Table Of Content

Topological transitions of gapless paired states in mixed-geometry lattices Dong-Hee Kim, Joel S. J. Lehikoinen, and Päivi Törmä∗ COMP Centre of Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland Fermion pairing may coexist with magnetism in pairingincontinuum[24,25], havebeentheoreticallyin- unconventional superconductors and nuclear matter, vestigated. Here we propose a fundamentally different stabilized by non-BCS mechanisms. Remarkable sce- type of asymmetry: a spin-dependent lattice where each narios of exotic pairing with Fermi surfaces mismatched spin component is separately loaded on a different un- 2 by chemical potential or mass difference have been derlying geometrical setup. 1 suggested [1–4], inspiring experiments in solid-state We propose a mixed-geometry setup of a two com- 0 materials [5, 6] and ultracold Fermi gases [7–10]. Here ponent Fermi gas where up- and down-spin components 2 we propose fermionic species to be confined in lattices areloadedonhoneycombandtriangularlattices, respec- p of different geometry and show how such distortion tively, as shown in Fig. 1a. We argue that this is one e of symmetry leads to unconventional quantum states. of the simplest yet nontrivial mixed-geometry configura- S In particular, we consider pairing of one species in tions. It is a superlattice of two spin-dependent sublat- 7 a graphene-like lattice with another in a triangular tices A and B. Tuned on short-ranged interactions, this sublattice. We find a rich phase diagram of multiband configurationcanrealizetheHubbardmodelwiththeon- ] s pairing with gapped and gapless excitations at zero site interaction selectively applied at A sites. The setup a temperature, implying that gapless superfluids can be could be experimentally approached by generalizing the g - stabilized in these systems. Quantum phase transitions techniquesin[11–15]orusingtwodifferentatomicspecies nt between topologically distinct phases are predicted. [19–21]. a Realization of mixed-geometry systems, e.g. using We consider the attractive Hubbard model for a u spin-dependent optical lattices [11–15], provides a new fermion pairing problem in the mixed-geometry lattices. q dimension to the ultracold quantum simulator for AsdescribedinMethods,themean-fieldHamiltoniancan . t exploring unconventional quantum states of matter. be written in momentum space as a m - The standard BCS theory considers a perfectly sym- HMF =(cid:88) (cid:88) ξα((cid:126)k)cˆ†α(cid:126)kcˆα(cid:126)k (1) d metric pairing in up- and down-spin particles with op- (cid:126)k α∈{1,2,3} n posite momenta lying on identical noninteracting Fermi +(cid:88)[g ((cid:126)k)cˆ† cˆ† +g ((cid:126)k)cˆ† cˆ† +h.c.] , o surfaces. One of the fundamental issues in condensed 1 1(cid:126)k 3,-(cid:126)k 2 2(cid:126)k 3,-(cid:126)k c (cid:126)k [ matter physics is what would happen if this setting was broken, for instance, with mismatched Fermi surfaces which reveals a multiband system where up-spin parti- 1 between the two spin components. The Fulde-Ferrell– clesinbandα=1,2wouldpairwithdown-spinparticles v 1 Larkin-Ovchinnikov(FFLO) state suggestsCooper pairs in band α = 3. The lattice effects of the mixed geom- 7 carrying finite center-of-mass momentum, causing a spa- etry are encoded in the noninteracting band dispersion 5 tially oscillating order parameter [1, 2]. The Sarma or ξ ((cid:126)k) sketched in Fig. 1b and in the coupling g ((cid:126)k). α 1,2 1 breached-pair (BP) state describes polarized superfluid With fixed interaction and hopping strengths, two con- . 9 of BCS-type zero-momentum pairs separated from un- trolparameterscangovernthepairinginthissystem: 1) 0 paired excess particles forming a Fermi sea [3, 4]. These tuning chemical potentials can lead to the Fermi surface 2 scenarios remain elusive but inspiring as possible uncon- mismatch; 2) the onsite energy modulation (cid:15)˜controls a 1 ventional pairing mechanisms. relative strength of the couplings g and g . : 1 2 v Ultracold Fermi gases are potentially an ideal testbed Figure 1d shows the zero-temperature phase diagram i X of quantum many-body physics because of their un- for the case of the graphene-like up-spin bands ((cid:15)˜= 0). precedented controllability [16]. Loaded in optical lat- We find that the diagram is divided into three main ar- r a tices, fermionic atoms with tunable interparticle inter- eas indicating the normal, gapped, and gapless phases. action can construct the repulsive or attractive Hub- While the normal phase is simply indicated by a vanish- bard model [17, 18]. One remarkable feature of ultra- ing order parameter ∆ = 0, a paired state with nonzero cold gases is the tunability of asymmetry between the ∆ is discriminated by its single-particle excitation spec- (pseudo-)spins associated with atomic internal states. trum to be in the gapped or gapless phases. The gapped Typical forms of spin asymmetry are population [7–10] phaseisafullypairedstate, similartotheBCSstate. In ormass[19–21]imbalancebetweenthespincomponents, the gapless phases, we find an exotic paired state which orspin-dependentopticallatticesrealizedforbosons[11– wecalltheincomplete-breached-pair(iBP)statebecause 15]. The effects of anisotropic hopping or onsite poten- of a partial breach found in momentum distribution. tials in a lattice [22, 23], as well as mixed-dimensional Intriguingly, the iBP state can be further divided into 2 different subclasses by the topological arrangement of suppressed. Figure 3 displays the phase diagrams for one or two Fermi surfaces, leading to Lifshitz transi- (cid:15)˜ = −1.0,−1.4,−1.8, indicating that the gapless phase tions [26]. The iBP states undergo two types of tran- becomes less robust as |(cid:15)˜| increases. This apparent con- sitions where the Fermi surface is either vanishing (2-FS nectionbetweenthecouplingimbalanceandthestability to 1-FS) or transforming to a topologically different sur- of the gapless phase implies that a well-balanced inter- face (Γ-centered to K-centered). In this mixed-geometry band contribution may be an important factor for our system, while the former is a continuous transition, it mixed-geometry lattice in stabilizing the iBP phase. turns out that the latter can be continuous or discontin- One of the main findings is the chemical-potential- uous transitions which meet at a multicritical point (see driven emergence of the paired states with zero, one, Figs. 1a,2a). This end point of the discontinuous transi- and two Fermi surfaces. The importance of the multi- tion may be explained as the marginal quantum critical bandcontributiontotheformationoftheseexoticpaired point of a Lifshitz transition in a two-dimensional inter- states can be intuitively understood in the band struc- acting system [27]. tureofthemixed-geometrysystem. Revisitingourmean- The nature of the iBP states can be characterized field procedures (see Methods), as illustrated in Fig. 4, by momentum distributions n ((cid:126)k) and a momentum- there are three cases of zero, one, and two crossings al- ↑,↓ resolved order parameter ∆ shown in Fig. 2. While lowed between the noninteracting particle-like (ξ ) and k ↑ the jumps in n ((cid:126)k) and ∆ indicate the presence of hole-like (−ξ ) dispersions with varying chemical poten- ↑,↓ k ↓ the Fermi surfaces, the breach surrounded by these dis- tials. The interaction (g ) causes band repulsion and 1,2 continuities shows clear contrast to that of the conven- gaps at crossings, leading to the two different configu- tional BP state in momentum space. It turns out that rations of quasiparticle energy dispersions. Depending the momentum-space breach in our iBP state shows a on the chemical potentials, the zero excitation level in- mixtureofpairedparticleswithfinite∆ coexistingwith dicates the formation of zero, one, or two Fermi surfaces k unpaired excess majority species, rather than a perfect (see Fig. 4b). phaseseparationsuggestedbytheconventionalBPstate. The simple band picture can also clarify the arrange- However, the Luttinger’s theorem still holds for a con- ments of the distinct iBP states in the phase diagrams. served quantity (here n −n ) in the broken symmetry For instance, considering the iBP state with two Fermi ↑ ↓ phase[28],explainingtheplateauofn −n =1observed surfaces, Fig. 4b indicates that there are two multiband ↑ ↓ between the Fermi surfaces. configurations forming such states, the one in the lower Controlled by chemical potentials or densities, topo- bandside(d)andtheotherintheupperbandside(f). In logically different iBP states arise with distinct Fermi contrast, the state with one Fermi surface occurs only in surface topologies. It is not only that the number of the the upper band side (e). This classification in the band Fermi surfaces essentially changes between one and two picture is consistent with the phase boundaries in the as observed for instance at µ = 0, but also the shape phase diagrams. In addition, the band picture indicates ↑ of the Fermi surface undergoes a topological transition. that the iBP states with two Fermi surfaces have a dif- Figure 2 shows the topologically different Fermi surfaces ferent nature depending on whether they originate from foundinthissystemandthephasediagramofthecorre- the upper or lower band side. Indeed, for instance, the spondingdistinctiBPstates. Notethatthetransitionsin momentum distributions found at (cid:15)˜=−1.8 show a fully ourphasediagramaredrivenbydensitiesorchemicalpo- polarizednormalareaofexcessmajorityparticlesaround tentials, which is in contrast to the BEC-BCS crossover K, which is in contrast to the fully paired area around K forimbalancedtwo-componentgaseswherethetransition observed in its counterpart found at (cid:15)˜=0 (see Fig. 3c). intotheBPstatewithoneFermisurfacemayoccurinthe Systematic study of finite temperature effects and quan- BEC regime driven by interaction [29, 30]. In addition, tumfluctuationsisbeyondthescopeofthiswork. While compared with a noninteracting case where a topologi- the Fermi surface is strictly defined at zero temperature, cal change of the Fermi surface occurs simply at the van we have tested that the main features are present at fi- Hovesingularityat|µ|=1(seeFig.1c),itisnotablethat nite temperatures within the mean-field theory, critical thetransitionlinesintheinteractingsystemareverydif- temperatures being typically of the order of a tenth of ferent from |µ| = 1 as indicated by our diagram of the the hopping amplitude. phases with Γ- and K-centered Fermi surfaces. Mixed-geometrylatticesystemsasweproposehereex- The characteristics of the gapless phase can be also tend the realm of pairing and strong correlation physics. tunedbyonsiteenergymodulation(cid:15)˜. Theeffectsoffinite TheattractiveHubbardmodelinthemixedhoneycomb- (cid:15)ãre two-fold. First, it opens a band gap in the nonin- triangular lattice that we have considered reveals a new teractingup-spinbands. Second, itcreatesanimbalance exotic multiband paired state, the incomplete-breached- between the interband couplings g and g (see Eq. (1)) pair state, and shows a rich phase diagram of topolog- 1 2 as described in Methods. Consequently, for a large neg- ically distinct phases. We have found that multiband ative (cid:15)˜ giving g (cid:29) g , the contribution of the up-spin pairing plays a key role in stabilizing the coexistence of 2 1 upper band (α = 1) to the pairing can be significantly pairing and magnetization, which may have an interest- 3 ing connection to multiband effects and magnetic fluctu- up-spin particles at different sites (A or B) are cou- ationsinterplayinginsolid-statesuperconductorssuchas pled. A more natural form of the noninteracting Hamil- iron-based materials and MgB . We argue that realizing tonian is obtained by choosing a new basis set that di- 2 mixed-geometry lattices would open a new dimension to agonalizes H(0). The transformed fermionic operators (cid:126)k the search for novel quantum states with ultracold gases cˆ† and cˆ with the index α ∈ {1,2,3} then represent as quantum simulators. α(cid:126)k α(cid:126)k the noninteracting energy bands ξ ((cid:126)k), where 1 and 2 α denote the upper and lower bands of the up-spin compo- METHODS nent, and 3 indicates the down-spin band. In the basis Ψ˜† ≡ (cˆ† ,cˆ† ,cˆ ), the full mean-field Hamiltonian is (cid:126)k 1(cid:126)k 2(cid:126)k 3−(cid:126)k Attractive Hubbard model. The Hubbard Hamil- rewrittenasH =(cid:80) Ψ˜†H Ψ˜ +(cid:80) ξ (−(cid:126)k)+|∆|2/U MF (cid:126)k (cid:126)k (cid:126)k (cid:126)k (cid:126)k 3 tonian in the mixed-geometry lattice system considered with the matrix representation becomes   ξ ((cid:126)k) 0 g∗((cid:126)k) H=−t↑ (cid:88) (aˆ†i↑ˆbj↑+h.c.)+(cid:15)a↑(cid:88)nâi↑+(cid:15)b↑(cid:88)nˆbi↑ H(cid:126)k = 10 ξ2((cid:126)k) g21∗((cid:126)k) , (2) (cid:104)i,j(cid:105)∈L↑ i i g ((cid:126)k) g ((cid:126)k) −ξ (−(cid:126)k) 1 2 3 (cid:88) (cid:88) −t (aˆ† aˆ +h.c.)+(cid:15)a nâ ↓ i↓ j↓ ↓ i↓ where the diagonal terms are the noninteracting band −µ (cid:104)(cid:88)i,j(cid:105)∈(nLˆ↓a +nˆb )−µ (cid:88)nâ −i U(cid:88)nâ nâ , dispersions ξ1,2((cid:126)k) = ±(cid:113)(cid:15)˜2+|h↑((cid:126)k)|2−µ↑ and ξ3((cid:126)k) = ↑ i↑ j↑ ↓ i↓ i↑ i↓ ξ ((cid:126)k), and the interband couplings are derived as i,j i i ↓ where aˆ† (aˆ) and ˆb† (ˆb) are fermionic creation (anni-  12 ∆ (cid:15)˜ hilation) operators in the sublattices A and B, respec- g1,2((cid:126)k)=−√ 1± (cid:113)  . tively, and the density operators nâ and nˆb denote aˆ†aˆ 2 (cid:15)˜2+|h↑((cid:126)k)|2 and ˆb†ˆb. The spin-dependent hopping occurs between This gives the Hamiltonian of Eq. 1. The quasiparticle nearest-neighboringsites(cid:104)i,j(cid:105)intheup-spinhoneycomb energies E ((cid:126)k) where α ∈ {1,2,3} can be obtained by lattice L and the down-spin triangular lattice L , while α ↑ ↓ diagonalizing H , and from the grand partition function the hopping strengths t↑ and t↓ are set to be unity for Z = Trexp[−βH(cid:126)k ], the grand potential Ω ≡ −1 lnZ simplicity. Introducing the control parameter (cid:15)˜ for on- MF β is calculated as site energy modulations, the onsite terms are chosen as (cid:15)a↑ =(cid:15)˜/2(cid:15)b↑ =−(cid:15)˜/2,and(cid:15)a↓ =−3withoutlossofgeneral- Ω(∆)= |∆|2 +(cid:88)ξ (−(cid:126)k)+ 1 (cid:88)ln(1+e−βEα((cid:126)k)). ity since other possible tunings of onsite energies can be U 3 β absorbed into the spin-dependent chemical potentials µ (cid:126)k (cid:126)k,α ↑ and µ↓. The onsite interaction is chosen as U =5. The ground state is self-consistently obtained by the Mean-field theory. The mean-field order parameter order parameter ∆ minimizing the grand potential Ω, is defined as ∆=U(cid:104)aî↓aî↑(cid:105). Neglecting fluctuations, the whichinprincipleisequivalenttosolvingagapequation interactiontermcanbeapproximatedas−U(cid:80) nâ nâ ≈ but finds only stable solutions. i i↑ i↓ −(cid:80) (∆aˆ† aˆ† + ∆∗aˆ aˆ ) + |∆|2/U. Possible Hartree i i↑ i↓ i↓ i↑ shifts can be incorporated into the chemical potentials and the onsite energy modulation (cid:15)˜. Performing the Fourier transformation, f˜(cid:126)kσ = (cid:80)ie−i(cid:126)k·(cid:126)xifîσ where f ∗ paivi.torma@aalto.fi is a or b, the noninteracting part of the Hamiltonian [1] Fulde, P. & Ferrell, R. A. Superconductivity in a strong can be written in the basis of Φ˜(cid:126)†k ≡ (a˜(cid:126)†k↑,˜b(cid:126)†k↑,a˜(cid:126)†k↓) as spin-exchange field. Phys. Rev. 135, A550–A563 (1964). H0 =(cid:80) Φ˜†H(0)Φ˜ with the matrix representation [2] Larkin, A. I. & Ovchinnikov, Y. N. Inhomogeneous (cid:126)k (cid:126)k (cid:126)k (cid:126)k state of superconductors. Sov. Phys. JETP 20, 762–769 (1965).   (cid:15)˜−µ h ((cid:126)k) 0 [3] Sarma, G. On the influence of a uniform exchange field ↑ ↑ H(0) =h∗((cid:126)k) −(cid:15)˜−µ 0 , actingonthespinsoftheconductionelectronsinasuper- (cid:126)k  ↑ ↑  conductor. J. Phys. Chem. Solids 24, 1029–1032 (1963). 0 0 ξ ((cid:126)k) ↓ [4] Liu,W.V.&Wilczek,F.Interiorgapsuperfluidity.Phys. Rev. Lett. 90, 047002 (2003). where h↑((cid:126)k) = −t↑[ei√k√x3 + 2e−2i√k3x cosk2y] √and ξ↓((cid:126)k) = [5] Radovan, H. et al. Magnetic enhancement of supercon- −t [cosk +cos1(k + 3k )+cos1(k − 3k )−3]− ductivityfromelectronspindomains.Nature425,51–55 ↓ y 2 y x 2 y x (2003). µ . Accordingly, the order parameter can be rewrit- ↓ [6] Kenzelmann, M. et al. Coupled superconducting and (cid:80) ten with the momentum resolution as ∆ ≡ ∆ = (cid:80) (cid:126)k (cid:126)k magnetic order in CeCoIn5. Science 321, 1652–1654 U (cid:126)k(cid:104)aˆ-(cid:126)k↓aˆ(cid:126)k↑(cid:105). We see that, in momentum space, the (2008). 4 [7] Zwierlein,M.W.,Schirotzek,A.,Schunck,C.H.&Ket- [29] Pao,C.-H.,Wu,S.-T.&Yip,S.-K.Superfluidstabilityin terle, W. Fermionic superfluidity with imbalanced spin theBEC-BCScrossover.Phys.Rev.B73,132506(2006). populations. Science 311, 492–496 (2006). [30] Sheehy, D. E. & Radzihovsky, L. BEC-BCS crossover [8] Partridge,G.B.,Li,W.,Kamar,R.I.,Liao,Y.&Hulet, in “magnetized” Feshbach-resonantly paired superfluids. R. G. Pairing and phase separation in a polarized Fermi Phys. Rev. Lett. 96, 060401 (2006). gas. Science 311, 503–505 (2006). [9] Nascimbéne, S. et al. Collective oscillations of an imbalanced Fermi gas: axial compression modes and polar on effective mass. Phys. Rev. Lett. 103, 170402 (2009). [10] Liao,Y.etal.Spin-imbalanceinaone-dimensionalFermi gas. Nature 467, 567–569 (2010). [11] LeBlanc,L.J.&Thywissen,J.H.Species-specificoptical lattices. Phys. Rev. A 75, 053612 (2007). [12] Catani,J.et al.Entropyexchangeinamixtureofultra- cold atoms. Phys. Rev. Lett. 103, 140401 (2009). [13] Lamporesi,G.etal.Scatteringinmixeddimensionswith ultracold gases. Phys. Rev. Lett. 104, 153202 (2010). [14] Soltan-Panahi,P.etal.Multi-componentquantumgases in spin-dependent hexagonal lattices. Nature Phys. 7, 434–440 (2011). [15] Soltan-Panahi, P., Lu¨hmann, D.-S., Struck, J., Wind- passinger,P.&Sengstock,K.Quantumphasetransition to unconventional multi-orbital superfluidity in optical lattices. Nature Phys. 8, 71–75 (2012). [16] Bloch,I.,Dalibard,J.&Zwerger,W.Many-bodyphysics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008). [17] Jo¨rdens, R., Strohmaier, N., Gu¨nter, K., Moritz, H. & Esslinger, T. A Mott insulator of fermionic atoms in an optical lattice. Nature 455, 204–207 (2008). [18] Schneider, U. et al. Metallic and insulating phases of re- pulsivelyinteractingfermionsina3Dopticallattice.Sci- ence 322, 1520–1525 (2008). [19] Wille, E. et al. Exploring an ultracold Fermi-Fermi mix- ture: interspecies Feshbach resonances and scattering properties of 6Li and 40K. Phys. Rev. Lett. 100, 053201 (2008). [20] Voigt, A.-C. et al. Ultracold heteronuclear Fermi-Fermi molecules. Phys. Rev. Lett. 102, 020405 (2009). [21] Kohstall, C. et al. Metastability and coherence of repul- sivepolaronsinastronglyinteractingFermimixture.Na- ture 485, 615–618 (2012). [22] Feiguin, A. E. & Fisher, P. A. Exotic paired states with anisotropic spin-dependent Fermi surfaces. Phys. Rev. Lett. 103, 025303 (2009). [23] Zapata, I., Wunsch, B., Zinner, N. T. & Demler, E. π phases in balanced fermionic superfluid on spin- dependent optical lattices. Phys. Rev. Lett. 105, 095301 (2010). [24] Nishida, Y. & Tan, S. Universal Fermi gases in mixed dimensions. Phys. Rev. Lett. 101, 170401 (2008). [25] Iskin, M.&Suba¸si, A.L.CooperpairingandBCS-BEC evolution in mixed-dimensional Fermi gases. Phys. Rev. A 82, 063628 (2010). [26] Lifshitz, I. M. Anomalies of electron characteristics of a metal in the high pressure region. Sov. Phys. JETP 11, 1130–1135 (1960). [27] Yamaji, Y., Misawa, T. & Imada, M. Quantum and topological criticalities of Lifshitz transition in two- dimensional correlated electron systems. J. Phys. Soc. Jpn 75, 094719 (2006). [28] Sachdev, S. & Yang, K. Fermi surfaces and Luttinger’s theorem in paired fermion systems. Phys. Rev. B 73, 174504 (2006). 5 a t↓ b t↑ 0) = t↑ n (U 2|~є| ↑-spin o ersi p -U dis gy ↓-spin -U er n e K Γ A c K M A B Γ d 0.5 Normal 0.4 Forbidden n atio 0.3 Forbidden z Gapless iBP ari Pol 0.2 2-FS-(Γ,K) 1-FS-(Γ) 0.1 2-FS-(K,K) 1-FS-(K) Normal Fully occupied down-spin band Gapped 0.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 up-spin density (majority) FIG. 1. The mixed-geometry lattice proposed and the phase diagram of exotic paired states at zero temperature. a. Up-spinanddown-spincomponentsareselectivelyloadedonhoneycomb(up-spin,sublatticesAandB)andtriangular (down-spin, sublattice A) lattices. The attractive Hubbard model is considered in the mixed-geometry lattice. Noninteracting banddispersionsaresketchedinb,wherethegapinup-spinbandsistunedbytheonsiteenergymodulation(cid:15)˜(seeMethods). c. The Γ-centered and K-centered Fermi surfaces are sketched. The high symmetry points Γ, K, and M are indicated in the first Brillouin zone. In a noninteracting system, a transition between the Γ- and K-centered shapes occurs at the van Hove singularity at |µ| = 1 where the Fermi surface is a hexagon (dotted line). d. The phase diagram is plotted as a function of up-spin density n and polarization P =(n −n )/(n +n ) for the case of (cid:15)˜=0. The gapped and gapless paired phases and ↑ ↑ ↓ ↑ ↓ thenormalphaseareidentified. ThegaplessstatesarefurthercharacterizedbytheFermisurfacetopologydenotedasz-FS-(X) indicatingzFermisurfacesbeingcenteredatthehighsymmetrypoints(X). Continuous(dottedlines)anddiscontinuous(solid lines) Lifshitz transitions are found, and the empty circle indicates the multicritical point. 6 2-FS-(Γ,K) 2-FS-(K,K) 1-FS-(K) 1-FS-(Γ) b c e f 1 a max ∆ 1 2-FS-(K,K) Normal ∆/k c e,g 1-FS-(K) 0 d 0 µ↓-1 Gappe Gapfless iBP d n 1.4 FS FS g n FS -2 1-FS-(Γ) butio 1.2 n↑−n↓=1 butio 1.5 n n =1 b,d stri 1.0 stri 1.0 ↑− ↓ -3 2-FS-(Γ,K) Normal di 0.8 di m 0.6 m u u -0.5 0.0 0µ.5 1.0 1.5 ent0.4 n (k) ent 0.5 n (k) ↑ m 0.2 n↑(k) m n↑(k) o o m0.0 ↓ m0.0 ↓ Γ K Γ K FIG. 2. Characterization of paired states in the gapless phase. Representative Fermi surface configurations are shown with momentum-resolved order parameter over the first Brillouin zone (dashed line) in b, c, e, and f sampled from topologicallydistinctstatesgivenina. Thephasediagram(a)isplottedasafunctionofthechemicalpotentialsµ andµ for ↑ ↓ (cid:15)˜=0. Thedottedandsolidlinesindicatecontinuousanddiscontinuoustopologicaltransitionswhichmeetatthemulticritical point indicated by the empty circle. The multicritical point corresponds to the fully filled down-spin band (µ = 0) in a ↓ noninteracting system showing a step-function-like singularity in the density of states. Momentum distributions in d and g further characterize the gapless paired states found in this system with 2-FS and 1-FS. The existence of plateaus of n −n ↑ ↓ in the shaded area enclosed by the Fermi surfaces is an indication of Luttinger’s theorem. This mixed-geometry system shows pairing coexisting with polarization in momentum space in the gapless phase, in contrast to the phase separation in the usual breached-pair state. 7 a 0.5 2-FS-(K,K) 0.0 Gapped 1-FS-(K) Gapless iBP ↓-0.5 µ 1-FS-(Γ) -1.0 2-FS-(Γ,K) -1.5 1-FS-(K) 2-FS-(K,K) ˜†=−1.0 b 0.5 1-FS-(K) 0.0 Gapless iBP Gapped 1-FS-(Γ) ↓-0.5 µ 2-FS-(K,K) 2-FS-(Γ,K) -1.0 ˜†= 1.4 − -1.5 c 0.5 1-FS-(Γ) 2 0.0 n (k) n (k) ↑ − ↓ 1 µ↓-0.5 2-FS-(Γ,K) 2-FS-(Γ,K) Gapped Gapless 0 iBP Γ K -1.0 ˜†= 1.8 − -1.5 -0.5 0.0 0.5 1.0 1.5 2.0 µ ↑ FIG. 3. Effects of the onsite energy modulation. The phase diagrams are examined for the onsite energy modulation (cid:15)˜= −1.0 (a), −1.4 (b), and −1.8 (c). While the gapped phase is insensitive to (cid:15)˜, the gapless phase (shaded) shows different Fermi surface topologies as (cid:15)˜changes. Compared with the phase diagram for (cid:15)˜= 0 in Fig. 2a, while the 1-FS-(Γ) state is relatively robust, the areas of other incomplete-breached-pair states at (cid:15)˜= 0 are largely changed. In particular, a dominant phase at a large (cid:15)˜is found to be the 2-FS-(Γ,K) state which turns out to be a different state from the one dominant at (cid:15)˜=0. This2-FS-(Γ,K)stateatfinite(cid:15)correspondstocasef inFig.4bratherthancased,showingadifferentmomentumdistribution with a fully polarized area as schematically shown in the inset of c. 8 a b 8 ξ ( k) − ↓ − c d e f 6 )Γ ξ (k) E=0d n (K− ↑ c ) 4 o ω si − er k, 2 p ( s ↓ di A y f 0 erg e gapped 2-FS 1-FS 2-FS n e -2 noninteracting interacting -4 K Γ K Γ K Γ K Γ FIG.4. SchematicsofpairinggapopeningandFermisurfaceformation. Thebanddispersionsandspectralfunctions areshownalongtheK−ΓlineinthefirstBrillouinzone. a. Thenoninteractingparticle-likeup-spinbands(ξ )andthehole-like ↑ down-spin band (-ξ ) are illustrated, possible points for level repulsion and gap formation are shown by empty boxes. b. The ↓ resultingquasiparticlebandswithinteractionturnedonaresketchedwithpossiblelocationsofthezeroexcitationlevel(E =0), showingtheformationofzero(c),one(e),andtwoFermisurfaces(d,f). Thearrowsindicatethepossibleevolutionsfromthe noninteractingbandstothequasiparticlebands. Theactualspectralfunctionsofthedown-spincomponent,A (k,−ω),forall ↓ typical cases are selected for display in c-f.