Abstract

We adapt the method of direct scaling analysis developed earlier for single-particle Anderson models, to the fermionic multiparticle models with finite or infinite interaction on graphs. Combined with a recent eigenvalue concentration bound for multiparticle systems, the new method leads to a simpler proof of the multiparticle dynamical localization with optimal decay bounds in a natural distance in the multiparticle configuration space, for a large class of strongly mixing random external potentials. Earlier results required the random potential to be IID.

1. Introduction

1.1. The Model: A Brief Historical Overview

Analysis of localization phenomena in multiparticle quantum systems with nontrivial interaction in a random environment is a relatively new direction in the Anderson localization theory, where during almost half a century, since the seminal paper by Anderson [1], most efforts were concentrated on the study of disordered systems in the single-particle approximation, that is, without interparticle interaction. While in numerous physical models such an approximation seems fairly reasonable, it was pointed out already in the first works by Anderson that the multiparticle models presented a real challenge.

The number of results on multiparticle localization obtained in the physical and mathematical communities still remains rather limited. We do not review here results obtained by physicists (cf. [2, 3]), based on the methods of theoretical physics and considered as firmly established by the physical community. The rigorous mathematical results on multiparticle localization obtained so far (cf., e.g., [4–7]) apply to -particle systems with arbitrary, but fixed , and the range of parameters (such as the amplitude of the disorder and/or the proximity to the edge(s) of the spectrum) is rapidly degrading as .

In this paper, we study spectral properties of the multiparticle random Hamiltonians of the formdescribing fermionic particles in the configuration space assumed to be a finite or countable graph. For clarity, we present first our method in the particular case where and then describe how to adapt the arguments to more general graphs with polynomial growth of balls (cf. Section 8). In (1), is a finite-difference operator representing the kinetic energy, for example, the nearest-neighbor discrete negative Laplacian , is the operator of multiplication by the interaction potential , and is the operator of multiplication by the random function where is a random field relative to some probability space . is a parameter measuring the amplitude of disorder. (All notations are explained in detail in Section 2.)

We consider only the restriction of to the fermionic subspace. The bosonic subspace can be treated essentially in the same way.

1.2. Motivation for This Work

The two most popular methods of the rigorous Anderson localization theory, the Multiscale Analysis developed in [8–11] and the Fractional Moments Method developed in [12, 13], start with the analysis of the Green functions (GFs) of a given random operator, and then one has to translate the upper bounds on the GFs obtained for a fixed energy (cf. [8, 11–13]) or in an energy interval (cf. [8, 10]) into the language of the eigenfunctions (EFs) and eigenfunction correlators (EFCs). In a prior work [14] we proposed an alternative approach, called the Direct Scaling Analysis of eigenfunctions, which focuses essentially on the eigenfunctions in finite domains of the configuration space, from the very beginning. It keeps some important advantages of the MSA, namely, a greater tolerance than in FMM to lower regularity of the disorder, and is close in spirit to the KAM (Kolmogorov-Arnold-Moser) approach, which often requires a tedious inductive scheme. In fact, work [14] has been inspired by the KAM approach to quasi-periodic Hamiltonians proposed earlier by Sinai [15]. Compared to the KAM method, the DSA has a simpler structure of the scale induction, similar to that of the MSA, although it also inherits somewhat weaker probabilistic bounds typical of the MSA.

Furthermore, as was said, there are two kinds of the MSA: with a fixed energy (FEMSA) and with a variable energy (VEMSA); the latter is often referenced to as the energy-interval MSA, for it treats the decay properties of the Green functions in an entire interval of energies at once. The FEMSA is slightly simpler than VEMSA in the single-particle case, but in the realm of multiparticle random Hamiltonians with a nontrivial interaction, the variable-energy variant of the (multiparticle) MSA is considerably more complex. Some of the components of the VEMPMSA are simply eliminated with the (MP)DSA approach, and the counterparts of some other components become quite elementary. A good example of such kind is Lemma 21 (see Section 4.3), the proof of which is very simple and fits in a few lines, while its analog in the variable-energy MPMSA (cf., e.g., [4, Lemma 3]) is substantially more involved.

This constatation has been the principal motivation for the present work. A reader familiar with the techniques of the papers [4, 16, 17] can see that the multiparticle variant of the DSA (MPDSA) from the present paper is much closer in its logical structure to the FEMSA, yet it leads directly to the VEMSA-type decay bounds on the entire eigenbases in arbitrarily large finite domains.

In preprint [18], which has been a starting point for the present paper, we gave the first proof of exponential decay of multiparticle eigenfunctions in presence of an infinite-range interaction decaying at infinity at a fractional-exponential rate, , . In a recent work [7], Aizenman and Warzel, further developing the techniques of [6], proved fractional-exponential decay of EF correlators (EFCs) for subexponentially decaying interactions of infinite range. Due to the logical structure of the FMM, the decay analysis of the eigenfunctions is subordinate to that of the EFCs, and this explains why [7] established only a subexponential decay of eigenfunctions. The DSA is free from this limitation, and this allows us to prove a genuine exponential decay of the multiparticle EFs even in the case where the interaction decay is subexponential.

1.3. Novelty

Apart from simplifications of the Multiparticle Multiscale Analysis (MPMSA) developed in [4], the novelty of this paper is that the external random potential is not necessarily IID (with independent and identically distributed values) but is strongly mixing. Earlier publications, including [19], required the common external random potential acting on the particles to be IID.

1.4. Structure of the Paper

The structure of the paper is as follows:(i)The principal assumptions and main results are stated in Sections 2.4–2.6.(ii)The deterministic techniques of scaling analysis are presented in Section 3.(iii)In Section 4, we treat first the case of finite-range interactions, in order to present the logical structure and the methodological advantages of the MPDSA in the most transparent way.(iv)In Section 7, our analysis is adapted to the interactions of infinite range.(v)Sections 5 and 6 are devoted to the derivation of spectral and dynamical localization from the final results of the MPDSA.

2. Preliminaries, Assumptions, and Main Results

2.1. Configurations of Indistinguishable Particles in

In the first part of this paper, we work with configurations of quantum particles in the one-dimensional lattice, . In quantum mechanics, particles of the same kind are considered indistinguishable; more precisely, depending on the nature of the particles, the wave functions describing particles have to be either symmetric (Bose-Einstein quantum statistics) or antisymmetric (Fermi-Dirac quantum statistics). We choose here the fermionic case; this gives rise to slightly simpler notations and constructions.

For clarity, we use often boldface symbols for the objects related to the multiparticle systems.

Quantum states of an -particle fermionic system in are elements of the Hilbert space , formed by all square-summable antisymmetric functions , with the inner product inherited from the Hilbert space . In this particular case where the “physical” configuration space is one-dimensional, admits a simple representation which we will use.

First, note that any antisymmetric function vanishes on all hyperplanes , . Further, any function defined on the “positive sector” admits a unique antisymmetric continuation to . An orthonormal basis in can be chosen in the usual form where with and is the indicator function of the single-point set . Here is the symmetric group acting in by permutations of the particle positions, , and denotes the parity of the permutation . It is readily seen that is unitarily isomorphic to the Hilbert space of square-summable functions on ; equivalently, one can consider square-summable functions on the set vanishing on the boundary . Indeed, the isomorphism is induced by the bijection between the orthonormal bases and : The subspace is invariant under any operator commuting with the action of the symmetric group .

An advantage of the above representation is that inherits its explicit, natural graph structure from . For more general graphs , one has to resort to the general construction of the so-called symmetric powers of graphs; see the details in Section 8.

2.2. Fermionic Laplacians

Any unordered finite or countable connected graph with the set of vertices and the set of edges is endowed with the canonical graph distance (defined as the length the shortest path over the edges) and with the canonical graph Laplacian : here denotes a pair of nearest neighbors, and is the coordination number of the point in , that is, the number of its nearest neighbors.

In particular, one can take (or a subgraph thereof) with the edges formed by vertices , with . In other words, the vector norm induces the graph distance on . The Laplacian on , which we will now denote by , can be written as follows: where is the identity operator acting on the th variable and is the one-dimensional negative lattice Laplacian in the th variable: The restriction to the subspace of antisymmetric functions can be equivalently defined in terms of functions supported by the positive sector , hence vanishing on its border . Indeed, the matrix elements of in the basis can be nonzero only for pairs , with , so that, for some , , while for all , . If , then . If, say, , then the respective matrix element of the Laplacian’s restriction to with Dirichlet boundary conditions on vanishes, and so does the function (which is no longer an element of the basis in ). Therefore, up to a constant factor, the restriction of the -particle Laplacian to the fermionic subspace is unitarily equivalent to the standard graph Laplacian on . From this point on, we will work with the latter.

It will be convenient in the course of the scaling analysis to use a different kind of metric on , the max-distance defined by and to work with the balls relative to the max-distance,Here .

In the positive sector , the above factorization of -balls is subject to the condition that the RHS of (10) is itself a subset of the positive sector (but the inclusion “” always holds true). Considering configurations as subsets of , one can define the distance between two configurations and in a usual way:

Lemma 1. Let be a union of two subconfigurations and , such that . Then the following identity holds true:

The proof is straightforward and will be omitted.

The graph distance will be useful in some geometrical constructions and definitions, referring to the graph structure inherited from .

Given a subgraph , we define its internal, external, and the so-called graph (or edge) boundary, in terms of the canonical graph distance (below stands for the complement of ):

We also define the occupation numbers of a site in the -particle configuration space relative to a configuration . Namely, define a function by , .

2.3. Multiparticle Fermionic Hamiltonians

Given a finite subset , let be the spectrum of the operator and its resolvent. The matrix elements of resolvents for , in the canonical delta-basis, usually referred to as the Green functions, are denoted by . In the context of random operators, the dependence upon the element will be often omitted for brevity, unless required or instructive. Similar notations will be used for infinite .

Given a real-valued function , we identify it with the operator of multiplication by and consider the fermionic -particle random Hamiltonian where is a second-order finite-difference operator, for example, the negative graph Laplacian . Furthermore, in our model the potential energy is random: where is the external random potential energy of the form of amplitude , and is a random field on relative to a probability space . The expectation relative to the measure will be denoted by . Our assumptions on are listed below (cf. (W1)–(W3)). See also Remark 4 where how (W1)–(W3) can be relaxed is explained.

Further, is the interaction energy operator; for notational brevity, we assume that it is generated by a two-body interaction potential , satisfying one of hypotheses (U0)-(U1) (see below).

Note that, without loss of generality, if and is a.s. bounded, then the random operator has a.s. bounded norm; hence its spectrum is a.s. covered by some bounded interval . In Remark 3, we explain that the boundedness of the potential is not crucial for the spectral localization, although the statement of the result on the dynamical localization is to be slightly modified for unbounded potentials.

One can also consider higher-order finite-difference operators ; this requires only minor technical modifications.

2.4. Assumptions on the Random Potential

We formulate the assumptions in the general situation where the -particle configuration space is a graph with graph-distance ; this includes the case .

We assume that the random field is (possibly) correlated, but strongly mixing; this includes of course the IID potentials treated earlier in [4, 6]. Let be the marginal probability distribution functions (PDF) of the random field and the conditional distribution functions (CDF) of the random field given the sigma-algebra generated by random variables . Our assumptions on correlated potentials are as follows:

()The marginal CDFs are uniformly Lipschitz continuous: for some and all  The LHS of (20) is a conditional probability, hence, a random variable, so (20) holds -a.s.

To formulate the next assumption, introduce the following notation: given a subset , we denote by the sigma-algebra generated by the values of the random potential .()(Rosenblatt strong mixing) For any pair of subsets , with , any events , and some

One can easily check that, for any , , some , , and large enough,The rate of decay of correlations indicated in the RHS of (22) is required to prove dynamical localization bounds with decay rate of EF correlators faster than polynomial, and it can be relaxed to a power-law decay, if one aims to prove only a power-law decay of EF correlators.

Assumptions (W1)-(W2) are sufficient for the proof of spectral and strong dynamical localization in multiparticle systems. In particular, the role of (W1) is to guarantee the eigenvalue concentration (EVC) estimates used in the MPMSA scheme. However, it was discovered in [4, 6] that the traditional EVC estimates do not provide all necessary information for efficient decay bounds on the eigenfunctions of multiparticle operators. More precisely, conventional EVC bounds seem so far insufficient for the proof of the exponential decay of eigenfunctions and EF correlators with respect to a norm in the configuration space of -particle systems, starting from . For this reason, we proposed earlier [20] a new method for comparing spectra of two strongly correlated multiparticle subsystems and improved the EVC estimate from [4]. For the new method to apply, one needs an additional assumption on the random potential field which we will describe now.

First, introduce the following notations. Given a finite subset , we denote by the sample mean of the random field over , and define the “fluctuations” of relative to the sample mean, , . Denote by the sigma-algebra generated by .

We will assume that the random field fulfills the following condition:()There exist such that for any finite subset of cardinality and any -measurable function and all

In the particular case of a Gaussian IID field, for example, with zero mean and unit variance, it is well-known from the standard courses of probability that is independent of the “fluctuations” , so it is Gaussian with variance (if has unit variance) that its probability density is bounded, although as . Property (W3) has been recently proven for a larger class of IID potentials (cf. [21]).

Specifically, (W3) has been established in [21] for IID random potentials with marginal probability measure which admits a representation of the form , where is an arbitrary probability measure on and is supported by some bounded interval and admits probability density which fulfills the following condition: A prototypical example is the uniform distribution on .

2.5. Assumptions on the Interaction Potential

We assume that the interaction potential generating the interaction energy satisfies one of the following decay conditions:()There exists such that ()There are some , , and such that

Naturally, (U0) (U1). We will prove the multiparticle localization first under the strongest assumption (U0) (leading to a simpler proof), to illustrate the general structure of the DSA procedure, and then extend the proof to the infinite-range interactions satisfying (U1).

2.6. Main Results

Theorem 2. Assume that a bounded random field fulfills conditions (W1)–(W3), and the interaction fulfills one of the conditions (U0) and (U1). There exists such that for the following holds true. (A)With probability one, the fermionic operator has pure point spectrum and all its eigenfunctions are exponentially decaying at infinity: for each , some , and all with large enough,(B)For some and any  Consequently, for any finite subset  where is the operator of multiplication by the function .

Remark 3. The assumption of boundedness of the random potential is inessential for the proof of spectral localization (exponential decay of all eigenfunctions) and for the analog of the decay bound (29) on eigenfunction correlators in arbitrary large but finite subset of the -particle configuration space (cf. Corollary 27). Without any condition on the rate of decay of the tail probabilities as , the propagator in (29)-(30) has to be replaced with , where is an arbitrary bounded interval, and is the spectral projection for on .

Remark 4. Exponential spectral localization can be proved without condition (W3), with (W1) relaxed to Hölder continuity. Indeed, one can use the EVC bound from [4, Lemma 2], valid for the pairs of balls distant in the so-called Hausdorff pseudo-metric in the multiparticle configuration space, defined as follows: for the configurations , , where In this case, the decay bounds on the eigenfunction correlators would also be established with respect to replacing the max-distance :Observe that the RHS bound is not uniform as in (29) but depends upon ; this is an artefact of the EVC bounds based on the Hausdorff distance (cf. [4, 6, 7, 17]). Of course, the roles of and in the LHS of (33) are similar, so the factor in the RHS can be replaced by or, more precisely, by some function . As to the proofs, they require only minor notational modifications, for they adapt easily to any kind of pseudometric in the -particle configuration space for which an EVC bound of the general form (64) can be proved, with some , , and .

3. Deterministic Bounds

3.1. Geometric Resolvent Inequality

The most essential part of the Direct Scaling Analysis concerns the finite-volume approximations of the random Hamiltonian , acting in finite-dimensional spaces , with , . We define where the indicator function is identified with the operator of multiplication by . is a bounded, finite-dimensional Hermitian operator, so its spectrum is real.

Operator can be represented as follows: and is the Kronecker symbol.

Recall that we denote by the resolvent of and by the matrix elements thereof in the standard delta-basis (the Green functions).

The so-called Geometric Resolvent Inequality (GRI) for the Green functions can be easily deduced from the second resolvent identity (we omit for brevity):whereSee, for example, [22]. Clearly, (36) implies the inequalitySometimes we use another inequality stemming from (36):Similarly, for the solutions of the eigenfunction equation one has, with ,provided that is not an eigenvalue of the operator .

3.2. Dominated Decay Bounds for the Green Functions and Eigenfunctions

The analytic statements of this subsection apply indifferently to any discrete Schrödinger operators, on fairly general graphs, regardless of their single- or multiparticle structure of the potential energy. To avoid any confusion, in this subsection we denote the underlying graph by ; in applications to -particle models in , one has to set .

Definition 5. Fix an integer and a number . Consider a finite connected subgraph and a ball . A function is called -dominated in if for any ball one has

Lemma 6. Let be given integers and a number . If on a finite connected graph is -dominated in a ball , then

Proof. The claim follows from (41) by induction (“radial descent”):

Lemma 7. Let be a finite connected graph and , , a function which is separately -dominated in and in , with . Then