L.D. Landau ( 1937 )

A second order phase transition is generally well described phenomenologically if one identifies:

a. The order parameter field

b. Symmetry group G and its spontaneous breaking pattern.

General phenomenological approach to second order phase transitions

( )i xφ

TheoryTheory

The order parameter field and spontaneous symmetry breaking

Rest of degrees of freedom are “irrelevant” sufficiently close to the critical temperature Tc. Later, the “relevant” part, namely the symmetry breaking pattern and dimensionality was termed “the universality class”.

0M S≡ >rr

T=0

well ordered

0r

0M S= =rr

An example: XYAn example: XY-- (anti) ferromagnet(anti) ferromagnet

( , )x yM M M≡r

x yM iMφ = +or, using complex numbers,

a. Order parameter : magnetization.a. Order parameter : magnetization.

b . Symmetry : 2D rotations

cos sin 'sin cos '

x x y x

y x y y

M M M MM M M M

χ χ

χ χ

→ + ≡

→− + ≡

( ) ( ')F M F M=r r

Symmetry means that the energy of the rotated state is the same as that of the original (not rotated) one.

ie χφ φ→Using complex numbers the symmetry transformation becomes U(1):

χ

2( )* ( ) * ( * )2

D b TF d x a Tφ φ φ φ φ φ⎡ ⎤= ∇ ∇ + +⎢ ⎥⎣ ⎦∫r r

Higher order terms

Effective free energy near the phase transitionEffective free energy near the phase transition

Most general functional symmetric under

and space rotations, with lowest possible powers ofie χφ φ→

φ

( )2 3* ,( * ) ,..φ φ φ φ∇ ∇are expected to be smaller close enough to Tc.

and lowest number of gradients is

( ) ( ) ...,( ) ...

ca T T Tb T

αβ

= − += +

The remaining coefficients can be expanded around Tc:

F

φ

cT T<

F

φ

cT T>Now we apply this general considerations to the superconductor – normal metal phase transition.

( ) ( )( ) ( ), ( )x x k c k c k↑ ↓Ψ ∝∆ ∆ ∝ −

The complex order parameter is “amplitude of the Cooper pair center of mass”

which is “the gap function” of BCS or any other (no matter how “unconventional”) microscopic theory .

a. Order a. Order parameterparameter

b. Symmetryb. SymmetryThe broken symmetry is charge U(1) mathematically the same symmetry as that of the XY magnet.Without external magnetic field the free energy near transition therefore is:

23 2[ ] * ( ) * ( * )

2 * 2* 2

c

e

F d x T Tm

m m

βα⎡ ⎤

Ψ = ∇Ψ ∇Ψ + − Ψ Ψ + Ψ Ψ⎢ ⎥⎣ ⎦

=

∫r rh

Ginzburg and Landau (1950) postulated a reasonable way to generalize this to the case of arbitrary magnetic field :( )B x

r

One is using the “principle” of local gauge invariance of electrodynamics.

( )( ) ( )* 2

( ) ( ) ( )*

i xx e xe ecA x A x x

e

χ

χ

⎧Ψ → Ψ⎪ =⎨

→ + ∇⎪⎩

r r rh

Electrodynamics is invariant under local gauge transformations:

( )xχ

*eD i Ac

∇ → = ∇ −r r rr

h

Leading to “minimal substitution”:

ScalesScalesξ

characterizes variations of order parameter, while the penetration depth λ characterizes variations of magnetic field

GL equations possess two scales. Coherence length

12

λκξ

= >

Abrikosov (1957)

(the type II superconductivity ) there exist “topologically nontrivial” solutions – the Abrikosov vortices.

Properties of solutions crucially depend on the GL parameter. When

x

ψ

H

Js

ξ

λ

Js

quantum of flux hc/2e

H

λ ξ

Abrikosov vorticesAbrikosov vortices linear topological defect.

coreξ λ

Jr

…… vorticesvortices……

Union City tornado

Spiral galaxy M-100 as seen by Hubble Telescope

Abrikosovvortices can be viewed as magnetic tornados: the core surrounded by encircling electrons

( )*

,ie x tt t

∂ ∂→ − Φ

∂ ∂ h

Time dependent GLTime dependent GL

The friction dominated dynamics is described by the TDGL

( ) ( )2

* , ,2 ,x t F

m t x tγ δψ ψ ψ

δψ∗

∗

∂ ⎡ ⎤= − ⎣ ⎦∂h

The driving force is introduced via (homogeneous far from Hc1) electric field

The vortex dynamics then can be simulated

where is the normal state inverse diffusion constantγ

Magnetic field Order parameter

Creating of vortices and anti-vortices at the hot spot

Disintegration of the magnetic flux at the normal line into vortices at type II SC

Same in type I SC

Near , neglecting fluctuations, Abrikosov found a hexagonal lattice solution

Ginzburg – Landau energy and the Abrikosov lattice solution

2

( 1)( , ) exp 24

1 2exp2

e

l l xx y i la

lya

ϕ π

π

∞

=−∞

⎧ ⎫−⎡ ⎤∝ +⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎡ ⎤⎛ ⎞− −⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

∑

Second order transition at

22 22 23

24

( )2 2

( )

2 8

2cab c

ab c

G d x T Tm m

B H

iec

ψ ψ α ψ

βψ

π

= + −

− + +

⎛ ⎞− +⎜ ⎟⎝ ⎠∫

Ah hh

∇∇

2cH

2

1 ;c c

B Tb t b tH T

+ = = =

0=Ψ

/ ct T T=

1

2 2/ /c cb B H H H≈=

1

normal

lattice0≠Ψ

0=Ψ

/ ct T T=

1

2 2/ /c cb B H H H≈=

1

normal

lattice0≠Ψ

London appr. for infinitely thin linesconstant order parameter

For vortices are well separated and have very thin cores

For vortex cores almost overlap. Instead of lines one just sees array of superconducting “islands”

Lowest Landau level appr. forconstant magnetic induction B

Normal

Mixed

Meissner

H

Hc2

Hc1

Tc T

Two complementary theoretical approaches to the mixed stateTwo complementary theoretical approaches to the mixed state

1cH H>>

2cH H

B

Homoheneity of magnetic induction B is a result of overlap of magnetic fields of roughly

Homogeneity of magnetic fieldHomogeneity of magnetic field

2

2c

BH

κmagnetic fields of individual vorticesMagnetization (although inhomogeneous) is small ( ) and one replaces B(r)=H

22cHκ

−

Ginzburg – Landau energy and a systematic expansion for the Abrikosov lattice solution

22 2

2 2 43*

* ( )2 2 2zxy ccG d x T T

m m

iec

βα= Ψ + Ψ − Ψ + Ψ

⎛ ⎞− +⎜ ⎟

⎝ ⎠∫ A

h h

h∇∇

GL equations (using as unit of length, )

212 2

bH D=− −

2 0hG H aδ ψ ψ ψ ψ

δψ= − + =

ξ ( )2 2 20/ 2ψ = Ψ Ψ

t

1

b

1

normal

lattice

2 ha

2

1, ,2hc c

T B t bt b aT H

− − = = =

Variational solution

One can look for a state of minimal energy on the LLL subspace

The operator H has a discrete spectrum:

1/ 2 2( ) ( / ) exp ( / )2

ikxNk N

br e H b y k b y k bφ ⎡ ⎤⎡ ⎤∝ − − −⎣ ⎦ ⎢ ⎥⎣ ⎦

NE Nb=

0,k N kkcψ φ == ∫

Symmetry leads to the hexagonal lattice superposition

( )20 2( ) exp 2 le ar C i l rπ

πϕ φ∞

∆=−∞

⎡ ⎤= ⎢ ⎥⎣ ⎦∑

-4 -2 0 2 4

-4

-2

0

2

4

Normalization is fixed by the minimization of GL energy:

It is not clear how to extend this variational method to dynamicswith dissipation, described by TDGL,

since there generally is no obvious functional to be minimized. One also would like to estimate how good is the approximation.

1/ 24

0 1/ 2 ;h

AA

aC β ϕβ

= =

( )2

* ' ,2 xt xti G

m tδγ γ ψ ψ ψ

δψ∗

∗

∂ ⎡ ⎤+ = − ⎣ ⎦∂h

The generalization can be achieved, if the variational result is presented as the leading approximant in a perturbation expansion.

Lascher. PR A140, 523 (65)

( )1/2 20 1 2 ..h h ha a aψ ψ ψ ψ= + + +The leading ( ) order equation gives the lowest LLL restriction, already motivated in the variational approach:

0 0 00H Cψ ψ ϕ ∆= ⇒ =

equation is:

1/ 2ha

3/ 2hawith normalization undetermined. The next to leading order, ,

22 *1 0 0 0 0H C C Cψ ϕ ϕ ϕ− + =

haPerturbation theory in

Multiplying it with and integrating one obtains *ϕ2* 2 *

1 0 0 0 0ce ll

H C C Cϕ ψ ϕ ϕ ϕ⎡ ⎤− + =⎣ ⎦∫1 / 2

0 AC β−=2 401 0C ϕ− + = ⇒∫

Multiplying the GL equation by ,and integrating over the unit cell, then using orthonormality relations one obtains:

* , 0N Nϕ >

Higher order correction will include the higher Landau level contributions

Corrections

1 10

NN

N

Cψ ϕ∞

=

= ∑

( )2* *1 1 0 03/2

201

1NN N

NN

period

H N bC C C

CCN b

ϕ ψ ϕ ϕ ϕ

ϕ ϕ ϕ∗

= = +

=

∫ ∫

The LLL component is found from the order etc.5/ 2

ha

All the corrections are very small numerically partially due to factors of 1/N with multiples of 6 contributing due to the hexagonal symmetry

(6) (12)1 1

0.279 0.025;6 12

C Cb b

= − =

leading next to leading

0+6th LL 0+6+12 LL

2/ 0.5 / 0.1 0.2c c hT T B H a= = =

haTherefore the perturbation theory in is useful up to surprisingly

LLL is by far the leading contribution above this line.

low fields and temperatures, roughly above the line

( )1 113

b t= −

Li, B.R. PRB60, 9704 (99)

2 3 4

2 3 2

0.044 0.0562 6 6

h h h

A

a a afb bβ

= − − +

When all the vortices are pinned there might be current without dissipation. The order parameter configuration cannot belong to LLL, since for general LLL configuration

Persistent current in magnetic fieldPersistent current in magnetic field

( )22i ij jeJm

ε∗

∗= ∂ Ψh

-4 -2 0 2 4

-4

-2

0

2

4

whole current LLL subtracted-4 -2 0 2 4

-4

-2

0

2

4

0 10.02ψ ϕ ϕ= +

Affleck, Brezin. NPB257, 451

(85)

Field driven flux motion probed by STM on NbSe2

FluxonsFluxons are light and can move. The motion is generally friction are light and can move. The motion is generally friction dominated with energy dissipated in vortex cores. The current dominated with energy dissipated in vortex cores. The current ““inducesinduces”” flux flow, causing voltage via flux flow, causing voltage via ““phase slipsphase slips””..

Electric field is present and, due to Electric field is present and, due to superposition between vortices is also superposition between vortices is also homogeneous in sufficiently dense homogeneous in sufficiently dense vortex matter vortex matter

Troyanovsky et al (04)

B

J

F

Flux flow

Hu, Thompson, PRL27, 1352 (75)

( )*

tieD x

t∂

= − Φ∂ h

Time dependent GL in the presence of electric fieldTime dependent GL in the presence of electric field

The friction dominated dynamics is described by the TDGL2

* ,2 t xt xtD F

mγ δψ ψ ψ

δψ∗

∗ ⎡ ⎤= − ⎣ ⎦h

The driving force is introduced via cov. derivative (unit of time )

where is the normal state inverse diffusion constant/ 2γ

$

$

2

2

2

| | 0

2

h

t

L a

L D Hb

ψ ψ ψ ψ− + =

Ε= + +

( )2 21 1 /2ha t b b= − − − − Ε

2GLt γξ=

*

4 GLGL GL

EEe t Eξ

= Ε=h

is not Hermitean.

Bifurcation point perturbation theoryBifurcation point perturbation theory

The perturbation in can be applied again. One first looks for eigenfunctions of the linear part of the equation$

Np Np NpL ω ω ωφ φ= Θ

( ) ( )2( ) 1/ 2e / exp /2

i kx tNp N

bH b y k b iv y k b ivωωφ− ⎡ ⎤⎡ ⎤= − + − − +⎣ ⎦ ⎢ ⎥⎣ ⎦

The right eigenfunctions are:

( )Np Nb i vkω ωΘ = + −

Note the “wave” exponential despite absence of Galileo invariance (due to microscopic disorder tied to the rest frame)

ha

Electric field therefore is an additional pair breaker. The critical line beyond which just a trivial normal solution exists is

2 21 / 0t b b− − −Ε = ⇒ ( ) ( )2 22 2, 1 /c cH T E H t b= − − Ε

3/ 2v b−= Ε

Hu, Thompson, PRL27, 1352 (75)

', ,( , , ) ( , , ) ( ') ( )N k NNNkx y t x y t x y t k kωωφ φ πδ δ δ ω ω′ ′ ′ ′= − −∫

The orthonormality relations take a form:

Within the bifurcation method one uses scalar products. In the present case these should be formed with the left eigenfunctions:

( )2( ) 1/ 2 *e ( / ) exp /2

i kx tNk N Nk

bH b y k b iv y k b ivωω ωφ φ− − ⎡ ⎤⎡ ⎤= − + − − + ≠ ⎣ ⎦ ⎢ ⎥⎣ ⎦

$0 0Lψ =

Assuming, as in statics, an expansion , ( )1/2 0 1 ..h ha aψ ψ ψ= + +

0,N vkω= =

the leading, , order equation is the LLL constraint

implying

1/ 2ha

For each moving lattice symmetry one gets normalization from the order:

( ) ( )1 / 2 * 20 ;A AC v vβ β ϕϕ ϕ−= =3/2ha

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

dissipation superfluid density

( )2 212i j ieJ v

m

∗

∗⎡ ⎤= ∂ Ψ + Ψ⎢ ⎥⎣ ⎦

hLLL supercurrent density:

22

*2 tp E J D

mγ ψ= ⋅ = h

Shape of the moving lattice, IShape of the moving lattice, I--VV

As in statics, one should maximally use the symmetries available, but now there are less due to electric field.

Flux flow

Generally there are the following options:

Conductivity also has small non – Ohmic corrections due to HLL

Li, Malkin, B.R. PRB70, 214528

(04)

Thermal fluctuations are taken into account via statistical sum

Thermal fluctuations on the mesoscopic level in Thermal fluctuations on the mesoscopic level in the flux latticethe flux lattice

2 2 43 1 12 2z T

G d x aT

ψ ψ ψ⎡ ⎤= ∂ − +⎢ ⎥⎣ ⎦∫

[ ]/( ) ( ) G TZ D x D x e ψψ ψ∗ −= ∫

With only one dimensionless parameter: the LLL scaled

temperature ( )2/3

1T

t bat b Gi

− −≡ −

For not very small fields one can consider effective lowest Landau level (LLL) only in which case the energy simplifies

There are two types of the fluctuation modes in expansion aroundthe Abrikosov solution:

Supersoft phonons in vortex solid near Hc2

. ,

( , , ) ( , ) ( , ) ( )zz

ik zk k k

k B Z k

x y z x y x y e O iAψ ϕ ϕ∈

= + +∫

*

22 4

*( ) ( )

( ) 1 2 .12 | |

m f O k k A k kk

k kA z T

A AzT

F F e k O O e k A A AAA AAAA

e k a k ka kβ γβ β

= + + + +

⎛ ⎞= − + + ≈ +⎜ ⎟

⎝ ⎠

∫

Substituting this into energy and diagonalizing quadratic part of free energy one obtains:

Eilenberger, PR 164, 622 (1967); Maki&Takayama, PTP 46, 1651 (1971)

Naively higher order contributions to energy are hopelessly divergent:

IR divergences

2log L→ 4L→

Is there a thermodynamic solid state for T>0 or experimentally observed vortex lattice is just a finite size effect?

A question of principle

Since experimentally the corrections are small one can speculatethat it is not analytic. The perturbation theory was abandoned. Is this correct?

( )2 2 22 2 2. , .1 1 log

z x yk B Z k k B Zx y z

Lk kk k k∈ ∈

= =++ +

∫ ∫

Answers

21/ 2 2.42.848

2T

sol TA T

af aaβ

= − + +

Even at melt ( ) the precision is 0.1%. From this one calculates magnetization, specific heat …

B.R. PRB 60,4268 (1999)

1. All the IR divergences exactly cancel like “spurious IR divergences” in the theory of critical phenomena. There exist a stable crystalline state

2. For the free energy, which is invariant under translation, one gets to the two loop order:

9.5mTa = −

spinodal

Metastable solid state becomes unstable at a spinodal point :

Appears not far from the melting line in weakly fluctuating superconductors like NbSe2

5.5spinodalTa = −

Xiao et al PRL92, 227004 (04)

Metastable solid and spinodal point

Thakur et al

PRB72, 134524

(05)

To get to negative one can perform “bubble resummation” also called Hartree – Fock, gaussian, renormalized…

Vortex liquidPerturbation theory at high temperatures and the

Gaussian resummation (mean field)

Standard high temperature perturbation theory is defined only only for with the excitation energy and is therefore useless for the experimentally interesting region around melting.

0Ta >

Ta

Ta

The gap equation for the excitation energy3/ 2 1/ 2 4 0Te a e− − =

Thouless, PRL 34, 946 (1975)

has a solution for any temperature.

It shows that overcooled liquid is metastable all the way down toT=0 with excitation energy vanishing as

Brezin et al, PRL65,1949 (1990)

4 / Te a≈ −Perturbations around Gaussian state were pushed to the 9th order. Unfortunately the series are asymptotic and can be used only for 2Ta > −

We constructed the optimized gaussian series, which are convergent rather than asymptotic

Beyond the renormalized perturbation theory

However it allowed us to verify unambiguously the validity of the Borel-Pade method which provided a convergent scheme everywhere down to T=0. It uses the fact that for repelling objects there exists a pseudo-critical fixed point at T=0.

DP Li, B.R. PRL86, 3618 (01)

Precision of the BP is finally good enough (0.1%) to study melting quantitatively

4.5Ta = −Radius of convergence was found to be still a bit short of melting.

Lortz et al PRB74, 227004 (06)

5.0 5.5 6.0 6.5 7.0

-40

-20

0

20

40

60

13.0 13.5 14.0 14.5 15.0 15.5

45

50

55

60

65

data 6 T data 6 T with 'vortex shaking' LLL vortex liquid LLL supercooled liquid LLL vortex lattice LLL overheated lattice

T(K)

C/T

(mJ

gat-1

K -2) Tclean

m

T cleansp

TmT=14 K

M (m

J ga

t-1T

-1)

H (T)

The melting point:

Melting line, discontinuities at melting

The magnetization jump:

Specific heat jump:

1.8 %sol

MM∆

=

22 20.75 %mf

C b tC t∆ − +⎛ ⎞= ⎜ ⎟

⎝ ⎠

9.5mTa = −

Spinodal point

Nishizaki et al, Physica C341,957 (00)

Welp et al, PRL76,4809 (1996)

Willemin et al, PRL81,4236 (1998)

fully oxidized YBCO

Effects of the quenched disorder

In the framework of GL a point like disorder leads to a local random modification of coefficients in the free energy. For example the “ “ disorder is described by a random field

with certain phenomenologically determined variance

2 ( ) ( ) ( )ab cx yV V n x yξ ξ δ= −

To find the influence of the disorder on the melting line in some cases perturbation theory in n is sufficient, but to find irreversibility line, study dynamics nonperturbative methods like the replica method or the dynamical MSR are needed.

( )1 V xα α→ +⎡ ⎤⎣ ⎦

cT∆

Disorder and thermal fluctuations in dynamics Disorder and thermal fluctuations in dynamics

The thermal fluctuations are introduced into the time dependent GL by the Langeven thermal noise for each degree of freedom

( ) ( ) ( )

( ) ( ) ( ) ( )

2

* , , , ,2 ,

, ', ' 2 ' '

x t F V x tm t x t

x t x t T x x t t

γ δψ ψ ψ ζδψ

ζ ζ γδ δ

∗∗

∂ ⎡ ⎤= − + ⎣ ⎦∂

= − −

h

The LLL approximation can also be developed in this case. Gaussian approximation allows calculation of the correlator and the response function in both ergodic and non-ergodic phases and subsequently the calculation of current.

Direct simulation becomes impractical and refined field theoretical methods of disorder averaging are necessary. The generalization of the Martin – Siggia – Rose method to GL is straightforward, but complicated.

Replica method applied to the GL model with disorder show that below the following line there is continuous RSB.

ResultsResultsThe irreversibility line and disorder effect on the

crystalline – homogeneous transition

Liquid gains more than solid from pinning. When disorder (and thermal fluctuations) is strong it creates a Kauzmann point in which the entropy jump vanishes.

( )214 2

nr t bGitπ

= − −

Shibauchi et al, PRB57, R5622(1998)

2 2- ( - ) [ ( ) ] kappa BEDT TTF Cu N CN BrmH

( ) ( )2 3 2, 2 3 ;gTa t b r r/ ⎛ ⎞= −⎜ ⎟

⎝ ⎠

Dynamical (bulk) irreversibility surface

Critical current in the homogeneous pinned state

( ),cJ T B

consistent with the replica results for static irreversibility line

0*

0 1/ 2

/

16(2 )

c

ab c

j J J

e TJGi ξ ξ

=

=h

Critical current

( ) ( ) ( )4 32 / 3

4 / 3 1/ 332 , ,bt gGi

c T Trj a t b a t b

π

/

⎡ ⎤= −⎣ ⎦

Dependence of the critical current in the homogeneous state on magnetic field, is different from the monotonic dependence in the crystalline state. As a result the stable state current jumps upon crossing the ODO line.

NbSe2 Kokubo et al PRL95,167004

(05)

Peak effect

ODO glass

Hc2

Banerjee et al Physica C355,39 (01)

The LLL part is dissipative, but does not have the persistent current part which appears as a correction due to first LL

I I –– V curvesV curves

( )

( ) ( )

0

1/32

32

1/21 35 6 2 5 1 6

2 13

RLLL

d

t Gijb

j b t rGi

π

π

Ε

// /

⎛ ⎞= Ε⎜ ⎟

⎝ ⎠

⎛ ⎞−=⎜ ⎟

⎝ ⎠

( ) ( )

( ) ( ) ( )3 20 0

2 / 32 2

4 1 1 0

1 /

T

T

r R a R

a t b G i b t b−

− − − Ε + =

Ε = − − − − Ε

Where the response function is solution of

0 *0

16 c LLL d nT EE j j j j

e Eξ= Ε = = + +

LaSCO

Divakar et al, PRL92,237004 (04)

Vortex matter phase diagram of a HTSC

Hg Vortex

liquid A.lattice

Bragg glass

vortex glass

ODO

ODO

(melting)M

agne

tic fi

eld

Temperature

Tc

BSCCO

Beidenkopf et al, PRL95,257006 (05)

Li, B.R, V.VinokurJS&M90,167 (07)