Dephasing and Many-body Delocalization in Strongly Disordered System

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Chapter 1

Dephasing and Many
-
body
Delocalization in Strongly
Disordered System


Alexander L. Burin

Department of Chemistry, Tulane University

New Orleans, LA 70118

aburin
@
tulane.edu

1.1. Introduction

The irrevers
ibility of dynamics in physical or chemica
l
systems is associated with
energy
current
s

flow
ing infinitely far in space. The
s
e

currents result in sharing the
local excitation energy with the whole thermal bath formed by other excitations
occupying macroscopically large phase volume
,

so the probabil
ity for the excitation
to return back is negligibly small.
At low temperatures i
n condensed media
energy
currents are carried by
delocalized long
-
living low energy excitations.

In
the regime
of strong disordering caused by structural
fluctuations

(in amorp
hous solids) or
impurities (in doped semiconductors) the relevant excitations get localized according
to the Anderson theorem [Anderson 1958
].
In the localization regime there is no
irreversible

relaxation and
the system is non
-
ergodic.

This consideration
is applicable
to
an almost zero temperature
s when
each
localized
excitation
can be treated as
independently of

others. At finite temperature
s
local excitations

even in strongly
disordered media

are capable to

talk to each other

. Formally they cannot be t
reated
independently and their interaction (similar
ly

to inelastic scattering) can give rise to
the irreversible system dynamics

that is essentially of the many
-
body nature
.





In this paper I consider delocalization and
irreversibility induced by the
long
-
range
interaction
U(r)
=U
0
/r
a

in a

strongly disordered system of interacting local excitations
,
which can be modeled by pseudospins ½
operators
S
z
,

having random local energies
and located randomly in space.



The

l
ocalization
is sensitive to the
in
teraction at long distances characterized by the
exponent
a
. When
a

3
,

all excitations are delocalized at arbitrary
strong
disorder

Dephasing and M
any
-
body Delocalization in Strongly Disordered System


2

[Anderson 1958;

Levitov, 1990].
This can be understood using the scaling
arguments
for
characteristic excitation energies ha
ving
with increasing
the system size
R
.
Consider the single spin in the excited state in the box of the size
R
. If this spin
undergoes the flip
-
flop transition involving the second spin the change in the system
energy is given by the difference in spins ex
citation energies. Because we do have the
total number of spins
N

scaling as
N~R
3

the smallest energy of the possible flip
-
flop
transitions scales as


min
~1/N~R
-
3
.

On the
other
hand the minimum value of
a

transition amplitude scales as the interaction at l
argest possible distance

U
min
~R
-
a
.

W
hen
a<3

U
min

always exceeds

min

at sufficiently large system size
R
. Thus

for
R



the flip
-
flop transition amplitude
of
any

excited spin with some of other spins
in
its

ground state always exceeds their energy differenc
e at sufficiently l
arge system
size leading to
inevitable
energy sharing and the
delocalization of all excitations.


The finite temperature creates new objects which are the flip
-
flop pairs formed by
the excited spin and the spin in the ground state. Be
cause of the Pauli statistics of
excitations these pairs
(i, j)

can be treated as
the new spin entities with respect to the
pair transition
2
/
1
,
2
/
1
2
/
1
,
2
/
1







z
j
z
i
z
j
z
i
S
S
S
S

[Burin,
Natelson, Osheroff
,
Kagan,
1998]
.
At finite temperature the number of such pairs
scales

as the squared number of spins
N
p
~N
2
~R
6
, while the characteristic minimum
energy difference for them is
.
~
6
1


R
N
p

Using the interaction
a
R
U

~
min
to
estimate the
resonant coupling of these new entities we see that when the
characteris
tic exponent
a

is less than
6

the inevitable delocalization of excitations
takes place at finite temperature. This crude estimate will be confirmed by the
analysis
given
below.



This study is the extension of previous works [Burin, Natelson, Osheroff,
Kagan,
1998; Burin, Kagan, Maksimov, Polishchuk, 1998; Burin, 1995] where the relaxation
of tunneling defects in amorphous solids caused by their
1/R
3

interaction

has been
investigated
.
The
results are also applicable to other systems including electrons i
n
doped semiconductors and interacting localized electronic and nuclear spins.
The
long
-
range interaction
in these and other systems has a fundamental origin related to
electrostatic or elastic forces caused by the virtual exchange of long
-
wavelength
photo
ns or phonons [Yu and Leggett, 1998].
The dephasing problem is very important
for the performance of quantum computers [
Ustinov, 2003
] where
pseudo
spins ½
of
different physical nature are used as quantum bits (qubits). A

long
-
range interaction
between them

can lead to the irreversible relaxation breaking the coherency
needed
for calculations
and
thus
restricting the computati
onal capability of these important

devices.


During the consideration below
I
will
ignore the relaxation of excitations induced
by

spontaneous emission or a
bsorption of long
-
wave
length phonons or photons
assuming that the involved energies are so small that the rate
s

of these
processes are

too slow compared to the interac
tion stimulated relaxation. The latter

assumption is
justified
for amorphous solids at temperatures below 0.1K
[Burin, Natelson, Osheroff,
Kagan, 1998]
,

while in other systems

the restrictions can be
possibly
made
weaker
.


Dephasing and M
any
-
body Delocalization in Strongly Disordered System


3

1.2. Delocalization Mechanisms

1.
2
.1. Model

Consider the system of interacting spins at fin
ite low temperature T (for the sake of
simplicity we set
k
B
=

=1
). The relevant group of spins involved into the dynamics is
formed by

th
ermal”

spins,
possessing the energy

i
/2

in the excited state and

-

i
/2

in
the ground state

with

i
~T
. They

can be des
cribed by the Hamiltonian
.



i
z
i
i
S

Let

the
density of such thermal spins be

n
T
.
Interaction
U(r
)=U
0
/r
a

represents
both
the
flip
-
flop
transition amplitude leading to the energy ho
pping between two spins

(S
+
S
-
)
and the characteristic
longitudinal

interaction (S
z
S
z
)

(see e. g. [
Burin,
Kagan, 199
4
])
.

1.2
.2
.
Resonant pairs and Anderson localization

Consider the excited spin

i
.
The problem of interest is whether i
t can share its
excitation with one of the neighboring spins
j

initially located in t
he ground state
.

This requires
the change in the “diagonal” part of
spin
energy
|

i
-

j
|

to be

small
compared to the flip
-
flop transition amplitude
U(R
ij
)





.
0
a
ij
j
i
R
U








(
1
)

Two spins
i

and
j

satisfying the resonance condition (
1
)
fo
rm

a
resona
nt pair
.
In this
pair the excitation of the spin
i

is shared between spins
i

and
j

and it bounces
back and
forth between them back
.


The c
oncept of resonant pairs can be used

to study the delocalization

of excitations.
E
ach excited spin has the certai
n number of resonant neighbors
N
r
. Each of the
resonant neighbors in average participates in the same number of resonances. One can
consider the
multi
-
step

resonance energy transfer from the initial spin
i

to one of its
resonance neighbors
j

and then from
j

to one of its neighbors
k
and so on. The
se

coupled
resonances form

a

percolation

network. When the average number of
resonance neighbors is large enough




,
1

r
N





(
2
)

resonant spins form the infinite cluster promoting energy transport
infinitely far from
the initial excited spin. This leads to the
irreversible spin dynamics
.


Let us calculate t
he average
number of resonant neighbors
N
r

for the given excited
spin with the energy

~T
.
This number is defined by the probability
for the
given pair
of spins

(i, j)

to be in the resonance summed over all possible neighbors
j
. The
probability of resonance condition (
1
) to be satisfied is given by the size of the
resonant
energy domain

)
/
,
/
(
0
0
a
ij
i
a
ij
i
R
U
R
U




divided by the width of the
energy
dis
tribution
of involved thermal spins
given by the thermal energy
T




).
/(
0
a
ij
res
ij
TR
U
P





(
3
)

The total number of resonances can be
estimated

averaging the sum of probabilities
(
3
) over possible locations of a neighboring spin
j


Dephasing and M
any
-
body Delocalization in Strongly Disordered System


4



),
(
)
(
)
/
(
)
/
(
3
/
1
0
R
L
n
R
R
d
T
n
U
N
T
a
T
r







R

(
4
)

where
the
step function
s


(x) restrict

the minimum value of
the interspin distances by
the

characteristic average distance
between them
3
/
1

T
n

(
to avoid the
irrelevant
contribution of rare configurations when two spins are too close to e
ach other
)

and the
maximum of
an

interspin distance by the system size
L
.
For
a<3

the integral increases
with the system size as
L
3
-
a

and it diverges in the limit
L


.
This means the
inevitable delocalization for
a<3

in accordance with
Refs.

[Anderson, 195
8; Levitov,
1990]. If
a=3

the divergence
with the size
L

is logarithmic giving




),
ln(
)
/
(
3
/
1
0
T
T
r
Ln
T
n
U
N




(
5
)

We are interested in the r
egime of strong disordering where

the prefactor in Eq. (
5
) is
very small

3
=U
0
n
T
/T<<1. For example in amorphous solids t
his prefactor is
as small
as

10
-
3

e. g. [
Burin and Kagan, 1994]. In this case the logarithmic factor cannot make
the number
N
r

greater than
1

at any reasonable system size. On the other hand
in the
case


3

<< 1

even for the large factor
N
r

(
5
)

the delocali
zation shows up
weakly
[Levitov, 1990] that it cannot
compete
with the many
-
body mechanism described
below at any reasonable temperature.


In the case
a>3

the integral in Eq. (
4
) is defined by the lower limit





.
/
3
/
0
T
n
U
N
a
T
a
r






(
6
)



We a
re interested in the regime of strong disordering where

a
<<1

and
a

3
. In
this case the Anderson localization of all single particle excitations takes place.


How to describe the system dynamics in this regime of strong disordering? The
vast majority

of spins in this case do not belong to the resonant pairs and they can be
treated quasi
-
statically. The
small
fraction
of spins

a
<<1

belongs to the resonant
pairs (
1
)
having the concentration

a
n
T
. They are
involved into qu
antum beat
s of the
excitation s
hared between two spins. The ensemble of resonant pairs can be classified
by the energy of
each pair. The

pair
of the size
R

is characte
rized by the single energy

E
R
~U
0
/R
a

describing its characteristic flip
-
flop transition amplitude, which
is close to

the
energy difference of involved spins
Eq.
(
1
). The inverse characteristic energy
gives the
period of the
coherent oscillation
of populations
inside the pair




.
/
~
0
1
U
R
E
a
R
R








(
7
)


We can classify
resonant

pairs by their energy rang
e starting with the maximum
energy
3
/
0
a
T
n
U

and reducing it
each step
by the scaling parameter

a

with
some
scaling factor


1.
The
subsystem of pairs is separated into an
infinite number of
energy d
omains enumerated by
integer numbers
k
=
0,
1,2,
3,4,..

and defined as
).
/
,
/
(
3
/
0
)
1
(
3
/
0
ak
a
T
k
a
a
T
n
U
n
U



This scaling corresponds to the range
s
of
pair
sizes

.
1
3
/
1
3
/
1





k
T
k
T
n
R
kn






(
8
)

The
concentration of resonant pairs
within each domain can be defined using the
integral expression (
4
) with upper and lower limits d
efined by the domain boundaries
(
8
)
,

multiplied by the density of excited spins
n
T
,


Dephasing and M
any
-
body Delocalization in Strongly Disordered System


5




.
/
~
)
3
(

a
k
a
T
p
k
n
n






(
9
)

In the case of dipole
-
dipole interaction
a=3

the density of resonant pairs is scale
invariant because of the logarithmic behavior of the int
egral. We will use
below
the
scaling (
8
), (
9
) to study the interaction of resonant pairs.
The results are not sensitive
to the choice of scaling parameters.

1.2.3

Delocalization induced by the interaction of resonant pairs

Q
uantum
oscillations

in
different reso
nant pairs interact

with each other.
This
behavior contrasts to non
-
interacting Bose or Fermi systems where different
excitations can be treated independently.
The

constraints for possible states of spin ½
permitting only two possible states for each spin
in addition to commut
ation of
different spin operators gives rise to finit
e “scattering” matrix elements
f
or

two
resonant pair interaction leading to simultaneous flip
-
flop transitions in both pairs

[Burin, Kagan 199
4
]
.
This interaction can create the coll
ective dynamics in the
ensemble of resonant pairs.


Each resonant pair of spins
i, j

can be treated as the new spin
½ entity,
characterized by the resonant doublet of

two possible “flip
-
flop” states
S
z
i
=
-
S
z
j
=1/2

and
-
S
z
i
=
S
z
j
=1/2
,

which are strongly cou
pled by the
interaction of spins (see Eq. (1
)).
Two other states
, where
bo
th spins have

identical values


1/2
, are separated f
r
o
m the
resonant doublet by
energy gaps of order of the thermal

energy
. T
he interaction is too
weak to mix the resonant doublet wi
th two those states in the strong disordering
regime

a
<<1

Eq. (6
).
The interaction of two resonant pairs falls off with the distance
between them identically to the original interaction
U
0
/R
a

(cf. [Burin, Natelson,
Kagan, Osheroff, 1998])
.



Thus

we ha
ve the ensemble of resonant pairs s
eparated into scaling domains (8),
with each domain characterized by the certain pair size
R
, energy
U
0
/R
a

and density
(9
), given by
.
)
/
(
~
)
(
3
3
/
1


a
T
a
T
p
R
n
n
R
n


How do resonant pairs of different
sizes affect each other? The con
sideration shoul
d begin with the
pairs having the
largest energy and the
smallest
possible
size
.
3
/
1

T
n

These pairs ha
ve the largest
frequency of
coherent oscillations

so other pairs can be treated quasi
-
statically

with
respect to them
. Their
interaction
p
U
min
can be estimated using the average minimum
distance between them
3
/
1
3
/
1
min
))
(
(
~

T
p
p
n
n
r
as




.
~
)
/(
~
3
/
3
/
min
0
min
a
a
a
T
o
a
p
p
n
U
r
U
U



(1
0
)

This
interpair

interaction is less than the
ir energy
3
/
a
T
o
n
U
by the
factor
1
3
/

a
a


in the strongly disordered system.
The
refore they do not influence each other.


To invest
igate the interaction of larger (slower)

pairs we will
subsequently

increase
the characteristic pair size by the scaling factor


and repeat the above considerat
ion
for the new range of pair energies and sizes.
If

the interaction of pairs is much weaker
than their characteristic energy
then
we can treat them independently as the
e
xcitations affecting

only
the static distribution of
other spin
parameters. The
chara
cteristic interaction energy
V(R)

within the strip of
pairs with the certain size
R

Dephasing and M
any
-
body Delocalization in Strongly Disordered System


6

(9) scales with this size as the characteristic interaction at the average distance
between pairs belonging to that strip



.
)
(
1
~
)
(
~
)
(
3
/
)
3
(
3
/
1
3
/
0
3
/
3
/
0

a
a
T
a
T
a
a
a
p
R
n
n
U
R
n
U
R
V



(1
1
)

This interaction sh
ould be compared with the characteristics pair energy
U
0
/R
a

(cf. Eq.
(7
))

to examine the strength of interpair interaction
. It is c
lear that if the interaction
(11
) decreases with the distance
R

slower than the
pair
energy
,
3
/
3
/
)
3
(
a
a
a



then at suff
iciently large size R the flip
-
flop interaction of pairs exceeds their energy
disordering leading to the delocalization of excitations within the ensemble of
resonant pairs. Resolving the
above
inequality we obtain the condition for the
interaction to prov
ide the energy delocalization



.
6

a







(12
)

This result is universal and does not depend on the strength of disordering.
The
perturbation theory analysis of the contribution of more complex resonant clusters
,

including for example trip
les and quartets
,

shows that it can be ignored for
a>6
,
while for
a<6

it becomes significant when the interaction of resonant pairs becomes

significant (see below Eqs. (13), (14
)). Therefore at
a<6

we can use the resonant pairs
to study the irreversible r
elaxation, while for
a>6

the perturbation theory approach
leads to the strong localization even for many
-
body excitations. One cannot exclude
a
possible asymptotic divergence of

the accurate perturbation
series including

large
clusters
. This divergence

sho
uld lead to exponentially slow relaxation. The crossover
regime
a=6

requires the special study that is beyond the scope of this paper.


In the case of interest
a<6

t
he characteristic size of pairs corresponding to the
delocalization transition

can be
estimated setting th
e energy of pair interaction (11
) to
be
approximately
equal to the pair energy
U
0
/R
a

in agreement with the delocalization
criterion
(2
).

This yields



.
~
)
6
/(
1
3
/
1
*
a
a
T
n
R










(13
)


The interaction energy at this

cros
sover distance defines the rate of
the
irreversible
relaxation in the subsystem of resonant pairs



.
~
~
/
1
)
6
/(
3
/
0
*
0
*
a
a
a
a
T
a
n
U
R
U







(14
)

Below we will apply this result to study dephasing and relaxation rates in the whole
system of spins.

1.2.4

Dephasing and relaxation

rates of spins

for
a<6

As was defined before the interaction of spins leads not only to the flip
-
flop
transitions
due to its flip
-
flop
S
+
S
-

component
but also to
the change in energy of
surrounding spins induced by irreversible transitions of resonant pai
rs due to
the
longitudinal interaction

S
z
S
z
. Therefore the relaxation in the ensemble of resonant
pairs gives
rise

to the spectral diffusion of all spin energies and
,

consequently
,

dephasing
.
The characteristic change

E

in energies of non
-
resonant spins d
uring the
time
*


of
irreversible flip
-
flop transition
s

(1
4
)

is given by the
interaction with the
nearest resonant pair of the size
.
*
R
According to our definition of the delocalization
Dephasing and M
any
-
body Delocalization in Strongly Disordered System


7

point (14
) this interaction coincide
s with the characteristic pair energy (1
4
) so the
energy fluctuation
occurring at the time
*


is given by
*
/
1


(1
4
).


T
he dephasing rate can be estimated as the time



when the phase fluctuation


=

E



becomes of ord
er of unity,
see e. g.
[
Burin, Kagan, Maksimov, Polishchuk,
1998]

(remember that we set

=1)
.

Since the energy fluctuation during the time
*


has the value
*
/
1


the phase fluctuation during this time is around one. Therefo
re
Eq. (16) defines the dephasing rate in the system of interacting spins.


The spectral diffusion changes energies of all spins including those forming
resonant pairs. Therefore during the system relaxation this spectral diffusion will
break some reso
nance pairs and form the new ones. After some time this process will
include all

spins with the thermal energy. In other words each spin has
a
very small
probability to be involved into resonant pairs

at the given time
.
After some
waiting
time t
he spectr
al

diffusion forms its resonance with some other spin. Under resonant
conditions the flip
-
flop transition takes place and after that the resonance pair gets
broken

by the spectral diffusion
. The spin waits the
vast
majority of time to be
involved into the
ne
w
resonance. This waiting time

1

defines t
he
longitudinal
relaxation
rate of
spin
s.
It is given by the quasi
-
period of
the
spectral diffusion.



How to fin
d the spectral diffusion period?

T
he process under consideration is of
the self
-
consistent natur
e
so the spin transitions induces the

spectral diffusion
,

which
itself induces the spin transitions
. In this regime the possibility of a long term memory
effects in the spectral diffusion can be significant.

Let us ignore such a memory and
then verify this

assumption self
-
consistently.


D
uring the time
*


the spins
with the
density
)
(
*
*
R
n
n
p

,
belonging to
resonant pairs of the size
*
R
,

make their transitions
. The
s
e

transitions lead to the
spin
energy fluct
uation
*
/
1


removing spins of the resonant subsystem out of

the
resonance and creating
*
n

of
new resonant spins.
The m
ultiple repetition of this
process leads to growing the
density

of spins making transitions with
in

the t
ime
range
(0,

t
)

as

*
*
/
)
(

t
n
t
n

. At the
time
t

equal to the
longitudinal relaxation

time


1

this
concentration approaches the total spin concentration
n
T

and all spins make in average

one transition.
Then

the longitudinal relaxation rate can be es
timated as



.
~
)
/
(
/
1
/
1
)
6
/(
)
3
(
3
/
0
*
*
1
a
a
a
a
T
T
n
U
n
n








(1
5
)


The
absence of the significant memory in

the spectral diffusion can be justified
considering time dependent energy fluctuations.
The energy fluctuation i
nduced by
transitions of all spins in the subsystem of con
centration
n

can be estimated as the
characteristic interaction with the closest spin from this subsystem, given by
U
n
a/3
.
This fluctuation increases with the time
t

as
3
/
*
*
)
/
)(
/
1
(
~
)
(
a
t
t
E



. Thus we are
dealing with the super
-
diffusion having the proba
bili
ty to return back defined by the
integral

3
/
/
a
t
dt
convergin
g

in the long time limit for
a>3
.

This makes the
probability of correlated transitions within the same resonant pair

negligible. The
special case
a=3

requires the separate study. The ana
lysis, made in
Ref.
[Burin,
Dephasing and M
any
-
body Delocalization in Strongly Disordered System


8

Kagan, 199
4
] shows that for two level defects in amorphous solids the
memory
effects can also

be
approximately
neglected.

1.3.

Conclusion


It was

shown that

the long
-
range interaction
U(R)~R
-
a

with
a<6

always leads to the
irreversi
ble relaxatio
n and dephasing in
strongly disordered system
s at any

finite
temperature
. Relaxation and dephasing rates are computed.
Applications of the above
consideration to
1/R
3

interacting
tunneling systems in amorphous solids have leaded
to
a

certain p
rogress in understanding low temperature
relaxation

experiments
see e.g.
[
Enss, 2002;
Burin, Natelson, O
sheroff, Kagan, 1998
].
The results can be applied

to
other
problems

including
interacting q
ubits in quantum computers.


Dephasing can cruciall
y aff
ect the coherent evolution of q
ubit
s

in
a

quantum
c
omputation

process. One possible way to reduce the interaction stimulated dephasing
is to use low dimensional systems
,

where
1/R
3

interaction will be much less efficient.
In fact, t
h
is effect
has been rece
ntly demonstrated in
acoustic measurements in glasses

[
Ladieu, Le Cochec, Pari, Trouslard, Ailloud,

200
3].


This work is partially supported by TAMS GL fund (account no. 211043) through
the Tulane University. Author is grateful to his advisors and
co
lleagues

Yuri
Kagan,
Leonid Maksimov, Il’
ya Polishchuk,
Douglas Osheroff
and Clare Yu
for many useful
ideas, discussions and suggestions

kindly provided by them
.

References

Anderson, P. W., 1958, Absence of diffusion in certain random lattices,
Physical R
eviews
,
1492
,
109
.

Burin, A.

L., Natelson, D.
, Osheroff, D.

D., Kagan, Yu,

1998, Interaction between Tunneling
Defects
,
in
Tunneling Systems in Amorphous and Crystallline Solids
,
edited by P. Esquinazi,
Springer
-
Verlag (Berlin, Heidelberg, New York), 223.


Levitov, L., S., 1990,
Delocalization of vibrational modes caused by electric dipole interaction
,
Physical Review Letters
,
547
,
64
.

Ladieu
, F.,
Le Cochec, J., Pari, P., Trouslard, P., Ailloud, P.,
200
3
,
Physical Review Letters
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2
05501
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90.

Burin, A.
L., Kagan, Yu.
,

1994
,
Low Energy Collective Excitations in Glasses. New Relaxation
Mechanism at Ultralow Temperatures
,

Journal of Experimental and theoretical Physics
,
633
, 106.


Burin, A. L., Kagan, Yu., Maksimov, L.A., Polishchuk, I. Y., 1998
,
Dephasing

rate in dielectric
glasses at ultralow temperatures
,
Physical Review Letters
,
2945
,

80.


Burin,
A. L.,
1995
,

Dipole gap effects in low energy excitations spectrum of amorphous solids


theory for dielectric relaxation,

Journal of Low Temperature Physics
,

309
, 100.

Enss C. C., 2002,
Beyond the tunneling model: quantum phenomena in ultracold glasses
,
Physica B
,
12
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Yu, C. C., Leggett, A. J., 1998,
Low
-
temperature prop
erties of amorphous materials:


through a glass darkly
,

Comments
on
Condense
d Matter Physics

231
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Ustinov, A.,
2003,
Quantum Computing Using Superconductors
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Nanoelectronics and
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Waiser, Wiley
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(Berlin),

463.