Unitary Fermi Supersolid: The Larkin-Ovchinnikov Phase

Aurel Bulgac and Michael McNeil Forbes*

Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA(Received 21 April 2008; revised manuscript received 14 August 2008; published 18 November 2008)

We present strong theoretical evidence that a Larkin-Ovchinnikov (LOFF/FFLO) pairing phase is

favored over the homogeneous superfluid and normal phases in three-dimensional unitary Fermi systems.

Using a density functional theory (DFT) based on the latest quantum Monte Carlo calculations and

experimental results, we show that this phase is competitive over a large region of the phase diagram. The

oscillations in the number densities and pairing field have a substantial amplitude, and a period some 3 to

10 times the average interparticle separation. Within the DFT, the transition to a normal polarized Fermi

liquid at large polarizations is smooth, while the transition to a fully paired superfluid is abrupt.

DOI: 10.1103/PhysRevLett.101.215301 PACS numbers: 67.80.K�, 03.75.Hh, 03.75.Ss, 05.30.Fk

The BCS mechanism for fermionic superfluidity isrooted in the notion of pairing: Can superfluidity survivein polarized systems with unequal numbers? This questionremains largely unanswered, even though it is fundamentalto many forms of matter, including superconductors, nu-clear matter, and high density QCD. The prospect of ob-serving exotic polarized superfluids has been revived intwo-component cold-atoms gases with s-wave interac-tions, especially in the unitary limit where the scatteringlength diverges: jaj ! 1. Here, the physics is universal,and inherently strongly coupled, depending solely on thedensities. These systems are experimentally tenable (seeRef. [1] for a review), and exhibit a remarkable diversity ofpolarized phases.

Clogston and Chandrasekhar [2] noted that the normalphase competes with BCS superfluidity when the chemicalpotential difference between the species becomes compa-rable to the energy gap. Kohn and Luttinger [3], however,showed that interactions render Fermi surfaces unstable atsufficiently low temperatures, suggesting pairing of higherangular momenta. This effect is exponentially suppressedin weak coupling, but may be strong enough in unitarygases to support symbiotic p-wave superfluids [4]. Anotherproposal by Fulde and Ferrell (FF) [5], and Larkin andOvchinnikov (LO) [6]—anisotropic or inhomogeneous po-larized superfluids, widely referred to as LOFF or FFLOstates—have also been vigorously sought (see [7] for re-views), but firm results have been sparse: Experimentallythere have been claims of quasi–two-dimensional FFLOstates in heavy-fermion superconductors [8], but no 3Drealizations have been reported. Other proposals includedeformed Fermi surfaces [9] and gapless (breached pair)superfluids [10].

We present here strong evidence that an inhomogeneousLarkin-Ovchinnikov (LO) state [6] may be realized in coldpolarized unitary Fermi gases. Our approach is novel inseveral respects: (1) it is the first calculation to find acompletely self-consistent LO solution in three dimen-sions; (2) the calculation is based on a density functional

theory (DFT) incorporating the best Monte Carlo calcula-tions and measurements of the unitary Fermi gas; and(3) includes both pairing and self-energy correlations.Previous calculations of LO states have not been fully

self-consistent, often relying on approximate forms ofspatial variations, or uncontrolled Ginzburg-Landau ex-pansions (see, e.g., Refs. [7,11,12]). Furthermore, self-consistent treatments are typically based on mean-field orBogoliubov–de Gennes (BdG) calculations, which do notproperly account for many-body effects such as theGorkov–Melik-Barkhudarov corrections [13] that lead tosignificant decreases in the pairing gap. Finally, mostcalculations account for only the pairing condensationenergy, which is exponentially suppressed in weak cou-pling, while the LO state has density variations that cansignificantly change the unsuppressed normal correlationenergy (‘‘Hartree’’ terms). Mean-field and BdG calcula-tions neglect these crucial correlation contributions:Without them, LO states are not competitive at unitarity.According to the theorems of Hohenberg and Kohn, a

DFT exists for any system of fermions. At unitarity, thestructure of the functional is strongly constrained by di-mensional arguments, and thus its determination is greatlysimplified. The remarkable accuracy of this approach forsymmetric systems—as demonstrated in Ref. [14]—givesus the confidence to extend the approach to polarizedsystems. To model the polarized unitary Fermi gas, weuse an asymmetric (ASLDA) generalization of the super-fluid local density approximation (SLDA) employed inRef. [14], expressed in terms of the following densities:

naðrÞ ¼XEn<0

junðrÞj2; nbðrÞ ¼X0<En

jvnðrÞj2;

�aðrÞ ¼XEn<0

jrunðrÞj2; �bðrÞ ¼X0<En

jrvnðrÞj2;

�ðrÞ ¼ 1

2

XEn

sgnðEnÞunðrÞv�nðrÞ;

where unðrÞ, vnðrÞ, and En are the quasiparticle wave

PRL 101, 215301 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending

21 NOVEMBER 2008

0031-9007=08=101(21)=215301(4) 215301-1 � 2008 The American Physical Society

functions and energies [14]. We use the same functionalform as Ref. [14], but allow the parameters to depend onthe local asymmetry xðrÞ ¼ nbðrÞ=naðrÞ. The resultingASLDA energy density EðrÞ has the form

EðrÞ ¼ @2

2m½�aðrÞ�aðrÞ þ �bðrÞ�bðrÞ� þ geffðrÞj�ðrÞj2

þ 3ð3�2Þ2=3@210m

½naðrÞ þ nbðrÞ�5=3�½xðrÞ�; (2)

where �aðrÞ ¼ �½xðrÞ� and �bðrÞ ¼ �½1=xðrÞ� are the in-verse effective masses in units of m�1 and defined in termsof the single function �ðxÞ, �ðxÞ ¼ �ð1=xÞ parametrizes

the normal interaction, and ðna þ nbÞ1=3=� ¼ 1=geff þ�defines the effective coupling geff that is regulated with thecutoff � as described in Refs. [14,15]. The forms of �ðxÞand �ðxÞ are well constrained by Monte Carlo data, asdescribed below and in Ref. [15]. The equations for thequasiparticle wave functions unðrÞ and vnðrÞ follow byminimizing the grand-canonical functional

� ¼ �Z

d3rP ðrÞ ¼Z

d3r½EðrÞ ��anaðrÞ ��bnbðrÞ�;

where �a;b are the chemical potentials corresponding to

the two fermion species, and P ðrÞ is the local pressure.We must now specify the forms of �ðxÞ and �ðxÞ. First

we analyze the symmetric superfluid phase as described inRefs. [14,15]. By matching the Monte Carlo values [16–19] for the parameters � ¼ ESF=EFG ¼ 0:40ð1Þ, ¼�="F ¼ 0:504ð24Þ, and the single quasiparticle excitationspectrum, we determine the inverse effective mass �ð1Þ �1:09ð2Þ, the constant ��1 ¼ �0:091ð8Þ, and the impliedenergy of the symmetric normal phase �N ¼ EN=EFG ¼�ð1Þ þ �ð1Þ ¼ 0:57ð2Þ. In Ref. [14] it was erroneouslystated that �ð1Þ could be extracted from the values of �and alone. A more careful analysis shows that thequasiparticle dispersion [17] must also be fit, resulting inthe modified SLDA parameters presented here. The inversemasses �a;b are also known for the fully-polarized gas. The

majority species is unaffected, whereas the minority spe-cies has the effective ‘‘polaron’’ mass m=m� � 1:04ð3Þ[20,21], constraining the endpoints �ð0Þ ¼ 1 and �ð1Þ ¼0:96ð3Þ. To determine the function �ðxÞ, we consider theenergy of the interacting normal state, setting � � 0. Thishas been well constrained by Monte Carlo calculations[16,20], and, along with the parametrization of �ðxÞ, a fitto this data uniquely specifies the function �ðxÞ (seeFig. 1).

In this Letter, we consider only the simplest LO states,with spatial modulations in a single direction (z). Unlikethe FF state [5], the LO state [6] does not break time-reversal invariance, and thus belongs to a different sym-metry class, as was emphasized by Yoshida and Yip [11].We do not consider FF states here as they are typically notcompetitive with the LO states. (FF states break time-

reversal invariance, and require additional terms in (2) torestore Galilean invariance [14,15].)The self-consistency equations are solved by discretiz-

ing the Hamiltonian along z with a discrete variable rep-resentation (DVR) basis [22] of period L, while integratingover the perpendicular momenta and the Bloch states. Ourquasiparticle wave functions thus satisfy the conditionsunðx; y; zþ LÞ ¼ einunðx; y; zÞ, vnðx; y; zþ LÞ ¼einvnðx; y; zÞ, and are plane waves in the (x, y) plane.We minimize the truncation error due to the finite DVRbasis set by using a smoothed version of the hybrid strategy[23], summing discrete states with energies less than acutoff Ec, while integrating over the remaining higher-energy semiclassical states.We start by specifying chemical potentials �a and �b,

and an ansatz for �ðzÞ / sinð2�z=LÞ containing a node atz ¼ 0, and then use a Broyden iteration scheme [24] to finda self-consistent solution. The choice of basis and iterationpreserve the node at z ¼ 0, converging to either a self-consistent LO state, or degenerating to a homogeneousnormal state with �ðzÞ ¼ 0 everywhere.The resulting self-consistent states depend on the exter-

nal parameter L. To find the physical LO state, we vary L tofind the spontaneously chosen length scale L ¼ LLO thatminimizes the potential � (maximizes the average pres-sure P ). The search is aided by the relationshipL@P=@L ¼ 2E � 3P between L, the average pressure,and the energy density [15], ensuring the unitary relation-ship P ¼ 2

3 E is satisfied by the physical state.

At unitarity, one may fully characterize all stable phasesby the single parameter y ¼ �b=�a as described inRef. [25]. We use the grand-canonical ensemble, where

x = nb/na

g(x

)

0.9

1.0

1.0

1.1

0.0 0.2 0.4 0.6 0.8

FIG. 1 (color online). The dimensionless convex function gðxÞ[25] that defines the energy density Eðna; nbÞ ¼ 3

5@2

2m ð6�2Þ2=3 �½nagðxÞ�5=3. The points with error-bars (blue) are theMonte Carlo data from Ref. [20]. The fully paired solutiongð1Þ ¼ ð2�Þ3=5 is indicated to the bottom right, and the recentMIT data [26] are shown (light �) for comparison. The phaseseparation discussed in Ref. [20] is shown by the Maxwellconstruction (thin black dashed line) of the 1st-order transitiony ¼ yN-SF in Fig. 2. The LO state (thick red curve) has lowerenergy than all pure states and phase separations previouslydiscussed. The Maxwell construction of the weakly 1st-ordertransition y ¼ yLO-SF in Fig. 2 is shown by the thick dashed line(red).

PRL 101, 215301 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending

21 NOVEMBER 2008

215301-2

only pure phases appear, and which properly accounts forthe phase separation that occurs at 1st-order transitions(kinks in hðyÞ, discontinuities in h0ðyÞ: see Fig. 2). We startby describing the homogeneous and isotropic states sup-ported in the ASLDA functional: For y < y0 [25], thesystem is a fully-polarized noninteracting Fermi gas(Na); between y0 < y< yN-SF the highest pressure corre-sponds to a partially polarized two-component Fermi gas;and above yN-SF < y < 1, the fully paired superfluid (SF)has the highest pressure. The point yN-SF, where the pres-sures of the partially polarized normal and fully pairedsuperfluid states are equal, is where the phase separationdiscussed in Ref. [20] would occur. Here, the competitionto LO from the normal and superfluid states is minimized,and the LO state is most likely to occur. For y > 1, thepicture is reversed with the species a $ b exchanged. OurASLDA parametrization does not admit any stable homo-geneous gapless superfluid (breached pair) states [10].

As shown in Fig. 2, we find competitive LO solutions fora large range of the parameter y 2 ðyLO-N; yLO-SFÞ withfinite periods in the range LLO-N � L � LLO-SF. At yLO-N ,the transition appears to be second order, with

maxfj�ðzÞjg ! 0 vanishing smoothly from the LO phaseto the normal phase, while at yLO-SF, the order parameterabruptly looses its spatial oscillations at a finite periodLLO-SF. Because of the presence of a node in �ðzÞ, theonly possibility for a smooth transition here would be forthe period to diverge LLO-SF ! 1; thus, this transitionappears to be weakly first order. The remaining normalstates between y0 < y< yLO-N would be susceptible to theKohn-Luttinger instability, and are candidates for the sym-biotic p-wave superfluids discussed in Ref. [4]. To studythis possibility requires an extension of the ASLDA.Figure 3 shows the typical structure of a LO state. The

pairing amplitude increases smoothly from zero at yLO-N,where the profile is almost sinusoidal, to a critical valueslightly less than �0 at yLO-SF, while the minority compo-nent exhibits large oscillations that break translation in-variance, giving the LO state the crystalline properties of aquantum solid. The majority component exhibits muchsmaller oscillations because the larger local kinetic energydensity suppresses gradients. These fluctuations inducelarge oscillation in the mean-field potentials (not shown),and have a significant impact on the normal correlationenergy. For this reason, all the terms in the energy densityfunctional are critical for a proper description of the LOphase.If no other phases compete, one should observe that the

LO-SF transition coincides with the termination point y1 ofthe partially polarized phases PPa [25]: yLO-SF ¼ y1. Amore complicated crystalline LO state with modulationsin all three directions may further increase the averagepressure, making y1 > yLO-SF. Current errors of the currentMonte Carlo calculations and experiments do not allow usto distinguish between these two cases. These results aresummarized in Fig. 2, where it can be seen that theseperiodic LO solutions occupy a substantial portion of thephase diagram, and lead to a significant increase in the

y = µb/µa

h(y)

y0 y1yN−SF

yLO−N yLO−SF

∆ /∆0

L/l0

1.0

1.1

1.2

1.3

0.0−0.2−0.4

1

0

5

10

FIG. 2 (color online). The dimensionless convex function hðyÞ[25] that defines the average pressure P ð�a;�bÞ ¼ 2

5 ð2m@2 Þ3=2 �½�ahðyÞ�5=2=ð6�2Þ is constrained to the thin dotted triangularregion [25]. The interacting normal state pressure [20] definingour ASLDA functional constrains this further (thin blue line),and displays a 1st-order transition at yN-SF where normal and SFphases could coexists (Maxwell construction in Fig. 1). The LOstate has an even higher pressure (thick red line), replacing muchof this region, including the former yN-SF transition. The ydependence of the amplitude of the pairing field � ¼maxfj�ðzÞjg and the period L are shown inset. Sample profilesfor the states marked � are shown in Fig. 3. Units are fixed interms of �� ¼ ð�a ��bÞ=2 [28].

∆(z

)/∆

0n(

z)/n

0

z/ l 0

-1

1

0.0

0.5

1.0

-5

0

0 5

FIG. 3 (color online). A single LO period showing the spatialdependence of the pairing field �ðzÞ (top) and the numberdensities of the majority (dotted) and minority (solid) species(bottom) at the values of y 2 ðyLO-N; yLO-SFÞ marked by � inFig. 2. Units are fixed in terms of �� [28].

PRL 101, 215301 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending

21 NOVEMBER 2008

215301-3

average pressure. The Legendre transformed results havebeen included in Fig. 1 to facilitate comparisons with theMonte Carlo data [20] and recent experiments [26].

In conclusion, we have shown that the ASLDA providesa valuable tool for quantitatively evaluating inhomogene-ous phases. By incorporating the latest nonperturbativedata about unitary Fermi gases, we have presented strongevidence that a new form of matter, such as a crystallineLO phase, is waiting to be found in the partially polarizedregime of cold unitary Fermi gases. This would be the firstexample of a Fermi supersolid at unitarity, and with thelarge pairing gap, there is a good chance of successfullystudying this state with cold atoms. In the experiments todate, the shells where LO phases may exist are too thin toallow for a complete LO period L. However, traps can beadjusted so that the LO phase will occupy a larger spatialregion, allowing for several LOFF oscillations to occur.Unlike LO in weak coupling, the amplitude of the densityfluctuations in the minority component is large and com-parable to that of vortices [27] in unitary gases. This willprovide the most direct signature of unitary supersolidmatter, and an clean way to study the LOFF phase.

M.M. F. would like to thank R. Sharma for help with thecode, and acknowledges the YITP workshop ‘‘NewFrontiers in QCD’’. The authors would also like to thankS. Giorgini and Y. Shin for providing the data in Fig. 1, andthe US Department of Energy for support under GrantsNo. DE-FG02-97ER41014 and No. DE-FC02-07ER41457.

*mforbes@alum.mit.edu[1] W. Ketterle and M.W. Zwierlein, arXiv:0801.2500.[2] A.M. Clogston, Phys. Rev. Lett. 9, 266 (1962); B. S.

Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962).[3] W. Kohn and J.M. Luttinger, Phys. Rev. Lett. 15, 524

(1965).[4] A. Bulgac, M.M. Forbes, and A. Schwenk, Phys. Rev.

Lett. 97, 020402 (2006); A. Bulgac and S. Yoon (unpub-lished).

[5] P. Fulde and R.A. Ferrell, Phys. Rev. 135, A550 (1964).[6] A. I. Larkin and Y.N. Ovchinnikov, Zh. Eksp. Teor. Fiz.

47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1965)].[7] R. Combescot, arXiv:cond-mat/0702399; R. Casalbuoni

and G. Nardulli, Rev. Mod. Phys. 76, 263 (2004).[8] K. Kakuyanagi et al., Phys. Rev. Lett. 94, 047602 (2005);

A. Bianchi et al., Phys. Rev. Lett. 89, 137002 (2002);H. A. Radovan et al., Nature (London) 425, 51 (2003);C. F. Miclea et al., Phys. Rev. Lett. 96, 117001 (2006); M.Nicklas et al., J. Phys. Soc. Jpn. 76, 128 (2007); B.-L.

Young et al., Phys. Rev. Lett. 98, 036402 (2007);

S. Yonezawa et al., J. Phys. Soc. Jpn. 77, 054712 (2008).[9] H. Muther and A. Sedrakian, Phys. Rev. Lett. 88, 252503

(2002).[10] W.V. Liu and F. Wilczek, Phys. Rev. Lett. 90, 047002

(2003); E. Gubankova, W.V. Liu, and F. Wilczek,

Phys. Rev. Lett. 91, 032001 (2003); M.M. Forbes,

E. Gubankova, W.V. Liu, and F. Wilczek, Phys. Rev.

Lett. 94, 017001 (2005).[11] N. Yoshida and S.-K. Yip, Phys. Rev. A 75, 063601

(2007).[12] S. Matsuo, S. Higashitani, Y. Nagato, and K. Nagai,

J. Phys. Soc. Jpn. 67, 280 (1998).[13] L. Gorkov and T. Melik-Barkhudarov, Sov. Phys. JETP 13,

1018 (1961); H. Heiselberg, C. J. Pethick, H. Smith, and

L. Viverit, Phys. Rev. Lett. 85, 2418 (2000).[14] A. Bulgac, Phys. Rev. A 76, 040502(R) (2007).[15] A. Bulgac and M.M. Forbes, arXiv:0808.1436.[16] J. Carlson, S. Y. Chang, V. R. Pandharipande, and K. E.

Schmidt, Phys. Rev. Lett. 91, 050401 (2003).[17] J. Carlson and S. Reddy, Phys. Rev. Lett. 95, 060401

(2005).[18] A. Bulgac, J. E. Drut, P. Magierski, and G. Wlazlowski,

arXiv:0801.1505; A. Bulgac, J. E. Drut, and P. Magierski,

Phys. Rev. A 78, 023625 (2008).[19] S. Zhang, K. E. Schmidt, and J. Carlson (private commu-

nication).[20] C. Lobo, A. Recati, S. Giorgini, and S. Stringari, Phys.

Rev. Lett. 97, 200403 (2006); R. Combescot, A. Recati,

C. Lobo, and F. Chevy, Phys. Rev. Lett. 98, 180402

(2007); S. Pilati and S. Giorgini, Phys. Rev. Lett. 100,030401 (2008).

[21] N. Prokof’ev and B. Svistunov, Phys. Rev. B 77, 020408(R) (2008).

[22] R. G. Littlejohn, M. Cargo, T. Carrington, Jr., K. A.

Mitchell, and B. Poirier, J. Chem. Phys. 116, 8691 (2002).[23] J. Reidl, A. Csordas, R. Graham, and P. Szepfalusy, Phys.

Rev. A 59, 3816 (1999).[24] A. Baran et al., Phys. Rev. C 78, 014318 (2008).[25] A. Bulgac and M.M. Forbes, Phys. Rev. A 75, 031605(R)

(2007).[26] Y.-I. Shin, Phys. Rev. A 77, 041603 (2008).[27] A. Bulgac and Y. Yu, Phys. Rev. Lett. 91, 190404 (2003);

M.W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H.

Schunck, and W. Ketterle, Nature (London) 435, 1047(2005).

[28] Our units are chosen so that �� is held fixed as is for

trapped systems. Thus, �þ ¼ ��ð1þ yÞ=ð1� yÞ. We

normalize everything in terms of the density n0 ¼najy¼yLO-SF , interparticle spacing l0 ¼ n�1=3jy¼yLO-SF , and

pairing gap �0 of the SF phase at the 1st-order transition

y ¼ yLO-SF where the LO phase ends.

PRL 101, 215301 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending

21 NOVEMBER 2008

215301-4