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Stable Sarma State in Two-band Fermi Systems Lianyi He and Pengfei Zhuang Physics Department, Tsinghua University, Beijing 100084, China We investigate fermionic superconductivity with mismatched Fermi surfaces in a general two- band system. The exchange interaction between the two bands changes significantly the stability structure of the pairing states. The Sarma state with two gapless Fermi surfaces which is always unstable in single-band systems, can be the stable ground state in two-band systems. To realize a 9 visible mismatch window for the stable Sarma state, two conditions should be satisfied: a nonzero 0 inter-bandexchange interaction and a large asymmetry between the two bands. 0 2 PACSnumbers: 74.20.-z,03.75.Kk,05.30.Fk n a I. INTRODUCTION ity. Forbes et.al.16 proposed that, a stable Sarma state J is possibleina modelwith finite rangeinteractionwhere 4 the momentum dependence of the pairing gap cures the 2 The Cooper pairing with mismatched Fermi surfaces, instability. Ontheotherhand,whentheattractiveinter- whichhasbeeninvestigatedmanyyearsago1,2,promoted ] action becomes strong enough which can be realized in n newinterestinthestudyofnewsuperconductingmateri- ultracoldfermionexperiments,thestabilityoftheSarma o alsinstrongmagneticfieldandultracoldfermionsdueto state can be changed. While the homogeneous Sarma c the realization of superfluidity in resonantly interacting - state is always unstable at the BCS side of the BCS- Fermigases. Thewell-knowntheoreticalresultfors-wave r BEC crossover, it becomes stable in the deep BEC re- p weak coupling superconductors is that, at a critical mis- gion9. However,this stable Sarma state at the BEC side u match, called Chandrasekhar-Clogston limit (CC limit) s is not the original “interior gap” or “breached pairing” h = 0.707∆ where ∆ is the zero temperature gap, a . c 0 0 state with two gapless Fermi surfaces proposed by Liu t a first order phase transition from the gapped BCS state and Wilczek15. Since the fermion chemical potential be- m to the normal state occurs3. Further theoretical stud- comes negative in the BEC region, the Sarma state in iesshowedthattheinhomogeneousFulde-Ferrell-Larkin- - this case possesses only one gapless Fermi surface, and d Ovchinnikov(FFLO)state2maysurviveinanarrowwin- the matter behaves like a Bose-Fermi mixture17. n dow between h and h = 0.754∆ . However, since c FFLO 0 o In this paper, we focus on how the multi-band struc- the thermodynamic critical field is much smaller than c the CC limit due to strong orbit effect3, it is hard to turewhichmaybe realizedinsolidmaterialsandoptical [ observe the CC limit and the FFLO state in ordinary latticeschangesthestabilityoftheSarmastate. Wecon- 3 superconductors. In recentyears,some experimental ev- siderageneraltwo-bandFermisystem,andshowthatthe v inter-band exchange interaction can cure the Sarma in- idence for the FFLO state in heavy fermion supercon- 9 ductors4, hightemperaturesupercondutors5 andorganic stability and the Sarma state can be the stable ground 9 6 superconductors6werefound. Morerecently,inthestudy state in visible parameter regions. 4 of ultracold atoms, Fermi superfluidity with population The multi-band theory of BCS superconductivity was . imbalance was realized by MIT and Rice groups inde- firstlyintroducedby Suhletal.18 in1959to describe the 5 pendently7. The ultracold fermion experiments has pro- possible multiple band crossings at the Fermi surface. 0 8 motedalotoftheoreticalworks8,9,10,11 onthesuperfluid- The two-band model has been applied to the study of 0 ity mechanism and the phase diagrams for the crossover high-T superconductors19toeffectivelydescribethepar- c : from Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein ticular crystalline and electronic structure. Recently, it v i Condensation(BEC)12. Theproblemofimbalancedpair- isfoundthat, the materialMgB2 isa standardtwo-band X ingisalsorelatedtothestudyofcolorsuperconductivity superconductor20 and many experimental data can be r and pion superfluidity in dense quark matter13,14. explained by the two-band model of BCS superconduc- a tivity. Multi-bandFermisystemsmayberealizedexperi- While most of the theoretical works focus on the in- mentallywithultracoldatomsinopticallattice21. Forex- homogeneous FFLO state, we in this paper are inter- ested in the homogeneousand gaplessSarma state1. For ample, if we confine the cold atoms in a one dimensional periodic external potential, the band structure will form weak coupling superconductors, the Sarma state is lo- intheconfineddirection,andthemattercanberegarded cated at the maximum of the thermodynamic potential as a multi-band system in two-dimensions. In this case, ofthesystem,andthereforecannotbethestableground state. ThiswascalledSarmainstabilitymanyyearsago1. by adjusting the coupling strength, one can study the possible BCS-BEC crossoverin multi-band systems19,22. The thermodynamic instability of the Sarma state can Theinter-bandphysicsinopticallatticesisrecentlystud- be traced to the existence of gapless fermion excitations ied23, and the multi-gap superfluidity is also possible in which cause a very large density of state at the gapless Fermi surfaces1,2. To realize a stable Sarma state, nuclear matter24. one should have some mechanism to cure the instabil- The paper is organized as follows. In Section II we 2 give an introduction to the Sarma state in single-band WhethertheZeemanenergyimbalancehorthespinden- systems. We discuss the stability ofthe Sarmastatein a sity imbalance δ is experimentally adjusted depends on generaltwo-bandmodel in Section III and summarize in detailed systems. For cold atoms, the spin density im- Section IV. balance δ is directly tuned, but in superconductors, the Zeeman splitting h is adjusted via an external magnetic field. II. SARMA STATE IN SINGLE-BAND MODEL Before discussing the stability of Sarma state in two- A. Stability Analysis bandsystems,weinthissectiongiveabriefintroduction totheSarmastateinsingle-bandsystems. Westartfrom If a solutionof the gapequation is the groundstate of the following standard BCS-type Hamiltonian the system,itshouldbe the globalminimumofthe ther- 2 modynamic potential Ω16,26. The condition for a local H = d3r ψσ†(r) −2∇m −µσ ψσ(r) minimum of Ω is that Z (cid:20) σ (cid:18) (cid:19) X ∂Ω(∆) ∂2Ω(∆) − U ψ↑†(r)ψ↓†(r)ψ↓(r)ψ↑(r) . (1) ∂∆ =0, ∂∆2 >0. (6) (cid:21) Weconstrainourselvestodiscusssystemsatzerotemper- The first condition corresponds to the gap equation and ature where the BCS mean field theory can be applied thesecondorderderivativeI =∂2Ω(∆)/∂∆2canbeeval- even at strong coupling12. In the mean field approxima- uated as tion, the Hamiltonian is approximated by d3k ∆2 Θ(E↓) I = k δ(E↓) . (7) Hmf = Z d3r"Xσ ψσ†(r)(cid:18)−2∇m2 −µσ(cid:19)ψσ(r) LetusstudytZhe(S2aπr)m3 Eaks2t(cid:20)ateEwkith−h>∆kw(cid:21)hichinduces Φ(r)2 a nonzero spin density imbalanceδ. At weak coupling, I + Φ(r)ψ†(r)ψ†(r)+H.c.+ | | , (2) ↑ ↓ U can be approximately evaluated as # where Φ(r) = U ψ (r)ψ (r) is the order parameter π2I hΘ(h ∆) − h ↓ ↑ i 2mµ 1 − (8) field of superconductivity. For homogeneous supercon- m ≃ − √h2 ∆2 (cid:20) − (cid:21) ductivity, the thermodynamic potential Ω can be ob- p tained by using the standard diagonal method18. It can which shows that ∂2Ω(∆)/∂∆2 is always negative and be expressed as therefore the Sarma state is unstable. ToachievetheBCS-BECcrossover,werenormalizethe ∆2 d3k Ω= + (ξk Ek)+ EkσΘ( Ekσ) (3) couplingconstantwiththetwo-bodyscatteringlengthas, U (2π)3 − − Z (cid:20) σ (cid:21) X m 1 d3k m withthe definitionofenergydispersionsξk =k2/(2m) = + . (9) µ,Ek = ξk2+∆2,Ek↑ =Ek+handEk↓ =Ek h,wher−e 4πas −U Z (2π)3k2 − µ = (µ↑ +µ↓)/2 and h = (µ↑ µ↓)/2 are, respectively, In this case, we first solve the coupled gap and number p − the averaged and mismatched chemical potentials, and equations at fixed total density n = k3/(3π2). The re- F ∆ is the modulus of Φ(r). sult canbe expressed9 as a function of the dimensionless Without loss of generality, we set h 0. The possible couplingparameterg =1/(k a )andthepopulationim- ≥ F s groundstateofthe systemcorrespondsto the stationary balance P =δ/n. The numericalcalculations show that, point of the thermodynamic potential Ω. This gives the the key quantity I is always negative at the BCS side so-called gap equation of the resonance(a < 0,g < 0) where the Sarma state s has two gapless Fermi surfaces, but the Sarma state can 1 d3k Θ(E↓) k ∆=0. (4) be a stable ground state in the strong coupling BEC re- U − (2π)3 2Ek ! gion (roughly for g > 2.2) where the chemical potential Z µ become negative. However, this Sarma state has only Toproperlyachievestrongcoupling,thechemicalpoten- onegaplessFermisurfaceandisdifferentfromthe Fermi tials should be renormalized by the number equations. surface topology of the so-called breached pairing state. The number density n and spin density imbalance δ can be evaluated as n = n +n = d3k 1 ξk Θ(E↓) , B. Solution at Weak Coupling ↑ ↓ (2π)3 − Ek k Z (cid:20) (cid:21) d3k Atweakcouplingwherethechemicalpotentialµiswell δ = n n = Θ( E↓). (5) ↑− ↓ (2π)3 − k approximated by the Fermi energy ǫF, the gap equation Z 3 can be approximated by III. SARMA STATE IN TWO-BAND MODEL 1 Λ Θ( ξ2+∆2 h) We in this section turn to the two-band model. Since dξ − ∆=0, (10) "UN −Z0 p ξ2+∆2 # thegoalofthispaperistosearchforthepossibilityofsta- ble Sarma state in general two-band Fermi systems, we where N is the density opf state for eachspin state at the consideracontinuumHamiltonianandneglectthedetails Fermisurface,andΛistheenergycutoffwhichplaysthe of the band structure in different systems. We will show roleof Debye energy~ωD in solids. After the integration thatthe keypointis the inter-bandscatteringwhichcan and using the condition ∆ Λ, we find maketheSarmastatestableinmulti-bandsystems. The ≪ possible complicated lattice structure in various materi- 1 2Λ h+√h2 ∆2 als and optical lattices will not qualitatively change our ln +Θ(h ∆)ln − ∆=0. "UN − ∆ − ∆ # conclusion. The obtained conclusion is generic and may be usefulfor the study ofsuperconducting materialsand (11) ultracold atom gases. There are three possible solutions to the gap equation The continuum Hamiltonian of the two-band model (11) for h=0. The first is the trivial normal phase with 6 can be written as18 ∆ = 0. The second corresponds to the ordinary fully N gapped BCS solution satisfying ∆>h, 2 H = d3r ψ† (r) ∇ µ ψ (r) ∆BCS =∆0 =2Λe−1/(UN). (12) Z (cid:20) ν,σ νσ (cid:18)−2mν − νσ(cid:19) νσ X Thethirdsolution,i.e.,thegaplessSarmastatesatisfying U ψ† (r)ψ† (r)ψ (r)ψ (r) , (18) ∆ < h, can be analytically evaluated via the comparing − νλ ν↑ ν↓ λ↓ λ↑ ν,λ (cid:21) with the BCS solution. It is25 X where ν,λ = 1,2 denote the band and σ = , the di- ∆S = ∆0(2h ∆0). (13) rection of fermion spin. In superconductors↑, t↓he band − degreesoffreedomusuallycomefromtheparticularcrys- Usingtheweakcoupplingapproximation,thegrandpo- talline and electronic structure of the materials. In ul- tential Ω for various solutions can be expressed as tracold atom gases, these degrees of freedom may come from different hyperfine states or different atom species ∆2 Λ Ω = +2N dξ ξ ξ2+∆2 ortheexternalperiodiclatticepotential. Ingeneralcase, U − Z0 h p the effective fermion mass depends only on the band in- + ( ξ2+∆2 h)Θ(h ξ2+∆2) . (14) dex, but the chemical potential is related to both the − − band and spin indexes due to the existence of external p p i magnetic field or population imbalance. The constants Performing the integral over ξ, and using the condition U U and U U are the intra-band couplings, ∆ Λ as well as the gap equation to cancel the cutoff 11 1 22 2 ≡ ≡ ≪ and U =U J is the inter-band exchange coupling. dependence, we have 12 21 ≡ ForvanishingJ,themodelisreducedtoasimplesystem N with two independent bands. In the following we focus Ω= ∆2 Θ(h ∆)Nh h2 ∆2. (15) − 2 − − − on how the inter-band coupling J changes the stability p of the Sarma state. Note that we have set the grand potential of the normal We first calculate the thermodynamic potential of the stateath=0tobezero,Ω (h=0)=0. Toseewhythe N two-bandHamiltonian. Inthemeanfieldapproximation, Sarma state is always thermodynamically unstable, one the Hamiltonian is approximated by should calculate the grand potential differences between Sarma and other two states25, 2 H = d3r ψ† (r) ∇ µ ψ (r) ΩS−ΩBCS = N(∆0−h)2, mf Z (cid:26)Xν,σ νσ (cid:18)−2mν − νσ(cid:19) νσ ΩS ΩN = N(∆0 2h)2, (16) + Φν(r)ψν†↑(r)ψν†↓(r)+H.c. − 2 − Xν h i 1 which confirm that the Sarma state always has higher + U Φ (r)2+U Φ (r)2 2 1 1 2 potential than the BCS and normal states. As a conse- G | | | | h quence, there exists a first order phase transition from J(Φ∗(r)Φ (r)+Φ∗(r)Φ (r)) , (19) BCS state to normal state. From the result − 1 2 2 1 i(cid:27) N ΩBCS−ΩN = 2 (2h2−∆20), (17) where Φν(r) = − λUνλhψλ↓(r)ψλ↑(r)i are two order parameter fields of the superconductivity, and G is de- the transition occurs at the CC limit of BCS supercon- fined as G=U U PJ2. For homogeneoussuperconduc- 1 2 − ductivity, h =∆ /√2. tivity, the thermodynamic potential Ω of this two-band c 0 4 model can be obtained by using the standard diagonal two-bandsystems. The firsttype (type I) is the solution method18. It can be expressed as where both mismatches are larger than the corresponding pairing gaps, namely h > ∆ and h > ∆ . For 1 1 2 2 1 Ω = U ∆2+U ∆2 2J∆ ∆ cos(ϕ ϕ ) (20) this type, there exist gapless excitations in both bands. G 2 1 1 2− 1 2 1− 2 The second type (type II) is the solution where only one + (cid:2) (d2π3k)3 (ξkν −Ekν)+ EkσνΘ(−Ekσ(cid:3)ν) mhis>ma∆tchainsdlahrge<r t∆hanotrhehco<rre∆spoannddinhg p>air∆ing. gFaopr, ν Z (cid:20) σ (cid:21) 1 1 2 2 1 1 2 2 X X this type, gapless excitations exist only in one band. We with the definition of energy dispersions ξkν = willshowinthefollowingthatthestabilitiesofthesetwo k2/(2mν) − µν, Ekν = ξk2ν +∆2ν, Ek↑ν = Ekν + hν types of Sarma states are quite different. hand=E(k↓µν = Eµkν)−/2haνr,ew,phreesrpeecµtνiv=ely,(µtνh↑e+avµerνa↓g)/e2d aanndd meAntnuismsehroicwanl eixnamFipgl.e1wfohricthwsoupsypmormtsettrhiecabbaonvdesawrgituh- ν ν↑ ν↓ mismatched−chemical potentials, and ∆ν the modulus U1 = U2. For the sake of simplicity, in our numer- of Φ and ϕ their phases through the definition Φ = ical examples presented here, we assume the same ef- ν ν ν ∆ eiϕν. Without loss of generality, we take h >0. For fective masses, chemical potentials and mismatches for ν ν J > 0, the choice of ϕ1 = ϕ2 is favored, otherwise there the two bands, i.e., m1 = m2 ≡ m, µ1 = µ2 ≡ µ is ϕ1 =ϕ2+π. We assume J >0 and set ϕ1 =ϕ2. and h1 = h2 ≡ h, this means that only the total den- The possible ground state of the system corresponds sity n and total spin density imbalance δ can be ad- to the stationary point of the thermodynamic potential justed. We write U1 and U2 in terms of the s-wave scat- Ω. This gives the so-called gap equations tering length aν with a momentum cutoff k0, Uν−1 = m/(4πa ) + d3k/(2π)3m/k2. Our qualitative − ν |k|<k0 U d3k Θ(E↓ ) J conclusions do not depend on the used regularization ν¯ν¯ kν ∆ ∆ =0 (21) R G −Z (2π)3 2Ekν ! ν − G ν¯ escqhueamtieo.nsInartehedicsatsreiboufteUd1s=ymUm2,etthriecasolllyutinionthseof∆the g∆ap 1 2 − with ν¯ = 1 for ν = 2 and ν¯ = 2 for ν = 1. The gap plane. BesidesthefamiliarBCSandnormalstateswhich equations are essentially the same as derived in18. To are, respectively, the global minimum and a local mini- properlyachievestrongcoupling,thechemicalpotentials muminFig.1,wehavesomeSarmastatesinthepotential should be renormalized by the number equations. The contour. The Sarma states C and D are of type I, and numberequationsforthefermiondensityn anddensity C is the global maximum and D indicates two saddle ν imbalance δ for the ν-th band can be evaluated as points. The type II Sarma states are marked by A and ν B, corresponding, respectively, to two local minima and n = n +n = d3k 1 ξkν Θ(E↓ ) , two saddle points. ν ν↑ ν↓ Z (2π)3 (cid:20) − Ekν kν (cid:21) Ifasolutionofthegapequationsisthegroundstateof d3k the system,itshouldbe the globalminimumofthe ther- δν = nν↑−nν↓ = (2π)3Θ(−Ek↓ν), (22) modynamic potential Ω16,26. The condition for a local Z minimum of Ω is that the matrix and the total density n and total density imbalance δ of ∂2Ω(∆1,∆2) ∂2Ω(∆1,∆2) the system are defined as n=n1+n2 and δ =δ1+δ2. = ∂∆21 ∂∆1∂∆2 (23) M  ∂2Ω(∆1,∆2) ∂2Ω(∆1,∆2)  ∂∆2∂∆1 ∂∆22 A. Stability Analysis   should have only positive eigenvalues, namely det > M 0 and Tr > 0. The second order derivatives can be Let us now discuss qualitatively what happens when evaluatedMas the mismatch h increases. For vanishing mismatch, the ν asynsdte∆m is i∆n a,faunlldytghaepsppeidn dBeCnSsitsytaimtebwaliatnhce∆δ1 i≡s z∆er1o0. ∂2Ω(∂∆∆12,∆2) = 2GJ ∆∆ν¯ +Iν, 2 ≡ 20 ν ν Withincreasinghν,whiletheBCSstateisstillasolution ∂2Ω(∆ ,∆ ) 2J 1 2 of the gap equations, there may appear another solution = (24) ∂∆ ∂∆ − G (Sarma) where at least one of the pairing gap ∆ is less ν ν¯ ν than the corresponding mismatch, namely hν > ∆ν. In with the quantities Iν defined as this state, the dispersion of the quasi-particle E↓ be- kν comesgaplessandthe systemhasanonzerospindensity d3k ∆2 Θ(E↓ ) I = ν kν δ(E↓ ) . (25) cimonbdaelannsaceteδi.sNalowtaeytshaatsotlhuetinoonrmofatlhsetagtaepweiqtuhavtiaonnisshainndg ν Z (2π)3Ek2ν " Ekν − kν # becomesthe groundstatewhenbothh1 andh2 arelarge For vanishing inter-band coupling J =0, the stability enough. condition becomes Different from the conventional Sarma state in single- band models, we may have two types of Sarma states in I I >0, I +I >0. (26) 1 2 1 2 5 on. Suppose the solution of the gap equations satisfies 120 ∆ ∆ and ∆ is not quite close to h , which cor- 1 2 1 1 ≪ 100 A B BCS 1 rteersmponinds(2t8o)tihselacragseebwuitththlearmgeodpuolluasriozafttihoensδe1c,otnhdetfierrmst is relatively small, and therefore the stability condition 80 0.8 can be satisfied, like the point A at the up-left cornel in Fig.1. However, on the other hand, for ∆ . h which D2 60 D C B 0.6 corresponds to the case with small polariza1tion 1δ1 → 0, theabsolutevalueofI isverylarge,andtheSarmastate 1 0.4 maybe unstable, which corresponds to the saddle point 40 B in the upper part of Fig.1. 0.2 We conclude that, in two-band Fermi systems with 20 non-zero inter-band pairing interaction, the Sarma state A 0 can become at least the local minimum of the thermo- D Normal dynamic potential, and therefore should be a potential 0 0 20 40 60 80 100 120 candidate of the ground state. D 1 However, the condition J = 0 is not sufficient for us 6 to have a real stable Sarma state. For the case with FIG. 1: The thermodynamic potential contour Ω(∆1,∆2) two symmetric bands shown in Fig.1, we found that the for two symmetric bands with U1 = U2. A proper unit global minimum is always the BCS or normal state for is chosen such that the Fermi energy ǫF = 200. The val- ues of the other parameters are k0 = 100kF, J = 10−4U0 any mismatch h, which means that the Sarma state can with U0 = 4π/(mkF), (kFa1)−1 = (kFa2)−1 = 0.5 and not be the ground state even though it can be a local h = 75, where kF √2mµ is the Fermi moment−um. The minimum. However, this can be significantly changed if band on the right ≃shows the relative strength of Ω corre- someasymmetrybetweenthetwobands,suchasunequal sponding to different colors. For the parameter setting, we couplings U = U , is turned on. In Fig.2 and Fig.3, we 1 2 6 haveU1 =U2 ≃0.0156U0, and henceJ ≪U1,U2. show the potential contour with U1 6= U2 for three val- ues of h. In this case, the number of Sarma solutions is largely suppressed due to the asymmetry, especially the state C inFig.1 asthe globalmaximumofΩ disappears. Note that the properties of the functions I and I are 1 2 Without regard to the saddle points which are impossi- the same as the function I defined in the last section. ble to be stable solutions, the only Sarma state marked Thusatthe BCSside,namelyfora <0anda <0,the 1 2 in Fig.2 and Fig.3 appears to be the global minimum of Sarma state is unstable. the system, when the mismatch h is in a suitable region. Nowwediscusshowtheinter-bandcouplingJ modifies From the top to the bottom in Fig.2 and Fig.3, when the Sarma instability at the BCS side. For J = 0, the 6 the mismatch h increases, the global minimum changes stability condition reads from the BCS state to the Sarma state and then to the 2J ∆1 ∆2 normalstate. Incontrasttotheconventionalsingleband I + I +I I >0, 1 2 1 2 G ∆ ∆ model where only one first order phase transition from (cid:18) 2 1 (cid:19) 2J ∆ ∆ the BCS to normal state is predicted, we have in this 1 2 + +I1+I2 >0. (27) two band system two first order phase transitions when G ∆ ∆ (cid:18) 2 1(cid:19) h increases: The first is from the BCS to Sarma state, For the type I Sarma state with h > ∆ and h > and the second is from the Sarma to normal state. The 1 1 2 ∆ , we have I < 0 and I < 0. In this case, we can firstorderphasetransitionfromBCStoSarmastatewas 2 1 2 exactly prove that the above two inequalities can not found in16 by considering the momentum structure of be satisfied simultaneously. This type of Sarma state the pairing gap. For ultracold atom gases, the chemi- shouldcorrespondtothemaximumorsaddlepointofthe cal potential mismatch h should be replaced by the spin thermodynamic potential and is hence unstable, like the population imbalance δ. However, the phase structure pointsC andD inFig.1. However,the situationchanges should be essentially independent of the assembles one for the type II Sarma state. Without loss of generality, used16,26, we here do not consider the case with fixed δ. let us discuss the case with h1 > ∆1 and h2 < ∆2. In LetuscomparethenumericalresultspresentedinFig.2 this case,onlythe firstbandis gapless,andhenceI1 <0 and Fig.3. In Fig.3, the coupling asymmetry is much and I2 >0. From I2 >0, the above two inequalities are largerthanthatinFig.2,wehave∆10/∆20 1.5inFig.2 equivalent to the following one and∆ /∆ 4inFig.3. Wefindthatthe≃hwindowfor 10 20 ≃ the Sarma state is wider when the coupling asymmetry 2J ∆ 2J ∆ 2 1 +I1 1+ >0. (28) becomes larger. In Fig.2, the window for Sarma state is G ∆ GI ∆ 1 (cid:18) 2 2(cid:19) roughlyfromh=52toh=71,andtheCClimitisabout Even though I <0, this condition can be satisfied, pro- h ∆ . In Fig.3, this window is roughly from h = 20 1 c 20 ≃ vided that a nonzero inter-band coupling J is turned to h=70, and the CC limit is about h 2.7∆ . c 20 ≃ 6 80 40 BCS 60 30 BCS D240 D220 20 Sarma 10 Sarma Normal Normal 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 D D 1 1 80 40 BCS 60 30 D240 D220 20 10 Sarma Sarma Normal Normal 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 D D 1 1 80 40 60 30 D240 D220 20 10 Normal Sarma Normal Sarma 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 D D 1 1 FIG.2: ThethermodynamicpotentialcontourΩ(∆1,∆2)for FIG.3: ThethermodynamicpotentialcontourΩ(∆1,∆2)for two different bands with (kFa1)−1 = 0.5 and (kFa2)−1 = two different bands with (kFa1)−1 = 0.5 and (kFa2)−1 = − − 0.8 at h = 45 (top), 60 (middle) and 75 (bottom). The 1.5 at h = 20 (top), 25 (middle) and 70 (bottom). The − − other parameters are thesame as that in Fig.1. other parameters are the same as that in Fig.1. and express the gap equations of our two band model in B. Solutions at Weak Coupling terms of it, At weak coupling, the same tricks used in Section II U2 J F(∆ ,h ) ∆ ∆ =0, 1 1 1 2 canbeemployed. Forconvenience,wedefinehereafunc- GN − − GN (cid:20) 1 (cid:21) 1 tion U J 1 F(∆ ,h ) ∆ ∆ =0, (30) 2 2 2 1 GN − − GN (cid:20) 2 (cid:21) 2 Λ Θ( ξ2+∆2 h) F(∆,h) = dξ − (29) where N1 and N2 are the densities of state at the Fermi Z0 p ξ2+∆2 surfacesforthetwobands. Unlikethesinglebandmodel, 2Λ p h+√h2 ∆2 the above coupled gap equations can not be solved ana- ln Θ(h ∆)ln − lytically. With the numerical solutions ∆ and ∆ , the ≃ ∆ − − ∆ 1 2 7 thermodynamic potential can be evaluated as N 1 Ω= ν∆2 +Θ(h ∆ )N h h2 ∆2 .(31) (a) − 2 ν ν − ν ν ν ν − ν Xν (cid:20) p (cid:21) 0.8 (b) For the sake of simplicity, we consider the case h1 = 20 h2 = h which corresponds to the realistic two-band su- ∆/20.6 ∆ perconductors in strong magnetic field. Let us assume = (c) y U N > U N which leads to ∆ > ∆ . According 0.4 1 1 2 2 1 2 to the stability analysis, we have three possible ground states: 1)The normal state with ∆ = ∆ = 0; 2)The 0.2 1 2 gapped BCS state with energy gaps ∆ ∆ > h and 1 10 ≡ ∆2 ∆20 > h; 3)The gapless Sarma state where only 00.4 0.6 0.8 1 1.2 1.4 ∆1 ≡> h but ∆2 < h. We focus here on the case with x=h/∆20 ∆ ∆ and J √U U . In this case, the solution of 2 1 1 2 ≪ ≪ ∆1 is approximately independent of h and is given by FIG. 4: The solution y = ∆2/∆20 to the gap equations as a function of x = h/∆20. The dashed line denotes the BCS ∆ =∆ 2Λe−1/(U1N1), (32) solutiony=1intheregion0<x<1. Thesolidlines(a)and 1 10 ≃ (b) are the Sarma solutions for J = 0. In the calculations, 6 and the Sarma solution for ∆2 is determined by the fol- we take N1/N2 =1.5 and J/√U1U2 =0.07. For (a) we take lowing equation (U1N1)/(U2N2) = 3 which leads to ∆10/∆20 9, and for ≃ (b) we have (U1N1)/(U2N2)=1.67 and hence ∆10/∆20 3. ≃ ∆20 J∆10 1 1 Thedot-dashedline(c)istheconventionalSarmasolutionfor ln = , (33) h+ h2−∆22 U1U2N2 (cid:18)∆20 − ∆2(cid:19) J =0, y=√2x−1 in theregion 0.5<x<1. where ∆ pis obtained by the equation 20 1 2Λ J ∆ much smaller than N . In this case, the BCS solution is 10 2 ln = . (34) U N − ∆ U U N ∆ absent and the Sarma state is the stable ground state. 2 2 20 1 2 2 20 This argument confirms our conclusion from the numer- We have numerically checked that the above approxi- ical results in Fig.2 and Fig.3: The h window for stable mation is sufficiently good for the scaled solution y = Sarma state is wider when the asymmetry between the ∆2/∆20 asafunction ofx=h/∆20. Note thatforJ =0 twobandsbecomeslarger. ThismeansthattheCClimit the conventional Sarma solution y =√2x 1(0.5<x< of such a two-band superconductor can be much higher − 1)isrecovered,but forJ =0the Sarmasolutionis qual- than the conventional value h =0.707∆ . 6 c 20 itatively changed: y =0 can not be a solution and there exist solutions for x>1. The solutions for both cases of J = 0 and J = 0 are illustrated in Fig.4. We find that 6 IV. SUMMARY for J = 0 the Sarma solution is quite different from the 6 conventional result. Unlike the well-known Sarma solu- We have studied the stability of Sarma state in two- tionwhichexistsinthe region0.5<x<1,forJ =0the 6 band Fermi systems via both the stability analysis and Sarma state exists almost in the region x>1 where the analytical solution at weak coupling. From the stabil- BCS solution y = 1 disappears. Obviously, in a narrow ity analysis,the Sarma state can be the minimum of the region x . 1 there exists a branch of the conventional thermodynamicpotentialandhenceapossiblecandidate type which is unstable, and the multi-value behavior of of the ground state, if the inter-band exchange interac- y means a first order BCS-Sarma phase transition at a tion is turned on. Both numerical and analytical studies critical field x which is slightly smaller than 1. 1 show that, a large asymmetry between the two bands To discuss the thermodynamic stability of the Sarma or the two pairing gaps is an important condition for state, we then needto compareit with the normalstate. thermodynamic stability of the Sarma state. When the In the case of ∆ <h, we find 2 condition is satisfied, two first order phase transitions N will occur when the mismatch increases, one is from the Ω Ω = 1(2h2 ∆2) S− N 2 − 1 BCS to Sarma state at a lower mismatch and the other N is fromthe Sarmato normalstate at a highermismatch. + 22 2h2−∆22−2h h2−∆22 . (35) Our predictions could be tested in multi-band supercon- (cid:18) q (cid:19) ductors and ultracold atom gases, and such a gapless Some analytical estimations can be made. For large superconductormayhavemanyunusualproperties,such asymmetry ∆ ∆ , around the h-window h ∆ as magnetism and large spin susceptibility27,28. 2 1 20 ≪ ∼ but h ∆ for the Sarma state, the sign of the quan- Acknowledgments: We thank W.V.Liu and M.Iskin 10 ≪ tity Ω Ω is dominated by the first term, if N is not for useful communications and Y.Liu and X.Hao for the S N 1 − 8 help in numerical calculations. 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