Geometry and superfluidity of the flat band in a non-Hermitian optical lattice
GGeometry and superfluidity of the flat band in a non-Hermitian optical lattice
Peng He, Hai-Tao Ding, and Shi-Liang Zhu
2, 3, ∗ National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,School of Physics and Telecommunication Engineering,South China Normal University, Guangzhou 510006, China Guangdong-Hong Kong Joint Laboratory of Quantum Matter,Frontier Research Institute for Physics, South China Normal University, Guangzhou 510006, China (Dated: January 5, 2021)We propose an ultracold-atom setting where a fermionic superfluidity with attractive s-waveinteraction is uploaded in a non-Hermitian Lieb optical lattice. The existence of a real-energyflat band solution is revealed. We show that the interplay between the skin effect and flat-bandlocalization leads to exotic localization properties. We develop a multiband mean-field descriptionof this system and use both order parameters and superfluid weight to describe the phase transition.A relation between the superfluid weight and non-Hermitian quantum metric of the quantum statesmanifold is built. We find non-monotone criticality depending on the non-Hermiticity, and the skineffect prominently enhances the phase coherence of the pairing field, suggesting ubiquitous criticalbehavior of the non-Hermitian fermionic superfluidity.
I. INTRODUCTION
A flat band is a dispersionless Bloch band withconstant energy and heavy degeneracy for all quasi-momentum. The flat band arises as a result of destruc-tive interference in bipartite quantum systems imposedby certain symmetries [1]. Systems with flat bands hostintriguing features which are entirely governed by thequantum geometry and topology of the bands [2–8], facil-itating the correlation in superconductivity [9–11], frac-tional quantum Hall states [12, 13], and frustration in fer-romagnetism [14, 15]. For instance, the flat band yields amaximal critical temperature within the mean-field the-ory of superconductivity [16, 17], and plays an importantrole in the cuprates [18] and twisted-bilayer graphene [19–21]. The flat bands have been recognized and experimen-tally observed in the various contexts of condensed mat-ter and synthetic quantum matter. Specifically, the Flatbands in Lieb lattice and Kagome lattice have been re-alized with the ultracold gases [22–25], photonic crystals[26–28], and electronic systems [30]. However, it is stillan interesting direction to test flat-band physics in manyvarieties and classifications of states and phases meetingnew progress of experiments.In recent years, non-Hermitian states of matter haveattracted considerable attention both theoretically andexperimentally [31–33]. The systems described by non-Hermitian Hamiltonians are usually non-conserved, suchas solids with finite quasi-particle lifetimes [34–37], ar-tificial lattice [38] with gain and loss or nonreciprocity[39–44], and etc. Recent developments have revived in-terest in various physical aspects. The flat band has alsobeen proposed in non-Hermitian systems [45–48], but stillremains largely unexplored. ∗ Electronic address: [email protected]
In this work, we study fermionic superfluidity withattractive s-wave interaction in an optical Lieb lattice.With the inclusion of both atom loss and inelastic colli-sion, together with an auxiliary lattice, the system is ef-fectively described by a non-Hermitian Hamiltonian withnon-reciprocal hopping amplitudes and complex interac-tion strength. The Lieb lattice features a diabolic singleDirac cone intersected with a flat band [49, 50]. In thepresence of non-hermiticity, the Dirac point will extendinto a pair of exceptional points. As suggested by previ-ous works, the non-Hermitian system with non-reciprocalhopping amplitudes will exhibit skin effect of which allbulk bands are pumped at the boundaries, as a manifes-tation of point gap topology associating with the excep-tional points [51–56]. In this work, we demonstrate thatthe skin effect is forbidden by the geometric frustrationof particle motion for the flat band. Furthermore, we findthat the skin effect prominently enhances the phase stiff-ness of the fermionic superfluidity. We adopt mean-fielddescription of the fermionic superfluidity by generaliz-ing the non-Hermitian Bardeen-Cooper-Schrieffer (BCS)theory developed in Ref. [57] to the multiband case, lead-ing to a non-Hermitian BdG Hamiltonian. In general, theemergence of the superconducting phase does not onlyrequire the form of the pairing field, but also the phasecoherence of the pairs, which enables the Meissner effect[58]. We solve the gap equation and calculate the super-fluid weight. We show that the superfluid weight for thenon-Hermitian superfluidity is related with the integral ofnon-Hermitian metric tensor of the quantum state man-ifold over the Brillouin zone, which is a manifestation ofnontrivial flat-band effect.This paper is organized as follows. In Section II, wepropose an ultracold-atom-based setup and present aneffective non-Hermitian Hamiltonian. In section III, theband structures and the localization properties of thestates are addressed in terms of interplay between theskin effect and destructive interference. Section IV devel- a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n ops a multiband mean-field description of the fermionicsuperfluid, and solve the gap equation. And in sectionV, we compute the superfluid weight and elucidate theunique criticality of the phase transition. Finally, a shortconclusion is given in Sec. VI. FIG. 1: (a) Schematic illustration of a proposed experimentalsetup with ultracold atoms in an optical lattice. (b) The Lieblattice with non-reciprocal hopping, which has three inequiv-alent sites in one unit cell. (c) The cross-sections of (a) along(i) the x direction and (ii) the y direction, with sublattices1,2 and 2,3 respectively. κ is the on-site decay rate in theauxiliary lattice. (d) The real part of the energy spectrum for δ/t = 0 . II. MODEL HAMILTONIAN
We consider a gas of fermionic atoms in a two-dimensional optical lattice V prm . with coherent couplingto the auxiliary degrees of freedom, as illustrated in Fig. 1. The primary lattice has a line-centered geometry,which could be experimentally realized by superimposingthree pairs of laser beams with the following formation[22, 49, 50], V prm . ( x, y ) = − V ( x )long cos ( k L x ) − V ( y )long cos ( k L y ) − V ( x )short cos (2 k L x ) − V ( y )short cos (2 k L y ) − V ( x )diag cos ( k L ( x − y )) − V ( y )diag cos ( k L ( x + y )) , (1)where k L = 2 π/λ is the wave number of the lattice and λ the wavelength of the lasers. The potential amplitudes V ( x,y )long , V ( x,y )short and V ( x,y )diag could be tuned by adjustingthe laser intensities, φ x,y and ϕ are the phases of laserbeams. For simplicity we choose ( V ( x ) long , V ( x ) short , V ( x ) diag ) =( V ( y ) long , V ( y ) short , V ( y ) diag ) = ( V long , V short , V diag ). To ensurethe nearest-neighbor sites are coupled, the potentialstrength of the auxiliary lattice is tuned opposite to thatof the primary lattice. The auxiliary lattice with decayis included to induce asymmetric hopping between thecoupled sites, which generates the skin effect as we willsee below.Two equally populated magnetic sublevels of the hy-perfine ground-state manifold are uploaded, to mimic theinteracting spin-up and spin-down electrons moving inthe lattice. Following the Ref. [29, 59, 61, 62], we pro-pose to engineer a collective loss from a combination ofon-site one-body losses of auxiliary lattice and two-bodydissipations of the primary states. In experiment, theone-body loss could be generated by applying a radiofrequency pulse to resonantly transfer the atoms to anirrelevant excited state, while the two-body dissipationcould be induced by inelastic collisions with photoasso-ciation process [63].The full open-system dynamics in the rotating frameof reference can be written as˙ ρ t = − i [ H + Ω2 ( (cid:88) i,j,σ a † i,j, ,σ ( c i,j, ,σ + ic i,j, ,σ ) + a † i,j, ,σ ( c i,j, ,σ + ic i +1 ,j, ,σ ) + a † i,j, ,σ ( c i,j, ,σ + ic i,j, ,σ )+ a † i,j, ,σ ( c i,j, ,σ + ic i +1 ,j, ,σ ) + H . c . ) , ρ t ] + (cid:88) i , j ,α D [ a i , j ,α,σ , c i , j ,α,σ ] ρ t (2)where a i,j,α,σ ’s ( a † i,j,α,σ ’s) denote the annihilation (cre-ation) operators of the particles with spin σ = ↑ , ↓ in theauxiliary orbital centered at r i,j,α ( α is the sublattice la-bel), c i,j,α,σ ’s ( c † i,j,α,σ ’s) denote corresponding annihila-tion (creation) operators of primary degrees of freedom,Ω is the coupling strength between the primary latticeand auxiliary lattice, H is the Hamiltonian for the pri- mary lattice, and D [ L ] ρ ≡ LρL † − { L † L, ρ } is the Lind-blad superoperator.In the regime that the on-site decay rate κ (cid:29) Ω, onecan adiabatically eliminate the decay modes in the aux-iliary lattice [59]. Thus the effective dynamics is welldescribed by ˙ ρ t = − i [ H , ρ t ] + D [ L ] ρ t , (3) H = − (cid:88) (cid:104) i α, j β (cid:105) ,σ t σ i α, j β c i ασ c † j βσ − ˜ U (cid:88) i α n i α ↑ n i α ↓ , (4) L = (cid:88) < i α, j β>,σ (cid:112) γ ( c i ασ + ic j βσ ) + (cid:88) i α √ Γ c i α ↓ c i α ↑ , (5)where γ = Ω / (2 κ ) and Γ is the two-body loss rate inthe primary lattice, we only consider the tight-bindingregime thus (cid:104) i α, j β (cid:105) runs over all nearest-neighbor sites[see Fig. 1 (b)], and we redefine i = ( i x , i y ) = ( i, j ).If we only consider the dynamics over a short time,the quantum jump term γLρ t L † is negligible. Thuswe have an effective non-Hermitian Hamiltonian H eff = H − ( i/ L † L [57, 64]. With Fourier transformation c k ασ = 1 / √ N c (cid:80) i e − i k · r i α c i ασ , the Hamiltonian in mo-mentum space reads H eff = H kin + H int − µN . The ki-netic term H kin is given by H kin = (cid:80) k Ψ † k H k Ψ k , withΨ k = [ c k α ↑ , c k α ↓ ] T , and H k ↑ = H k ↓ = a k λ + b k λ , (6)where a k = 2 t cos( k x / − iδ sin( k x / b k =2 t cos( k y /
2) + 2 iδ sin( k y /
2) ( δ ≡ γ ), and λ i ’s are the i-thGell-Mann matrices. The interaction term is given by H int = − U (cid:80) kk (cid:48) ,α c † k α ↑ c †− k α ↓ c − k (cid:48) α ↑ c k (cid:48) α ↓ , where interac-tion strength U ≡ ˜ U + i Γ / III. FLAT BAND LOCALIZATION AND SKINEFFECT
FIG. 2: The swapping of energy eigenvalues along the loopencircling the EP. The red line shows a path from θ = 0 to θ = 2 π . θ ∈ [0 , π ] parametrizes the loop L . Let us start by considering the interaction-free Hamil-tonian H k . The eigen-equation is given by H k | g n k (cid:105) = ε n k | g n k (cid:105) with the Bloch functions | g n k (cid:105) and the energy dispersion ε n k ( n = ± , ε ± = ± (cid:112) a k + b k [as il-lustrated in Fig. 1 (d,i)], while the middle band is purelyreal and strictly flat ε = 0, as in the Hermitian limit.This is a result of the chiral symmetry of the Hamil-tonian preserved under the non-Hermitian perturbationˆ CH k ˆ C = − H k , with ˆ C ≡ diag[1 , − , H NH = − iγ diag[1 , , − H k has a pair of EPs at K ± = ± (arccos( δ − t δ + t ) , arccos( δ − t δ + t )), which are con-nected by a bulk Fermi arc, as shown in Fig. 1 (d,i).As a bipartite quantum system from the chiral uni-versality classes, the Lieb lattice admits destructive in-terference of states on flat band, which guarantees co-herent cancellation of net particle flows thus leads tothe localization of states [1, 65]. A single-particle state | Φ (cid:105) = (cid:80) r ,α P r ,α c † r ,α | (cid:105) satisfies H L | Φ (cid:105) = 0 if and only if (cid:88) (cid:104) ( r ,α ) , ( r (cid:48) ,β ) (cid:105) P r (cid:48) ,β = 0 , ∀ r , α , (7)where H L is the lattice Hamiltonian in real space with U = 0 and | (cid:105) is the vacuum state. We call the statesatisfying condition Eq.(7) a ring mode state. To see this,we consider straightforward manipulations on Wannierstates, H L c † i | (cid:105) = [( t − δ ) c † i − a i x , + ( t + δ ) c † i + a i x , ] | (cid:105) , (8) H L c † i | (cid:105) = [( t + δ ) c † i − a i y , + ( t − δ ) c † i + a i y , ] | (cid:105) , (9)thus we have a ring-mode state with ε = 0, | Φ (cid:105) = 12 [ t c † i,j, − t c † i,j, + t c † i +1 ,j +1 , − t c † i +1 ,j +1 , ] | (cid:105) , (10)where t = (cid:113) t − δt + δ , t = (cid:113) t + δt − δ . Furthermore, any lin-ear superposition of the single-plaquette ring-mode statesgiven by Eq.(10) is also a solution, which gives arise toa large subspace with ε = 0. As revealed by Mielke [15],this can also be understood by the line graph theory.A line graph L ( G ) of a graph G is constructed with itsvertex set the edge set of G ( V, E ). The incidence ma-trix B ( G ) = ( b ve ) | V |×| E | (with unit entry b ve = 1 if thevertex v is incident to the edge e and zero otherwise) isrelated with the adjacency matrix A L of the line graphby B T B = 2 I + A L , due to the fact that every edge isalways connected with two vertexes. If the number ofvertexes n V is smaller than that of the edges zn V / A L will have a highly degenerate subspace with eigenvalue −
2, such that B ( G ) has large zero space.Due to the asymmetric hopping amplitudes, bulkstates of non-Hermitian systems exhibit anomalousboundary localization, dubbed the skin effect. As shownin Fig. 1 (d,ii), all states with nonzero energy are pumpedtowards the direction with stronger hopping amplitude,which give rise to corner-state distributions. This is remi-niscent of the second order skin effect studied in Ref. [60],but only happens for the two dispersive bands. The skineffect occurs as manifestation of the nontrivial point-gaptopology signaling with the presence of the EPs, whichis assigned with a winding number ν = 12 π (cid:73) L ∇ q arg[ ε + − ε − ] · d q , (11)where L is a closed loop encircling the EP. Each EP isassociated with ν = ± , as illustrated in Fig. 2. Thebands with a point gap (finite interior in the complexspectra) have nonzero direct current J = (cid:72) n ( E, E ∗ ) dE with n ( E, E ∗ ) the energy distribution, which gives rise tononreciprocal pumping, i.e. , the skin effect [53]. In con-trast, The flat band is purely real, and does not acquirea point gap in the complex plane, thus does not have theskin effect. IV. MEAN-FIELD DESCRIPTION
FIG. 3: Numerical solution of the order parameters as thefunction of Γ for ˜ U = 1 . β = ∞ . The red dashed lineindicates the real part of the order parameters at strong inter-action strength ˜ U = 10, while the blue dashed line indicatesthe imaginary part. We adopt a mean-field description within the frame-work of BCS theory and decouple the interaction termas H int ≈ (cid:88) k ,α ∆ α c † k α ↑ c †− k α ↓ + ¯∆ α c − k α ↑ c k α ↓ , (12)where ∆ α = − U (cid:80) k (cid:104)(cid:104) c − k α ↓ c k α ↑ (cid:105)(cid:105) and ¯∆ α = − U (cid:80) k (cid:104)(cid:104) c † k α ↑ c †− k α ↓ (cid:105)(cid:105) , and here we use the inner product in the biorthonormal basis (cid:104)(cid:104)O(cid:105)(cid:105) = (cid:104) ˜Ψ |O| Ψ (cid:105) , with | Ψ (cid:105) ( (cid:104) ˜Ψ | ) the right (left) ground eigenstate of the Hamilto-nian H eff and O any operator. In Nambu representationΨ k = [ c k α ↑ , c †− k α ↓ ] T , The mean-filed Hamiltonian reads H MF = (cid:80) k Ψ † k H ( k )Ψ k , with the Bogoliubov-de Gennes(BdG) Hamiltonian H ( k ) = (cid:20) H k ↑ − µ ∆∆ − ( H k ↓ + µ ) (cid:21) , (13)where is the 3 × , ∆ , ∆ ], and we have chosen the gauge for which H † MF = H MF , such that ∆ = ¯∆.To obtain the superfluid weight, we consider the sys-tem subject to a uniform gauge field A . By Peierls substi-tution k to k − q ( q = e A ) and diagonalizing the kineticblock, the Hamiltonian reads¯ H k ( q ) = (cid:20) ε k − q − µ ˜ G k − q ∆ G k + q ˜ G k + q ∆ G k − q − ( ε k + q − µ ) (cid:21) , (14)where ε k = diag[ ε n k ], and G k =[ | g + k (cid:105) , | g k (cid:105) , | g − k (cid:105) ] , ˜ G k =[ (cid:104) ˜ g + k | , (cid:104) ˜ g k | , (cid:104) ˜ g − k | ] T . (15)The Hamiltonian (14) can be diagonalized as ¯ H k = (cid:80) k E k (¯ γ k ↑ γ k ↑ + ¯ γ − k ↓ γ − k ↓ ) − (cid:80) k E k , where γ k σ =[ γ k ασ ] T , E k = diag[E + k , E k , E − k ], E ± k = (cid:112) ε k + ∆ s ± ∆ d , E k = ∆ , with ∆ d ≡ (∆ − ∆ ) / s ≡ (∆ + ∆ ) /
2. The quasiparticle operators are given by¯ γ k ↑ = u k c † k ↑ − v k c − k ↓ , ¯ γ − k ↓ = v k c k ↑ + u k c †− k ↓ ,γ k ↑ = u k c k ↑ − v k c †− k ↓ ,γ k ↓ = v k c † k ↑ + u k c − k ↓ , (16)where the coefficients u k and v k take values of u k = 1 √ cos φ k − cos φ k φ k φ k ,v k = 1 √ sin φ k − sin φ k φ k φ k , (17)with cos φ k = √ (cid:113) (cid:15) k √ (cid:15) +∆ , and sin φ k = √ (cid:113) − (cid:15) k √ (cid:15) +∆ . From Eq. (16)-(17), the gap equationis obtained,∆ = U N c (cid:88) k [ t + , k sin φ k + t − , k ] + U β ∆ , ∆ = U N c (cid:88) k [ t + , k sin φ k − t − , k ] , where t ± , k = 12 (tanh βE + , k ± tanh βE − , k , (18)and N c is the number of unit cells and we also have∆ = ∆ as a result of the sublattice (chiral) symmetryof the Hamiltonian. Note that the flat-band contributiononly enters in the gap equations for the order parameter∆ (∆ ), due to the destructive interference on sublattice-2 [4].We numerically solve the gap Eq. (18). The results areshown in Fig. 3. Here we only focus on the zero temper-ature solutions since the Fermi distribution does not giveoccupation probabilities due to the complex eigenener-gies thus the temperature is not well defined for the non-Hermitian systems. Fig. 3 (a) and (b) show the pairingorders as functions of two-body loss Γ. Both Re∆ andRe∆ are suppressed by the two-body loss, and vanishafter crossing a critical point. However, when the two-body loss is large enough, Re∆ and Re∆ acquire non-trivial values again and are even enhanced as the lossincreases, which can be attributed to the localization-enhanced pairing due to the quantum Zeno effect forstrong dissipation. This is reminiscent of the behaviorof the pairing field in a single-band BCS system [57].Although the pair field is suppressed by the loss, thegap ∆ d in the mean-field spectrum is instead enhanced,due to slower decrement occurring in the pair field Re∆ for sublattice-1 than for Re∆ , as shown in Fig. 3 (d).This implies that the superfluidity is more robust againstthe two-body loss perturbation for the flat bands. V. SUPERFLUID WEIGHT
To compute the superfluid weight, we expand the freeenergy F ( A ) to second order F ( A ) ≈ F + V [ D s ] ij A i A j ,where V is the system area, D s is the superfluid weight,and F ( A ) = − β ln[ e − β ¯ H k ]. Thus the superfluid weight isgiven by [ D s ] i,j = 1 V (cid:126) ∂ F ∂q i ∂q j (cid:12)(cid:12)(cid:12)(cid:12) µ, q =0 , (19)where i, j = x, y are spatial indices.For multiband superconducting phase, geometricalcontribution enters the superfluid weight in the form ofFubini-Study metric integral of the quantum state man-ifold, beyond the conventional Landau-Ginzburg formal-ism. Specifically, For Hamiltonian Eq. (14), the super-fluid weight reads D i,js = 1 A (cid:126) (cid:88) k [ − t + , k cos φ k ∂ k i ∂ k j (cid:15) k − t − , k E + , k − E − , k ∂ k i (cid:15) k ∂ k j (cid:15) k + 2∆ (tanh β ∆ t + , k sin φ k + t − , k )( (cid:104) ∂ k i ˜ s k | ∂ k j s k (cid:105) + (cid:104) ∂ k j ˜ s k | ∂ k i s k (cid:105) ) − ∆ (cid:104) ∂ k i ˜ s k | s k (cid:105)(cid:104) ˜ s k | ∂ k j s k (cid:105) f ( k ) − ∆ ( (cid:104) ∂ k i ˜ s k | c k (cid:105)(cid:104) ˜ c k | ∂ k j s k (cid:105) + ( i ↔ j )) g ( k )] . (20)The flat band has contribution,[ D s ] i,j | f . b = ∆ π (cid:126) tanh β ∆2 B ij | f . b , (21)where B ij = (2 π ) − (cid:82) B . Z . d k G ij is the Brillouin-zone in-tegral of the tensor, with its real part the quantum met-ric. The quantum metric is defined as M ij = Re G ij ,where G ij = 12 [ (cid:104) ∂ k i ˜ g k | ∂ k j g k (cid:105) + (cid:104) ∂ k j ˜ g k | ∂ k i g k (cid:105)−(cid:104) ∂ k i ˜ g k | g k (cid:105)(cid:104) ˜ g k | ∂ k j g k (cid:105)−(cid:104) ∂ k j ˜ g k | g k (cid:105)(cid:104) ˜ g k | ∂ k i g k (cid:105) ] . (22)In Fig. 4, we plot the geometric contribution and theconventional contribution respectively. Here we only fo-cus on the diagonal components of the superfluid weighttensor [ D s ] xx = [ D s ] yy ≈ D s , since the off-diagonalcomponents [ D s ] xy/yx is small. In the absence of non-Hermiticity, the superfluid weight D s is enhanced as˜ U increases in the weak-coupling regime. But D s de-creases after peaking at ˜ U ∼ t with ˜ U further increases. The non-monotonicity of D s can be understood fromthe viewpoint of pseudopotential theory and a crossoverfrom unbound Cooper pairs to performed fermions (orBose liquid). Here we give a rough estimation. Thes-wave scattering length a s is rebuilt from the relation˜ U = 4 π (cid:126) a s /M with M the mass of atoms. Taking (cid:126) = M r = 1, and the Fermi momentum k F ∼ π , we have1 /k F a s ∼ − − ˜ U = −
4. After reaching the unitarityregime ( − < /k F a s < D s . Inthe strong coupling regime, the superfluid weight is morerobust to the non-Hermitian perturbations, as shown inFig. 4.Notably, we find that the skin effect enhances the su-perfluid weight in the weak-coupling regime, as shown inFig. 4 (b) for the non-reciprocal BdG Hamiltonian. Asrevealed in Sec. III, in the presence of non-reciprocity,compact localized states for all bands have large den-sity due to the skin effect and the destructive interfer- (a) (b)(c) (d) FIG. 4: (a) and (b) The geometric and conventional con-tribution in the superfluid weight at zero temperature withonly non-reciprocity respectively. (c) and (d) The geometricand conventional contribution in the superfluid weight at zerotemperature with only two-body loss. The inset in (d) showsthe superfluid weights as functions of two-body loss strengthΓ at ˜ U = 1 . ence, which enhances the overlapping of the Cooper pairsthus the coherence of the superconducting phase is pro-moted for the conventional part. In contrast, the geo-metric part decreases as non-reciprocity increases in theweak-coupling regime. On the other hand, for the Hamil-tonian only with two-body loss, the superfluid weight isdecreased by the loss as shown in Fig. 4 (d), since thetwo-body loss does not induce the skin effect.Furthermore, we also find that the overall behavior ofthe geometric part mostly depends on the gap parameter∆ d after comparing the results in Fig. 4 and ∆ d undersame system parameters. VI. DISCUSSION AND CONCLUSION
In two dimension, the superfluid weight is related withthe Berezinskii-Kosterlitz-Thouless transition. A largersuperfluid weight at zero temperature usually implies ahigher transition temperature [19]. Thus the superfluidweight can be estimated by the critical temperature inthe experiment.In conclusion, we have proposed a cold atomic setupwith uniform s-wave interaction and atom loss, which iscaptured by a non-Hermitian Hamiltonian. We startedby elucidating unique localization properties of the Blochbands as an interplay of skin effect and the flat-banddestructive interference. Then with the inclusion ofcomplex s-wave interaction, we revealed novel critical- ity of the non-Hermitian superfluidity related with theflat band within the framework of a mean-field BCS the-ory, in terms of both order parameters and the super-fluid weight. We built a relation between the superfluidweight and the geometry of the Bloch states manifold inthe non-Hermitian case. And we have also showed thatthe skin effect would optimize the superfluid weight. Ourwork provides an example that the skin effect can bringimportant physical implications.
Acknowledgments
We thank Dan-Bo Zhang and Yu-Guo Liu for use-ful discussions. This work was supported by the Key-Area Research and Development Program of Guang-Dong Province (Grant No. 2019B030330001), the Na-tional Natural Science Foundation of China (GrantsNo. 12074180 and No. U1801661), the Key Projectof Science and Technology of Guangzhou (Grants No.201804020055 and No. 2019050001), and the Na-tional Key Research and Development Program of China(Grants No. 2016YFA0301800).
Appendix A: Quantum metric tensor
In this section, we introduce the non-Hermitian quan-tum metric tensor, following the definition in Ref. [66].For the set of the quantum states {| g n k (cid:105)} on the BZ torus,the density matrix is defined as, ρ n ( k ) ≡ | ˜ g n k (cid:105)(cid:104) g n k | . (A1)The fidelity between ρ n ( k ) and ρ n ( k + δ k ) is given by F ( ρ n ( k ) , ρ n ( k + δ k )) = tr (cid:113) | ρ / n ( k ) ρ n ( k + δ k ) ρ / n ( k ) | . (A2)The Fubini-Study metric defines a distance of nearbystates on the BZ torus, ds := 2[1 − F ( ρ n ( k ) , ρ n ( k + δ k ))] . (A3)By expanding the states | g n k + δ k (˜ g n k + δ k ) (cid:105) to second or-der, we have ds = 12 Re( (cid:104) ∂ µ ˜ g n k | ∂ ν g n k (cid:105) + (cid:104) ∂ ν ˜ g n k ( λ ) | ∂ µ g n k (cid:105)− (cid:104) ∂ µ ˜ g n k | g n k (cid:105)(cid:104) ˜ g n k | ∂ ν g n k (cid:105) ) dk µ dk ν . (A4)Here to shorten notations we define ∂ µ ≡ ∂ k µ .After noting that (cid:104) ∂ µ ˜ g n k | g n k (cid:105)(cid:104) ˜ g n k | ∂ ν g n k (cid:105) = (cid:104) ∂ ν ˜ g n k | g n k (cid:105)(cid:104) ˜ g n k | ∂ µ g n k (cid:105) , we can rewrite the quan-tum metric tensor as M ij = 12 Re[ (cid:104) ∂ k i ˜ g n k | ∂ k j g n k (cid:105) + (cid:104) ∂ k j ˜ g n k | ∂ k i g n k (cid:105)−(cid:104) ∂ k i ˜ g n k | g n k (cid:105)(cid:104) ˜ g n k | ∂ k j g n k (cid:105)−(cid:104) ∂ k j ˜ g n k | g n k (cid:105)(cid:104) ˜ g n k | ∂ k i g n k (cid:105) ] . (A5) Appendix B: Derivation details of the superfluidweight
By virtual of the biorthogonal representation, we de-rive the superfuild weight of the non-Hermitian super-fluidity, following the method originally developed in theRefs. [3, 4, 67]. The first derivative of the free energyΩ( q ) is the current density, J ( q ) = − V (cid:126) (cid:88) k Tr (cid:2) sign ( E k ( q )) W T k ( q ) ∂ q H k ( q ) W k ( q ) (cid:3) , (B1) − ∂ q H k ( q ) = (cid:20) ∂ k ε k − q ∂ q D k ( q ) − ∂ q D k ( − q ) ∂ k ε k + q (cid:21) , (B2)Here we define D k ≡ ˜ G k − q ∆ G k + q , and W k ≡ (cid:20) u k − v k v k u k (cid:21) . (B3)Straightforward calculation shows that,˜ G k ∆ G k = ∆ A (cid:104) ˜ s k | s k (cid:105) √ (cid:104) ˜ s k | c k (cid:105) (cid:104) ˜ s k | s k (cid:105)√ (cid:104) ˜ c k | s k (cid:105) (cid:104) ˜ c k | c k (cid:105) √ (cid:104) ˜ c k | s k (cid:105)(cid:104) ˜ s k | s k (cid:105) √ (cid:104) ˜ s k | c k (cid:105) (cid:104) ˜ s k | s k (cid:105) + ∆ B −
10 0 0 − . (B4)Here we denote k = k − q and k = k + q , and we intro-duce a two-component spinor | s k (cid:105) (together with (cid:104) ˜ s k | )and its partner | c k (cid:105) = iσ y | s k (cid:105) (together with (cid:104) ˜ c k | ), | s k (cid:105) = 1 (cid:112) a k + b k (cid:20) a k b k (cid:21) , | c k (cid:105) = 1 (cid:112) a k + b k (cid:20) b k − a k (cid:21) , (B5) (cid:104) ˜ s k | = [ a k , b k ] / (cid:113) a k + b k , (cid:104) ˜ c k | = [ b k , − a k ] / (cid:113) a k + b k . (B6)Note that (cid:104) ˜ s k (˜ c k ) | is not the Hermitian conjugate of | s k ( c k ) (cid:105) . The derivate of the current density gives riseto the superfluid weight after setting q = 0,[ D s ] ij = 1 V (cid:126) ∂ F ∂q i ∂q j (cid:12)(cid:12)(cid:12)(cid:12) q =0 = [ D s , conv ] i,j + [ D s , geom ] i,j , (B7)[ D s , conv ] ij = 2 V (cid:126) (cid:88) k Tr[( v k e − βE k +1 v T k + u k e βE k +1 u T k ) ∂ k i ∂ k j ε k ] , (B8)[ D s , geom ] ij = 1 V (cid:126) { (cid:88) k Tr[( u k v T k − u k e βE k +1 v T k − v k e βE k +1 v T k ) ∂ q i ∂ q j D k ( q = 0)] − (cid:88) k (cid:88) a,b [ T k ] a,b [ N k ,i ] a,b [ N k ,j ] b,a } , (B9)where N k ,i = W T k ( q = 0) ∂ q i ¯ H k ( q = 0) W k ( q = 0),and the off-diagonal components of T k [ T k ] a,b =(tanh( βE k / a,a − [tanh( βE k / b,b ) / ([ E k ] a,a − [ E k ] b,b ), and the diagonal components [ T k ] a,a =[ β/ ( βE k / a,a . [1] Z. Liu, F. Liu, and Y. S. Wu, Exotic electronic states inthe world of flat bands: From theory to material, Chin. Phys. B , 077308 (2014).[2] K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Nearly Flatbands with Nontrivial Topology, Phys. Rev. Lett. , 236803 (2011).[3] S. Peotta and P. T¨orm¨a, Superfluidity in topologicallynontrivial flat bands, Nat. Commun. 6, 8944 (2015).[4] A. Julku, S. Peotta, T.-I. Vanhala, D.-H. Kim, and P.T¨orm¨a, Geometric Origin of Superfluidity in the Lieb-Lattice Flat Band, Phys. Rev. Lett. , 045303 (2016).[5] P. T¨orm¨a, L. Liang, and S. Peotta, Quantum metric andeffective mass of a two-body bound state in a flat band,Phys. Rev. B , 220511(R) (2018).[6] J. Rhim, K. Kim, and B. Yang, Quantum distance andanomalous Landau levels of flat bands, Nature. , 59(2020).[7] C. Danieli, A. Andreanov, and S. Flach, Many-body flat-band localization, Phys. Rev. B , 041116(R) (2020).[8] Y. Kuno, T. Orito, and I. Ichinose, Flat-band many-bodylocalization and ergodicity breaking in the Creutz ladder,New J. Phys. , 964(2014).[10] R. Mondaini, G.-G. Batrouni, and B. Gr´emaud, Pairingand superconductivity in the flat band: Creutz lattice,Phys. Rev. B , 155142 (2018).[11] J. Mao, S.P. Milovanovi´c, M. Andelkovi´c, et al., Evidenceof flat bands and correlated states in buckled graphenesuperlattices, Nature , 215 (2020).[12] T. Neupert, L. Santos, C. Chamon, and C. Mudry, Frac-tional Quantum Hall States at Zero Magnetic Field,Phys. Rev. Lett. , 236804 (2011).[13] E. J. Bregholtz and Z. Liu, Topological flat band modelsand fractional Chern insulators, Int. J. Mod. Phys. B ,1330017 (2013).[14] E. H. Lieb, Two Theorems on the Hubbard Model, Phys.Rev. Lett. , 1201 (1989).[15] A Mielke, Ferromagnetic ground states for the Hubbardmodel on line graphs, J. Phys. A: Math. Gen. , L73(1991).[16] N. B. Kopnin, T. T. Heikkil?, and G. E. Volovik,Hightemperature surface superconductivity in topologi-cal flatband systems, Phys. Rev. B , 220503(R) (2011).[17] V. I. Iglovikov, F. H´ebert, B. Gr´emaud, G. G. Batrouni,and R. T. Scalettar, Superconducting transitions in flat-band systems, Phys. Rev. B , 094506 (2014).[18] V. J. Emery, Theory of High- T c Superconductivity in Ox-ides, Phys. Rev. Lett. , 2794 (1987).[19] F. Xie, Z. Song, B. Lian, and B. A. Bernevig,Topology-Bounded Superfluid Weight in Twisted BilayerGraphene, Phys. Rev. Lett. , 167002 (2020)[20] X. Hu , T. Hyart, D. I. Pikulin, and Enrico Rossi, Geo-metric and Conventional Contribution to the SuperfluidWeight in Twisted Bilayer Graphene, Phys. Rev. Lett. , 237002 (2019).[21] A. Julku, T. J. Peltonen, L. Liang, et al., Super-fluid weight and Berezinskii-Kosterlitz-Thouless transi-tion temperature of twisted bilayer graphene, Phys Rev.B , 060505(R) (2020).[22] S. Taie, H. Ozawa, T. Ichinose, T. Nishio, S. Naka-jima and Y. Takahashi, Coherent driving and freezingof bosonic matter wave in an optical Lieb lattice, Sci.Adv. , 1500854 (2015).[23] T.-H. Leung, M. N. Schwarz, S.-W. Chang, C. D.Brown1, G. Unnikrishnan , and D. Stamper-Kurn, Interaction-Enhanced Group Velocity of Bosons in theFlat Band of an Optical Kagome Lattice, Phys. Rev.Lett. , 133001 (2020).[24] G.-W. Chern, C.-C. Chien, and M. Di Ventra, Dynam-ically generated flat-band phases in optical kagome lat-tices, Phys. Rev. A , 013609 (2014).[25] G.-B. Jo, J. Guzman, and C. K. Thomas, UltracoldAtoms in a Tunable Optical Kagome Lattice, Phys. Rev.Lett. , 045305 (2012).[26] D. Guzm´an-Silva, C. Mej´ia-Cort´es, M. A. Bandres, M.C. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Sza-meit, and R. A. Vicencio, Experimental observation ofbulk and edge transport in photonic Lieb lattices, NewJ. Phys. , 063061 (2014).[27] R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, etal., Observation of Localized States in Lieb Photonic Lat-tices, Phys. Rev. Lett. , 245503 (2015).[28] S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman,P. ¨Ohberg, E. Andersson, and R. R. Thomson, Observa-tion of a Localized Flat-Band State in a Photonic LiebLattice, Phys. Rev. Lett. , 245504 (2015).[29] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi-gashikawa, and M. Ueda, Topological phases of non-Hermitian systems, Phys. Rev. X , 031079 (2018).[30] M. Slot, T. Gardenier, P. Jacobse, et al., Experimentalrealization and characterization of an electronic Lieb lat-tice, Nature Phys , 672C676 (2017).[31] A. Ghatak and T. Das, New topological invariants in non-Hermitian systems, J. Phys. D: Appl. Phys. , 263001(2019).[32] E. J. Bergholtz, J. C. Budich, and F. K. Kunst,Exceptional Topology of Non-Hermitian Systems,arXiv:1912.10048 (2019).[33] Y. Ashida, Z. Gong, and M. Ueda, Non-HermitianPhysics, arXiv:2006.01837 (2020).[34] V. Kozii and L. Fu, Non-Hermitian topological theoryof finite-lifetime quasiparticles: prediction of bulk Fermiarc due to exceptional point, arXiv:1708.05841 (2017).[35] H. Shen, B. Zhen, and L. Fu, Topological band theoryfor non-Hermitian Hamiltonians, Phys. Rev. Lett. ,146402 (2018).[36] M. Papaj, H. Isobe, and L. Fu, Nodal arc of disorderedDirac fermions and non-Hermitian band theory, Phys.Rev. B , 201107 (2019).[37] X. Shen, F. Wang, Z. Li, and Z. Wu, Landau-Zener-St¨uckelberg interferometry in PT-symmetric non-Hermitian models, Phys. Rev. A , 062514 (2019).[38] D. W. Zhang, Y. Q. Zhu, Y. X. Zhao, H. Yan, and S. L.Zhu, Topological quantum matter with cold atoms, Adv.Phys. , 253 (2018).[39] D.-W. Zhang, L.-Z. Tang, L.-J. Lang, H. Yan, and S.-L.Zhu, Non-Hermitian topological Anderson insulator, Sci.China-Phys. Mech. Astron. , 267062 (2020).[40] H. Jiang, L.-J. Lang, C. Yang, S.-L. Zhu, and S. Chen,Interplay of non-Hermitian skin effects and Anderson lo-calization in nonreciprocal quasiperiodic lattices, Phys.Rev. B , 054301 (2019).[41] J. Li, A. K. Harter, J. Liu, L. Melo1, Y. N. Joglekar, andL. Luo, Observation of parity-time symmetry breakingtransitions in a dissipative Floquet system of ultracoldatoms, Nat. Commun, , 855 (2019).[42] H. Zhou et al., Observation of bulk Fermi arc and polar-ization half charge from paired exceptional points, Sci- ence, , 1009-1012 (2018).[43] A. Cerjan et al., Experimental realization of a Weyl ex-ceptional ring, Nat. Photonics , 623-628 (2019).[44] L. Z. Tang, L. F. Zhang, G. Q. Zhang, and D.-W. Zhang,Topological Anderson insulators in two-dimensional non-Hermitian disordered systems, Phys. Rev. A , 063612(2020).[45] S. M. Zhang, and L Jin., Flat band in two-dimensionalnon-Hermitian optical lattices, Phys. Rev. A , 043808(2019).[46] L. Jin, Flat band induced by the interplay of syntheticmagnetic flux and non-Hermiticity, Phys. Rev. A ,033810 (2019).[47] D. Leykam, S. Flach, and Y. D. Chong, Flat bands inlattices with non-Hermitian coupling, Phys. Rev. B ,064305 (2017).[48] H. Ramezani, Non-Hermiticity-induced flat band, Phys.Rev. A , 011802(R) (2017).[49] R. Shen, L. B. Shao, B. Wang, and D. Y. Xing, SingleDirac cone with a flat band touching on line-centered-square optical lattices, Phys. Rev. B , 041410 (2010).[50] C. Weeks and M. Franz. Topological insulators on theLieb and perovskite lattices. Phys. Rev. B , 085310(2010).[51] S. Yao and Z. Wang, Edge states and topological invari-ants of non-Hermitian systems, Phys. Rev. Lett. ,086803 (2018).[52] F. Song, S. Yao, and Z. Wang, Non-Hermitian skin effectand chiral damping in open quantum systems, Phys. Rev.Lett. , 170401 (2019).[53] K. Zhang, Z. Yang, and C. Fang, Correspondencebetween Winding Numbers and Skin Modes in Non-Hermitian Systems, Phys. Rev. Lett. , 126402 (2020).[54] L. Li, C. H. Lee, and J. Gong, Topological Switch forNon-Hermitian Skin Effect in Cold-Atom Systems withLoss, Phys. Rev. Lett. , 250402 (2020).[55] N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato,Topological Origin of Non-Hermitian Skin Effects, Phys.Rev. Lett. , 086801 (2020).[56] L. Jin, Z. Song, Bulk-boundary correspondence in a non- Hermitian system in one dimension with chiral inversionsymmetry, Phys. Rev. B , 081103 (2019).[57] K. Yamamoto, M. Nakagawa, K. Adachi, K. Takasan,M. Ueda, and N. Kawakami, Theory of non-hermitianfermionic superfluidity with a complex-valued interac-tion, Phys. Rev. Lett. , 123601 (2019).[58] J. Corson, R. Mallozzi, J. Orenstein, J. Eckstein, andI. Bozovic,, Vanishing of phase coherence in underdopedBi Sr CaCu O δ . Nature , 221 (1999).[59] T. Liu, Y. Zhang, Q. Ai, Z. Gong, K. Kawabata, M.Ueda, and F. Nori, Second-Order Topological Phases inNon-Hermitian Systems. Phys. Rev. Lett. , 076801(2019).[60] C. H. Lee, L. Li, and J. Gong, Hybrid Higher-Order SkinTopological Modes in Nonreciprocal Systems, Phys. Rev.Lett. , 016805 (2019).[61] M. M¨uller, S. Diehl, G. Pupillo, and P. Zoller, EngineeredOpen Systems and Quantum Simulations with Atomsand Ions, Adv. At. Mol. Opt. Phys. , 1 (2012).[62] S. Diehl, E. Rico, M.A. Baranov, and P. Zoller, Topologyby Dissipation in Atomic Quantum Wires, Nat. Phys. ,971 (2011).[63] T. Tomita, S. Nakajima, I. Danshita, Y. Takasu, andY. Takahashi, Observation of the Mott insulator to su-perfluid crossover of a driven-dissipative Bose-Hubbardsystem, Sci. Adv. , e1701513 (2017).[64] J.F. Poyatos, J.I. Cirac, and P. Zoller, Quantum Reser-voir Engineering with Laser Cooled Trapped Ions, Phys.Rev. Lett. , 4728 (1996).[65] Z. H. Yang, Y. P. Wang, Z. Y. Xue, W.-L. Yang, Y. Hu,J.-H. Gao, and Y. Wu, Circuit quantum electrodynamicssimulator of flat band physics in a Lieb lattice, Phys.Rev. A , 062319 (2016).[66] D.-J. Zhang, Q.-H. Wang, and J. Gong, Quantum ge-ometric tensor in PT -symmetric quantum mechanics,Phys. Rev. A , 042104 (2019).[67] L. Liang, T. I. Vanhala, S. Peotta, T. Siro, A. Harju, andP. T¨orm¨a, Band geometry, Berry curvature, and super-fluid weight, Phys. Rev. B95