Interaction-driven Lifshitz transition with dipolar fermions in optical lattices
E. G. C. P. van Loon, M. I. Katsnelson, L. Chomaz, M. Lemeshko
IInteraction-driven Lifshitz transition with dipolar fermions in optical lattices
E. G. C. P. van Loon, M. I. Katsnelson, L. Chomaz,
2, 3 and M. Lemeshko ∗ Radboud University, Institute for Molecules and Materials, NL-6525 AJ Nijmegen, The Netherlands Institut f¨ur Experimentalphysik, Universit¨at Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria Institut f¨ur Quantenoptik und Quanteninformation,¨Osterreichisches Akademie der Wissenschaften, 6020 Innsbruck, Austria IST Austria (Institute of Science and Technology Austria), Am Campus 1, 3400 Klosterneuburg, Austria
Anisotropic dipole-dipole interactions between ultracold dipolar fermions break the symmetry ofthe Fermi surface and thereby deform it. Here we demonstrate that such a Fermi surface deformationinduces a topological phase transition – so-called Lifshitz transition – in the regime accessible topresent-day experiments. We describe the impact of the Lifshitz transition on observable quantitiessuch as the Fermi surface topology, the density-density correlation function, and the excitationspectrum of the system. The Lifshitz transition in ultracold atoms can be controlled by tuning thedipole orientation and – in contrast to the transition studied in crystalline solids – is completelyinteraction-driven.
PACS numbers: 67.85.-d, 67.85.Lm, 71.10.Fd, 71.27.+a
I. INTRODUCTION
The concept of Fermi surface plays a central role in thedescription of electronic systems. Several physical prop-erties, such as the electrical conductivity and the absorp-tion spectrum of the system are determined by the shapeof the Fermi surface as well as the electrons’ dispersionrelation in the vicinity of this surface. Stationary pointsof the dispersion relation correspond to the Van Hovesingularities (VHS) . If a VHS occurs close to the Fermisurface, it can dramatically alter the properties of theelectron gas. For example, in two-dimensional systemsthe density of states exhibits a logarithmic divergence atthe VHS . If one deforms the Fermi surface such thatit crosses a VHS, there occurs an electronic topologicaltransition – the Lifshitz transition .The Lifshitz transition has been explored in a vari-ety of systems, from high-temperature copper-oxide and iron-based superconductors to superfluid helium .In condensed matter systems, the change in Fermi sur-face at the Lifshitz transition affects observable quanti-ties such as resistivity and thermoelectric power , lat-tice dynamics, elastic moduli and related thermal proper-ties such as heat capacity and thermal expansion .In some cases, it determines the peculiarities of phasediagrams of metals under pressure as well as metal al-loys . In such settings, there is a strong and com-plicated interplay between the electrons experiencing thetransition and the underlying ionic lattice. In isotropicsystems, it is challenging to induce the Lifshitz transi-tion using a tunable interaction, since Luttinger’s the-orem combined with the symmetry of the systemstrongly constrains the Fermi surface. Therefore, usu-ally the Lifshitz physics is studied by changing the single-particle properties of the system, such as the chemicalpotential or the electrons’ kinetic energy .Experiments with ultracold atomic and molecularFermi gases in optical lattices pave the way to unravelthe properties of strongly-correlated condensed-matter systems using “clean” and highly tunable setups , ex-emplifying the concept of a quantum simulator as intro-duced by Feynman . For instance, it became possibleto prepare a fermionic Mott insulator and study theproperties of the repulsive Fermi-Hubbard model ,probe the BEC-BCS crossover in lattices , study short-range magnetism and multiflavor spin dynamics , aswell as to realize artificial graphene sheets and the topo-logical Haldane model .Typical ultracold fermion experiments deal with short-range isotropic interparticle interactions. Recent ex-perimental efforts, however, have been devoted to ex-ploit particles possessing a large electric or magneticdipole moment. Ultracold fermionic molecules, such as K Rb and Na K , have been prepared intheir absolute ground states, while ultracold gases ofmagnetic atoms such as Dy , Er , and Cr ,have been brought to Fermi degeneracy. As opposedto the conventional condensed matter systems wherethe Coulomb interaction between electrons is screenedby the ionic crystal, these systems allow to realizetruly long-range interactions between the trapped par-ticles. One further advantage of ultracold gases com-pared to condensed matter systems is their high tun-ability. For instance, the relative strength of the long-and short-range interactions can be controlled via Fes-hbach resonances and control over the long-rangeinteraction via time-dependent dipole orientation orstate-dressing . The anisotropic and long-range char-acter of the dipole-dipole interactions (DDI) is predictedto give rise to novel many-body Hamiltonians , someof which have already been realized in laboratory .In this work we demonstrate that dipolar quantumgases trapped in optical lattices offer a unique oppor-tunity to study the physics associated with Lifshitz tran-sitions. First, the ultracold experimental setups allow totune the properties of the fermions and underlying lat-tice independently, which is rather challenging to realizein crystalline solids. Second, the anisotropic nature of a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y θφ aa r jk j kx y Figure 1. Single-component dipolar fermions on a square op-tical lattice. Due to the symmetry, the orientation of thedipoles is given by two angles, φ ∈ [0 , . π ] and θ ∈ [0 , . π ]. DDI breaks the spatial symmetry of the system, man-ifesting itself in Fermi surface deformations , as re-cently observed in experiment . Here we show that, inthe context of lattice systems, such deformations can beused to generate a Lifshitz transition, which, in turn, hasa strong impact on the correlations in the system. Sincethe orientation of the dipoles can be controlled by an ex-ternal field, dipolar fermions provide a convenient wayto study such a transition experimentally. Juxtaposed tothe Lifshitz transition observed in solids, the one studiedhere is interaction-driven , i.e. it occurs solely due to thetwo-particle terms of the Hamiltonian.A similar scenario has been investigated theoreti-cally in coupled quasi-1D chains of ultracold atoms .There, the interchain hopping was used as the tuning pa-rameter and the external field was oriented to rule outintrachain interactions. In contrast, the transition stud-ied in this paper occurs in an isotropic lattice, and thedipolar character of the fermions is truly essential. II. DIPOLAR FERMIONS ON AN OPTICALLATTICE
We expect the physics of the Lifshitz transition to bequalitatively similar for any Hubbard-like model in theFermi liquid phase. Therefore, without loss of general-ity, we restrict ourselves to the single-component dipolarfermion model on a square two-dimensional lattice, asschematically illustrated in Fig. 1. Furthermore, ultra-cold atomic gases of fully polarized fermions are readilyavailable in experiment , and allow to avoid dealingwith complex spin preparation protocols and dipolar re-laxation effects between spin components that modifiesthe initial spin preparation .The model’s Hamiltonian is given by: H = − t (cid:88) (cid:104) jk (cid:105) c † j c k + 12 (cid:88) jk V djk n j n k , (1)where c † j ( c j ) creates (annihilates) a fermion on site j , and n j = c † j c j counts whether there is a fermion on site j . Hopping with an amplitude t occurs between pairs (cid:104) jk (cid:105) of nearest neighbors. The dipole-dipole interaction V djk = c d (cid:104) − r jk · ˆ d ) (cid:105) / ( r jk /a ) depends on the vector r jk connecting sites j and k , ˆ r jk = r jk /r jk , a is the latticeconstant, and c d sets the strength of the DDI. An externalfield orients the dipoles along the direction ˆ d , given bythe spherical angles θ , φ , see Fig. 1.Many-body effects have a significant effect on the Lif-shitz transition, resulting e.g. in its two-side characterin three dimensions and interaction-driven band flat-tening in two dimensions , and therefore need to beproperly taken into account. In order to achieve thisgoal, we employ the dual boson approach to stronglycorrelated systems , since it is capable of accountingfor many-body effects in the strongly-interacting regime.This method has previously been applied to the dipo-lar Fermi-Hubbard model (DFH) , see Appendix A foradditional computational details.In order to observe the Lifshitz transition, we startwith a system below but close to half-filling, such thatthe Fermi surface is close to the VHSs and that evenmoderate deformations suffice to cross them. We use adensity n = 0 . ± .
01. Our simulations use the grandcanonical ensemble and therefore operate at fixed chemi-cal potential. As a result, the particle density cannot becompletely fixed. However, by subtracting the Hartreecontributions to the chemical potential, the changes indensity can be made negligible.Three relevant energy scales of this problem are givenby the hopping amplitude, t , the temperature of thefermions, T , and the dipolar interaction strength, c d .In order to observe a relatively sharp Fermi surface, themagnitude of k B T , with k B Boltzmann’s constant, hasto be small compared to the hopping bandwidth of 8 t .The dipolar coupling, in turn, determines the magnitudeof the anisotropic effects. For experiments with highlymagnetic lantanide atoms, t can be tuned over a widerange from hundreds of mHz to hundreds of Hz, whilethe dipolar coupling c d is set by the atomic species andthe lattice spacing a selected. For Erbium with a = 266nm, c d was measured to be 40Hz .A single component Fermi gas of highly magneticatoms also offers an unprecedented and highly efficientcooling mechanism as DDI ensures a finite scatteringcross-section and thus allows thermalization betweenatoms, while the Pauli principle forbids short-range s-wave scattering and thus suppresses losses caused byinelastic three body collisions . Efficient cooling iscrucial when simulating condensed matter systems sincethe Fermi temperature T F changes from the Kelvin scalein solid state systems to the nano-Kelvin scale in theatomic gas. In the bulk, temperatures down to ≈ . In the presence of a periodicpotential, the Fermi temperature is set by half the band-width T F ≈ t . By minimizing heating effects, one canexpect to keep T /T F nearly constant during the ramp-ing up of the optical potential while in the Fermi Liquidregime .Below, we exemplify the calculations by considering t = 100 Hz, T = 20 Hz, and c d = 50 Hz, with all ener-gies given in units of t . While this order of magnitudeof T /T F has been achieved in experiments with ultra-cold Er in a harmonic trap , heating effects will needto be minimized in order to achieve a similar tempera-ture in a lattice. In general, the lower the temperature,the sharper the Fermi surface and the clearer the Lifshitztransition can be observed. III. THE LIFSHITZ TRANSITION
In a 2D square lattice, the Brillouin Zone (BZ) definesquasi-momenta k x , k y ∈ [ − π, π ] in units of the inverselattice spacing. The stationary points of the dispersion t k = − t [cos( k x ) + cos( k y )] are at the points ( ± π,
0) and(0 , ± π ) on the edge of the BZ (dots in Fig 2), correspond-ing to the VHSs of the non-interacting system. The dipo-lar interaction does not affect the location of the VHSs.In the absence of interactions, the x and y directionsof the system are identical the Fermi surface resemblesa diamond with rounded corners. When the DDI isturned on, with dipoles oriented along the xy -diagonal( φ = 0 . π , any θ ), the Fermi surface preserves thisshape, see Fig. 2 (red line).However, orienting the dipoles along the x -axis ( φ = 0, θ = 0 . π ), breaks the symmetry between the x and y directions. As a consequence, the Fermi surface loses itssymmetry as well. The resulting deformation leads tothe Lifshitz transition: the Fermi surface now enclosesthe VHSs at X = ( q x = ± π, q y = 0). Furthermore,the Lifshitz transition changes the topology of the Fermisurface, which now connects neighboring Brillouin Zonesin the horizontal direction, as can be inferred from theperiodic continuation of Fig. 2.An additional insight into the Lifshitz transition canbe obtained by studying the properties of the spectralfunction, A ( E, k ), which describes the energies and mo-menta of the single-particle excitations in the system. Inorder to highlight the anisotropy due to the DDI, we cal-culate the spectral function along two distinct paths inthe Brillouin Zone, Γ-X-M and Γ-Y-M, where Γ = (0 , π, π ) is the corner of the BrillouinZone, as shown in Fig. 2.Fig. 3 shows the sum of the two spectral functionsalong these paths for two different dipole orientations,with the Fermi energy at E = 0 (white horizontal line).In the left panel of Fig. 3 we show the “symmetric” case φ = 0 . π with n = 0 . < .
5. In the absence of in-teractions, the Van Hove singularity crosses the Fermisurface exactly at half-filling. In Fig. 3(a), the VHS isclearly visible as a very flat dispersion at the X and Y k x k y Γ MXY
Figure 2. Fermi surfaces. The red curve corresponds to φ =0 . π , where the Fermi surface is independent of the angle θ .At φ = 0, θ = 0 . π (blue line), the Fermi surface is deformedanisotropically due to the DDI. This breaks the symmetrybetween the X and Y points in the Brillouin Zone. In the k x direction, the Fermi surfaces of neighboring Brillouin Zonesare connected and the VHS (black dots) is enclosed by theFermi surface. In the k y direction, the Fermi surfaces are notconnected and the VHS is outside of the Fermi surface. points, however it is now located above the Fermi energy E = 0. Fig. 3(b) shows the spectral function at φ = 0,on the other side of the Lifshitz transition. Here, the dis-persion has two branches corresponding to the X pointand Y point respectively. The branches remain flat, cor-responding to two VHSs, one above and one below theFermi energy, see Appendix B for additional details.A naive estimate of the energy difference between the X and Y points can be obtained using Hartree-Fock the-ory. As shown in Appendix B, the contribution of theFock diagram lowers (raises) the energy of the VHS atthe X (Y) point by ∆ E ≈ . c d in the limit of zero tem-perature and taking into account only nearest-neighborinteraction. This correctly predicts the order of magni-tude of the splitting observed in Fig. 3. The interactionstrength c d determines the scale of the anisotropy, there-fore the energy-resolved measurements need a resolutionof the same order to be able to detect these effects. Insituations where the temperature is substantially largerthan c d , all effects are likely to be thermally smeared out.The static susceptibility (cid:104) nn (cid:105) q , which is defined as theFourier transform of the density-density correlation func-tion to momentum space, provides an alternative way toinvestigate the system. Compared to the spectral func-tion, which contains information on the single-particle ex-citations, the susceptibility gives access to the collectiveexcitations, in particular, to the charge density waves.Thus, the susceptibility reveals whether the system isin a charge-ordered state. Along with deforming theFermi surface and altering the spectral function, the DDIalso affect the susceptibility, as demonstrated by Fig. 4.For φ = 0 . π and dipoles perpendicular to the latticeplane, panel (a), we observe an isotropic susceptibilitywith maxima close to the M = ( ± π, ± π ) points, whichcorresponds to a checkerboard pattern in real space asthe interaction is isotropically repulsive in plane. As thedipoles get oriented parallel to the lattice plane whilekeeping φ = 0 . π , the symmetry between the two di-agonals is broken, reflecting the asymmetry between thedirection φ = ± . π introduced by the anisotropy of ΓΓ X/YX/Y MM ΓΓ E F − t +5 t φ = 0 . π φ = 0 . s p ec tr a l f un c t i o n ( a r b . un i t s ) . Figure 3. Spectral function at φ = 0 . π (left) and φ = 0(right), both for θ = 0 . π . The sum of the spectral functionalong the Γ-X-M and Γ-Y-M paths (see Fig. 2) is shown. At φ = 0 . π , the spectral functions along these paths are identi-cal, whereas at φ = 0 the X and Y points are distinguishabledue to the anisotropic interaction. This corresponds to thesplitting into two bands. DDI. For large θ , a maximum starts to appear at small q and long wavelength, which is reminiscent of the sus-ceptibility observed in the ultralong-range ordered phaseof the dipolar Fermi-Hubbard model . Note that thisevolution of the susceptibility is completely interaction-driven and happens while the Fermi surface remains un-perturbed, as shown in Fig. 2. In panel (b), the effectof rotation in plane is illustrated, going from dipoles ori-ented along the diagonal ( φ = 0 . π ) to dipoles point-ing along the x -axis ( φ = 0). We observe that the lineof maxima in the susceptibility follows the dipole orien-tation angle, and the Fermi surface is deformed in thisprocess, cf. Fig. 2. Finally, in Fig. 4c, we consider thepath backwards to the dipoles aligned out of plane, nowkeeping φ = 0 constant. As for fixed φ = 0 . π , the sus-ceptibility evolves from anisotropic to isotropic. Howeverthe orientation of the line of maxima is now rotated andin contrast to fixed φ = 0 . π , this evolution is associ-ated to a deformation of the Fermi surface: at θ = 0 it isisotropic whereas at θ = 0 . π it is deformed.Let us have a more detailed look onto the specific caseof φ = 0, θ = 0 .
5, where the Fermi surface deformationis the largest. In Fig. 5(a), we show a cross-section at q x = 0 of the susceptibility (cid:104) nn (cid:105) q . The green line corre-sponds to the same density, n = 0 .
40, as in Fig. 2. Thesusceptibility changes, however, if one changes the den-sity. At the lowest density shown, n = 0 .
37 (blue) there isa clear maximum in the susceptibility. This is the Kohnanomaly corresponding to excitations from the flat topof the Fermi surface to the bottom of the next Fermisurface, as also shown in Fig. 5(b) (blue arrow).As the density increases, the Fermi surface expands(Fig. 5(b)) and the Kohn anomaly shifts to slightly lowermomentum. When reaching a given critical density (here between n = 0 .
38 and 0 . X -point in the Brillouin Zone. As a result,excitations with small momentum transfer are possible,as illustrated in Fig. 5(b) (arrows), and the susceptibil-ity at small q y is greatly enhanced. This time, insteadof a sharp peak, there is a much broader enhancement.Since the X -point is a VHS, the single-particle energyclose to X only depends weakly on momentum, and sodoes the occupation n k . This means that the densityprofile is relatively flat near the Fermi surface here andthe corresponding excitations are less sharply peaked. IV. CONCLUSIONS
In this work we demonstrated that ultracold dipolarfermions in an optical lattice can be used as an effi-cient quantum simulation platform to study topologicalLifshitz transitions. As their crucial property, the Lif-shitz transitions predicted in ultracold quantum gasesoccur solely due to the anisotropic interparticle inter-actions, and therefore can be observed in an isotropicoptical lattice. It was shown that the transition can bedetected by measuring the Fermi-surface deformations,the spectral function, and the static susceptibility. Thus,several complimentary experimental techniques can beused. The Fermi surface deformation can be deter-mined using adiabatic mapping time-of-flight measure-ments . The spectral function can be revealed us-ing momentum-resolved radiofrequency spectroscopy or momentum-resolved Bragg scattering . Lattice-modulation spectroscopy can show the energies ofthe available states, however without the momentum res-olution. The splitting of the Van Hove singularity intotwo energies associated with the X and Y point can be in-vestigated in this way. The static susceptibility (cid:104) nn (cid:105) q canbe accessed by two-body correlation analysis of the time-of-flight density distribution, so called noise-correlationmeasurement . These observation techniques willhave to be integrated into the experimental set-up re-quired for the Lifshitz transition. h nn i q q x /πq x /πq x /πq x /πq x /π q y / π q y / π q y / π fixed φ fixed φ fixed θφ = 0 . πθ = 0 . πφ = 0(a)(b)(c) θ = 0 . θ = 0 . π θ = 0 . π θ = 0 . π θ = 0 . πφ = 0 . π φ = 0 . π φ = 0 . π φ = 0 . π φ = 0 . πθ = 0 . π θ = 0 . π θ = 0 . π θ = 0 . π θ = 0 . π Figure 4. Static density-density correlation function in mo-mentum space. The dipolar angles correspond to those ofFig. 1. n = 0 . n = 0 . q y /π q y q x s u s ce p t i b ili t y (a) (b) Figure 5. (a) q x = 0 cross section of the static susceptibil-ity. The arrows indicate the momenta corresponding to thetransitions illustrated on the right. (b) Fermi surface (onlythe top right quadrant of the Brillouin Zone is shown), thegreen Fermi surface corresponds to Fig. 2. Different lines cor-respond to the densities n = 0 .
37, 0 .
38, 0 .
40, 0 .
42 and 0 . φ = 0, θ = 0 . π . In order to observe the interaction-induced Lifshitztransition, the fermion density needs to be close to theVan Hove filling, which for nearest-neighbor hopping oc-curs at half-filling. Furthermore, since the phase transi-tion point depends on the local density, the confinementpotential has to be be sufficiently flat to simultaneouslyinduce the Lifshitz transition in a large part of the trap.Furthermore, Fermi surface deformations are most nat-urally observed in momentum space, so observation ishelped by homogeneity. Novel techniques such as boxtraps and anticonfinement potentials may help inreducing inhomogeneous trapping effects. Other tech-niques such as single-site adressing or super-latticeengineering/tuning may help both in preparing re-gions of controlled filling and give access to original prob-ing schemes .While ultracold magnetic atoms are the primarycandidates to observe the interaction-induced Lifshitztransitions, similar measurements can be performedwith high-density samples of ultracold heteronuclearmolecules . Furthermore, the phenomenon is ex-pected to occur for other types of anisotropic interpar-ticle interactions, such as quadrupole-quadrupole cou-plings or interactions induced by far-off-resonantlaser fields . ACKNOWLEDGMENTS
We thank Francesca Ferlaino, Koen Reijnders and JanKaczmarczyk for useful discussions. E.G.C.P.v.L. andM.I.K. acknowledge support from ERC Advanced Grant338957 FEMTO/NANO. L.C. acknowledges support bythe FWF through SFB FoQuS and START grant underProject No. Y479-N20.
Appendix A: The dual boson approach
We use the dual boson formalism to strongly cor-related systems . Here, we give a short syn-opsis of the method. The main idea of the ap-proach is to separate the interaction effects into twoparts: momentum-independent mean-field effects andmomentum-dependent corrections.The first stage of the computation is the determinationof the effective mean-fields. This is achieved by introduc-ing an auxiliary single-site problem with dynamical , localfields ∆ ν , Λ ω , that replace the non-local terms t jk and V jk of the original system. In the action formulation, thisauxiliary problem is defined as S = − (cid:88) ν c ∗ ν [ iν + µ − ∆ ν ] c ν + 12 (cid:88) ω Λ ω n ω n ω , (A1)where ν and ω are fermionic and bosonic Matsubarafrequencies respectively. This single-site problem canbe solved numerically exactly, and the Green’s func-tion and two-particle correlation functions can be deter-mined. Our numerical solution of the auxiliary single-site problem is based on the ALPS libraries . Fromthe Green’s function and susceptibility of the auxiliarymodel, we then obtain an approximation for the Green’sfunction and susceptibility of the lattice model. Thefields ∆ ν and Λ ω are chosen self-consistently, by requiringthe local Green’s function and susceptibility of the latticemodel to be identical to those of the auxiliary problem.The second stage consists of momentum-dependent“dual” corrections to the mean-field solution. Theseare crucial for studying Fermi surface deformations,since that is an essentially momentum-dependent phe-nomenon. The associated diagrams are shown in Fig. 6,we refer the reader to Ref. 80 for explicit formulas. Inthese diagrams, the lines with arrows describe fermionpropagation, the wiggly lines the propagation of den-sity fluctuations and the (filled) triangles the (ladder-renormalized ) interaction between the fermions and thedensity fluctuations. The numerical values of these ele-ments are determined from the auxiliary model.The Fock-like diagram in Fig. 6(a) is essential to theFermi surface deformation. Due to the DDI, the wig-gly line is anisotropic and as a result, the self-energy isalso anisotropic and the Fermi surface deforms. The dualdiagrammatic technique was applied until (inner) self-consistency was achieved (usually 10 iterations weresufficient), to allow for feedback of the Fermi surfacedeformation on the susceptibility via diagram 6(b) andback. Finally, the nonlocal corrections from the dual dia-grammatic technique are applied to the original fermions.The calculations were performed on a 64 ×
64 squarelattice. The Fermi surface is determined from the Green’sfunction at the point where the occupation n k crosses1 /
2. There is a small discretization uncertainty due tothe finite momentum resolution. The spectral functionof Fig. 3 was obtained from the Green’s function at Mat-subara frequencies using Pad´e approximants . (a) (b)Figure 6. Feynman diagrams employed in the dual bosonapproach. Diagram (a) renormalizes the fermion propagatorand diagram (b) renormalizes the susceptibility. The Fermisurface deformation occurs due to the anisotropy of diagram(a), coming from the anisotropic DDI. Γ MXY
Figure 7. For a non-interacting, half-filled system, the redpart of the Brillouin Zone lies within the Fermi surface. Theblack and gray dots denote the Van Hove singularities (sta-tionary points and end points of the dispersion respectively),and we define the points Γ = (0 , π, , π ) and M = ( π, π ). Appendix B: Estimates at zero temperature
In order to get a feeling for the expected magnitudeof the effects, here we perform a perturbative analysis.It is most convenient to do this at T = 0 and close tohalf-filling, where the integrals over the Brillouin Zonecan be drastically simplified.The energy of the non-interacting system is given bythe Fourier transform, t k , of the hopping, H = (cid:88) k E k n k E k = t k = − t [cos( k x ) + cos( k y )]The Fermi surface is determined by the condition of E k = µ , with µ the chemical potential. Half-filling occursat µ = 0, and the resulting Fermi surface the diamondshown in Fig. 7. In the main text, we studied a systembelow half-filling at n ≈ .
4, where the Fermi surface isslightly smaller than the diamond.The Van Hove singularities (black dots) are found asthe stationary points of the dispersion, ∇ k E k = 0. Twosaddle points occur at the center of the sides of the Bril-louin Zone. The global minimum and maxima of thedispersion are shown as the gray dots, at the origin andthe corners of the Brillouin Zone respectively. Account-ing for the periodicity, there are two saddle points, oneminimum and one maximum per Brillouin Zone, the min-imum number of critical points predicted by Van Hove .Let us now consider an anisotropic interaction. Forsimplicity, we take into account only the nearest-neighborcouplings and set the dipoles’ orientation along the x -axis ( θ = π/ φ = 0). In momentum space, that interactionis given by V q = 2 c d [ − q x ) + cos( q y )] . (B1)Now, we estimate the self-energy of the fermion usingthe Hartree-Fock approximation. The expectation valuewith respect to H is denoted by (cid:104)·(cid:105) . The Hartree con-tribution to the self-energy is independent of k and onlyleads to a change in the chemical potential, which canbe ignored. The Fock contribution, on the other hand,induces anisotropy :Σ Fock k = − (cid:88) q V q (cid:104) n k + q (cid:105) . (B2)The sums over momenta in equation (B2) should be un-derstood as normalized integrals over the Brillouin Zone.Performing this calculation explicitly for the high-symmetry points Y and X , i.e., k = (0 , π ) and k = ( π, Y point withrespect to the Fermi energy is given by:Σ Fock k = Y = − π ) (cid:90) k + q ∈ Fermi volume V q d q =12 c d /π ≈ . c d (B3)On the other hand, for the X point, Σ Fock k = X ≈ − . c d , andthe dispersion is pushed below the Fermi energy. Herewe used that (cid:104) n (cid:105) is zero outside of the Fermi surface andunity inside. These estimates of the energy splitting be-tween the X and Y points match the order of magnitudeof the results in Fig. 3. Note that an exact match is notexpected, since the results of Fig. 3 are obtained at finitetemperature, away from half-filling and with interactionbeyond nearest neighbors.In Sec. III, the numerical results showed that the VHSsdo not move in the presence of interaction. This canbe demonstrated perturbatively in the zero-temperaturelimit. Let us show that the energy (B2) is stationary atthese points. The first term, t k , is stationary since thesepoints are the VHSs of the non-interacting system. Then,we have to determine the gradient of (cid:104) n k + q (cid:105) . Since thedensity is a step function, its derivative is a delta functionon the Fermi surface. ∇ k (cid:0) Σ Fock k (cid:1) = − ∇ k (cid:88) q V q (cid:104) n k + q (cid:105) = (cid:90) k + q ∈ Fermi surface − V q d q (B4) ∇ k (cid:0) Σ Fock k (cid:1) | X, Y = 0Since the gradient of the Fock self-energy is zero at the
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