A ‘Pair-hopping’ terms and the marginal-Fermi liquid

Magnetotransport in a model of a disordered strange metal

Abstract

We engineer a microscopic model of two-dimensional conduction electrons locally and randomly scattering off impurity sites which are described by Sachdev-Ye-Kitaev (SYK) models. For a particular choice of the scattering interaction, this model realizes a controlled description of a diffusive marginal-Fermi liquid (MFL) without momentum conservation, which has a linear-in- resistivity and a specific heat as . By tuning the strength of the scattering interaction relative to the bandwidth of the conduction electrons, we can additionally obtain a finite- crossover to a fully incoherent regime that also has a linear-in- resistivity. We describe the magnetotransport properties of this model. We then consider a macroscopically disordered sample with domains of such MFLs with varying electron and impurity densities. Using an effective-medium approximation, we obtain a macroscopic electrical resistance that scales linearly in the magnetic field applied perpendicular to the plane of the sample, at large . The resistance also scales linearly in at small , and as at intermediate . We consider implications for recent experiments reporting linear transverse magnetoresistance in the strange metal phases of the pnictides and cuprates.

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I Introduction

Essentially all correlated electron high temperature superconductors display an anomalous metallic state at temperatures above the superconducting critical temperature at optimal doping Keimer et al. (2015); Kasahara et al. (2010); Sachdev and Keimer (2011). This metallic state has a ‘strange’ linearly-increasing dependence of the resistivity, , on temperature, ; it can also exhibit bad metal behavior with a resistivity much larger than the quantum unit (in two spatial dimensions) Emery and Kivelson (1995). More recently, strange metals have also been demonstrated to have a remarkable linear-in- magnetoresistance, with the crossover between the linear-in- and linear-in- behavior occurring at Hayes et al. (2016); Giraldo-Gallo et al. (2017).

This paper will present a model of a strange metal which exhibits the above linear-in- and linear-in- behavior. The model builds on a lattice array of ‘quantum dots’, each of which is described by a Sachdev-Ye-Kitaev (SYK) model of fermions with random all-to-all interactions Sachdev and Ye (1993); Kitaev (2015). A single SYK site is a 0+1 dimensional non-Fermi liquid in which the imaginary-time () fermion Green’s function has the low ‘conformal’ form Sachdev and Ye (1993); Parcollet and Georges (1999); Faulkner et al. (2011); Sachdev (2015)

(1)

where is a parameter controlling the particle-hole asymmetry. As was recognized early on Sachdev and Ye (1993), such a Green’s function implies a ‘marginal’ Varma et al. (1989) susceptibility, , with a real part which diverges logarithmically with vanishing frequency () or . Specifically, in the all-to-all limit of the SYK model, vertex corrections are sub-dominant, and leads to the spectral density

(2)

whose Hilbert transform leads to the noted logarithmic divergence. The form (2) is consistent with recent electron scattering observations Mitrano et al. (2017). A linear-in- resistivity now follows upon considering itinerant fermions scattering off such a local susceptibility, and the itinerant fermions realize a marginal Fermi liquid (MFL) with a self energy Varma et al. (1989); Sachdev and Ye (1993); Sachdev (2010); Faulkner et al. (2013).

A specific model for a bulk strange metal was provided by Parcollet and Georges Parcollet and Georges (1999). They considered a doped Mott insulator described by a random - model at hole density , where is the root-mean-square (r.m.s.) electron hopping, and is the r.m.s. exchange interaction. At low doping with , they found strange metal behavior in the intermediate regime , where the coherence energy . This strange metal is more properly identified as an ‘incoherent metal’ (IM) (rather than a MFL), because the electron Green’s function has the local form in Eq. (1). Bad metal behavior was found with a resistivity .

Another model of an IM appeared in the recent work of Song et al. Song et al. (2017). They considered a lattice of SYK sites, with r.m.s. on-site interaction , and r.m.s. inter-site hopping . As in Ref. Parcollet and Georges, 1999, they found an IM in the intermediate regime , with a local electron Green’s function as in Eq. (1), and a bad metal resistivity . Their coherence scale was . (This lattice SYK model should be contrasted from earlier studies Gu et al. (2017); Davison et al. (2017), which only had fermion interaction terms between neighboring SYK sites: the latter models realize disordered metallic states without quasiparticle excitations as , but have a -independent resistivity.)

Figure 1: (a) A cartoon of our microscopic model. Conduction electrons (green) hop around on a lattice (black). At each lattice site, they interact locally and randomly with SYK impurities (blue) through an interaction (orange) that independently conserves the numbers of conduction and impurity electrons. (b) Finite-temperature regimes of the model. When the conduction electron bandwidth is large enough, it realizes a disordered marginal-Fermi liquid (MFL) for the conduction electrons for all temperatures (Sec. III.1). For a finite bandwidth, there can be a finite-temperature crossover to an ‘incoherent metal’ (IM), in which all notion of electron momentum is lost, if the coupling is large enough (Sec. III.2). Note that we always have and .

In this paper, we consider a lattice of ‘impurity’ SYK sites coupled to a separate band of itinerant electrons. Our model is in the spirit of effective Kondo lattice models which have been proposed as models of the physics of the disordered, single-band Hubbard model Milovanović et al. (1989); Bhatt and Fisher (1992); Potter et al. (2012). Other two band models of itinerant electrons coupled to SYK excitations have been considered in Refs. Burdin et al., 2002; Ben-Zion and McGreevy, 2017. Our model exhibits MFL behavior as , with a linear-in- resistivity, and a specific heat. For an appropriate range of parameters, there is a crossover at higher to an IM regime, also with a linear-in- resistivity. The itinerant electrons have a non-random hopping , the SYK sites have a random interaction with r.m.s. strength , and these two sub-systems interact with a random Kondo-like exchange of r.m.s. strength : see Fig. 1a for a schematic illustration. Fig. 1b illustrates the regimes of MFL and IM behavior in our model.

The magnetotransport properties of this model will be a significant focus of our analysis. In the MFL regime, we find that the longitudinal and Hall conductivities can be written as scaling functions of , as shown in Eq. (36); in contrast, the dependence is much less singular in the IM regime. We then consider a macroscopically disordered sample with domains of MFLs with varying electron and impurity densities; employing earlier work on classical electrical transport in inhomogeneous ohmic conductors Stroud (1975); Guttal and Stroud (2005); Parish and Littlewood (2003, 2005); Ramakrishnan et al. (2017); Dykhne (1971); Song et al. (2015), we obtain the observed linear-in- magnetoresistance with a crossover scale at .

This paper is organized as follows: In Sec. II, we introduce our basic microscopic model of a disordered MFL, and determine its single-electron properties and finite-temperature crossovers in Sec. III. In Sec. IV, we solve for transport and magnetotransport properties of this basic model exactly in various analytically-tractable regimes. In Sec. V, we introduce the effective-medium approximation and apply it to a macroscopically disordered sample containing domains of the basic model, obtaining analytical results for the global magnetotransport properties for certain simplified considerations of macroscopic disorder. We summarize our results and place them in the context of recent experiments in Sec. VI.

Ii Microscopic model

We consider flavors of conduction electrons, , hopping on a lattice that are coupled locally and randomly to impurities on each lattice site (Fig. 1a). The impurities contain flavors of valence electrons, , which interact among themselves in such a way that they realize SYK models. The hamiltonian for our system is given by

(3)

We will take the limits of and , but we will be interested in values of that are at most . We choose and as independent complex Gaussian random variables, with and and all other ’s being zero, where denotes disorder-averaging. The disorder-averaged action then is

(4)

where we have followed the usual strategy for SYK models Sachdev (2015); Davison et al. (2017) and introduced the auxiliary fields corresponding to Green’s functions and self-energies of the and fermions respectively at each lattice site. In the limit, the integrals over the fields enforce the definitions of at each lattice site . The large , saddle-point equations are obtained by varying the action with respect to these and fields

(5)

We define chemical potentials such that half-filling occurs when . The impurities are not capable of exchanging electrons with the Fermi sea, so there is no reason a priori to have , or even for impurities at different sites to have the same . However, for convenience we will keep the of all the impurities the same. A real system operates at fixed densities, and and will appropriately renormalize as the mutual coupling is varied, in order to keep the densities of and individually fixed. However, as we shall find, the half-filled case always corresponds to regardless of . We will always have in this work, and also , so whenever an ultraviolet (UV) energy cutoff is required, we use . A sketch of the phases realized by our model as a function of temperature is shown in Fig. 1b.

Iii Fate of the conduction electrons

iii.1 The case of infinite bandwidth

We first consider the case of infinite bandwidth, or equivalently . It doesn’t matter then precisely where is as long as its magnitude is not infinite, as the conduction electrons float on an effectively infinitely deep Fermi sea. Then, we can use the standard trick for evaluating integrals about a Fermi surface, and we have

(6)

where is the density of states at the Fermi energy. Within this approximation, we will also have .

We take the lattice constant to be . This makes dimensionless by redefining to be . The energy dimension of then comes from the inverse band mass. The density of states then has the dimension of 1/(energy) (on a lattice , where is the bandwidth).

Within this approximation, we will also have , so

(7)

with other intervals obtained by applying the Kubo-Martin-Schwinger (KMS) condition . At , we have

(8)

We consider to begin with. Then, the electrons are not affected by the electrons, and their Green’s functions are exactly those of the SYK model, which, in the low-energy limit, are given by Sachdev (2015)

(9)

where is a function of with for small . Other intervals are again obtained by the KMS condition . The zero-temperature limit of this, and similar expressions appearing later, can be straightforwardly taken Sachdev (2015)

(10)

We have

(11)

Fourier transforming with a cutoff of at and gives

(12)

where is the digamma function and is the Euler-Mascheroni constant. This choice of cutoff is justified by the fact that the short-time divergences are generated by the singularities in the conformal form of the SYK Green’s functions. As foreseen, this satisfies on the fermionic Matsubara frequencies. For

(13)

Hence we have manufactured a non-translationally-invariant MFL out of the conduction electrons. Since the large and limits are taken at the outset, this is stable even as . For finite and , the coupling is irrelevant in the infrared (IR) Ben-Zion and McGreevy (2017), and the model reduces to a theory of non-interacting electrons as , with the MFL existing only above a temperature scale whose magnitude is exponentially suppressed in .

Upon analytically continuing , we get the inverse lifetime for the conduction electrons defined by

(14)

Since the coupling of the conduction electrons to the SYK impurities is spatially disordered, this rate also represents the transport scattering rate up to a constant numerical factor. The scattering of electrons off the impurities requires the electrons inside the impurities to move between orbitals. Hence vanishes when the impurities are flooded or drained by sending respectively, say, by doping them.

If we do not have , the SYK Green’s function will be affected as there is a back-reaction self-energy to the SYK impurities. To see what this does when we perturbatively turn on , we compute it with the Green’s functions with a cutoff of at and

(15)

If , then as , which is sub-leading to , so the SYK character of the impurities survives in the IR.

If but is small, then for , . In contrast . Therefore the frequency-dependent part of is still subleading. Hence, in the IR we may still assume that all that happens to the SYK impurities is that their chemical potential gets renormalized. By solving , we obtain the corrected relation. At small , this is

(16)

The total particle number on each impurity, , commutes with . Since the SYK particle density is a universal function of , independent of and , (16) just implies a renormalization of the nonuniversal UV parts of the SYK Green’s function and the impurity chemical potential, while the particle density remains fixed. Similarly, the vanishing of the zero-frequency real part of (12) regardless of implies that there is no renormalization of either the density or chemical potential of the conduction electrons in this infinite-bandwidth limit, since their number is independently conserved as well. For a finite bandwidth, the chemical potential of the conduction electrons renormalizes in such a way that their density remains fixed.

In Appendix A, we consider the effects of adding a ‘pair-hopping’ term to (3),

(17)

with , and . This term has identical power-counting to the term, but can trade electrons for electrons and vice-versa. Since the numbers of and electrons are no longer independently conserved in this case, there is only one chemical potential, and . We find that this term also generates an MFL as long as the bandwidth of the electrons is large.

As is well known, the marginal-Fermi liquid self-energy we obtained (1213) also leads to the leading low-temperature contribution to the specific heat scaling as  Crisan and Moca (1996). Note that the entropy has a non-vanishing limit from the contribution of the SYK impurities Georges et al. (2001), but this does not contribute to the specific heat.

iii.2 The case of a finite bandwidth

If the bandwidth (and hence Fermi energy) of the conduction electrons is sizeable compared to the couplings, then the local Green’s function is no longer independent of the details of the self energy . We consider two spatial dimensions, with the isotropic dispersion , and a bandwidth . Since is dimensionless, the band mass has dimensions of . The density of states is then just , at all energies , and we implicitly make use of this fact while simplifying and rewriting certain expressions. On a lattice, .

The momentum-integrated conduction electron Green’s function is

(18)

We still expect . The chemical potential must now take an appropriate value to reproduce the correct density of conduction electrons. The conduction band filling is given by

(19)

for the exact solution to , which can be found by the MATLAB code ggc.m Patel (2017) (The low-energy ‘conformal-limit’ solutions described below are not valid at the short times , and do not display this property).

In general, the Dyson equations can now only be solved numerically, which the MATLAB code ggc.m Patel (2017) does, albeit by holding the chemical potentials and , rather than densities, fixed. In an extreme limit where far exceeds the bandwidth for all , which can happen only at a finite temperature, we have a simplification

(20)

This then leads to an SYK solution in the low-energy conformal limit for both and , realizing a fully incoherent metal. We use the trial solutions

(21)

is universally related to the conduction band filling, with at half filling, and when the band is full or empty respectively. When , there is no back-reaction to the impurities, and is given by (9). We use the conditions and to determine , and also in terms of the fixed . Cutting off integrals in the Fourier transforms at a distance from singularities, we have

(22)

with no feedback on the SYK impurities. For (20) to derive from (18), this requires or

(23)

Furthermore, for (9) and (21) to hold, we also need and , implying . For , we go back to the MFL.

Turning on a small but finite , we have to additionally use the conditions and simultaneously to determine a renormalized and renormalized , while keeping fixed as before. We again cut off integrals in the Fourier transforms at a distance from singularities. This gives

(24)

and we do not show the nonuniversal relations because they are rather uninsightful and the physics is better described in terms of which universally represent the conserved densities.

When , we enter non-universal regimes at finite temperature regardless of the bandwidth, where the impurity Green’s functions are not given by simple conformally-invariant solutions. However, deep enough in the IR, we always recover the MFL, due to the back-reaction self energy being irrelevant, and the conduction electron self energy also vanishing at the lowest energies.

Since on a lattice, fine-tuning makes the scattering rate (14) ‘Planckian’, i.e. an number times , since it is given by ratios of large quantities. The MFL doesn’t break down if we do this; In (18), , so the infinite-bandwidth result (14) is still applicable. The crossover to the IM doesn’t occur either, since , and finally, the part of the back-reaction self-energy to the SYK impurities that does not renormalize their chemical potentials is which is , i.e. the part of the internal self-energy of the SYK impurities that doesn’t renormalize chemical potential, as long as is not , so the SYK character of the impurities also survives.

In the IM regime, since both the conduction and impurity electrons have local SYK Green’s functions, the specific heat scales as , with no logarithmic corrections Davison et al. (2017).

Iv Transport in a single domain

In this section we again consider two spatial dimensions, again with the isotropic dispersion . In our double large and limit, if , the only vertex corrections to the uniform conductivities that aren’t trivially killed by this limit are the ones that involve uncrossed vertical ladders of propagators in the current-current correlator bubbles (First diagram of Fig. 2b). However, since the propagators are purely local and independent of momentum, these diagrams vanish due to averaging of the vector velocity in the current vertices over the closed fixed-energy contours in momentum space, as the scattering of the conduction electrons is isotropic, just like in the textbook problem of the non-interacting disordered metal Ėfros and Pollak (1985). Unlike the non-interacting disordered metal, there is no localization in two dimensions as the crossed-ladder ‘Cooperon’ diagrams are suppressed by the large limit. Hence, the relaxation-time-like approximation of keeping only self-energy corrections is valid.

If is nonzero but or smaller, then certain 3-loop and higher order ladder insertions (Such as Fig. 2c) also contribute extensively in to the current-current correlation. However, these diagrams again vanish due to the averaging of the vector velocity mentioned above. All this happens regardless of the values of , and for both energy and electrical currents.

Figure 2: (a) The uniform current-current correlation bubble used to compute conductivities. The current vertices are black squares and the black lines are conduction electron () propagators. (b) and (c) Additional diagrams forming ladder series, with ladder units of up to loops, that contribute to the conductivities and are not immediately suppressed by the large and limits. The red lines are impurity fermion () propagators that do not carry momentum. The dashed blue lines carry momentum and come from disorder averaging of the non-translationally invariant coupling . These diagrams however vanish upon momentum integration in the loops containing the current vertices, for reasons mentioned in the main text.

iv.1 Marginal-Fermi liquid

We first discuss the MFL regime. For simplicity, we consider infinite bandwidth and an infinitely deep Fermi sea. The uniform current-current correlation bubble (Fig. 2a) is given by, for an isotropic Fermi surface,

(25)

where is the Fermi velocity (on a lattice , since the lattice constant is set to ). Using the spectral representation, this can be converted to give the DC conductivity

(26)

Inserting the self energy, we can scale out and numerically evaluate the integral, giving

(27)

If we want , we must have , implying a crossover into the IM regime. Thus the MFL is never a true bad metal, but its resistivity can still numerically exceed the quantum unit , depending on parameters.

The ‘open-circuit’ thermal conductivity , which is defined under conditions where no electrical current flows, is given by

(28)

where is the ‘closed-circuit’ thermal conductivity in the presence of electrical current, and is the thermoelectric conductivity. The thermoelectric conductivity vanishes when the temperature is much smaller than the bandwidth and Fermi energy, due to effective particle-hole symmetry about the Fermi surface, so . The Lorenz ratio is then given by

(29)

which smaller than for a Fermi liquid.

In the presence of a uniform transverse magnetic field, we can use the following improved relaxation-time linearized Boltzmann equation (which incorporates an off-shell distribution function) for a temporally slowly-varying and spatially uniform applied electric field Kamenev (2011); Nave and Lee (2007), since there are no Cooperons in the large- limit, and hence none of the typical localization-related corrections Altshuler et al. (1980) to the conductivity tensor. The Boltzmann equation reads (here, is time, not the hopping amplitude, and is a dimensionless version of the magnetic field which shall be explained below)

(30)

where is the Fermi distribution, is the change in the distribution due to the applied electric field, the conduction electrons are negatively charged, and the magnetic field points out of the plane of the system. This equation is derived in Appendix B from the Dyson equation on the Keldysh contour, and can be solved by the ansatz .

In the DC limit, the effective mass enhancement does not matter Nave and Lee (2007) (the effective mass enhancement is important for AC magnetotransport and affects the frequency at which the cyclotron resonance occurs; it shifts the cyclotron resonance from the cyclotron frequency defined by the bare mass to the one defined by the effective mass. The enhanced effective mass also appears in the specific heat Crisan and Moca (1996) and Lifshitz-Kosevich formula Pelzer (1991) of MFLs). We then have

(31)

We note that in (31), is dimensionless in our choice of units. Since the quantities we set to were the magnitude of the electron charge , the lattice constant , and and , we have

(32)

i.e. the flux per unit cell in units of .

Substituting into (31), we obtain

(33)

Using the current density

(34)

we get the longitudinal and Hall conductivities

(35)

Note that these can be immediately written as

(36)

The asymptotic forms of the functions and are

(37)

This scaling between magnetic field and temperature in the orbital magnetotransport of the MFL at a microscopic level translates into a scaling between magnetic field and temperature in the global magnetoresistance of a sample with additional macroscopic disorder as discussed in Sec. V.

In (35), for the ‘Planckian’ choice of parameters described at the end of Sec. III.2, becomes ‘large’ (i.e., the cyclotron term in the denominators overwhelms for , causing to start decreasing with increasing ), when . Using reasonable values of the lattice constant and the hopping , the above inequality can also roughly be written as , where is the Bohr magneton, since for these parameters.

In the analysis of the IM regime to follow, there is no such notion of ‘large’ magnetic fields; regardless of the value of , the field-dependent corrections to the conductivity tensor remain much smaller than its zero-field value.

iv.2 Incoherent metal

In the IM regime we have

(38)

The spectral function is independent of in the IM, and we decoupled the momentum integral implicit in the above equation, generating a prefactor of . For simplicity we consider in this subsection. A small finite only rescales , as shown by (24, 21), and hence leads to no qualitative difference in any of the following results. We have