Translationally invariant non-Fermi liquid metals with critical Fermi-surfaces: Solvable models

We construct examples of translationally invariant solvable models of strongly-correlated metals, composed of lattices of Sachdev-Ye-Kitaev dots with identical local interactions. These models display crossovers as a function of temperature into regimes with local quantum criticality and marginal-Fermi liquid behavior. In the marginal Fermi liquid regime, the dc resistivity increases linearly with temperature over a broad range of temperatures. By generalizing the form of interactions, we also construct examples of non-Fermi liquids with critical Fermi-surfaces. The self energy has a singular frequency dependence, but lacks momentum dependence, reminiscent of a dynamical mean field theory-like behavior but in dimensions $d<\infty$. In the low temperature and strong-coupling limit, a heavy Fermi liquid is formed. The critical Fermi-surface in the non-Fermi liquid regime gives rise to quantum oscillations in the magnetization as a function of an external magnetic field in the absence of quasiparticle excitations. We discuss the implications of these results for local quantum criticality and for fundamental bounds on relaxation rates. Drawing on the lessons from these models, we formulate conjectures on coarse grained descriptions of a class of intermediate scale non-fermi liquid behavior in generic correlated metals.

A number of strongly correlated materials with a metallic parent state exhibit a variety of non-Fermi liquid (NFL) properties. Some of the best known examples of such behavior occur in the ruthenates [1][2][3][4], cobaltates [5,6], iron-based superconductors [7] and heavy-fermion materials [8], amongst others. Some of these materials display striking non-Fermi liquid behavior over a broad range of temperatures above an emergent low energy scale but develop Fermi liquid-like properties and well defined Landau quasiparticles below this scale, while others remain non-Fermi liquid-like down to the lowest temperatures. Perhaps the most striking example of the latter behavior occurs in the "strange-metal" regime [9,10] of the cuprate superconductors and some quantum critical heavy-Fermion systems [8,11].
One of the most dramatic properties associated with many of these materials is a linear dependence of the dc resistivity on temperatures without any sign of saturation. In the cuprates, much of the phenomenology of the normal state is apparently well described by the "marginal Fermi liquid" (MFL) model [12], which postulates the existence of marginally defined quasiparticles, whose scattering rate is comparable to their energy.
Broadly speaking, a few theoretical frameworks have been proposed to explain the phenomenology of strange metals: (i) Quantum critical fluctuations of a bosonic degree of freedom coupled to a Fermi-surface leading to a non-Fermi liquid ground state, which dominates the properties of the system in a range of temperatures above the critical point. Concrete examples of such theories involve the situation where an order-parameter field (such as a nematic or antiferromagnetic order-parameter) at its critical point couples to an electronic Fermi-surface [11]. Much progress has been made in understanding the properties of this class of metallic quantum critical points in recent years [13]. (ii) A distinct class of non-Fermi liquids arise at a critical point driven by electronic fluctuations associated with the destruction of the Fermi surface. Examples include a Kondo breakdown transition 1 in a heavy Fermi liquid [14][15][16][17][18] and a Mott transition between a metal and a quantum disordered insulator [18,19]. Such non-Fermi liquid quantum critical points have been argued [19,20] to possess a critical Fermi surface -i.e., the electronic excitations at the critical point are characterized by the presence of a sharply defined Fermi surface but with no sharp Landau quasiparticles. 2 Currently known concrete low-energy theories for such quantum critical points involve fractionalized degrees of freedom and associated dynamical gauge fields. Theoretical progress has been possible on a few examples of such theories [18][19][20][21][22][23]. While these concretely tractable examples are extremely useful, much more remains mysterious about the general theory of quantum critical points associated with the 'death' of a Fermi surface. 3 (iii) Instead of appearing just at a critical point, a non-Fermi liquid can arise as a stable zero temperature phase, as has been observed for instance in numerical studies of lattice models [24]. A classic example of such non-Fermi liquid behavior occurs in a two-dimensional electron gas under high magnetic fields, when a compressible metallic phase is realized at a filling of ν = 1/2 [25]. Indication of such non-Fermi 1 The onset of antiferromagnetism (as a function of some tuning parameter) in several heavy Fermi liquids is known to have a striking non Hertz-Millis character and is accompanied by a dramatic change in the Fermi-surface volume. 2 Critical Fermi surfaces are also expected to occur at some quantum critical points driven by fluctuations of a Landau order parameter associated with ordering at zero momentum. 3 In particular, in all the examples so far, there is a remnant 'ghost' Fermi surface of fractionalized degrees of freedom once the electronic Fermi surface dies. It is not known if continuous quantum phase transitions can occur to phases where there is no such ghost.
liquid quantum phases have also been reported in correlated mixed-valence materials [26,27]. (iv) Finally, in the limit of sufficiently strong interactions and at intermediate temperatures, it is possible that strange metal behavior arises generically without tuning to the vicinity of a quantum critical point. However, the ground state is a Landau Fermi liquid or some other conventional state (e.g. a superconductor) and the strange metal regime appears only as a crossover at higher temperatures.
Despite all this progress in the theory of non-Fermi liquids, there is no clear mechanism that produces a linear in T resistivity over a broad range of temperature in quantum critical or other non-Fermi liquids in translationally invariant models as a result of strong local electronic interactions.
The phenomenological "marginal Fermi liquid" theory assumes coupling to a bosonic fluctuating mode that gives linear resistivity [12]; however, it is not clear how to derive such a bosonic spectrum from a microscopic model. The results of recent quantum Monte Carlo (QMC) simulations of an Ising nematic transition [28] are consistent with a linear behavior of the resistivity at the quantum critical point. 4 There is currently no theoretical understanding of these results.
Empirically, it is likely that these different routes to non-Fermi liquid physics are realized in different materials. Our focus in this paper is on route (iv) above. In a number of different systems (for example, in some cobaltates [5,6] and ruthenates [29,30]) it is indeed seen that there is a wide intermediate temperature T U V T T coh where strange metallic transport is observed, including non-Fermi liquid temperature dependent resistivity with values exceeding the Mott-Ioffe-Regel limit. As the temperature drops below a low 'coherence scale' T coh there is a crossover to more conventional behavior. Importantly, it does not appear that T coh can be pushed close to zero by tuning some parameter, 5 suggesting that it may be fundamentally impossible to stabilize such NFL states at zero temperature. In other words, the intermediate-T NFL physics of these systems may not in principle be controlled by T = 0 Infra-Red (IR) fixed points with a finite number of relevant perturbations. We call such intermediate-T non-Fermi liquid states as examples of "IRincomplete" states of matter (see Ref. [31] for a possibly useful exposition). By themselves, they cannot be the deep IR theory of any state of matter and hence require IR-completion. 4 These results are subject to uncertainties associated with analytical continuation from imaginary to real time.
From the imaginary time data, one can extract "resistivity proxies" that coincide with the dc resistivity under certain assumptions, such as the absence of sharp features in the frequency-dependent conductivity over a scale ω T . The validity of these assumptions is hard to assess from imaginary-time data, and has to be checked independently. 5 It is worth pointing out that this is likely not the situation for the cuprate strange metal and in some heavy electron materials like YbRh2Si2 [11]. In both these systems by tuning one parameter it has been possible to stabilize the NFL physics to ultra-low T suggesting that T coh can, in principle, be tuned to zero.
Examples include electron-phonon systems above their Debye temperature [32], lattice models with bounded kinetic energy at high T [33,34], spin-incoherent Luttinger liquids [35], electrons coupled to a lattice of bound-states [36], holographic non-Fermi liquids [37,38], and some states found in DMFT calculations at finite temperature [39,40]. Common to many of these examples of IR-incomplete theories is that they have extensive residual low-T entropy (i.e. the entropy extrapolated to T = 0 from the regime in which the theory applies is non-zero) which is then relieved below T coh leading to a crossover to a conventional state.
Progress in understanding strongly interacting IR-incomplete non-Fermi liquids has been hindered by the lack of suitable controlled theoretical techniques. The Sachdev-Ye-Kitaev (SYK) model [41][42][43][44][45][46][47][48][49], consisting of a large number of degrees of freedom coupled via a random all-to-all interaction, provides a window into the behavior of strongly coupled systems with no quasiparticles. The model is (0 + 1)−dimensional, and thus it does not contain information about transport.
Higher dimensional generalizations of the model have been considered [43,[50][51][52][53][54][55]. Refs. [43,50] studied lattice models of itinerant fermions coupled to spins with a long-ranged all-to-all interactions. Refs. [51,52,54] considered lattice models with an SYK dot placed in every site, with a random short ranged inter-site coupling. The charge and thermal transport properties have been computed. The solution of these models have many appealing characteristics, such as a locally quantum critical, non-Fermi liquid crossover regime where the resistivity is linear in temperature and quasi-particles are destroyed.
In all of the above models, translational symmetry is strongly broken, raising a number of questions: (i) Does quenched disorder play an essential role in the behavior of strange metals as suggested in Ref. [56], or could it be realized even in a perfectly crystalline system? (ii) Can a non (or marginal-)Fermi liquid with a critical Fermi-surface (to be defined below) appear within this class of models, and what are its transport and other related properties? (iii) Does a non-Fermi liquid with a critical Fermi surface show quantum oscillations in an external applied magnetic field?
In order to address these questions, in this work we construct a set of translationally invariant models that can be solved exactly in the large N limit, where N is the number of fermion flavors (or "orbitals") per site, coupled by a frustrated on-site interaction. Our construction is therefore different from other constructions of higher-dimensional generalizations of SYK-type models at a fundamental level. The crucial new ingredient, namely the exact translation symmetry (instead of a statistical symmetry) at the level of each realization will allow us to address many interesting questions beyond the scope of previous works. Specifically, we will address questions related to the possibility of obtaining non-Fermi liquid behavior in models without disorder, the existence of a sharp Fermi surface (or lack thereof) in translation invariant non-Fermi liquids, the fate of quantum oscillations due to critical Fermi surfaces beyond semiclassical quantization of quasiparticle-based theories and other related issues. Our paper will also lead to new insights into a class of non-Fermi liquid metals, namely the "IR-incomplete" NFLs (of which there are numerous examples, as highlighted later), and will potentially be useful for future developments in the field.
Within our construction, if there is a single band of bandwidth W , and the typical interaction strength is U , we find that the system crosses over at a temperature T ∼ W 2 /U (≡ Ω * ) from a lowtemperature Landau Fermi liquid ground state to locally quantum critical non-Fermi liquid state, where the Fermi surface is completely destroyed, but there still is a well-defined Fermi energy.
The resistivity crosses over from ρ ∼ T 2 at T Ω * to ρ ∼ T at T Ω * ; the value of the resistivity at the crossover scale (T ∼ Ω * ) is ρ ≈ h/N e 2 . In addition, the two salient features of the one band model are as follows: (i) At strong coupling (i.e. U W ) and at low temperatures compared to Ω * , the momentum dependence of the electron self-energy becomes parametrically smaller in (W/U ) compared to the frequency dependence. The resulting Fermi liquid has a sharp Fermi surface but the self-energy is momentum independent. At temperatures higher than Ω * , this sharp Fermi surface is lost and the electronic excitations become incoherent. (ii) In the incoherent regime, even though the system is translationally invariant, as a result of the locally critical structure of the correlation functions and strong momentum dissipation on the lattice, the previously established mechanism for incoherent transport in disordered SYK-like models [43,54] continues to be applicable to our one-band model. In the Fermi liquid regime, the resistivity is finite and arises from umklapp scattering. Our results for the translationally invariant one-band model shed interesting light on the validity of 'locally critical' theories in a microscopic setting, where the self-energy is allowed to be momentum dependent apriori but becomes unimportant in the large−N and strong coupling regime. Within the multi-band setup, we also consider models where the on-site interactions for one of the bands involves q > 4-body terms, which allows us to obtain non Fermi liquids with a singular self-energy and a critical Fermi-surface. Interestingly, upon applying a magnetic field, both the marginal Fermi liquid and the non-Fermi liquid regimes are characterized by quantum oscillations of the magnetization as a function of the inverse of the field. The period of the oscillations is the same as that of an ordinary Fermi liquid, but the temperature dependence of their amplitude is different from that of a Fermi liquid.
It has been proposed that transport in the strange metal regime [30] can be understood in terms of the conjectured "Planckian" bound on relaxation rates, 1/τ k B T / [57,58]. It is interesting to examine our results in the context of this proposal; however, there is no unique definition for a "transport scattering rate". One can naively choose to define it from the dc conductivity by fitting it to a 'Drude-like' form σ = ne 2 τ dc /m * , where m * is the effective mass of the low-temperature Fermi liquid state, and expect a bound on τ dc (∼ 1/T ). 6 In the two-band non-Fermi liquid state described in Sec. V, we find that 1/τ dc has a non-Planckian form: 1/τ dc ∼ T α with α < 1. Alternatively, a natural way of defining the transport scattering rate is to use the temperature dependent crossover frequency scale, 1/τ opt , across which the optical conductivity crosses over from its high frequency regime to the dc limit. For the models considered below, we find that τ opt satisfies a Planckian-type bound with τ −1 opt ≤ ak B T / , where a is an O(1) number. 7 Thus, the question of the existence of a bound requires a sharp definition of what one means by the "scattering rate." In light of the phenomenologically appealing features of the solution of these models, it is interesting to ask about lessons we might learn and apply to real correlated materials described by some generic model. Restricting to IR-incomplete non-Fermi liquids, it is interesting to consider the structure of a coarse-grained description. We expect that there will be a few distinct universality classes of such non-Fermi liquids with different coarse grained descriptions. The models studied in this paper suggest one possible universal route to non-Fermi liquid behavior. Specifically we propose that in a class of generic systems that show intermediate-T NFL physics, there is an 6 This is the definition used in Ref. [30]. One we may alternatively define a scattering rate by expressing σ ∝ κv * 2 F τ d where κ is the compressibility, or σ ∝ ω 2 p τp where ωp is the plasma frequency. 7 There are examples of models that violate this bound on τopt, however. See, e.g., Ref. [59]. emergent large length scale a (the microscopic scale) such that within patches of size the system is maximally chaotic (in the sense that it obeys the chaos bound of Ref. [60]; see Appendix J for details) though globally, i.e. at longer scales it may not be so. Further we expect that the assumption of maximal chaos severely restricts the structure of correlators within such a patch.
A coarse-grained description of the macroscopic physics -appropriate at scales much longer than -can then be built by coupling together maximally chaotic bubbles with generic interactions.
Note that the (0 + 1)-dimensional SYK models are well known to be maximally chaotic. Thus the models we study may be viewed as a concrete example of such a coarse grained effective model. In general the appropriate description of a maximally chaotic bubble in such a metal will not likely be an SYK-like model, and will in the future have to be replaced by a better theory that takes into account spatial locality within each bubble. Nevertheless these solvable models point to the importance of maximally chaotic intermediate scale bubbles as a possible universal route to a class of non-Fermi liquids.
The rest of this paper is organized as follows: we introduce our model of a strongly interacting translationally invariant one-band metal in section II and compute the fermion Green's function, thermodynamic and transport properties in sections II A, II B and II C respectively. In section III we provide a very simple qualitative understanding of these one-band models which demystifies their properties and provides a complementary approach to analyzing the key features of the model.
We introduce an additional band with a parametrically smaller bandwidth and study the effect of inter-band interactions in section IV. We compute the fermion Green's function in section IV A and find a regime with a marginal Fermi liquid behavior. We explore the thermodynamic and transport properties associated with the MFL in sections IV B and IV C respectively. The two band model is generalized in section V, where we find a regime with non-Fermi liquid behavior and a singular self-energy with a variable exponent; the thermodynamic and transport behavior are discussed in section V B and V C. For the generalized model, we explore the "2K F " singularities and quantum oscillations in the magnetization as a function of an external magnetic field as a result of the presence of the critical Fermi surface in section V D and V E, respectively. On the basis of our study of all the models with locally critical degrees of freedom, we propose some general constraints on models with local quantum criticality in section VI. Finally, in section VII we conclude with a summary of our results and their relation to other recent works. In section VII C 2 we also present our conjectures for intermediate scale non Fermi liquid physics in generic strongly correlated models and explore their consequences for the phenomenology of a wide variety of non-Fermi liquid metals. We study the toy problem with q = 2 (i.e. a random-matrix) in the presence of uniform hopping terms as an interesting exercise, which can be solved exactly, in Appendix A in order to shed some light on issues related to transport. A number of accompanying technical details appear in the appendices.

II. ONE-BAND MODEL
Let us begin with a microscopic model in d−dimensions on a hypercubic lattice (d = 2 will be of primary interest) with N orbitals per site and fermionic operators defined by, c † r, , c r, , ( = 1, ..., N ). The fermions satisfy usual anti-commutation algebra {c r, , c † r , } = δ δ rr . We assume that there is a global U (1) symmetry corresponding to a single conserved density (V ≡volume), The value of 0 < Q c < 1 can be tuned by a chemical potential µ c . The Hamiltonian is given by where the hopping terms between sites r and r , t c rr , are diagonal in the orbital subspace and depend only on |r − r | (assumed to be identical for all orbitals). The interaction term, U c ijk , is purely on-site and is properly antisymmetrized with U c ijk = −U c jik = −U c ij k and U c ijkl = U c klij . The values of U c ijk are assumed to be independent of the site-label, r (see Fig. 1(a) for a caricature of the model; Fig. 1(b) elucidates the structure of interactions within each site). The model can be viewed as a lattice of Sachdev-Ye-Kitaev (SYK) [41][42][43][44][45][46] quantum dots with identical on-site interactions, connected by orbital-diagonal, translationally invariant hopping matrix elements. 8 The model (1) is difficult to solve. However, just as in the SYK model, if we consider the interaction terms U c ijk to be random, independent variables with a zero mean, and take the limit N → ∞, then it is possible to compute properties of the model averaged over realizations of U c ijk . It is important to note that we are not only assuming that the coupling constants on different sites have the same distribution; rather, in every realization they are identical to each other, and hence the Hamiltonian defined in Eq. 1 is translationally invariant. For convenience, we set the distribution of the coupling constants to be Gaussian. The distribution satisfies U c ijk = 0 and (U c ijk ) 2 ≡ U 2 c , where U c characterizes the strength of the interactions. The other energy scale in our problem is the free electrons' bandwidth, which we denote by W c .
It is believed that the properties of the SYK model are self-averaging, in the sense that the correlation functions of a typical realization are close to those of the mean, up to 1/N corrections. In   Appendix B, we demonstrate that the standard deviations and higher cumulants of the correlation functions in our model are suppressed by powers of 1/N . We therefore expect that the correlation functions in our model are self-averaging in the large N limit, as in the single-site SYK model.

A. Fermion Green's Function
The fermion Green's function can be analyzed diagrammatically, such that the large-N saddlepoint solution reduces to studying the following set of equations self-consistently, where´k ≡´d d k/(2π) d and ε k is the dispersion for the c−band. Formally, the above set of equations corresponds to resumming an infinite class of 'watermelon-diagrams', as shown in Fig. 2. One can arrive at the same set of saddle-point equations by starting from the path-integral formulation, as described in Appendix C. In Sec. III, we provide a simple alternate derivation of the results for the one band model using scaling-type arguments which provide much physical insight.
As we shall now show, the fermionic spectral function has qualitatively different behavior at different temperatures. When the temperature is much lower than the characteristic crossover scale Ω * c ≡ W 2 c /U c , the spectral function has a Fermi-liquid like form. In the interesting case U c W c , there is a second regime defined by Ω * c T U c , where the spectral function has an incoherent, local form without any remnant of a Fermi-surface. To make this statement more precise, we can take the limit of U c → ∞ keeping W c finite (such that Ω * c collapses to zero), and then take the limit of T → 0, thus obtaining a compressible phase of electronic matter without quasiparticleexcitations in a clean system, lacking any sharp momentum-space structure. We refer to this state as a local incoherent critical metal (LICM).
To analyze the equations (2a-2c), we focus on the two extreme limits of T (or ω) that are either much larger or much smaller than Ω * c . In the limit T Ω * c , we find that the system follows Fermi liquid behavior at sufficiently low frequencies. To show this, let us use a Fermi liquid-like ansatz for the fermionic self energy. At low frequencies we assume that Σ c has the following form near the Fermi surface: where Z is the quasiparticle residue, to be determined self-consistently, k = |k − k F | (k F is the Fermi momentum),ṽ F (v F ) are the renormalized (bare) Fermi-velocities with the renormalization v F /v F = A to be determined self-consistently, and the . . . denote higher power terms in an expansion in ω, k. We stress thatṽ F is different from the effective Fermi velocity v * F = Zṽ F , which is the physical speed with which quasi-particles propagate. For simplicity, we have dropped the constant term, which can be absorbed in the chemical potential. Inserting this form into the self-consistency equations (2a-2c), we obtain after a standard computation (see Appendix D for details) Here, ν 0 ∼ k d F /W c is the bare density of states at the Fermi energy. (We set the units of length such that the lattice spacing a = 1.) In Eq. (4) we have taken into account the contribution of the quasi-particle poles of the Green's functions at iω = Zε k , and ignored the additional branch cut singularities, that turn out not to change the final result qualitatively. Next, we feed Eq. (4) back into (2b), giving where α, ζ are numerical factors of order unity that depends on the geometry of the Fermi surface (Appendix D). The factor of ln Zṽ F k F |ω| in (5) is special to d = 2; it is absent in higher dimensions.
Equating this to Eq. (3), we get that In particular, in the weak coupling limit, ν 0 U c 1, we get that Z ≈ 1 − (ν 0 U c ) 2 . In the opposite limit, ν 0 U c 1, we get to logarithmic accuracy that Z = 1/(ν 0 U c ), and A is O(1). In this strong coupling limit, even though the electronic self-energy is allowed to be apriori momentum dependent, the frequency dependence is parametrically larger in (U c /W c ) compared to the momentum dependence. Hence, the ground state is a Fermi liquid for any coupling strength; in the strong coupling limit, the quasi-particle weight becomes small, and the effective mass increases as where m is the bare mass while the momentum dependence of the self-energy is independent of U c . This state is therefore a heavy Fermi-liquid. Moreover, since the self-energy is only weakly dependent on the momentum but strongly frequency dependent, the resulting state is reminiscent of a DMFT description [39] of a heavily renormalized Fermi liquid. Note, however, that while DMFT is exact in the limit of infinite dimension, in our case d is finite; instead, we have to take the large N and strong coupling limits.
In particular, we get that at frequencies smaller than U c , Σ c (ω) ∼ isgn(ω) U c |ω| [41,43,44]. Extrapolating Σ c (ω) from high to intermediate frequencies, we reproduce the result that Σ c (ω) W c for ω W 2 c /U c , consistent with the extrapolation from low frequencies. To find the residual momentum dependence of the Green's function in the strong coupling incoherent regime, we expand the self-consistent equations 2c in powers of ε k 9 . To leading order, we where G 0 (ω) and Σ 0 (ω) are the Green's function and the self-energy of the single site SYK model, respectively (see Appendix E for details). Importantly, we see that although the momentum dependence of the Green's function decreases with increasing frequency, the correlation length over which G c (r, ω) decays (obtained by taking the fourier transform of G c (k, ω)) remains frequency-independent and is determined by the spatial extent of the hopping parameters, t c rr . To summarize, we get that for strong coupling, G c (k, ω) has the following form in the two extreme frequency limits: where Z ∼ 1/(ν 0 U c ), and α is a number of order unity. B(ω) is a constant independent of frequency for both ω > 0 and ω < 0 though its precise value is different for the two signs of ω. Indeed it is a direct descendant of the "spectral asymmetry" that characterizes the Green's function of a single SYK island [41,43].
At low frequencies, there is a Fermi surface with well-defined, albeit strongly renormalized quasi- The ω 2 term in the denominator of G c becomes the imaginary part of the self-energy after an analytic continuation to real frequency.
It can be written in a revealing form: Σ (ω) ∼ ω 2 ln W * c |ω| /W * c . At finite temperatures, the zero-frequency imaginary part is Σ (0, T ) ∼ π 2 T 2 ln W * c T /W * c . Note that, upon extrapolating this form to the crossover scale, Σ (0, T ∼ Ω * c ) ∼ W * c , i.e. at this scale, the scattering rate of quasiparticles is comparable to the effective bandwidth, and we expect the quasi-particle picture to break down.
At energies much higher than the renormalized bandwidth, the Fermi surface is destroyed, and the single-particle spectral function has no sharp features in momentum space. Instead, it is well approximated by A c (k, ω) ∼ 1/ U c max(|ω|, T ). This is the LICM regime.

B. Thermodynamic Properties
We now turn to discuss the thermodynamic properties of the one-band model. As we saw in the previous subsection, at sufficiently low temperatures, T Ω * c , the system is well described by Fermi-liquid theory. This implies, in particular, that the entropy per unit cell follows a linear At temperatures much higher than Ω * c , we can calculate the thermodynamic properties perturbatively in the inter-site hopping 10 . Then, the entropy is given by that of a single SYK dot, up to a correction of the order of (W c /U c ) 2 . The entropy takes the form S(T Ω * c ) = N (S 0 + γ 0 T ), where S 0 and γ 0 are known constants [43]. At temperatures of the order of Ω * c , we expect the entropy to interpolate between these two behaviors. Based on our analysis of the saddle point equations in this section, as well as our simpler understanding using scaling in Sec. III below where we study the perturbative effects of the relevant hopping terms as a function of decreasing energy starting from the decoupled SYK-like regime, we find a strong indication of a single crossover separating the two regimes at the coherence-scale Ω * c . All of the thermodynamic quantities, as well as the frequency dependent self-energies, evolve smoothly through this crossover without any associated phase transitions; we have checked this explicitly by solving the saddle-point equations numerically for small system sizes (results not shown). These aspects of our results are thus qualitatively similar to the results reported in Ref. [54] for the disordered version of the one-band model.
Next, we turn to discuss the compressibility, given by N κ = (∂n/∂µ), where n = r, c † r, c r, is the total density for all the orbitals. We begin by noting that each site (i.e. SYK island) has a finite compressibility which is given by κ 0 ∼ 1/U c [52,54]. As a result of the finite hopping and bandwidth, there is a correction to this result and at strong coupling we obtain where c 0 is a constant of order unity. As discussed earlier, in this regime the mass enhancement

C. Transport
Let us now discuss both the optical conductivity and the dc resistivity of the metallic phases introduced above. The real part of the optical conductivity is given by the Kubo formula where Π ret Jx (ω) is the retarded current-current correlation function for the current in x direction. The total current operator is given by with J i denotes the current from orbital i and v i k = ∇ k ε i k . For the previously assumed identical dispersions for all the orbitals, the velocities are also the same. The leading diagrams which contribute to Π ret Jx (ω) are shown in Fig. 3. In Fig. 3 (a), we show the leading graph without vertex corrections.
In the high temperature (T Ω * c ) regime, the vertex corrections ( Fig. 3b) are subleading. To see this, note that the electron velocity is odd in momentum, while all the Green's functions are momentum independent to lowest order in W c /U c . Then each of the loops over orbital i and j vanish individually. We therefore consider only the diagram in Fig. 3

(a), and this results in
where A k (ω) is the electron spectral function and f (...) represents the Fermi-Dirac distribution function. In the high temperature SYK-like regime (T Ω * c ), the optical conductivity clearly satisfies (ω/T ) scaling. At frequencies much higher than the temperature (i.e. by arranging Focusing on the dc conductivity in this regime, we find (in units of e 2 /h): where v 2 (∼ t 2 c ) represents an average over the Fermi-surface. As a result of the ω/T scaling, the crossover scale from the high-frequency to the dc limit is of order T . We note that in this incoherent regime, once the electronic correlation functions become locally critical, the previously established mechanism for incoherent transport in disordered SYK-like models [43,54] continues to be applicable as a result of the strong momentum dissipation in the lattice model.

III. A SIMPLE VIEW ON THE ONE-BAND RESULTS
In this section, we provide a simple alternate understanding of the physics of the one-band model that does not require a detailed analysis of the saddle-point equations in Eq. (2a-2b). We carry out a simple scaling analysis for the extension of the usual SYK model (as defined above in Eq. 1), as well as an extension of the model that involves higher than 2−body interactions. The latter will be used later in Sec. V to obtain a non-Fermi liquid with a critical Fermi surface. We begin by considering the limit where the hopping t c U c . When t c = 0 the different SYK islands are decoupled from each other. Further we know that within each island the electron has power law correlations in time with a scaling dimension ∆ c = 1 4 . For small hopping t c , we can study the relevance/irrelevance of the hopping term in the decoupled SYK theories. In the action, the hopping term becomes Clearly then under a scaling transformation τ → τ = τ s , t c → t c = t c s 1 2 so that the hopping is relevant. To study the system at a non-zero temperature T , we run the scaling until a scale With decreasing temperature we will stay in the regime of weak hopping until a temperature such that Uc which matches exactly with the coherence scale identified in section II A.
For T T coh the physics will be that of weakly coupled SYK islands and we can calculate physical properties in perturbation theory in t c . For T T coh it is natural to expect that the coupling between the different islands leads to a Fermi liquid phase.
We can now understand the thermodynamics and transport through simple physical arguments.
First we recall that for the (0 + 1)-dimensional SYK model, the entropy is known [43,47] to obey The ground state entropy S 0 is nonzero in the limit N → ∞, and then T → 0. As argued in Sec. II B, in the limit t c = 0 this is obviously the entropy per site of the lattice model. When t c = 0 and at sufficiently high temperature such that T T coh , both the entropy and the compressibility only get small corrections when we perturb in t c . For T T coh , however, the ground state entropy of the decoupled limit is relieved, and S(T →0) γT . An estimate for γ can be obtained by matching this entropy extrapolated to T = T coh with the residual entropy of the high temperature phase. This In Fermi liquid theory the γ coefficient directly gives the quasiparticle effective mass m * ∼ Uc t 2 c a 2 . (a is the lattice spacing.) Note that the "bare" mass determined from the hopping Hamiltonian is m ∼ 1 tca 2 . Therefore the mass enhancement m * m ∼ Uc tc 1 in exact agreement with the solution of the self consistency equations in section II A. The behavior of the compressibility in both the high-T and low-T limits have already been described in section II B.
Let us now turn to transport. In the high-T regime in perturbation theory in t c , the conductivity where t c (s T ) is the effective renormalized hopping at a temperature T introduced above. We therefore get This is again in exact agreement with the calculations in section II C. For T T coh , if the Fermi surface is big enough to allow umklapp scattering of the low energy quasiparticles, we will get a resistivity ρ(T ) =Ã N T 2 . To estimateÃ, we require that when extrapolated to T = T coh this matches the extrapolation of the high T result down to T coh . This leads toÃ ∼ h e 2 1 T 2 coh . Note that in the low-T Fermi liquidÃ ∼ γ 2 thereby obeying the Kadowaki-Woods relationship [62].
The understanding above readily generalizes to the physics of coupled SYK models where the onsite interaction is composed of q (q ≥ 4 and even) fermion operators [63]; we studied a generalized two-band version of this model in section V. Specifically consider the model of just a single band of electrons with the Hamiltonian As before we take U i 1 i 2 ...iq and the hopping t c to be translationally invariant and U i 1 i 2 ...iq = 0, and We focus on the small t c regime. For general q, the scaling dimension of the fermion is ∆(q) = 1/q. It follows that a small t c is relevant at the decoupled fixed point and scales Following the discussion above, we determine that the physics will be that of weakly coupled islands In the high-T regime, the entropy and compressibility have the same qualitative behavior as for q = 4. Importantly, there is a residual entropy S 0 (with a linear T correction) and a finite non-zero compressibility. At T T coh we again expect a Fermi liquid. The residual entropy is relieved, and the low-T heat capacity coefficient is γ ∼ S 0 T coh . This can be converted into an estimate for the quasiparticle effective mass in the Fermi liquid.
The electrical resistivity in the high-T regime, estimated as above, is of the form Note that ρ dc increases faster than linearly with T , but slower than T 2 . Thus the high-T linear resistivity is not a generic property of coupled SYK models and requires q = 4. As before (with umklapp scattering) the low-T resistivity is ρ(T ) =ÃT 2 withÃ ∼ γ 2 .

A. Explicit transport calculation at high-temperature
It is instructive to explicitly calculate the conductivity in the high-T regime in perturbation theory in the hopping, taking special care with issues regarding disorder averaging. As the leading temperature dependence is ∝ t 2 c , a second order perturbative calculation should give the exact answer for this leading term. The imaginary frequency current-current correlator is readily related to the electron Green's function of each SYK island (details of such perturbative calculations are straightforward -see for example Appendix E of Ref. [64]): Note that we have not carried out the disorder averaging yet. G ij (r, r; iω ν ) is the frequency dependent fermion Green's function within the SYK island at site r and r is the site neighboring to r in the positive x direction.
We now wish to average this over disorder realizations. If the SYK interactions were independently random at different sites (like in the models studied in Ref. 54), then obviously upon disorder averaging (indicated with an overline) the products G ij G ji that appear above can be replaced by G ij G ji for any N . In our translation invariant models, the SYK interactions are the same at every site. Thus strictly speaking we must instead take G ij G ji . Fortunately (as shown in Appendix B) for q ≥ 4 in the large-N limit 11 , the property holds and we can continue to make this replacement in the products entering the correlation function. Further we know that when N → ∞, only G ii is O(1) and G ij for i = j is suppressed. 12 Therefore we will henceforth replace all Green's functions by their averages (and drop the overlines).
Implicitly this has been done in all of the discussions in this paper. Analytically continuing Eq. (21) to real frequencies we get the familiar form for the real part of the conductivity Here A(ω) is the spectral function for the Green's function within a single SYK island. For SYK q (with q ≥ 4) this satisfies ω/T scaling, 11 The q = 2 case is special and will be discussed in detail in Appendix A. with F q (...) a known universal function. It follows from Eq. (23) that the conductivity itself satisfies ω/T scaling. We get with S q (...) a universal function determined in terms of F q (...) by Eq. (23). In particular in the dc limit we reproduce the temperature dependence previously obtained for general q ≥ 4. As a result of the ω/T scaling, it is easy to see that the frequency scale over which σ xx (ω) reaches its , with a an O(1) number. Moreover, the scaling function S q (x) ∼ 1/x 2−4/q at large x. Therefore, at frequencies much larger than the temperature, the conductivity has the form σ xx (ω) ∼ 1/ω 2−4/q .

IV. TWO-BAND MODEL -MARGINAL FERMI LIQUID
In the previous section, we saw an example of a crossover from a Fermi-liquid to an incoherent metal, without any remnant of a Fermi-surface, in a one-band model. It is interesting to ask if a critical Fermi-surface [19] can emerge in the general class of translationally invariant models that are being considered here. Before proceeding further, it is useful to define precisely what we mean by a critical Fermi-surface. Within our definition, the criticality is associated with the gapless single-particle excitations of physical electrons over the entire Fermi surface, which remains sharply defined 13 . However there are no Landau quasiparticles across the critical Fermi surface and the quasiparticle residue Z is zero. We describe two classes of models in the next two sections that host such a critical Fermi surface.
Let us begin with a model where we introduce an additional band of f −fermions with operators f † r, , f r, ( = 1, ..., N ) and an associated conserved U (1) charge density, Q f that may be tuned by a chemical potential µ f , which we set to zero. The modified where H c is as described in Eq. (18), and H f is defined in an identical fashion with translationally invariant hoppings t f rr and on-site interactions U f ijk . The form of the inter-band interaction is chosen to be where the coefficients, V ijkl , are chosen to be identical at every site with U f ijkl = V ijkl = 0, and where the distribution of the couplings satisfy ( We now assume that t f r,r t c r,r , i.e. the bandwidth for the f −fermions is much smaller than the bandwidth for the c−fermions (W f W c ). The model described by (26) therefore has some similarity to models for 'heavy-Fermion' systems, with a specific form of interaction terms, and where the direct hybridization term, H hyb = r,ij M ij c † ri f rj has been set to zero. To leading order in 1/N , the saddle point equations for the Hamiltonian defined in Eq. (26) are given by, We have introduced ξ k as the dispersion for the f fermions and Σ c , Π c are as defined earlier in Eqs. (2a-2c). The watermelon-diagrams for the self-energies are shown in Fig.4.
Based on our analysis for the one-band model in section II, we see immediately that if H cf = 0, the two decoupled subsystems have a Fermi-liquid to LICM crossover at frequencies or temperatures of the order of Ω * c and Ω * f respectively (Ω * f Ω * c ). In the high-temperature regime, T > Ω * c Ω * f , when both bands are in a LICM phase, adding H cf does not alter any of the features qualitatively and the resulting state is thus still described by a LICM phase. Similarly, at low-temperatures, scale). 14 In order to have a sharp meaning to the notion of a non-Fermi liquid with a critical Fermi surface, it is useful to also send the scale Ω * f to zero. This is conveniently done by setting t f = 0 while keeping U f finite. In this limit, we can pose sharp questions about the presence or absence of quasiparticles and Fermi surfaces in the T → 0 limit.

A. Fermion Green's Function
In the window of intermediate energies, where the f fermions enter the LICM regime, while the c fermions do not, one may find a Fermi surface formed by the lighter bands, with an anomalous single-particle lifetime due to scattering off the heavy band. As we show below, it is precisely in this regime that we obtain a marginal Fermi liquid regime with a critical Fermi surface of the c fermions. In the next section, we will generalize the model to obtain a critical Fermi surface of the c fermions with a singular frequency dependent self-energy with variable exponents.
In order to obtain the structure of the solution in the intermediate frequency regime, Ω * f < ω < Ω * c , we begin by considering the effect of the inter-band interaction perturbatively. Later, we will check that the behavior we find is self-consistent. As emphasized earlier, our conclusions hold in this regime for a finite U c , but we set U c = 0 for simplicity. We assume that the f fermions which is the familiar form in the SYK model [41,43,44]. We ignore here the weak-momentum dependent correction to the f Green's function in the LICM phase (as discussed in section II A) by considering the limit of W f /U f → 0, with U f fixed (the crossover scale Ω * f also goes to zero in this limit). Then, Π f (q, ω) has the momentum independent form 15 , Inserting Π f (ω) into Eq. (28d) we get for the c self-energy in Fig.4(a) (see Appendix F) The self-energy of the c fermions then has a 'marginal Fermi liquid' (MFL) [12] form. It is important to note that the above result is valid at most up to scales at which the self-energy becomes of the order of the bandwidth, i.e. Σ cf ∼ W c . This scale can be easily seen to be In order to check that the form of Σ cf (iω) in (30) is self consistent, we need to verify that it does not change qualitatively if it is evaluated using the full Green's functions. Moreover, we also need to evaluate Σ cf (ω) (the self-energy for the f fermions due to coupling to the c fermions; Fig. 4b) using the renormalized G c , and verify that its behavior is sub-leading to that of Σ f (ω) ∼ U f ω.
We demonstrated that this solution is indeed self-consistent in Appendix G. In particular, focusing for simplicity on the case where U cf W c , the contribution to the f fermion self-energy due to the inter-species interaction consists of two contributions -an analytic correction, which is given by that renormalizes the bare 'iω' term, and a singular (but subleading) correction (see Appendix F) In the limit U cf W c , the MFL regime extends over a frequency (or temperature) window the MFL extends only up to temperatures of the order of Ω * cf . We leave the behavior above Ω * cf for a future study.
To conclude, we find a broad temperature regime above Ω * f where the f fermions behave as a LICM, while the c fermions follow MFL behavior, with a well-defined Fermi surface and marginally defined fermionic quasiparticles. As we discuss in section V below, a generalized version of the twoband model gives a critical Fermi surface where quasiparticles are not even marginally defined (a full-fledged non-Fermi liquid). In Sec. V we analyze the density response at the "2K F " wavevector and quantum oscillations in magnetization of such a critical Fermi surface.
Interestingly, in both the marginal Fermi liquid of the present section and in the non-Fermi liquid of Sec. V, the structure of the c−fermion self-energy is such that it is singular in the limit of ω → 0 for all momenta, even far away from the Fermi-surface. This is a consequence of the fact that the fluctuations of the f fermions are critical for all momenta. We analyze this structure more carefully in Appendix G.

B. Thermodynamic Properties
Let us begin by analyzing the specific heat in the MFL regime, where the entropy density has contributions from the critical Fermi-surface of the c fermions as well as the f fermions which are in an SYK-like regime. The total extrapolated zero temperature entropy is finite as a result of the finite entropy density from the SYK sector. However, this excess entropy is relieved as a result of the crossover to the Fermi-liquid below Ω * f . In order to extract the contribution from the critical Fermi-surface, we can compute the free energy at a finite temperature using the standard Luttinger-Ward (LW) analysis (Appendix C and H). Let us consider the different contributions to the free energy, written as where Φ LW [G c , G f ] is the Luttinger-Ward functional, which depends on the exact Green's functions of the c and f electrons. The first two terms, with the form of the MFL self energy in Eq. (30), gives rise to a low temperature singular logarithmic correction [65] to the linear in T specific heat, i.e. it has a T ln(1/T ) form. This feature is reminiscent of the results of Refs. [12,65]. However, we note that the self-energy alone does not fix the thermodynamic properties. In particular, the other contribution to the free energy arises from the LW term, [The derivation of (34) follows closely the derivation in the single-band case, outlined in Appendix C.] Given the local character of the f Green's function, we only need the local form of the c fermion bubble above (which are the same as in a Fermi liquid). At low temperature, the first term in the LW functional above is proportional to T , and the second is proportional to T 2 .
Hence, the LW term does not lead to any singular modification of the results for the specific heat, The MFL has a critical Fermi surface that satisfies Luttinger's theorem: where A FS is the area of the Fermi surface and n c is the density of c fermions. This follows from a Luttinger-Ward analysis, applied to the c−fermion Green's function: , where χ 0 is the non-interacting compressibility. The LW analysis for the conserved f fermion density has been carried out in Ref. [44].

C. Transport
Next, we consider the charge transport properties in the MFL regime. The arguments below will also apply to the discussion of transport for the non-Fermi liquid metals described in section V.
We set U c = 0 and only consider the effects of inter-band scattering (U cf ) when the f electrons are in an incoherent SYK-like regime.
We are interested in the real part of the charge conductivity, which is given by where Π ret Jx (Ω) is the retarded correlation function of the x component of the current at a non-zero frequency. In particular, it is important to explore the role of vertex corrections of the current. associated with transport of the c and f fermions. Here we will be interested in the conductivity due to the c fermions. The conductivity due to the f fermions will be essentially identical to the discussion in the one-band model and we will not elaborate further on it here.
To leading order in 1/N , σ xx (Ω) is given by the sum over the set of ladder diagrams shown in Fig. 5(a), where the self-consistent equation for the current vertex Γ Jx (ω, Ω), is described diagramatically in Fig. 5(b): where v x k = ∂ε k /∂k x is the 'velocity' along x and we have assumed an identical dispersion for all the orbitals. It is important to recall that in a system that preserves inversion symmetry, the velocity (or equivalently, the current) vertex itself is odd with respect to the momentum label, k, . At the same order in 1/N , there is another contribution to the set of ladder diagrams and the current vertex 16 , as shown in Fig. 6. However, this correction vanishes as a result of the local structure of the f fermions, as explained below. In general, the above self-consistent equation is difficult to solve for the full current vertex.
However, the ladder insertions in the above series of diagrams, Π f , have a simple local structure that greatly simplifies the problem. At leading order in ξ k /U f , as we have discussed above, Π f (q, ω) is independent of the momentum q and has the form shown in Eq. the Green's function, G f . If we ignore the momentum dependence of Π f in Eq. (36) above, it is straightforward to see that the momentum integral in the second term vanishes, as the integrand is odd in l. The correction in Fig. 6 vanishes for the same reason. In this limit, we can therefore ignore the vertex corrections altogether, such that the conductivity is given only by the diagram in Fig. 5(a) without any rungs, which reduces the expression for the conductivity to where A k (ω) is the spectral function for the c fermions and f (...) represents the Fermi-Dirac distribution function. At frequencies much higher than the temperature (Ω T ), this leads to: where v 2 is the average of (v x k ) 2 over the Fermi surface. Let us now focus on the dc limit. We find that the scattering rate determined from the dc resistivity is determined by the single-particle scattering rate of the c−fermions. This result is not surprising, since in the regime that is being considered here where the f −fermions provide a momentum independent scattering channel, providing an effective 'momentum sink' for the c−fermions 17 . Therefore, the resistivity (in units of h/e 2 ) is given by, 17 One might be tempted to associate the momentum relaxing scattering in the clean system above with 'umklapp' scattering. However, we note that in the regime of interest here, there is no restriction on the respective c or f −fermion densities.
We can now estimate the frequency scale at which the high-frequency form of the optical conductivity matches the low-frequency dc result. A simple analysis immediately reveals the crossover scale (in units of k B / ) to be Note that the coefficient of the T −linear term in the scattering rate need not be O(1) due to the ln 2 (1/T ) term in the denominator.
In the regime T Ω * cf = U f (W c /U cf ) 2 that we are considering here, the dc resistivity is always smaller than the Mott-Ioffe-Regel limit, ρ dc h/(N e 2 ). At higher temperatures, we expect the MFL behavior to break down. We shall not treat this regime here, leaving it to a future investigation. Fermi-surface and a more singular (and variable) self-energy.
Let us reintroduce the f −electron operators f i , f † i with i = 1, .., N orbitals, as before. We generalize the interaction terms to have a q−fold term [63] (with q even; the models considered thus far correspond to q = 4). The Hamiltonian is still given by H = H c +H f + H cf , where the modified Hamiltonian for the f −electrons is given by, The hopping matrix-elements t f r,r are translationally invariant and diagonal in orbital-space. The on-site inter-orbital interactions, U f i 1 ...iq , are assumed to be random with U i 1 i 2 ...iq = 0, (U i 1 i 2 ...iq ) 2 = U 2 f and taken to be identical on every site. The model is therefore a translationally invariant generalization of the SYK q model [63] with uniform hoppings. Moreover, since we have already discussed the special case of q = 4 in the previous section, we shall only consider the case of q > 4 from now onwards.

A. Fermion Green's Function
As before, we are interested in the regime above a crossover-scale, Ω * f (q), where the f band realizes an incoherent metallic state without any remnant of a Fermi-surface. This crossover scale for the q−fold interactions is given by Ω , which reduces to the standard expression for q = 4. In this regime, the scaling dimension of the f −operators is ∆(q) = 1/q, such that the Greens function has the form G Let us now address the self-energy of the c fermions as a result of the quartic inter-band scattering in H cf . The bubble, Π f (q, ω), has a momentum-independent form, Solving for the c fermion self-energy self-consistently (see appendix G), we obtain which has a strong non-Fermi liquid form with an exponent 4∆(q) < 1 for q > 4. This behavior is valid to scales up to which the self-energy becomes of the order of bandwidth, which immediately gives Ω * cf (q) ∼ U f (W c /U cf ) 2/4∆(q) . Once again, for simplicity we restrict our attention to the case where U cf W c (which implies Ω * cf (q) U f ). Just like in the case of the MFL, a natural question to ask is if the feedback of the c fermions on the f fermions as a result of the inter-band scattering modifies the SYK form of their self-energy. There will be an analytic correction that renormalizes the bare 'iω' term, with a coefficient (W c /U f ) 2∆(q) (U cf /W c ) 2 . However this correction can be made small compared to the bare iω term if U f W c (U cf /W c ) 1/∆(q) . In addition, an explicit computation of the singular (but subleading) correction to the f self-energy as a result of the inter-species interaction leads to which, as before, is subleading to Σ f (iω) at frequency (or temperature) scales small compared to Ω * cf (q). We therefore conclude that in the intermediate regime between the scales Ω * f (q) and min(U f , W c ), the f fermions have a local SYK q form of correlations, while the c fermions have a NFL character with a well-defined Fermi surface but no sharply defined Landau quasiparticles.
Here, S f (T ) is the entropy of a single SYK q model (where S 0,q and γ q have been computed in Ref. [52]), S c (T ) comes from the first and second terms in Eq. (33), and S int (T ) originates from the inter-species interaction term δΦ LW ∝ U 2 cf´d τ G 2 f G 2 c in the LW functional. The extrapolated zero temperature entropy S(T → 0) = S 0,q is finite in the above regime. However, as described earlier, there is a crossover to a Fermi-liquid regime below the scale Ω * f , where the excess entropy is relieved. In the non-Fermi liquid regime, the specific heat scales as c V = T ∂S/∂T ∼ T 4∆(q) .
The compressibility associated with the conserved densities for both species of fermions follows from our discussion in section IV B. In particular, the c−fermions continue to satisfy Luttinger's theorem, such that their density is given by the area inside the critical Fermi-surface (see Appendix H). Moreover, the compressibility for the c−fermions, that are scattering off the incoherent f fermions, is a non-singular function of U cf . For U c = 0, the compressibility is given by that of non-interacting c-fermions, up to a correction of the order of U 2 cf .

C. Transport
The transport properties of the non-Fermi liquid regime considered here follow from a straightforward generalization of our results in section IV C. As a result of the completely local form of the Green's function, G f , at temperatures above the crossover scale (Ω * f ), we can continue to ignore the vertex corrections to the current vertex in Fig. 5(a). The optical conductivity for the c fermions is then given by Eq. (37). It is clear from the form of the spectral function in the non-Fermi liquid regime that the optical conductivity satisfies Ω/T scaling. At frequencies much higher than the temperature (ω T ), which is determined by the single-particle scattering rate of the c fermions. Following the discussion in section IV C, the dc resistivity is given by, which reduces to the marginal Fermi liquid form for q = 4. However note that just as we discussed in the context of the MFL in Sec.IV C above, in the regime T Ω * cf (q) that we consider here, the dc resistivity is always smaller than the Mott-Ioffe-Regel limit. As a result of the Ω/T scaling that holds in the same window of temperature and frequency scales, the conductivity can be expressed as, where H(...) is a universal scaling function.
The dc resistivity in Eq. (50) displays a strong departure from the "Planckian" form, since ρ dc ∼ T 4∆(q) with 4∆(1) < 1. However the scattering rate associated with the temperature dependent frequency scale that determines the crossover from the high frequency to the dc behavior, 1/τ opt , saturates the Planckian bound: 1/τ opt ∼ ak B T / with a = O(1).

D. 2K F Singularities
The sharp structure associated with a Fermi-surface in momentum space in a conventional Fermi-liquid leads to a singular 2K F response. A natural question to ask here is if the non-Fermi liquid states considered in this paper have a 2K F response that is different from other known examples of (non-)Fermi liquids [66,67]? In this section, we analyze the modification to the singularity for the non-Fermi liquid regime discussed above and find that it is fixed by the scaling dimension of the fermions, ∆(q). We assume that the temperature, while higher than the crossover scale Ω * f , continues to be much smaller than the Fermi-energy for c Fermions such that effects of thermal smearing can be ignored.
We are interested in studying the response of the critical Fermi-surface to an external source field that couples to the fermion density (for any orbital i) at a 2K F momentum. Naively, we expect there to be a suppression of the 2K F response as a result of the smearing of the Landau quasiparticle due to scattering off the 'local', incoherent f −electrons. Moreover, we have here a situation where the vertex corrections from the interactions, which are usually important in determining the 2K F response, are only weakly momentum dependent (since Π f (q, ω) has a weak dependence on q).
Let us then consider an external source field that couples to the 2K F fermion density, i.e. a particle at K F and a hole at −K F , through the following term in the action (it will be sufficient to consider a pair of antipodal patches for this purpose), where R and L correspond to the two antipodal patches at ±K F . The 2K F operator is defined and we are interested in the long range behavior of the correlation function C 2K F (x, τ ) = ρ * 2K F (x, τ ) ρ 2K F (0, 0) . We can obtain the singular structure for the correlation function by scaling, when the low energy physics is scale invariant under the following scaling transformation (z is the dynamical exponent), Let us suppose that u = u b φ (φ = 1 under the above rescaling, but we allow for a general φ to account for the possible singular renormalization from corrections to be considered below). Then the 2K F operator satisfies, where The fourier transform then satisfies the scaling, where p in this case represents the deviation of the full momentum away from 2K Fx , i.e. p x (p y ) is the direction perpendicular (parallel) to the Fermi-surface. Then we can immediately write the scaling form, For a conventional Fermi liquid, φ = 1 and z = 2, which leads to the famous √ ω singularity in the 2K F correlations. For the non-Fermi liquid considered above, if we ignore vertex corrections (which will be shown to be negligible below), φ = 1 and z = 1/(2∆(q)), which leads to the singular which is what we would naively obtain by computing the density-density response ∼´k´Ω G c (p + k, ω + Ω) G c (k, Ω) with the above self-energy in Eq. (43).
Let us now compute the one-loop vertex correction (Fig. 7), which may apriori change the singular structure. For simplicity, we set all the external momenta and frequencies to zero. The expression for the diagram is then given by, where ε ± k = ±vk x + k 2 y and v denote the dispersions and Fermi-velocities near the R/L patches; we have set the curvature to unity. Then, The above k y integral is convergent and leads to, We may now include the effect of this vertex on the density-density response, in order to compute the correction δC 2K F (p, ω) ∼´k´Ω δu G c (p + k, ω + Ω) G c (k, Ω) ∼ ω. This is clearly less singular than the result obtained from scaling (or equivalently, the bare correlation function without vertex corrections). This is simple to understand within our model, where the completely local form of the fluctuations associated with the incoherent f −fermions leads to scattering at all momenta for the c−fermions, and in particular no additional singularities arise as a result of any special scattering across the anti-podal patches. Note, however, that the vertex correction will be important for density correlations near q ≈ 0, and indeed are needed to obtain the finite non-zero compressibility that we argued characterizes these states. It is useful to treat the problem of oscillations in three-dimensions, where all of our previously obtained results for the self-consistent solutions to the saddle point equations continue to be true.

E. Quantum Oscillations
We focus on the example of the non-Fermi liquid (with q > 4). We note that strictly speaking, we should work at fixed density and account for the oscillations of the chemical potential as a function of the magnetic field. However, our calculations below will be done at a fixed chemical potential rather than a fixed density. In three dimensions, this is a justified approximation as an expansion in leading powers of the ratio of the magnetic field to the cross-sectional area of the Fermi-surface. At a fixed density the chemical potential has an oscillatory correction to its value at zero field whose amplitude vanishes linearly in field [72]. We do not include the effect of such chemical potential oscillations, that are subleading in powers of the ratio described above.
Let us now study the effect of a uniform magnetic field, B, along the z−direction through its orbital coupling to the c fermions (we assume that there is no orbital coupling to the U (1) charge associated with the f fermions, which is explicitly true when we set W f → 0). We analyze the structure of the saddle-point equations in the presence of the magnetic field in Appendix I 1.
A key property of the solution is that even at B = 0 both the c and f -self energies are completely local in space. In the NFL regime, the f fermions continue to be described in terms of the (0 + 1) dimensional SYK model and the self-energy for the c fermions as a result of the coupling to the f fermions can be written as, We study the effect of the first term above (independent of B) on all of the oscillatory phenomena; the effects arising from the explicit dependence ofΣ cf on B are of higher order in (ω c /µ c ), where ω c = eB/m * is the cyclotron frequency 18 . The magnetic field leads to a singular modification of the kinetic energy of the c fermions into Landau "bands" in three dimensions that disperse along the direction of the field 19 .
The Green's function for the c fermions in the LL basis is given by, We are interested in the oscillatory contribution to two quantities: (i) the spectral density of states, and, (ii) the (orbital) magnetization. The oscillatory component of the spectral density of states in the limit of ω → 0 at a finite T is of the form, Let us now focus on the oscillatory component of the orbital magnetization, M osc , which is a thermodynamic quantity. It is possible to write down the oscillatory component of the free energy and compute the magnetization by taking appropriate derivatives. Instead, we compute the magnetization in a different manner here by noticing that the dependence on B enters only through the kinetic energy of the c fermions (as already described above). The magnetization density defined per unit area then is given by, where only the kinetic part of H c in the presence of magnetic field enters the above expression: where after Peierls' substitution h rr = t c rr e iA rr (A rr ≡vectorpotential corresponding to uniform B along z−direction). In the LL basis, the magnetization is then given by, where n labels the LL index, α denotes all of the degenerate states within each LL and φ nα (r) is the LL wave function. The latter sum over r, r can be carried out to yield, In the above equation n (p z ) is as denoted in Eq. (65b). Equivalently, this can be obtained directly by writing the Hamiltonian in the LL basis as, We then have, Using the Poisson resummation formula, the oscillatory component of the magnetization is then, . (72) After some standard manipulations, details of which appear in Appendix I 3, we obtain, Here, A(...) is a purely real amplitude for the oscillation of the k th −harmonic [an explicit expression for the amplitude appears in Eq. (I23)]. We find that the period of oscillations is determined by the cross-sectional area of the Fermi-surface and remains unaffected by the form of the self-energy.
The amplitude, on the other hand, is affected by the non-Fermi liquid form of the self-energy and has a non Lifshitz-Kosevich form 20 .
The universal scaling structure for the temperature dependence of the amplitude of the oscillations can be determined to be as follows (see Appendix I 3) where R(x) is a scaling function of ω c ) that decays exponentially at large x. The scale for damping of the amplitude for any given harmonic at any finite temperature is then given by

VI. GENERAL CONSTRAINTS ON LOCAL CRITICALITY
Both the incoherent metal regime in the single band model (Sec. II) and the marginal/non Fermi liquid regime in the two-band model (Sec. IV,V) display "local quantum critical" behavior. By that, we mean that the temporal correlation functions decay as power laws (up to a correlation time ξ τ ∼ 1/T ), whereas the spatial correlations decay exponentially over a temperature-independent length-scale of a few lattice constants. 21 In both of our models, the local quantum critical regime is unstable at sufficiently low temperatures: below a certain "coherence temperature," a crossover to a different, more conventional behavior occurs. This is consistent with the fact that in both models, the entropy in the local quantum critical regime extrapolates to a non-zero value in the limit T → 0, violating the third law of thermodynamics. Instead, the one-and two-band models cross over to a Fermi liquid regime below the energy scales Ω * c and Ω * f , respectively, and relieve the excess entropy.
This raises the question whether, in generic lattice models with a finite number N of degrees of freedom per unit cell, local quantum critical behavior can be stable down to T = 0 (either as a quantum phase or at a quantum critical point). On scaling grounds, it has been argued that local quantum criticality must be accompanied by a finite entropy density in the limit T → 0 [74], although some caveats have been pointed out [75]. 22 Here, we argue that in any translationally 20 Non Lifshitz-Kosevich forms for the amplitude of magnetization oscillations have been obtained in earlier holographic calculations [73]. 21 Note that this is different from the scenario where the system has long range spatial correlations in addition to the power law correlations in time, but only the frequency dependent correlations have anomalous dimensions [17] -a situation also referred to as "local quantum criticality." This behavior has been invoked in the context of heavy Fermion quantum criticality [16]. Here, we refer only to the situation where the spatial correlations are strictly local. 22 For a hyperscaling-violating theory in d spatial dimensions, the entropy density scales as S ∼ T (d−θ)/z , where θ is the hyperscaling violation exponent. Naively, z → ∞ implies a finite ground state entropy density. Ref. [75] pointed out that this can be avoided if θ → −∞.
invariant lattice model with finite N , local quantum criticality is not possible down to arbitrarily low temperature. More generally, systems with a weaker form of quantum criticality, where the correlation length diverges sub-polynomially in 1/T (as in Refs. [76]), must have an entropy that scales as a power of the linear dimension L in the limit T → 0. We expect this large residual entropy to lead to an instability at sufficiently low temperature, resulting in a lower entropy state.
First, consider a translationally invariant system where the correlation time ξ τ ∼ 1/T , while the correlation length ξ is independent of T . In a finite cluster of linear size L = αξ, the temporal correlation functions of local operators approach their values in the thermodynamic limit for sufficiently large α. Hence, the temporal correlations decay as a power law up to times of the order of ξ τ . Since the system is finite, ξ τ cannot exceed the inverse of the mean level spacing near the ground state, ξ τ ≤ 1/δ(L); i.e., δ(L) ≤ T . Therefore, in a generic system with a finite N , the local quantum critical behavior cannot persist to arbitrarily low T , otherwise δ(L) → 0.
Next, we consider systems with a weaker version of local quantum criticality in which the correlation time, ξ τ , grows faster than polynomially as a function of the correlation length, ξ. The dynamical critical exponent, z (defined via ξ τ ∼ ξ z ) is still infinite. Repeating the argument above for a finite cluster of linear size L = αξ at T = 0, we get that ξ τ cannot exceed the inverse of the level spacing near the ground state, ξ τ ≤ 1/δ(L).
Hence, δ(L) must decrease faster than polynomially in L. In contrast, the level spacing near the ground state in generic many-body systems with local interactions is expected to depend polynomially on the system size [77].
The anomalously small level spacing near the ground state has consequences for the entropy in the limit T → 0. In the microcanonical ensemble, the low-temperature entropy scales as S(T → 0) ∼ log[∆E/δ(L)], where ∆E is a sub-extensive energy shell. As a concrete example, suppose that ξ ∼ log(ξ τ ), as proposed in Refs. [76,78] for certain quantum critical points. In this case, following the considerations above, δ(L) ≤ e −L/α . Therefore, we find that S(T → 0) ∼ L/α. Even though such behavior does not violate the third law of thermodynamics in spatial dimension d > 1, we do not expect it to hold down to T = 0. The high density of low energy states generically leads to an instability that lifts the near-degeneracy of the ground state. Similarly, if the correlation time scales as ξ τ ∼ [log(ξ)] γ , we get that S(T → 0) ∼ L 1/γ . We note some interesting exceptions to this rule. The disorder-averaged correlations of disordered systems at infinite randomness fixed points [79] are known to display z = ∞ behavior. This behavior comes from rare regions where the correlation time is much longer than the typical one.
However, we do not expect such rare region effects in generic translationally invariant systems. An-other exception is found in certain three-dimensional topologically ordered states, called "fracton states" [80][81][82], that have S(T → 0) ∼ L without any fine tuning. However, this property probably does not lead to quantum critical behavior of local correlation functions, since local operators have vanishingly small matrix elements between the topologically distinct near-degenerate states that are responsible for the low-temperature entropy.

VII. DISCUSSION
In this work, we have defined a class of translationally invariant models that can be solved in the large N limit. Even though the ground states of these models are conventional (although strongly renormalized) Fermi liquids, they exhibit a crossover at an intermediate energy scalewhich can be parametrically smaller than the microscopic coupling constants -into a non-Fermi liquid regime. This regime is characterized by local quantum critical scaling of certain correlation functions -i.e., the correlation time diverges as ξ τ ∼ 1/T , while the correlation length is nearly temperature-independent.
Interestingly, many of the properties of the non-Fermi liquid regimes are reminiscent of those seen in different quantum materials. In the one-band model of Sec. II, the resistivity grows linearly with temperature, and does not saturate at the Mott-Ioffe-Regel limit. The two-band version of the model (Sec. IV,V) exhibits a regime where the light band has a critical Fermi surface -either a marginal Fermi liquid or a non-Fermi liquid, depending on the precise nature of the interactions between the heavy and light bands. The resistivity grows as ρ ∝ T in the MFL and as ρ ∝ T 4/q with an exponent q > 4 in the NFL.
In this section, we will put these results in the context of previous work, and discuss their possible implications either to more generic models (in particular, ones that do not involve the limit of a large number of degrees of freedom per unit cell), as well as to strongly correlated materials.

A. Relation to other work
Several models composed of lattices of coupled SYK dots have been studied recently [51,52,54,[83][84][85][86]. Of these, the one-band model we introduce here is closest to the model solved by Song, Jian, and Balents [54], who studied a lattice of SYK dots coupled by single-particle hopping. The main difference between this work and the present one is that the model studied here is translationally invariant, whereas the model of Ref. [54] is strongly disordered -both the interactions and the hopping matrix elements vary from site to site. The translational invariance allows us to address the properties of the Fermi surface in the low-temperature Fermi liquid regime. In the strong coupling limit, we find a strongly renormalized Fermi-liquid with a momentum independent selfenergy. Interestingly, however, the properties of the high-temperature (T W 2 c /U c ) LICM phase are similar in the two models. This is a consequence of the fact that, in this regime, the correlations become short-range in space; hence, the presence of translational invariance does not modify the properties of the system in a fundamental way. For example, even with translational symmetry, there is no remnant of a Fermi surface, and the resistivity is linear in temperature in both cases. In our model, the resistivity scales as T 2 in the low temperature regime (T W 2 c /U c ), as expected in a Fermi liquid; in contrast, in the model studied in Ref. [54], we expect the resistivity to saturate to a temperature-independent constant, due to the presence of strong disorder.
Earlier work [43]  Finally, our results are -not surprisingly 23 -similar to those found in strongly coupled theories that can be solved using holographic dualities [37,38,87,88]. These models give locally quantum critical behavior associated with a non-vanishing entropy in the limit T → 0. Upon coupling the locally quantum critical degrees of freedom to itinerant fermions, marginal Fermi liquid and non-Fermi liquid states can result (see also Ref. [84]). As in the case of lattices of SYK dots, these models involve taking the limit of a large number of local degrees of freedom. Moreover, as in our model, the locally quantum critical regime is unstable at low energies to the formation of either long-range ordered states or a heavy Fermi liquid. 23 It is the possibility of a simple holographic description of the 0 + 1-D SYK model that has partly contributed to the tremendous recent interest in this model. Our two-band model is roughly similar in spirit to the "semi-holographic" theory in Ref. [87], although of course the details are very different.

B. Bounds on transport
It is interesting to discuss our results in the context of possible "universal bounds" on transport coefficients. It has been proposed [57,58] that the relaxation time (or "dephasing time" [57]) is bounded by the Planckian time, 1/τ P ≤ ak B T / , where a is an unknown constant of order unity.
Following this idea, a number of bounds on transport coefficients have been proposed [89][90][91][92]. An interesting conjectured bound on the heat and charge diffusion constants in Ref. [91] involved the many-body 'chaotic' properties of the system (see Appendix J). However explicit calculations [93,94] have demonstrated violation of such bounds in different settings (at present there is no known counterexample to the bound proposed in Ref. [92]). Empirically, the transport lifetime of many metals where the resistivity is linear in T has been found to be not far from /(k B T ) [30].
As we already described in the introduction, there is no unique choice of a transport scattering rate that may have an associated universal bound. In order to compare with the procedure adopted in Ref. [30], where the scattering rate was extracted by fitting the transport data to a Drude-like form, let us focus on the case of the MFL and NFL states discussed in sections IV,V above. For the model in section IV, we may extract the renormalized mass m * /m ∼ (ν 0 U 2 cf /U f ) ln(1/T coh ) from the low temperature FL regime (as measured in quantum oscillations) below T coh ∼ Ω * f . Using σ = ne 2 τ dc /m * to define τ dc in the MFL regime at high temperatures leads to 1/τ dc ∼ T , which satisfies a Planckian bound for the particular choice of the dc scattering rate. Note, however, that the resistivity in the MFL is ρ ∝ T with no logarithmic corrections, i.e., it is not simply proportional to 1/τ dc . A similar procedure adapted to the NFL regime of the two band model with q > 4 in section V leads to a lifetime with a strongly non-Planckian form, However it is still true that 1/τ dc < T (since in the NFL regime, T > T coh ).
It is interesting to point out that in all the cases studied here, the "optical scattering rate," 1/τ opt , defined as the frequency scale at which the high frequency optical conductivity approaches its dc value, satisfies 1/τ opt < ak B T / with a = O(1). In the incoherent regime of the oneband model and in the two-band non-Fermi liquid, 1/τ opt ∼ T ; in the two-band MFL, 1/τ opt ∼ T / ln 2 (1/T ) [see Eq. (40)]. Thus, 1/τ opt satisfies a Planckian-type bound, but the temperature dependence of the dc resistivity does not necessarily follow that of 1/τ opt .

C. Implications for generic models
Clearly, the models (1,26) are fine-tuned in many ways. In particular, the number of local degrees of freedom, N , is taken to be large, and the interactions U ijk are taken to be independent, random variables whose average is precisely zero. It is thus important to ask which of the properties of the solution are peculiar to these models, and which are expected to hold more generically, even in less fine-tuned models with a finite number of degrees of freedom per unit cell.
Here, we will discuss possible implications of our results for generic models (with a finite number of degrees of freedom per unit cell). In particular, we describe how local quantum critical behavior may arise in an intermediate temperature window in systems where the coherence scale (e.g., the effective Fermi energy or the Bose condensation temperature) is much smaller than the microscopic scale. We then formulate a conjecture for an effective "coarse grained" description of non-Fermi liquid states in generic models, inspired by the models constructed in this work, based on notions of many-body quantum chaos.

Local quantum criticality in generic models
The local quantum critical behavior found in some regimes of our models is unlikely to be stable in generic models down to zero temperature. A diverging correlation time without a corresponding diverging correlation length clearly requires an infinite number of local degrees of freedom.
Moreover, as we have argued in Sec. VI, even a correlation length that diverges sub-polynomially with the correlation time implies a divergent (although not necessarily macroscopic) entropy in the T → 0 limit. Hence an instability is likely to occur at a sufficiently low temperature.
Nevertheless, we speculate that local quantum critical behavior (with a correlation time that scales as /T and a nearly temperature-independent correlation length) can appear generically in strongly correlated metals, over a finite but broad temperature window. To see how such behavior can arise, consider a model with a metallic (Fermi-liquid) ground state. At low temperature, the single-particle lifetime scales as 1/τ ∼ T 2 /Ω * , where Ω * is a non-universal "coherence scale" that depends on the strength and form of the inter-particle interactions. If the interactions are sufficiently strong, Ω * may be much smaller than the microscopic coupling constants of the model, such as the hopping or the interaction strength. We expect that are the renormalized Fermi energy and Fermi velocity, respectively 24 . This is indeed the case in the 24 This is certainly not universally the case; for example, in the vicinity of a metallic quantum critical point with one-band model of Sec. II. In the Fermi liquid regime, temporal correlations decay exponentially over a timescale ξ τ ∼ 1/T . Spatial correlations decay over the thermal length, ξ T ∼ v * F /T . Thus, crudely extrapolating to T ∼ Ω * , where the Fermi liquid behavior starts to break down, we get that the correlation length at the crossover temperature becomes ξ T ∼ v * F /Ω * ∼ λ F , implying that the correlation length reaches the microscopic length scale set by the Fermi wavelength. (In a typical metal, this is of the same order of magnitude as the lattice spacing.) On the other hand, the correlation time at this temperature is of the order of 1/T . If the renormalized Fermi energy is much smaller than the microscopic energy scales (set by the interaction strength and the hopping), then at T ∼ Ω * , the correlations extend over a time which is much longer than the inverse of the "bare" Fermi energy. 25 What happens at temperatures higher than Ω * ? The spatial correlations already decay over a microscopic length scale at T ∼ Ω * , so it is natural to assume that the correlation length is not strongly temperature dependent in this regime. We argued above that the correlation time at T ∼ Ω * is ξ τ ∼ 1/T . Further, we assume that the "scrambling rate" (discussed in Appendix J) at this temperature is close to saturating the bound [60], λ L ∼ T . Therefore, one can guess that the bound remains nearly saturated at T > Ω * . The natural appearance of a Planckian time scale implies that the correlation time ξ τ remains of the order of 1/T even above Ω * . Hence, if the temperature window between the renormalized Fermi energy and the bare one can be made very large, then we expect this window to exhibit some form of "local quantum criticality." 26 a Q = 0 order parameter, Ω * and v * F kF are parametrically different. Here, we are assuming that there is a single energy scale Ω * , which is small not because of the proximity to a quantum critical point, but due to strong microscopic interactions. 25 A classic example of a Fermi system with a low coherence temperature is the normal state of 3 He; the renormalized Fermi energy is significantly smaller than the bare one, due to the strong inter-particle interactions. The temperature window above the renormalized Fermi energy, where Fermi liquid behavior breaks down but the system is still quantum mechanical, has been termed a "semi-quantum liquid" [95]. 26 Interestingly, the scenario discussed here is similar to the behavior found in the "spin-incoherent Luttinger liquid" regime [35]; above the spin coherence temperature, the single-particle Green's function decays in space over a length scale set by the inter-particle spacing, but displays power-law behavior in time, up to τ ∼ 1/T . However, in this case, the spatial correlations of other operators -such as the density -still decay as a power law. In contrast, we are assuming that not only the single particle correlation functions become short ranged at T ∼ Ω * , but all correlation functions do.

Towards a "coarse-grained" description of non-Fermi liquid behavior in correlated materials
As outlined in the introduction, there is a zoo of materials that display non-Fermi liquid behavior, in terms of their single-particle properties and transport, over a broad range of temperatures.
However, even amongst all of these materials there is a varying degree to which the non-Fermi liquid behavior persists down to the lowest temperatures. A general observation across the wide variety of systems displaying non-Fermi liquid properties are as follows: 1. In many correlated metals, the dc resistivity is often linear in temperature 27 , i.e. ρ dc ∼ T , and persists over a broad intermediate range of temperatures with a temperature independent slope. Moreover, it shows no sign of saturation and exceeds the Mott-Ioffe-Regal limit.
2. In a number of materials where the above is true, there is a low coherence scale below which there is a departure from the non-Fermi liquid behavior and a crossover to more conventional Fermi liquid type behavior (and possibly to other ordered phases). Moreover, the extrapolated zero temperature entropy from the finite temperature non-Fermi liquid regime is finite and has been reported in certain members of the ruthenates family [4] and in the cobaltates [96]. This excess entropy is relieved below the coherence scale associated with the low temperature Fermi liquid.
There are a number of outliers to the above description, most prominent amongst them being the optimally doped cuprates and certain quantum critical heavy-Fermion materials, where the non-Fermi liquid behavior observed at intermediate temperatures persists down to the lowest temperatures without any changes or characteristic crossovers. Similarly, the extrapolated zero temperature entropy in the non-Fermi liquid regime is zero (see e.g. Ref. [97] for the cuprates).

The intermediate scale behavior is remarkably similar in a wide variety of these systems, in
spite of the microscopic details being totally distinct. This is particularly surprising, since it appears that there is an emergent universal behavior and the details of the microscopic physics are somehow not important. However, e.g. the coefficient of the T −linear transport scattering rate can generically be different and dependent on the underlying details.
The above experimental observations pose an interesting theoretical challenge. In particular the apparent universality of the phenomena suggests that the explanation does not rely too much 27 Other power-laws have also been reported, e.g. in some families of the ruthenates [4]. on the precise microscopic details of any single system but instead is generic to strong correlations between the electrons at the lattice scale. The theoretical models studied by us in this paper are consistent with a number of these empirical observations. Is it then possible to draw some general lessons from this exercise in order to bridge the gap between a realistic description of materials and the solvable models considered by us?
Below we will consider the possibility of a coarse-grained description over scales much longer than any microscopic scale in the problem with a few key assumptions, that allows us to reproduce the features described above. We propose one possible route that allows us to give such a coarsegrained description of non-Fermi liquid metals in a general setting below. This will allow us to place the specific models studied in this paper in context within a conceptual framework that applies to generic strongly correlated materials.
The apparent universality of intermediate scale non-Fermi liquid physics in diverse correlated systems naturally leads to the possibility that there is a universal coarse-grained description. After all if the macroscopic behavior is universal it makes sense that the universality has set in at some finite length/time scale large compared to microscopic scales. This length/time scale will itself be non-universally related to the microscopic scales but the subsequent behavior at even longer scales will be universal. There will thus be a universal coarse grained description (much like in hydrodynamics or other theories of universal macroscopic phenomena). We will use the notion of 'many-body' quantum chaos to formulate our conjectures below (see Appendix J for a brief exposition to the subject).
• Conjecture 1 (C1)-For systems that display non-Fermi liquid behavior over a wide range of temperatures above a low crossover scale (Ω * ), there is an intermediate emergent lengthscale , with a L (a ≡lattice spacing and L ≡system size), such that a subsystem defined within a region of size is maximally chaotic. The entire system may or may not be maximally chaotic globally (on scales ∼ L).
• Conjecture 2 (C2)-For a patch of size the assumption of maximal chaos severely restricts the structure of general n-point correlators, i.e, it restricts them to a set of universality classes of possibilities.
Let us state the first conjecture a bit more sharply. Consider the squared commutator for generic local operators, W and V , where depends on r. The statement of Conjecture C1 is that for "normal" non-fermi liquid systems, there is a length scale a (a = microscopic length scale) such that for a |r| , and for times /v B t |r|/v B the Lyapunov exponent λ L = 2πT thereby saturating the chaos bound. These time scales are long enough for two local operators at x, x within a patch to mix but short enough that information has not moved between patches. On the other hand, for |r| , the system need not be maximally chaotic with λ L ≤ 2πT .
Conjecture C2 simply says that the physics of a maximally chaotic bubble is restricted to some universality classes.
A coarse grained description of the system would then consist of "islands" of typical size that are maximally chaotic, which are coupled to each other by generic hopping and interaction terms.
Why might we expect these conjectures to be true? Let us start with C1. We have already noted that it is natural that there exists a long length scale at which universality first emerges in a "normal" NFL system. Sufficiently complex and generic strong local interactions may make it natural that the dynamics is maximally chaotic at these length scales (with no guarantee of course that maximal chaos persists out to macroscopic scales). We regard this as roughly analogous to the assumption of molecular chaos in the kinetic theory of gases.
As for Conjecture C2, given the existence of a bound on the Lyapunov exponent it is again natural that systems that saturate the bound are very special and have universal properties.
Further inspiration for these conjectures comes from current ideas on strongly coupled continuum quantum field theories and their relationship to quantum black holes. Consider a UV field theory with a conserved global U (1) symmetry that is sufficiently strongly coupled that it has a classical gravity dual. We assume the theory is at a non-zero density of the global U (1) charge.
This UV theory will flow under the RG to some IR behavior that in general will describe different physics. As the temperature is decreased there will be a change from a regime controlled by the strongly coupled UV theory to whatever IR theory emerges under the RG flow. In the high temperature regime, in the gravity description of the UV theory we should include a charged black hole. It is well known that this black hole has a residual zero temperature entropy. Thus the corresponding high temperature behavior of the boundary quantum field theory is IR-incomplete and has an extrapolated ground state entropy. Now, it is believed that black holes are the "fastest scramblers" [98], i.e they saturate the chaos bound. Thus in the high-T regime the quantum field theory we are considering will satisfy the chaos bound. However there is no guarantee that this will continue to be the case as the temperature is decreased. The restricted behavior of maximally chaotic systems can then be plausibly related to the different universality classes of systems captured holographically by charged black holes.
This situation mimics the situation we envisage for generic, complex, strongly coupled lattice models. Of course the presence of the lattice (and the concomitant finite number of degrees of freedom/unit cell) requires that maximal chaos can only develop on some length scale much bigger than a.
We of course leave for the future explorations of these conjectures and their development into a useful coarse grained description of non-fermi liquids. Here they provide a conceptual context within which we can place the solvable models studied in this paper. Each SYK island is a specific example of a maximally chaotic system. Thus we can view our models as a toy example of a macroscopic system made out of coupling maximally chaotic bubbles. We note however that a future development of a universal coarse-grained NFL description will need to be more refined than simply modeling each bubble by an SYK island 28 . The refinement will need to include spatial locality within each bubble. Further it will need to include microscopic lattice symmetries as effective internal symmetries at scale . Finally it will have to incorporate the right microscopic Note added: As this paper was being completed for submission, we became aware of a related work [99] that studies disordered higher dimensional generalizations of the SYK model. Our point of strongest overlap is in the discussion of the two-band model where both constructions find a marginal Fermi liquid (in addition, we also obtain a critical Fermi surface in this example). In Ref. [99], the authors analyze the magnetotransport properties of such a disordered metallic regime; we study the fate of such a critical Fermi surface under the effect of a magnetic field, that gives rise to quantum oscillations even in the absence of quasiparticles. are different. This is unlike the interacting case, where we expect no δ−function singularity in the spectral function except at the Fermi surface, even within a single realization.
Since every realization of Eq. (A1) is a free electron model with translational invariance, we expect the real part of the frequency-dependent conductivity to contain a Drude-like contribution, where D is a temperature-dependent Drude weight. Moreover, since the current operator is diagonal in orbital space, σ xx (ω) = σ xx,Drude (ω) for all realizations of J c ij ; i.e., there is no "regular" background in the optical conductivity. Let us demonstrate this for the zero-temperature case. The real part of the conductivity at non-zero frequency is given by σ xx (Ω) = ImΠ ret Jx (Ω)/Ω, where Π ret Jx (ω) is the retarded correlation function of the x component of the current. To leading order in 1/N , σ xx (Ω) is given by the sum over the set of ladder diagrams shown in Fig. 8(a). To compute this sum, it is useful to first solve the self-consistent equation for the current vertex Γ Jx (iω, Ω), shown in Fig. 8 where v x k = ∂ε k /∂k x is the band dispersion along x. Solving this equation and inserting Γ Jx into the bubble in the last equality in Fig. 8(a), we get that that the Matsubara frequency Π Jx (iΩ) is given by Evaluating the ω integral gives that Π ret Jx (iΩ) = 0 for any Ω = 0. To show this, we analytically continue the integrand to the complex plane, iω → z. A little bit of algebra shows that the denominator of the integrand in Eq. (A6) does not vanish for any z unless J = 0 or Ω = 0. [This is shown using the explicit form of G(k, z) from Eqs. (A2,A3).] Therefore, the only singularities in the integrand are the branch cuts of G(k, z)G(k, z + iΩ), shown in Fig. 9. Since the integrand The contour can be deformed into the two closed contours shown in blue above, going around the branch cuts of the integrand. decays as |z| −2 at |z| → ∞, we can deform the integration contour into a pair of contours that enclose the parts of the branch cuts to the right of the imaginary z axis (shown in blue in Fig. 9).
The integral then becomes The integrand is purely imaginary, as can be seen by noting that the fourth term in the square brackets is the complex conjugate of the first, and the third term is the complex conjugate of the second. [This follows from the fact that G(k, z * ) = G(k, z) * .] On the other hand, the original integral in Eq. (A6) is a real function of Ω, as can be seen performing a change of variables, ω → −Ω − ω. Therefore, the integral in Eq. (A6) vanishes.
The discussion above shows that Π Jx (iΩ) = 0 for any Ω = 0; analytically continuing to real frequency, we get that Π ret Jx (Ω) = 0 for Ω = 0. This is a direct consequence of the fact that, for any realization of our model, the current is an exactly conserved quantity. Therefore, σ(Ω = 0) = 0.
According to the conductivity f-sum rule, where D is the Drude weight. We conclude that σ xx = Dδ(Ω), as expected.
The calculation above was at T = 0. However we expect that the zero frequency delta function in the conductivity actually holds at all temperatures. To see this explicitly consider the calculation of the conductivity at high temperature from the standpoint of the perturbation theory in the hopping described in Sec. III A. In contrast to the SYK q models with q ≥ 4, at q = 2, we cannot replace G ij G ji by G ij G ji : This can be checked by explicit calculation of both sides. It is readily seen that the correct averaging G ij G ji leads to the expected δ(ω) peak in σ xx in the high temperature limit.
Appendix B: Self averaging of the correlation functions at large N In this Appendix, we show that in the large N limit, the correlation functions of a single realization of the model (18) are essentially the same as the averaged correlation functions over realizations of U ijk . Consider, for example, the orbital-diagonal single-particle Green's function G c,ii (k, iω). We define δG c,ij (k, iω) = G c,ij (k, iω) − G c,ij (k, iω) as the deviation of the Green's function of a single realization from the mean, where the overline denotes averaging over realizations of the interaction U ijk . The variance of G c,ij is given by This quantity can be represented as a sum of all the diagrams with two Green's functions connected by at least one interaction line. [The disconnected terms are subtracted off by the last term in Eq. (B1).] We examine some of the leading order diagrams that contribute to C 2 in Fig. 10(a,b).
The lowest-order contribution, Fig. 10(a), scales as 1/N 2 . Therefore, we conclude that the standard deviation of the Green's function is much smaller than the average in the large N limit. Similar considerations hold for any correlation function.
Moreover, we can estimate higher cumulants of the Green's function. Consider, for example, the fourth order cumulant, . The leading order diagram for C 4 is shown in Fig. 10(b). As can be seen from the figure, C 4 ∼ 1/N 6 . Similarly, one can show that the nth cumulant of δG c,ij , C n ∼ 1/N 2n−2 . Hence, all the higher cumulants C n decrease rapidly with n, and we expect the distribution of the Green's function to become approximately Gaussian in the large N limit.

Appendix C: Path integral formulation
Here we briefly describe the path integral formulation of our translationally invariant models.
This gives an alternate view on the self-consistency equations as a saddle point approximation (which becomes exact in the N → ∞ limit) to the path integral. It leads immediately to the Luttinger-Ward functional used in many places in the paper. Our discussion will closely follow the treatment described in detail in previous work on SYK models [41,46,60]. We will mainly emphasize the minor differences arising from the translation invariant form of the SYK interactions in our models. We begin with the one-band model. The partition function is given by the imaginary To deal with the SYK interactions we should average over their probability distribution. Strictly speaking this should be done using replicas. However as is well known from previous SYK studies (and as we demonstrated in Appendix B due to the self-averaging property of our version of the model), the physical propeties of interest can be extracted by averaging a single replica, i.e.
by directly averaging the partition function. We therefore just study Z (and drop the overline henceforth). After disorder averaging we find Most importantly, the independence of U c ijkl on r leads to a sum over all pairs of lattice sites r, r in S int . If we had instead chosen U c to be independent random variables at different sites, S int would have only involved on-site interactions. Now define the function G(r , τ ; r, τ ) through Inserting the following identity into the path integral, we rewrite the delta function as The disordered averaged interaction can be expressed directly in terms of G as where θ is the angle between q and k = k F , and we therefore get = Zν 0 1 − |Ω| whereṽ F is the renormalized Fermi-velocity and ν 0 is the density of states at the Fermi-energy.
The contribution to the polarization function from high-energy electron-hole excitations has a completely local q−independent form and is given by

Electronic self-energy
Having calculated the polarization function, it is straightforward to calculate the electron selfenergy Σ c (k, iω) in the Fermi-liquid regime. Once again, we separate the self-energy into two components, arising from Π 1 c (q, iΩ) and Π 2 c (q, iΩ): Σ c (k, iω) = Σ 1 c (k, iω) + Σ 2 c (k, iω). (D8) Let us first evaluate the self-energy at k = k F and finite ω. The important contribution comes from small momentum scattering, |q| k F ; for such wavevectors, we can approximateε k F +q ≈ v F q cos(θ), with θ being the angle between q and k F . We thus get sign(ω + Ω) (ω + Ω) 2 + (Zṽ F q) 2 1 − |Ω| The integrand vanishes for Ω Zṽ F q; we thus consider only the limit where Ω, ω Zṽ F q: The ω 2 term displays the well-known logarithmic divergence in the imaginary part which occurs in the calculation of the self-energy of two-dimensional Fermi liquids; this can be fixed by recalling that in the q integral, the expression is valid only for q much larger than ω/Zṽ F . Introducing the appropriate IR cutoff leads to, with α is a number of order unity.
We now focus on the contribution from the high-energy excitations that arise from the component Π 2 c : which at strong coupling, where Z ∼ 1/ν 0 U c , is comparable to Σ 1 c (k F , ω), up to logarithmic factors. Let us now evaluate the self-energy at ω = 0 and finite (but small) k = |k−k F |. The contribution from the low lying excitations is, We are interested in the contribution from the second term, which gives the leading k dependence.
In particular, the real part is given by, where in the Ω integral, we have assumed that Ω < Zṽ F |q|. Consider now the external momentum k = (k x , 0). The typical internal momenta q x q 2 y /k F with q y < k F . Restricting ourselves to momenta close to the Fermi-surface, The term linear in k x has a contribution from two terms above, = − Z 2 U 2 c ν 0 k Fˆk F 0 dq y k x (q y − q y log q y − q y ) ≈ −ζZ 2 U 2 where ζ is an O(1) number that depends on the fermi-surface geometry. for the MFL and NFL models considered above in appendix G, and find results that are consistent with ones obtained below.

Self energy of the c−electrons
In the regime where the f −electrons form a LICM (i.e. T > Ω * f ), we use the following momentum independent form of the polarization function, Using this to calculate the self-energy of the c-electrons leads to, We again approximate ε k F +q ≈ v F q cos(θ), with θ the angle between q and k, and perform the integral over θ to obtain The most singular contribution comes from the region where Ω Zv F q, and thus for small ω < U f , this becomes the marginal Fermi liquid form, The self-energy of interest is given by, It is then straightforward to see that taking H(x) to be a constant and integrating over ε leads to a self-consistent solution for Σ cf . The exponent, φ = 4∆(q), is then fixed by the Ω dependence of Π f (Ω) (determined by ∆(q) = 1/q). The key difference from the quantum-critical metals is that this singular frequency dependence persists everywhere in momentum-space, and arises from the locally critical 'bath' that is coupled to the c fermions.
The above set of equations can be solved self-consistently by a completely local form of the self-energy (and Green's function) for the f fermions with Σ f (r, τ ; r , τ ) = Σ f (τ − τ )δ rr and G f (r, τ ; r , τ ) = G f (τ − τ )δ rr ; as discussed before Σ f (τ − τ ) then has the usual SYK-like form.
The self-energy for the c fermions then also has a local character As a result, in the presence of a magnetic field, when we express the equations in the LL basis, the self-energy for the c fermions has a B−independent piece (i.e. which does not depend explicitly on the LL index) and has the usual marginal Fermi-liquid character (for q = 4) and non-Fermi liquid character (for q > 4) as discussed earlier. The oscillations therefore arise from the effect of the magnetic field on the kinetic energy of the c fermions (through the formation of Landau bands).
The above simplification arises from the absence of a kinetic energy term for the f fermions.
Even as in the presence of coupling to the c fermions, the local structure of the f fermion Green's function survives.

Density of states oscillations
We are only interested in the oscillatory component of the spectral density of states and so let us begin by considering, where the Green's function is of the form shown in Eq. (65a). Upon using the Poisson summation formula this yields, A osc (iω m ) = B 2πˆ∞ −∞ dp z 2π .
We can extract the universal scaling structure of the above integral from the following simple arguments. For the NFL model, the spectral function where S(...) andS(...) are universal scaling functions. The amplitude then has a scaling form, as in Eq. (74).
Appendix J: Many-body quantum chaos This appendix serves as a self-contained resource for some of the key aspects of many-body quantum chaos. We use these ideas to formulate our conjectures for a universal description of non-Fermi liquid metals in Section VII C 2 above. It has been proposed in recent years that the spread of information (or information scrambling [102]) can be diagnosed by studying special correlation functions which involve squared (anti-)commutators of local operators [42,98,103].
Such correlators were considered decades ago in a different context [104] and have been employed more recently in a variety of different settings. They have been shown to diagnose quantum chaos in black hole physics [42,48,103,105], which are supposed to be the fastest scramblers in nature [106].
The squared anti-commutators for local Fermionic operators can be defined as, where ρ = e −βH is the density matrix at a temperature T = β −1 and c x,i (t) = e iHt c x,i e −iHt ; r = x − x . For spatially well separated operators, these (anti-)commutators start out small and then grow at late times. For generic non-integrable systems and in systems with a large number of local degrees of freedom, the growth is expected to be of the form, C(t, r) ∼ e λ L t , where in general depends on t, r and on the number of degrees of freedom in the system and the growth rate is denoted the 'Lyapunov exponent' (λ L ). There is a fundamental limit on how large λ L (≤ 2πk B T / ) can be and black-holes are known to saturate the bound. An interesting feature of the (0 + 1)−dimensional SYK model in the large N limit [42,47,48,107], as well as its higherdimensional generalizations that preserve the SYK form of the interactions [51,52], is that they are also maximally chaotic with λ L = 2πk B T / . Such correlation functions have been computed