In this paper, the local iterative Lie–Schwinger block-diagonalization method, introduced and developed in our previous work for quantum chains, is extended to higher-dimensional quantum lattice systems with Hamiltonians that can be written as the sum of an unperturbed gapped operator, consisting of a sum of on-site terms, and a perturbation, consisting of bounded interaction potentials of short range multiplied by a real coupling constant t. Our goal is to prove that the spectral gap above the ground-state energy of such Hamiltonians persists for sufficiently small values of |t|, independently of the size of the lattice. New ideas and concepts are necessary to extend our method to systems in dimension d > 1: As in our earlier work, a sequence of local block-diagonalization steps based on judiciously chosen unitary conjugations of the original Hamiltonian is introduced. The supports of effective interaction potentials generated in the course of these block-diagonalization steps can be identified with what we call minimal rectangles contained in the lattice, a concept that serves to tackle combinatorial problems that arise in the course of iterating the block-diagonalization steps. For a given minimal rectangle, control of the effective interaction potentials generated in each block-diagonalization step with support in the given rectangle is achieved by exploiting a variety of rather subtle mechanisms, which include, for example, the use of weighted sums of paths consisting of overlapping rectangles and of large denominators, expressed in terms of sums of orthogonal projections, which serve to control analogous sums of projections in the numerators resulting from the unitary conjugations of the interaction potential terms involved in the local block-diagonalization step.
I. MODELS OF GAPPED QUANTUM LATTICE SYSTEMS AND SURVEY OF RESULTS
In this paper, we introduce and study a family of quantum lattice systems describing insulating materials in two or more dimensions. We are interested in analyzing the low-energy spectrum of the Hamiltonians of these systems and, in particular, in showing that the ground-state energies of these Hamiltonians are separated from the rest of their energy spectrum by a strictly positive gap.1 Our analysis is based on a novel method consisting in iteratively block-diagonalizing the Hamiltonians with respect to the ground-state subspace. The block-diagonalization is accomplished by a sequence of unitary conjugations of the Hamiltonians. Our analysis is motivated, in part, by recent interest in characterizing “topological phases” (see, e.g., Refs. 2–4) and, more specifically, by studying Hamiltonians of a class of “topological insulators” whose ground-state energy is separated from the higher-lying spectrum by a strictly positive energy gap. However, the scope of our techniques is actually more general.
To be concrete, we consider tight-binding models of electrons hopping on a lattice , with Hamiltonians that are given as the sum of an unperturbed operator, K0, and a perturbation, KI, consisting of a sum of bounded interaction potentials. The operator K0 can be written as a sum of terms, Hi, only depending on the degrees of freedom located at single sites , while the interaction potentials contributing to KI only couple degrees of freedom located on subsets of the lattice of strictly bounded diameter. We focus our attention on unperturbed operators K0 with a unique ground-state, Ω, and a positive energy gap above their ground-state energy. However, our methods can be extended to families of operators with degenerate ground-state energies. Indeed, in Ref. 6, our scheme has been employed to deal with small perturbations of the Hamiltonian of the Kitaev chain, which has a degenerate ground-state. The extension to unbounded operators for quantum chains (e.g., the relativistic, massive ϕ4 model on a one-dimensional lattice) was discussed in Refs. 7 and 8, where we specify the general structure of the degenerate ground-state subspace that allows us to implement our block-diagonalization scheme (see Remark 1.1 of Ref. 7). Under the same assumption on the ground-state subspace, our method works in any dimensions.
Our aim is to iteratively construct an anti-self-adjoint operator , such that the ground state of the operator is given again by Ω, and the spectrum of the restriction of this operator to the subspace orthogonal to Ω lies strictly above the ground-state energy, provided that the absolute value of the coupling constant t is small enough. Our method to construct the operator S = S(t) is inspired by a novel technique introduced in Ref. 6, which, in its original form, has been limited to chains, i.e., to one-dimensional systems. This technique represents an interesting example of multi-scale, iterative perturbation theory: it consists in successively block-diagonalizing the Hamiltonians associated with sequences of bounded, connected subsets of the lattice. In one dimension, such subsets are intervals. However, for d > 1, the number of connected subsets of a given cardinality, R, containing a fixed point of the lattice grows exponentially in R, and this causes certain difficulties that make it necessary to refine the methods in Ref. 6 in a rather subtle way; see Sec. II.
We remark that the procedure described here is amenable to be extended to analogous lattice systems but with unbounded interactions.9
It is appropriate to comment on earlier work addressing problems closely related to the ones treated in our paper. Methods somewhat similar to those developed in this paper, but much simpler, have been applied to the analysis of phase diagrams of quantum lattice systems, see Refs. 5, 11, and 17. Our main results are similar to some that can be found in the literature. In Ref. 10, fermionic path integral methods have been used for the same purpose, and in Refs. 12–14, quasi-adiabatic flows have been constructed to establish results related to ours. In Refs. 15 and 16, similar results have been obtained by using cluster expansions based on operator methods. (Reference 15 is the only one among the quoted papers where unbounded interactions are considered.) More recent results concern developments of the technique introduced in Refs. 3 and 4 in order to prove gap stability for the so-called lattice fermions (see Refs. 18 and 19) and for insulators in the presence of edge states (see Refs. 20 and 21). In Refs. 22 and 23, criteria for SPT (Symmetry-Protected-Topological) phases in two dimensions have been introduced.
It is primarily the mathematical methods used in our analysis that are novel. Compared to previous work, our algorithm is extremely robust: it applies with both self-adjoint and non-self-adjoint Hamiltonians; it can be used to control interactions of both fermions and bosons; and it can cope with some general degeneracy structure of the ground-state subspace. In contrast to some other work on related problems, our paper is self-contained. Indeed, when applied to fermion systems, our methods require no other preliminaries than elementary linear algebra.
A. A family of quantum lattice systems
denotes the rectangle in with sides of lengths k1, k2, …, kd, respectively, whose 2d corners are the sites given by (q1 + ɛ1k1, …, qd + ɛdkd), ɛj = 0 or 1, for j = 1, …, d. [Note that , where N − 1 = (N − 1, …, N − 1) and 1 = (1, …, 1)].
- k ≡ |k| denotes the circumference (= sum of the side lengths) of a rectangle Jk,q, i.e.,(1.6)
The range of the interaction potentials, namely, the integer with the property that rectangles Jk,q appearing in (1.5), is arbitrary, but fixed and N-independent.
- is a symmetric matrix on with the property thatfor all k, q, with , as in (iii) (and whenever ). The rectangle Jk,q is called the “support” of .(1.7)
is a coupling constant independent of N.
B. Main result
Our main result is the following theorem proven in Sec. V (see Theorem 5.3).
There exists some td > 0 independent of N such that for any coupling constant with |t| < td and for all N < ∞, we have the following.
KN(t) has a unique ground-state and
the energy spectrum of KN(t) has a strictly positive gap, , above the ground-state energy.
Results similar to this theorem have appeared in the literature; see, e.g., Ref. 10. The main novelty of our paper is the method of proof.
Organization of this paper. In Sec. II, we explain the formal aspects of our construction. In Sec. II A, we introduce the notion of “minimal rectangles” that will play an important role in our analysis. In Sec. II B, we describe the local (so-called Lie–Schwinger) conjugations of the Hamiltonian associated with minimal rectangles. Next, in Sec. II C, we introduce an algorithm that describes the flow of effective interactions determined by the iterative conjugations of the Hamiltonian used to block-diagonalize it. Moreover, we outline the new features and the complications of our strategy arising in dimensions d ≥ 2, as compared to the one used in Ref. 6 for chains.
In Sec. III, we describe a scheme of re-expansions of collections of effective interaction potentials and a method to derive estimates on the norms of these operators that involve keeping track of paths of connected rectangles.
In Sec. V, the proof of convergence of our construction of the operator SN(t) is presented with a few technicalities deferred to the Appendix. Theorem 5.1 is the core result in our proof of convergence, enabling us to control the norms of the effective interactions by using a composite strategy combining different mechanisms, depending on the regime of the growth processes of rectangles; see Sec. II C. From Theorem 5.1, the final result of this paper, Theorem 5.3, follows.
For chains, i.e., d = 1, the rectangles Jk,q coincide with the connected one-dimensional graphs, Ik,q, , used in Ref. 6, with k edges connecting the k + 1 vertices q, 1 + q, …, k + q, which can also be seen as “intervals” of length k whose left end-point coincides with q.
- We use the same symbol for the operator Oj acting on and the corresponding operatoracting on for any j ∈ Jk,q. Similarly, with a slight abuse of notation, we do not make a distinction between an operator acting on and the corresponding operator acting on the whole Hilbert space , which is obtained out of by tensoring by the identity matrix operator on all the remaining sites.
With the symbol “⊂,” we denote strict inclusion; otherwise, we use the symbol “⊆.”
The multiplicative constant implicit in the symbol can depend on the spatial dimension d.
II. OUTLINE OF THE PROOF STRATEGY
The conjugations used to block-diagonalize the Hamiltonian in (1.5) determine a flow of effective Hamiltonians. These operators are expressed in terms of effective interaction potentials with supports that can be represented as connected unions of the rectangles Jk,i labeling interaction terms in formula (1.5). Whereas for chains, d = 1, when starting from a family of intervals (i.e., Ik,q ≡ Jk,q with k = k and 1 ≤ q ≤ N − k), the connected sets associated with the new interaction potentials are again intervals, the situation is much more complicated in higher dimensions, d > 1, because connected sets of arbitrary shape arise in the flow. The control of growth processes giving rise to each fixed shape that can appear in our construction is crucial in order to accomplish the block-diagonalization of the Hamiltonian. For an arbitrary connected set of a fixed shape, the number of growth processes scales factorially in the number of edges of the set. This crude estimate is, however, not good enough to control the norms of the interaction potentials associated with a given shape since the expected prefactor, tn, in the norm of the interaction potential labeled by a connected set of cardinality n with a fixed shape arising from all possible growth processes terminating in the given shape cannot compensate the number, , of such growth processes when n tends to ∞ (here, t is the coupling constant). Hence, in our estimates, we cannot simply count all growth processes giving rise to each fixed shape since some of them are, in fact, forbidden by the ordering encoded in the block-diagonalization procedure. In this paper, we circumvent this problem with a strategy outlined in Sec. II C, which involves the notion of “minimal rectangles” introduced in Subsection II A.
A. Minimal rectangles
We recall that the symbol denotes a rectangle in whose sides have lengths k1, k2, …, kd and that |k| denotes the sum of these lengths, i.e., . The coordinates of the 2d corners of Jk,q are d-tuples of integers given by either qj or qj + kj at the jth position for all 1 ≤ j ≤ d with qj ≤ N − kj.
or, if , , for some 1 ≤ j ≤ d, with , ∀l > j;
or, if , for all l, , for some 1 ≤ j ≤ d, with , ∀l > j.
As will become clear from our description of the block-diagonalization flow in Sec. II B, the ordering among rectangles must ensure that rectangles with larger circumference |k| succeed those of smaller circumference. With this requirement fulfilled, the ordering chosen here is convenient, but it is definitely not the only possible ordering.
With the symbols (k,q)+j and (k,q)−j, we denote the jth successor and the jth predecessor of (k, q), respectively, in the ordering introduced above. The initial step is (0, N) because the “potentials” associated with the degenerate rectangles consisting of a single point are the on-site terms, Hi, which are already block-diagonal with respect to the pair of projections defined in (2.20) and (2.21). The final step is (N − 1, 1), where N − 1 = (N − 1, …, N − 1) and 1 = (1, …, 1).
B. Effective Hamiltonians
The pairs are used to index all rectangles Jk′,q′ with |k′| = j.
For a fixed rectangle Jl,i, the corresponding potential term may change in each conjugation step of the block-diagonalization procedure until the step (k, q) = (l, i) is reached; hence, is the potential term associated with Jl,i arising in step (k, q) of the block-diagonalization, the superscript (k, q) keeping track of the changes in the potential term arising in step (k, q). The operator depends on the coupling constant t, but this is not made explicit in our notation; it acts as the identity on the spaces for j ∉ Jl,i. A more precise description of how these operators arise in our procedure and an outline of the strategy to control their norms are deferred to Sec. II C.
- For all rectangles Jl,i with (k, q) ≻ (l, i) and for the rectangle Jl,i = Jk,q, the associated effective potential is block-diagonal with respect to the decomposition of the identity acting on into the sum of projections,(2.20)The effective Hamiltonian of (2.19) is obtained after the conjugation step labeled by (k, q). Starting from(2.21)(2.22)the conjugation step labeled by (k, q) is given by(2.23)where the anti-symmetric matrix is chosen in such a way that the interaction potential is block-diagonal; see Sec. IV. More precisely, following the Lie–Schwinger procedure, is built so as to block-diagonalize the local operator given by the sum of all terms in whose support is contained in Jk,q. In other words, is chosen in such a way that the conjugation in (2.24) renders the operator(2.24)block-diagonal, where(2.25)Here, “block-diagonalization” refers to the projections and corresponding to the decomposition of the Hilbert space into vacuum subspace and its orthogonal complement, respectively. The operator plays the role of the “unperturbed” operator since it is already block-diagonal with respect to the decomposition of the identity,(2.26)i.e.,The construction outlined here works because one can show inductively that the energy gap in the spectrum of the Hamiltonian above its ground-state eigenvalue is bounded away from zero, uniformly in the size of the rectangle Jk,q, when a suitable upper bound on the operators norms of the interaction potentials is imposed. The control of this gap (see Sec. IV) relies on the fact that all the effective potentials appearing in have been block-diagonalized already in the previous steps.(2.27)
These properties of the operator , combined with bounds on the norms of the effective potentials obtained at the previous conjugation step, enable us to construct the anti-symmetric matrix used at the next conjugation step and control the norms of the effective potentials obtained after conjugation with . This is described in more detail in Sec. II C.
C. The algorithm and the different regimes in the growth processes of rectangles
Our strategy to control the norms of the effective potentials is based on the following key ideas, which will give rise to a concrete algorithm.
The number of shapes of connected sets of lattice sites arising in our construction is limited by making use of “minimal rectangles” in such a way that, instead of two connected sets, only the minimal rectangle containing them will be recorded (i.e., the rectangle with the property that any rectangle of smaller size cannot contain the union of those sets). Only keeping track of minimal rectangles reduces the combinatorial divergence because the number of rectangles with a given circumference containing a specified site of the lattice grows polynomially in k, namely, like . We, then, lump together all effective potential terms whose support is contained in a given rectangle in such a way that no rectangle of smaller size can contain it. The sum of the norms of these terms is expected to be bounded above by , where c is a universal constant.
We will exploit some subtle mechanisms to identify and control the growth processes allowed by the algorithm introduced below. Depending on the relation between the size, k, of Jk,q and the size, r, of Jr,i, we will distinguish three different regimes for the growth processes that may give rise to the term in (2.31).
As implicitly indicated in (2.22) and (2.23) for the effective Hamiltonian , the potentials must be re-combined properly after each conjugation step (k, q) so as to determine a well-defined flow of operators, , for every fixed support Jr,i. This flow is obtained with the help of a specific algorithm described in Definition 2.4. In Theorem 4.1, we check that our algorithm is consistent with the conjugation in (2.24). This amounts to showing that the right-hand side of (2.24) has the form given in (2.22) and (2.23), with (k,q)−1 replaced by (k, q) and effective potentials as defined in Definition 2.4, formulated next.
Assuming that, at fixed (k, q)−1 with (k, q)−1 ≻ (0, N), for any r, i, the operators and [defined as in (4.51) and (4.52)] are well defined or assuming that (k, q) = (11, 1) [where 11 = (1, 0, …, 0) and 1 = (1, …, 1), respectively] and are well defined, then we define the following:
- if Jk,q ⊈ Jr,i,(2.29)
- if Jr,i = Jk,q,where is defined such as in (4.53) and diag means the diagonal part with respect to the projections and ;(2.30)
The rationale motivating the recombination of terms described in Definition 2.4 is explained in Sec. IV. Here, a remark on item (c) of Definition 2.4 may be helpful in order to understand the key ideas used to control the operator norms of the effective potentials.
The sum on the right-hand side of (2.31) accounts for all contributions to the term with support Jr,i that correspond to “growth processes” of rectangles, i.e., to processes where the union of a rectangle Jk′,q′ ≠ Jr,i and of the fixed rectangle Jk,q labeling the conjugation step in the block-diagonalization is a set with the property that Jr,i is the minimal rectangle associated with it, i.e., such that [Jk′,q′ ∪ Jk,q] ≡ Jr,i.
We recall that, for quantum chains, control of the norms relies on a feature of formula (2.31) that holds only in dimension d = 1: An interval can only grow at the two end-points and, hence, at a number of vertices independent of the size of the interval. However, in higher dimensions, d > 1, the number of terms in the sum in formula (2.31) labeled by rectangles, Jk′,q′, that intersect the rectangle Jk,q only at the boundary grows like a positive power of r (depending on the dimension d). This motivates the introduction of three different regimes, , and , enabling us to exploit a different mechanism to estimate the number of terms in each of the regimes, as outlined below; see also Fig. 1.
() The first regime deals with rectangles labeled by (k, q) that are “small” as compared to the rectangle labeled by (r, i), namely, with pairs (k, q) such that . In order to establish the desired estimate (2.32), we iterate the re-expansion of the potential by applying formulas (2.29) and (2.31). As a consequence, each potential term resulting from the re-expansion can, then, be associated with a connected sequence of rectangles labeling the operators , plus one labeling one of the potentials appearing in the Hamiltonian of definition (1.5) or a potential of the type (where ), with the property that Jr,i is the minimal rectangle associated with this sequence. Roughly speaking, the result, then, holds for the following reasons:
(1) At least rectangles are present in each connected set, and all the corresponding operators have norms of order ; apart from the resulting product of norms , which is also crucial in the argument, it is important that a total factor or smaller is gained from the re-expansion (due to the constraint that holds in this regime).
(2) Note that the rectangles contained in the considered connected set are ordered according to ≻, and consequently, only one growth process can yield each such a set. Due to this observation, the number of connected sets of rectangles resulting from the re-expansion, when each connected set is properly weighted in accordance with the inductive hypothesis on the norms of the potentials , provides an upper bound to . In fact, for |t| small enough but independent of N, this weighted number yields the sought bound (2.32) for .
() The second regime is associated with pairs (k, q) with the property that . In this regime, thanks to the upper bound on k, the size of the rectangles Jk′,q′ in formula (2.31) is so large that it is enough to carry out only one re-expansion step and to then use the inductive hypotheses, similarly to the treatment of chains in Ref. 6. In this regime, we use a basic mechanism involving the use of the denominator in the inductive estimate [see (2.32)] of the potential. If and are both large as it happens in this regime, we can still control the polynomially growing number of terms in the sum of formula (2.31).
() The third regime is associated with “large” rectangles (k, q) since the . In this regime, we exploit a mechanism based on large denominators. This means that we shall collect the contributions in (2.31) corresponding to potentials that are already block-diagonal and, then, estimate them in terms of a sum of projections controlled, through an induction, by the denominator appearing in the expression of [see formula (4.52)]; in the proof by induction for this regime, we make use of the auxiliary quantities displayed in (5.91).
III. TREE STRUCTURE AND PATHS OF RECTANGLES
In order to study regime , we shall re-expand the potentials using the recursive definition (Definition 2.4) repeatedly. The method we develop to single out the terms in the re-expansion contributing to a certain effective potential and to, then, count and weight them is of some independent interest, irrespective of the crucial role it will play in our analysis of regime . We, therefore, describe it carefully in this section.
The strategy can be summarized as consisting of the following steps.
Introducing tree diagrams, we show that every contribution, , to an effective potential—where stands for “branch-operator,” a notion that is motived by the tree structure described below—of the re-expansion resulting from (3.34)–(3.37) is determined by a set, , of rectangles that are ordered and whose union is connected.
We show that there is an injective map from to a set, , of paths of rectangles with certain properties.
By assigning suitable weights to the paths , we will be able to derive upper bounds on the norms of the contributions . This will allow us to estimate the norm by counting (weighted) paths belonging to the set .
A. Tree expansion
The levels of a tree used to identify the contributions to the re-expansion of a potential are labeled by (k′, q′) with (k′, q′) such that (k, q)⪰(k′, q′) ⪰ (0, N). We say that such a tree is rooted at level (k, q).
There is a single vertex at the top of a tree rooted at level (k, q); it is labeled by the symbol of the potential.
The vertices at level (k′, q′)−1 of a tree rooted at level (k, q) are determined by the vertices of the tree at level (k′, q′) in the following way: Each vertex at level (k′, q′), labeled by , is linked to two sets of descendants (vertices) at level (k′, q′)−1 with the following properties: The two sets of vertices are empty if (s, u) = (k′, q′); otherwise, we have the following:
the leftmost set of vertices actually consists of a single vertex, which is labeled by the potential ;
the rightmost set of vertices is empty if Jk′,q′ ⊄ Js,u; otherwise, it contains a vertex for each element Js′,u′ belonging to , and this vertex is labeled by .
Each vertex at level (k′, q′) is connected by an edge to its descendants at level (k′, q′)−1. Edges are labeled by rectangles, or carry no label, in the following way:
- the edge connecting a vertex at level (k′, q′) to its leftmost descendant at level (k′, q′)−1 has no label. It stands for the mapwhere is the potential labeling and labels its leftmost descendant at level (k′, q′)−1;
- each edge connecting the vertex at level (k′, q′) to other descendants at level (k′, q′)−1 is labeled by a rectangle Jk′,q′. It stands for the mapwhere labels the vertex and is the potential labeling the vertex connected to by the edge .
A leaf of the tree is a vertex at some level (k′, q′) that has no descendants, i.e., it is not connected to any vertex at level (k′, q′)−1 by any edge. Note that a leaf of the tree is labeled by a potential of the type for some (k″, q″) ≥ (0, N).
A branch of a tree rooted at (k, q) is an ordered connected set of edges with the following properties:
the first edge of a branch has the vertex at level (k, q) as an endpoint,
the last edge of a branch has a leaf at some level (k″, q″) as an endpoint (referred to as the leaf of the branch), and
there is a single edge connecting vertices at levels (k′, q′) and (k′, q′)−1 for every (k′, q′) with (k, q) ≥ (k′, q′) ≻ (k″, q″).
With each branch of a tree, we associate a set, , of rectangles consisting of (i) those rectangles labeling the edges of and (ii) the rectangle indicating the support of the potential labeling the leaf of .The set inherits the ordering relation (2.14), and hence, its elements can be enumerated by a mapwith (k(i), q(i)) ≻ (k(i+1), q(i+1)), where is the cardinality of the set . Note that is the rectangle associated with the potential labeling the leaf of .
- To every branch , we can associate the “branch operator,” also denoted by ,where is the potential labeling the leaf of ; can be either or .(3.40)
The set of branches whose corresponding branch operators are non-zero is denoted by .
We stress that the operators corresponding to most of the branches are actually zero, for example, when the corresponding leaf is an operator of the type with r > 1, which is zero by definition.
1. Properties of the branches
Definition 3.1 implies the following properties of the elements of the set defined above:
For , the cardinality, , of the set of rectangles is such that . This lower bound on is a consequence of the restriction imposed on k = |k| and required in regime [and it will turn out to be crucial to derive our estimate (5.105) and (5.106) in Theorem 5.1].
The set Jr,i is the minimal rectangle associated with for any branch . Furthermore, if we amputate a branch at some vertex by keeping only the descendants of that vertex (i.e., the lower part only), then the same property holds for the rectangle associated with the potential labeling the (new) root vertex of the amputated branch that has been created.
Two different branches are associated with two different (ordered) sets of rectangles and .
Sketch of proof:
The two branches must cross at some vertex.
Consider the first vertex (starting at the bottom of the tree) where they cross and the two (possibly) amputated branches corresponding to the two original branches that have this vertex as their root vertex.
Now, note that there are two alternatives: (3-i) either the rectangles associated with the two edges linked to the root vertex (the vertex where they cross) are different in the sense that one edge is associated with a rectangle and the other to none (3-ii) or some of the remaining rectangles in the amputated branches must differ due to property (P-iii) since the potentials labeling the vertices at the level just below the common root vertex are different.
Each term in the re-expansion is associated with a branch of the tree, and this correspondence is bijective by construction. Thus, by property (P-iv), two distinct non-zero terms in the re-expansion, corresponding to two different branches , are labeled by two different sets of rectangles, and , respectively.
B. Summing over the norms of branch-operator: Weights and paths,
In order to count the sets , we shall assign a path, , to each , where has the property to visit all the rectangles in the set . Since we must estimate the “weighted” number of sets , the paths must be weighted accordingly.
1. Paths of connected rectangles
The following definitions clarify what we mean by a path visiting rectangles.
A path Γ is a finite sequence of rectangles , for some , with the property that and for every i = 1 … n − 1.
Warning: In contrast to item 7 in Definition 3.1, no relation is assumed here between the ordering labeled by the index i and the ordering ≺.
- The set of ordered pairs,is called the set of steps of the path .
The length, lΓ, of the path is defined to be lΓ ≔ n − 1.
- The support, supp(Γ), of a path is defined to be
A path is closed if .
Each rectangle of the set contributes to weight (3.41) of through [except for that contributes through ], which (as it will be shown) decreases with the size of the rectangle. Thus, we have to make sure that the path does not visit small rectangles of , which have a “big” weight, repeatedly. This motivates the requirements imposed on the paths considered henceforth, in particular property (C) stated in Sec. III B 2.
2. Connected components, , of rectangles and definition of
(i.e., there is only one component for ρ = k0);
rectangles of the same size but belonging to different components do not overlap, i.e., for any ρ, for j ≠ j′.
the support of is ;
for each component consisting of the union of rectangles, at most steps are made [i.e., there are at most steps for which ];
there are at most two steps connecting rectangles in with rectangles of lower size: more precisely, for every connected component , there is at most one Js,u in such that with s′ < s, and at most one such that with s > s′, .
The precise construction is carried out by induction in k in Lemma A.5, combined with Lemma A.4, i.e., we assume that we have constructed a path , with k0+1 ≤ k′ ≤ k, fulfilling (A)–(C) for the set , which is connected by property (P-i). Starting from this path, we construct a new one, denoted by , with the desired properties.
3. Weighted sums of paths
The features specified by (A)–(C), above, are used to distribute the total weight available, as shown in (3.41), among the steps of the path , in a way that is optimal to derive suitable bounds. In fact, we will associate a weight with the steps of the paths described in Sec. III B 2, so as to estimate (3.41) in terms of a weighted sum of paths. The mechanism, which we shall illustrate below, is essentially the one used in Theorem 5.1 to control regime , with some modifications that we omit here in order not to obscure the key ideas, which are related to the proof by induction of Theorem 5.1.
A minor modification of the inequality provided in (3.44) will enable us to prove the result of Theorem 5.1 concerning regime .
IV. THE UNITARY CONJUGATION AND THE SPECTRAL GAP OF
- where “od” means the off-diagonal with respect to the decomposition of identity (4.46);(4.52)
- , and for j ≥ 2,We recall that(4.53)The algorithm described in Definition 2.4 can be motivated by inspecting the proof of the next theorem, which establishes the consistency property alluded to in Sec. II C before introducing Definition 2.4.(4.54)
The Hamiltonian can be written in the form given in (2.19) , where the terms are obtained from the terms according to the algorithm described in Definition 2.4.
- For all rectangles Jl,i such that Jl,i ∩ Jk,q = ∅, we have thatwhere the last identity is due to item (a) in Definition 2.4.(4.56)
- Regarding the terms constituting [see the definition in (2.26)], we note that if we add , we getwhere the first equation results from the Lie–Schwinger procedure and the second one follows from Definition 2.4, items (a) and (b).(4.57)
- For the terms with Jl,i ∩ Jk,q ≠ ∅, but Jk,q ⊈ Jl,i and Jl,i ⊈ Jk,q, we writewhere the first term on the right-hand side is by definition [see item (a) in Definition 2.4] and the second term contributes to the potential , where Jr,j ≡ [Jl,i ∪ Jk,q], along with analogous terms contained in the second sum on the right-hand side of formula (2.31) (where i is replaced by j) and with(4.58)Note that the term in (4.59) corresponds to the first term in (2.31) (where i is replaced by j).(4.59)
Observe that the number of shapes29 of rectangles Jl,i at fixed |l| = l is bounded above by (l + 1)d−1 = O(ld−1). As a consequence, we have the following:
the number of rectangles Jk,q ⊂ Jr,i with fixed circumference k is bounded by (r + 1)d(k + 1)d−1 = (rdkd−1),
the number of rectangles Jk′,q′ ⊂ Jr,i is, then, bounded by , and
the number of rectangles in is bounded by .30
Lemma 4.4 implies that under the assumption in (4.60), the Hamiltonian has a gap that can be estimated from below by for t > 0 sufficiently small but independent of N and (k, q), as stated in the corollary below.
V. CONTROL OF
The next theorem is the key result of this paper and is based on a lengthy analysis of the different regimes (outlined in Sec. II C) to control the potentials yielded, step by step, by the algorithm in Definition 2.4.
There exists td > 0 such that for 0 ≤ t < td, the Hamiltonians and are well defined, and for any rectangle Jr,i, with r = |r| ≥ 1, and for xd ≔ 20d, we have the following:
- Let (k, q)* ≔ (k*, q*) be defined for some (k*, q*) such that , where ⌊·⌋ is the integer part. If (k, q) ≺ (k,q)*, thenLet (k, q)** ≔ (k**, q**) be defined for some (k**, q**) such that . If (k, q)** ≻ (k, q) ≥ (k, q)*, then(5.89)If (r, i) ≻ (k, q) ⪰ (k,q)**, then(5.90)and(5.91)If (k, q) ⪰ (r, i), then(5.92)(5.93)
The proof is by induction in the diagonalization step (k, q). Hence, for each (r, i), we shall prove (S1) and (S2) from (k, q) = (0, N) up to (k, q) = (N − 1, 1) (note that in step (k, q), (S2) concerns the Hamiltonian , and it is not defined for (k, q) = (N − 1, 1)). That is, we assume that (S1) holds for all with (k′, q′) ≺ (k, q) and (S2) for all (k′, q′) ≺ (k, q). Then, we show that they hold for all and for . By Lemma A.3, this implies that and, consequently, that are well defined operators [see (4.54)].
At each stage of our proof, we choose t( ≥0) in an interval such that the previous stages and Lemma A.3 are verified. Hence, by this procedure, we may progressively restrict such an interval until we determine td > 0 for which all the stages hold true for 0 ≤ t < td.
Warning: Throughout the proof, several positive constants are introduced. We shall use the symbols c, C for those that are universal and the symbols cd, Cd for those that depend on the dimension d, and their value may change from line to line.
Induction step in the proof of (S1). Starting from Definition 2.4, we consider the following cases:
Case r = 1.Let k > 1(= r) or k = 1(= r) but Jr,i such that i ≠ q. Then, the possible cases are described in (a); see Definition 2.4, and we have thatLet k = 1, and assume that Jr,i is equal to Jk,q. Then, we refer to case (b) and find that(5.94)where we have following:(5.95)
the inequality holds for t(≥0) sufficiently small uniformly in q and N, thanks to Lemma A.3, which can be applied since we assume (S1) and (S2) in step (k, q)−1;
we use .
Inequality (5.91) follows trivially by using and .
Case r = 2.This case is not much different from the one corresponding to r = 1 with the exception thatalso must be used in the re-expansion, for some (k′, q′) with k′ = 1, and then iterated for the first term of the right-hand side of (5.96) if the conditions of (c) in Definition 2.4 are fulfilled. The second term in (5.96) is a reminder that, however, is produced along the re-expansion only for a finite number of steps, and this number is bounded by a constant independent of (k, q), i, and N. Note also that, for t > 0 sufficiently small, the norm of the last term in (5.96) can be bounded by a constant multiplied by a factor t using Lemma A.3 and inductive hypotheses (S1) and (S2) for r = 1. For t(≥0) sufficiently small, these observations suffice to state (S1) for rectangles with r = 2, provided that (S1) and (S2) hold for r = 1.(5.96)
Case r > 2.
As explained in Sec. II C, in order to control the norm , we distinguish three regimes depending on the relative magnitude between k = |k| and r = |r|. They are associated with (5.89)–(5.93), respectively. For the convenience of the reader, we recall how the inductive hypotheses are used in the following analysis of the three regimes. By assuming that (5.89)–(5.93) are true for the potentials associated with any rectangle Jl′,i′, with (l′, i′) ≤ (r, i), in steps (k′, q′) ≺ (k, q), we prove that, depending on the considered regime, (5.89)–(5.91) hold, respectively, in step (k, q) for the potential associated with Jr,i, but if (5.91) is verified, then, consequently, (5.92) and (5.93) also hold true [in step (k, q)].
A. Regime ()
Following the scheme described in Sec. III B, we estimate the weighted sum of sets in terms of a weighted sum of paths . Different from Sec. III B 3, here, we assign the weight to each step after extracting from (5.98) what is needed to provide the bound in (5.89). The overall control will be ensured by the pre-factor that is small enough due to the upper bound on k, , in regime . Indeed, the latter implies the lower bound .
the term Cd ⋅ r2d−1 is a bound31 on the number of possible initial rectangles of a fixed path ;
the sum over j is the sum over the number of steps of , which by construction is bounded from below by ⌊cd · r/k⌋.
B. Regime ()
(k*, q*) is the greatest rectangle of regime with respect to the ordering ≻, and by construction, ;
- the factoris an upper bound to the sum of the products of the type for some j, where the size of the rectangle associated with (k, q)−j is equal to s;
the multiplicative factor is an upper bound estimate (see Remark 4.2) to the number of rectangles Jk′,q′ ⊂ Jr,i such that [Jk′,q′ ∪ Jk,q] = Jr,i.
C. Regime ()
On the contrary, at each step, we estimate the terms of types (5.122) and (5.123) that are produced by the iteration, without further expanding the potentials analogous to and that are contained in them.
- the leading order term(5.138)where we have used ;(5.139)
- the remainder termIn order to estimate the leading order term (5.138), we make use of the inequality(5.140)(5.141)Now, we introduce the notation(5.142)and(5.143)We can write(5.144)(5.145)(5.146)(5.147)
When l = d, the estimate of (5.162) written above holds with the product replaced by 1.
- the quantityis bounded from above by a d-dependent constant;(5.170)
- for the considered k,since kj = rj for j = l + 1, …, d by assumption.(5.171)
bounds from above the number of rectangles overlapping with the rectangle Js,q′;
is the bound to , provided by the inductive hypotheses.
Higher order terms in (5.123)
We observe that we have the following:
in all these terms, there are either two factors Sk,q or two factors [see (5.137)] or Jk′,q′ is large such that (k′, q′) ≻ (k, q); thus, we get at least an extra factor ;
Complete estimate of (5.91)
We point out that we have the following:
regarding the expression in (5.189) and (5.190), the factor [coming from the inductive hypothesis used to estimate (5.142)] provides the expected behavior since in regime , and the rest can be made less than due to the definition of xd, as we explain below;