1 On the microscopic level temperature modifies properties of the electron gas and the pairing interaction responsible for the creation of Cooper pairs.

1

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

VI. INTRODUCTION to THERMAL FLUCTUATIONS in TYPE II SC. BKT

TRANSITION in 2D

x

A. Two scales of thermal fluctuations1. The microscopic thermal fluctuations’ place in

the GL description

2

One loosely describes the effective mesoscopic order parameter

/

iKXk K k thermal quantum

k a

X e c c

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”:

K

k

Xx a

3

* * *

/ /

1/* *

0

* *

D D D D D D

exp , , ,

1D D exp , , ,

k k k kk a K a

TiKX

L K L k k K KL

Z T c c c c X X

X e c c L c c c c

X X F X X T aT

Quantum effects on the mesoscopic level are usually small (only when temperature is very close to T=0 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.

x

4

are dependent on temperature expressing these “microscopic” thermal fluctuations. The dependence can, in principle, be derived from a microscopic theory (example: Gor’kov’s derivation from the BCS theory of “conventional superconductors). The coefficients also depend on UV cutoff or a, but we will see that this dependence can be “renormalized away”.

In practice the “constants” are also weakly (typically logarithmically) temperature dependent

*, ,m

2 2 2 4( )

2 * 2cF D T Tm

��������������

The coefficients of the GL energy

5

Normal

Mixed state

2cH

1cHMeissner

cT

22

( )' |

c

cc c

T T

dH TH T

dT

leading for example to “curving down” of the Hc2(T) line:

6

2. Two kinds of mesoscopic thermal fluctuations: “perturbative” and “topological”

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.

The mesoscopic fluctuations qualitatively are of two sorts: “perturbative” small ones and “topologically nontrivial” or vortex ones.

7

Broadening of the resistivity

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

Magnetization or conductivity are greatly enhanced in the “normal” region due to thermally generated “virtual” Cooper pairs.

TTC

Res

ista

nce

Nonfluctuating SCNormal Metal

Fluctuating SC

8

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

Since symmetry of the normal and liquid phases are same, the normal – liquid line becomes just a crossover.

Just crossover

NormalVortex liquid

2cH

1cHMeissner

FLL

H

TcT

U(1) breaking

E2 breaking

Two different symmetries are at play: the geometrical E(2) (including translations and rotations) and the electric charge U(1).

Symmetries broken at the two transitons

10

* *1D D exp ,Z x x F x x

T

1. Gaussian fluctuations around the Meissner state

2 22 2 2 4

* *( )

2 2 2z cab cx

F T Tm m

��������������

B. Nontopological excitations. The Ginzburg criterion.

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

11

Ginzburg number

In the new units the Boltzmann factor becomes:

/ ct T T

2 2 20/ ; / / ; / 2cr r z z z

It is convenient to use units of coherence length (which might be different in different directions), in units of and energy scale in units of critical temperature

2 * 22 2

2 * 2 *400

2 4

20

2 2 4

44

11

2 2

1 1 1 1

2 2 2

abzD

ab cD

c

c

D

m

m mFf d x

T T T T

T

td x

��������������

��������������

12

22 2

2 2

1 8

2ce T

Gic

2

400

2 29 2

2 2

2./ / 4

88.10 ;

D

T TGi t

coh vol g

eT A T K

c

Here an important dimensionless parameter characterizing strength of thermal fluctuations is

with the anisotropy parameter /c abm m

Where the temperature independent Ginzburg number characterizing he material was introduced:

13

To calculate the thermal effects via a somewhat bulky functional integral, the simplest method is the saddle point evaluation assuming that 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)

min

min

22

min *| ...

2

fl

fl

ff f

and then expands in “small” fluctuations”

min

0f

Perturbation theory

14

Now we consider the normal phase t>1 in which the saddle point value of the order parameter is trivial:

However the superfluid density

*,1 *D D 0

fx Z x e

The thermal average of the order parameter is still zero due to the U(1) symmetry:

min 0 0 fl flf f

2 *0x x x

due to fluctuations. First we calculate the fluctuation correction to free energy and thereby to specific heat which is impacted the most.

15

*,*D D fl flfffl fle Z e

From now on fl is dropped.

4* 12 4,

21

( ) ( ) ( ) ;2

1( )

2 2

x y xf x G x y y x f f

tG x x

The expansion of the partition function results in gaussian integrals:

2 4 2

22*

4 4D D 1 ...2

f f fZ e e f f

The fluctuation contribution to free energy

16

21 * *2 4

, ,

2 21 2

;2

integer, range from toa a

1

2 2 2

Dk k l m k l m

k k l m

f L G k f f

kL

k t kG k m

It is more convenient to perform the calculation in momentum space

which is presented graphically as “diagrams”:

G x x G x x

2

0

0

1 2

f

Z Z

Z e

17

since the basic gaussian integral in momentum space becomes a product

21 1log*

0/

kk k

G k G k

k kk a

Z d d e e

2 21 3

0 3 3

/ 2 2 2 3 2 3

3 2 2 3 30

3/ 23

2 3 3

22

2 2 2 2

2 2

k

a

k

c cc

T k mF T Log G k d k Log

T k dk k m T m mLog

a a

T T T tT

a a

Therefore the fluctuations contribution to the free energy density is:

The leading order

18

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.

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

2 2

333 2 22 3

21/ 2

2 3

21

2

14

df m d mdFS d k

dT dt k md m

d mt

a dt

Entropy and specific heat

19

33 2 2 33 2 2

1/ 2

2 3

2 11

82

8

dSC T d k

dT mk m

t

The second derivative, the fluctuation contribution to specific heat, is already finite

20

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.

2. Interactions of the excitations and “critical” fluctuations. Ginzburg criterion.

21

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:

2* *

2 20

22 2

1

ikx

k kx k

tG k e x

m t

The leading correction is:

2

0 2

2/

klG k G l G k a m

m

A measure of coherence in the Meissner phase (density of Cooper pairs) is

Correlator and its divergence at criticality

* 2 * 20 min min 0

12 2

2

tx x

22

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.

22

2 2

2 2m

m a m 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?

cT RcT

Renormalized perturbation theory

23

G k ...

RcT

One can try to improve on it by resumming some diagrams

At correlator decays slower:

2 32 2

2 2 2 2

2 2 2

...

1 1

1 / 2 2 / 2

2 2

R

R

G k G k Bub G k Bub

G k

G k Bub k m Bub k m

m m Bub m m

24

To first order in fluctuations criticality occurs at2 20 2 1 /

2

RR c c

Rc c c

m m T T

T T T

cT 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”.R

cT

2R Rc c cT T T

Ginzburg criterion After renormalization we return to the perturbation

theory applicability test:2

22 2

2 21m m t Gi

m m

25

This might be compared to the jump between Meissner and normal phases before mesoscopic fluctuations are taken into account:

2

0

11 1/

4M M

tf g units c

The condition that fluctuations do not become “stronger” than the mean field effect is

1 12fl Mc c m

m

This is known as Ginzburg criterion. Plugging correct constants one obtains:

28cT TGi

T

26

22 4*1 1

[ ]2 2

Df d x a

Re

Im

F

leads to a nontrivial minimum

a v

The negative coefficient of the quadratic term

1

2

ta

3. Fluctuations in the Meissner phase. Goldstone modes.

27

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

min( ) ( ) ( ) ( ) ( )flx x x v O x iA x

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

2 4 2 21 1[ ] [ ( ) ( ) ]

2 O AF av v O e O A e A

-v ( + ) +1/2 ( + 2 + )

Where the “masses” of excitations are

Feynman rules with shift and the Goldstone mode.

28

Goldstone mode, “acoustic”

2 2

2

2

3 2 0

0

( )

O

A

A

e a v v

e a v

e k k

( )A Ae k k

Dispersion relation of the A mode is that of acoustic phonons:

Regular massive mode, “optic”

And in the confuguration space the correlator is:

1 0

log 0

x

x L

x

L

x

in 2D

in 3D

2

1( ) (0) D ik xA x A d k e

k

29

The field A itself is not the order parameter since it does not transforms linearly under the symmetry transformation. The order parameter is ( )i xe

( ) ( / ) /x arctg A O A v Where the phase of is related to the fields by:

( ) (0) ( ) (0)( ) (0)

1

log

1

1 0

x A x Ai x i

x

L

x

e e e e

e

ex

in 2D

in 3D

Its correlator is:

30

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!

( ) (0)

x

i x ie e e

The 2D correlator in the “ordered” phase decays albeit slowly (as a power rather than exponential in the disordered phase

31

4. Destructions caused by IR divergencies in D2 and the MWC theorem.

The energy to the one loop level is:

2 4 2 21 1 1[ ] [ log( ) log( )]

2 2 O AF av v Tr e Tr e

The corrected value of v is found by minimizing it perturbatively in “loops”: 2 2 2v a v Noting that

22 2

1log( )

k

Tr ev k e

32

one obtains a logarithmically divergent correction to VEV:

To higher orders the logs can be resummed:

2 3v

22

1logd k L

k

2 2 2 log 1 (1 log log ...) 0Lv a L L a e

L

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.

33

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)

O(2) invariant quantities

There also is correction due to change in v:

(34corrF

2)

2 32

F

2 3 6a 2 ][ ] [

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?

2 2 23 2 1log log log 0

2 2 4

aL L L

a

2log L

log L

It is hard to say generally, but at least in extreme case of 1D the answer is clear.

35

For D=1 the model is equivalent to QM of particle on a plane with Mexican hat potential

Physics below the lower critical dimension

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)

O

QM ground state

Pert. ground state

A

36

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

* * * 2[ ( ) ]2

D bF d x a

0L

L

37

Its energy in various dimensions is

2DE L

L

L

1

1L

for 1D

2D

3D

no order

order

D=2 is a border case in which there exists “almost long range order”

38

)()( ixx

C. Topological excitations. The dual picture

It is therefore advantageous in such a case to reformulate the theory in terms of vortex degrees of freedom only:

/

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 ):

39

Minimal excitation (Cooper pair) = Smallest vortex ring

min2 ( 0)T L

The Feynman-Onsager excitation and the “spaghetti” vacuum.

The normal phase is reinterpreted as a proliferation od loops:

SCNormal

40

1. In 3D “non ligh Tc” materials is not very large.

2. Vortex loop in 3D is a complicated object: infinity of degrees of freedom (unlike in 2D)

This point of view is not commonly accepted or used with one notable exception: the “BKT” transition in thin films. Reasons:

The dual picture was nevertheless advanced after the discovery of high Tc. Its extentions to include external magnetic field were unsuccessful so far.

41

2. KT transition in thin films or layered SC

deff

22

Pearl (1964)

may be very large .

Interaction between 2D fluxons in logr up to scale

eff

SC - normal phase transition in thin films is of a novel type: the Berezinskii –Kosterlitz-Thouless continuous type (71).

eff

42

Dipole unbinding triggers the proliferation

The basic picture is just 2D “slice” of the fluxon proliferation 3D picture : instead of vortex loop - dipoles.

Dual dipoles: SC

Free “dual charges”: normal

-Kosterlitz, Thouless (72)

43

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 ie

The magnetic flux symmetry

Leads to charge conservation

i i

dJ

dt

Other examples: rotations – angular momentum…

44

Similarly one can interpret the Maxwell equation

;i ii ij j

dB E E E

dt

The symmetry is unbroken in the SC, while spontaneously breaks down (photon is a Golstone boson) in the normal phase

As a conservation law of the magnetic flux, yet another global U(1) symmetry, sometimes calle “inverted” or “dual” U(1) .

The order parameter field was constructed and the GL theory in terms of it was established

-Kovner, Rosenstein, PRL (92)

45

The analogy of the charge U(1) and the magnetic flux U(1) is as follows:

charge vortex

duality

dualityii ij j

duality

Q B

J E E

-Nelson, Halperin, PRB (81)

The dual picture

46

3. A brief history of phase transitions with continuous symmetry in 2D.

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 Landau’s “postulate” –no phase transition.

a. They do not exist

The correlator is a power decay at low temperatures

)(* 1

)()0( Tx

x -Berezinskii(71)

47

b. The high temperature expansion: something happens in between.

lx

ex )()0( *

High temp. expansion gives an exponential

Some qualitative change should happen in between!

This “something” is unbinding of vortices.

48

Number of states:

2 logr

E qa

2

2logR R

Sa a

c. The energy – entropy heuristic argument

-Kosterlitz, Thouless (72)

+

-

Energy of a pair of size r:

49

Where a is the core size. This becomes negative for

22

22

R RF q Log T Log

a a

2

4KT

qT

c. The energy – entropy argument

-

The free energy of a pair therefore is:

which means that the pair proliferate and superconductivity is lost

50

d. Systematic expansions and exact solution.

Kosterlitz (74) and Young (79) developed a heuristic renormalization group (RG) approach to account for differences between pair’s 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)

Recent experiments on BSSCO and other layered

high Tc superconductors found new area of

applications for KT.

Starting with Zamolodehikov’s (80) exact results

were obtained that confirm approximate ones.Nowadays that KT theory is one of the most solid

in theoretical physics.

51

4. The RG theory of the BKT transition

The energy of the KT pair neglecting interactions with other pairs is

20 02

rU r q Log

a

Let us assume that screening can be represented by the dielectric function (r).

2

2 0qq r

r

52

It takes into account the polarization of the pairs due to smaller ones.

The energy therefore gets reduced:

22

0q rq

F rr r r

2

'

'2 '

r

r a

rU r q r Log

a

r

53

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:

1 4r r

To calculate the polarization due consider constant electric field we first write the polarizability of a single dipole:

E

00

4 00

0

1cos |

2

cos1cos exp |

2

TE

E

dp r q r

dE

U r Eq rq r da

n r dE T

r

54

Therefore one gets

'

2 2'30

4 '

1 4 2 ' ' '

41 exp ' /

r

r a

r

r a

r r n r p r

qr U r T

Ta

004

2 2 2 20 0

4

cos1cos exp

2

exp2 2

q r U rq rp r

n r a T T

U rq r q r

Tn r a T T

since the number density of such pairs is

4 expU r

n r aT

55

Thereby we have derived an integral form of the RG eqs. for U and Differentiating it with respect to the pair size one obtains a differential form of the RG eqs.

2 2

304

4exp /

d qr r U r T

dr Ta

with initial conditions

20d q r

U r Logdr r a

1

2

r a

U r a

56

The equations can be turned into a set of autonomous ones by going to a log scale

; lrl Log r a e

a

2 20 0d d dr q q

U l U r rdl dr dl l r l

2 2

404

4exp /

d d dr ql r r U l T

dl dr dl Ta

4 4/ exp[ / ]N l r a U l T

and rescaling the density variable

57

4 4 2 2

/0 04 4

4 4U l Td r r q qN l e N l

dl a a T l T l

2 2

04d ql N l

dl T

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

2 20 04 0

4KT

q qT T

T l l

58

Let us slightly redefine again new variables to make the solution evident

2y l N l

With x small near the transition

20

41

T lx l

q

Exact solution near the KT temperature

22 2

1

dy l y l x l y l

dl x l

24d

x l y ldl

59

It is clear that parabolas are the flow lines

The first equation now can be integrated

2 2 2 2 2 20 00 0C x y x l y l x y

10 0

cothfor

csch

2 cosh /

KT

x l C AC T T

y l C A

A Cl x y

22d

x l x l Cdl

With the result

60

0

0

2

2 for KT

xx l

l x C T T

y l x l

1

0, 1c cl T T x T

The critical value of dielectric constant is finite

10 0

tanfor

sec

2 sin /

x l C AC

y l C A

A Cl x y

61

With criticality of the very weak KT type

Density of bound pairs is

1 1

exp c

KT c

TLog

T T T

U r

0

cl

cn l n lWith the result

Singularities at transition

62

It is clear that parabolas are the flow lines

The first equation now can be integrated

2 2 2 20 0C x y x l y l

10 0

coth

csch

2 cosh /

x l C AC

y l C A

A Cl x y

22d

x l x l Cdl

With the result

63

5. Phenomenology of the BKT transition

The energy of the KT pair neglecting interactions with other pairs is

20 02

rU r q Log

a

Let us assume that screening can be represented by the dielectric function (r).

2

2 0qq r

r