Metal-insulator transition from combined disorder and interaction effects in Hubbard-like electronic lattice models with random hopping

# Metal-insulator transition from combined disorder and interaction effects in Hubbard-like electronic lattice models with random hopping

Matthew S. Foster Department of Physics, Columbia University, New York, NY 10027    and Andreas W. W. Ludwig Department of Physics, University of California, Santa Barbara, CA 93106
August 28, 2019
###### Abstract

We uncover a disorder-driven instability in the diffusive Fermi liquid phase of a class of many-fermion systems, indicative of a metal-insulator transition of first order type, which arises solely from the competition between quenched disorder and interparticle interactions. Our result is expected to be relevant for sufficiently strong disorder in spatial dimensions. Specifically, we study a class of half-filled, Hubbard-like models for spinless fermions with (complex) random hopping and short-ranged interactions on bipartite lattices, in . In a given realization, the hopping disorder breaks time reversal invariance, but preserves the special “nesting” symmetry responsible for the charge density wave instability of the ballistic Fermi liquid. This disorder may arise, e.g., from the application of a random magnetic field to the otherwise clean model. We derive a low energy effective field theory description for this class of disordered, interacting fermion systems, which takes the form of a Finkel’stein non-linear sigma model (FNLM) [A. M. Finkel’stein, Zh. Eksp. Teor. Fiz. 84, 168 (1983), Sov. Phys. JETP 57, 97 (1983)]. We analyze the FNLM using a perturbative, one-loop renormalization group analysis controlled via an -expansion in dimensions. We find that, in dimensions, the interactions destabilize the conducting phase known to exist in the disordered, non-interacting system. The metal-insulator transition that we identify in dimensions () occurs for disorder strengths of order , and is therefore perturbatively accessible for . We emphasize that the disordered system has no localized phase in the absence of interactions, so that a localized phase, and the transition into it, can only appear due to the presence of the interactions.

###### pacs:
71.30.+h, 71.10.Fd, 72.15.Rn

## I Introduction

In many experimental situations, the effects of both static (‘quenched’) disorder and interparticle interactions may play comparatively important roles. These issues have, for example, come again to the forefront of debate in view of discussions centered around the fascinating, yet still controversial “metal-insulator transition” observed in semiconductor inversion layers.2DMIT Unfortunately, theoretical descriptions of quantum many-particle systems incorporating both disorder and interactions are typically quite challenging, and difficult to analyze reliably.

With this state of affairs in mind, we apply in this paper a powerful analytical technique, known as the Finkel’stein non-linear sigma model (FNLM) formalism,Finkelstein ; BK to a model of spinless lattice fermions subject simultaneously to both, static disorder and short-ranged interactions. The isolated effects of disorder or interactions upon the lattice model that we consider are already well-understood, yet still non-trivial. Our goal is to gain insight into the interplay possible between the Fermi liquid, Mott insulating, and what we term “Anderson-Mott” footnote-a insulating phases of interacting many-body Fermi systems in spatial dimensions . The FNLM formalism admits a renormalization group analysis controlled via an -expansion in dimensions, permitting us to address the following general questions within the context of our model: in the simultaneous presence of both, disorder and interactions, (i) does a conducting phase occur in , and (ii) is there a metal-insulator transition (MIT) in dimensions ? If yes, what is its nature?

The primary purpose of this paper is to present a detailed derivation and a thorough discussion of results previously announced briefly in Ref. AIIIshortpaper, . A short summary of these results and an outline of this work appears below in Sec. I.3.

### i.1 Interactions and sublattice symmetry

We study a class of “Hubbard-like” modelsAuerbach for spinless fermions at half filling on bipartite lattices, possessing short-ranged interparticle interactions and quenched disorder. We work in spatial dimensions throughout. For concreteness, we consider in this introduction the hypercubic lattice. By definition, any bipartite lattice may be subdivided into two interpenetrating sublattices, which we will distinguish with the labels ‘’ and ‘.’ The dimensional example of the square lattice is depicted in Fig. 1.

Our starting point is the clean (zero-disorder), generalized Hubbard-like Hamiltonian

 H0= −t∑⟨ij⟩c†AicBj+H.c.+V∑⟨ij⟩δ^nAiδ^nBj +U(∑⟨⟨ii′⟩⟩δ^nAiδ^nAi′+∑⟨⟨jj′⟩⟩δ^nBjδ^nBj′), (1)

where and are creation and annihilation operators for spinless fermions on the and sublattices of the bipartite lattice, respectively. Here, and respectively index the and sublattice sites, and the sums on run over all nearest neighbor - lattice bonds, while the sums on and run over all next-nearest neighbor (same sublattice) pairs of sites. The homogeneous hopping amplitude is taken to be real. The operators denote deviations of the local sublattice ( or ) fermion densities from their value at half filling. Finally, the interaction strengths and appearing in Eq. (I.1) couple to nearest neighbor and next-nearest neighbor density-density interactions, respectively.

The model at half-filling, given by Eq. (I.1), possesses the following special symmetry, which we refer to here as “sublattice” symmetry (SLS) [this symmetry is termed “chiral” in the classification scheme of Ref. Zirnbauer, (see also Refs. 1DChiral, ; Gade, ; LeeFisher, ; Furusaki, ; FC, ; GLL, ; RyuHatsugai, ; MDH2D, ; BocquetChalker, ; MRF, ; BDIpaper, )]: the Hubbard-like Hamiltonian in Eq. (I.1) is invariant under the transformation

 cAi→c†Ai,cBj→−c†Bj, (2)

where we simultaneously complex conjugate all scalar terms in the Hamiltonian. [This transformation, like that of time-reversal, is antiunitary. In the presence of time-reversal invariance (TRI), SLS is equivalent to the usual particle-hole symmetry.] As is well known, the Fermi surfacefootnote-b of the half-filled, non-interacting model, Eq. (I.1) with , possesses perfect “nesting,”

 ε(k+KN)=−ε(k), (3)

where is the non-interacting bandstructure, and is a nesting wavevector. For the hypercubic lattice with lattice spacing , takes the form

 KN≡π(n1,n2,…,nd), (4)

where the numbers , with . Fig. 2 depicts the Brillouin zone (BZ) associated with the square lattice shown in Fig. 1. The set of nesting wavevectors defined in Eq. (4) span the sublattice Brillouin zone (sBZ), appropriate to the and sublattices of the composite bipartite lattice. For the special case of the square lattice, the boundary of the sBZ also serves as the Fermi line at half filling, shown in Fig. 2.

Nesting and SLS are tied together. Under the transformation given by Eq. (2), the hopping part of the Hamiltonian in Eq. (I.1) transforms as

 ∫BZddk(2π)dε(k)c†kck →∫BZddk(2π)dε∗(k)ck−KNc†k−KN (5)

where is a nesting wavevector as in Eq. (4), and

 ck≡∑i∈Ae−ik⋅RicAi+∑j∈Be−ik⋅RjcBj. (6)

Fermi surface nesting is, in a sense, the defining property of Hubbard-like models for interacting lattice fermions in dimensions. It is the nesting condition which makes the ballistic Fermi liquid phase at half filling in such models unstable to Mott insulating order in the presence of generic, arbitrarily weak interparticle interactions.Hirsch ; GSST ; Shankar “Nesting” instabilities can occur for microscopically repulsive interparticle interaction strengths, and arise through the exchange of the nesting momenta through the particle-hole channel of these interactions. Such models may also exhibit the BCS superconducting instability,Schrieffer which exists for any TRI Fermi liquid with (effectively) attractive pairing interactions. The ground state of such a half-filled, Hubbard-like model with weak, but non-vanishing interactions is typically a Mott insulator, with lattice translational symmetry spontaneously broken at the nesting wavelength, a superconductor, or a mixture of these, such as a “supersolid.”Hirsch ; SLGMWSSD ; MSSWB ; Shankar

A simple calculationGSST using the random phase approximation (RPA) predicts that the Fermi liquid phase of the (not disordered) model defined by Eq. (I.1) is unstable to charge density wave (CDW) order for any . The CDW state is a Mott insulator, in which a greater proportion of the fermion density resides on one sublattice than the other. For the case of the square lattice, the RPA calculation predicts a transition to the CDW state at a temperaturefootnote-c

 Tc∼2texp[−π√tV−U], (7)

in the weak coupling limit . Alternatively, a one-loop renormalization group (RG) calculation,Shankar performed upon a low-energy effective field theory description of the model given by Eq. (I.1), shows that the effective CDW interaction strength, which we define here as

 Wc∼12(U−V), (8)

(corresponding to a staggered charge density) grows to large negative values under renormalization if its initial value , i.e. was negative to begin with. This run off to strong interaction coupling is taken to signal the onset of CDW formation.footnote-d These and analogous results regarding the Néel ground state of the half-filled, spin- Hubbard model in are well established, and the latter have been further confirmed with extensive numerical work. (See e.g. Refs. Hirsch, ; MSSWB, ; 3DHubbard, .) The spinless Hubbard-like Hamiltonian in Eq. (I.1), on the other hand, has received less attention in the numerical literature; early Monte Carlo studiesGSST of the version of this Hamiltonian, with , show the existence of a CDW ground state for positive , but were unable to access the weak coupling limit .

Sublattice symmetry plays a crucial role in establishing the CDW ground state of the model in Eq. (I.1) (for ). Nesting can also occur in the absence of SLS, since most aspects of Fermi surface geometry typically depend strongly upon microscopic details. Under the repeated application of an appropriate renormalization group transformation,Shankar however, this geometry is expected to deform continuously. SLS protects the nesting condition even as other details (such as the presence Fermi surface van Hove singularities) mutate through the RG process. The persistence of nesting and unbroken SLS, up to the onset of Mott or supersolid order, allows arbitrarily weak repulsive interparticle interactions to destabilize the Fermi liquid phase.

### i.2 Quenched disorder

Now we turn to the incorporation of quenched (static) disorder into the model given by Eq. (I.1). First, note that the addition of on-site (or: “diagonal”) randomness, characterized by a strength , breaks sublattice symmetry [Eq. (2)] in every realization of disorder. Thus, turning on diagonal disorder is expected to destroy the CDW ground state,VUV ; UV at least for sufficiently weak interactions. [Keeping the disorder strength fixed, for example, we expect the absence of zero temperature CDW order in a window of (small) interaction strengths, e.g. for , with )]. The proximity of a Mott insulating phase to the non-interacting (ballistic) Fermi liquid is the most essential characteristic of the Hubbard-like lattice model defined in Eq. (I.1). We conclude that this characteristic is lost upon the incorporation of diagonal disorder. Studies of Hubbard-like models subject to diagonal disorder include those listed in Refs. VUV, ; UV, ; BKAHB, and Refs. Ma, ; ScalettarD, ; HeidarianTrivedi, for spinless and spin- lattice fermions, respectively.

Instead, we consider the Hamiltonian in Eq. (I.1), weakly perturbed by purely “off-diagonal” randomness, which is taken to occur only in the intersublattice hopping amplitudes. Such a model is invariant under the sublattice symmetry (SLS) transformation [Eq. (2)] for each and every static realization of the disorder. We consider complex random (nearest-neighbor) hopping that breaks TRI. [Without interactions, the model is in the “chiral” symmetry class AIII of Ref. Zirnbauer, .] The full lattice Hamiltonian is given by , where was defined in Eq. (I.1), and

 δH=−∑⟨ij⟩δti,jc†AicBj+H.c. (9)

Here, the random part of the hopping matrix element, , is taken to be a Gaussian complex random variable with zero mean, independent on different lattice links. For our system of spinless fermions, this is consistent with the application of a random magnetic field to the otherwise clean model. For non-interacting spin- fermion systems, with nearest neighbor hopping on bipartite lattices at half filling, in random orbital and/or Zeeman magnetic fields, a general classification is (briefly) mentioned in Sec. II.2, with more details being provided in Appendix C. We will conclude from this classification that the results of this paper also apply to a related Hubbard-like model of spin- fermions, subject to an orbital magnetic field and to (homogeneous or random) spin-orbit coupling. Independent work on the effects of interparticle interactions upon these and several other classes of disordered spin- fermions has very recently appeared in Ref. DellAnna, .

Our principal motivation for studying the model in Eqs. (I.1) and (9) is that, due to the presence of SLS, we expect both disorder and interparticle interactions to play important roles in the description of the low-energy physics. Because random hopping preserves the special SLS, our disordered model retains the “nesting” CDW instability of the associated clean system. This instability can therefore compete with the unusual localization physics arising from SLS in the disordered, but non-interacting model (see below). The further assumption of broken TRI guarantees that we do not have to confront an additional superconducting instability.BK ; JengLudwigSenthilChamon ; Hirsch ; GSST ; Shankar We note that the effects of hopping disorder upon the Néel ground state of the (slightly more complex) spin- Hubbard model at half filling were studied numerically in Refs. ScalettarOD1–BDI, and ScalettarOD2–BDI+C, , although these studies were limited to .

A second motivating factor is that, interestingly, and as alluded to above, the presence of SLS radically changes the localization physics of the disordered, non-interacting random hopping model [Eqs. (I.1) and (9) with ]. SLS enables the random hopping model (RHM) to evade the phenomenon of Anderson localization. Specifically, the non-interacting system exhibits a critical, delocalized phase at the band center (half filling) in one, two, and three dimensions for finite disorder strength, with a strongly divergent low-energy density of states in .LeeFisher ; 1DChiral ; Gade ; Furusaki ; FC ; GLL ; RyuHatsugai ; MDH2D ; BocquetChalker ; MRF In particular, there is no MIT and no Anderson insulating phase in (in the absence of interactions). The essential features of random hopping model physics are summarized in Figs. 3 and 4. These figures apply, for example, to spinless lattice fermions with sublattice symmetry, with or without TRI (classes BDI and AIII, respectively),Zirnbauer in . Upon approaching the band center () in , both the density of states and the localization length strongly diverge, as indicated in Fig. 3. By contrast, RHMs in support a diffusive metallic phase, Gade ; FC characterized by a band of delocalized states with energies , represented by the shaded regions in Fig. 4. ( are particle and hole mobility edges.) While is expected to decrease monotonically with increasing disorder, as indicated by the arrows along the dashed boundaries of the shaded region in Figs. 4(a) and 4(b), it is believed that in , there remains always a region of delocalized states of finite thickness in energy, centered around , for any finite disorder strength in a non-interacting system with SLS.

Indeed, random hopping models have been of significant theoretical interest in the recent past, both because of the unusual delocalization physics described above, but also because these models have proven amenable to a variety of powerful analytical techniques in , with many exact and/or non-perturbative features now understood.1DChiral ; GLL ; MDH2D ; MRF This situation should be contrasted with our understanding of the conventional non-interacting (“Wigner-Dyson”) MIT, which is based largely upon perturbative analytical results in , using the -expansion.LeeRamakrishnan

### i.3 Summary of results and outline

In this work, we analyze the stability of the diffusive Fermi liquid phase of the Hubbard-like model defined by Eqs. (I.1) and (9), in the simultaneous presence of both disorder and interactions. We derive a low-energy, continuum field theory description of this system, which takes the form of a (class )Zirnbauer Finkel’stein non-linear sigma model (FNLM).Finkelstein ; BK We employ the Schwinger-KeldyshKELDYSH method in order to ensemble average over realizations of the hopping disorder. The FNLM contains parameters which specify the random hopping disorder and the interparticle interaction strengths. We compute the one-loop renormalization group flow equations for these parameters, using a Wilsonian frequency-momentum shell background field methodology.Finkelstein We then discuss the physics of our model in (i) and (ii) dimensions. We now briefly summarize our results:

For the case, we find that the conducting phase of the disordered, non-interacting system is destabilized by the interactions. By contrast, such a phase, a metallic ‘diffusive Fermi liquid,’ does exist (trivially) in ; we identify what we call an “Anderson-Mott” disorder-driven instability of this metallic ‘diffusive Fermi liquid.’ This instability arises solely from the competition between disorder and short-ranged interactions, and is perturbatively controlled via an -expansion in . The instability that we find is indicative of a first-order metal-insulator transition (MIT); it occurs for disorder strengths of order , and is therefore perturbatively accessible for . We expect our result to be relevant for the Hubbard-like model in Eqs. (I.1) and (9) in three spatial dimensions () for sufficiently strong disorder, and we stress that this “Anderson-Mott” instability is clearly distinct from the pure Mott “nesting” instability, which is driven solely by interactions (and not by disorder), although the latter also appears in the phase diagram of our model (see Fig. 23, below). The non-interacting random hopping model, Eqs. (I.1) and (9) with , has no localized phase with disorder in the absence of interactions (see e.g. Section V.1.1); therefore, a localized phase can only appear due to the presence of the interactions. The discovery of this disorder-driven, interaction-mediated diffusive Fermi liquid instability was previously announced in Ref. AIIIshortpaper, . In this paper, we present a derivation of this result, a detailed analysis of the phase diagram of the model, and we discuss our findings in view of previously known results for other, related disordered and interacting fermion systems.

We note that the -expansion is employed in this work as a technical tool in the continuum FNLM description, and should be thought of as a controlled approximation scheme to access the physics of the disordered, interacting Hubbard-like Hamiltonian in Eqs. (I.1) and (9) in . Clearly, we cannot easily define a bipartite lattice fermion model in a fractional number of dimensions; instead, we work with a continuum field theory, the FNLM, argued to capture the low energy physics of the lattice model in both () and (), whose internal structure is constrained by the crucial SLS [Eq. (2)]. The field theory action of the continuum FNLM [displayed in Eqs. (57)–(60), Section II.1.4] can be analytically continued between integer dimensions in the usual way. The internal structure of the FNLM, and thus SLS, is preserved under this continuation; it is SLS, then, that gives meaning to such an interpolation between lattice models in disparate integer dimensions. The “Anderson-Mott” instability of the diffusive Fermi liquid identified in this work occurs for perturbatively accessible, weak disorder strengths only for , i.e. in . We conjecture that this instability also exists in the FNLM, and should therefore be found (i.e. via numerics) in the Hubbard-like lattice fermion model. This conjecture cannot be directly proven here, since the instability of the FNLM, if it exists, would occur in the strong coupling regime.

The organization of this paper is as follows:

In Sec. II, we formulate a Schwinger-KeldyshKELDYSH path integral representation for the lattice model given by Eqs. (I.1) and (9); we then derive the low-energy, continuum FNLM description. The technical content of our work appears in Sections III and IV. We set up our one-loop, frequency-momentum shell renormalization group calculation in Sec. III, specifying the parameterization of the FNLM and stating the necessary diagrammatic Feynman rules. The actual one-loop calculation is chronicled in Sec. IV. We analyze and discuss our results in Sec. V. The reader less interested in calculational details may skip Secs. III and IV entirely, and proceed from the end of Sec. II directly to Sec. V.

A variety of elaborations, extensions, and technical details are relegated to the Appendices. In Appendix A, we show how the structure of the continuum FNLM for the model in the absence of interparticle interactionsGade follows immediately from a symmetry analysis of the non-interacting Keldysh action for the random hopping model [Eqs. (I.1) and (9), with ]. Appendix B details an expansion used in Sec. II, while Appendix C describes the random matrix symmetry classification (along the lines of Ref. AltlandZirnbauer, ) of disordered, bipartite lattice models for spin- electrons. [See also Sec. II.2]. Appendix D provides a surprising alternative interpretation of the class AIII Finkel’stein NLM, studied in this paper, in terms of the spin- quasiparticles of a spin-triplet, p-wave superconductor. This quasiparticle system may be defined directly in the continuum, without reference to a “microscopic” lattice model or an additional sublattice symmetry. Finally, Appendix E collects the loop integrals required in the RG calculation presented in Sec. IV.

## Ii FnlσM formulation

In this section we derive the class AIII Finkel’stein non-linear sigma model (FNLM) description of the Hubbard-like lattice model given by Eqs. (I.1) and (9), using the Schwinger-KeldyshKELDYSH method to perform the disorder averaging. The results of the derivation are provided below in Eqs. (57), (58), and (59), and interpreted in the discussion following these equations.

### ii.1 Class AIII Finkel’stein NLσM

#### ii.1.1 Schwinger-Keldysh path integral

To begin, we envisage a zero temperature, -dimensional real time-ordered (-ordered) path integral for the model defined in Eqs. (I.1) and (9). As usual, we need to normalize this path integral to unity in order to perform the ensemble average over realizations of the hopping disorder. We employ the Schwinger-Keldysh (Keldysh) method,KELDYSH which exploits the identity , where is an anti-time ordered (-ordered) path integral for the same model. We write the Keldysh generating functionKELDYSH

 Z≡ZTZ¯T=∫D¯cDcei(S1+S2), (10)

where the non-interacting action is given by

 S1=∑a=1,2ξa∫ dω⎧⎪⎩∑⟨ij⟩ti,j¯caAi(ω)caBj(ω)+H.c. +∑i∈A¯caAi(ω)[ω+iξaηsgn(ω)]caAi(ω) +∑j∈B¯caBj(ω)[ω+iξaηsgn(ω)]caBj(ω)⎫⎪⎭, (11)

with . The interactions reside in

 S2=∑a=1,2ξa∫dt⎧⎪⎩ −V∑⟨ij⟩naAi(t)naBj(t) −U∑⟨⟨ii′⟩⟩naAi(t)naAi′(t) −U∑⟨⟨jj′⟩⟩naBj(t)naBj′(t)⎫⎪⎭. (12)

The generating functional defined by Eq. (10) is an integral over the Grassmann fields , , and , , defined on the and sublattices of the bipartite lattice. Note the locality of the non-interacting sector of the theory [ in Eq. (II.1.1)] in frequency , versus the locality of the interacting sector [ in Eq. (II.1.1)] in time . In Eqs. (II.1.1) and (II.1.1), indices and label and sublattice sites, respectively, while the “Keldysh” species index denotes the -ordered () and -ordered () branches of the theory. The number takes the values

 ξa={−1,a=1(T-ordered),−1,a=2(¯T-ordered). (13)

The factors in Eq. (II.1.1) are frequency integration pole prescriptions appropriate to - and -ordered correlation functions, with . Finally, the density fields in Eq. (II.1.1) are defined via .footnote-e

The FNLM that we are after is a matrix field theory; it therefore makes good sense to introduce compactifying matrix notation at this early stage. We think of the field () as a column (row) vector with a single “superindex,” which is a direct product of frequency and Keldysh indices: e.g. . In the Keldysh formalism, it is often useful to further divide frequency into a product of (discrete) sign and (continuous) modulus spaces: . Next, we introduce two commuting sets of Pauli matrices: the matrix acts in the space, while the matrix acts in the Keldysh species (/) space, with . Employing the conventional basis for all Pauli matrices, we identify and , with . [ was defined in Eq. (13).] Finally, we define the single particle energy matrix , with

We make a change of variables , and rewrite the non-interacting sector [Eq. (II.1.1)] of the Keldysh action as

 S1≡S01+δS1. (14)

The clean bipartite hopping model appears in the term

 S01= +[¯cAk¯cBk][^Σ3|^ω|00^Σ3|^ω|][cAkcBk]⎫⎪⎭ (15) ≡ ¯c[^Σ3|^ω|+iη^Σ3^ξ3−^σ1ε(^k)]c. (16)

The momentum integration in Eq. (15) is taken over the sublattice Brillouin zone (sBZ); is the clean energy band structure.footnote-f Equation (16) gives the most compact representation that we will use for the clean, non-interacting action. Here we have introduced a third set of Pauli matrices, , , acting in the sublattice flavor space, and we have also promoted momentum to an operator, . The column vector carries indices in the momentum , mod-energy , (), sublattice flavor (), and Keldysh () spaces, i.e.  with all indices displayed. Here, denotes the Keldysh species. The disorder is relegated to the perturbation

 δS1=∑⟨ij⟩(¯cAiδti,jcBj+¯cBjδt∗i,jcAi). (17)

#### ii.1.2 Disorder averaging and Hubbard Stratonovich decoupling

We now ensemble average over realizations of the complex random hopping amplitudes appearing in Eq. (17). In order to simplify the derivation of the FNLM, we employ the following artifice: we assume that of the nearest neighbors bonds surrounding a given site belonging to the sublattice of the bipartite lattice under study, only one such bond is disordered. ( is the coordination number.) We assume further that the same type of bond (specified by its orientation) is made random at each and every site, thus allowing a unique, orientationally homogeneous pairing (dimer covering) of the and sublattice sites of the bipartite lattice. An example of such a pairing is provided for the square lattice in Fig. 5. This seemingly pathological constraint upon the disorder distribution allows for the quickest derivation of the low-energy effective field theory.footnote-g

With such a pairing established between each sublattice site and its associated sublattice site , we have

 δti,j={δti,j(i),j=j(i)0,j≠j(i). (18)

We take the “dimerized” bond amplitudes to be Gaussian random variables, independent on different nearest-neighbor lattice links , and identically distributed with the following purely real mean and variance:

 ¯¯¯¯¯¯¯¯¯¯¯¯¯δti,j(i)=δtdim (19)

and

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯δt∗i,j(i)δti,j(i)=λ(m), (20)

respectively. The overbars in Eqs. (19) and (20) denote disorder averaging. Although we will be ultimately interested in the limit of zero mean bond dimerization, it will prove convenient in the interim to retain . The superscript “” appearing on the right-hand side of Eq. (20) stands for “microscopic,” indicating that the quantity is defined at the lattice scale. Using Eqs. (17)–(20), we define the disorder-averaged action via

 ei¯¯¯¯¯¯¯δS1 ≡∫|Dδt|2exp[−1λ(m)∑i∈A|δti,j(i)−δtdim|2+iδS1] =exp⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩λ(m)2∑i∈ATr[^A2i]+iδSdim⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭, (21)

with

 ^Ai≡⎡⎣0^Ai^A†j(i)0⎤⎦=^σ1+i^σ22^Ai+^σ1−i^σ22^A†j(i), (22)

where we have defined

 ^Ai →Aa,a′iω,ω′≡caAi(ω)¯ca′Ai(ω′), (23a) ^A†j(i) →A†a,a′jω,ω′≡caBj(ω)¯ca′Bj(ω′). (23b)

In Eq. (II.1.2), denotes a matrix trace over indices in the sublattice flavor (), frequency (), and Keldysh () spaces. The field , introduced in Eq. (II.1.2), is a matrix of fermion bilinears; Eq. (22) details the sublattice flavor space decomposition of in terms of the purely off-diagonal Pauli matrices and . Although they occupy the off-diagonal blocks of , the fields and actually describe fermion degrees of freedom residing entirely on the and sublattices, respectively, as shown in Eq. (23). and each carry frequency and Keldysh species indices. Finally, a non-zero mean bond dimerization [Eq. (19)] gives rise to the following homogeneous term in Eq. (II.1.2):

 δSdim= δtdim∑i(¯cAicBj(i)+¯cBj(i)cAi) = −¯c[^σ1ϕR(^k)+^σ2ϕI(^k)]c. (24)

The second line of Eq. (II.1.2) expresses in momentum space; here we have adopted the same compact notation employed in Eq. (16), wherein and act in sublattice flavor space, and is the (crystal) momentum operator. The functions and in Eq. (II.1.2) are real and imaginary components of the “dimerization” function

 ϕR(k)+iϕI(k)≡−δtdimeik⋅n. (25)

In this equation, is a unit vector pointing in the direction determined by the chosen bond dimerization [see Eqs. (18) and (19), and Fig. (5)].

We have adopted a suggestive notation in Eq. (23) to denote the pure sublattice fields and . Consider the following spatially uniform deformation of these fermion bilinears:

 ^Ai→^UA^Ai^U†B,^A†j(i)→^UB^A†j(i)^U†A, (26)

where and are independent unitary transformations in (frequencyKeldysh) space:

 ^U†A^UA=^U†B^UB=^1|ω|⊗Σ⊗ξ. (27)

The transformation in Eq. (26) is clearly a symmetry of the disorder-averaged action , as can be seen from Eqs. (II.1.2)–(II.1.2). In fact, there is a direct relationship between this transformation and the symmetry structure of the non-interacting Keldysh action [Eqs. (16) and (17)], in every fixed realization of the static disorder; the connection is articulated in Appendix A. Equation (26) suggests that we may regard as the “Hermitian adjoint” of its associated nearest neighbor , justified on length scales much larger than the lattice spacing. The identification implies the Hermiticity of the composite matrix field in Eq. (23): .

Next, we decouple all four fermion terms appearing in the disorder-averaged and interacting sectors of the theory with bosonic Hubbard-Stratonovich fields. In the disorder-averaged sector [Eq. (II.1.2)], we write

 exp⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩λ(m)2∑i∈ATr[^A2i]⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭=∫D^Qexp⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∑i∈ATr[−12λ(m)^Q2i+^Qi^Ai]⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭. (28)

The matrix field is taken to be Hermitian, and purely off-diagonal in sublattice flavor space, i.e.

 ^Qi≡⎡⎣0^Qi^Q†j(i)0⎤⎦=^σ1+i^σ22^Qi+^σ1−i^σ22^Q†j(i), (29)

with

 ^Qi→Qa,a′iω,ω′,^Q†j(i)∼^Q†i→Q†a,a′iω,ω′. (30)

[Compare Eqs. (22) and (23)].

Turning to the interacting sector, we re-write Eq. (II.1.1) as

 S2=−∑a=1,2ξa2∫dtNTa(t)^XNa(t), (31)

where superscript “” denotes the matrix transpose operation, and

 ^X≡[U^xV^yV^yTU^x] (32)

is a matrix in sublattice flavor space (), with elements involving the position space coupling functions and (defined below), while the fermion sublattice densities are encoded in the vector

 Na(t)→[naA(t)naB(t)]. (33)

We have suppressed all position space indices in Eqs. (31)–(33). In Eq. (32), the function () equals unity for pairs of nearest neighbor (next-nearest neighbor) lattice sites (), and vanishes otherwise. Now we decouple

 eiS2=∫Dρexp⎧⎪⎩∑a=1,2i∫dt[ ξa2ρTa(t)^X−1ρa(t) +ρTa(t)Na(t)]⎫⎪⎭, (34)

where

 ρa(t)→[ρaA(t)ρaB(t)]. (35)

Again, we have suppressed all position space indices in Eqs. (II.1.2) and (35).

#### ii.1.3 Saddle point and gradient expansion

Gathering together the homogeneous hopping [Eq. (16)], mean bond dimerization [Eq. (II.1.2)], disorder-averaged [Eq. (28)], and interacting [Eq. (II.1.2)] pieces of our theory, we perform the Gaussian integral over the fermion fields, and arrive finally at the following effective field theory:

 Z=∫D^QDρexp[−SQ−Sρ−SDET], (36)

where

 SQ=12λ(m)∑i∈ATr[^Q2i], (37)
 Sρ=∑a=1,2−iξa2∫dtρTa(t)^X−1ρa(t), (38)

and

 SDET=−Tr⎧⎪⎩ln[G−10+i^σ1^Q+ρ]⎫⎪⎭. (39)

The trace in Eq. (39) is performed over indices in the position, sublattice flavor (), mod-energy , (), and Keldysh species () spaces. The operator in Eq. (39) represents the inverse of the (Keldysh) single particle Green’s function for the clean, non-interacting hopping model described by Eq. (16), incorporating, in addition, the mean bond dimerization from Eq. (II.1.2):

 G−10=^Σ3|^ω|+iη^Σ3^ξ3−^σ1[ε(^k)+ϕR(^k)]−^σ2ϕI(^k). (40)

The functions and are real and imaginary parts of the “dimerization” function defined by Eq. (25).

We look for a spatially homogeneous saddle point solution to the action given by Eqs. (37) and (39), in terms of the matrix field , with in Eqs. (39) and (40) (i.e. ignoring the interparticle interactions, and considering the limit of zero mean bond dimerization). In the low-energy/long-time limit , the structure of the saddle point solution is determined by the pole prescription piece (the term proportional to ) in Eq. (40). We make the standard ansatz

 ^QSP≡12τ^σ1^Σ3^ξ3, (41)

with the elastic scattering lifetime due to the disorder. Then the saddle point condition reduces to the usual self-consistent Born approximation (SCBA) for the elastic decay rate :

 1λ(m)=∫sBZddk(2π)d[ε2(k)+1(2τ)2]−1. (42)

For the case of the square lattice (with its concomitant van Hove singularities at half filling), Eq. (42) gives

 12τ∼λ(m)2πtln(t2λ(m)), (43)

in the weak disorder limit .

Next we consider fluctuations about the saddle point solution given by Eq. (41). Dominant within the diffusive metallic phase are the long-wavelength, low-energy Goldstone (diffusion) modes that preserve the saddle point norm . These modes are generated by applying a slowly spatially varying generalization of the symmetry transformation in Eqs. (26) and (27) to . (See also Appendix A.) In terms of the sublattice flavor space decomposition [Eq. (29)], we define

 ^Q(r)≡^σ1+i^σ24τ^Q(r)+^σ1−i^σ24τ^Q†(r), (44)

where

 ^Q(r) =^UA(r)^QSP^U†B(r), (45a) ^Q†(r) =^UB(r)^QSP^U†A(r), (45b)

and

 ^QSP=^Σ3^ξ3. (46)

Alternatively, we note that Eqs. (27), (45), and (46) imply the unitary constraint

 ^Q†(r)^Q(r)=^1|ω|⊗Σ⊗ξ. (47)

Our FNLM will therefore be a field theory of the unitary matrix .

Finally, we assemble the action for the FNLM from the components given by Eqs. (37)–(39). The unitary constraint [Eq. (47)] renders in Eq. (37) an irrelevant constant, so we concentrate upon in Eq. (39). Keeping only the most relevant terms in a gradient expansion, one obtains

 SDET∼ i2τ∫ddrTr⎧⎪⎩(^Σ3|^ω|+iη^Σ3^ξ3)[^Q†(r)+^Q(r)]⎫⎪⎭ +(2¯¯¯vFτ)2∫ddrTr⎧⎪⎩∇^Q†(r)⋅∇^Q(r)⎫⎪⎭ +SdimDET+SI, (48)

where the terms and are defined below in Eqs. (49) and (II.1.3), respectively. Eqs. (II.1.3), (49), and (II.1.3) are derived in Appendix B.

Versions of the “energy” (symmetry breaking) and “stiffness” terms generic to localization sigma models, but specialized here to the Keldysh class AIII FNLM, appear on the first and second lines of Eq. (II.1.3), respectively. Here, is the average Fermi velocity (at half filling), while the elastic scattering lifetime is determined by the SCBA [Eq. (42)]. We emphasize that the structure of these two terms, written out explicitly in Eq. (II.1.3), is independentfootnote-h of the peculiar “dimerized” bond distribution [Eq. (18)] assumed for the random hopping.

The first term on the third line of Eq. (II.1.3) is a perturbation arising from the presence of a non-zero mean bond dimerization [Eq. (19)]:

 SdimDET≡cδtdimt∫ddrTr[^Q†(r)∇^Q(r)]⋅n, (49)

where the unit vector , originally introduced in Eq. (25), specifies the orientation of the mean bond dimerization, and is a constant. Eq. (49) is derived in Appendix B.

Consider the local vector operator

 i∇Φ(r)≡Tr[^Q†(r)∇^Q(r)], (50)

where is the U(1) phase of the unitary matrix field . Eq. (49) implies that provides a coarse-grained measure of the local bond dimerization, i.e. of the orientation of the strongest nearest neighbor hopping bonds within a neighborhood of size the elastic scattering length, in the sublattice symmetric random hopping model. This interpretation can be argued on very general symmetry grounds, at least for the case of the hypercubic lattice: here, a bond dimerization homogeneous in both magnitude and orientation preserves both SLS and sublattice translational symmetry, while breaking lattice rotation, reflection, and composite (intersublattice) translational symmetries: see e.g. Fig. 5. Eq. (49) is the most relevant term (in the sense of the renormalization group proximate to the diffusive metallic phase) consistent with these conditions. [A homogeneous background dimerization, as in Eqs. (II.1.2) and (25), breaks lattice reflection invariance through a lattice plane perpendicular to the direction of the dimerization vector ; a composite sublattice translation involves an exchange of sublattice degrees of freedom, represented by in the continuum theory.]

Given this interpretation, we can generalize Eq. (49) to the case of a perturbation involving a static random vector field :

 (51)

In Eq. (51), is taken to vary in both orientation and magnitude, and represents long-wavelength, quenched orientation fluctuations in the bond strength dimerization of the random hopping. Since Eq. (51) is consistent with the sublattice symmetry of the underlying lattice model, its effects should be included in the low energy effective theory upon the grounds of universality. We see that in order to obtain a final effective field theory that possesses rotational, translational, and reflection invariances on average, it is necessary to specify a second measure of the random hopping strength, in addition to the parameter , the latter of which appears in the SCBA, Eq. (42), and was introduced in Eqs. (20) and (II.1.2). We take to be a Gaussian random variable of vanishing mean, delta function correlated with the variance

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯nidim(r)njdim(r′)=λAδi,jδ(d)(r−r′). (52)

Using Eqs. (51) and (52), we define the disorder-averaged action viafootnote-i

 e−¯¯¯¯¯¯¯¯¯¯¯SdimDET =exp⎧⎪ ⎪ ⎪⎩λA2∫ddr[i∇Φ(r)]2⎫⎪ ⎪ ⎪⎭, (53)

where the operator was defined in Eq. (50). In the low energy effective field theory, then, the random hopping is characterized by the two parameters [Eq. (20)] and [Eq. (52)]; the former sets the elastic scattering lifetime and the conductance,Gade ; GLL ; FC and therefore reflects the “microscopic” structure of the random hopping up to small distance scales of order the Fermi wavelength (responsible for elastic backscattering events involving large crystal momentum transfers), while the latter characterizes the orientational fluctuations of the random hopping at larger distance scales (of order the mean free path).

The last term on the third line of Eq. (II.1.3) is due to the interactions, and is given by (see Appendix B)

 SI≡i2τ∑a=1,2∫dtddr[ ρaA(t,r)Qa,at,t(r) +ρaB(t,r)Q†a,at,t(r)]. (54)

Using Eq. (38), we can now perform the Gaussian integral over the auxiliary field , leading to the result

 (55)

where

 ΓU∼(2τ)2U,ΓV∼(2τ)2V (56)

[c.f. Eqs. (31) and (32)], and is the decay rate, Eqs. (42) and (43). In moving between Eqs. (38), (II.1.3), and (55), we have approximated the short-ranged position space functions and , introduced as elements of the sublattice matrix in Eq. (32) and defined below Eq. (33), as .

#### ii.1.4 The FNLσM and its coupling constants

Combining the results of the previous subsection, Eqs. (II.1.3), (II.1.3), and (55), we arrive at last to the final form of the Finkel’stein NLM (FNLM) description of the Hubbard-like model defined in Eqs. (I.1) and (9). The FNLM is given by the functional integral

 Z=∫D^Qe−SD−SI, (57)

where

 SD=12λ ∫ddrTr⎧⎪⎩∇^Q†(r)⋅∇^Q(r)⎫⎪⎭ +ih ∫ddrTr⎧⎪⎩(^Σ3|^ω|+iη^Σ3^ξ3)[^Q†(r)+^Q(r)]⎫⎪⎭ −