernel (shunt resistance is RN)
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Expt (Kravchenko, Coleridge,..)
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PHASE DIAGRAM
Callan Freed result (1992)
Mapping of the line a1 under z ? 1/(1 inz)
Proposed phase diagram (Callan Freed, 1992)
Arguments leading to this phase diagram based
mainly on duality, and 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.
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DIRECT CALCULATION of m (Chen Stamp)
We wish to calculate directly the time
evolution of the reduced density matrix of the
particle. It satisfies the eqtn of motion
The propagator on the Keldysh contour g is
The influence functional is written in the form
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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 few 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 we now see
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We find that we can get some 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 close dipoles, with the dipolar
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.
The behaviour on this circle should be testable
in experiments.
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Conclusions

• In the weak-coupling limit (with dimensionless
  couplings l ), the
• disentanglement rate for a set of N
  coupled qubits, is actually linear
• in N provided Nl lt 1
• In the coherence window, this is good for quite
  large N
• In the dissipative Hofstadter model duality
  apparently fails. There is
• actually a whole set of exact solutions
  possible on various circles.

It will be interesting to explore decoherence
rates for topological computation- note that the
bath couplings are local but one still has to
determine the couplings to the non-local
information
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THE END
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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 a 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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UCL 19
DYNAMICS of the DIPOLAR SPIN NET
NV Prokofev, PCE Stamp, PRL 80, 5794
(1998) JLTP 113, 1147 (1998) PCE Stamp, IS
Tupitsyn Rev Mod Phys (to be publ.).
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 (Dj tjx ej tjz)
Sij Vijdip tiz tjz
HNN(Ik) Hf(xq)
interactions For magnetic systems this
leads to the picture at right.
Almost all experiments so far are done in
the region where Do 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
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UCL 20
Quantum Relaxation of a single NANOMAGNET
Structure of Nuclear spin Multiplet ?
Our Hamiltonian
When D ltltEo (linewidth of the nuclear multiplet
states around each magbit level), the magbit
relaxes via incoherent tunneling. The nuclear
bias acts like a rapidly varying noise field,
causing the magbit to move rapidly in and out
of resonance, PROVIDED gmBSHo lt Eo
Tunneling now proceeds over a range Eo
of bias, governed by the NUCLEAR SPINmultiplet.
The relaxation rate is G D2/Eo
for a single qubit.
Fluctuating noise field
Nuclear spin diffusion paths
NV Prokofev, PCE Stamp, J Low Temp Phys 104,
143 (1996)
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UCL 30
The path integral splits into contributions
for each M. They have the effective action
of a set of interacting instantons
The effective interactions can be mapped to a
set of fake charges to produce an action having
the structure of a spherical model involving a
spin S
The key step is to then reduce this to a sum
over Bessel functions associated with each
polarisation group.