Fermi-surface reconstruction without symmetry breaking

We present a sign-problem free quantum Monte Carlo study of a model, which exhibits quantum phase transitions without symmetry breaking, along with 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 electron-like, gauge-neutral fermion excitation providing access to Fermi liquid phases. One of the phases of the model is a previously studied orthogonal semi-metal, which has $\mathbb{Z}_2$ topological order, and Luttinger-volume violating Fermi points with gapless orthogonal fermion excitations. We elucidate the global phase diagram of the model, which also contains a confining Fermi liquid, with a large Luttinger-volume Fermi surface. We present results for the electron spectral function, showing its evolution from the orthogonal semi-metal with spectral weight near momenta $\{\pm \pi/2, \pm \pi/2\}$, to a large Fermi surface.


I. INTRODUCTION
Quantum phase transitions involving a change in the volume enclosed by the Fermi surface play a fundamental role in correlated electron compounds.In the cuprates, there is increasing evidence of a phase transition from a low doping pseudogap metal state with small density of fermionic quasiparticles, to a higher doping Fermi liquid (FL) state with a large Fermi surface of electronic quasiparticles [1][2][3][4][5][6][7].In the heavy fermion compounds, much attention has focused on the transitions between metallic states distinguished by whether the Fermi volumes counts the localized electronic f moments or not [8][9][10][11].
Given the strong coupling nature of such transitions, quantum Monte Carlo simulations can offer valuable guides to understanding the consequences for experimental observations.As the transitions involve fermions at non-zero density, the sign problem is a strong impediment to simulating large systems.However, progress has been possible in recent years by a judicious choice of microscopic Hamiltonians which are argued to capture the universal properties of the transition, but are nevertheless free of the sign problem.Such approaches have focused on density wave ordering transitions [12][13][14][15][16][17][18][19][20][21][22][23][24][25], where spontaneous 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 has been taken into account.
Our paper will present Monte Carlo results for quantum phase transitions without symmetry breaking, accompanied by a change in the Fermi surface size from a non-Luttinger volume to a Luttinger volume.The phase with a non-Luttinger volume Fermi surface must necessarily have topological order and emergent gauge degrees of freedom [10,26].There have been a few quantum Monte Carlo studies of fermions coupled to emergent gauge fields [27][28][29][30][31][32][33][34], and our results are based on a generalization of the model of refs.[27,29], containing a Z 2 gauge field coupled to spinful 'orthogonal' fermions f α (spin index α =↑, ↓) carrying a Z 2 gauge charge hopping on the square lattice at half-filling.The parameters are chosen so that the f α experience an average π-flux around each plaquette; consequently when the fluctuations of the Z 2 gauge flux are suppressed in the deconfined phase of the Z 2 gauge theory, the fermions have the spectrum of massless Dirac fermions at low energies.There a T i e U 5 g n t T N S / G w n m S g 1 5 Y C Y 5 1 j 0 1 7 4 3 F / 7 x G r D v n z Y S K K N Y g y P S j T s w c H T r j Y p w 2 l U A 0 G x q C i a Q m q 0 N 6 W G K i T X 0 z l 9 o D G q k 0 9 c M 0 d s 6 U 5 M 1 X s k i q J 0 X P L X q 3 p 4 X S R V p X F h 2 g P D p C H j p D J X S N y q i C C H p C L + g V v V n P 1 r v 1 a X 1 N R z N W u r O P Z m B 9 / w L V z a l 6 < / l a t e x i t > Connecting lines are guide to the eye.are four species of two-component Dirac fermions, arising from the two-fold degeneracies of spin and valley each.In addition to the Z 2 gauge charges, the f α fermions also carry spin and global U(1) charge quantum numbers, and so are identified with the orthogonal fermions of ref. [35], and the deconfined phase of the Z 2 gauge theory is identified as an orthogonal semi-metal (OSM).
It is important to note that the OSM phase preserves all the symmetries of the square lattice Hamiltonian, and it realizes a phase of matter which does not have Luttinger volume Fermi surface.This is compatible with the topological non-perturbative formulation of the Luttinger theorem (LT) [36] because Z 2 flux has been expelled (about a π-flux background), and the OSM has Z 2 topological order [10,26].There is a Luttinger constraint associated with every unbroken global U(1) symmetry [37,38], stating that the total volume enclosed by Fermi surfaces of quasiparticles carrying the global charge (along with a phase space factor of 1/(2π) d , where d is spatial dimension) must equal the density of the U(1) charge, modulo filled bands.In Oshikawa's argument [36], this constraint is established by placing the system on a torus, and examining the momentum balance upon insertion of one quantum of the global U(1) flux: the Luttinger result follows with the assumption that the only low energy excitations which respond to the flux insertion are the quasiparticles near the Fermi surface.When there is Z 2 topological order, a Z 2 flux excitation (a 'vison') inserted in the cycle of the torus costs negligible energy and can contribute to the momentum balance: consequently, a non-Luttinger volume Fermi surface becomes possible (but is not required) in the presence of topological order [10,26].In the OSM, the orthogonal fermions f α carry a global U(1) charge and have a total density of 1: so in the conventional Luttinger approach, there must be 2 Fermi surfaces (one per spin) each enclosing volume (1/2)(2π) 2 .However, with Z 2 topological order, the OSM can evade this constraint, and the only zero energy fermionic excitations are at discrete Dirac nodes, and the Fermi surface volume is zero.
Earlier studies [27][28][29] presented indirect evidence for the existence of such a Luttingerviolating OSM phase, and we will present direct evidence here in spectral functions.
Our interest here is in quantum transitions out of the OSM, and in particular, into phases without Z 2 topological order.In the previous studies [27][28][29], the Z 2 confined phases broke either the translational or the global U(1) symmetry.In both cases, there was no requirement for a Fermi surface with a non-zero volume, and all fermionic excitations were gapped once Z 2 topological order disappeared.Here, we will extend the previous studies by including an Ising matter (Higgs) field, τ z , which also carries a Z 2 gauge charge.This allows us to define a gauge-invariant local operator with the quantum number of the electron [35]: The τ z matter fields do not carry global spin or U(1) charges.With this dynamic Ising matter field present, it is possible to have a Z 2 -confined phase which does not break any symmetries, and phase transitions which are not associated with broken symmetries.In particular, the Luttinger constraint implies that any Z 2 confined phase without broken symmetries must have large Fermi surfaces with volume (1/2)(2π) 2 for each spin.Our main new results are measurements of the c α spectral function with evidence for such phases and phase transitions.
One of the unexpected results of our Monte Carlo study is the appearance of an additional "Deconfined FL" phase: see the phase diagram in fig.1b.As we turn up the attractive force between the f α fermions and the Ising matter (Higgs) field τ z , we find a transition from the OSM to a phase with a large Luttinger-volume Fermi surface of the c α , but with the Z 2 gauge sector remaining deconfined.Only when we also turn up the Z 2 gauge fluctuations do we then get a transition to a "Confined FL" phase, which is a conventional Fermi liquid with a large Fermi surface.Within the resolution of our current simulations, we were not able to identify a direct transition from the OSM to the Confined FL, and we have indicated a multicritical point separating them in fig.1b.
It is interesting to note that the Deconfined FL phase has some features in common with 'fractionalized Fermi liquid' (FL*) phases used in recent work [7,39,40] to model the pseudogap phase of the cuprates.These phases share the presence of excitations with deconfined Z 2 gauge charges co-existing with a Fermi surface of gauge-neutral fermions c α .
There is a difference, however, in that the present Deconfined FL state has a large Fermi surface, while the FL* states studied earlier had a small Fermi surface.Both possibilities are allowed by the topological LT. [10,26].
In passing, we note the recent study of Chen et al. [34] which also examined a Z 2 gauge theory coupled to orthogonal fermions, f α , and an Ising matter field, τ z .However, their Z 2 deconfined phase is different from ours and earlier studies [28,29]: their phase has a large, Luttinger-volume Fermi surface of the f α fermions which move in a background of zero flux (such zero flux states were also present in the studies of ref. [27,31]).Consequently, their Z 2 confinement transition to the Fermi liquid does not involve a change in the Fermi surface volume, and this weakens the connection to finite doping transitions in the cuprate superconductors.
The rest of the paper is structured as follows.In section II we introduce a lattice realization of the Ising-Higgs gauge theory coupled to orthogonal fermions and discuss its global and local symmetries, in section III we determine the global phase diagram of our model using a sign-problem-free QMC simulation.In particular, we study the structure of the Fermi-surface and state of the gauge sector in the different phases, and comment on the nature of the numerically observed quantum phase transitions, in section IV we present a mean field calculation of the physical fermion spectral function in the OSM phase, and lastly in section V we summarize our results, discuss relations to experiments and highlight future directions.

II. ISING-HIGGS GAUGE THEORY COUPLED TO ORTHOGONAL FERMIONS A. Lattice Model
As a concrete microscopic model for orthogonal fermions, we consider the square lattice model depicted in fig.1a.The dynamical degrees of freedom are Ising gauge fields, σ z b = ±1, residing on the square lattice bonds b = {r, η}, with r being the lattice site and η = x/ŷ, and two types of matter fields: an Ising field τ z r = ±1 and a spinful orthogonal fermion f α,r , with α =↑, ↓ labeling the spin index.Both matter fields are defined on the lattice sites.The dynamics is governed by the Hamiltonian, H = H Z 2 + H τ + H f + H c comprising the lowest order terms that are invariant under local Ising gauge transformations, as we detail below.
The first two terms in H correspond to the standard Ising-Higgs gauge theory [41], In the above equations, the operators σ = {σ x , σ y , σ z } and τ = {τ x , τ y , τ z } are the conventional Pauli matrices, acting on the Hilbert spaces of the Ising gauge fields and Ising matter fields, respectively.The Ising magnetic flux term ,Φ = b∈ σ z b , in H Z 2 equals to the product of the Ising gauge field belonging to the elementary square lattice plaquettes, .H τ is a transverse field Ising Hamiltonian for the Ising matter field, where in order to comply with Ising gauge invariance the standard Ising interaction is modified to include an Ising gauge field, σ z b , along the corresponding bonds b.
Fermion dynamics is captured by the last two terms in H, Here, H f includes a gauge invariant nearest-neighbor hopping of orthogonal fermions and on-site Hubbard interaction between fermion densities, n f r,α = f † r,α f r,α .The last term, H c , defines nearest-neighbor hopping of physical (gauge-neutral) c r,α fermions as can be readily verified by substituting eq. ( 1).The model is tuned to half-filling (the chemical potential vanishes).
In relation to past works, the model considered here affords a non-trivial generalization of the ones studied previously in refs.[27][28][29].In particular, in contrast to prior studies, where the Ising matter fields τ z were infinitely massive (g → ∞ in eq. ( 2)), here, varying the transverse field, g, in eq. ( 2) controls the excitation gap for τ z particles.Consequently, the Ising matter fields and subsequently the physical (gauge-neutral) fermion c α = f α τ z are now dynamical degrees of freedom.This important extension provides access to a more generic phase diagram and observables that probe FL physics.

B. Global and local symmetries
We now turn to discuss the global and local symmetries of our model.H is invariant under a global SU s (2) symmetry of spin rotations.Furthermore, since we tune to half filling, particle-hole symmetry enlarges the U (1) symmetry, corresponding to fermion number conservation, to form a SU c (2) pseudo-spin symmetry [42].Physically, the SU c (2) symmetry generates rotations between the charge density wave and s-wave superconductivity order parameters.
The gauge structure of our model is manifest through the invariance of H under an infinite set of local Z 2 gauge transformations generated by the operators G r = (−1) n f r τ x r b∈+r σ x b .Here, n f r = α n f r,α and + r denotes the set of bonds emanating from the site r.Since [H, G r ] = 0 for all sites r, the eigenvalue Q r = ±1 of G r are conserved quantities.Physically, Q r is identified with the static on-site Z 2 background charge assignment.
To properly define a gauge theory, one must fix the background charge configuration, Q r .This procedure enforces an Ising variant of Gauss's law G r = Q r .Requiring a translationally invariant configuration, two distinct gauge theories may be defined: an even lattice gauge thoery with a trivial background (Q r = 1) and an odd lattice gauge thoery with a single Ising charge at each site (Q r = −1).For concreteness, in what follows, we will only consider the case of an odd lattice gauge thoery.As explained in ref. [29], at half-filling, the corresponding results for the even sector may be obtained by applying a partial particle-hole transformation acting on one of the spin species [42].
Our model is also invariant under discrete square lattice translations.The operators Tx and Ty generate translation by a lattice constant along the x and y directions, respectively.
When acting on fractionalized excitations, such as the matter fields τ z r and f r,α in our case, translations may be followed by a Z 2 gauge transformation.The symmetry operation then forms a projective representation.In the general case, this allows for a richer group structure then the standard (gauge-neutral) linear representation [43].
For the specific case of a Z 2 gauge symmetry on a square lattice, a projective implementation of translations is potentially non-trivial.In particular, lattice translation along the x and y directions may either commute or anti-commute [43], namely Physically, the former corresponds to trivial translations, whereas the latter defines a π-flux pattern threading each elementary plaquettes of the square lattice.While for every given choice of a gauge fixing condition, the π-flux lattice inevitably breaks lattice translations, as it leads to doubling of the unit cell, for fractionalized excitations, translational symmetry is restored by applying an Ising gauge transformation [43].This key observation allows for the OSM phase to violate LT without breaking of translational symmetry.

III. QUANTUM MONTE CARLO SIMULATIONS A. Methods
Our model is free of the numerical sign-problem for all fermion densities (the focus here is only on the half-filled case).We can therefore elucidate its phase diagram using an unbiased and numerically exact (up to statistical errors) QMC calculations.To control the Trotter discretization errors, we set the imaginary time step to satisfy ∆τ ≤ 1/(12|t|), a value for which we found that discretization errors are sufficiently small to obtain convergent results.
We explicitly enforce the Ising Gauss law using the methods introduced in ref. [27,29].
Additional details discussing the implementation of the auxiliary-field QMC algorithm and its associated imaginary time path-integral formulation are given in appendix A. Similar results were obtained without imposing the constraint and using the "ALF-library [44].

B. Observables
To track the evolution of the c electron Fermi surface, we will study the imaginary-time Z Tr e −βH O , with Z = Tr e −βH being the thermal partition function at inverse temperature β = 1/T .We emphasize that in contrast to the orthogonal f electron, for which, in the absence of a string operator, gauge invariance requires that the two-point Green's function must vanish for all non-equal space-time points, the c electron is a gaugeneutral operator and hence its associated spectral function may be non-trivial.
Determining the Fermi-surface structure requires knowledge of real time quantum dynamics.Therefore, some form of analytic continuation of the imaginary time QMC data to real frequency must be carried out.Quite generically, devising a reliable and controlled numerical analytic continuation technique is an outstanding challenge due to the inherent instability of the associated inversion problem [45].
To overcome this difficulty, we employ a commonly used proxy for the low frequency spectral response [46].More explicitly, by computing G β (k, τ ) (the spin index was omitted for brevity) at the largest accessible imaginary time difference τ = β/2, we obtain an estimate for the single particle residue Z.To see how to relate this quantity to real time dynamics, we consider the integral relation where , where S z r = n f ↑ − n f ↓ is the f electron spin polarization along the z axis.From χ S (k), we can compute the staggered magnetization M AFM = χ S (G AFM )/L 2 , with G AFM = {π, π} being the Bragg vector associated with AFM order.The zero temperature AFM order parameter is obtained by taking the thermodynamic limit ∆ AFM = lim L→∞ lim β→∞ M AFM (L, β).In practice, for a given system size, we monitored the convergence of the staggered magnetization toward its zero temperature value.Following that, we extrapolated the finite size data to the infinite system size value, using a polynomial fit in powers of 1/L.
In the presence of dynamical matter fields, determining whether the gauge sector is confined or deconfined is a particularly challenging task due to charge screening.Standard methods, relying on evaluating Wilson loops, no longer sharply distinguish between the two phases.Alternative methods based on extracting the topological contribution to the entanglement entropy [47,48] and the FredenhagenMarcu order parameter [49] are difficult to reliably scale with system size in fermionic systems.Instead, following refs.[27,29]

C. Numerical Results
To render the numerical computation tractable, we must restrict the relatively large parameter space spanned by the set of coupling constants appearing in H. Stating from either the deconfined (g K) or confined phase (g K) our goal is to probe the emergence of physical c fermions and the resulting formation of FL phases in the limit of large hopping amplitude, t.To that end, we numerically map out the phase diagram as a function of the transverse field, g, and physical fermion hopping amplitude, t.Throughout, we will consider to be sufficiently large compared to J in order to avoid condensation of Ising matter fields.
The resulting two-parameter phase diagram is depicted in fig.1b.
We begin our analysis by examining the limiting case t g.In this regime, together with the above choice of microscopic parameter, the Ising matter field τ z is gapped and consequently also the physical fermion, c, is expected to be gapped.The low energy physics then involves only orthogonal fermions coupled to a fluctuating Ising gauge field.This physical setting was already studied extensively in previous works [27][28][29].
In the context of our problem, we expect to find a similar structure of quantum phases in the above parameter regime: (i) A confining phase (g K), where the orthogonal fermions together with the on-site background static charge (we consider an odd lattice gauge theory) form a localized gauge-neutral bound state, leaving the electronic spin as the only dynamical degree of freedom.Subsequently, quantum fluctuations will generate an effective antiferromagnetic Heisenberg coupling, leading to AFM order at zero temperature.
(ii) In the deconfined phase (g K), on the other hand, the orthogonal fermions are free and their dispersion is determined by the background flux configuration.For the case of a π-flux lattice, the band structure consists of two gapless and linearly dispersing bands.
To numerically test the above reasoning, in fig.2, we fix t = 0.2 and plot the evolution of the flux susceptibility and staggered magnetization as a function of g.Indeed, in agreement with refs.[27][28][29], we can identify the aforementioned phases: a deconfined OSM phase for small g, and with increase in g, we observe a transition towards a confining phase accompanied with AFM order.We use a finite size scaling analysis to estimate the location of both the confinement and AFM symmetry breaking transitions.The flux susceptibility, see fig.2a, develops a peak at g c = 0.85(5) that increases with system size and marks the position of the confinement transition.Concomitantly, in fig.2b, we find that the AFM order parameter, ∆ AFM begins to rise at g c = 0.85 (5).
Due to the increased complexity of the model considered in this work, we found it challenging to reliably estimate universal data associated with the OSM confinement transition, such as critical exponents.Hence, we were unable to make a direct comparison with previous works.Nevertheless, the key signature of the OSM confinement transition, namely the non-trivial co-incidence of confinement and symmetry breaking [29] is fully consistent with the numerical data.
We now turn to address the main inquiry of this study, namely the emergence of low energy gauge-neutral c fermions and their associated spectral signatures.With that goal in mind, in figs.3a-3c, we depict our numerical estimate for the momentum resolved c electron residue, Z(k), at g = 0.4625 and for several increasing values of t, begining from the OSM phase.Remarkably, we find that, in the OSM phase, Z(k), comprises four maxima located at momenta k = {±π/2, ±π/2}.This result is at odds with the conventional LT, which, at half-filling and in the absence of topological order or translational symmetry breaking, predicts a large Fermi-surface encompassing half of the Brillouin zone.With increase in t, the spectral function continuously evolves into the standard diamond shaped Fermi-surface in compliance with LT.
It is tempting to identify the observed "Fermi-pockets" with the nodal points of the OSM.This explanation, however, is incorrect since the orthogonal f fermion is not a gauge invariant object and in particular the location of the Dirac nodes in momentum space is a gauge-dependent quantity.In section IV, we provide a simple explanation for this phenomena To complement the above analysis, we now test whether the appearance of a Fermisurface is accompanied with the development of AFM order or confinement in the gauge sector.In fig.4b, we track the evolution of the staggered magnetization as a function of the hopping amplitude, t (with g = 0.4625, as before).We do not find a numerical evidence for a finite AFM order, even when a large Fermi-surface is fully developed at large t.Moving to the gauge sector, in fig.4a, we probe χ B along the same trajectory as above.The flux susceptibility appears to cross the transition smoothly.We can, therefore, conclude that the Fermi-surface reconstruction does not involve neither translational symmetry breaking nor the loss of topological order.Thus, the phase at large t is a deconfined FL.
We remark that due to the perfect nesting condition of the half-filled square lattice, the ground state in expected to exhibits AFM order for arbitrarily small Hubbard interaction.This phenomenon was not observed in our simulations, since in the weak coupling regime U t the magnetization is exponentially small in the coupling constant and its detection is a notoriously difficult numerical task.
Next, we examine the path connecting the confined AFM state and the FL phase, by setting g = 1.25 and probing the evolution of the spectral function as a function of t.The results of this analysis are shown in figs.3d-3f.We observe a featureless flat spectrum deep in the confined AFM phase, as expected due to the absence of fermionic quasi-particles.By contrast, with increase in t, a large Fermi surface appears.
To better appreciate the above result, we note that in a confining phase, the low energy spectrum must contain solely gauge-neutral excitations.Indeed, in the confined AFM phase, we can identify these excitations with spin-waves.However, fractionalized orthogonal fermions, carrying U (1) electromagnetic charge, are localized.On the other hand, with increase in t the attractive force between τ z and f allows forming a gauge-neutral bound states of c fermions and a metallic state that supports both spin and charge excitations at low energies.
To further illustrate the emergence of fermionic quasi-particles, in fig.5a, we plot the spectral weight, Z(k), evaluated on the "anti-nodal" point k = {0, π} as a function of the hopping amplitude t for several values of g.As a starting point, at low t, We consider both the OSM and AFM phases.We find that for all g values the spectral weight vanishes at low t and rises continuously starting from a critical coupling t c (g).This analysis was used to determine the phase boundaries appearing in fig.1b.
Finally, we study the confinement transition along a path connecting the deconfined and confined FL phases.To detect the confinment transition, in fig.5b, we plot the flux susceptibility at t = 2.0 as a function of g.Indeed, we observe a divergence in χ B at g c = 0.85(5) marking the location of the confinement transition.The above result demonstrates that our model sustains FL phases in the background of either a confined or a deconfined gauge sector.

D. Quantum phase transitions
We now briefly remark on the theoretical expectations for the quantum phase transitions described above and appearing in Fig. 1b.
• The theory for the transition from the OSM to the AFM was discussed in some detail in ref. [29], and identified as a deconfined critical point with an emergent SO( 5) symmetry.
• The AFM to Confined FL transition is a conventional symmetry-breaking transition between two confining phases, and is expected to be described by Landau-Ginzburg-Wilson theory combining with damping from Fermi surface excitations, i.e.Hertz-Millis theory [50].
• The transition from the Deconfined FL to the Confined FL phase is a confinement transition without a change in size of the Fermi surface.It is therefore expected to described by the condensation of an Ising scalar, which can be viewed as representing either the 'vison' of the Ising gauge theory, or the Ising matter field τ z [51].The large Fermi surface of electron-like quasiparticles will damp the quasiparticles, but this damping is much weaker than that in Hertz-Millis theory: the resulting field theory was described in ref. [52].We note that this field theory applies also to the confinement transition in ref. [23], where the damping is due to a large Fermi surface of orthogonal fermions, and there is no change in the size of the Fermi surface across the transition.
• We do not have a theory for a direct transition from the OSM to the Deconfined FL with a large Fermi surface transition.Indeed, the possibility that this transition occurs via an intermediate phase, with Fermi surfaces of both electrons and orthogonal fermions [37,53], cannot be ruled out by the resolution of our present study.
< l a t e x i t s h a 1 _ b a s e 6 4 = " a Z T t P P a 7        However, the Ising matter field, τ z follows Bose statistics and hence will concentrate at the band minimum, {0/π, 0}.As a result, the integral over the internal momentum in fig.6c will be appreciable only at momentum transfer k = {±π/2, ±π/2} (black arrow in fig.6b connecting the f and τ z particle).In other words, the momentum-space splitting of the two flavors of matter fields is responsible for the unconventional spectral response of the OSM phase.

V. DISCUSSION AND SUMMARY
We have studied a lattice model of orthogonal-fermions coupled to an Ising-Higgs gauge theory.The absence of the sign-problem enabled us to determine its global phase diagram and explore related phase transitions using a numerically exact QMC simulation.A key ingredient of our study, which non-trivially distinguish it from previous works, is the introduction of an Ising matter field that together with the orthogonal fermion may form a physical fermion.This crucial feature of our model enabled access to the study of FL phases in the presence of Z 2 topological order.Notably, our model hosts both non-FL quantum states that violate LT due to Z 2 topological order, and also LT preserving FL states, where the gauge sector is either confined or deconfined.On tuning of microscopic parameters, we are able to cross quantum critical on a specific site i, for the Z 2 matter field we obtain the matrix element, e hτ x i e iπ where γ = log(tanh( h)).Physically, the above action corresponds to the gauge invariant Ising interaction along the temporal direction.Importantly, we must take h > 0 in order to avoid a sign problem.
Similarly to ref. [27], the constraint term associated with the Ising gauge field leads to a spatio-temporal plaquette term in the 3D Ising gauge theory, and the f fermions Green's function is modified by the introduction of a diagonal matrix P [λ i ] with diagonal elements t e x i t s h a 1 _ b a s e 6 4 = " o j h 1 Z d + a e w / G o / d J S n l 6 p M 3 G A O 4 = " > A A A C I n i c b V D L S s N A F J 3 4 r PV V d e k m W A Q X p S Q i q L u i G 5 c V 7 A O a G m 6 m N + 3 Q y Y O Z i V B D v s W N v + L G h a K u B D / G a R t E W y 8 M n D n n 3 n t m j h d z J p V l f R o L i 0 v L K 6 u F t e L 6 x u b W d m l n t y m j R F B s 0 I h H o u 2 B R M 5 C b C i m O L Z j g R B 4 H F v e 8 H K s t + 5 Q S B a F N 2 o U Y z e A f s h 8 R k F p y i 2 d O 5 M d q c c T z H w 3 F R U H e D y A 7 I c H O s w q + U 1 g L 3 M U J L f 3 r n B L Z a t q T c q c B 3 Y O y i S v u l t 6 d 3 o R T Q I M F e U g Z c e 2 Y t V N Q S h G O W Z F J 5 E Y a z f o Y 0 f D E A K U 3 X R i n J m H m u m Z f i T 0 C Z U 5 Y X 9 P p B B I O Q o 8 3 R m A G s h Z b U z + p 3 U S 5 Z 9 1 U x b G i c K Q T o 3 8 h J s q M s d 5 m T 0 m k C o + 0 g C o Y P q t J h 2 A A K p 0 q k U d g j 3 7 5 X n Q P K 7 a V t W + P i n X L v I 4 C m S f H J A j Y p N T U i N X p E 4 a h J I H 8 k R e y K v x a D w b b 8 b H t H X B y G f 2 y J 8 y v r 4 B 4 4 2 m V g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " o j h 1 Z d + a e w / G o / d J S n l 6 p M 3 G A O 4 = " > A A A C I n i c b V D L S s N A F J 3 4 r P V V d e k m W A Q X p S Q i q L u i G 5 c V 7 A O a G m 6 m N + 3 Q y Y O Z i V B D v s W N v + L G h a K u B D / G a R t E W y 8 M n D n n 3 n t m j h d z J p V l f R o L i 0 v L K 6 u F t e L 6 x u b W d m l n t y m j R F B s 0 I h H o u 2 B R M 5 C b C i m O L Z j g R B 4 H F v e 8 H K s t + 5 Q S B a F N 2 o U Y z e A f s h 8 R k F p y i 2 d O 5 M d q c c T z H w 3 F R U H e D y A 7 I c H O s w q + U 1 g L 3 M U J L f 3 r n B L Z a t q T c q c B 3 Y O y i S v u l t 6 d 3 o R T Q I M F e U g Z c e 2 Y t V N Q S h G O W Z F J 5 E Y a z f o Y 0 f D E A K U 3 X R i n J m H m u m Z f i T 0 C Z U 5 Y X 9 P p B B I O Q o 8 3 R m A G s h Z b U z + p 3 U S 5 Z 9 1 U x b G i c K Q T o 3 8 h J s q M s d 5 m T 0 m k C o + 0 g C o Y P q t J h 2 A A K p 0 q k U d g j 3 7 5 X n Q P K 7 a V t W + P i n X L v I 4 C m S f H J A j Y p N T U i N X p E4 a h J I H 8 k R e y K v x a D w b b 8 b H t H X B y G f 2 y J 8 y v r 4 B 4 4 2 m V g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " o j h 1 Z d + a e w / G o / d J S n l 6 p M 3 G AO 4 = " > A A A C I n i c b V D L S s N A F J 3 4 r P V V d e k m W A Q X p S Q i q L u i G 5 c V 7 A O a G m 6 m N + 3 Q y Y O Z i V B D v s W N v + L G h a K u B D / G a R t E W y 8 M n D n n 3 n t m j h d z J p V l f R o L i 0 v L K6 u F t e L 6 x u b W d m l n t y m j R F B s 0 I h H o u 2 B R M 5 C b C i m O L Z j g R B 4 H F v e 8 H K s t + 5 Q S B a F N 2 o U Y z e A f s h 8 R k F p y i 2 d O 5 M d q c c T z H w 3 F R U H e D y A 7 I c H O s w

FIG. 1 :
FIG. 1: (a) Lattice model of orthogonal fermions coupled to an Ising-Higgs lattice gauge theory.The matter fields f r,α (blue circle) and τ z r (red circle), reside on the square lattice sites and the Ising gauge field σ z r,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 main text.
0)] , where T denotes time ordering and the operator c † k,α = r f † r,α τ z r e ik•r creates a c fermion carrying momentum k and spin polarization α.Expectation values are taken with respect to the thermal density matrix, O = 1 is the c fermions spectral function.Since cosh(βω) −1 tends to unity for βω 1 and rapidly vanishes in the opposite limit βω 1 , G(k, τ = β/2) amounts to an integral over the spectral function A(k, ω) over a frequency window of order ∼ T .Further assuming a well-behaved spectral response for frequencies ω < T or equivalently no additional low energy excitations, we may use Z(k) = βG(k, τ = β/2) as an estimate for the c electron single particle residue Z = A β (k, ω = 0).To probe a potential instability towards an antiferromagnetic (AFM) order, we monitor the finite-momentum equal-time spin fluctuations χ S (k) = r e ik•r S z r 2

L = 6 , β = 6 L = 6 , β = 9 LFIG. 2 :
FIG. 2: Orthogonal semi-metal confinement transition.Evolution of the (a) flux susceptibility χ B and (b) staggered magnetization M AFM across the phase transition separating the OSM and the confined AFM as a function of the transverse field g for a fixed hopping amplitude t = 0.2.Different curves correspond to a set of increasing systems sizes and inverse temperature.∆ AMF was obtained through a extrapolation of the finite size data to the thermodynamic limit.

1 FIG. 3 :L = 6 , β = 6 L = 6 , β = 9 L
FIG.3: Momentum resolved Z(k), for a linear system size L = 16 and inverse temperature β = 14, as a function the hopping amplitude t along two parameter cuts: (i) In the top panels (a-c) we fix g = 0.4625 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 = {±π/2, ±π/2}.With increase in t, a large diamond shaped Fermi surface gradually appears upon approach to the deconfined FL phase (ii) In the bottom panels (d-e), on the other hand, we set g = 1.25 and monitor the appearance of a Fermi-surface, in the large t limit, starting from a featureless low-energy spectrum deap in the confined AFM phase.
FIG. 5: (a) Single particle residue Z(k) evaluated at the anti-nodal point k = {0, π} as a function of t.Different curves correspond to different values of g (b) Flux susceptibility χ B as a function of g for t = 2.0 along the path connecting between the confined and deconfined FL phases.
5 X 6 6 U m u g w y J p P U U E n m i 6 K U I x O j P A A 0 Y I o S w y c W M F H M 3 o r I C C t M j I 2 p Y k P w F r + 8 D O 2 z u m f 5 7 r z W u C 7 i K M M R H M M p e H A B D b i F J r S A Q A L P 8 A p v T u q 8 O O / O x 7 y 1 5 B Q z h / B H z u c P o V W R Z g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " a Z T t P P a7 I g n X S R X m D J x 9 O r F g / j E = " > A A A B 8 3 i c b Z D L S g M x F I b P 1 F u t t 6 p L N 8 E i u C o z I u h G K L p x W c F e o D O W T J p p Q 5 P M k G S E O v Q 1 3 L h Q x K 0 v 4 8 6 3 M d P O Q l t / C H z 8 5 x z O y R 8 m n G n j u t 9 O a W V 1 b X 2 j v F n Z 2 t 7 Z 3 a v u H 7 R 1 n C p C W y T m s e q G W F P O J G 0 Z Z j j t J o p i E X L a C c c 3 e b 3 z S J V m s b w 3 k 4 Q G A g 8 l i x j B x l q + r 9 l Q 4 I e n f n j l 9 a s 1 t + 7 O h J b B K 6 A G h Z r 9 6 p c / i E k q q D S E Y 6 1 7 n p u Y I M P K M M L p t O K n m i a Y j P G Q 9 i x K L K g O s t n N U 3 R i n Q G K Y m W f N G j m / p 7 I s N B 6 I k L b K b A Z 6 c V a b v 5 X 6 6 U m u g w y J p P U U E n m i 6 K U I x O j P A A 0 Y I o S w y c W M F H M 3 o r I C C t M j I 2 p Y k P w F r + 8 D O 2 z u m f 5 7 r z W u C 7 i K M M R H M M p e H A B D b i F J r S A Q A L P 8 A p v T u q 8 O O / O x 7 y 1 5 B Q z h / B H z u c P o V W R Z g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " a Z T t P P a 7 I g n X S R X m D J x 9 O r F g / j E = " > A A A B 8 3 i c b Z D L S g M x F I b P 1 F u t t 6 p L N 8 E i u C o z I u h G K L p x W c F e o D O W T J p p Q 5 P M k G S E O v Q 1 3 L h Q x K 0 v 4 8 6 3 M d P O Q l t / C H z 8 5 x z O y R 8 m n G n j u t 9 O a W V 1 b X 2 j v F n Z 2 t 7 Z 3 a v u H 7 R 1 n C p C W y T m s e q G W F P O J G 0 Z Z j j t J o p i E X L a C c c 3 e b 3 z S J V m s b w 3 k 4 Q G A g 8 l i x j B x l q + r 9 l Q 4 I e n f n j l 9 a s 1 t + 7 O h J b B K 6 A G h Z r 9 6 p c / i E k q q D S E Y 6 1 7 n p u Y I M P K M M L p t O K n m i a Y j P G Q 9 i x K L K g O s t n N U 3 R i n Q G K Y m W f N G j m / p 7 I s N B 6 I k L b K b A Z 6 c V a b v 5 X 6 6 U m u g w y J p P U U E n m i 6 K U I x O j P A A 0 Y I o S w y c W M F H M 3 o r I C C t M j I 2 p Y k P w F r + 8 D O 2 z u m f 5 7 r z W u C 7 i K M M R H M M p e H A B D b i F J r S A Q A L P 8 A p v T u q 8 O O / O x 7 y 1 5 B Q z h / B H z u c P o V W R Z g = = < /l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " a Z T t P P a 7 I g n X S R X m D Jx 9 O r F g / j E = " > A A A B 8 3 i c b Z D L S g M x F I b P 1 F u t t 6 p L N 8 E i u C o z I u h G K L p x W c F e o D O W T J p p Q 5 P M k G S E O v Q 1 3 L h Q x K 0 v 4 8 6 3 M d P O Q l t / C H z 8 5 x z O y R 8 m n G n j u t 9 O a W V 1 b X 2 j v F n Z 2 t 7 Z 3 a v u H 7 R 1 n C p C W y T m s e q G W F P O J G 0 Z Z j j t J o p i E X L a C c c 3 e b 3 z S J V m s b w 3 k 4 Q G A g 8 l i x j B x l q + r 9 l Q 4 I e n f n j l 9 a s 1 t + 7 O h J b B K 6 A G h Z r 9 6 p c / i E k q q D S E Y 6 1 7 n p u Y I M P K M M L p t O K n m i a Y j P G Q 9 i x K L K g O s t n N U 3 R i n Q G K Y m W f N G j m / p 7 I s N B 6 I k L b K b A Z 6 c V a b v 5 X 6 6 U m u g w y J p P U U E n m i 6 K U I x O j P A A 0 Y I o S w y c W M F H M 3 o r I C C t M j I 2 p Y k P w F r + 8 D O 2 z u m f 5 7 r z W u C 7 i K M M R H M M p e H A B D b i F J r S A Q A L P 8 A p v T u q 8 O O / O x 7 y 1 5 B Q z h / B H z u c P o V W R Z g = = </ l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " D r o l 6 x s m 4 J f H A G g 2 y i c s 9 m 9 e B s g = " > A A A B 6 H i c b Z B N S 8 N A E I Y n 9 a v W r 6 p H L 4 t F 8 F Q S E e q x 6 M V j C / Y D 2 l A 2 2 0 m 7 d r M J u x u h h P 4 C L x 4 U 8 e p P 8 u a / c d v m o K 0 v L D y 8 M 8 P O v E E i u D a u + + 0 U N j a 3 t n e K u 6 W 9 / Y P D o / s d m U b A j e 6 s n r 0 L 6 q e p a b 1 5 X 6 b R 5 H E c 7 g H C 7 B g x r U 4 R 4 a 0 A I G C M / w C m / O o / P i v D s f y 9 a C k 8 + c w h 8 5 n z / K 8 4 z q < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " D r o l 6 x s m 4 J f H A G g 2 y i c s 9 m 9 e B s g = " > A A A B 6 H i c b Z B N S 8 N A E I Y n 9 a v W r 6 p H L 4 t F 8 F Q S E e q x 6 M V j C / Y D 2 l A 2 2 0 m 7 d r M J u x u h h P 4 C L x 4 U 8 e p P 8 u a / c d v m o K 0 v L D y 8 M 8 P O v E E i u D a u + + 0 U N j a 3 t n e K u 6 W 9 / Y P D o / s d m U b A j e 6 s n r 0 L 6 q e p a b 1 5 X 6 b R 5 H E c 7 g H C 7 B g x r U 4 R 4 a 0 A I G C M / w C m / O o / P i v D s f y 9 a C k 8 + c w h 8 5 n z / K 8 4 z q < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " D r o l 6 x s m 4 J f H A G g 2 y i c s 9 m 9 e B s g = " > A A A B 6 H i c b Z B N S 8 N A E I Y n 9 a v W r 6 p H L 4 t F 8 F Q S E e q x 6 M V j C / Y D 2 l A 2 2 0 m 7 d r M J u x u h h P 4 C L x 4 U 8 e p P 8 u a / c d v m o K 0 v L D y 8 M 8 P O v E E i u D a u + + 0 U N j a 3 t n e K u 6 W 9 / Y P D o / d m U b A j e 6 s n r 0 L 6 q e p a b 1 5 X 6 b R 5 H E c 7 g H C 7 B g x r U 4 R 4 a 0 A I G C M / w C m / O o / P i v D s f y 9 a C k 8 + c w h 8 5 n z / K 8 4 z q < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " D r o l 6 x s m 4 J f H A G g 2 y i c s 9 m 9 e B s g = " > A A A B 6 H i c b Z B N S 8 N A E I Y n 9 a v W r 6 p H L 4 t F 8 F Q S E e q x 6 M V j C / Y D 2 l A 2 2 0 m 7 d r M J u x u h h P 4 C L x 4 U 8 e p P 8 u a / c d v m o K 0 v L D y 8 M 8 P O v E E i u D a u + + 0 U N j a 3 t n e K u 6 W 9 / Y P D o / d m U b A j e 6 s n r 0 L 6 q e p a b 1 5 X 6 b R 5 H E c 7 g H C 7 B g x r U 4 R 4 a 0 A I G C M / w C m / O o / P i v D s f y 9 a C k 8 + c w h 8 5 n z / K 8 4 z q < / l a t e x i t > ⌧ z < l a t e x i t s h a 1 _ b a s e 6 4 = " m u k U x 5 F P a 6 8 h c m r y d 3 f n 8 9 W n i 7 A = " > A A A B 7 X i c b Z B N S 8 N A E I Y n 9 a v W r 6 p H L 4 t F 8 F Q S E f R Y 9 O K x g v 2 A N p b N d t O u 3 W z C 7 k S o o f / B i w d F v P p / v P l v 3 L Y 5 a O s L C w / v z L A z b 5 B I Y d B 1 v 5 3 C y u r a + k Z x s 7 S 1 v b O 7 V 9 4 / a J o 4 1 Y w 3 r / 1 T o p h p d + J l S S I l d s / l G Y S o I x m Z 5 O + k J z h n J s g T I t 7 K 6 E D a m m D G 1 A J R u C t 3 j y M j T P q p 7 l 2 / N K 7 S q P o w h H c A y n 4 M E F 1 O A G 6 t A A B g / w D K / w 5 s T O i / P u f M x b C 0 4 + c w h / 5 H z + A L Y e j z Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " m u k U x 5 F P a 6 8 h c m r y d 3 f n 8 9 W n i 7 A = " > A A A B 7 X i c b Z B N S 8 N A E I Y n 9 a v W r 6 p H L 4 t F 8 F Q S E f R Y 9 O K x g v 2 A N p b N d t O u 3 W z C 7 k S o o f / B i w d F v P p / v P l v 3 L Y 5 a O s L C w / v z L A z b 5 B I Y d B 1 v 5 3 C y u r a + k Z x s 7 S 1 v b O 7 V 9 4 / a J o 4 1 Y w 3 r / 1 T o p h p d + J l S S I l d s / l G Y S o I x m Z 5 O + k J z h n J s g T I t 7 K 6 E D a m m D G 1 A J R u C t 3 j y M j T P q p 7 l 2 / N K 7 S q P o w h H c A y n 4 M E F 1 O A G 6 t A A B g / w D K / w 5 s T O i / P u f M x b C 0 4 + c w h / 5 H z + A L Y e j z Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " m u k U x 5 F P a 6 8 h c m r y d 3 f n 8 9 W n i 7 A = " > A A A B 7 X i c b Z B N S 8 N A E I Y n 9 a v W r 6 p H L 4 t F 8 F Q S E f R Y 9 O K x g v 2 A N p b N d t O u 3 W z C 7 k S o o f / B i w d F v P p / v P l v 3 L Y 5 a O s L C w / v z L A z b 5 B I Y d B 1 v 5 3 C y u r a + k Z x s 7 S 1 v b O 7 V 9 4 / a J o 4 1 Y w 3

FIG. 6 :
FIG. 6: (a) Ising Landau gauge fixing condition for the π-flux lattice, black (red) bonds corresponds to σ z b = 1(−1) (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 the physical fermion c propagator G β c (k, iω m ).The bubble diagram evaluates to a convolution between the Ising matter field φ z and 'orthogonal' fermion f σ propagators defined on the π-flux lattice (d) Finite temperature spectral function A(k, ω = 0) = 1 π ImG(k, iω m = i0+) computed by a numerical evaluation of the bubble diagram.