PHYSICAL REVIEW X 10, 041057 (2020)
Fermi Surface Reconstruction without Symmetry Breaking
Snir Gazit Racah Institute of Physics and The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Fakher F. Assaad Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany, and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
Subir Sachdev Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
(Received 10 August 2020; revised 5 October 2020; accepted 16 October 2020; published 21 December 2020)
We present a sign-problem-free quantum Monte Carlo study of a model that exhibits quantum phase transitions without symmetry breaking and associated changes in the size of the Fermi surface. The model is an Ising gauge theory on the square lattice coupled to an Ising matter field and spinful “orthogonal” fermions at half filling, both carrying Ising gauge charges. In contrast to previous studies, our model hosts an electronlike, gauge-neutral fermion excitation providing access to Fermi-liquid phases. One of the phases of the model is a previously studied orthogonal semimetal, which has Z2 topological order and Luttinger-volume-violating Fermi points with gapless orthogonal fermion excitations. We elucidate the global phase diagram of the model: Along with a conventional Fermi-liquid phase with a large Luttinger- volume Fermi surface, we also find a “deconfined” Fermi liquid in which the large Fermi surface coexists with fractionalized excitations. We present results for the electron spectral function, showing its evolution from the orthogonal semimetal with a spectral weight near momenta f π=2; π=2g to a large Fermi surface.
DOI: 10.1103/PhysRevX.10.041057 Subject Areas: Computational Physics, Condensed Matter Physics, Strongly Correlated Materials
I. INTRODUCTION Given the strong coupling nature of such transitions, Quantum phase transitions involving a change in the quantum Monte Carlo simulations can offer valuable volume enclosed by the Fermi surface play a fundamental guides to understanding the consequences for experimental role in correlated electron compounds. In the cuprates, observations. As the transitions involve fermions at non- there is increasing evidence of a phase transition from a zero density, the sign problem is a strong impediment to low-doping pseudogap metal state with small density of simulating large systems. However, progress has been fermionic quasiparticles to a higher-doping Fermi-liquid possible in recent years by a judicious choice of micro- (FL) state with a large Fermi surface of electronic quasi- scopic Hamiltonians which are argued to capture the particles [1–9]. In the heavy fermion compounds, much universal properties of the transition but are nevertheless attention has focused on the transitions between metallic free of the sign problem. Such approaches focus on density – states distinguished by whether the Fermi volume counts wave ordering transitions [14 27], where spontaneous the localized electronic f moments or not [10–13]. translational symmetry breaking accompanies the change in the Fermi volume: Consequently, both sides of the transition have a Luttinger-volume Fermi surface, after the expansion of the unit cell by the density wave ordering is taken into account. Published by the American Physical Society under the terms of Our paper presents Monte Carlo results for quantum the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to phase transitions without symmetry breaking, accompanied the author(s) and the published article’s title, journal citation, by a change in the Fermi surface size from a non-Luttinger and DOI. volume to a Luttinger volume. The phase with a
2160-3308=20=10(4)=041057(15) 041057-1 Published by the American Physical Society GAZIT, ASSAAD, and SACHDEV PHYS. REV. X 10, 041057 (2020) non-Luttinger-volume Fermi surface must necessarily have OSM phase, and we present direct evidence here in topological order and emergent gauge degrees of freedom spectral functions. [12,28,29]. There have been a few quantum Monte Carlo Our interest here is in quantum transitions out of the studies of fermions coupled to emergent gauge fields OSM and, in particular, into phases without Z2 topological [30–37], and our results are based on a generalization of order. In previous studies [30–32], the Z2-confined phases the model of Refs. [30,32], containing a Z2 gauge field break either the translational or the global U(1) symmetry. coupled to spinful “orthogonal” fermions fα (spin index In both cases, there is no requirement for a Fermi surface α ¼ ↑; ↓) carrying a Z2 gauge charge hopping on the with a nonzero volume, and all fermionic excitations are square lattice at half filling. The parameters are chosen so gapped once Z2 topological order disappears. Here, we that the fα experience an average π flux around each extend the previous studies by including an Ising matter z plaquette; consequently, when the fluctuations of the Z2 (Higgs) field τ , which also carries a Z2 gauge charge. This gauge flux are suppressed in the deconfined phase of the Z2 field allows us to define a gauge-invariant local operator gauge theory, the fermions have the spectrum of massless with the quantum number of the electron [38]: Dirac fermions at low energies. There are four species of z two-component Dirac fermions, arising from the twofold cα ¼ τ fα: ð1Þ degeneracies of spin and valley each. In addition to the Z2 gauge charges, the fα fermions also carry spin and global The τz matter fields do not carry global spin or U(1) U(1) charge quantum numbers and so are identified with charges. With this dynamic Ising matter field present, it is the orthogonal fermions of Ref. [38], and the deconfined possible to have a Z2-confined phase which does not break phase of the Z2 gauge theory is identified as an orthogonal any symmetries and phase transitions which are not semimetal (OSM). associated with broken symmetries. In particular, the It is important to note that the OSM phase preserves all Luttinger constraint implies that any Z2-confined phase the symmetries of the square lattice Hamiltonian, and it without broken symmetries must have large Fermi surfaces realizes a phase of matter which does not have a Luttinger- with volume ð1=2Þð2πÞ2 for each spin. Our main new volume Fermi surface. This phase is compatible with the results are measurements of the cα spectral function with topological nonperturbative formulation of the Luttinger evidence for such phases and phase transitions. theorem (LT) [39], because Z2 flux is expelled (about a One of the unexpected results of our Monte Carlo study π-flux background) and the OSM has Z2 topological order is the appearance of an additional “deconfined FL” phase: [12,28,29]. There is a Luttinger constraint associated with See the phase diagram in Fig. 1(b). As we turn up the every unbroken global U(1) symmetry [40,41], stating that attractive force between the fα fermions and the Ising the total volume enclosed by Fermi surfaces of quasipar- matter (Higgs) field τz, we find a transition from the OSM ticles carrying the global charge [along with a phase space to a phase with a large Luttinger-volume Fermi surface of d factor of 1=ð2πÞ , where d is the spatial dimension] must the cα but with the Z2 gauge sector remaining deconfined. equal the density of the U(1) charge, modulo filled bands. Such a phase does not contradict the topological arguments, In Oshikawa’s argument [39], this constraint is established which do not require (but do allow) a Luttinger-violating by placing the system on a torus and examining the Fermi surface in the deconfined state. Only when we also momentum balance upon insertion of one quantum of turn up the Z2 gauge fluctuations do we then get a transition the global U(1) flux: The Luttinger result follows with the to a “confined FL” phase, which is a conventional Fermi assumption that the only low-energy excitations which liquid with a large Fermi surface. Within the resolution of respond to the flux insertion are the quasiparticles near the our current simulations, we are not able to identify a direct Fermi surface. When there is Z2 topological order, a Z2 transition from the OSM to the confined FL, and we flux excitation (a “vison”) inserted in the cycle of the torus indicate a multicritical point separating them in Fig. 1(b). costs negligible energy and can contribute to the momen- It is interesting to note that the deconfined FL phase has tum balance: Consequently, a non-Luttinger-volume Fermi some features in common with “fractionalized Fermi-liquid” surface becomes possible (but is not required) in the (FL*) phases used in recent work [8,42,43] to model the presence of topological order [12,28,29]. In the OSM, pseudogap phase of the cuprates. These phases share the the orthogonal fermions fα carry a global U(1) charge and presence of excitations with deconfined Z2 gauge charges have a total density of 1: So, in the conventional Luttinger coexisting with a Fermi surface of gauge-neutral fermions approach, there must be two Fermi surfaces (one per spin) cα. There is a difference, however, in that the present 2 each enclosing volume ð1=2Þð2πÞ . However, with Z2 deconfined FL state has a large Fermi surface, while the topological order, the OSM can evade this constraint, FL* states studied earlier have a small Fermi surface. Both and the only zero-energy fermionic excitations are at possibilities are allowed by the topological LT [12,28]. discrete Dirac nodes, and the Fermi surface volume is In passing, we note the recent study of Chen et al. [37], zero. Earlier numerical studies [30–32] present indirect which also examines a Z2 gauge theory coupled to z evidence for the existence of such a Luttinger-violating orthogonal fermions fα and an Ising matter field τ .
041057-2 FERMI SURFACE RECONSTRUCTION WITHOUT SYMMETRY … PHYS. REV. X 10, 041057 (2020)
(a) (b)
FIG. 1. (a) Lattice model of orthogonal fermions coupled to an Ising-Higgs lattice gauge theory. The matter fields fr;α (blue circle) and z z τr (red circle) reside on the square lattice sites, and the Ising gauge field σr;η (green square) is defined on the lattice bonds. (b) Global phase diagram of our model [Eqs. (2) and (3)] as a function of the hopping amplitude t and transverse field g. The phase boundaries are determined by the location of the confinement transition and emergence of c fermions spectral weight; see the main text. Connecting lines are guides to the eye.
Z σz 1 However, their 2-deconfined phase is different from ours gauge fields, b ¼ , residing on the square lattice bonds and earlier studies [31,32]: Their phase has a large, b ¼fr; ηg, with r being the lattice site and η ¼ x=ˆ yˆ, and z Luttinger-volume Fermi surface of the fα fermions, which two types of matter fields: an Ising field τr ¼ 1 and a move in a background of zero flux (such zero flux states are spinful orthogonal fermion fα;r, with α ¼ ↑; ↓ labeling the also present in the studies of Refs. [30,34]). Consequently, spin index. Both matter fields are defined on the lattice their Z2 confinement transition to the Fermi liquid does not sites. The dynamics is governed by the Hamiltonian H H H H H involve a change in the Fermi surface volume, which ¼ Z2 þ τ þ f þ c comprising the lowest-order weakens the connection to finite doping transitions in the terms that are invariant under local Ising gauge trans- cuprate superconductors. formations, as we detail below. The rest of the paper is structured as follows. In Sec. II, The first two terms in H correspond to the standard we introduce a lattice realization of the Ising-Higgs gauge Ising-Higgs gauge theory [44]: theory coupled to orthogonal fermions and discuss its global and local symmetries. In Sec. III, we determine X Y X H − σz − σx the global phase diagram of our model using a sign- Z2 ¼ K b g b; problem-free quantum Monte Carlo (QMC) simulation. X□ b∈□ bX H − σz τzτz − τx In particular, we study the structure of the Fermi surface τ ¼ J r;η r rþη h r: ð2Þ and state of the gauge sector in the different phases and r;η r comment on the nature of the numerically observed quantum phase transitions. In Sec. IV, we present a In the above equations, the operators σ ¼fσx; σy; σzg and mean-field calculation of the physical fermion spectral τ ¼fτx; τy; τzg are the conventional Pauli matrices, acting function in the OSM phase, and, lastly, in Sec. V,we on the Hilbert spaces of the Ising gauge fields and Ising summarize our results, discuss relations to experiments, matterQ fields, respectively. The Ising magnetic flux term and highlight future directions. Φ σz H □ ¼ b∈□ b in Z2 equals the product of the Ising gauge field belonging to the elementary square lattice II. ISING-HIGGS GAUGE THEORY COUPLED plaquettes, □. Hτ is a transverse-field Ising Hamiltonian TO ORTHOGONAL FERMIONS for the Ising matter field, where, in order to comply with Ising gauge invariance, the standard Ising interaction is A. Lattice model σz modified to include an Ising gauge field b along the As a concrete microscopic model for orthogonal fer- corresponding bonds b. mions, we consider the square lattice model depicted in The fermion dynamics is captured by the last two terms Fig. 1(a). The dynamical degrees of freedom are Ising in H:
041057-3 GAZIT, ASSAAD, and SACHDEV PHYS. REV. X 10, 041057 (2020) X H − σz † f ¼ w r;ηfr;αfrþη;α þ H:c: As explained in Ref. [32], at half filling, the corresponding η α results for the even sector may be obtained by applying a r; ; X 1 1 partial particle-hole transformation acting on one of the U nf − nf − ; þ r;↑ 2 r;↓ 2 spin species [45]. Xr Our model is also invariant under discrete square lattice H − τz † τz ˆ ˆ c ¼ t rfr;α rþηfrþη;α þ H:c: ð3Þ translations. The operators Tx and Ty generate translation r;η;α by a lattice constant along the x and y directions, respec- tively. When acting on fractionalized excitations, such as Here, H includes a gauge-invariant nearest-neighbor z f the matter fields τr and fr;α in our case, translations may be hopping of orthogonal fermions and on-site Hubbard followed by a Z2 gauge transformation. The symmetry f † interaction between fermion densities, nr;α ¼ fr;αfr;α. operation then forms a projective representation. In the The last term, Hc, defines nearest-neighbor hopping of general case, this representation allows for a richer physical (gauge-neutral) cr;α fermions as can be readily group structure than the standard (gauge-neutral) linear verified by substituting Eq. (1). The model is tuned to half representation [46]. filling (the chemical potential vanishes). For the specific case of a Z2 gauge symmetry on a square In relation to past works, the model considered here lattice, a projective implementation of translations is affords a nontrivial generalization of the ones studied potentially nontrivial. In particular, lattice translation along previously in Refs. [30–32]. In particular, in contrast to the x and y directions may either commute or anticommute z prior studies, where the Ising matter fields τ are infinitely [46], namely, TxTy ¼ TyTx. Physically, the former cor- massive [h → ∞ in Eq. (2)], here, varying the transverse responds to trivial translations, whereas the latter defines a field h in Eq. (2) controls the excitation gap for τz particles. π-flux pattern threading each elementary plaquette of the Consequently, the Ising matter fields and, subsequently, square lattice. While, for every given choice of a gauge- z the physical (gauge-neutral) fermion cα ¼ fατ are now fixing condition, the π-flux lattice inevitably breaks dynamical degrees of freedom. This important extension lattice translations, as it leads to doubling of the unit cell, provides access to a more generic phase diagram and for fractionalized excitations, translational symmetry is observables that probe FL physics. restored by applying an Ising gauge transformation [46]. This key observation allows for the OSM phase to violate B. Global and local symmetries LT without breaking translational symmetry. We now turn to discuss the global and local symmetries III. QUANTUM MONTE CARLO SIMULATIONS of our model. H is invariant under a global SUsð2Þ symmetry of spin rotations. Furthermore, because we tune A. Methods to half filling, particle-hole symmetry enlarges the U(1) Our model is free of the numerical sign problem. We can, symmetry, corresponding to fermion number conservation, therefore, elucidate its phase diagram using unbiased and 2 to form a SUcð Þ pseudospin symmetry [45]. Physically, numerically exact (up to statistical errors) QMC calcula- 2 the SUcð Þ symmetry generates rotations between the tions. To control the Trotter discretization errors, we set the charge density wave and s-wave superconductivity order imaginary time step to satisfy Δτ ≤ 1=ð12jtjÞ, a value for parameters. which we find that discretization errors are sufficiently The gauge structure of our model is manifest through small to obtain convergent results. We explicitly enforce H Z the invariance of under an infinite set of local 2 the Ising Gauss law using the methods introduced in gauge transformationsQ generatedP by the operators Gr ¼ Refs. [30,32]. Additional details discussing the implemen- f f f −1 nr τx σx. Here, n n α and denotes the ð Þ r b∈þr b r ¼ α r; þr tation of the auxiliary-field QMC algorithm and its asso- set of bonds emanating from the site r. Because ½H;Gr ¼0 ciated imaginary-time path-integral formulation are given for all sites r, the eigenvalues Qr ¼ 1 of Gr are conserved in the Appendix A. Similar results are obtained without quantities. Physically, Qr is identified with the static on-site imposing the constraint and using the algorithms for lattice Z2 background charge assignment. fermions (ALF) library [47]. To properly define a gauge theory, one must fix the background charge configuration Qr. This procedure B. Observables enforces an Ising variant of Gauss’slawG ¼ Q . r r To track the evolution of the c electron Fermi surface, Requiring a translationally invariant configuration, two we study the imaginary-time two-point Green’s function distinct gauge theories may be defined: an even lattice β † Gαðk; τÞ¼−hT ½ck;αðτÞc αð0Þ i, where T denotes time gauge theory with a trivial background (Qr ¼ 1) and an k; P † † τz ik·r odd lattice gauge theory with a single Ising charge at ordering and the operator ck;α ¼ r fr;α re creates a each site (Qr ¼ −1). For concreteness, in what follows, c fermion carrying momentum k and spin polarization α. we consider only the case of an odd lattice gauge theory. Expectation values are taken with respect to the thermal
041057-4 FERMI SURFACE RECONSTRUCTION WITHOUT SYMMETRY … PHYS. REV. X 10, 041057 (2020) density matrix, hOi¼1=Z Tr½e−βHO , with Z ¼ Tr½e−βH and the current operator at site r along the x direction is being the thermal partition function at inverse temperature given by β 1 ¼ =T. We emphasize that, in contrast to the orthogonal X f electron, for which, in the absence of a string operator, z † z † z J −i wσ f αf ˆ α tτ f ατ f ˆ α − H:c: : ’ r;x ¼ r;x r; rþx; þ r r; rþxˆ rþx; gauge invariance requires that the two-point Green s α function must vanish for all nonequal space-time points, 7 the c electron is a gauge-neutral operator, and, hence, its ð Þ associated spectral function may be nontrivial. With the above definition, we can compute ρ as the limit: Determining the Fermi surface structure requires s knowledge of real-time quantum dynamics. Therefore, ρ ρ Π 0 ω 0 s ¼ lim sðqyÞ¼ lim xxðqx ¼ ;qy; m ¼ Þ: ð8Þ some form of analytic continuation of the imaginary-time qy→0 qy→0 QMC data to real frequency must be carried out. Quite generically, devising a reliable and controlled numerical It is convenient to express the superfluid stiffness in analytic continuation technique is an outstanding chal- terms of the orbital magnetic susceptibility χðqÞ¼ 2 lenge due to the inherent instability of the associated −ρsðqÞ=q . The singularity associated with the Dirac node inversion problem [48]. leads to a diverging magnetic response at low momenta To overcome this difficulty, we employ a commonly χðqÞ¼−gvgsvf=ð16jqjÞ [51], with gv (gs) being the valley used proxy for the low-frequency spectral response [49]. (spin) degeneracy and vf the Dirac fermion velocity. β τ More explicitly, by computing G ðk; Þ (the spin index is Consequently, at low momenta, ρsðqyÞ ∼ qy ∼ 1=L, so that omitted for brevity) at the largest accessible imaginary-time ρs exhibits a slow decay that is inversely proportional to difference τ ¼ β=2, we obtain an estimate for the single- the system size. By contrast, a Fermi liquid admits a particle residue Z. To see how to relate this quantity to real- finite diamagnetic response at zero momentum (Landau time dynamics, we consider the integral relation diamagnetism), such that the expected behavior is ρ ∼ 2 ∼ 1 2 Z sðqyÞ qy =L , which vanishes more rapidly than ∞ Aðk; ωÞ Gβðk; τ ¼ β=2Þ¼− dω ; ð4Þ before. −∞ 2 coshðβωÞ To probe a potential instability toward an antiferromag- netic (AFM) order, we monitor theP finite-momentum equal- β ω −1 π Gβ ω ω δ 2 where A ðk; Þ¼ = Im ðk; i m ¼ þ i Þ is the c time spin fluctuations χ ðkÞ¼hð eik·rSzÞ i, where Sz ¼ −1 S r r r fermion spectral function. Since coshðβωÞ tends to unity f f n − n is the f electron spin polarization along the z axis. for βω ≪ 1 and rapidly vanishes in the opposite limit ↑ ↓ χ βω ≫ 1, Gðk; τ ¼ β=2Þ amounts to an integral over the From SðkpÞ,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi we can compute the staggered magnetization χ 2 π π spectral function Aðk; ωÞ over a frequency window of the MAFM ¼ SðGAFMÞ=L , with GAFM ¼f ; g being order of approximately T. Further assuming a well-behaved the Bragg vector associated with AFM order. The spectral response for frequencies ω 041057-5 GAZIT, ASSAAD, and SACHDEV PHYS. REV. X 10, 041057 (2020) C. Numerical results fluctuations generate an effective antiferromagnetic To render the numerical computation tractable, we must Heisenberg coupling, leading to AFM order at zero temper- ≪ restrict the relatively large parameter space spanned by the ature. (ii) In the deconfined phase (g K), on the other set of coupling constants appearing in H. Starting from hand, the orthogonal fermions are free, and their dispersion either the deconfined (g ≪ K) or confined phase (g ≫ K), is determined by the background flux configuration. For the π our goal is to probe the emergence of physical c fermions case of a -flux lattice, the band structure consists of two and the resulting formation of FL phases in the limit of gapless and linearly dispersing bands. large hopping amplitude t. To that end, we numerically map To numerically test the above reasoning, in Fig. 2, we fix 0 2 out the phase diagram as a function of the transverse field g t ¼ . and plot the evolution of the flux susceptibility and and physical fermion hopping amplitude t. Throughout, we staggered magnetization as a function of g. Indeed, in – consider a negative Ising flux coupling constant K<0, for agreement with Refs. [30 32], we can identify the afore- which a π-flux lattice is energetically favorable in decon- mentioned phases: a deconfined OSM phase for small g, fined phases. and, with an increase in g, we observe a transition toward a More concretely, we fix the microscopic parameters confining phase accompanied with AFM order. We use a w ¼ −1, J ¼ 0.1, h ¼ 1.0, U ¼ 0.1, and K ¼ −1.All finite-size scaling analysis to estimate the location of both energy scales are measured in units of jKj. We note that the confinement and AFM symmetry-breaking transitions. we choose h to be sufficiently large compared to J in order The flux susceptibility [see Fig. 2(a)] develops a peak at 0 75 5 to avoid condensation of Ising matter fields. The resulting gc ¼ . ð Þ that increases with system size and marks the two-parameter phase diagram is depicted in Fig. 1(b). position of the confinement transition. Concomitantly, in Δ We begin our analysis by examining the limiting case Fig. 2(b), we find that the AFM order parameter AFM 0 75 5 t ≪ g. In this regime, together with the above choice of begins to rise at gc ¼ . ð Þ. microscopic parameters, the Ising matter field τz is gapped, Because of the increased complexity of the model and, consequently, also the physical fermion c is expected considered in this work, we find it challenging to reliably to be gapped. The low-energy physics then involves only estimate universal data associated with the OSM confine- orthogonal fermions coupled to a fluctuating Ising gauge ment transition, such as critical exponents. Hence, we are field. This physical setting was already studied extensively unable to make a direct comparison with previous works. in previous works [30–32]. Nevertheless, the key signature of the OSM confinement In the context of our problem, we expect to find a similar transition, namely, the nontrivial coincidence of confine- structure of quantum phases in the above parameter regime: ment and symmetry breaking [32], is fully consistent with (i) A confining phase (g ≫ K), where the orthogonal the numerical data. fermions together with the on-site background static charge We now turn to address the main inquiry of this study, (we consider an odd lattice gauge theory) form a localized namely, the emergence of low-energy gauge-neutral c gauge-neutral bound state, leaving the electronic spin as the fermions and their associated spectral signatures. With that only dynamical degree of freedom. Subsequently, quantum goal in mind, in Figs. 3(a)–3(c), we depict our numerical (a) (b) FIG. 2. Orthogonal semimetal confinement transition. Evolution of the (a) flux susceptibility χB and (b) staggered magnetization MAFM across the phase transition separating the OSM and the confined AFM as a function of the transverse field g for a fixed hopping 0 2 Δ amplitude t ¼ . . Different curves correspond to a set of increasing system sizes and inverse temperatures. AFM is obtained through a extrapolation of the finite-size data to the thermodynamic limit. 041057-6 FERMI SURFACE RECONSTRUCTION WITHOUT SYMMETRY … PHYS. REV. X 10, 041057 (2020) (a) (b) (c) (d) (e) (f) FIG. 3. Momentum-resolved Z˜ ðkÞ, for a linear system size L ¼ 16 and inverse temperature β ¼ 16, as a function of the hopping amplitude t along two parameter cuts. (i) In (a)–(c), we fix g ¼ 0.55 and cross the transition between the OSM and deconfined FL phases. Deep in the OSM phase, we find four maxima with a finite spectral weight centered about k ¼f π=2; π=2g. With an increase in t, a large diamond-shaped Fermi surface gradually appears upon approach to the deconfined FL phase. (ii) In (d) and (e), on the other hand, we set g ¼ 1.2 and monitor the appearance of a Fermi surface, in the large t limit, starting from a featureless low-energy spectrum deep in the confined AFM phase. estimate for the momentum-resolved c electron residue single-particle residue evaluated at the antinodal point, Z˜ ðkÞ at g ¼ 0.55 and for several increasing values of t, Z˜ ð0; πÞ, along the path connecting the OSM and decon- beginning from the OSM phase. Remarkably, we find that, fined FL phases as a function of t. Indeed, we observe that ˜ in the OSM phase, ZðkÞ comprises four maxima located at the spectral weight vanishes for t 041057-7 GAZIT, ASSAAD, and SACHDEV PHYS. REV. X 10, 041057 (2020) (a) (b) (c) (d) FIG. 4. Phase transition between the OSM and deconfined FL phases as a function of t for g ¼ 0.55. (a) Single-particle residue Z˜ ðkÞ at the antinodal points. (b) Critical finite-size scaling of the superfluid stiffness Lρs. (c) Staggered magnetization MAFM. ΔAFM is computed by an extrapolation to the thermodynamic limit. (d) Flux susceptibility χB. low-energy orthogonal fermions. The parameter regime Hubbard interaction. This phenomenon is not observed t>5 is beyond our current numerical capabilities. in our simulations, since in the weak coupling regime To further examine the transition, we test for the U ≪ t the magnetization is exponentially small in the development of AFM order or confinement in the gauge coupling constant, and its detection is a notoriously difficult sector. In Fig. 4(c), we track the evolution of the staggered numerical task. magnetization as a function of the hopping amplitude t Next, we examine the path connecting the confined AFM (with g ¼ 0.55, as before). We do not find numerical state and the FL phase, by setting g ¼ 1.2 and probing the evidence for a finite AFM order, even when a large evolution of the spectral function as a function of t. The Fermi surface is fully developed at large t. Moving to results of this analysis are shown in Figs. 3(d)–3(f).We the gauge sector, in Fig. 4(d), we probe χB along the same observe a featureless flat spectrum deep in the confined trajectory as above. The flux susceptibility appears to cross AFM phase, as expected due to the absence of fermionic the transition smoothly. We can, therefore, conclude that quasiparticles. By contrast, with an increase in t, a large the Fermi surface reconstruction involves neither transla- Fermi surface appears. tional symmetry breaking nor the loss of topological order. To better appreciate the above result, we note that, in a Thus, the phase at large t is a deconfined FL. confining phase, the low-energy spectrum must contain We remark that, due to the perfect nesting condition solely gauge-neutral excitations. Indeed, in the confined of the half filled square lattice, the ground state is AFM phase, we can identify these excitations with spin expected to exhibit AFM order for arbitrarily small waves. However, fractionalized orthogonal fermions, 041057-8 FERMI SURFACE RECONSTRUCTION WITHOUT SYMMETRY … PHYS. REV. X 10, 041057 (2020) (a) (b) FIG. 5. (a) Flux susceptibility χB as a function of g for t ¼ 3.0 along the path connecting the confined and deconfined FL phases. (b) Single-particle residue Z˜ ðkÞ evaluated at the antinodal point k ¼f0; πg as a function of t. Different curves correspond to different values of g. carrying U(1) electromagnetic charge, are localized. On the (i) The theory for the transition from the OSM to the other hand, with an increase in t, the attractive force AFM is discussed in some detail in Ref. [32] and between τz and f allows forming a gauge-neutral bound identified as a deconfined critical point with an state of c fermions and a metallic state that supports both emergent SO(5) symmetry. spin and charge excitations at low energies. (ii) Away from half filling, the AFM to confined FL We now turn to study the confinement transition along transition is a conventional symmetry-breaking tran- a path connecting the deconfined and confined FL phases. sition between two confining phases and is expected To detect the confinement transition, in Fig. 5(a),weplot to be described by Landau-Ginzburg-Wilson theory the flux susceptibility at t ¼ 3.0 as a function of g. Indeed, combining with damping from Fermi surface ex- we observe a divergence in χB at gc ¼ 0.85ð5Þ marking citations, i.e., Hertz-Millis theory [55]. the location of the confinement transition. The above (iii) The transition from the deconfined FL to the con- result demonstrates that our model sustains FL phases in fined FL phase is a confinement transition without a the background of either a confined or a deconfined change in the size of the Fermi surface. It is, gauge sector. therefore, expected to be described by the conden- Lastly, we summarize our spectral analysis in Fig. 5(b), sation of an Ising scalar, which can be viewed as where we plot the spectral weight Z˜ ð0; πÞ, evaluated on the representing either the vison of the Ising gauge “antinodal” point as a function of the hopping amplitude t theory or the Ising matter field τz [56]. The large for several values of g. As a starting point, at low t,we Fermi surface of electronlike quasiparticles damps consider both the OSM and AFM phases. We find that the quasiparticles, but this damping is much weaker for all g values the spectral weight is small at low t and than that in Hertz-Millis theory: The resulting field theory is described in Refs. [57,58]. We note that this rises continuously starting from the energy scale tcðgÞ. field theory also applies to the confinement tran- Operationally, we numerically estimate tcðgÞ by locating the hopping amplitude for which Z˜ ð0; πÞ > 0.01. This sition in Ref. [25], where the damping is due to a analysis is used to mark the phase boundaries appearing large Fermi surface of orthogonal fermions, and in Fig. 1(b). We remark again that at half filling the Fermi- there is no change in the size of the Fermi surface liquid phase is unstable toward the formation of AFM across the transition. (iv) We do not have a theory for a direct transition from order. Consequently, at strictly zero temperature Z˜ ð0; πÞ must vanish for all t, and the transition between the AFM the OSM to the deconfined FL with a large Fermi and Fermi-liquid phases is a smooth crossover. surface transition. Indeed, it may well be that this transition occurs via an intermediate phase, with gapless excitations of both electrons and orthogonal D. Quantum phase transitions fermions [40,59]. The presence of a regime within We now briefly remark on the theoretical expectations the deconfined FL region of the phase diagram for the quantum phase transitions described above and where we appear to observe coexistence of a c appearing in Fig. 1(b). Fermi surface (as measured by the c spectral density) 041057-9 GAZIT, ASSAAD, and SACHDEV PHYS. REV. X 10, 041057 (2020) with a ρs which vanishes as ∼1=L is evidence in IV. MEAN-FIELD CALCULATION OF THE support of such an intermediate phase. The Luttinger PHYSICAL FERMION SPECTRAL FUNCTION constraint requires that the orthogonal fermions IN THE OSM PHASE which are absorbed into the c Fermi surface must The numerical observation of a finite spectral weight be accounted for by those forming the Dirac nodes. centered about the four nodal points k ¼f π=2; π=2g is One way to preserve the Dirac nodes in an inter- surprising and requires further analytic understanding. mediate phase, and also maintain consistency with To that end, in this section, we present a simple and particle-hole symmetry, is to form electronlike and intuitive explanation using a mean-field calculation of the holelike pockets of an equal area of the c fermions; spectral function of c fermions in the OSM phase. In our the c hole pockets would then be in ancillary trivial π insulator, similar to Refs. [60,61]. Our resolution is calculation, we consider a static background -flux con- not sharp enough to resolve such an intricate Fermi figuration and neglect gauge field fluctuations. This surface evolution, which is likely present in the approximation is justified deep in the OSM phase, where intermediate phase, and we leave its study to such fluctuations are small due to the finite vison gap. In future work. this setting, we can model the dynamics of the f electron (a) (b) (c) (d) π σz 1 −1 FIG. 6. (a) Ising Landau gauge-fixing condition for the -flux lattice; black (red) bonds correspond to b ¼ ( ). (b) Energy contours of the lower-band Dirac spectrum on the π-flux lattice using the above gauge fixing. (c) Leading-order Feynman diagram for β z the physical fermion c propagator Gcðk; iωmÞ. The bubble diagram evaluates to a convolution between the Ising matter field ϕ and orthogonal fermion fσ propagators defined on the π-flux lattice. (d) Finite-temperature spectral function Aðk; ω ¼ 0Þ computed by a numerical evaluation of the bubble diagram. 041057-10 FERMI SURFACE RECONSTRUCTION WITHOUT SYMMETRY … PHYS. REV. X 10, 041057 (2020) (τz field) by a free fermion (scalar field) hopping in the τz follows Bose statistics and, hence, concentrates at the background of a π-flux lattice. band minimum f0=π; 0g. As a result, the integral over the To make progress, we must choose a concrete a gauge- internal momentum in Fig. 6(c) is appreciable only at fixing condition for the Ising gauge field. We take an Ising momentum transfer k ¼f π=2; π=2g [black arrow in ry z variant of Landau’s gauge, σr;xˆ ¼ð−1Þ and σr;yˆ ¼ 1; see Fig. 6(b) connecting the f and τ particle]. In other words, Fig. 6(a). We emphasize that, while the gauge-fixing the momentum-space splitting of the two flavors of matter procedure inevitably breaks both lattice translations and fields is responsible for the unconventional spectral π=2 rotations, expectation values of physical (gauge- response of the OSM phase. neutral) observables must preserve these symmetries. More concretely, we take the effective Hamiltonians V. DISCUSSION AND SUMMARY X HMF − † f ¼ tf tr;ηfr frþη þ H:c:; We study a lattice model of orthogonal fermions coupled r;η to an Ising-Higgs gauge theory. The absence of the sign X π2 ω2 X X problem enables us to determine its global phase diagram MF r m 2 2 Hϕ ¼ þ Δϕ þ ðϕ − t ηϕ ηÞ : and explore related phase transitions using a numerically 2m 2 r r r; rþ r r r;η exact QMC simulation. A key ingredient of our study, ð9Þ which nontrivially distinguishes it from previous works, is the introduction of an Ising matter field that together with the orthogonal fermion may form a physical fermion. This Here, ϕ is a scalar real field and π is its canonically r r crucial feature of our model enables access to the study of conjugate momentum, ½ϕr; πr0 ¼iδr;r0 . With the above FL phases in the presence of Z2 topological order. gauge choice, the hopping amplitudes are set to t η ¼ r; Notably, our model hosts both non-FL quantum states −1 ry δ δ ½ð Þ η;xˆ þ η;yˆ . The scalar field Hamiltonian is para- Z ω that violate LT due to 2 topological order and also meterized by the inertial mass m, oscillation frequency , LT-preserving FL states, where the gauge sector is and Δ, which allows controlling the single-particle exci- ϕ either confined or deconfined. On tuning of microscopic tation gap for particles. parameters, we are able to cross quantum critical points We tune Δ to work in a regime where, on the one hand, that separate these phases, some of which appear to be the scalar field has a finite gap to avoid condensation, but, continuous. on the other hand, it is sufficiently small to render the It is interesting to make a connection, even if suggestive, physical fermion gap small. The lowest-order diagram [62] between our numerical results and experimental signatures Gβ ω contributing to cðk; i mÞ is the bubble diagram shown in of Fermi surface reconstruction observed in cuprate mate- Fig. 6(c). Evaluating the diagram boils down to a con- rials. In particular, the OSM phase shares several properties Gβ ω volution of the Ising matter field propagator ϕðk; i mÞ and with the pseudogap phase: a strong depletion in the Gβ ω density of states at the antinodal points and a concentration the orthogonal fermion propagator fðk; i mÞ. Both propa- gators can be readily computed by diagonalizing the of finite fermionic spectral weight at the nodal points π 2 2 quadratic Hamiltonians in Eq. (9). Further details of this k ¼f = ; = g. Most importantly, there is experimental calculation are given in Appendix B. evidence that these phenomena occur in the absence of In Fig. 6(d), we depict the zero-frequency spectral translational symmetry breaking. We also describe phase β β þ transitions involving the appearance of a large Fermi function A ðk; ω ¼ 0Þ¼−1=π Im G ðk; iωm ¼ i0 Þ, evaluated for the microscopic parameters t ¼ m ¼ ω ¼ 1 surface from such a state, and these transitions have and Δ ¼ −1.1 and inverse temperature β ¼ 1=32. connections to phenomena in the cuprates near optimal – Remarkably, our simplified model displays a finite spectral doping [1 8]. weight located at the nodal points k ¼f π=2; π=2g,in Our study leaves some open questions. In particular, it agreement with the exact QMC calculation. As a nontrivial would be interesting to develop a field theory description to check of our computation, we observe that, unlike the the transition between the OSM and deconfined FL, which gauge-fixed hopping Hamiltonians of Eq. (9), the physical numerically appears to be continuous. Such a description spectral function respects the full square lattice C4 point will have to address the unusual finite-size scaling of the group symmetry. We note that, at strictly zero temperature, superfluid stiffness in the deconfined FL phase. From the the spectral weight must vanish due to the nonzero Ising numerical perspective, simulations on larger lattices are key matter field gap. in resolving the properties of this transition. In addition, it An intuitive understanding of this result may be derived would be interesting to study the fate of the OSM phase and by examining the Dirac band structure in the π-flux phase, neighboring phases away from half filling. This study can shown as a contour plot in Fig. 6(b). At low temperatures, be achieved, using a sign-problem-free QMC, at least for the orthogonal fermions occupy all k-space modes up to the the even lattice gauge theory case. We leave these interest- Dirac point at f π=2; π=2g. However, the Ising matter field ing questions for future studies. 041057-11 GAZIT, ASSAAD, and SACHDEV PHYS. REV. X 10, 041057 (2020) P ACKNOWLEDGMENTS identities in the σz and τz basis, 1 ¼ jτz; σzihτz; σzj, leading to We thank Ashvin Vishwanath, Aavishkar A. Patel, and Amit Keren for useful discussions. S. G. acknowledges X Y support from the Israel Science Foundation, Grant Z β τz σz ˆ −ϵH τz σz … ð Þ¼ Trfh 0; 0j Pi;λe j M−1; M−1i No. 1686/18. S. S. was supported by the National Science λ;τz;σz i Foundation (NSF) under Grant No. DMR-1664842. S. S. τz σz −ϵH τz σz was also supported by the Simons Collaboration on Ultra- × h 1; 1je j 0; 0i: ðA4Þ Quantum Matter, which is a grant from the Simons Foundation (No. 651440). F. F. A. thanks the Deutsche ˆ We first focus on the last time step, which contains P λ. Forschungsgemeinschaft collaborative research center i; σz τz SFB1170 ToCoTronics (Project No. C01) for financial In the Ising sector, the only off-diagonal (in the , basis) terms are the transverse field and the constraint. Focusing support as well as the Würzburg-Dresden Cluster of Z Excellence on Complexity and Topology in Quantum on a specific site i, for the 2 matter field we obtain the Matter ct.qmat (EXC 2147, Project-id No. 390858490). matrix element This work was partially performed at the Kavli Institute for Theoretical Physics (NSF Grant No. PHY-1748958). F. F. A. z iðπ=2Þð1−λ Þ½ð1−τxÞ=2 þϵhτx z hτ je i i i jτ i gratefully acknowledges the Gauss Centre for i;0 X i;M−1 π 2 1−λ 1−τx 2 ϵ τx τz ið = Þð iÞ½ð i Þ= þ h i τx τx τz Supercomputing (GCS) e.V. for funding this project by ¼ h i;0je j i ih i jj i;M−1i τx 1 providing computing time on the GCS Supercomputer i ¼ SUPERMUC-NG at the Leibniz Supercomputing Centre. 1 X ϵ τx π 1−τx 2 1−λ 2 1−τz 2 1−τz 2 h i ½ð i Þ= f½ð iÞ= þ½ð i;0Þ= þ½ð i;M−1Þ= g This research used the Lawrencium computational cluster ¼ 2 e i e τx 1 resource provided by the IT Division at the Lawrence i ¼ Berkeley National Laboratory (supported by the Director, 1 ϵ −ϵ π 1−λ 2 1−τz 2 1−τz 2 h h i f½ð iÞ= þ½ð i;0Þ= þ½ð i;M−1Þ= g Office of Science, Office of Basic Energy Sciences, of the ¼ 2 ðe þ e e Þ U.S. Department of Energy under Contract No. DE-AC02- ϵ λ τz τz 1 coshð hÞ i i;0 i;M−1 ¼ ; 05CH11231) and the Intel Labs Academic Compute ¼ ðA5Þ ϵ λ τz τz −1 Environment. sinhð hÞ i i;0 i;M−1 ¼ : — Note added. Recently, we became aware of a related work The effective Boltzmann weight is then by Chen et al. [37] studying a lattice model of orthogonal fermions in a parameter regime complementary to our γτz λ τz work, which we describe briefly in Secs. I and III D. λ τz τz ∝ i;0 i i;M−1 Wð i; i;0; i;M−1Þ e ; ðA6Þ APPENDIX A: PATH INTEGRAL FORMULATION where γ ¼ −1=2 log½tanhðϵhÞ . Physically, the above The partition function ZðβÞ at inverse temperature β is action corresponds to the gauge-invariant Ising interaction given by along the temporal direction. Importantly, we must take h>0 in order to avoid a sign problem. ˆ −βH ZðβÞ¼Tr½Pe ; ðA1Þ Similarly to Ref. [30], the constraint term associated with Q the Ising gauge field leads to a spatiotemporal plaquette ˆ where the projection operator P ¼ i Pi with term in the 3D Ising gauge theory, and the f fermion ’ Y Green s function is modified by the introduction of a 1 f λ λ ˆ 1 −1 ni τx σx diagonal matrix P½ i with diagonal elements Pii ¼ i. Pi ¼ 2 þð Þ i ij : ðA2Þ þ We rewrite the projection operator using a discrete APPENDIX B: DETAILS OF THE Lagrange multiplier as MEAN-FIELD CALCULATION X X P π 2 1−λ 1−σx 2 1−τx 2 f To evaluate the bubble diagram in Fig. 6(c), we first ˆ ˆ ið = Þð iÞf ½ð i;jÞ= þ½ð i Þ= þni g Pi ¼ Pi;λ ¼ e þ : express the mean-field Hamiltonians [Eqs. (2) and (3)] λ 1 λ 1 i i¼ in momentum space defined on a reduced Brillouin zone 0 2π 0 π ðA3Þ ( 041057-12 FERMI SURFACE RECONSTRUCTION WITHOUT SYMMETRY … PHYS. REV. X 10, 041057 (2020) X MF † H ¼ −t f f 0 ½2 cosðk Þδ 0 þ 2 cosðk Þδ 0 ˆ f f k k y k;k x k;k þπky k;k0 X 2 cos k 2 cos k f 0 − † † y x k; ¼ tf ð fk;0 fk;π Þ ; ðB1Þ 2 cos kx −2 cos ky f π 0 † † † † ϵ where fk;0 ¼ fk and fk;π ¼ f ˆ . Diagonalizing the above Hamiltonian, we obtain the fermionic eigenspectrum ðkÞ kþπky and eigenmodes fαðkÞ¼Vα;γðkÞfγðkÞ, where γ ¼ , α ¼ 0=π, and Vα;γ is the diagonalizing matrix. A similar analysis is carried out in order to diagonalize the scalar field ϕ mean-field Hamiltonian. Writing Eq. (9) in pffiffiffiffi P ϕ π 1 ϕ π ikr momentum space gives ( r= r ¼ = N k k= ke ) X π π ω2 X X H k −k m Δϕ ϕ ϕ ϕ 4 − 2 δ − 2 δ ϕ ¼ þ k −k þ k k0 ½ cosðkyÞ k;−k0 cosðkxÞ − 0 πˆ 2 2 k; k þ ky k k k;k0 X X π π ω2 X k;α −k;α m ϕ ϕ ¼ 2 þ 2 k;αKα;α0 ðkÞ −k;α0 ; ðB2Þ α 0 0 By diagonalizing Kα;α0 (the mass matrix is already diagonal), we obtain the eigenfrequencies ωκðkÞ and the normal modes ϕk;α ¼ Uα;κðkÞϕk;κ. As before, κ ¼ , and α ¼ 0; π. 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