A toolbox for elementary fermions with a dipolar Fermi gas in a 3D optical lattice
TTopological supersolids with tunable chern numbers of a dipolar Fermi gas in a three-dimensionalanisotropic optical lattice
Shuai Li,
1, 2
Biao Dong,
1, 2
Hongrong Li,
1, 2
Fuli Li,
1, 2 and Bo Liu
1, 2, ∗ Department of Applied Physics, School of Science, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China
Supersolids are among the most desirable unconventional many-body quantum states, which are linked to thefundamental issue of the coexistence of crystalline long-range order and off-diagonal long-range order (super-fluidity). The recent experimental breakthrough in implementation of tuning the dipole-dipole interaction witha rotating magnetic field in quantum gases [Phys. Rev. Lett. 120, 230401 (2018)] opens up a new thrust towardsdiscovering new types of supersolids, which have no prior analogs in previous studies. Here we report that var-ious fantastic topological supersolids emerge in a single-component dipolar Fermi gas trapped in an anisotropic3D optical lattice with an effective dipolar interaction engineered by a rotating external field. Through simplyadjusting the average filling of our proposed lattice system, these supersolids demonstrate novel features withhighly tunable chern numbers. The system undergoes phase transitions between a p -wave superfluid and differ-ent topological supersolids when increasing the dipolar interaction, whose experimental signatures include thehighly number-adjustable chiral edge modes as well as Majorana Fermions. Whether superfluidity and broken translational symmetrycan coexist, leading to supersolidity, is a long standing issuein condensed matter physics [1]. It has attracted tremendousinterests in both theoretical and experimental studies duringthe past years [2–6]. Supersolid is an exotic quantum phasecharacterized by two independent spontaneously broken sym-metries, i.e., U (1) and translation with corresponding super-fluidity and density order [7, 8]. It leads to a variety of fas-cinating phenomena such as the unusual reduction of the mo-ment of inertia (Non-classical Rotational Inertia) [1, 9] andother novel transport features [10, 11]. The fate of this con-ceptually important phase in solid helium remains, however,debatable and to prove its existence is still an open question inrecent experiments [12–15]. Besides the continuously grow-ing efforts in solids, there have been great interests in search-ing such a long-sought exotic state of matter via ultracoldgases in both experimental and theoretical studies, motivatedby the recent experimental advances to create tunable inter-acting ultracold atoms [16–30]. Supersolids were previouslypredicted to appear in polar molecules, magnetic and Rydbergatoms [31–36]. Along this line, there have been several excit-ing experimental breakthroughs, such as the recent observa-tion of supersolid orders in spin-orbit-coupled or cavity pho-ton assisted Bose-Einstein condensates [37, 38]. In particular,the tunable spin-orbit coupling or non-Abelian gauge fields ingeneral involved in supersolid phenomena would open up anew thrust on exploring the interplay between topology andsupersolidity. However, henceforth proposed cold-atom re-alizations of topological supersolids have required more so-phisticated experimental designs, such as the spin-dependentoptical lattices in spin-orbit coupled gases or a special tun-nelling scheme in a dipolar fermi gas [39, 40], which havenot yet been realized in experiments and remain a subject of asubstantial experimental effort [22, 41].Here we report the emergence of fantastic topological su-persolidity with highly tunable chern numbers in a single-component dipolar Fermi gas trapped in an anisotropic 3D optical lattice with an effective dipolar interaction engineeredby a rotating external field. The key idea here is to engineer adirection-dependent dipole-dipole interaction combined withthe density redistribution effect caused by the external poten-tial, i.e., optical lattices. We shall illustrate this idea with amagnetic dipolar Fermi gas composed of one hyperfine sate inan anisotropic 3D optical lattice. The direction of dipole mo-ments can be fixed by applying an external magnetic field. Letthe external field be orientated at a small angle with respectto the xy -plane and rotate fast around the z -axis. The time-averaged interaction between dipoles is isotropically attrac-tive in the xy -plane and repulsive along the z -direction. Sucha scheme has been realized in the experimental system of Dyrecently [42]. In general, the attraction is expected to causeCooper pairing instability in the xy -plane, while the repulsionshould restrict the pairing along the z -direction and leads toform certain density wave (CDW) pattern on the top of latticebackground. Their combined effect could give rise to superso-lidity with fascinating topological properties. Such a heuris-tically argued result is indeed confirmed by a self-consistentcalculation through the model to be introduced below. Thisidea is motivated by the recent rapid experimental progressin both magnetic dipolar atoms (such as Dy [43, 44] and Er [45, 46] atoms) and polar molecules [47, 48]. Tremen-dous interests have been attracted on exploring dipolar ef-fects in many-body quantum phases [41]. In particular, var-ious exotic quantum matter arising from the anisotropic effectof dipolar interaction has been predicted, such as a p -wavesuperfluid with the dominant p z symmetry, Weyl superfluid-ity [49], p + ip superfluids [50–52] in a single D plane andthe interlayer superfluidity or supersolids [53–56]. In the fol-lowing, we shall show that our proposed system can lead tounexpected topological supersolids. Effective model —
Consider a spinless dipolar Fermi gassubjected to an external rotating magnetic field [42] B ( t ) = B [ˆ z cos ϕ + sin ϕ (ˆ x cos Ω t + ˆ y sin Ω t )] ,where Ω is the ro-tation frequency, B is the magnitude of magnetic field, the a r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r rotation axis is z , and ϕ is the angle between the mag-netic field and z -axis. In strong magnetic fields, dipolesare aligned parallel to B ( t ) . With fast rotations, the effec-tive interaction between dipoles is the time-averaged interac-tion V ( r ) = d (3 cos ϕ − r (1 − θ ) ≡ d (cid:48) r (1 − θ ) ,where d (cid:48) ≡ d ϕ − with the magnetic dipole mo-ment d , r is the vector connecting two dipolar particles, and θ is the angle between r and z -axis. The effective attrac-tion, V ( r ) < , is created by making cos ϕ < (cid:112) / ,which is our focus here. We then further consider thesedipolar atoms loaded in a three dimensional optical lattice V opt ( r ) = − V [cos ( k Lx x ) + cos ( k Ly y )] − V z cos ( k Lz z ) ,where k Lx , k Ly and k Lz are the wave vectors of the laserfields and the corresponding lattice constants are defined as a x = π/k Lx , a y = π/k Ly and a z = π/k Lz along x, y and z directions respectively. Here V and V z are the lattice depthin xy -plane and z -direction respectively. In this work, we con-sider an anisotropic 3D lattice with a x = a y ≡ a < a z andthe lattice depth is large enough. Therefore, the system can bedescribed by the following lowest band Fermi-Hubbard modelin the tight binding regime: H = − (cid:88) α = x,y,z (cid:88) i t α ( c † i c i + e α + h.c. ) − µ (cid:88) i c † i c i + 12 (cid:88) i (cid:54) = j V i − j c † i c † j c j c i , (1)where t x = t y ≡ t and t z are the hopping amplitude de-scribing the tunneling along x, y and z directions respec-tively. i, j are the lattice site indices denoting the latticesites R i and R j , µ is the chemical potential and e α repre-sents the unit vector. The dipole-dipole interaction is given by V i − j = d (cid:48) | R i − R j | − i z − j z ) a z | R i − R j | .We apply the self-consistent Hartree-Fock-Bogoliubov(HFB) method, in which the Hamiltonian in Eq. (1) can beapproximated as H MF = − (cid:88) α = x,y,z (cid:88) i t α ( c † i c i + e α + h.c. ) + 12 (cid:88) i (cid:54) = j V i − j ( n j c † i c i + n i c † j c j ) − µ (cid:88) i c † i c i + 12 (cid:88) i (cid:54) = j V i − j [( (cid:104) c † i c † j (cid:105) c j c i + h.c. ) − ( (cid:104) c † i c j (cid:105) c † j c i + h.c. )] − E I , (2)where n i = (cid:104) c † i c i (cid:105) is the local density and E I = (cid:80) i (cid:54) = j V i − j ( n i n j − |(cid:104) c † i c j (cid:105)| + |(cid:104) c j c i (cid:105)| ) . To describe CDWorder, the density distribution can be expressed as n i = n + C cos( Q · R i ) and Q represents the periodicity of den-sity pattern. Here n = (cid:80) i (cid:104) c † i c i (cid:105) /N L is the average filling ofthe system and N L is the total lattice site. We also introducethe superfluid order parameter as ∆ ij = V i − j (cid:104) c j c i (cid:105) . Thenin the momentum space, we have H MF = (cid:80) k ξ k c † k c k + (cid:80) k ( ∆( k )2 c † k c †− k + h.c. ) + (cid:80) Q m = ± Q (cid:80) k ( δ Q m c † k c k + Q m + h.c. ) − E I , where ξ k = ε k + Σ k − µ with the band energy ε k = − t (cos k x a + cos k y a ) − t z cos k z a z and µ is thechemical potential. Σ k is the Hartree-Fock self-energy givenby Σ k = V (0) n − N L (cid:80) k (cid:48) V ( k − k (cid:48) ) n k (cid:48) with V (0) = (cid:80) n (cid:54) =0 V n , for example V (0) = − . Jt when considering a z /a = 3 . , where J ≡ | d (cid:48) / ( ta ) | capturing the strength ofdipolar interaction, and δ Q m = V ( Q m ) C/ , where V ( k ) = (cid:80) n (cid:54) =0 V n exp( − i k · r n ) . The mean-field Hamiltonian in themomentum space can be diagonalized by means of the Bo-goliubov transformation and the corresponding eigenenergiesare labeled as E n . Straightforward calculations lead to thefollowing form of the mean-field thermodynamic potential Ω MF = − k B T (cid:80) n log(1 + exp( − E n k B T )) + (cid:80) k ξ k − E I .The order parameters defined above can be obtained by min-imizing the mean-field thermodynamic potential via ∆( k ) , δ Q m , Σ k , and Q combined with the conservation of the to-tal atom number through the relation − N L ∂ Ω MF ∂µ = n . Topological supersolids with tunable chern numbers —
Through the method mentioned above, we can determine thephase diagram of our proposed system. Surprisingly, wefind that by simply adjusting the average filling of the lat-tice system here, various topological supersolids with tun-able chern numbers can be achieved under different dipo-lar interaction strength J . For example, as shown in Fig. 1,when the average filling n = 0 . , through minimizing themean-field thermodynamic potential, we find that Q is lo-cated at Q = (0 , , π/a z ) (see details in SupplementaryMaterials). And the corresponding mean filed Hamiltonianup to an innocuous additive constant can be expressed as H πSS = (cid:80) k ψ † k (cid:18) ξ k σ z + ∆ × k δ × δ × ξ k + Q σ z + ∆ × k + Q (cid:19) ψ k ,where ∆ × k = (cid:18) k )∆ ∗ ( k ) 0 (cid:19) , δ × = (cid:18) δ Q − δ − Q (cid:19) , σ z is the Pauli matrix, and ψ † k = ( c † k , c − k , c † k + Q , c − k − Q ) .Through numerics, we also find that superfluid order pa-rameter ∆( k ) behaves like a in-plane p + ip superfluid,i.e., ∆( k ) = ∆( k z )(sin( k x a ) + i sin( k y a )) , where ∆( k z ) records very slight variation for different k z when consider-ing the case as shown in Fig. 1(a) (see details in Supplemen-tary Materials). Therefore, we can rewrite superfluid gap as ∆(sin( k x a ) + i sin( k y a )) . Since it is shown that δ Q = δ − Q here, we thus define δ ≡ δ Q to describe CDW order along z -direction. Through analyzing the behavior of these orderparameters, we can map out the phase diagram as shown inFig. 1(a). There are five different phases in the phase dia-gram, which consists of a p -wave superfluid and four differ-ent supersolids. A threshold strength of dipolar interactionseparates the p -wave superfluid and supersolids. Below thatthreshold, the ground state is a p -wave superfluid, character-ized by ∆ (cid:54) = 0 and δ = 0 . When increasing the strengthof dipolar interaction, different supersolids characterized with ∆ (cid:54) = 0 and δ (cid:54) = 0 , indicating the coexistence of superfluid andCDW orders, will be favored. As shown in Fig. 1(a), for cou-pling strength J > J ≈ . , one type of supersolid (SS-A)will be the ground state of the system. While when interacting SS-G SS-FSS-Hp-waveSF SS-E
SS-C SS-Bp-waveSF SS-D SS-A ∆ /t δ /t ( E - E ) / E J0.04 20.040.0200.5 1 1.5 2 31.50 ∆ /t δ /t π/2 k y ak y ak y a k y ak y ak y a ππ−π/2 −π−π1.07 1.15 1.296 1.298 2.633 2.640.08−0.08 0.03−0.030.02−0.010.01−0.029.88.6 9.79.6 ×10 −3 −3 ×10 −3 J (d)(b) (c)(e)(a) (g)(f) (h) J ( E - E ) / E J FIG. 1: (a)(b) Zero-temperature phase diagram as a function of dipolar interaction strength when considering average filling n = 0 . and n = 0 . , respectively. The dashed and solid lines stand for the pairing gap and CDW order parameter respectively. Inset shows that in theweak interacting region (below the threshold of J ) the system favors a p -wave superfluid, since the ground state energy E of a superfluid issmaller than that E of a normal state. Above the threshold of J , different supersolids marked by various colored regions emerge, characterizedby the coexistence of superfluid and CDW orders. (c-e) Energy spectrum of various supersolids with open (periodic) boundary conditions inthe x ( y ) directions for a fixed k z . In (c), there is one pair of chiral edge states marked by blue-dashed-line, when considering SS-A phase at J = 2 , n = 0 . and k z a z = − π/ . In (d), there are doubly degenerate chiral edge modes labeled by blue-dashed-line and red-solid-linerespectively, when considering SS-D phase at J = 0 . , n = 0 . and k z a z = − π/ . In (e), the blue, red and green branches correspond tothree pairs of chiral edge modes respectively, when considering SS-H phase at J = 1 . , n = 0 . and k z a z = 0 . (f),(g) and (h) zoom in onthe part near by the boundaries between the edge and bulk modes in (e). Other parameters are t z /t = 0 . , a z /a = 3 . . strength J ≈ . < J < J , there are three different typesof supersolids (SS-B SS-C and SS-D) to appear.Now let us discuss the differences among these four su-persolids via their distinguished topological properties. Sincethe Brillouin zone in the z -direction is folded by CDW or-der along z -axis, by choosing a certain k z a z ∈ [ − π π ) in H πSS , we can define an effective two-dimensional Hamilto-nian H effπSS in ( k x , k y ) -plane. The SS-A and SS-C phases arecharacterized by an uniform chern number (see details in Sup-plementary Materials), when k z a z ∈ [ − π π ) . We find thatthe chern numbers of SS-A and SS-C phases are C = 1 and C = 0 respectively, indicating their distinguished topologicalproperties. While for SS-B and SS-D phases, they are quitedifferent. There are two different topological regions along k z -axis. For example, for the SS-B phase in Fig. 1(a) with J = 1 . , when − k cSS − B < k z a z < k cSS − B ≈ . π , thechern number of SS-B is C = 1 . However, in other regions − k cSS − B > k z a z ≥ − π or k cSS − B < k z a z < π , the topolog- ical invariance vanishes. The SS-D phase shows similar fea-tures compared to SS-B, such as when considering J = 0 . inFig. 1(a), for the region − k cSS − D < k z a z < k cSS − D ≈ . π ,the topological invariance vanishes. However, distinguishedfrom all the other three kinds of supersolids above, when − k cSS − D > k z a z ≥ − π or k cSS − D < k z a z < π , a new typesupersolid with higher chern number C = 2 can be achieved.These supersolids show the highly tunable chern numbers andwould lead to some unique features which will be discussed inthe following. Although as shown in Fig. 1(a), we only showthe phase diagram with a certain average filling n = 0 . , itshould be emphasized that there is a region of the average fill-ing, i.e., . ≤ n < n A ≈ . , where Q is also locatedat Q = (0 , , π/a z ) determined by the minimization of themean-field thermodynamic potential and the similar phase di-agram as shown in Fig. 1(a) will be obtained in such a region.Furthermore when further increasing the average filling ofthe proposed system, more fascinating supersolids can berealized. As shown in Fig. 1(b), when the average filling n = 0 . , it is shown that Q is located at Q = (0 , , π/ a z ) determined via minimizing the mean-field thermodynamicpotential. Through numerics, we find that the superfluidorder parameter here can also be expressed as ∆( k ) =∆(sin( k x a ) + i sin( k y a )) due to its slight variation along k z -axis for the case considered as shown in Fig. 1(b). The or-der parameter δ ≡ δ Q can also be introduced since δ Q = δ − Q here. Using the same method as the case with Q =(0 , , π/a z ) described above, we can obtain the phase dia-gram when considering average filling n = 0 . . As shownin Fig. 1(b), there is a threshold of dipolar interaction strengthseparating a p-wave superfluid and four different supersolids.Below that threshold, it is shown that ∆ (cid:54) = 0 and δ = 0 ,indicating the ground state is a p-wave superfluid. For cou-pling strength J > J (cid:48) ≈ . , one type of supersolid (SS-E)will be energetically favored. While when interacting strength J (cid:48) ≈ . < J < J (cid:48) , there are three different types of su-persolids (SS-H SS-G and SS-F) to appear. These four kindsof supersolids can also be distinguished by different chernnumbers via the same analysis as discussed above. The SS-E and SS-G phases are characterized by an uniform chernnumber in the folded Brillouin zone along z -direction, i.e., k z a z ∈ [ − π π ) , which are C = 0 and C = 1 for SS-E and SS-G respectively, indicating their notable differencesof topological properties. For SS-F phase in Fig. 1(b) with J = 1 . , when − k cSS − F < k z a z < k cSS − F ≈ . π ,the topological invariance vanishes. However, in the region − k cSS − F > k z a z ≥ − π or k cSS − F < k z a z < π , thechern number is C = 1 . In SS-H phase when consider-ing J = 0 . , in the region − k cSS − H > k z a z ≥ − π or k cSS − H ≈ . π < k z a z < π , the chern number is C = 1 .However, distinguished from all the supersolids mentionedabove, when − k cSS − H < k z a z < k cSS − H , a new type su-persolid with more higher chern number C = 3 is realized.It should also be emphasized that the similar phase diagramas shown in Fig. 1(b) will be obtained for the average filling n A < n < n B ≈ . , since in such a region all the Q arelocated at Q = (0 , , π/ a z ) .To further demonstrate the fantastic topological nature ofvarious supersolids discussed above, we shall show that thehighly tunable exotic chiral edge modes as well as MajoranaFermions are supported in these states. To see this, for a fixed k z , we consider a cylinder geometry in xy -plane, where theopen (periodic) boundary condition is chosen in the x ( y ) di-rection respectively. The energy spectrum in Fig. 1 is labeledby the momentum k y . In SS-A, as shown in Fig. 1(c), allthe bulk modes are gapped and there is one pair of chiral edgestates located at the two outer edges of the system respectively,since here the Chern number is C = 1 satisfying the so-calledbulk-edge correspondence. Furthermore, the two zero-energyedge states turn out to be a pair of Majorana fermions locatedat the two edges respectively. For SS-D phase, when choosinga fixed k z within the region where Chern number C = 2 , asshown in Fig. 1(d), there are doubly degenerate chiral edgemodes located at the two outer edges of the system respec- k B T /t k B T /t (b)(a) T CDWMF T BKT T SFMF T CDWMF T BKT T SFMF
FIG. 2: (a)(b) Finite-temperature phase diagram as a function ofdipolar interaction strength when considering average filling n =0 . and n = 0 . , respectively. The blue and green lines show themean-filed transition temperature of superfluids and CDW respec-tively. The red line indicates the BKT transition temperature of ourproposed aniostropic 3D lattice system. Other parameters are thesame as in Fig. 1. tively and four zero-energy edge states are two pairs of Ma-jorana fermions. While in SS-H, when considering a fixed k z within the region where Chern number C = 3 , as shown inFig. 1(e), there are three pairs of chiral edge modes located atthe two outer edges of the system respectively and six zero-energy edge states are three pairs of Majorana fermions. Asdescribed above, our proposed supersolids possess highly tun-able chern numbers and thus can control the numbers of chi-ral edge modes as well as that of Majorana fermions, whichwould be very useful to the potential application in quantumcomputation and quantum information.Another important quantity is the superfluid density, whichcan demonstrate the novel transport properties of both super-fluid and supersolid phases, such as supporting dissipation-less currents. The superfluid density can be understood asthe stiffness of the system responding to the phase twists.To simplify the calculation here, the exchange interaction en-ergy − V i − j (cid:104) c † i c j (cid:105) in Eq. (2) is considered only between thenearest neighbors and the corresponding self-energy is de-noted by Σ α = x,y,z . Then from the response function of phasetwists, the anisotropic superfluid fraction ρ αs can be obtainedas ρ αs = (2 t α − Σ α ) / (2 N t α ) (cid:80) k [ n k cos( k α a α ) − f α ] (seedetails in Supplementary Materials) with total number of par-ticles N . Since here we consider an anisotropic 3D latticewith a x = a y < a z and t z < t x = t y , it can be approximatedas a layered superconducting system. The finite temperaturephase transition can be interpreted as the Kosterlitz-Thouless(KT) type and the critical temperature T BKT is determinedby the relation [57–59] k B T BKT = π ¯ J ( k B T BKT ) , where ¯ J = ( J x + J y )2 and J α = n αs, d / m α with m α = 1 / t α a α . n αs, d = n αs a z is the effective 2D areal superfluid density and n αs = ρ αs N/N L a a z . As shown in Fig. 2(a), when consider-ing the average filling n = 0 . , in the region with weak dipo-lar interaction, T BKT is approximately equal to the mean-fieldcritical transition temperature of superfluids T SFMF . At largerinteracting coupling region, CDW phase can survive highertemperatures as indicating by the mean-field transition tem-perature T CDWMF . However, there is a large derivation of T BKT and T SFMF as expected, for the reason that mean field analysisunderestimates fluctuation effects. As shown in Fig. 2(a), be-low T BKT the supersolid region is quite sizable and can be un-ambiguously identified. The finite temperature phase diagramfor the case with the average filling n = 0 . is also shownin Fig. 2(b). In the current experiments, for example, Dyatom’s magnetic dipole moment is µ B , the BKT transitiontemperature can reach around . nK, when considering Dyin a lattice with a = 225 nm and J = 3 . Furthermore, tak-ing advantage of recent experimental realization of Feshbachresonance in magnetic lanthanide atoms such as Er [60], thedipole-dipole interaction is highly tunable. The BKT transi-tion temperature can be estimated to reach around nK oreven higher, making it promising to obtain the various super-solids in experiments. Conclusion —
In summary, we have demonstrated that var-ious fascinating topological supersolids can be realized in aspinless dipolar Fermi gas loaded in an anisotropic 3D opticallattice. The crucial ingredient of our model, i.e., the direction-dependent effective attraction between dipoles generated bya rotating external field, has been achieved in recent experi-ments, solving the main experimental difficulty of implement-ing our proposed setup and make it even easier to be real-ized in realistic system. The appearance of chiral edge modesas well as Majorana Fermions with highly number-tunabilityare predicted as the concrete experimental signatures for thesenovel states.
Acknowledgment —
This work is supported by the Na-tional Key Research and Development Program of China(2018YFA0307600) and NSFC Grant No. 11774282. H. L.is supported by NSFC Grant No. 11774284. F. L. is sup-ported by NSFC Grant No. 11534008 and the National KeyR&D Project (Grant No. 2016YFA0301404). ∗ Electronic address: [email protected][1] A. J. Leggett, Phys. Rev. Lett. , 1543 (1970).[2] M. Boninsegni and N. V. Prokof’ev, Rev. Mod. Phys. , 759(2012).[3] S. Sasaki, R. Ishiguro, F. Caupin, H. J. Maris, and S. Balibar,Science , 1098 (2006).[4] E. Kim and M. H. W. Chan, Phys. Rev. Lett. , 115302 (2006).[5] L. Pollet, M. Boninsegni, A. B. Kuklov, N. V. Prokof’ev, B. V.Svistunov, and M. Troyer, Phys. Rev. Lett. , 135301 (2007).[6] N. Prokof’ev, Advances in Physics , 381 (2007).[7] E. P. Gross, Phys. Rev. , 161 (1957).[8] E. Gross, Annals of Physics , 57 (1958).[9] B. K. Clark and D. M. Ceperley, Phys. Rev. Lett. , 105302(2006). [10] M. W. Ray and R. B. Hallock, Phys. Rev. Lett. , 235301(2008).[11] M. 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CHERN NUMBERS OF VARIOUS SUPERSOLIDS
The distinguished topological properties of various supersolids can be characterized by the distinct chern numbers. Whenconsidering average filling n = 0 . , through minimizing the mean-field thermodynamic potential, we find that Q is locatedat Q = (0 , , π/a z ) . Then the system can be described by a mean-field Hamiltonian H πSS as defined in the main text. Sincethe Brillouin zone in the z -direction is folded by CDW order along z -axis, by choosing a certain k z a z ∈ [ − π π ) in H πSS , wecan define an effective two-dimensional Hamiltonian H effπSS in the ( k x , k y ) momentum plane. The topological properties can becharacterized by the chern number defined as C = 12 π (cid:88) n ∈ occupied (cid:90) BZ dk x dk y F ( n ) ij ( k ) , (S1)where F ( n ) ij ( k ) = (cid:15) ij ∂ k i A nj ( k ) and A ni ( k ) = i (cid:104) φ n ( k ) | ∂ k i φ n ( k ) (cid:105) . Here | φ n ( k ) (cid:105) is the eigenstate of H effπSS , which satisfiesthe relation H effπSS ( k ) | φ n ( k ) (cid:105) = E n ( k ) | φ n ( k ) (cid:105) . For the case with Q located at Q = (0 , , π/ a z ) , a similar method can beapplied. SUPERFLUID DENSITY
The superfluid density can be understood as the stiffness of the system responding to the phase twists. To simplify thecalculation here, the exchange interaction energy − V i − j (cid:104) c † i c j (cid:105) in Eq. (2) is considered only between the nearest neighborsand the corresponding self-energy is denoted by Σ α = x,y,z . Then from the response function of phase twists, the anisotropicsuperfluid density ρ αs can be obtained ρ αs = (2 t α − Σ α )2 N t α (cid:88) k [ n k cos( k α a α ) − f α ] (S2)where f α = − (2 t α − Σ α ) k B T (cid:88) k (cid:48) (cid:90) β (cid:90) β dτ dτ (cid:48) sin( k α a α ) sin( k (cid:48) α a α )[ − δ k , − k (cid:48) F ( k , τ, τ (cid:48) ) F † ( k (cid:48) , τ, τ (cid:48) ) − δ k , − k (cid:48) − Q ˜ F ( k , τ, τ (cid:48) ) ˜ F † ( k (cid:48) , τ, τ (cid:48) ) + δ k , k (cid:48) G ( k , τ, τ (cid:48) ) G † ( k (cid:48) , τ, τ (cid:48) ) + δ k , k (cid:48) + Q ˜ G ( k , τ, τ (cid:48) ) ˜ G † ( k (cid:48) , τ, τ (cid:48) )] (S3) -2.6-2.4-2.2 Q z a z Q z a z π π π π -1.9-2.1-2.3 0M Г X M M Г X M (a) (b)