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

BREAKING TRANSLATIONAL INVARIANCE

BY POPULATION IMBALANCE:

THE FULDE

FERRELL LARKIN OVCHINNIKOV

STATES

Gertrud Zwicknagl

Institut f¨ur Mathematische Physik,

Technische Universit¨at Braunschweig, 38106 Braunschweig, Germany

Jochen Wosnitza

Hochfeld-Magnetlabor Dresden,

Forschungszentrum Dresden-Rossendorf, 01314 Dresden, Germany

An overview is given of our present understanding of superconductivity

with spontaneously broken translation symmetry in polarized Fermi sys-

tems. The existence of “crystalline” superconducting phases is considered

in a wide range of systems, prominent examples being conduction electrons

in metals, ultra-cold atoms in a trap, nuclear matter and dense quark sys-

tems. The underlying physics is delineated and theoretical approaches to

the inhomogeneous phases and their properties are discussed. From the ex-

perimental side, it is argued that superconductivity with imbalance-induced

order parameters is realized in layered organic compounds and potentially

in heavy-fermion systems.

1. Introduction

Superﬂuid phases with unequal numbers of particles forming Cooper pairs

have been the focus of interest for many decades. Of particular interest has

been the conjecture by Fulde and Ferrell (FF)

and Larkin and Ovchinnikov

(LO)

that at low temperatures, superﬂuid states with spontaneously broken

translational symmetry should form. These FFLO states have been under

theoretical investigation for a variety of systems including metals,

3,4

ultra-

cold atoms,

nuclear matter and dense quark systems.

In the present review, we shall restrict ourselves to non-relativistic

two-component Fermi systems, well-known examples being the conduction

electrons in metals. In an imbalanced Fermi liquid there will be a majority

337

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338 G. Zwicknagl and J. Wosnitza

and a minority component which we shall denote as ↑ and ↓, respectively.

The system is characterized by the corresponding densities n

↑

≥ n

↓

. For

independent particles with no interconversion, the densities of the majority

and minority degrees of freedom, n

↑

and n

↓

, can alternatively be speciﬁed

by chemical potentials µ

↑

and µ

↓

. The latter are expressed by the average

value and the diﬀerence

↑

= µ + δµ ; µ

↓

= µ − δµ . (1)

In spin-1/2-fermion systems, such as the conduction electrons in metals, the

diﬀerence in the chemical potentials, δµ, is tuned by an external magnetic

ﬁeld. In trapped atomic systems, on the other hand, imbalance can be

generated by preparing mixtures with diﬀerent numbers of partners and the

natural variable is the polarization, i.e. the diﬀerence in the corresponding

particle numbers.

Due to the formation of Cooper pairs the susceptibility of the supercon-

ductor is reduced as compared to the normal state. For high diﬀerences in

the chemical potentials the polarized normal state has a lower energy than

the superconducting phase. Qualitatively, the transition from the paired

superconducting to the polarized normal phase will occur when the polar-

ization energy equals the condensation energy of the Cooper pairs. This

condition deﬁnes the limiting chemical-potential mismatch, δµ

(T ), beyond

which homogeneous superconductivity will not exist. The low-temperature

limit δµ

(0) for a conventional superconductor was ﬁrst given by Chan-

drasekhar

and Clogston

δµ

(0) =

∆

√

where ∆

is the energy gap of the uniformly balanced reference system. The

δµ-T phase diagram for homogeneous superconductors was studied in de-

tail by Sarma.

9,10

In addition to the normal phase and the homogeneous

superconductor, he considered a compromise state which embodies “phase

separation in momentum space”. The breached-pair (BP) states

are char-

acterized by the coexistence of paired superﬂuid and normal polarized com-

ponents which reside in diﬀerent regions of momentum space. Unfortunately,

these gapless states turn out to be unstable in the “classical” situation of a

superconductor in an external magnetic ﬁeld acting on the spins of the elec-

trons. The interest in BP states, however, has been recently rekindled since

they could be realized in dense quark systems or ultra-cold atoms. FFLO

predicted a spatially inhomogeneous partially gapped compromise state with

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

Breaking Translational Invariance by Population Imbalance 339

both paired superﬂuid and normal polarized regions to be the stable ground

state in a (narrow) range above the Chandrasekhar–Clogston limit.

The idea of forming a textured state with paired and normal regions

closely parallels the concept of the vortex phase which appears in type-II

superconductors below the upper critical ﬁeld. This phase gains its stability

from the fact that it allows magnetic ﬂux to penetrate into the supercon-

ductor. Like the FFLO state, it was predicted theoretically by Abrikosov.

Shortly afterwards, it was conﬁrmed by numerous experiments employing a

variety of techniques. Although the existence of inhomogeneous compromise

states in imbalanced superﬂuids seems plausible and despite intensive search,

these phases have not yet been identiﬁed unambiguously. There are, how-

ever, indications that they are realized in some highly anisotropic materials,

as will be discussed below.

There are several reasons for the diﬃculties in the experimental observa-

tion. First, the FFLO state requires very stringent conditions which strongly

restrict the appropriate classes of superconducting materials. Soon after the

FFLO phases were conjectured, it became clear that impurities act as pair

breakers suppressing the corresponding transition temperatures.

A nec-

essary criterion for the realization of an FFLO state is a long mean-free

path exceeding the superconducting coherence length. An additional tech-

nical diﬃculty encountered in metals is related to the fact that electrons are

charged particles. An external magnetic ﬁeld required to create the imbal-

ance in the conduction electron system will inevitably act on their orbits and

suppress superconductivity. To observe imbalance-induced high-ﬁeld phases

the orbital pair breaking must be strongly reduced as pointed out by Maki.

The early search for the FFLO phases concentrated on thin superconducting

ﬁlms in parallel magnetic ﬁelds.

Although polarization eﬀects were clearly

seen, it was not possible to stabilize the FFLO state. Promising candidates

are recently discovered heavy-fermion compounds and the layered organic

superconductors. Second, there are no simple criteria which could help to

identify the phases. The order parameter as such is a quantum mechanical

object and hence cannot be observed directly. The generally accepted proce-

dure in this situation is to calculate observable properties which depend on

the order parameter and hence reﬂect its structure. It turns out, however,

that the properties of FFLO states are non-universal depending sensitively

on the normal state, i.e. the dispersion of the quasiparticles and their inter-

actions. The latter strongly aﬀect the transition line separating the normal

and the superﬂuid phases as well as the order of the phase transitions. In ad-

dition, the anisotropy of the order parameter in the homogeneously balanced

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

340 G. Zwicknagl and J. Wosnitza

phase which is related to the presence or absence of low-energy excitations

plays an important role.

Considering these complications, the present article does not attempt

at providing a ﬁnal answer to the existence and potential realization of

imbalanced-induced spontaneous breaking of translational symmetry. The

central goal is to summarize basic concepts and discuss the present state-of-

the-art emphasizing what has been solved and what should be solved.

The article is organized as follows: in Sec. 2, we brieﬂy review the ground-

state properties of a superconductor with population imbalance. The prop-

erties of inhomogeneous phases are summarized in Sec. 3. In Sec. 4, we com-

ment on the experimental situation with an emphasis on condensed-matter

systems. The article closes with a summary and an outlook in Sec. 5.

2. Theoretical Background

2.1. Order parameter

The highly correlated superﬂuid phases in a multi-component Fermi liq-

uid are characterized by long-range order in the two-particle density matrix

which allows us to deﬁne order parameters

(x, x

) = hψ

(x)ψ

)i. (2)

The indices A and B denote (pseudo) spins in the case of electron systems or

He, for ultra-cold atoms they refer to hyperﬁne states, whereas they include

color, spin, and ﬂavor indices in quark condensates.

In the two-component systems to be considered here, the pair ampli-

tudes Eq. (2) depend upon the fermion spins and positions which are con-

veniently expressed in terms of the center-of-mass and relative coordinates

R =

(x + x

) and r = x−x

, Ψ

σσ



R +

r, R −



. We focus on the BCS

regime where the characteristic time and length scales in superconducting

phenomena, set by the inverse gap function ∆

−1

and the coherence length

π∆

, respectively, are very large on the atomic scale determined by the

inverse Fermi energy E

and Fermi wavelength λ

. This separation of length

and time scales allows us to classify the order parameter of inhomogeneous

systems in close analogy to their homogeneous counterparts.

In homogeneous superconductors, the order parameter is a function of

the relative variable only. It varies on the scale of the Fermi wavelength

which follows from the short-ranged character of the attractive quasiparticle

interaction. Performing a Fourier transformation with respect to r with

σσ

(r) → Ψ

σσ

(k) and restricting the wave vector to the Fermi surface k