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.

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 Zd,d2, 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 iZd, 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 SS(t)=S(t)*,tR, such that the ground state of the operator eSK0+tKIeS 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.

Ideas sharing some similarities with the ones presented in our paper were used in Ref. 24 for purposes analogous to ours and in Refs. 25 and 26 for a partial analysis of many-body localization in one dimension.

We consider a finite, d-dimensional lattice, ΛNdZd, with sides consisting of N vertices, where N < is arbitrary (but fixed). Each vertex in ΛNd is labeled by a multi-index i ≔ (i1, …, id) with ij ∈ (1, …, N), j = 1, …, d. The Hilbert space of pure state vectors of the quantum lattice systems studied in this paper is given by
H(N)iΛNdHiwith HiCM,iΛNd,
where M is an arbitrary, but finite, N-independent integer. Let H be a non-negative M × M matrix with the properties that 0 is an eigenvalue of H corresponding to an eigenvector ΩCM, and
where 1 is the identity matrix.
We define
where 1j is the identity matrix on Hj. By PΩi, we denote the orthogonal projection onto the subspace,
ΛNdjiHj{CΩ}ithslotH(N) andPΩi1PΩi.
We study quantum systems on the lattice ΛNd with Hamiltonians of the form
where we have the following:
  • Jk,qJk1,,kd;q1,,qd denotes the rectangle in ΛNd 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 ΛNdJN1,1, where N1 = (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.,
  • The range of the interaction potentials, namely, the integer k̄< with the property that |k|k̄, rectangles Jk,q appearing in (1.5), is arbitrary, but fixed and N-independent.

  • VJk,q is a symmetric matrix on H(N) with the property that
    VJk,q acts as the identity on jΛNd,jJk,qHjandVJk,q1
    for all k, q, with |k|k̄<, as in (iii) (and VJk,q=0 whenever |k|>k̄). The rectangle Jk,q is called the “support” of VJk,q.
  • tR is a coupling constant independent of N.

Our main result is the following theorem proven in Sec. V (see Theorem 5.3).


Under the assumption that (1.4) and (1.7) hold for an arbitrary, but fixed finite range k̄<, the Hamiltonian KN(t) defined in (1.5) has the following properties.

There exists some td > 0 independent of N such that for any coupling constant tR 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, ΔN(t)12, 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.

We define
which is the orthogonal projection onto the ground-state subspace of the unperturbed operator K0,NKN(t=0)=iΛNdHi. We will construct an anti-symmetric matrix SN(t) = −SN(t)* acting on H(N) (so that exp±SN(t) are unitary matrices) with the property that, after conjugation, the operator
is “block-diagonal” with respect to the pair Pvac,Pvac1Pvac of projections in the sense that Pvac projects onto the ground-state of K̃N(t),
with ΔN(t)12 for |t| < td, uniformly in N.
The Hamiltonian we will study in the following has the special form,
i.e., the range of the interaction potentials is k̄=1. We could study potentials with an arbitrary finite range. However, in order to keep our exposition as transparent as possible, we restrict our attention to the nearest neighbor “hopping terms.” For simplicity, we also assume that the coupling constant is positive, i.e., t > 0.

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. IV, we recall how to provide a lower bound on the spectral gap ΔN(t) for sufficiently small values of the coupling constant t following the same procedure as in Ref. 6.

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.


  1. For chains, i.e., d = 1, the rectangles Jk,q coincide with the connected one-dimensional graphs, Ik,q, kN, 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.

  2. We use the same symbol for the operator Oj acting on Hj and the corresponding operator
    acting on iJk,qHi for any jJk,q. Similarly, with a slight abuse of notation, we do not make a distinction between an operator OJl,i acting on HJl,ijJl,iHj and the corresponding operator acting on the whole Hilbert space H(N), which is obtained out of OJl,i by tensoring by the identity matrix operator on all the remaining sites.
  3. With the symbol “⊂,” we denote strict inclusion; otherwise, we use the symbol “⊆.”

  4. The multiplicative constant implicit in the symbol O() can depend on the spatial dimension d.

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,qJk,q with k = k and 1 ≤ qNk), 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, O(n!), 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.

We recall that the symbol Jk,qJk1,,kd;q1,,qd denotes a rectangle in ΛNd whose sides have lengths k1, k2, …, kd and that |k| denotes the sum of these lengths, i.e., |k|i=1dki. 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 ≤ jd with qjNkj.

The rectangles Jk,q play the role of the intervals Ik,q in the one-dimensional case considered in Ref. 6. Similarly to the one-dimensional case, the pairs (k, q) label the block-diagonalization steps, which are ordered according to the ordering relation  “≻”  defined as follows:27 
  • j=1dkj>j=1dkj;

  • or, if j=1dkj=j=1dkj, kj>kj, for some 1 ≤ jd, with kl=kl, ∀l > j;

  • or, if kl=kl, for all l, qj>qj, for some 1 ≤ jd, with ql=ql, ∀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 (N1, 1), where N1 = (N − 1, …, N − 1) and 1 = (1, …, 1).

Definition 2.1.
Given an arbitrary rectangle Jk,q of sites in ΛNd, we define

Definition 2.2.
Consider two rectangles, Jk,q and Jk′,q, with nonempty intersection. The minimal rectangle associated with Jk,qJk′,q is defined to be the smallest rectangle containing Jk,q and Jk′,q. Note that its corners are the 2d numbers with either
at the jth position. The minimal rectangle associated with Jk,q and Jk′,q is denoted by

Definition 2.3.
Let Jk,qJl,i. We define a family, GJl,i(k,q), of rectangles by

Each conjugation step in the block-diagonalization of the original Hamiltonian is labeled by a rectangle Jk,q and, consequently, by a pair (k, q). In the effective Hamiltonian arising from a conjugation step, a potential term, VJl,i(k,q), is associated with each rectangle Jl,i. More precisely, after the conjugation step (k, q), the effective Hamiltonian reads
where we have the following.
  1. The pairs (k(j),q) are used to index all rectangles Jk′,q with |k′| = j.

  2. 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, VJl,i(k,q) 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 VJl,i(k,q) depends on the coupling constant t, but this is not made explicit in our notation; it acts as the identity on the spaces Hj for jJl,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.

  3. For all rectangles Jl,i with (k, q) ≻ (l, i) and for the rectangle Jl,i = Jk,q, the associated effective potential VJl,i(k,q) is block-diagonal with respect to the decomposition of the identity acting on H(N) into the sum of projections,
    The effective Hamiltonian KΛNd(k,q) of (2.19) is obtained after the conjugation step labeled by (k, q). Starting from
    the conjugation step labeled by (k, q) is given by
    where the anti-symmetric matrix SJk,q is chosen in such a way that the interaction potential VJk,qk,q is block-diagonal; see Sec. IV. More precisely, following the Lie–Schwinger procedure, SJk,q is built so as to block-diagonalize the local operator given by the sum of all terms in KΛNd(k,q)1 whose support is contained in Jk,q. In other words, SJk,q is chosen in such a way that the conjugation in (2.24) renders the operator
    block-diagonal, where
    Here, “block-diagonalization” refers to the projections PJk,q() and PJk,q(+) corresponding to the decomposition of the Hilbert space iJk,qHi into vacuum subspace and its orthogonal complement, respectively. The operator GJk,q plays the role of the “unperturbed” operator since it is already block-diagonal with respect to the decomposition of the identity,
    The construction outlined here works because one can show inductively that the energy gap in the spectrum of the Hamiltonian GJk,q 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 GJk,q have been block-diagonalized already in the previous steps.

These properties of the operator GJk,q, combined with bounds on the norms of the effective potentials obtained at the previous conjugation step, enable us to construct the anti-symmetric matrix SJk,q used at the next conjugation step and control the norms of the effective potentials obtained after conjugation with exp[SJk,q]. This is described in more detail in Sec. II C.

Our strategy to control the norms of the effective potentials VJr,i(k,q) 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 k(i=1dki) containing a specified site of the lattice grows polynomially in k, namely, like O(kd1). 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 O(tck), 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 VJr,i(k,q) in (2.31).

As implicitly indicated in (2.22) and (2.23) for the effective Hamiltonian KΛNd(k,q)1, the potentials must be re-combined properly after each conjugation step (k, q) so as to determine a well-defined flow of operators, VJr,i(k,q), 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 VJl,i(k,q) as defined in Definition 2.4, formulated next.

The algorithm is supposed to enable us to iteratively determine effective potentials VJr,i(k,q) in terms of the potentials obtained at the previous step (k, q)−1, starting from
VJ0,i(0,N)Hi,VJ1j,q(0,N)VJ1j,q,and VJk,i(0,N)=0for|k|2.

Definition 2.4.

Assuming that, at fixed (k, q)−1 with (k, q)−1 ≻ (0, N), for any r, i, the operators VJr,i(k,q)1 and SJk,q [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 SJ11,1 are well defined, then we define the following:

  • if Jk,qJr,i,
  • if Jr,i = Jk,q,
    where (VJr,i(k,q)1)jdiag is defined such as in (4.53) and diag means the diagonal part with respect to the projections PJr,i() and PJr,i(+);
  • if Jk,qJr,i,
    where ad is defined in (4.49) and (4.50). We observe that the set GJr,i(k,q) [see (2.18)] is not empty only if the rectangle Jk,q has a nonempty intersection with the boundary of the rectangle Jr,i.

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.

Remark 2.5.

The sum on the right-hand side of (2.31) accounts for all contributions to the term VJr,i(k,q) with support Jr,i that correspond to “growth processes” of rectangles, i.e., to processes where the union of a rectangle Jk′,qJr,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′,qJk,q] ≡ Jr,i.

To control the operator norms of the effective potentials, we begin by observing that, by construction, the potential VJr,i(k,q) does not change anymore whenever (k, q) ≻ (r, i). Using this observation, we will prove by induction that, for every pair (r, i), an upper bound of the form
holds true at all steps (k, q) up to step (r, i) (included), where Cj and the exponent ρjρj(d) > 0 (d being the space dimension) depend on the regime Rj introduced below, and the different regimes, R1,R2, and R3, depend on the relative magnitude of the circumferences k = |k| and r = |r|.

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, R1,R2, and R3, 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.

  • (R1) 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 kr14. In order to establish the desired estimate (2.32), we iterate the re-expansion of the potential VJr,i(k,q) 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 Jk,q labeling the operators Sk,q, plus one labeling one of the potentials appearing in the Hamiltonian of definition (1.5) or a potential of the type VJk,q(k,q) (where kr14), 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 O(r/r14) rectangles Jk,q are present in each connected set, and all the corresponding operators Sk,q have norms of order |t|Vk,q(k,q)1; apart from the resulting product of norms Vk,q(k,q)1, which is also crucial in the argument, it is important that a total factor |t|O(r/r14) or smaller is gained from the re-expansion (due to the constraint kr14 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 VJk,q(k,q)1, provides an upper bound to VJr,i(k,q). In fact, for |t| small enough but independent of N, this weighted number yields the sought bound (2.32) for VJr,i(k,q).

  • (R2) The second regime is associated with pairs (k, q) with the property that r14krr14. 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 rρ2 in the inductive estimate [see (2.32)] of the potential. If kρ2 and (rk)ρ2 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).

  • (R3) The third regime is associated with “large” rectangles (k, q) since the rr14kr. 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 VJk,q(k,q)1 that are already block-diagonal and, then, estimate them in terms of a sum of projections PJk,q(+) controlled, through an induction, by the denominator appearing in the expression of (SJr,i)1 [see formula (4.52)]; in the proof by induction for this regime, we make use of the auxiliary quantities displayed in (5.91).

FIG. 1.

Examples of configurations of R1,R2,R3, respectively.

FIG. 1.

Examples of configurations of R1,R2,R3, respectively.

Close modal

In order to study regime R1, we shall re-expand the potentials VJr,i(k,q) 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 R1. We, therefore, describe it carefully in this section.

For the purpose of re-expanding VJr,i(k,q) using Definition 2.4, we observe that, for r ≫ 1, case (b) of Definition 2.4 can occur only after many steps of the re-expansion because kr14 in regime R1. In order to streamline our formulas, we introduce the notation
Depending on the relative position between Jk,q and Jr,i, we are instructed to use either
corresponding to cases (a) and (c) in Definition 2.4, respectively. We will use formulas (a) and (c) of Definition 2.4 iteratively for the potentials on the right-hand side of (3.34)–(3.37) when they apply, if it is the case all the way down to step (0, N), but do not re-expand potentials of the type VJk,q(k,q) when they appear (i.e., we stop the re-expansion), which corresponds to case (b) of Definition 2.4.

The strategy can be summarized as consisting of the following steps.

  • Introducing tree diagrams, we show that every contribution, b, to an effective potential—where b 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, Rb, of rectangles that are ordered and whose union is connected.

  • We show that there is an injective map from {Rb} to a set, {Γb}, of paths of rectangles with certain properties.

  • By assigning suitable weights to the paths Γb, we will be able to derive upper bounds on the norms of the contributions b. This will allow us to estimate the norm VJr,i(k,q) by counting (weighted) paths belonging to the set {Γb}.

In order to find an efficient description (see Definition 3.1) of the structure of contributions emerging from the re-expansion of VJr,i(k,q), we study the type of terms we get after a few re-expansion steps. For example, if we assume that the relative positions of Jk,q and Jr,i are such that the first re-expansion step is of type (c), followed by a re-expansion step of type (a), then we get
Note that in (3.33) and, consequently, in (3.39), we interpret the sum over n as a single contribution. The re-expansion of every potential term alluded to above, iterated down either to the first level where case (b) of Definition 2.4 applies or, if this does not happen, to level (0, N), can be described using an upside-down tree structure (see the first three levels in Fig. 2), following the list of prescriptions described in the next definition.
FIG. 2.

Example of a tree associated with the first two steps of the re-expansion of VJr,i(k,q).

FIG. 2.

Example of a tree associated with the first two steps of the re-expansion of VJr,i(k,q).

Close modal

Definition 3.1.

  1. The levels of a tree used to identify the contributions to the re-expansion of a potential VJr,i(k,q) 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).

  2. There is a single vertex at the top of a tree rooted at level (k, q); it is labeled by the symbol VJr,i(k,q) of the potential.

  3. 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 vvVJs,u(k,q) at level (k′, q′), labeled by VJs,u(k,q), 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 VJs,u(k,q)1;

    • the rightmost set of vertices is empty if Jk′,qJs,u; otherwise, it contains a vertex for each element Js′,u belonging to GJs,u(k,q){Js,u}, and this vertex is labeled by VJs,u(k,q)1.

  4. Each vertex v 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 v at level (k′, q′) to its leftmost descendant at level (k′, q′)−1 has no label. It stands for the map
      where VJs,u(k,q) is the potential labeling v and VJs,u(k,q)1 labels its leftmost descendant at level (k′, q′)−1;
    • each edge e connecting the vertex v at level (k′, q′) to other descendants at level (k′, q′)−1 is labeled by a rectangle Jk′,q. It stands for the map
      where VJs,u(k,q) labels the vertex v and VJs,u(k,q)1 is the potential labeling the vertex connected to v by the edge e.

  5. 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 VJk,q(k,q) for some (k″, q″) ≥ (0, N).

  6. 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″).

  7. With each branch b of a tree, we associate a set, Rb, of rectangles consisting of (i) those rectangles labeling the edges of b and (ii) the rectangle Jk,q indicating the support of the potential labeling the leaf of b.

    The set Rb inherits the ordering relation (2.14), and hence, its elements can be enumerated by a map
    with (k(i), q(i)) ≻ (k(i+1), q(i+1)), where |Rb| is the cardinality of the set Rb. Note that Jk(|Rb|),q(|Rb|) is the rectangle associated with the potential labeling the leaf of b.
  8. To every branch b, we can associate the “branch operator,” also denoted by b,
    where VLb is the potential labeling the leaf of b; VLb can be either VJk(|Rb|),q(|Rb|)(k(|Rb|),q(|Rb|)) or VJs,u(0,N).

The set of branches whose corresponding branch operators are non-zero is denoted by BVJr,i(k,q).

Remark 3.2.

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 VJr,i(0,N) with r > 1, which is zero by definition.

1. Properties of the branches bBVJr,i(k,q)

Definition 3.1 implies the following properties of the elements of the set BVJr,i(k,q) defined above:

  • For bBVJr,i(k,q), the set
    is connected due to (3.40), although Jk(i),q(i)Jk(i+1),q(i+1) might be empty for some i. Likewise, for any fixed n1|Rb|, the set ni|Rb|Jk(i),q(i) is connected. Indeed, for any operator O and for any m, AJk(m),q(m)(O)=0 whenever the supports of O and SJk(m),q(m) have empty intersection; see formula (3.33).
  • For bBVJr,i(k,q), the cardinality, |Rb|, of the set Rb of rectangles is such that |Rb|O(rk)O(r34). This lower bound on |Rb| is a consequence of the restriction imposed on k = |k| and required in regime R1 [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 i1,,|Rb|Jk(i),q(i) for any branch bBVJr,i(k,q). 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 b,bBVJr,i(k,q) are associated with two different (ordered) sets of rectangles Rb and Rb.

    Sketch of proof:

    1. The two branches must cross at some vertex.

    2. 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.

    3. 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 b 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 b1,b2BVJr,i(k,q), are labeled by two different sets of rectangles, Rb1 and Rb2, respectively.

Our task is to estimate the norms of the potentials VJr,i(k,q), which can be accomplished by taking the re-expansion of the potentials into account according to the prescriptions of Definition 3.1. More precisely, each potential VJr,i(k,q) can be expressed as the sum bBVJr,i(k,q)b, where b are the branch operators defined in point 8 of Definition 3.1. Therefore, we are led to estimating the sum over the norms of branch operators to wit
This can be done by assigning a “weight” to every set Rb of rectangles, the weight being proportional to the product of operator norms of the potentials associated [in step (k,q)−1] with each rectangle Jk,q in the set Rb, i.e.,
where VLb is the potential labeling the leaf of b since a factor (ct)VJk(i),q(i)(k(i),q(i))1 is associated with the map AJk(i),q(i); here, c > 0 is a universal constant.

In order to count the sets Rb, we shall assign a path, Γb, to each b, where Γb has the property to visit all the rectangles in the set Rb. Since we must estimate the “weighted” number of sets Rb, the paths must be weighted accordingly.

1. Paths of connected rectangles

The following definitions clarify what we mean by a path visiting rectangles.

Definition 3.3.

  • A path Γ is a finite sequence of rectangles {Js(i),u(i)}i=1n, for some nN, with the property that Js(i),u(i)Js(i+1),u(i+1) and Js(i),u(i)Js(i+1),u(i+1) 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 ΓJs(i),u(i)i=1n.
  • The length, lΓ, of the path Γ{Js(i),u(i)}i=1n is defined to be lΓn − 1.

  • The support, supp(Γ), of a path Γ{Js(i),u(i)}i=1n is defined to be
  • A path Γ{Js(i),u(i)}i=1n is closed if Js(1),u(1)=Js(n),u(n).

Each rectangle Jk(i),q(i) of the set Rb contributes to weight (3.41) of Rb through ctVJk(i),q(i)(k(i),q(i))1 [except for Jk(|Rb|),q(|Rb|) that contributes through cVLb], which (as it will be shown) decreases with the size of the rectangle. Thus, we have to make sure that the path Γb does not visit small rectangles of Rb, which have a “big” weight, repeatedly. This motivates the requirements imposed on the paths Γb considered henceforth, in particular property (C) stated in Sec. III B 2.

2. Connected components, Zρ(j), of rectangles and definition of Γb

Since the weight of a rectangle is a function of its size, it is convenient to write the connected set i{1,,|Rb|}Jk(i),q(i) as the union
where {Zρ(j),j=1,,jρ} are distinct connected components of (unions of) rectangles of a given size ρ, k0ρk, starting from the lowest one k0 ≥ 1, with the following properties:
  1. jk0=1 (i.e., there is only one component for ρ = k0);

  2. rectangles of the same size but belonging to different components do not overlap, i.e., for any ρ, Zρ(j)Zρ(j)= for jj′.

We call supp(Zρ(j)), ρ = k0, …, k, j = 1, …, jρ, the set of rectangles of Zρ(j), i.e.,
Starting from a branch bBVJr,i(k,q), we shall inductively construct a path, Γb, of length lΓb bounded by
where nρ(j)|supp(Zρ(j))|, with the following properties:
  • the support of Γb is Rb;

  • for each component Zρ(j) consisting of the union of nρ(j) rectangles, at most 2nρ(j)2 steps are made [i.e., there are at most 2nρ(j)2 steps σSΓb for which σsupp(Zρ(j))×supp(Zρ(j))];

  • there are at most two steps connecting rectangles in supp(Zρ(j)) with rectangles of lower size: more precisely, for every connected component Zρ(j), there is at most one Js,u in supp(Zρ(j)) such that (Js,u,Js,u)SΓb with s′ < s, Js,uRb and at most one Js,usupp(Zρ(j)) such that (Js,u,Js,u)SΓb with s > s′, Js,uRb.

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 Γb(k1), with k0+1 ≤ k′ ≤ k, fulfilling (A)–(C) for the set ρ=k0k1j=1jρZρ(j), which is connected by property (P-i). Starting from this path, we construct a new one, denoted by Γb(k), 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 Γb, in a way that is optimal to derive suitable bounds. In fact, we will associate a weight with the steps of the paths Γb 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 R1, 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.

We observe that there are nρ(j) rectangles in the set supp(Zρ(j)) and that, for the paths Γb, there are at most 2nρ(j)2 steps between these rectangles; see property (B). In addition, there are at most two steps, from rectangles of lower size and back, to be taken into account; see property (C). Consequently, to each step σ=(Js(i),u(i),Js(i+1),u(i+1))SΓb, we can assign the weight
where t is sufficiently small such that (c+1)tVJk(i),q(i)(k(i),q(i))1<1, and the following estimate holds:
The previous inequality is true because, if we denote by SZρ(j) the set of at most 2nρ(j)2 steps between rectangles of suppZρ(j) and the additional at most two steps from rectangles of lower size and back, then we have
Finally, we use the estimate
where cdrk is a lower bound for |Rb| and Cd · r2d−1 is an upper bound on the possible positions of the rectangle Jk(|Rb|),q(|Rb|) of the path, where cd, Cd are d-dependent constants; finally,
where Cd is a d-dependent constant, is an upper bound on the number of possible directions of a path Γ={Js(i),u(i)}i=1n, extended by one more step as specified here: given the path Γ={Js(i),u(i)}i=1n, the number of paths Γ+={Js(i),u(i)}i=1n+1 of length lΓ+=n, whose first n elements agree with Γ (i.e., {Js(i),u(i)}i=1n={Js(i),u(i)}i=1n) and for which s(n+1)s′ and s(n)s, is bounded from above by Ds,s.

A minor modification of the inequality provided in (3.44) will enable us to prove the result of Theorem 5.1 concerning regime R1.

The operator eSJk,q is constructed so as to block-diagonalize the Hamiltonian GJk,q+tVJk,q(k,q)1 with respect to the decomposition of the identity
The operator
is already block-diagonal with respect to (4.46).
For this construction, we refer the reader to the notation and the results in Secs. 2 and 3 of Ref. 28. We add the definition of EJk,q, which is, in fact, the ground-state energy of the operator GJk,q,
We recall that
where A and B are bounded operators, and for n ≥ 2,
To carry out the block-diagonalization step (k, q), the operator SJk,q is defined by the series
  • (SJk,q)jad1GJk,q((VJk,q(k,q)1)jod)1GJk,qEJk,qPJk,q(+)(VJk,q(k,q)1)jPJk,q()h.c.,
    where “od” means the off-diagonal with respect to the decomposition of identity (4.46);
  • (VJk,q(k,q)1)1VJk,q(k,q)1, and for j ≥ 2,
    We recall that
    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.

Theorem 4.1.

The Hamiltonian K Λ N d ( k , q ) e S J k , q K Λ N d ( k , q ) 1 e S J k , q can be written in the form given in (2.19) , where the terms { V J l , i ( k , q ) } are obtained from the terms { V J l , i ( k , q ) 1 } according to the algorithm described in Definition 2.4.

In the expression
we observe that we have the following:
  • For all rectangles Jl,i such that Jl,iJk,q = ∅, we have that
    where the last identity is due to item (a) in Definition 2.4.
  • Regarding the terms constituting GJk,q [see the definition in (2.26)], we note that if we add VJk,q(k,q)1, we get
    where the first equation results from the Lie–Schwinger procedure and the second one follows from Definition 2.4, items (a) and (b).
  • For the terms VJl,i(k,q)1 with Jl,iJk,q ≠ ∅, but Jk,qJl,i and Jl,iJk,q, we write
    where the first term on the right-hand side is VJl,i(k,q) by definition [see item (a) in Definition 2.4] and the second term contributes to the potential VJr,j(k,q), where Jr,j ≡ [Jl,iJk,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
    Note that the term in (4.59) corresponds to the first term in (2.31) (where i is replaced by j).

In the remainder of this section, we reproduce a key result, established in Ref. 6, which enables us to estimate the spectral gap above the ground-state energy of the Hamiltonian GJk,q. The proof is included for the convenience of the reader, but the arguments are essentially identical to those used in Ref. 6. As for chains (d = 1; see Ref. 6), it is not difficult to prove that under the assumption that
the Hamiltonian GJk,q has a gap ΔJk,q12 for all t ∈ [0, td), where td depends on the lattice dimension but is independent of (k, q) and N. The main ingredients for the proof can be found in Lemma A.1 and Corollary A.2, namely,

Remark 4.2.

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,qJr,i with fixed circumference k is bounded by (r + 1)d(k + 1)d−1 = O(rdkd−1),

  • the number of rectangles Jk′,qJr,i is, then, bounded by (r+1)dk=1r(k+1)d1=O(r2d), and

  • the number of rectangles in GJr,i(k,q) is bounded by 2d(r+1)d1k=1r(k+1)d1=O(r2d1).30 

Remark 4.3.
Our block-diagonalization procedure relies on the following crucial property: If VJl,i(k,q) is block-diagonal with respect to the decomposition of the identity into
i.e., if
then we have that
for Jl′,i with Jl,iJl′,i. This is seen by using
in the first term and
combined with
in the second term.

Lemma 4.4.
Assuming (4.60), the following bound on the operator GJk,q holds:
for t ∈ [0, td), with td independent of (k, q) and N, where Cd is the d-dependent constant implicit in the estimate of the number of shapes in Remark 4.2.

We observe that, due to Remark 4.3, for 1 ≤ j ≤ |k| − 1, we can write
Furthermore, we can estimate
where we have used the following:
  1. the bound in (4.60) for the step from (4.70) to (4.71) and

  2. the property in (4.62) combined with Remark 4.2 for the step from (4.71) to (4.72).

Hence, we can combine the inequality [due to (1.4)]
with (4.68)–(4.72), and we get
Next, we use the identity
in the right-hand side of (4.75), and we get
By invoking the obvious bound
we finally get
where Lemma A.1 is used for the last inequality and t(>0) is assumed small enough such that

Lemma 4.4 implies that under the assumption in (4.60), the Hamiltonian GJk,q has a gap that can be estimated from below by 12 for t > 0 sufficiently small but independent of N and (k, q), as stated in the corollary below.

Corollary 4.5.
Assuming Lemma 4.4 for t > 0 sufficiently small, dependent on d but independent of N and (k, q), the Hamiltonian GJk,q has a gap ΔJk,q12 above the ground state energy
corresponding to the ground state vector iJk,qΩi due to the identity

Remark 4.6.
Estimates (4.73) and (4.69)–(4.72) show that, after implementing our block-diagonalization procedure and subtracting the ground-state energy, the interaction terms of the transformed Hamiltonian are form-bounded by the unperturbed Hamiltonian uniformly in the size of the region Jk,q, provided that the coupling constant t > 0 is small enough. Spectral calculus, then, implies that, for z outside the spectrum of the unperturbed Hamiltonian, iJk,qHi, i.e., for δdist(z;σ(iJk,qHi))>0, the resolvent
is well defined on PJk,q(+)H(N), provided that 0 < t < tδ,z, where the constant tδ,z only depends on δ and z but is independent of Jk,q.

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.

Theorem 5.1.

There exists td > 0 such that for 0 ≤ t < td, the Hamiltonians GJk,q and KNd(k,q) 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 |k*|=r14, where ⌊·⌋ is the integer part. If (k, q) ≺ (k,q)*, then
    Let (k, q)** ≔ (k**, q**) be defined for some (k**, q**) such that |k**|=rr14. If (k, q)** ≻ (k, q) ≥ (k, q)*, then
    If (r, i) ≻ (k, q) ⪰ (k,q)**, then
    If (k, q) ⪰ (r, i), then
  • GJ(k,q)+1 has spectral gap ΔJ(k,q)+112 above its ground state energy, where GJk,q is defined in (2.26) for |k| ≥ 2, and
    provided that (1j,q)+1 is of the form (1j, q′) for some jand q; (1j, q) is defined in (1.13).


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) = (N1, 1) (note that in step (k, q), (S2) concerns the Hamiltonian GJ(k,q)+1, and it is not defined for (k, q) = (N1, 1)). That is, we assume that (S1) holds for all VJr,i(k,q) with (k′, q′) ≺ (k, q) and (S2) for all (k′, q′) ≺ (k, q). Then, we show that they hold for all VJr,i(k,q) and for GJ(k,q)+1. By Lemma A.3, this implies that SJk,q and, consequently, that KΛNd(k,q) are well defined operators [see (4.54)].

For (k, q) = (0, N), (S1) can be verified by direct computation because
and VJr,i(0,N)=0 otherwise; (S2) holds trivially since, by definition, (0, N)+1 = (11, 1) and GJ11,1=HJ11,1(0) [recall that 1j is defined in (1.13)].

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:

  1. Case r = 1.

    Let k > 1(= r) or k = 1(= r) but Jr,i such that iq. Then, the possible cases are described in (a); see Definition 2.4, and we have that
    Let k = 1, and assume that Jr,i is equal to Jk,q. Then, we refer to case (b) and find that
    where we have following:

    • the inequality VJk,q(k,q)2VJk,q(k,q)1 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 VJk,q(k,q)1=VJk,q(k,q)2==VJk,q(0,N)1.

    Inequality (5.91) follows trivially by using 1jJr,iPΩj+11 and PJr,i(#)VJr,i(k,q)PJr,i(#̂)VJr,i(k,q).

    Case r = 2.

    This case is not much different from the one corresponding to r = 1 with the exception that
    also 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.

    Case r > 2.

    As explained in Sec. II C, in order to control the norm VJr,i(k,q), 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)].

Here, we apply the argument explained in Sec. III B in order to show that (S1) holds for VJr,i(k,q) with (k, q) belonging to the first regime, provided that (S1) and (S2) hold for all potentials in step (k′, q′) ≺ (k, q). Given the assumption, we can exploit (A7) in Lemma A.3 so as to conclude that for any (bounded) operator V,
if (k″, q″) ⪯ (k, q), where c is a universal constant. We recall that, as explained in Sec. III, the strategy is to re-expand the potential VJr,i(k,q) according to the prescriptions of Definition 3.1. Consequently, the potential can be expressed as the sum bBVJr,i(k,q)b, where b are the branch operators defined in point 8 of Definition 3.1. Due to property (P-v) in Sec. III A 1, the number of summands coincides with the number of sets Rb that are associated with VJr,i(k,q). Furthermore, in order to estimate the norm of the sum of the operators resulting from the re-expansion, it is enough to use (5.97) repeatedly, i.e.,
where VLb is the potential labeling the leaf of b, and compute the “weighted” number of sets {Jk(i),q(i),i{1,,|Rb|}}, weighted in the sense that each rectangle Jk(i),q(i) is given the weight ctVJk(i),q(i)(k(i),q(i))1 except for the one associated with the leaf of the branch, that is, given the weight VLb.

Following the scheme described in Sec. III B, we estimate the weighted sum of sets Rb in terms of a weighted sum of paths Γb. 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 (ct)|Rb|1 that is small enough due to the upper bound on k, kr14, in regime R1. Indeed, the latter implies the lower bound |Rb|cdr/k.

In detail, concerning the powers of t, note that from the product
we get at least tr13 due to (1) the requirement that Jr,i is the minimal rectangle associated with i{1,,|Rb|}Jk(i),q(i) and (2) borrowing a power t23 from each factor t in (ct)|Rb|1. Hence, in product (5.99), we can factor out tr13 and keep a power t13 for each rectangle of Rb except the one associated with the leaf of the branch. This also means that we can assign at least a factor
to each rectangle of size ρ in Rb.
Consider the rectangles of the set supp(Zρ(j)) (see Sec. III B 2): there are nρ(j) such rectangles, and for the constructed paths Γb, there are at most 2nρ(j)2 steps between them. In addition, there are at most two steps, from rectangles of lower size and back, to be taken into account. To each step SΓbσ=(Js(i),u(i),Js(i+1),u(i+1)) we assign the weight
with sσ ≔ max{s(i), s(i+1)}, with wσ < 1 for t > 0 sufficiently small.
From the considerations regarding (5.99) and (5.100), we get the first inequality in the following formula:
whereas for the second inequality, we use the following observation: if we denote by SZρ(j) the set consisting of at most 2nρ(j)2 steps between rectangles of suppZρ(j) and the additional at most two steps from rectangles of lower size and back, then we have
since wσ, σSZρ(j), coincides with (c+1)t16ρxd12<1 and |SZρ(j)|2nρ(j), by construction.
Hence, the total weighted number of rectangles
can be bounded from above by estimating the number of weighted paths Γb as follows:
where we have the following:
  • ρ=1k(c+1)t1/6(max{ρ,ρ})xd1/2Dρ,ρ
    accounts for all the weighted directions for a step from a rectangle of size ρ, where Dρ,ρ is defined in (3.45); note that the weight for the number of directions is due to the restriction of the class of paths used in the argument that culminates in (5.101);
  • the term Cdr2d−1 is a bound31 on the number of possible initial rectangles of a fixed path Γb;

  • the sum over j is the sum over the number of steps of Γb, which by construction is bounded from below by ⌊cd · r/k⌋.

Next, we bound
where t ≥ 0 has been chosen small enough such that (recall kr14)
For (k, q) in this regime, starting from the inequality
we only keep expanding the first potential on the right-hand side. Then, using inductive hypotheses (5.89), (5.90), (5.92), and (5.93), for t ≥ 0 sufficiently small, we can estimate
where we have the following:
  • (k*, q*) is the greatest rectangle of regime R1 with respect to the ordering ≻, and by construction, k*=r14;

  • the factor
    is an upper bound to the sum of the products of the type AJk,qj(VJk,q(k,q)j1) for some j, where the size of the rectangle associated with (k, q)j is equal to s;
  • the multiplicative factor O(r2d1) is an upper bound estimate (see Remark 4.2) to the number of rectangles Jk′,qJr,i such that [Jk′,qJk,q] = Jr,i.

Now, for any s with r14srr14, we have
since rr14r2. However, then, using the inductive hypothesis for VJr,i(k,q)*,
since xd = 20d and t ≥ 0 is small enough.

Proof of (5.91).
For (k, q)** ≺ (k, q) ≺ (r, i), we first consider
We recall that for (k, q) ≺ (r, i), the two types of re-expansion that have to be considered correspond to (a) and (c) in Definition 2.4. Note that the re-expansion of type (a) is trivial since it does not change the potential. Using the re-expansion of type (c), which is associated with formulas (3.34)–(3.36), we get
We shall keep re-expanding the terms analogous to VJr,i(k,q)1 in (5.121) from (k, q)−1 down to (k**, q**). The pair (k**, q**) represents the greatest rectangle with respect to the ordering ≻ in regime R2 and by construction has k**=rr14.

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 VJr,i(k,q)1 and VJk,q(k,q)1 that are contained in them.

Estimate of (5.122).
Concerning (5.122), we observe that using inductive hypotheses (5.90)–(5.93) along with Lemma (A.3), we can bound
At fixed k, the number of contributions of type (5.122) can be estimated from above by O(rdkd1); see Remark 4.2. Being krr14 in regime R3, the power tk13 will be used to control the number of these types of contributions produced along the way down to (k**, q**).

Estimate of (5.123).
It is convenient to split the corresponding term, (5.123), into
In (5.126), we distinguish Jk′,q small and large depending on whether (k′, q′) ≺ (k, q) or (k, q) ⪯ (k′, q′), respectively, and denote by (GJr,i(k,q))small the subset formed by the small Jk′,q belonging to the set GJr,i(k,q). We call
respectively, the corresponding contributions to (5.126). Next, we study some commutators that enter the expression (5.126)small estimated below. We observe that
where we have exploited that VJk,q(k,q)1 is block-diagonalized since, by definition, small means (k′, q′) ≺ (k, q). We also observe that PJk,q(+)PJr,i()=0 since Jk′,qJr,i by construction; hence,
We recall that for j ≥ 1,
and, from Lemma A.3, we get
for t ≥ 0 sufficiently small. Hence, we split (5.126)small into two contributions:
  1. the leading order term
    where we have used PJk,q(+)PJr,i()=0;
  2. the remainder term
    In order to estimate the leading order term (5.138), we make use of the inequality
    Now, we introduce the notation
    We can write

Leading terms in (5.123) : Contribution proportional to (5.146)

We observe that for Jk,qJk,q=, we have
On the contrary, we note that
even if Jk,qJk,q. Indeed,
Hence, we can estimate
where in the step from (5.159) and (5.160) we have used [PJk,q(+),PJk,q(+)]=0.
Now, suppose that there are 1 ≤ ld components of k different from the corresponding ones in r. Without loss of generality, we can assume that they are the first l components; for ld − 1, we get
=s:u with Js,uGJr,i(k,q)u:Js,uGJr,i(k,q)2VJs,u(k,q)1PJs,u(+)jJr,iPΩj+1
(we call s1, …, sd the components of k′) where in the step from (5.163) and (5.164) we use the following:
  • an upper bound for VJs,u(k,q)1 that is independent of u by means of inductive hypothesis (5.93),
  • the fact that, for fixed k′, if kjrj for j = 1, …, l, then q1,,ql are uniquely32 determined by the condition [Jk′,qJk,q] = Jr,i;

  • the estimate
    that can be proved following the same reasoning of Corollary A.2.

When l = d, the estimate of (5.162) written above holds with the product j=l+1d(sj+1) replaced by 1.

Next, for j = 1, …, l, we set
and we observe that since sjrjkj for j = 1, …, l and sj ≥ 0 for j = l + 1, …, d, we have
Hence, we can estimate
where we have exploited the following:
  • the quantity
    is bounded from above by a d-dependent constant;
  • for the considered k,
    since kj = rj for j = l + 1, …, d by assumption.

Leading terms in (5.123) : Contribution proportional to (5.147)

By the Schwarz inequality and the trivial bound aba22+b22, we estimate [recall the notation ∑′ in (5.144)]
Since the expression in (5.173) is symmetric under the permutation of Jk′,q with Jk,q, we can write
With steps similar to (5.163)–(5.169), assume that there are 1 ≤ ld components of k different from the corresponding ones in r (without loss of generality, we identify them with the first l components). Then, we can bound (5.175) as [warning: for l = d, j=l+1d(sj+1),j=l+1dsj must be replaced by 1 in (5.176) and related formulas]
due to the estimate
where we have the following:

  • O(j=1dsjwd1) bounds from above the number of rectangles Jw,q overlapping with the rectangle Js,q;

  • O(tw13wxd) is the bound to VJw,q(k,q)1, provided by the inductive hypotheses.

Next, using the definition in (5.166) and arguments as in (5.168) and (5.169), we write
Now, we multiply the right-hand side of (5.177) by
and we get
where, in the step from (5.179) and (5.180), we have used that xdd + 1 and that all the following quantities are bounded from above by a d-dependent constant:
  • (r1k1++rlkl)xddw=rkrwd11wxd
  • sl+1=0rsd=0rj=l+1dsjtj=l+1dsj/3j=l+1d(sj+1)
  • ρ1=0ρl=0tj=1lρj3j=1lρj+rjkj(r1k1++rlkl).

Leading terms in (5.123) : Contribution proportional to (5.142)

Finally, we estimate (5.142) by exploiting the inequality
where the first factor can be estimated to be less than 3, provided that td is small enough, by using the bound in (4.67) (see Lemma 4.4) that holds due to (S2) in the previous step; for the second factor, we invoke the inductive hypothesis in (5.91).
Hence, we conclude that at fixed k and with l components different from the corresponding components of r,

Higher order terms in (5.123)

In order to show the bound in (5.91), with regard to (5.123), we have still to estimate the following:

  • remainder (5.140) [coming from the study of (5.126)small] and those corresponding to (5.126)large, i.e., proportional to terms with Jk′,q such that (k,q)k,q;

  • the contribution due to (5.127).

We observe that we have the following:

  • in all these terms, there are either two factors Sk,q or two factors (VJk,q(k,q)1)1 [see (5.137)] or Jk′,q is large such that (k′, q′) ≻ (k, q); thus, we get at least an extra factor O(tr2r1/413);

  • the bound from above, O(r2d1), of the number of the elements of GJr,i(k,q) (see Remark 4.2). Hence, just using inductive hypotheses (5.92) and (5.93), we can estimate
    At fixed k, there are at most O(rdkd1) contributions of type (5.185).

Complete estimate of (5.91)

Finally, by the re-expansion outlined above and due to the estimates of (5.122), (5.138), (5.140), (5.126)large, and (5.127) that have been derived [see (5.124), (5.184), and (5.185)], we can conclude that
where Θ is the characteristic function of R+; indeed, krr14 in regime R3. In addition to summand (5.188) that is smaller than 2tr13rxd+2d by the inductive hypothesis (5.90), on the right-hand side of the estimate above, we have three summands that we shall discuss in detail. Prior to this discussion, we explain why the final estimate in (5.205) works.

Remark 5.2.

We point out that we have the following:

  • regarding the expression in (5.189) and (5.190), the factor 1kxd+2d [coming from the inductive hypothesis used to estimate (5.142)] provides the expected behavior since krr14 in regime R3, and the rest can be made less than tr133 due to the definition of xd, as we explain below;

  • regarding the expressions in (5.191) and (5.192), we exploit the extra powers tk13 and tr2r1413, respectively, in order to control the sum over k and provide the desired behavior.

As for (5.189) and (5.190), we first observe that we have