VI. INTRODUCTION to THERMAL FLUCTUATIONS in TYPE II SC. BKT TRANSITION in 2D

1
VI. INTRODUCTION to THERMAL FLUCTUATIONS in TYPE
II SC. BKT TRANSITION in 2D
A. Two scales of thermal fluctuations
1. The microscopic thermal fluctuations place in
the GL description
On the microscopic level temperature
modifies properties of the electron gas and the
pairing interaction responsible for the creation
of Cooper pairs. When we integrated out the
microscopic (electronic) degrees of freedom to
obtain the effective mesoscopic GL theory in
terms of the distributions of the order parameter
with ultraviolet (UV) cutoff a

2
One loosely describes the effective
mesoscopic order parameter
as a classical field of Cooper pairs with center
of mass at x. More mathematically, the remaining
mesoscopic part of the statistical sum obeys the
scale matching
3
Quantum effects on the mesoscopic level are
usually small (only when temperature is very
close to T0 they might be of importance) and
will be neglected. In this case the mesoscopic
classical field is independent of time.
Dynamical generalization of the GL approach will
be introduced later.
4
The coefficients of the GL energy
are dependent on temperature expressing
these microscopic thermal fluctuations. The
dependence can, in principle, be derived from a
microscopic theory (example Gorkovs derivation
from the BCS theory of conventional
superconductors). The coefficients also depend on
UV cutoff L or a, but we will see that this
dependence can be renormalized away.
In practice the constants
are also weakly (typically logarithmically)
temperature dependent
5
leading for example to curving down of the
Hc2(T) line
Normal
Mixed state
Meissner
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2. Two kinds of mesoscopic thermal fluctuations
perturbative and topological
The mesoscopic fluctuations qualitatively are of
two sorts perturbative small ones and
topologically nontrivial or vortex ones.
Since under magnetic field the order parameter
takes a form of vortices, the mesoscopic
fluctuations can be qualitatively viewed as
distortions of a system of vortices or thermal
motion vibrations, rotations, waves.
7
Major thermal fluctuations effects include
broadening of the resistivity drop
(paraconductivity) and diamagnetism in the normal
phase and melting of the vortex solid into a
homogeneous vortex liquid state
Resistance
Broadening of the resistivity
Nonfluctuating SC
Normal Metal
Fluctuating SC
T
TC
Magnetization or conductivity are greatly
enhanced in the normal region due to thermally
generated virtual Cooper pairs.
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Influence of the fluctuations on the vortex
matter phase diagram
Due to the thermally induced vibrations lattice
can melt into vortex liquid and the vortex
matter phase diagram becomes more complicated.



9
Symmetries broken at the two transitons
Since symmetry of the normal and liquid phases
are same, the normal liquid line becomes just a
crossover.
Two different symmetries are at play the
geometrical E(2) (including translations and
rotations) and the electric charge U(1).
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B. Nontopological excitations. The Ginzburg
criterion.
1. Gaussian fluctuations around the Meissner state
We ignore the thermal fluctuations of the
magnetic field which turn out to be very small
even for high Tc SC in magnetic field. The effect
of thermal fluctuations on the mesoscopic scale
is determined by
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Ginzburg number
It is convenient to use units of coherence
length (which might be different in different
directions), Y in units of Y0 and energy scale in
units of critical temperature
In the new units the Boltzmann factor
becomes
12
with the anisotropy parameter
Here an important dimensionless parameter
characterizing strength of thermal fluctuations is
Where the temperature independent Ginzburg number
characterizing he material was introduced
13
Perturbation theory
To calculate the thermal effects via a
somewhat bulky functional integral, the simplest
method is the saddle point evaluation assuming
that w is small.
One minimized the free energy around the
classical or nonfluctuating solution (which we
found in the preceding parts also for the SC
phase)
and then expands in small fluctuations
14
Now we consider the normal phase tgt1 in
which the saddle point value of the order
parameter is trivial
The thermal average of the order parameter
is still zero due to the U(1) symmetry
However the superfluid density
due to fluctuations. First we calculate the
fluctuation correction to free energy and thereby
to specific heat which is impacted the most.
15
The fluctuation contribution to free energy
From now on fl is dropped.
The expansion of the partition function
results in gaussian integrals
16
which is presented graphically as diagrams
It is more convenient to perform the
calculation in momentum space
17
since the basic gaussian integral in
momentum space becomes a product
The leading order
Therefore the fluctuations contribution to
the free energy density is
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The fact that free energy depends on the UV
cutoff a means that it is not a directly
measurable mesoscopic quantity energy
differences or derivatives are.
Entropy and specific heat
The mesoscopic Cooper pairs contribution to the
entropy density is less dependent on the
division of degrees of freedom into micro and
meso just a constant
19
The second derivative, the fluctuation
contribution to specific heat, is already finite
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2. Interactions of the excitations and critical
fluctuations. Ginzburg criterion.
Closer than that to Tc the perturbation
theory cannot be used due to IR divergencies.
Physics in critical region is therefore dominated
by fluctuations. A nonperturbative method like RG
required.
The fluctuation contribution outside the
critical region is called gaussian since
fluctuations were considered to be
non-interacting. Corrections already do depend on
interactions. They are smaller by a factor at
least. It is more instructive to see this on
example of the correlator.
21
Correlator and its divergence at criticality
A measure of coherence in the Meissner phase
(density of Cooper pairs) is
A measure of the SC correlation (or virtul
density of Cooper pairs) in the normal phase is
Fourier transform of correlator and small wave
vector
The leading correction is
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Renormalized perturbation theory
This expression is a small perturbation only
when
and therefore cannot be applied close to Tc.
Perturbation theory seems to be useless due to UV
divergencies. However it is natural to assume
that quantities measurable on the mesoscopiclevel
should not depend on cutoff a.
It is reasonable to assume that when the
mesoscopic fluctuations are switched on the
superconducting correlations are destroyed at
below which takes into account only the
microscopic ones. How to quantify it?
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At correlator decays slower
One can try to improve on it by resumming some
diagrams
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To first order in fluctuations criticality
occurs at
Since usually is not known
theoretically and is not a quantity of interest,
one expresses it via measured critical
temperature used as an input parameter.
Ginzburg criterion
After renormalization we return to the
perturbation theory applicability test
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This is known as Ginzburg criterion.
Plugging correct constants one obtains
This might be compared to the jump between
Meissner and normal phases before mesoscopic
fluctuations are taken into account
The condition that fluctuations do not
become stronger than the mean field effect is
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3. Fluctuations in the Meissner phase. Goldstone
modes.



The negative coefficient of the quadratic term
leads to a nontrivial minimum
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Feynman rules with shift and the Goldstone mode.

Since one of many possible shifts was chosen it
is convenient to present the real and imaginary
parts of the fluctuation separately


The energy in terms of two real fields O and A
becomes


)
-v (

) 1/2 (
2

Where the masses of excitations are
28

Regular massive mode, optic
Goldstone mode, acoustic
Dispersion relation of the A mode is that of
acoustic phonons


And in the confuguration space the correlator is
in 3D
in 2D
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The field A itself is not the order parameter
since it does not transforms linearly under the
symmetry transformation. The order parameter is
Where the phase of is related to the
fields by
Its correlator is


in 3D
in 2D
30

The 2D correlator in the ordered phase decays
albeit slowly (as a power rather than exponential
in the disordered phase
Fluctuations due to Goldstone bosons in 2D
destroy perfect order. Such a phase is called
quasi long range order phase (or
Berezinski-Kosterlitz-Thouless phase). We started
from the assumption of nonzero VEV. It seems that
fluctuations destroy this assumption!
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4. Destructions caused by IR divergencies in Db2
and the MWC theorem.



The energy to the one loop level is
The corrected value of v is found by minimizing
it perturbatively in loops
Noting that
32

one obtains a logarithmically divergent
correction to VEV


To higher orders the logs can be resummed
The VEV decays do not diverges, indicating that
order is slowly restored. This is
Hohenberg-Mermin-Wagner theorem in 2D continuous
symmetry is not broken. More importantly this
does not mean the perturbation theory is useless.
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O(2) invariant quantities

For such quantities the collective coordinates
method simplifies into perturbation theory around
broken vacuum. All the IR divergencies cancel.
Let us see this for the energy to two loops
order. Jevicki, (1987)


There also is correction due to change in v
34

The leading IR divergencies are
easy to evaluate


Subleading divergencies also cancel
although it is much less obvious. Cancellations
occur to all orders F.David (1990) in loop
expansion. What is the mechanism behind this
cancellation of spurious divergencies?
It is hard to say generally, but at least in
extreme case of 1D the answer is clear.
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Physics below the lower critical dimension

For D1 the model is equivalent to QM of particle
on a plane with Mexican hat potential


Pert. ground state
O
A
QM ground state
Ground state is O(2) invariant but is very far
from the origin (0,0) pert. Ground state is bad,
but theory corrects it using IR divergent
matrix elements Kao,B.R.,Lee
PRB61, 12652 (2000)
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5. Heuristic argument about destruction of order
by Goldstone bosons.
For the XY model (same universality class as
GL) a Typical excitation is a wave of phase
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Its energy in various dimensions is
D2 is a border case in which there exists
almost long range order
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C. Topological excitations. The dual picture
1. An extreme point of view topological
fluctuations dominate thermodynamics.
Vortices are the most important degrees of
freedom in an
extremely type II SC (even in the absence of
magnetic field )
It is therefore advantageous in such a case to
reformulate the theory in terms of vortex degrees
of freedom only
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The Feynman-Onsager excitation and the
spaghetti vacuum.
Minimal excitation (Cooper pair) Smallest
vortex ring
The normal phase is reinterpreted as a
proliferation od loops
SC
Normal
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This point of view is not commonly accepted
or used with one notable exception the BKT
transition in thin films. Reasons
1. In 3D non ligh Tc materials k is not very
large. 2. Vortex loop in 3D is a complicated
object infinity of degrees of freedom (unlike in
2D)
The dual picture was nevertheless advanced
after the discovery of high Tc. Its extentions to
include external magnetic field were unsuccessful
so far.
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2. KT transition in thin films or layered SC
Pearl (1964)
may be very large .
Interaction between 2D fluxons in logr up to
scale
SC - normal phase transition in thin films is of
a novel type the Berezinskii Kosterlitz-Thouless
continuous type (71).
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Dipole unbinding triggers the proliferation
The basic picture is just 2D slice of the
fluxon proliferation 3D picture instead of
vortex loop - dipoles.

• Kosterlitz, Thouless (72)

Dual dipoles SC
Free dual charges normal
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The magnetic flux symmetry
Noeter theorem ensures that (if the symmetry is
not spontaneously broken) any continuous symmetry
has a corresponding conservation law. Examples
the electric charge global U(1) symmetry
Leads to charge conservation
Other examples rotations angular momentum
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Similarly one can interpret the Maxwell equation
As a conservation law of the magnetic flux, yet
another global U(1) symmetry, sometimes calle
inverted or dual U(1) .
The symmetry is unbroken in the SC, while
spontaneously breaks down (photon is a Golstone
boson) in the normal phase

• Kovner, Rosenstein, PRL (92)

The order parameter field was constructed and the
GL theory in terms of it was established
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The dual picture
The analogy of the charge U(1) and the magnetic
flux U(1) is as follows

• Nelson, Halperin, PRB (81)

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3. A brief history of phase transitions with
continuous symmetry in 2D.
a. They do not exist
The Hohenberg-Mermin-Wagner theorem
demonstrates that fluctuations destroy long range
order. According to the dual picture a continuous
magnetic flux symmetry should be spontaneouly
broken. This seems to be impossible in 2D, hence
according to Landaus postulate no phase
transition.
The correlator is a power decay at low
temperatures

• Berezinskii(71)

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b. The high temperature expansion something
happens in between.
High temp. expansion gives an exponential
Some qualitative change should happen in between!
This something is unbinding of vortices.
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c. The energy entropy heuristic argument
Energy of a pair of size r

• Kosterlitz, Thouless (72)

-

Number of states
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c. The energy entropy argument
The free energy of a pair therefore is
Where a is the core size. This becomes negative
for
-
which means that the pair proliferate and
superconductivity is lost
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d. Systematic expansions and exact solution.
Kosterlitz (74) and Young (79) developed a
heuristic renormalization group (RG) approach to
account for differences between pairs sizes. We
will follow this approach.
The XY model or the Coulomb gas maps onto
the sine-Gordon field theory for which
perturbation theory exists
Wiegman (78)
Starting with Zamolodehikovs (80) exact results
were obtained that confirm approximate ones.
Nowadays that KT theory is one of the most solid
in theoretical physics.
Recent experiments on BSSCO and other layered
high Tc superconductors found new area of
applications for KT.
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4. The RG theory of the BKT transition
The energy of the KT pair neglecting interactions
with other pairs is
Let us assume that screening can be represented
by the dielectric function e(r).
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It takes into account the polarization of the
pairs due to smaller ones.
r
The energy therefore gets reduced
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The dielectric function itself is created by the
polarization due to dependence of the Boltzmann
weight on the orientation of the thermally
created KT dipoles
E
To calculate the polarization due consider
constant electric field we first write the
polarizability of a single dipole
q
r
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since the number density of such pairs is
Therefore one gets
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Thereby we have derived an integral form of the
RG eqs. for U and e. Differentiating it with
respect to the pair size one obtains a
differential form of the RG eqs.
with initial conditions
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The equations can be turned into a set of
autonomous ones by going to a log scale
and rescaling the density variable
57
Exercise 5 Plot the vector field of this
autonomous system. Solve this system of
differential equations either numerically or
approximately analytically using the separatrix
method.
It is clear that the character of solution
changes at
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Exact solution near the KT temperature
Let us slightly redefine again new variables to
make the solution evident
With x small near the transition
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It is clear that parabolas are the flow lines
The first equation now can be integrated
With the result
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The critical value of dielectric constant is
finite
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Singularities at transition
With criticality of the very weak KT type
Density of bound pairs is
With the result
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It is clear that parabolas are the flow lines
The first equation now can be integrated
With the result
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5. Phenomenology of the BKT transition
The energy of the KT pair neglecting interactions
with other pairs is
Let us assume that screening can be represented
by the dielectric function e(r).