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Breaking Translational Invariance by Population Imbalance 351

requires the calculation of the free energy for general inhomogeneous order

parameters.

3. Properties of the Inhomogeneous States

At suﬃciently low temperatures, the polarized normal state of an unbal-

anced Fermi liquid can become unstable against an inhomogeneous super-

conducting state. The modulation vector minimizing the free energy, how-

ever, is not unique. Usually, there are several equivalent modulation vectors

, with m = 1, . . . , M , in agreement with the point-group symmetry of

the underlying crystal. In the vicinity of the second-order transition, the

corresponding plane-wave order parameters ∼e

·R

are equivalent. In

idealized isotropic systems, all plane-wave states ∼e

iq·R

with the same

wavelength are degenerate at the transition. In the fully developed super-

ﬂuid phase below the transition this degeneracy is lifted by nonlinear terms

in the free-energy as shown by Larkin and Ovchinnikov.

Forming linear

combinations of the equivalent plane-wave states results in an order param-

eter whose amplitude is periodically modulated which results in “crystalline”

superconductivity.

Determining the optimal spatial structure of the periodic order parameter

and predicting the properties of the corresponding superconducting phase is

a problem which has been solved only partially. An adequate microscopic

analysis of the nonuniform FFLO phases requires simultaneous selfconsistent

calculations of the following properties which are intimately intertwined:

(i) the inhomogeneous order parameter structure, (ii) the local excitation

spectrum, and (iii) the quasiparticle selfenergy accounting for the renor-

malizations and Fermi-liquid interactions. The technical diﬃculties are as-

sociated with the fact that often there is a great number of diﬀerent yet

energetically almost degenerate states.

3.1. Mean-ﬁeld results: quasiclassical theory

A suitable framework for a numerical study of the complete phase diagram

of inhomogeneous superconductors is the quasiclassical theory of supercon-

ductivity developed by Eilenberger, Larkin and Ovchinnikov and Eliashberg

(ELOE).

24–26

The quasiclassical theory of superconductivity is valid on the

coarse-grained length and time scales of the superconducting phenomena set

by the coherence length ξ

= ~v

/π∆

and the inverse gap frequency ∆

−1

At the outset, properties determined by the Fermi wavelength k

−1

and the

Fermi energy 

are explicitly eliminated from the theory. We would like

September 14, 2010 15:18 World Scientiﬁc Review Volume - 9.75in x 6.5in ch14

352 G. Zwicknagl and J. Wosnitza

to emphasize that, since the traditional BCS theory is restricted in accu-

racy to terms of order T

/

an advantage of the quasiclassical method is

that it acknowledges this fact and eliminates as many intermediate steps as

possible. For the special case of a superconductor close to the transition

point a quasiclassical description was derived by deGennes.

The quasiclassical theory of ELOE is based on formal many-body pertur-

bation theory, expressed in terms of the thermodynamic (imaginary time)

Green’s function G(x, x

; iω

). Here G is a 4 × 4 Nambu–Gorkov matrix

Green’s function

28,29

which contains the “anomalous” Green’s functions in

the oﬀ-diagonal quadrants. In what follows, only the static limit will be

important, and, consequently, only one frequency, the energy variable, ap-

pears as an argument. In the ﬁrst step, center-of-mass and relative variables

are introduced and a Fourier transformation with respect to the latter is

performed G(R +

r, R −

r; iω

) −→ G(k, R; iω

). The central quan-

tity of the theory is the “quasiclassical” or “ξ-integrated” Green’s function

g(k

, R; iω

) which is derived from the full propagator G(k, R; iω

) by av-

eraging over the kinetic energy of the quasiparticles. The equations deter-

mining g(k

, R; iω

) take the form of transport-like equations which are

supplemented by a normalization condition. For a comprehensive review,

see Ref. 30.

The quasiclassical method was successfully used to study the evolution of

FFLO states characterized by one-dimensional stripe patterns for the order

parameter. For quasi-2D s-wave superconductors, Burkhardt and Rainer

examined selfconsistently the FFLO state including Fermi-liquid interac-

tions. An important result is that the transition from the inhomogeneous

FFLO state to the uniform superconducting phase should be of second order.

The authors showed that both the critical value of the chemical-potential dif-

ference as well as the order of the normal-to-superconducting phase transi-

tions sensitively depend upon the isotropic exchange interaction among the

quasiparticles. Variation with temperature of the order of the normal-to-

superconducting transition was also found for isotropic 3D s-wave supercon-

ductors by Matsuo et al.

and conﬁrmed by Combescot and Mora.

The

full phase diagram for a d

−y

-wave order parameter in an isotropic quasi-

2D superconductor was studied by Vorontsov et al.

In unconventional

superconductors, the orientation of the stripe pattern relative to the nodal

direction of the gap can vary discontinuously with temperature.

23,34–36

The

change in the modulation vector q is found below T

∗

' 0.06T

. Concerning

the identiﬁcation of FFLO phases, Vorontsov et al.

suggest to measure the

Andreev resonance spectrum for which they predict characteristic features.

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Breaking Translational Invariance by Population Imbalance 353

0 0.5 1

0.5

T / T

hexagonal

square

triangular

BCS (uniform)

+ +

)

Fig. 5. Phase diagram of an s-wave superconductor with cylindrical Fermi surface.

The chemical-potential mismatch δµ = gµ

H is due to the Zeeman splitting of

the quasiparticle states in a magnetic ﬁeld applied perpendicular to the axis of

the Fermi-surface cylinder. With decreasing temperature several types of periodic

structures are predicted in the FFLO state.

3.2. Mean-ﬁeld results: Ginzburg

Landau approach

While the solution of the full quasiclassical equations in the full δµ and T

range could be solved only for stripe textures, more insight can be gained in

restricted regions of the phase diagram. In the vicinity of the second-order

transition from the normal to the superconducting phase a (generalized)

Ginzburg–Landau expansion of the free energy in terms of the supercon-

ducting order parameter were derived from the microscopic theory. The

central focus of these studies is “pattern formation” associated with the

higher-order terms contributions. The latter lift the degeneracy of the pri-

mary plane-wave order parameters ∼ e

·R

and induce higher harmonics.

Neglecting the admixture of higher harmonics, Larkin and Ovchinnikov

compared three cases, i.e. the single-vector state discussed in the previous

section, the striped phase, and the simple cubic crystal and concluded that

among these candidates the striped phase has the lowest energy. Shimahara

predicts a series of phase transitions between diﬀerent periodic structures as

shown in Fig. 5.

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354 G. Zwicknagl and J. Wosnitza

Recently, a detailed study was presented by Bowers and Rajagopal

who analyzed 23 diﬀerent crystalline structures. The coeﬃcients of the free-

energy functional derived from the microscopic Gorkov equations

allow for

a determination of the stable structures within weak-coupling theory. The

name of the multiple-wave conﬁgurations refers to the polyhedron inscribed

in a sphere of radius |q| whose vertices correspond to the wave vectors q

Among the structures which are stable within sixth-order expansion the

octahedron gives the lowest free energy. Special attention should be paid to

the cube structure for which the coeﬃcients of the quadratic, quartic and

sextic terms in the GL expansion are negative. In real space, ∆(R) exhibits a

periodic face-centered cubic structure. These candidate states are displayed

schematically in Fig. 6. To actually discuss the free-energy one has to go

to the eighth order. Bowers and Rajagopal argue that this structure might

correspond to a rather low value of the free energy.

-1 0 1

Fig. 6. Unit cell of candidate crystalline FFLO states in isotropic 3D supercon-

ductors. The octahedron state (left panel) gives the lowest energy within sixth-

order Ginzburg–Landau expansion. Qualitative considerations suggest that the

cube structure (right panel) should be the most favorable structure. The contours

correspond to ∆(R)/∆

max

= −0.9, −0.5, −0.3, 0, 0.3, 0.5, 0.9. The lengths of

the cubes are 2π/ |q| and 2π/ |q|

√

3 for the octahedron and the cube structure,

respectively.

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Breaking Translational Invariance by Population Imbalance 355

3.3. Fluctuations

Having discussed the mean-ﬁeld predictions, we next turn to the supercon-

ducting ﬂuctuations. In this context we have to distinguish between the

thermal ﬂuctuations at ﬁnite temperatures T > 0 and the T = 0 quantum

ﬂuctuations. While the former are predominantly classical, the dynamic

nature of the latter is essential.

Thermal ﬂuctuations in superconductors have been studied for a long

time (for a review, see Ref. 38). The role of phase ﬂuctuations of the mod-

ulated order parameter is most prominent in isotropic systems where the

FFLO states are inﬁnitely degenerate. The situation is diﬀerent in the

so-called “generic” case where the degeneracy manifold is reduced to iso-

lated points in momentum space either by anisotropies in the normal-state

quasiparticle properties or the node structure of the superconducting order

parameter.

Quantum ﬂuctuations at T = 0 have received little attention. Samokhin

and Marenko

calculated the corrections to the spin susceptibility and the

quasiparticle-decay rate of fermionic quasiparticles for both the isotropic and

the generic case. The results reﬂect the singular behavior of the mean-ﬁeld

ground-state energy displayed in Fig. 3. Diener et al.

found Fermi-liquid

interaction corrections to thermodynamic properties as a consequence of

quantum ﬂuctuations.

3.4. Microscopic studies

Microscopic model studies beyond the mean-ﬁeld approximation have been

performed for quasi-1D systems. The exact ground states of two-component

Fermi liquids with attractive contact interaction and unequal densities are

known from Bethe ansatz. Three phases appear at T = 0: the paired su-

perﬂuid state, the polarized normal Fermi liquid and an intermediate par-

tially polarized phase which exhibits FFLO-type correlations. These ﬁndings

were recently conﬁrmed by accurate Density Matrix Renormalization Group

(DMRG) and Quantum Monte Carlo (QMC) calculations (see Refs. 41–44

and references therein).

In the weak-coupling limit the phase diagram is well described by mean-

ﬁeld theory.

4. Search for Experimental Realization

Shortly after the theoretical predictions for the possible existence of an

inhomogeneous superconducting state at high magnetic ﬁelds and low

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356 G. Zwicknagl and J. Wosnitza

temperatures, ﬁrst experimental searches started. It took, however, more

than 30 years until ﬁrst claims for the observation of the FFLO state ap-

peared. Yet, those proved to be not well founded. Only in the last decade,

better justiﬁed evidences have been reported for homogeneous materials

(heavy-fermion and organic superconductors) as well as for heterogeneous

materials (artiﬁcial superlattices).

4.1. Homogeneous materials

In real superconducting materials two main conditions have to be fulﬁlled

in order to allow the FFLO state to occur. First, the orbital critical

ﬁeld, H

orb

, should be suﬃciently larger than the Pauli-paramagnetic limit,

= ∆

√

2µ

), where ∆

is the superconducting energy gap at T = 0.

Or more precise, the Maki parameter,

α =

√

orb

, should be larger

than 1.8.

Second, the superconductor needs to be in the clean limit with a

mean-free path ` much larger than the coherence length ξ. These are rather

rigorous conditions that are not satisﬁed by many superconductors. Likely

candidates for a possible realization of the FFLO state are heavy-fermion or

layered superconductors. In heavy-fermion materials the high eﬀective mass,

eﬀ

, results in a small Fermi velocity, v

, which in turn leads to a short co-

herence length (ξ ∝ v

) and, consequently, a large orbital critical ﬁeld. With

` in high-quality heavy-fermion crystals much larger than ξ, the FFLO state

may well be expected. Layered, or quasi-two-dimensional superconductors,

such as some cuprates, many organic superconductors, or artiﬁcial superlat-

tices may be even better suited for exhibiting the FFLO state.

3,6

For such

materials, the orbital pair breaking can be greatly suppressed when applying

the magnetic ﬁeld parallel to the superconducting layers. At the same time,

samples of very high quality are available with suﬃciently large mean-free

paths.

Nevertheless, even up to now only very few condensed-matter realizations

exist that are reported to show indications for an FFLO-like state. First

reports appeared in the 1990s, in which evidence for such an inhomogeneous

state was suggested to exist in the heavy-fermion compounds UPd

CeRu

and UBe

These claims, however, later had to be revised or

are inconclusive. See Refs. 3, 6 and 50 for more details.

In the ﬁrst decade of the 21st century, new experimental work strongly

suggested the existence of an FFLO state in novel superconducting materials.

These are the heavy-fermion compound CeCoIn

51,52

and the quasi-two-

dimensional organic superconductors. Whereas for CeCoIn

some recent

experimental results

cast some doubt on the genuine realization of the

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Breaking Translational Invariance by Population Imbalance 357

FFLO state, the organic materials make a strong case in favor for such a

state. In the following, the experimental status quo for these materials will

be discussed in more detail.

4.1.1. Heavy-fermion systems

A recent prime candidate for the realization of an FFLO state is CeCoIn

It is a heavy-fermion superconductor with T

= 2.3 K and tetragonal crys-

tal structure.

It is a clean type-II superconductor with a large orbital

critical ﬁeld, a suﬃciently large Maki parameter, and an unusual ﬁeld-

temperature phase diagram.

51,52,55

Therefore, all conditions to observe the

FFLO state are fulﬁlled. And indeed, speciﬁc-heat data

51,52

as well as

thermal-conductivity

and penetration-depth measurements using a self-

resonant tank circuit

showed clear evidence for a phase transition within

the superconducting state, as expected for the FLLO state.

The speciﬁc-heat data of Bianchi et al.

for magnetic ﬁelds aligned along

the [110] direction is shown in the left panel of Fig. 7. First of all, above

about 10 T and below about 1 K, the phase transition from the normal

into the superconducting state changes from second to ﬁrst order. This can

be realized by the pronounced sharpening of the anomalies as well as by

the occurrence of hysteresis in the speciﬁc-heat data collected by use of the

relaxation method for rising and falling temperatures. Further to that, a

Fig. 7. (left panel) Speciﬁc heat of CeCoIn

in magnetic ﬁelds aligned parallel to

the [110] direction. A Schottky-like nuclear contribution has been subtracted. The

inset shows the anomalies, T

(originally called T

FFLO

), at 10.6 and 11 T. (right

panel) Contour plot of the electronic part of the speciﬁc heat divided by temperature

for H ||[110]. The grey lines denote T

and T

. After Ref. 51.

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358 G. Zwicknagl and J. Wosnitza

small but clear anomaly is resolved at lower temperatures around 300 mK,

labeled T

(originally labeled T

FFLO

) in the inset of Fig. 7. This anomaly

is most prominent for in-plane ﬁeld orientations, but a smaller anomaly was

found as well for the ﬁeld aligned along the c axis.

The phase diagram over a narrow region in ﬁeld and temperature is shown

in the right panel of Fig. 7 as a contour plot. The data were collected by use of

the relaxation method.

Besides the phase transition from the normal to the

superconducting state at T

, a second phase transition at low temperatures

and high ﬁelds within the superconducting region is clearly evident (T

Such kind of phase diagram, conﬁrmed by other methods as well, is very

suggestive for an FFLO state. There are, however, some features in contrast

to the expected behavior. Especially, there is no saturation of the upper

critical ﬁeld, H

, due to the Pauli limiting observed and there is no upturn

of H

at low temperatures at which the additional phase appears.

In order to get a deﬁnitive proof for the existence of an FFLO state,

the spatial modulation of the superconducting order parameter should be

measured. The method of choice for that purpose is small-angle neutron

scattering. By that, the expected periodic oscillation of the magnetic-ﬁeld

strength along the applied ﬁeld may in principle be detected. Indeed, in 2008

Kenzelmann et al. used high-ﬁeld neutron diﬀraction to search for the mag-

netic response of CeCoIn

in the suggested FFLO phase.

Clear evi-

dence of magnetic Bragg peaks with an incommensurate ordering vector

Q = (q, q, 0.5) was found. This Bragg peaks are not present outside, what

is now called, the Q phase. In the FFLO state, one would expect the

Cooper pairs to carry a momentum that depends on the magnetic ﬁeld via

|q| = 2µ

H/(~v

). In the experiment, however, q = 0.442 was found to be

independent of the magnetic ﬁeld.

This is in stark contradiction to the

usual picture of an FFLO state. Anyway, the ﬁnding that antiferromag-

netism appears only inside the superconducting state of CeCoIn

at high

ﬁelds and low temperatures is remarkable. It strongly suggests that the su-

perconducting condensate carries a ﬁeld-independent momentum. A similar

behavior seems to occur for magnetic ﬁelds applied parallel to the c direc-

tion. Recent µSR data hint to a ﬁeld-induced coupled superconducting and

spin-density-wave order for this ﬁeld orientation.

The ﬁndings of Kenzelmann et al.

have triggered numerous theoretical

eﬀorts trying to explain the nature of the highly unusual Q state. Recently,

Mierzejewski et al. discussed the possibility of a mutual enhancement of

an incommensurate magnetic order and FFLO superconductivity.

Before

that, Yanase and Sigrist suggested that the observed antiferromagnetism

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Breaking Translational Invariance by Population Imbalance 359

might be stabilized by the FFLO state.

On the basis of Bogoliubov–de

Gennes equations this was argued to be caused by the appearance of An-

dreev bound states localized around the zeros of the FFLO order parameter.

In other words, the antiferromagnetic order should occur in regions where

the superconducting order parameter vanishes, as opposed to Ref. 59 where

the two orders mutually enhance each other. Anyway, in both cases the

antiferromagnetic order might occur in the FFLO state even when it is nei-

ther stable in the normal state nor in the BCS state. In agreement with this

prediction, recent nuclear magnetic resonance (NMR) experiments indicated

that the incommensurate magnetic order in CeCoIn

is likely to coexist with

a modulated superconducting phase, that may well be the FFLO state.

The authors further found some indication for a new region below the Q

phase in the phase diagram, which could be an FFLO phase without mag-

netic order. It is evident that further work is necessary to settle all the open

issues in this highly fascinating material.

4.1.2. Organic superconductors

Besides the heavy-fermion materials, the quasi-two-dimensional (2D) organic

superconductors are prime candidates for exhibiting the FFLO state. When

applying the magnetic ﬁeld exactly parallel to the highly conducting planes,

the orbital critical ﬁeld will be greatly enhanced. In a number of diﬀer-

ent experiments, some indications for an FFLO state have been reported.

In most of the experiments, however, no thermodynamic proof was given.

Only recently, clear thermodynamic evidence for a narrow intermediate su-

perconducting state was presented.

One of the ﬁrst reports claiming evidence for an FFLO state in a 2D or-

ganic superconductor was presented for κ-(BEDT-TTF)

Cu(NCS)

, where

BEDT-TTF is bisethylenedithio-tetrathiafulvalene. This conclusion was

based on measurements sensitive to the loss in vortex stiﬀness.

The ob-

served rather broad features in the signal and the derived phase diagram are,

however, largely at odd with the thermodynamic evidences for this material

presented below.

For another 2D organic superconductor, λ-(BETS)

GaCl

, where BETS

is bisethylenedithio-tetraselenafulvalene, thermal-conductivity, κ, data in-

dicate a possible FFLO state.

Tanatar et al. investigated samples of

diﬀerent quality, which was determined by the resistivity ratio or by the

relative thermal-conductivity maximum. For low-quality samples (2 and 5 in

Fig. 8), the rather sharp increase of κ/T at H

might indicate a broadened

ﬁrst-order transition. For high-quality samples (1 and 6 in Fig. 8), this

September 14, 2010 15:18 World Scientiﬁc Review Volume - 9.75in x 6.5in ch14

360 G. Zwicknagl and J. Wosnitza

0.05

0.10

0.15

/T(Wm

-1

-2

) + offset

Magnetic field (T)

(T)

Temperature (K)

Fig. 8. (left) Field dependence of the thermal conductivity of four diﬀerent λ-

(BETS)

GaCl

samples at 0.3 K. The ﬁeld was aligned parallel to the conducting

layers. The curves for sample 6 and 5 are shifted up and down by 0.04 Wm

−1

−2

respectively. (right) Phase diagram for sample 2. Up and down triangles are from

the resistive transitions for parallel and perpendicular ﬁeld orientations, respec-

tively. Open circles are from thermal conductivity. The inset shows the low-

temperature part of the phase diagram for sample 1 (closed circles) and sample

2 (open circles), respectively. The diamonds are from the low-ﬁeld slope change in

κ/T (left arrows in the left panel) for sample 1. After Ref. 64.

feature is replaced by two distinct slope changes that are much farther apart

than in the low-quality samples. The magnetic phase diagram for samples 1

and 2 are shown in the right panel of Fig. 8. For sample 2, a clear satura-

tion of H

is observed below about 1 K, probably due to the Pauli-limiting

eﬀect. For the high-quality sample 1, the H

line continues to grow even

below 1 K (inset of Fig. 8). When interpreting the low-ﬁeld slope change as

another phase transition (diamonds in the inset of Fig. 8), a phase diagram

as expected for a superconductor with an FFLO state evolves. Although

these experimental results give no deﬁnite proof for such an inhomogeneous

state in this organic superconductor, all observed features are in accord with

such a conclusion.

Another isostructural material, λ-(BETS)

FeCl

, where just gallium is

replaced by magnetic iron, has become famous because of its ﬁeld-induced

superconducting state.

65,66

Thereby, after removing a zero-ﬁeld antiferro-

magnetic state by the application of an in-plane magnetic ﬁeld, superconduc-

tivity appears in the ﬁeld range between 17 and 45 T. This behavior is well

understood by the Jaccarino–Peter compensation mechanism.

Thereby,