Download Quantum Phase Transitions

Quantum Phase Transitions
G. H. Lai
May 5, 2006
Abstract
A quantum phase transition (QPT) is a zero-temperature, generically continuous transition tuned by a parameter in the Hamiltonian at
which quantum fluctuations of diverging size and duration (and vanishing energy) take the system between two distinct ground states [4].
This short review discusses the characteristic aspects of QPT and illustrates possible applications in physics and biology.
Contents
1 What are quantum phase transitions?
1
2 Features of a quantum phase transition
3
3 Relevance in experimental Physics and Biology
5
4 Further work
9
1
What are quantum phase transitions?
A phase transition is a fundamental change in the state of a system when
one of the parameters of the system (the order parameter) passes through its
critical point. The states on opposite sides of the critical point are characterized by different types of ordering, typically from a symmetric or disordered
state, which incorporates some symmetry of the Hamiltonian, to a brokensymmetry or ordered state, which does not have that symmetry, although
the Hamiltonian still possesses it.
1
As we approach the phase transition, the correlations of the order parameter become long-ranged. Fluctuations of diverging size and duration (and
vanishing energy) take the system between two distinct ground states across
the critical point. When are quantum effects significant? Surprisingly, all
non-zero temperature transitions are considered “classical”, even in highly
quantum-mechanical systems like superfluid helium or superconductors. It
turns out that while quantum mechanics is needed for the existence of an
order parameter in such systems, it is classical thermal fluctuations that govern the behaviour at long wave-lengths. The fact that the critical behaviour
is independent of the microscopic details of the actual Hamiltonian is due
to the diverging correlation length and correlation time: close to the critical
point, the system performs an average over all length scales that are smaller
than the very large correlation length. As a result, to correctly describe universal critical behaviour, it should suffice to work with an effective theory
that keeps explicitly only the asymptotic long-wavelength behaviour of the
original Hamiltonian (for instance, the phenomenological Landau-Ginsburg
free-energy functional).
So far, we have shown that classical theory suffices. However, consider
what happens when the temperature around the critical point is below some
characteristic energy of the system under consideration [2]. For example,
the characteristic energy of an atom would be the Ryberg energy. We see
a characteristic frequency ωc and it follows that quantum mechanics should
be important when kb T < ~ωc . In the same spirit, if kb T >> ~ωc close
to the transition, the critical fluctuations should behave classically. This
argument also shows that zero-temperature phase transitions, where Tc = 0,
are qualitatively different and their critical fluctuations have to be treated
quantum mechanically.
In such zero-temperature or quantum phase transitions (QPT), instead
of varying the temperature through a critical point, we tune a dimensionless
coupling constant J in the controlling Hamiltonian H(J). Generically [1], the
ground state energy of H(J) will be a smooth, analytic function of g for finite
lattices. However, suppose we have the Hamiltonian H(J) = H0 +JH1 , where
H0 and H1 commute. This means that H0 and H1 can be simultaneously
diagonalized and the eigenstates of the system will remain the same even
though the eigen-energies will vary with J. A level crossing is possible where
an excited level crosses the ground energy level at some coupling constant Jc ,
creating a point of non-analyticity in the ground state energy as a function
of J, which manifests as a quantum phase transition.
2
11
Quantum Phase Transitions
10
D.BELITZ AND T.R.KIRKPATRICK
critical region
T
T
PM
Tc (J)
FM
classical
PM
FM
QM
Jc
QM
Jc
J
(A)
J
(B)
Figure
Schematic phase diagram as in Fig. 1 showing the vicinity of the quantum
Figure 1. Schematic phase diagram showing a paramagnetic (PM)
and2.a ferromagnetic
critical apoint
(J = phase
Jc , T = 0). Indicated are the critical region, and the regions dominated
(FM) phase. The path indicated by the dashed line path represents
classical
by the
classical and quantum mechanical critical behavior, respectively. A measurement
transition, and the one indicated by the solid line a quantum phase
transition.
along the path shown will observe the crossover from quantum critical behavior away
from the transition to classical critical behavior asymptotically close to it.
Figure 1: (A) Schematic phase diagram showing a paramagnetic (PM) and
ferromagnetic
(FM) phase. The dotted path represents a classical phase
1.3.a AN
EXAMPLE: THE PARAMAGNET-TO-FERROMAGNET
TRANSITION
transition while the solid line indicates
a quantum phase transition. (B)
speaking, related to the classical one in a different spatial dimensionality,
Let us illustrate the concepts introduced above by means of a concrete
and
since we
usually
Vicinity
of the we
quantum
critical
point
(Jknow
= that
Jc , changing
T = Tthe
Indicated
aremeans
thechangc ).dimensionality
example.
For definiteness,
consider a metallic
or itinerant
ferromagnet.
ing the universality class, we expect the critical behavior at this quantum
5
critical region, as well as the regions
by the
clasical
andatquantum
criticaldominated
point to be different
from the
one observed
any other point on
the coexistence
curve.
mechanical
critical behaviour(QM)
[2].
1.3.1.
The phase diagram
This brings
to the question of how continuity is guaranteed when one
Figure 1 shows a schematic phase diagram in the T -J plane,
with Justhe
along the coexistence curve. The answer, which was found by Suzuki
strength of the exchance coupling that is responsible moves
for ferromagnetism.
[9], also
explains
why the behavior at the T = 0 critical point is relevant
The coexistence curve separates the paramagnetic phase
at large
T and
for observations
atJ,
small but non-zero temperatures. Consider Fig. 2, which
small J from the ferromagnetic one at small T and large
J. For a given
an enlarged
there is a critical temperature, the Curie temperature shows
Tc , where
the phasesection of the phase diagram near the quantum critical point.
The critical
transition occurs. This is the usual situation: A particular
material
has a region, i.e. the region in parameter space where the
critical
laws
given value of J, and the classical transition is triggered
by power
lowering
thecan be observed, is bounded by the two dashed lines.
It then
out that
the critical region is divided into two subregions,
temperature through Tc . Alternatively, however, we can
imageturns
changing
J
by ‘QM’ and ‘classical’ in Fig. 2, in which the observed critical
at zero temperature (e.g. by alloying the magnet withdenoted
some non-magnetic
behavior is predominantly
quantum mechanical and classical, respectively.
material). Then we will encounter the paramagnet-to-ferromagnet
transiThetransition
division between
tion at the critical value Jc . This is the quantum phase
we are these two regions (shown as a dotted line in Fig. 2)
is not sharp
neither is the boundary of the critical region), but rather
interested in. Since we have seen that the quantum transition
is,(and
loosely
2
Features of a quantum phase transition
To illustrate the concepts, we consider the example of a metallic or itinerant
ferromagnet [2]. Figure 1A shows a schematic phase diagram in the T − J
plane, with T the temperature and J the strength of the exchange coupling
that is responsible for ferromagnetism. The coexistence curve separates the
paramagnetic
phasewe(large
Tconsider
, small
J)spins,
from
the ferromagnetic phase (small
For the purposes of this subsection
might as well
localized
but the
theory discussed in Sec. 2 below applies to itinerant magnets only.
T , large J). For any given J, there is an associated Curie temperature Tc
and classical phase transition occurs if we vary the temperature through Tc .
On the other hand, imagine changing J at zero temperature, for instance,
by alloying/doping the magnet with some non-magnetic material. Here, we
encounter a quantum phase transition between the paramagnet and the ferromagnet at the critical value Jc .
The behaviour at the T = 0, J = Jc is very different from T 6= 0. In order
to understand what happens in the vicinity of a QPT, a special mathematical
trick which allows us to connect QPT with classical statistical mechanics is
needed [3]. Consider the partition function of a d-dimension classical system
governed by a Hamiltonian H,
5
3
Z(β) = Tr e−β Ĥ
(1)
which can be written, via Feynman’s path integral formulation, as a function integral of the form (generalized for quantum many-body systems)
Z
Z = D[ψ̄, ψ] eS[ψ̄,ψ] .
(2)
Here, we let Ĥ be the Hamiltonian operator and S is the action of the
system,
Z 1
kB T
∂
dτ ψ̄(x, τ )[− +µ]ψ(x, τ )−
dτ H(ψ̄(x, τ ), ψ(x, τ )).
S[ψ̄, ψ] = dx
∂τ
0
0
(3)
ψ̄ and ψ are the conjugate fields isomorphic to the creation and annihilation
operators in the second quantized formulation of the Hamiltonian. We have
sneaked in the trick of Wick rotation: analytically continuing the inverse
temperature β into imaginary time τ via τ = −i~β, so that the operator
density matrix of Z, eβ Ĥ , looks like the time-evolution operator e−iĤτ /~ . The
end result is seen in the action, which looks like that of a d + 1 Euclidean
space-time integral, except that the extra temporal dimension is finite in
extent (from 0 to β). As T → 0, we get the same (infinite) limits for a d + 1
effective classical system. This equivalent mapping between a d-dimension
quantum system and a d + 1-dimensional classical system allows for great
simplifications in our understanding of QPT.
Since we know that the quantum transition is related to a classical analog
in a different spatial dimensionality, and since changing the dimensionality
usually means changing the universality class, the critical behaviour at the
quantum critical point Jc should be different from that observed at any other
point along the coexistence curve in Figure 1.
Also, another interesting aspect of QPT is that dynamics and thermodynamics(i.e. statics, as thermodynamics is very much a misnomer) cannot
be independently analyzed, unlike the case for classical statistical mechanics. This loss of freedom is due to the non-commutability of coordinates
and momenta in the quantum problem. As a result, both the form of the
Hamiltonian as well as the equations of motion are required, meaning that
one cannot solve the thermodynamics without also solving the dynamics –
Z
Z
1
kB T
4
a feature that makes quantum statistical mechanics much harder to solve.
Hence, even though it was mentioned in the earlier paragraph that a quantum
transition can, in some fashion, be mapped into a classical thermodynamic
transition, information about the correlations in excited states which drive
the system out of the ground state cannot be extracted from this map.
3
Relevance in experimental Physics and Biology
Quantum phase transitions attract intense attention because they are relevant to a host of experimental issues, despite the fact that T = 0 cannot
be achieved experimentally. This is because associated with the quantum
critical point, there is a sizable region where quantum critical behaviour is
observable (labeled QM in Fig. 1B). Some examples are:
• Anderson-Mott models and metal-insulator transitions [4],
• superconductor-insulator (SI) transition in granular superconductors [5],
• transitions between quantum hall states [3],
• the physics of vortices in the presence of columnar disorder [6].
Besides the above, knowledge of dissipative effects [7] on quantum coherence and QPT is essential to assess the reliability of mesoscopic quantum
devices in performing tasks that strongly depend on their ability to maintain
entanglement (for instance, in quantum computation). Among quantum devices widely used, many are based on a collection of regularly arranged or
single small Josephson junctions (JJ). A Josephson junction comprises two
superconductors linked by a very thin insulating oxide barrier and the current that tunnels through the barrier is the Josephson current. Josephson
junction arrays also constitute a particularly attractive testing ground for
the superconducting-insulating (SI) transition, because all parameters are
well under control and are widely tunable. Cooper pairs of electrons are able
to tunnel back and forth between grains and hence communicate about the
quantum state on each grain. If the Cooper pairs are able to move freely
from grain to grain throughout the array, the system behaves like a superconductor. If the grains are very small, a large charging energy is incurred to
5
Sondhi et al.: Continuous quantum phase317
transitions
Sondhi 318
et al.: Continuous quantum phase transitions
rameter on the metallic elements connected by the
junctions7 and their conjugate variables, the charges (excess Cooper pairs, or equivalently the voltages) on each
grain. TheSondhi
intermediate
state ! m ! , at time
" j #j
$ " , that
FIG. 1. Schematic representation of a 1D Josephson-junction
317
et al.: Continuous jquantum
phase
transitions
enters
the quantum
partition function of Eq. (5) can
array. The crosses represent the tunnel junctions
between
superconducting segments, and . i are the phases
of the
thus
be superdefined by specifying the set of phase angles
conducting order parameter in the latter.
% & ( " j ) ' # ( & 1 ( " j ), & 2 ( " j ), . . . , & L ( " j ) ) , where & i ( " j ) is the
phase angle of the ith grain at time " j . Two typical paths
A. Partition functions and path integrals
or time histories on the interval ( 0,* + ) are illustrated in
FIG. 1. Schematic representation
1D Josephson-junction
Figs. of
2 aand
3, where the orientation of the arrows
array.
represent for
the Z.
tunnel
junctions between suLet us focus for
nowThe
oncrosses
the expression
Notice
(‘‘spins’’)
indicates
the local phase angle of the order
segments,
.same
phases of the superthat the operatorperconducting
density matrix
e ! ! H isand
the
as
the
i are the
parameter. The statistical weight of a given path, in the
conducting
order, parameter
in
latter.the
time-evolution operator
e !iHT/"
providedsum
wethe
assign
in Eq. (5), is given by the product of the matrix
imaginary value T"!i" ! to the time elements
interval over
A. Partition
and path when
integrals
which the system
evolves. functions
More precisely,
the
3. Typical path or time history of a 1D JosephsonFIG. 2. Typical path or time history of a FIG.
1D JosephsonH(B)
(C)
e " . 1/* / $ "array.
!%&. " j /'! ,
(6)
trace is written in terms of (A)
a complete set of states,
- % & . " j!1 / ' !junction
junction
array
one of the
con-in the insulating phase, where correlations fall
Let us focus for now on the j expression for
Z. Notice Note that this is equivalent to off
exponentially
figurations of a 1+1D classical XY model. The long-range
cor- in both space and time. This corresponds to
!!H
that! !the
operator density where
matrix e
is the same as the
the disordered
relations shown here are typical of the superconducting
phasephase in the classical model.
Z# ! $"
! e H! n ' ,
(3)
& ntime-evolution
operator e !iHT/" , provided
we
assign the
C
n
of the 1D
array
ˆ "
ˆ or, equivalently, of the
H#
V 2j "E
(7)ordered phase of the
imaginary value T"!i" ! to
the2 time
interval
J cos. &over
j & j!1 /
classical model.
j
easily identified, finding the classical Hamiltonian for
Z takes the formwhich
of a sum
imaginary-time
transition
the of
system
evolves. More
precisely, when the
FIG.
2. Typical path or time
history
of a is1D
Josephsonthis X-Y
model
somewhat
more work and requires an
isstate
quantum
Josephson-junction
amplitudes for the
system
to start
some
! n ' andsetHamiltonian
icontrace
is written
in in
terms
of
athe
complete
of states, of the
junction
array.
Note
that
this
is equivalent
to oneofofthe
the matrix
explicit
evaluation
elements, which interreturn to the same state after an imaginaryarray.
time Here
interval
&ˆ j is the
the
phase
willoperator
observerepresenting
thatfigurations
the expression
forof
the quantum
partiof a 1+1D
classical
XY
model.
The
long-range
cor8
readers
can find in the Appendix.
!i" ! . Thus we see that calculating the
thermodynamics
parameter
on
jth
grain,
tionorder
function
(5) the
has
the here
formare
oftypical
a ested
classical
partirelations
shown
of
the
superconducting
phase
Z# ! $"
(3) in Eq.
& n ! e ! ! H ! nthe
' , superconducting
9
It expressed
is shown in the Appendix that, in an approximation
of a quantum system is the same
as calculating
transition 1 / 1tion
V j #"i(2e/C)(
& j ) isfunction,
conjugate
the
phase
andequivalently,
is
i.e.,
sum
over
n
oftoa
the
1D
array configurations
or,
of the ordered phase of the
preserves the universality class of the problem,10 the
amplitudes for its evolution in imaginary the
time,
with the
voltage
on the in
jthterms
junction,
E J is matrix,
the
Josephson
of a and
transfer
if we thinkthat
of imaginary
classical
model.
Z
takes
the
form
of
a
sum
of
imaginary-time
transition
product
of matrix
elements in Eq. (6) can be rewritten in
total time interval fixed by the temperature
of interest.
coupling
energy. The
two
in the Hamiltonian
then In
time
as terms
an additional
spatial dimension.
particular,
if
for happens
the system
to start
in some state ! n ' and
thethe
form
e "H XY , where the Hamiltonian of the equivaThe fact that theamplitudes
time interval
torepresent
be
imaginary
the charging
energy ofsystem
each grain
Joour quantum
lives and
in dthe
dimensions,
expresreturn
to
the
same
state
after
an
imaginary
time
interval
will
observe
that
the
expression
for
the
quantum
lent
classical
X-Y
model partiis
is not central. The key idea we hope to transmit
the
sephson to
coupling
of the
across function
the junction
sion
forphase
its partition
looksbelike a classical parti. Thusevoke
we seeanthat
calculating
the thermodynamics
function
Eq. (5)
has the form
of a classical partireader is that Eq.!i"
(3) !should
image
of quantum
tween
grains.
tion function fortion
a system
withind#1
dimensions,
except
1
of a quantum
system is the same as calculating transition
tion the
function,
i.e.,in
aextent—"
sum overH
dynamics and temporal
propagation.
As indicated previously,
candimension
map
quantum
sta(8)
that the we
extra
is finite
! configurations
in #
units cos. &expressed
XY
i" & j /
amplitudes
for its
evolution
ina imaginary
time, with the
in the
terms
of statistical
a transfer
matrix,
we Kthink
- ij ! of imaginary
This way of looking
at things
can
be given
particutistical
mechanics of
of time.
the array
onto
classical
As T→0
system
size in this
extraif ‘‘time’’
total
time
interval
fixed
by
the
temperature
of
interest.
time
as
an
additional
spatial
dimension.
In
particular,
if
larly beautiful and practical implementation
in the lanmechanics
by identifying
statistical weight
of a getand athe sum
direction thediverges,
and
we
trulyruns over nearest-neighbor points in the
The fact
that the time
interval of
happens
to be imaginary
quantum
system
livestwo-dimensional
in d dimensions,(space-time)
the expres- lattice.11 The nearestguage of Feynman’s
path-integral
formulation
quanspace-time
path in d#1-dimensional,
Eq. (6) with the our
Boltzmann
weight
ofsystem.
effective
classical
is
not
central.
The
key
idea
we
hope
to
transmit
to
the
sion
for
its
partition
function
looks
like a classical
tum
mechanics
(Feynman,
1972). a two-dimensional
Feynman’s
spatial
of abetween
classical asysneighborquancharacter
of theparticouplings identifies the classiSinceconfiguration
this equivalence
d-dimensional
reader
is that
Eq. (3)amplitude
should evoke
an image of quantum
function
a system
with
d#1asdimensions,
except
prescription is that
the net
transition
between
tem. In
this case the
thereforefor
a twocal
model
model, extensively studied in
tumclassical
systemsystem
and tion
a is
d#1-dimensional
classical
system
isthe 2D X-Y
andbe
temporal
propagation.
that
the
extra
dimension
is
finite
in
extent—"
!
in
units
two states of thedynamics
system can
calculated
by
summing
dimensional X-Y model,
its degreeselse
of freedom
crucial i.e.,
to everything
we haveare
to say the
and context
since Eq.of superfluid and superconducting films
way of
looking
at things
canThe
be given a particuof
time.
As
T→0
the
system
size
in
this
extra
‘‘time’’
amplitudes for allThis
possible
paths
between
them.
planar
spins,
specified
by
angles
&
,
that
live
on
a
two(Goldenfeld,
1992;
Chaikin
and Lubensky, 1995). We
(5) is probably not
i very illuminating for readers not used
larlysystem
beautiful
and practical
implementation
in the landirection
diverges,
and
wevery
getwhilea thetruly
path taken by the
is defined
by dimensional
specifying
the
squaretolattice.
that
temperatures
that,
straightforward identification
a daily(Recall
regimen
ofattransfer
matrices, itemphasize
will be
guage
of
Feynman’s
path-integral
formulation
of
quaneffective
classical
system.
state of the system at a sequence of finelyabove
spacedzero
interthe lattice
a finited#1-dimensional,
width
* + / $example
" in the
of the
of freedom of the classical model in this
usefulhas
to consider
a specific
in order
to degrees
be able
tumFormally
mechanics
1972).
Feynman’s
Since
equivalence
between
quan- of the resulting classimediate time steps.
we write(Feynman,
temporal direction.)
While thewhat
degrees
of this
freedom
are
exampleaisd-dimensional
robust, this simplicity
to visualize
Eq.
(5)
means.
prescription is that the net transition amplitude between
tum
system
and
a
d#1-dimensional
classical
system isof a minor miracle.
cal
Hamiltonian
is
something
!!H
! # 1/" $ ) * H N
" ( e two states
(4)
e
+ , of the system can be calculated
by summing
crucial to everything else we
to say to
and
since
It have
is essential
note
thatEq.
the dimensionless coupling
Example:
Josephson-junction
arrays readers not used
amplitudes
allispossible
betweenB.them.
The One-dimensional
(5) is
not very
interval6for
that
small7It
onpaths
time
where ) * is a time
constant K for
in H XY , which plays the role of the temperais the
believed
that neglecting fluctuations
in probably
the magnitude
of illuminating
taken where
by the,system
is ultraviolet
defined
by specifying the
to a daily
regimenand
of transfer
matrices,
it will
be very
scales of interestpath
( ) * ""/,,
is some
the order
parameter is a good approximation
(see Bradley
ture
in
the
classical
model,
depends
on the ratio of the
Consider
a one-dimensional
array
comprising
a large
the system
at achosen
sequence
finely
spaced
useful to consider
a specific
example
in orderenergy
to be able
Doniach,
1984;
Wallin
et al.,inter1994).
cutoff) and N state
is aof large
integer
so of
that
capacitive
charging
E C #(2e) 2 /C to the Josephnumber
L
of
identical
Josephson
junctions
as
illustrated
mediate
time
steps.
Formally
we
write
to visualize what Eq. (5) son
means.
8
N ) * "" ! . We then insert a sequence of sums
over
com- here is that % & ( " ) ' refers
couplingbyE J in the array,
Our
notation
to thehave
configuration
in Fig. 1. Such arrays
recently been studied
$ ) * Hthe
plete sets of intermediate
into
expression
(4)
e ! ! H "states
( e ! # 1/"
+ N , of
the entirefor
set of Haviland
angle variables
at time (1996).
slice " . A
The
&ˆ ’s
and Delsing
Josephson
junction
a J,
!EisC /E
K2
(9)
Z( ! ):
appearing
Hamiltonian
in Eq. (7)
angulartwo
coordinate
B.areExample:
One-dimensional
Josephson-junction
arrays
6
tunnel
connecting
superconducting
metallic
that in
is the
small
on thejunction
time
where ) * is a time interval
operators, and j is a site label. The state at time slice " is an
and
has
nothing
to
do
with
the
physical
temperature
(see
grains.
Cooper pairs of electrons
are able to tunnel back
scales of interest! #(1/"
) *$ )""/,,
where , is some
ultraviolet
*H
array comprising
a large
of and
these
operators:
cos(grains
&ˆ j"&ˆ a
)and
!%&(")hence
the
appendix).
The physics
here is that a large JosephZ# ! $"
n!e
! meigenfunction
'!
j!1one-dimensional
1'
forth
betweenConsider
the
communicate
and& N is a large
integer chosen
so
that
n mcutoff)
1 ,m 2 , . . . ,m N
#cos(&j(")"&j!1("))!%&(")'!.
number L of identical Josephson
junctions
as illustrated
son coupling
produces
a small value of K, which favors
information
N ) * "" ! . We then insert a9 sequence
of sums
over com- about the quantum state on each grain. If
! # 1/" $ ) * H
in
Fig.
1.
Such
arrays
have
recently
been
studied
by
It
is
useful
to
think
of
this
as
a
quantum
rotor
model.
The
-& m 1plete
! e sets of
! mintermediate
'
the Cooper
states
into the expression
forpairs are able to move freely from grain to
2 '& m 2 ! ••• ! m N
state with wave function e im j & j has m j units
angular
and
Delsing
(1996).
A Josephson junction is a
10
grain throughoutHaviland
theofarray,
themomensystem is
a superconducZ( !!):
That
is,
the
approximation
is
such that the universal aspects
# 1/" $ ) * H
tum representing
pairs on
grain j. The
-& m N ! e
!n'.
(5) mtor.
junction
superconducting
metallic
j excess Cooper tunnel
If the grains are very
small, connecting
however,
it two
costs
a large
of
the critical
behavior, such as the exponents and scaling funcCooper-pair number operator in this
representation
isof electrons
grains.
Cooper
pairs
are
able
to
tunnel
back
1/" $ ) * H
charging
energy
to
excess
Cooper
pairwill
onto
tions,
be agiven without error. However, nonuniversal
This rather messy Z
expression
actually nhas
! mWallin
# ! $"
& n ! ae !1#rather
/ 1 & j ) (see
et al.,
1994).and
Themove
cosinean
term
in Eq.
1'
j #"i(
forth
between
thethe
grains
andpairs
hence
communicate
quantities,
such
as the
critical coupling, will differ from an exn Formally
m 1 ,m 2 , . . . inclined
,m N
grain.
If transfers
this energy
enough,
Cooper
simple physical interpretation.
(7) is a readers
‘‘torque’’ term
which
unitsisoflarge
conserved
anc termsJare irrelevant
information
about
the
quantum
state on
each grain.
If
act
evaluation.
Technically,
the neglected
fail
to
propagate
and
become
stuck
on
individual
grains,
(Cooper pairs) from site to site. Note that
H
2
-& m 1 ! e ! # 1/" $ ) *gular
! m 2momentum
m
!
•••
!
m
'& 2
'
the
Cooper
pairs
are ableatto
from grain
themove
fixed freely
point underlying
thetotransition.
Nwhich
system
to be
an insulator.
the potential
energy
of thecauses
bosonsthe
is represented,
somewhat
c
grain
throughout
the array,11the
system
is a superconduc6
! # 1/"
* H so that the
The essential
degrees
of freedom
system
arecrucial
Notice
this
change in notation from Eq. (7), where
paradoxically,
by the kinetic
energy
the
quantum
rotors andin this
For convenience we have chosen
) *! eto
be$ )real,
-& m N
!n'.
(5) of tor.
If the grains
are very jsmall,
however,
a large
vice is
versa.
refers
to a papoint it
incosts
1D space,
not 1+1D space-time.
the phases of the complex
order
small interval of imaginary time that it represents
!i ) * .
J superconducting
charging energy
to move an excess Cooper pair onto a
This rather messy expression actually has a rather
grain. If this energy is large enough, the Cooper pairs
simple physical interpretation. Formally inclined readers
Rev. Mod. Phys., Vol. 69, No. 1, January 1997
Rev. Mod. Phys., Vol. 69, No. 1, January 1997
fail to propagate and become stuck on individual grains,
which causes the system to be an insulator.
6
The essential degrees of freedom in this system are
For convenience we have chosen ) * to be real, so that the
the phases of the complex superconducting order pasmall interval of imaginary time that it represents is !i ) * .
,
%
Figure 2: (A) Representation
of a 1D Josephson Junction array. Crosses
0
represent junctions between superconducting segments, and θ are the phases
of the% superconducting order parameter. (B) Typical path or time history
of the 1D JJ array. Notice it is equivalent to the configuration of a 1+1D
classical XY model. The long-ranged correlations are typical of the ordered
(superconducting for 1D JJ) phase. (C) Typical path or time history of the
0 for 1+1D XY) [3].
1D JJ array in the insulating phase (or disordered phase
move an excess Cooper pair onto a grain. When this energy is large enough,
Cooper pairs cannot propagate and becomes confined on individual grains,
leading to the quenching of the collective superconducting phase.
As an example, we consider a one-dimensional array of identical JJs (Fig.
2). The essential degrees of freedom are the phases of the complex superconducting order parameter on the metallic segments connected by the junctions
and their conjugate variables, the charges (excess Cooper pairs, or equivalently the voltages) on each grain. Even without doing further math, from
% Fig.
%
2B and 2C, one can draw an analogy between our system and a 2-D
classical XY model. It turns out that, in an approximation that preserves the
universality class of the system, we can indeed map our 1D JJ array
p into a 2D
%
%
X-Y system [3], with our dimensionless coupling constant K ∼ E /E playing the role of the temperature in the classical analog, where E = (2e) /C is
the capacitive charging energy and E the Josephson coupling in the array.
This equivalence generalizes to d-dimensional arrays and d + 1-dimensional
classical X-Y models. Among the systems studied, two-dimensional ones are
most interesting as no true long-range order is possible at finite temperatures
while
occurs at T = 0. Moreover, 2D samples can be easily
Rev.
Mod. Phys.,a
Vol.genuine
69, No. 1, JanuaryQPT
1997
fabricated and experimentally characterized.
6
al symmetry and that at the transition the polyelectrolyte bundle adopts a universal response
ar.
numbers: 61.20.Qg, 87.15.-v, 74.40.+k
ged DNA molecules in aqueous soludense hexagonal bundles (or tori) in
w concentrations of positively charged
or “counter-ions” [1, 2, 3]. This
as interesting applications in biology
bited as well by other highly charged
attracted significant theoretical interclear disagreement with the classical
n mean-field theory of aqueous electrol [4] and analytical [5, 6] studies of
mitive model”–where the biopolymer is
, charged rod and the counter-ions as
cate that the attraction between two
fact a fundamental feature of aqueous
interaction, which has a short range,
ut-of-phase” correlations between orarrays on the two rods. The purpose
show that the problem of the finite
er-ion freezing, or “Wigner crystallizaxagonal polyelectrolyte bundle can be
= 0 quantum phase transition, and furping leads to surprising predictions for
olyelectrolyte bundles.
this claim, we begin with a single rod
ed charge per unit length, −eρ0 , plus
ribution of mobile polyvalent ions of
ensed onto the rod. The rod is placed
saline aqueous solution with Debye pae absence of thermal fluctuations, the
-ions form a one-dimensional “lattice”
= Z/ρ0 . The modulated part of the
he rod can be expressed as a sum over
ce vectors of the Wigner crystal:
#
$
%
!"
|ρn |ein Gz+φ(z) + cc. .
(1)
placement of the of the counter-ion lattice with respect
to the rod. The phase variable obeys an effective Hamiltonian,
&
C L ' ∂φ (2
,
(2)
dz
Frod [φ] =
2 0
∂z
where CG2 is the one-dimensional, T = 0 compression
modulus of the lattice–roughly proportional to |ρ1 |2 e−κd –
and L is the length of the rods. Due to phase fluctuations
at finite temperatures, the contribution from the nth term
in eq. (1) to the thermal expectation value "δρ(z)# van2
ishes as e−n L/βC and the distribution has only liquidlike correlations, "δρ(0)δρ(z)# ∼ e−|z|/ξz cos Gz, with
ξz = 2βC. Below, we include only the lowest order terms,
ẑ
ŷ
x̂
(a)
(b)
FIG. 1: The two degenerate, chiral ground states of the anti-
Figure 3: Correspondence
between
and
ferromagnetic XY
model onpolyelectrolyte
a triangular latticebundle
are shown
in T = 0 2D JJ
Thedegenerate,
helical ordering
around
a triangular
plaquette
the
array. (A) The(a).two
chiral
ground
states
of theofantiferromagnetic
s the reciprocal lattice vector, |ρn | an
lattice corresponding to one of these ground states is
XY model on bundle
a triangular
lattice. The helical ordering around a triangular
shown in (b). The dark black lines depict the polymer backned by the charge distribution of a sinplaquette
of
the
bundle
lattice
shown
where the
dark black lines
bone while the spheres is
depict
peaksinin(B),
the condensed
charge
nd φ(z) a phase variable restricted to
density.backbone while the spheres depict peaks in the condensed
depict
Specifically, φ(z)G−1 is the
local the
dis- polymer
charge density [9].
=0
7
The electromagnetic properties of a Josephson array,
such as conductivity and diamagnetic susceptibility, are
determined by the response of the array to an applied
vector potential. Specifically, the conductivity can be
expressed by the Kubo relation as σ(ω) = ImCxx (q =
TABLE I: Correspondence between the d = 3, classical polyelectrolyte system (left) and the d = 2+1, quantum frustrated
Josephson array system (right).
free-energy functional, βF [φ]
height along polymer, z
longitudinal phonon
(or phase) stiffness, βC
electrostatic, inter-rod
coupling, βEij
shear strain, 2G$⊥z
elastic response, Gxzxz (qz )
imaginary-time action, h̄−1 S[φ]
imaginary time, τ = it
inverse grain charging
energy, h̄Ec−1
Josephson inter-grain
coupling, h̄−1 EJ
vector potential, (2π/Φ0 )a⊥
current response, −Cxx (−iω)
sponse to uniform sliding deformation
Gxzxz (q⊥ , qz ) ∝ (kB T G2 )qz2 ξ(α). H
columnar liquid crystal with a bendin
the bundle which diverges at the crit
correlation length, ξ(α). Turning next
ducting phase (α < αc ) of the Josephso
zero-frequency conductivity is infinite t
current-current correlation function is p
superfluid density, so that Cxx (q⊥ = 0
[14]. Therefore, the shear modulus of t
bundle, Cxzxz = Gxzxz (q = 0) ∝
is finite in the low-temperature, char
and vanishes at the critical point. T
ulus corresponding to the uniform ela
pictured in Fig. 2. Finally, right at
(α = αc ) the Josephson is predicted to
i.e. to have a finite zero-frequency c
Cxx (q⊥ , −iω) = σ ∗ |ω|. Moreover, th
tance is expected to be a universal q
e2 /2h̄ [10]. The corresponding strain
tion, Gxzxz (q⊥ , qz ) ∝ (kB T G2 )|qz |, wo
Figure 4: Correspondence between d = 3 classical polyelectrolyte system
(left) and d = 2 + 1 quantum frustrated JJ array system (right) [9].
In biology, nature has a small range of temperatures to play with, due to
the protein nature of life chemistry, and most phase transitions occur with
the variation of a parameter other than temperature. I hypothesize that
it may be possible to draw analogies with QPTs, where the temperature is
fixed (at T = 0) and some other parameter is varied, and hence tap into the
immerse theoretical and experimental work that has been done on JJ arrays.
Although i do not have anything solid to back up my statement, the field of
polyelectrolyte condensation may show some promise. Polyelectrolyte chains
naturally repel each other, but will nevertheless form condensed bundles
in the presence of oppositely charged counterions beyond a certain valency
(depending on the nature of the polyelectrolyte). Examples include DNA
and F-actin. In fact, polyelectrolytes condense above a certain counterion
concentration and dissociate above another counterion concentration. It has
become clear that this condensation results from some form of organization of
the counterions, either dynamical (correlated charge density fluctuations) or
essentially static, in the form of a counterion lattice (positional correlations
between condensed counterions).
Some theoretical work on the counterion ‘melting’ transition suggest that
the melting transition (i.e. the bundle-to-individual polyelectrolyte transition) is continuous and can be shown to be in the universality class of the
three-dimensional XY model [8]. In another work [9], researchers managed to
establish a mapping between a model for hexagonal polyelectrolyte bundles
and a two-dimensional, frustrated Josephson-junction array (Fig.3). They
found that the T = 0 SI transition of the quantum system corresponds to
a continuous liquid-to-solid transition of the condensed charge in the finite
8
temperature classical polyelectrolyte system. Moreover, the role of the vector
potential in the JJ system is played by elastic strain in the classical system
(Fig. 4). The general conclusion of this work relates the elastic constants
of polyelectrolyte bundles to the phase behaviour of the counterions, which
is significant as elastic constants are easier to measure than the scattering
amplitudes of counterion charge modulation.
4
Further work
Due to constraints, much experimental and theoretical details have been
left out. There are interesting instances of quantum phase transitions in
two dimensional quantum magnets which cannot be predicted with an order parameter using the GLW (Ginzburg-Landau-Wilson) formalism that we
have adopted in this study [10]. Also, studies on the dynamics of QPTs
are still active and a quantum counterpart for the classical Kibble-Zurek
mechanism of second order thermodynamic phase transitions was recently
proposed [11, 12, 13]. It appears that both experimentalists and theorists
have their work cut out for them!
References
[1] S. Sachdev, Quantum Phase Transitions (Cambridge Uni. Press, Cambridge, England, 1999)
[2] D. Belitz, T. R. Kirkpatrick, cond-mat/9811058 (1998), contribution to
Models and Kinetic Methods for Non-equilibrium Many-Body Systems
(editor: J. Karkheck)
[3] S. L. Sondhi, S. M. Girvin, J. P. Carini, D. Shahar, Rev. Mod. Phys.
69, 315 (1997)
[4] H. -L. Lee, J. P. Carini, D. V. Baxter, W. Henderson, G .Grüner, Science
28 (287): 633-636 (2000)
[5] Kuper et. al., Phys. Rev. Lett. 56, 278 (1986); Paalanen et. al., Phys.
Rev. Lett. 65, 927 (1990); Garno et. al., Phys. Rev. Lett. 69, 3567 (1992);
Kapitulnik et. al., Phys. Rev. Lett. 74, 3037 (1995)
9
[6] D. R. Nelson, V. M. Vinokur, Phys. Rev. B 48, 13060 (1993)
[7] Vaia et. al., Phys. Rev. Lett. 94, 157001 (2005)
[8] J. Rudnick, D. Jasnow, cond-mat/0207651
[9] G. M. Grason, R. F. Bruinsma, cond-mat/0601462
[10] T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, M. P. A. Fisher,
Phys. Rev. B 70, 144407 (2004)
[11] J. Dziarmaga, A. Smerzi, W. H. Zurek, A. R. Bishop, Phys. Rev. Lett.
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10