A Model for translationally invariant random matrix (SYK{}_{2}) with uniform hoppings

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

Abstract

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 . 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.

I Introduction

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 Hussey et al. (1998); Schneider et al. (2014); Klein et al. (1996); Allen et al. (1996), cobaltates Li et al. (2004); Wang et al. (2003), iron-based superconductors Shibauchi et al. (2014) and heavy-fermion materials Stewart (2001), 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 Takagi et al. (1992); Keimer et al. (2015) of the cuprate superconductors and some quantum critical heavy-Fermion systems Löhneysen et al. (2007); Stewart (2001).

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 Varma et al. (1989), 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 Löhneysen et al. (2007). Much progress has been made in understanding the properties of this class of metallic quantum critical points in recent years Lee (2017). (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 transition1 in a heavy Fermi liquid Schröder et al. (2000); Coleman et al. (2001); Si et al. (2001, 2003); Senthil et al. (2004) and a Mott transition between a metal and a quantum disordered insulator Senthil (2008); Senthil et al. (2004). Such non-Fermi liquid quantum critical points have been argued Senthil (2008, 2004) 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 Senthil et al. (2004); Senthil (2008, 2004); Barkeshli and McGreevy (2012); Nandkishore et al. (2012); Chowdhury and Sachdev (2015). 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 Jiang et al. (2012). 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 Halperin et al. (1993). Indication of such non-Fermi liquid quantum phases have also been reported in correlated mixed-valence materials Nakatsuji et al. (2008); Matsumoto et al. (2011). (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 resistivity over a broad range of temperature in quantum critical or other non-Fermi liquids. The phenomenological “marginal Fermi liquid” theory assumes coupling to a bosonic fluctuating mode that gives linear resistivity Varma et al. (1989); 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 Lederer et al. (2017) 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 Li et al. (2004); Wang et al. (2003) and ruthenates Tyler et al. (1998); Bruin et al. (2013)) it is indeed seen that there is a wide intermediate temperature 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’ there is a crossover to more conventional behavior. Importantly, it does not appear that 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- NFL physics of these systems may not in principle be controlled by Infra-Red (IR) fixed points with a finite number of relevant perturbations. We call such intermediate- non-Fermi liquid states as examples of “IR-incomplete” states of matter (see Ref. Senthil (2011) for a possibly useful exposition). By themselves, they cannot be the deep IR theory of any state of matter and hence require IR-completion.

Examples include electron-phonon systems above their Debye temperature Ziman (1960), lattice models with bounded kinetic energy at high  Lindner and Auerbach (2010); Mukerjee et al. (2006), spin-incoherent Luttinger liquids Fiete (2007), holographic non-Fermi liquids Liu et al. (2011); Faulkner et al. (2011), and some states found in DMFT calculations at finite temperature Georges et al. (1996); Deng et al. (2016). Common to many of these examples of IR-incomplete theories is that they have extensive residual low- entropy (i.e. the entropy extrapolated to from the regime in which the theory applies is non-zero) which is then relieved below 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 Sachdev and Ye (1993); Kitaev (); Parcollet and Georges (1999); Georges et al. (2001); Fu and Sachdev (2016); Sachdev (2015); Maldacena and Stanford (2016); Kitaev and Suh (2017); Banerjee and Altman (2017), 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 dimensional, and thus it does not contain information about transport. Higher dimensional generalizations of the model, in which an SYK “quantum dot” is placed on every lattice site and the different sites interact via a random coupling 6, have been considered Gu et al. (2017a); Davison et al. (2017); Song et al. (2017). In particular, in Ref. Song et al. (2017) the charge and thermal transport properties of such a model have been computed. The solution of the model has 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. Sachdev (), 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 limit, where is the number of fermion flavors (or “orbitals”) per site, coupled by a frustrated on-site interaction. If there is a single band of bandwidth , and the typical interaction strength is , we find that the system crosses over at a temperature from a low-temperature 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 at to at ; the value of the resistivity at the crossover scale is . If there are multiple bands with parametrically different bandwidths (or an itinerant band coupled to localized electrons, as in a Kondo lattice), a richer behavior is observed. In addition to the low temperature Fermi liquid and the high temperature incoherent regime, we find an intermediate range of temperatures where the correlations in the narrow band are locally quantum critical, while the band with the larger bandwidth forms a marginal Fermi liquid, with a single particle inverse lifetime proportional to , where is the energy. This region realizes the marginal Fermi liquid phenomenological model proposed in Ref. Varma et al. (1989), with the density (or flavor) fluctuations of the narrow, incoherent band playing the role of the critical bosonic degree of freedom. Within the multi-band setup, we also consider models where the on-site interactions for one of the bands involves -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 Bruin et al. (2013) can be understood in terms of the conjectured “Planckian” bound on relaxation rates,  Sachdev (2011); Zaanen (2004). 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 , where is the effective mass of the low-temperature Fermi liquid state, and expect a bound on .7 In the two-band non-Fermi liquid state described in Sec. V, we find that has a non-Planckian form: with . Alternatively, a natural way of defining the transport scattering rate is to use the temperature dependent crossover frequency scale, , across which the optical conductivity crosses over from its high frequency regime to the dc limit. For the models considered below, we find that satisfies a Planckian-type bound with , where is an number.8 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- NFL physics, there is an emergent large length scale (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. Maldacena et al. (2016a); 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 -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.1, II.2 and II.3 respectively. In section III we provide a very simple qualitative understanding of these one-band models which demystifies their properties. 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.1 and find a regime with a marginal Fermi liquid behavior. We explore the thermodynamic and transport properties associated with the MFL in sections IV.2 and IV.3 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.2 and V.3. For the generalized model, we explore the “” 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.4 and V.5, 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.3.2 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 (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 dimensions on a hypercubic lattice ( will be of primary interest) with orbitals per site and fermionic operators defined by, , , (). The fermions satisfy usual anti-commutation algebra . We assume that there is a global symmetry corresponding to a single conserved density (volume), . The value of can be tuned by a chemical potential . The Hamiltonian is given by

(1)

where the hopping terms between sites and , , are diagonal in the orbital subspace and depend only on (assumed to be identical for all orbitals). The interaction term, , is purely on-site and is properly antisymmetrized with and . The values of are assumed to be independent of the site-label, (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) Sachdev and Ye (1993); Kitaev (); Parcollet and Georges (1999); Georges et al. (2001); Fu and Sachdev (2016); Sachdev (2015) quantum dots with identical on-site interactions, connected by orbital-diagonal, translationally invariant hopping matrix elements.9

The model (1) is difficult to solve. However, just as in the SYK model, if we consider the interaction terms to be random, independent variables with a zero mean, and take the limit , then it is possible to compute properties of the model averaged over realizations of . 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 and , where characterizes the strength of the interactions. The other energy scale in our problem is the free electrons’ bandwidth, which we denote by .

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 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 . We therefore expect that the correlation functions in our model are self-averaging in the large limit, as in the single-site SYK model.

Figure 1: (a) A two-dimensional lattice where each site contains orbitals (represented by different colors). The hoppings, , between any neighboring sites (colored arrows) are diagonal in orbital-index. Each site is identical and the system is translationally invariant. (b) The internal structure of a single site with orbitals. The on-site interactions, , are quartic in the fermion operators, with all orbital indices unequal.

ii.1 Fermion Green’s Function

The fermion Green’s function can be analyzed diagrammatically, such that the large- saddle-point solution reduces to studying the following set of equations self-consistently,

(2a)
(2b)
(2c)

where and is the dispersion for the 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.

Figure 2: The self-energy diagram, , for fermions with orbital index in the single-band model due to . The solid black lines represent fully dressed Green’s functions, ; see Eq. (2a). The dashed line corresponds to contraction and carries no frequency/momentum.

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 , the spectral function has a Fermi-liquid like form. In the interesting case , there is a second regime defined by , 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 keeping finite (such that collapses to zero), and then take the limit of , thus obtaining a compressible phase of electronic matter without quasiparticle-excitations 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 (or ) that are either much larger or much smaller than . In the limit , 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 has the following form near the Fermi surface:

(3)

where is the quasiparticle residue, to be determined self-consistently, ( is the Fermi momentum), are the renormalized (bare) Fermi-velocities with the renormalization to be determined self-consistently, and the denote higher power terms in an expansion in . We stress that is different from the effective Fermi velocity , 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)

(4)

Here, is the bare density of states at the Fermi energy. (We set the units of length such that the lattice spacing .) In Eq. (4) we have taken into account the contribution of the quasi-particle poles of the Green’s functions at , 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

(5)

where are numerical factors of order unity that depends on the geometry of the Fermi surface (Appendix D). The factor of in (5) is special to ; it is absent in higher dimensions. Equating this to Eq. (3), we get that

(6)
(7)

In particular, in the weak coupling limit, , we get that . In the opposite limit, , we get to logarithmic accuracy that , and is . 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 is the bare mass while the momentum dependence of the self-energy is independent of . 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 Georges et al. (1996) of a heavily renormalized Fermi liquid. Note, however, that while DMFT is exact in the limit of infinite dimension, in our case is finite; instead, we have to take the large and strong coupling limits.

Next, we turn to the behavior of at high frequencies. We focus on the strong coupling limit, . In this regime, exceeds the Fermi energy for sufficiently large . Extrapolating from Eq. (3) with , we get that this occurs at frequencies larger then . Then, to zeroth order, we can neglect relative to in Eq. (2a). In this limit, the self-consistent equations (2a-2b) reduce to those of the single site SYK model Sachdev and Ye (1993); Kitaev (); Parcollet and Georges (1999); Georges et al. (2001); Fu and Sachdev (2016); Sachdev (2015); Maldacena and Stanford (2016). In particular, we get that at frequencies smaller than ,  Sachdev and Ye (1993); Parcollet and Georges (1999); Georges et al. (2001). Extrapolating from high to intermediate frequencies, we reproduce the result that for , 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 10. To leading order, we get that , where and 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 decays remains frequency-independent.

To summarize, we get that for strong coupling, has the following form in the two extreme frequency limits:

(8)

where , and is a number of order unity. is a constant independent of frequency for both and though its precise value is different for the two signs of . Indeed it is a direct descendant of the “spectral asymmetry” that characterizes the Green’s function of a single SYK island Sachdev and Ye (1993); Parcollet and Georges (1999).

At low frequencies, there is a Fermi surface with well-defined, albeit strongly renormalized quasiparticles. The renormalized bandwidth is . The term in the denominator of becomes the imaginary part of the self-energy after an analytic continuation to real frequency. It can be written in a revealing form: . At finite temperatures, the zero-frequency imaginary part is . Note that, upon extrapolating this form to the crossover scale, , 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 . This is the LICM regime.

ii.2 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, , the system is well described by Fermi-liquid theory. This implies, in particular, that the entropy per unit cell follows a linear temperature dependence, , where . At temperatures much higher than , we can calculate the thermodynamic properties perturbatively in the inter-site hopping11. Then, the entropy is given by that of a single SYK dot, up to a correction of the order of . The entropy takes the form , where and are known constants Parcollet and Georges (1999). At temperatures of the order of , we expect the entropy to interpolate between these two behaviors.

Next, we turn to discuss the compressibility, given by , where 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 Davison et al. (2017); Song et al. (2017). As a result of the finite hopping and bandwidth, there is a correction to this result and at strong coupling we obtain

(9)

where is a constant of order unity. As discussed earlier, in this regime the mass enhancement factor . This can be reconciled within the Fermi-liquid description of the state if one introduces a large dimensionless ‘Landau-parameter’, .

ii.3 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

(10)

where is the retarded current-current correlation function for the current in direction. The total current operator is given by

(11)

with denotes the current from orbital and . For the previously assumed identical dispersions for all the orbitals, the velocities are also the same. The leading diagrams which contribute to are shown in Fig. 3. In Fig. 3 (a), we show the leading graph without vertex corrections.

Figure 3: Current-current correlation function for evaluating the conductivity in the one-band model. Wiggly line denotes the insertion of the current operator. The solid lines represent the fully dressed propagators. (a) Feynman diagram without vertex corrections. (b) The lowest order vertex correction diagram, which is subleading in the high temperature (LICM) regime. The dashed line represents a contraction, as before.

In the high temperature () 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 . Then each of the loops over orbital and vanish individually. We therefore consider only the diagram in Fig. 3(a), and this results in

(12)

where is the electron spectral function and represents the Fermi-Dirac distribution function. In the high temperature SYK-like regime (), the optical conductivity clearly satisfies scaling. At frequencies much higher than the temperature (i.e. by arranging ), we find . Focusing on the dc conductivity in this regime, we find (in units of ):

(13)

where represents an average over the Fermi-surface. As a result of the scaling, the crossover scale from the high-frequency to the dc limit is of order .

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 begin by considering the limit where the hopping . When 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 . For small hopping , we can study the relevance/irrelevance of the hopping term in the decoupled SYK theories. In the action, the hopping term becomes

(14)

Clearly then under a scaling transformation , so that the hopping is relevant. To study the system at a non-zero temperature , we run the scaling until a scale . The effective renormalized hopping at this scale is then . With decreasing temperature we will stay in the regime of weak hopping until a temperature such that . This corresponds to a temperature scale which matches exactly with the coherence scale identified in section II.1.

For the physics will be that of weakly coupled SYK islands and we can calculate physical properties in perturbation theory in . For 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 -dimensional SYK model, the entropy is known Parcollet and Georges (1999); Maldacena and Stanford (2016) to obey

(15)

The ground state entropy is nonzero in the limit , and then . As argued in Sec. II.2, in the limit this is obviously the entropy per site of the lattice model. When and at sufficiently high temperature such that , both the entropy and the compressibility only get small corrections when we perturb in . For , however, the ground state entropy of the decoupled limit is relieved, and . ( is the total number of sites). In the low- Fermi liquid we expect . An estimate for can be obtained by matching this entropy extrapolated to with the residual entropy of the high temperature phase. This gives

(16)

In Fermi liquid theory the coefficient directly gives the quasiparticle effective mass . ( is the lattice spacing.) Note that the “bare” mass determined from the hopping Hamiltonian is . Therefore the mass enhancement in exact agreement with the solution of the self consistency equations in section II.1. The behavior of the compressibility in both the high- and low- limits have already been described in section II.2.

Let us now turn to transport. In the high- regime in perturbation theory in , the conductivity will be . In , is dimensionless in units of . We thus expect that for , where is the effective renormalized hopping at a temperature introduced above. We therefore get

(17)

This is again in exact agreement with the calculations in section II.3. For , if the Fermi surface is big enough to allow umklapp scattering of the low energy quasiparticles, we will get a resistivity . To estimate , we require that when extrapolated to this matches the extrapolation of the high result down to . This leads to . Note that in the low- Fermi liquid thereby obeying the Kadowaki-Woods relationship Kadowaki and Woods (1986).

The understanding above readily generalizes to the physics of coupled SYK models where the on-site interaction is composed of ( and even) fermion operators Gross and Rosenhaus (2017); 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 and the hopping to be translationally invariant and , and . We focus on the small regime. For general , the scaling dimension of the fermion is . It follows that a small is relevant at the decoupled fixed point and scales as

(19)

Following the discussion above, we determine that the physics will be that of weakly coupled islands until a coherence scale . In the high- regime, the entropy and compressibility have the same qualitative behavior as for . Importantly, there is a residual entropy (with a linear correction) and a finite non-zero compressibility. At we again expect a Fermi liquid. The residual entropy is relieved, and the low- heat capacity coefficient is . This can be converted into an estimate for the quasiparticle effective mass in the Fermi liquid.

The electrical resistivity in the high- regime, estimated as above, is of the form

(20)

Note that increases faster than linearly with , but slower than . Thus the high- linear resistivity is not a generic property of coupled SYK models and requires . As before (with umklapp scattering) the low- resistivity is with .

iii.1 Explicit transport calculation at high-temperature

It is instructive to explicitly calculate the conductivity in the high- regime in perturbation theory in the hopping, taking special care with issues regarding disorder averaging. As the leading temperature dependence is , 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. Zou and Senthil (2016)):

(21)

Note that we have not carried out the disorder averaging yet. is the frequency dependent fermion Green’s function within the SYK island at site and is the site neighboring to in the positive 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. Song et al., 2017), then obviously upon disorder averaging (indicated with an overline) the products that appear above can be replaced by for any . In our translation invariant models, the SYK interactions are the same at every site. Thus strictly speaking we must instead take . Fortunately (as shown in Appendix B) for in the large- limit12, the property

(22)

holds and we can continue to make this replacement in the products entering the correlation function. Further we know that when , only is ) and for is suppressed. 13 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

(23)

Here is the spectral function for the Green’s function within a single SYK island. For SYK (with ) this satisfies scaling,

(24)

with a known universal function. It follows from Eq. (23) that the conductivity itself satisfies scaling. We get

(25)

with a universal function determined in terms of by Eq. (23). In particular in the dc limit we reproduce the temperature dependence previously obtained for general . As a result of the scaling, it is easy to see that the frequency scale over which reaches its dc value is (in units of ), with an number. Moreover, the scaling function at large . Therefore, at frequencies much larger than the temperature, the conductivity has the form .

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 Senthil (2008) 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 defined14. However there are no Landau quasiparticles across the critical Fermi surface and the quasiparticle residue 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 fermions with operators , () and an associated conserved charge density, that may be tuned by a chemical potential , which we set to zero. The modified Hamiltonian (with a symmetry) is

(26)

where is as described in Eq. (LABEL:hc), and is defined in an identical fashion with translationally invariant hoppings and on-site interactions . The form of the inter-band interaction is chosen to be

(27)

where the coefficients, , are chosen to be identical at every site with , and where the distribution of the couplings satisfy . We now assume that , i.e. the bandwidth for the fermions is much smaller than the bandwidth for the fermions (). 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, has been set to zero.

To leading order in , the saddle point equations for the Hamiltonian defined in Eq. (26) are given by,

(28a)
(28b)
(28c)
(28d)
(28e)
(28f)

We have introduced as the dispersion for the fermions and , are as defined earlier in Eqs. (2a-2c). The watermelon-diagrams for the self-energies are shown in Fig.4.

Figure 4: The self-energy diagrams in the two-band model due to inter-band scattering for (a) fermions, , and, (b) fermions, , with orbital index . The solid black (red) lines represent fully dressed Green’s functions, ; see Eq. (28a) and 28b. The dashed lines correspond to contractions respectively and carry no frequency/momentum.

Based on our analysis for the one-band model in section II, we see immediately that if , the two decoupled subsystems have a Fermi-liquid to LICM crossover at frequencies or temperatures of the order of and respectively (). In the high-temperature regime, , when both bands are in a LICM phase, adding does not alter any of the features qualitatively and the resulting state is thus still described by a LICM phase. Similarly, at low-temperatures, , when both species are in a Fermi-liquid phase, a finite does not modify the qualitative aspects. There are two Fermi-surfaces for the and fermions consistent with their individual Luttinger count; adding a finite hybridizes the two Fermi-surfaces, breaking the two independent symmetries down to a single conserved charge corresponding to the total fermion density. The low-energy description of the Fermi-liquid phase is similar to our considerations from the previous section. The key question that remains is what is the fate of the system in the intermediate regime .

For the purpose of our subsequent discussion, we can set , such that the band is uncorrelated and the scale is pushed out to infinity (the bandwidth, , is the physical UV scale).15 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 to zero. This is conveniently done by setting while keeping finite. In this limit, we can pose sharp questions about the presence or absence of quasiparticles and Fermi surfaces in the limit.

iv.1 Fermion Green’s Function

In the window of intermediate energies, where the fermions enter the LICM regime, while the 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 fermions. In the next section, we will generalize the model to obtain a critical Fermi surface of the 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, , 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 , but we set for simplicity. We assume that the fermions are in the LICM regime, such that for imaginary times , which is the familiar form in the SYK model Sachdev and Ye (1993); Parcollet and Georges (1999); Georges et al. (2001). We ignore here the weak-momentum dependent correction to the Green’s function in the LICM phase (as discussed in section II.1) by considering the limit of , with fixed (the crossover scale also goes to zero in this limit). Then, has the momentum independent form16,

(29)

Inserting into Eq. (28d) we get for the self-energy in Fig.4(a) (see Appendix F)

(30)

The self-energy of the fermions then has a ‘marginal Fermi liquid’ (MFL) Varma et al. (1989) 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.