Local iterative block-diagonalization of gapped
                    Hamiltonians: A new tool in singular perturbation theory

Del Vecchio, Simone; Fröhlich, Jürg; Pizzo, Alessandro; Rossi, Stefano

doi:10.1063/5.0084552

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 $Z^{d}, d \geq 2$ ⁠, with Hamiltonians that are given as the sum of an unperturbed operator, K₀, and a perturbation, K_I, consisting of a sum of bounded interaction potentials. The operator K₀ can be written as a sum of terms, H_i, only depending on the degrees of freedom located at single sites $i \in Z^{d}$ ⁠, while the interaction potentials contributing to K_I 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 $S \equiv S (t) = - S {(t)}^{*}, t \in R$ ⁠, such that the ground state of the operator $e^{S} (K_{0} + t \cdot K_{I}) e^{- S}$ 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.

A. A family of quantum lattice systems

We consider a finite, d-dimensional lattice,

Λ_{N}^{d} \subset Z^{d}

⁠, with sides consisting of N vertices, where N < ∞ is arbitrary (but fixed). Each vertex in

Λ_{N}^{d}

is labeled by a multi-index i ≔ (i₁, …, i_d) with i_j ∈ (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)} ≔ ⨂_{i \in Λ_{N}^{d}} H_{i} with H_{i} ≃ C^{M}, \forall i \in Λ_{N}^{d},

(1.1)

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

Ω \in C^{M}

⁠, and

H ↾_{{C Ω}^{⊥}} \geq 1,

where

1

is the identity matrix.

We define

H_{i} ≔ (⨂_{Λ_{N}^{d} ∋ j \neq i} 1_{j}) ⨂ \underset{\underset{i^{t h} slot}{↑}}{H},

(1.2)

where

1_{j}

is the identity matrix on

H_{j}

⁠. By

P_{Ω_{i}}

⁠, we denote the orthogonal projection onto the subspace,

(⨂_{Λ_{N}^{d} ∋ j \neq i} H_{j}) ⨂ \underset{\underset{i^{t h} slot}{↑}}{{C Ω}} \subset H^{(N)} and P_{Ω_{i}}^{⊥} ≔ 1 - P_{Ω_{i}} .

(1.3)

Then,

H_{i} = P_{Ω_{i}}^{⊥} H_{i} P_{Ω_{i}}^{⊥} + P_{Ω_{i}} H_{i} P_{Ω_{i}}

with

P_{Ω_{i}} H_{i} P_{Ω_{i}} = 0 and P_{Ω_{i}}^{⊥} H_{i} P_{Ω_{i}}^{⊥} \geq P_{Ω_{i}}^{⊥} .

(1.4)

We study quantum systems on the lattice

Λ_{N}^{d}

with Hamiltonians of the form

K_{N} \equiv K_{N} (t) ≔ \underset{K_{0}}{\underset{︸}{\sum_{i \in Λ_{N}^{d}} H_{i}}} + t \cdot \underset{K_{I}}{\underset{︸}{\sum_{J_{k, q} \subset Λ_{N}^{d}, k \leq \bar{k}} V_{J_{k, q}}}},

(1.5)

where we have the following:

$J_{k, q} \equiv J_{k_{1}, \dots, k_{d}; q_{1}, \dots, q_{d}}$ denotes the rectangle in $Λ_{N}^{d}$ with sides of lengths k₁, k₂, …, k_d, respectively, whose 2^d corners are the sites given by (q₁ + ɛ₁k₁, …, q_d + ɛ_dk_d), ɛ_j = 0 or 1, for j = 1, …, d. [Note that $Λ_{N}^{d} \equiv J_{N - 1, 1}$ ⁠, where N − 1 = (N − 1, …, N − 1) and 1 = (1, …, 1)].
k ≡ |k| denotes the circumference (= sum of the side lengths) of a rectangle J_k,q, i.e.,
$k \equiv | k | ≔ \sum_{i = 1}^{d} k_{i} .$
(1.6)
The range of the interaction potentials, namely, the integer $\bar{k} < \infty$ with the property that $| k | \leq \bar{k}, \forall$ rectangles J_k,q appearing in (1.5), is arbitrary, but fixed and N-independent.
$V_{J_{k, q}}$ is a symmetric matrix on $H^{(N)}$ with the property that
$V_{J_{k, q}} acts as the identity on ⨂_{j \in Λ_{N}^{d}, j \notin J_{k, q}} H_{j} and ‖ V_{J_{k, q}} ‖ \leq 1$
(1.7)
for all k, q, with $| k | \leq \bar{k} < \infty$ ⁠, as in (iii) (and $V_{J_{k, q}} = 0$ whenever $| k | > \bar{k}$ ⁠). The rectangle J_k,q is called the “support” of $V_{J_{k, q}}$ ⁠.
$t \in R$ 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).

Theorem.

Under the assumption that (1.4) and (1.7) hold for an arbitrary, but fixed finite range $\bar{k} < \infty$ , the Hamiltonian K_N(t) defined in (1.5) has the following properties.

There exists some t_d > 0 independent of N such that for any coupling constant $t \in R$ with |t| < t_d and for all N < ∞, we have the following.

K_N(t) has a unique ground-state and
the energy spectrum of K_N(t) has a strictly positive gap, $Δ_{N} (t) \geq \frac{1}{2}$ , 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

P_{vac} ≔ ⨂_{i \in Λ_{N}^{d}} P_{Ω_{i}},

(1.8)

which is the orthogonal projection onto the ground-state subspace of the unperturbed operator

K_{0, N} \equiv K_{N} (t = 0) = \sum_{i \in Λ_{N}^{d}} H_{i}

⁠. We will construct an anti-symmetric matrix S_N(t) = −S_N(t)^* acting on

H^{(N)}

(so that exp

[\pm S_{N} (t)]

are unitary matrices) with the property that, after conjugation, the operator

e^{S_{N} (t)} K_{N} (t) e^{- S_{N} (t)} = : {\tilde{K}}_{N} (t)

(1.9)

is “block-diagonal” with respect to the pair

(P_{vac}, P_{vac}^{⊥} ≔ 1 - P_{vac})

of projections in the sense that P_vac projects onto the ground-state of

{\tilde{K}}_{N} (t)

⁠,

{\tilde{K}}_{N} (t) = P_{vac} {\tilde{K}}_{N} (t) P_{vac} + P_{vac}^{⊥} {\tilde{K}}_{N} (t) P_{vac}^{⊥}

(1.10)

and

infspec (P_{vac}^{⊥} {\tilde{K}}_{N} (t) P_{vac}^{⊥} ↾_{P_{vac}^{⊥} H^{(N)}}) \geq infspec (P_{vac} {\tilde{K}}_{N} (t) P_{vac} ↾_{P_{vac} H^{(N)}}) + Δ_{N} (t)

(1.11)

with

Δ_{N} (t) \geq \frac{1}{2}

for |t| < t_d, uniformly in N.

The Hamiltonian we will study in the following has the special form,

K_{N} (t) ≔ \sum_{i \in Λ_{N}^{(d)}} H_{i} + t \sum_{j = 1}^{d} \sum_{q_{1} = 1}^{N} \dots \sum_{q_{j} = 1}^{N - 1} \dots \sum_{q_{d} = 1}^{N} V_{J_{1_{j}, q}},

(1.12)

where

(1_{j}, q) ≔ (0, \dots, k_{j} = 1, \dots, 0; q_{1}, \dots, q_{d}),

(1.13)

i.e., the range of the interaction potentials is

\bar{k} = 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 Δ_N(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 S_N(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.

Notation.

For chains, i.e., d = 1, the rectangles J_k,q coincide with the connected one-dimensional graphs, I_k,q, $k \in N$ ⁠, 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 O_j acting on $H_{j}$ and the corresponding operator
$O_{j} ⨂ 1_{J_{k, q} \ {j}}$
acting on $⨂_{i \in J_{k, q}} H_{i}$ for any j ∈ J_k,q. Similarly, with a slight abuse of notation, we do not make a distinction between an operator $O_{J_{l, i}}$ acting on $H_{J_{l, i}} ≔ ⨂_{j \in J_{l, i}} H_{j}$ and the corresponding operator acting on the whole Hilbert space $H^{(N)}$ ⁠, which is obtained out of $O_{J_{l, i}}$ 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 $O (\cdot)$ 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 J_k,i labeling interaction terms in formula (1.5). Whereas for chains, d = 1, when starting from a family of intervals (i.e., I_k,q ≡ J_k,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, $O (n!)$ ⁠, 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.

A. Minimal rectangles

We recall that the symbol $J_{k, q} \equiv J_{k_{1}, \dots, k_{d}; q_{1}, \dots, q_{d}}$ denotes a rectangle in $Λ_{N}^{d}$ whose sides have lengths k₁, k₂, …, k_d and that |k| denotes the sum of these lengths, i.e., $| k | ≔ \sum_{i = 1}^{d} k_{i}$ ⁠. The coordinates of the 2^d corners of J_k,q are d-tuples of integers given by either q_j or q_j + k_j at the jth position for all 1 ≤ j ≤ d with q_j ≤ N − k_j.

The rectangles J_k,q play the role of the intervals I_k,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

(k^{'}, q^{'}) ≻ (k, q) iff

(2.14)

$\sum_{j = 1}^{d} k_{j}^{'} > \sum_{j = 1}^{d} k_{j}$ ⁠;
or, if $\sum_{j = 1}^{d} k_{j}^{'} = \sum_{j = 1}^{d} k_{j}$ ⁠, $k_{j}^{'} > k_{j}$ ⁠, for some 1 ≤ j ≤ d, with $k_{l}^{'} = k_{l}$ ⁠, ∀l > j;
or, if $k_{l}^{'} = k_{l}$ ⁠, for all l, $q_{j}^{'} > q_{j}$ ⁠, for some 1 ≤ j ≤ d, with $q_{l}^{'} = q_{l}$ ⁠, ∀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, H_i, 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).

Definition 2.1.

Given an arbitrary rectangle J_k,q of sites in

Λ_{N}^{d}

⁠, we define

H_{J_{k, q}} ≔ ⨂_{i \in J_{k, q}} H_{i} .

(2.15)

Definition 2.2.

Consider two rectangles, J_k,q and J_k′,q′, with nonempty intersection. The minimal rectangle associated with J_k,q ∪ J_k′,q′ is defined to be the smallest rectangle containing J_k,q and J_k′,q′. Note that its corners are the 2^d numbers with either

\min {q_{j}, q_{j}^{'}} or \max {q_{j} + k_{j}, q_{j}^{'} + k_{j}^{'}}

(2.16)

at the jth position. The minimal rectangle associated with J_k,q and J_k′,q′ is denoted by

[J_{k, q} \cup J_{k^{'}, q^{'}}] .

(2.17)

Definition 2.3.

Let J_k,q ⊂ J_l,i. We define a family,

G_{J_{l, i}}^{(k, q)}

⁠, of rectangles by

G_{J_{l, i}}^{(k, q)} ≔ \{J_{k^{'}, q^{'}} | J_{k^{'}, q^{'}} \neq J_{l, i} and [J_{k, q} \cup J_{k^{'}, q^{'}}] = J_{l, i}\} .

(2.18)

B. Effective Hamiltonians

Each conjugation step in the block-diagonalization of the original Hamiltonian is labeled by a rectangle J_k,q and, consequently, by a pair (k, q). In the effective Hamiltonian arising from a conjugation step, a potential term,

V_{J_{l, i}}^{(k, q)}

⁠, is associated with each rectangle J_l,i. More precisely, after the conjugation step (k, q), the effective Hamiltonian reads

\begin{matrix} K_{Λ_{N}^{d}}^{(k, q)} & = & \sum_{i \in Λ_{N}^{(d)}} H_{i} + t \sum_{k_{(1)}^{'}, q^{'}} V_{J_{k_{(1)}^{'}, q^{'}}}^{(k, q)} + t \sum_{k_{(2)}^{'}, q^{'}} V_{J_{k_{(2)}^{'}, q^{'}}}^{(k, q)} + \dots + t \sum_{k_{(| k |)}^{'}, q^{'}} V_{J_{k_{(| k |)}^{'}, q^{'}}}^{(k, q)} \\ + t \sum_{k_{(| k | + 1)}^{'}, q^{'}} V_{J_{k_{(| k | + 1)}^{'}, q^{'}}}^{(k, q)} + \dots + t V_{J_{N - 1, 1}}^{(k, q)}, \end{matrix}

(2.19)

where we have the following.

The pairs $(k_{(j)}^{'}, q^{'})$ are used to index all rectangles J_k′,q′ with |k′| = j.
For a fixed rectangle J_l,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, $V_{J_{l, i}}^{(k, q)}$ is the potential term associated with J_l,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 $V_{J_{l, i}}^{(k, q)}$ depends on the coupling constant t, but this is not made explicit in our notation; it acts as the identity on the spaces $H_{j}$ for j ∉ J_l,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 J_l,i with (k, q) ≻ (l, i) and for the rectangle J_l,i = J_k,q, the associated effective potential $V_{J_{l, i}}^{(k, q)}$ is block-diagonal with respect to the decomposition of the identity acting on $H^{(N)}$ into the sum of projections,
$P_{J_{l, i}}^{(-)} ≔ ⨂_{j \in J_{l, i}} P_{Ω_{j}},$
(2.20)
$P_{J_{l, i}}^{(+)} ≔ {(⨂_{j \in J_{l, i}} P_{Ω_{j}})}^{⊥} .$
(2.21)
The effective Hamiltonian $K_{Λ_{N}^{d}}^{(k, q)}$ of (2.19) is obtained after the conjugation step labeled by (k, q). Starting from
$K_{Λ_{N}^{d}}^{{(k, q)}_{- 1}} = \sum_{i \in Λ_{N}^{(d)}} H_{i} + t \sum_{k_{(1)}^{'}, q^{'}} V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} + t \sum_{k_{(2)}^{'}, q^{'}} V_{J_{k_{(2)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} + \dots + t \sum_{k_{(| k |)}^{'}, q^{'}} V_{J_{k_{(| k |)}^{'}, q^{'}}}^{{(k, q)}_{- 1}}$
(2.22)
$+ t \sum_{k_{(| k | + 1)}^{'}, q^{'}} V_{J_{k_{(| k | + 1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} + \dots + t V_{J_{N - 1, 1}}^{{(k, q)}_{- 1}},$
(2.23)
the conjugation step labeled by (k, q) is given by
$e^{S_{J_{k, q}}} K_{Λ_{N}^{d}}^{{(k, q)}_{- 1}} e^{- S_{J_{k, q}}} = : K_{Λ_{N}^{d}}^{(k, q)},$
(2.24)
where the anti-symmetric matrix $S_{J_{k, q}}$ is chosen in such a way that the interaction potential $V_{J_{k, q}}^{(k, q)}$ is block-diagonal; see Sec. IV. More precisely, following the Lie–Schwinger procedure, $S_{J_{k, q}}$ is built so as to block-diagonalize the local operator given by the sum of all terms in $K_{Λ_{N}^{d}}^{{(k, q)}_{- 1}}$ whose support is contained in J_k,q. In other words, $S_{J_{k, q}}$ is chosen in such a way that the conjugation in (2.24) renders the operator
$G_{J_{k, q}} + t V_{J_{k, q}}^{{(k, q)}_{- 1}},$
(2.25)
block-diagonal, where
$G_{J_{k, q}} ≔ \sum_{i \subset J_{k, q}} H_{i} + t \sum_{J_{k_{(1)}^{'}, q^{'}} \subset J_{k, q}} V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} + \dots + t \sum_{J_{k_{(| k | - 1)}^{'}, q^{'}} \subset J_{k, q}} V_{J_{k_{(| k | - 1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} .$
(2.26)
Here, “block-diagonalization” refers to the projections $P_{J_{k, q}}^{(-)}$ and $P_{J_{k, q}}^{(+)}$ corresponding to the decomposition of the Hilbert space $⨂_{i \in J_{k, q}} H_{i}$ into vacuum subspace and its orthogonal complement, respectively. The operator $G_{J_{k, q}}$ plays the role of the “unperturbed” operator since it is already block-diagonal with respect to the decomposition of the identity,
$1 = P_{J_{k, q}}^{(+)} + P_{J_{k, q}}^{(-)},$
i.e.,
$G_{J_{k, q}} = P_{J_{k, q}}^{(+)} G_{J_{k, q}} P_{J_{k, q}}^{(+)} + P_{J_{k, q}}^{(-)} G_{J_{k, q}} P_{J_{k, q}}^{(-)} .$
(2.27)
The construction outlined here works because one can show inductively that the energy gap in the spectrum of the Hamiltonian $G_{J_{k, q}}$ above its ground-state eigenvalue is bounded away from zero, uniformly in the size of the rectangle J_k,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 $G_{J_{k, q}}$ have been block-diagonalized already in the previous steps.

These properties of the operator $G_{J_{k, q}}$ ⁠, combined with bounds on the norms of the effective potentials obtained at the previous conjugation step, enable us to construct the anti-symmetric matrix $S_{J_{k, q}}$ used at the next conjugation step and control the norms of the effective potentials obtained after conjugation with $\exp [S_{J_{k, q}}]$ ⁠. 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 $V_{J_{r, i}}^{(k, q)}$ 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 $k (≔ \sum_{i = 1}^{d} k_{i})$ containing a specified site of the lattice grows polynomially in k, namely, like $O (k^{d - 1})$ ⁠. 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 $O (t^{c \cdot k})$ ⁠, 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 J_k,q and the size, r, of J_r,i, we will distinguish three different regimes for the growth processes that may give rise to the term $V_{J_{r, i}}^{(k, q)}$ in (2.31).

As implicitly indicated in (2.22) and (2.23) for the effective Hamiltonian $K_{Λ_{N}^{d}}^{{(k, q)}_{- 1}}$ ⁠, the potentials must be re-combined properly after each conjugation step (k, q) so as to determine a well-defined flow of operators, $V_{J_{r, i}}^{(k, q)}$ ⁠, for every fixed support J_r,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 $V_{J_{l, i}}^{(k, q)}$ as defined in Definition 2.4, formulated next.

The algorithm is supposed to enable us to iteratively determine effective potentials

V_{J_{r, i}}^{(k, q)}

in terms of the potentials obtained at the previous step (k, q)₋₁, starting from

V_{J_{0, i}}^{(0, N)} ≔ H_{i}, V_{J_{1_{j}, q}}^{(0, N)} ≔ V_{J_{1_{j}, q}}, and V_{J_{k, i}}^{(0, N)} = 0 for | k | \geq 2 .

(2.28)

Definition 2.4.

Assuming that, at fixed (k, q)₋₁ with (k, q)₋₁ ≻ (0, N), for any r, i, the operators $V_{J_{r, i}}^{{(k, q)}_{- 1}}$ and $S_{J_{k, q}}$ [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 $S_{J_{1_{1}, 1}}$ are well defined, then we define the following:

if J_k,q ⊈ J_r,i,
$V_{J_{r, i}}^{(k, q)} ≔ V_{J_{r, i}}^{{(k, q)}_{- 1}};$
(2.29)
if J_r,i = J_k,q,
$V_{J_{r, i}}^{(k, q)} ≔ \sum_{j = 1}^{\infty} t^{j - 1} {(V_{J_{r, i}}^{{(k, q)}_{- 1}})}_{j}^{d i a g},$
(2.30)
where ${(V_{J_{r, i}}^{{(k, q)}_{- 1}})}_{j}^{d i a g}$ is defined such as in (4.53) and diag means the diagonal part with respect to the projections $P_{J_{r, i}}^{(-)}$ and $P_{J_{r, i}}^{(+)}$ ⁠;
if J_k,q ⊂ J_r,i,
$V_{J_{r, i}}^{(k, q)} ≔ e^{S_{J_{k, q}}} V_{J_{r, i}}^{{(k, q)}_{- 1}} e^{- S_{J_{k, q}}} + \sum_{J_{k^{'}, q^{'}} \in G_{J_{r, i}}^{(k, q)}} \sum_{n = 1}^{\infty} \frac{1}{n!} a d^{n} S_{J_{k, q}} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}}),$
(2.31)
where ad is defined in (4.49) and (4.50). We observe that the set $G_{J_{r, i}}^{(k, q)}$ [see (2.18)] is not empty only if the rectangle J_k,q has a nonempty intersection with the boundary of the rectangle J_r,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 $V_{J_{r, i}}^{(k, q)}$ with support J_r,i that correspond to “growth processes” of rectangles, i.e., to processes where the union of a rectangle J_k′,q′ ≠ J_r,i and of the fixed rectangle J_k,q labeling the conjugation step in the block-diagonalization is a set with the property that J_r,i is the minimal rectangle associated with it, i.e., such that [J_k′,q′ ∪ J_k,q] ≡ J_r,i.

To control the operator norms of the effective potentials, we begin by observing that, by construction, the potential

V_{J_{r, 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

‖ V_{J_{r, i}}^{(k, q)} ‖ \leq C_{j} \frac{t^{\frac{r - 1}{3}}}{r^{ρ_{j}}}, j = 1, 2, 3,

(2.32)

holds true at all steps (k, q) up to step (r, i) (included), where C_j and the exponent ρ_j ≡ ρ_j(d) > 0 (d being the space dimension) depend on the regime

R j

introduced below, and the different regimes,

R 1, R 2

⁠, and

R 3

⁠, 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, J_k′,q′, that intersect the rectangle J_k,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, $R 1, R 2$ ⁠, and $R 3$ ⁠, 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.

(⁠ $R 1$ ⁠) 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 $k \leq ⌊ r^{\frac{1}{4}} ⌋$ ⁠. In order to establish the desired estimate (2.32), we iterate the re-expansion of the potential $V_{J_{r, i}}^{(k, q)}$ 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 $J_{k^{''}, q^{''}}$ labeling the operators $S_{k^{''}, q^{''}}$ ⁠, plus one labeling one of the potentials appearing in the Hamiltonian of definition (1.5) or a potential of the type $V_{J_{k^{'}, q^{'}}}^{(k^{'}, q^{'})}$ (where $k^{'} \leq ⌊ r^{\frac{1}{4}} ⌋$ ⁠), with the property that J_r,i is the minimal rectangle associated with this sequence. Roughly speaking, the result, then, holds for the following reasons:
- (1) At least $O (r / ⌊ r^{\frac{1}{4}} ⌋)$ rectangles $J_{k^{''}, q^{''}}$ are present in each connected set, and all the corresponding operators $S_{k^{''}, q^{''}}$ have norms of order $| t | \cdot ‖ V_{k^{''}, q^{''}}^{{(k^{''}, q^{''})}_{- 1}} ‖$ ⁠; apart from the resulting product of norms $‖ V_{k^{''}, q^{''}}^{{(k^{''}, q^{''})}_{- 1}} ‖$ ⁠, which is also crucial in the argument, it is important that a total factor $| t |^{O (r / ⌊ r^{\frac{1}{4}} ⌋)}$ or smaller is gained from the re-expansion (due to the constraint $k \leq ⌊ r^{\frac{1}{4}} ⌋$ 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 $V_{J_{k^{''}, q^{''}}}^{{(k^{''}, q^{''})}_{- 1}}$ ⁠, provides an upper bound to $‖ V_{J_{r, i}}^{(k, q)} ‖$ ⁠. In fact, for |t| small enough but independent of N, this weighted number yields the sought bound (2.32) for $‖ V_{J_{r, i}}^{(k, q)} ‖$ ⁠.
(⁠ $R 2$ ⁠) The second regime is associated with pairs (k, q) with the property that $⌊ r^{\frac{1}{4}} ⌋ \leq k \leq r - ⌊ r^{\frac{1}{4}} ⌋$ ⁠. In this regime, thanks to the upper bound on k, the size of the rectangles J_k′,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 $r^{ρ_{2}}$ in the inductive estimate [see (2.32)] of the potential. If $k^{ρ_{2}}$ and ${(r - k)}^{ρ_{2}}$ 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).
(⁠ $R 3$ ⁠) The third regime is associated with “large” rectangles (k, q) since the $r - ⌊ r^{\frac{1}{4}} ⌋ \leq k \leq r$ ⁠. 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 $V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}}$ that are already block-diagonal and, then, estimate them in terms of a sum of projections $P_{J_{k^{'}, q^{'}}}^{(+)}$ controlled, through an induction, by the denominator appearing in the expression of ${(S_{J_{r, i}})}_{1}$ [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.

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Examples of configurations of $R 1, R 2, R 3$ ⁠, respectively.

III. TREE STRUCTURE AND PATHS OF RECTANGLES

In order to study regime $R 1$ ⁠, we shall re-expand the potentials $V_{J_{r, i}}^{(k, q)}$ 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 $R 1$ ⁠. We, therefore, describe it carefully in this section.

For the purpose of re-expanding

V_{J_{r, 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

k \leq ⌊ r^{\frac{1}{4}} ⌋

in regime

R 1

⁠. In order to streamline our formulas, we introduce the notation

\sum_{n = 1}^{\infty} \frac{1}{n!} a d^{n} S_{J_{k, q}} (\dots) = : A_{J_{k, q}} (\dots) .

(3.33)

Depending on the relative position between J_k,q and J_r,i, we are instructed to use either

V_{J_{r, i}}^{(k, q)} = V_{J_{r, i}}^{{(k, q)}_{- 1}}

(3.34)

+ A_{J_{k, q}} (V_{J_{r, i}}^{{(k, q)}_{- 1}})

(3.35)

+ \sum_{J_{k^{'}, q^{'}} \in G_{J_{r, i}}^{(k, q)}} A_{J_{k, q}} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}})

(3.36)

or

V_{J_{r, i}}^{(k, q)} = V_{J_{r, i}}^{{(k, q)}_{- 1}},

(3.37)

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 ()–() 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

V_{J_{k^{''}, 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, $R_{b}$ ⁠, of rectangles that are ordered and whose union is connected.
We show that there is an injective map from ${R_{b}}$ to a set, ${Γ_{b}}$ ⁠, of paths of rectangles with certain properties.
By assigning suitable weights to the paths $Γ_{b}$ ⁠, we will be able to derive upper bounds on the norms of the contributions $b$ ⁠. This will allow us to estimate the norm $‖ V_{J_{r, i}}^{(k, q)} ‖$ by counting (weighted) paths belonging to the set ${Γ_{b}}$ ⁠.

A. Tree expansion

In order to find an efficient description (see Definition 3.1) of the structure of contributions emerging from the re-expansion of

V_{J_{r, 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 J_k,q and J_r,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

V_{J_{r, i}}^{(k, q)} = e^{S_{J_{k, q}}} V_{J_{r, i}}^{{(k, q)}_{- 2}} e^{- S_{J_{k, q}}} + \sum_{J_{k^{'}, q^{'}} \in G_{J_{r, i}}^{(k, q)}} A_{J_{k, q}} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 2}})

(3.38)

= V_{J_{r, i}}^{{(k, q)}_{- 2}} + A_{J_{k, q}} (V_{J_{r, i}}^{{(k, q)}_{- 2}}) + \sum_{J_{k^{'}, q^{'}} \in G_{J_{r, i}}^{(k, q)}} A_{J_{k, q}} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 2}}) .

(3.39)

Note that in () and, consequently, in (), 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.

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Example of a tree associated with the first two steps of the re-expansion of $V_{J_{r, i}}^{(k, q)}$ ⁠.

Definition 3.1.

The levels of a tree used to identify the contributions to the re-expansion of a potential $V_{J_{r, i}}^{(k, q)}$ 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).
There is a single vertex at the top of a tree rooted at level (k, q); it is labeled by the symbol $V_{J_{r, i}}^{(k, q)}$ 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 $v \equiv v_{V_{J_{s, u}}^{(k^{'}, q^{'})}}$ at level (k′, q′), labeled by $V_{J_{s, u}}^{(k^{'}, q^{'})}$ ⁠, 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 $V_{J_{s, u}}^{{(k^{'}, q^{'})}_{- 1}}$ ⁠;
- the rightmost set of vertices is empty if J_k′,q′ ⊄ J_s,u; otherwise, it contains a vertex for each element J_s′,u′ belonging to $G_{J_{s, u}}^{(k^{'}, q^{'})} \cup {J_{s, u}}$ ⁠, and this vertex is labeled by $V_{J_{s^{'}, u^{'}}}^{{(k^{'}, q^{'})}_{- 1}}$ ⁠.
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
  $V_{J_{s, u}}^{(k^{'}, q^{'})} \to V_{J_{s, u}}^{{(k^{'}, q^{'})}_{- 1}},$
  where $V_{J_{s, u}}^{(k^{'}, q^{'})}$ is the potential labeling $v$ and $V_{J_{s, u}}^{{(k^{'}, q^{'})}_{- 1}}$ 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 J_k′,q′. It stands for the map
  $V_{J_{s, u}}^{(k^{'}, q^{'})} \to A_{J_{k^{'}, q^{'}}} (V_{J_{s^{'}, u^{'}}}^{{(k^{'}, q^{'})}_{- 1}}),$
  where $V_{J_{s, u}}^{(k^{'}, q^{'})}$ labels the vertex $v$ and $V_{J_{s^{'}, u^{'}}}^{{(k^{'}, q^{'})}_{- 1}}$ is the potential labeling the vertex connected to $v$ by the edge $e$ ⁠.
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 $V_{J_{k^{''}, q^{''}}}^{(k^{''}, q^{''})}$ 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 $b$ of a tree, we associate a set, $R_{b}$ ⁠, of rectangles consisting of (i) those rectangles labeling the edges of $b$ and (ii) the rectangle $J_{k^{''}, q^{''}}$ indicating the support of the potential labeling the leaf of $b$ ⁠.

The set $R_{b}$ inherits the ordering relation (2.14), and hence, its elements can be enumerated by a map
$i \in \{1, \dots, | R_{b} |\} \to J_{k^{(i)}, q^{(i)}} \in R_{b}$
with (k⁽ⁱ⁾, q⁽ⁱ⁾) ≻ (k⁽ⁱ⁺¹⁾, q⁽ⁱ⁺¹⁾), where $| R_{b} |$ is the cardinality of the set $R_{b}$ ⁠. Note that $J_{k^{(| R_{b} |)}, q^{(| R_{b} |)}}$ is the rectangle associated with the potential labeling the leaf of $b$ ⁠.
To every branch $b$ ⁠, we can associate the “branch operator,” also denoted by $b$ ⁠,
$b ≔ A_{J_{k^{(1)}, q^{(1)}}} (A_{J_{k^{(2)}, q^{(2)}}} (\dots A_{J_{k^{(| R_{b} | - 1)}, q^{(| R_{b} | - 1)}}} (V_{L_{b}}) \dots)),$
(3.40)
where $V_{L_{b}}$ is the potential labeling the leaf of $b$ ⁠; $V_{L_{b}}$ can be either $V_{J_{k^{(| R_{b} |)}, q^{(| R_{b} |)}}}^{(k^{(| R_{b} |)}, q^{(| R_{b} |)})}$ or $V_{J_{s, u}}^{(0, N)}$ ⁠.

The set of branches whose corresponding branch operators are non-zero is denoted by $B_{V_{J_{r, i}}^{(k, q)}}$ ⁠.

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 $V_{J_{r, i}}^{(0, N)}$ with r > 1, which is zero by definition.

1. Properties of the branches $b \in B_{V_{J_{r, i}}^{(k, q)}}$

Definition 3.1 implies the following properties of the elements of the set $B_{V_{J_{r, i}}^{(k, q)}}$ defined above:

For $b \in B_{V_{J_{r, i}}^{(k, q)}}$ ⁠, the set
$⋃_{i \in \{1, \dots, | R_{b} |\}} J_{k^{(i)}, q^{(i)}}$
is connected due to (3.40), although $J_{k^{(i)}, q^{(i)}} \cap J_{k^{(i + 1)}, q^{(i + 1)}}$ might be empty for some i. Likewise, for any fixed $n \in \{1 \dots | R_{b} |\}$ ⁠, the set $⋃_{n \leq i \leq | R_{b} |} J_{k^{(i)}, q^{(i)}}$ is connected. Indeed, for any operator $O$ and for any m, $A_{J_{k^{(m)}, q^{(m)}}} (O) = 0$ whenever the supports of $O$ and $S_{J_{k^{(m)}, q^{(m)}}}$ have empty intersection; see formula (3.33).
For $b \in B_{V_{J_{r, i}}^{(k, q)}}$ ⁠, the cardinality, $| R_{b} |$ ⁠, of the set $R_{b}$ of rectangles is such that $| R_{b} | \geq O (\frac{r}{k}) \geq O (r^{\frac{3}{4}})$ ⁠. This lower bound on $| R_{b} |$ is a consequence of the restriction imposed on k = |k| and required in regime $R 1$ [and it will turn out to be crucial to derive our estimate (5.105) and (5.106) in Theorem 5.1].
The set J_r,i is the minimal rectangle associated with $⋃_{i \in \{1, \dots, | R_{b} |\}} J_{k^{(i)}, q^{(i)}}$ for any branch $b \in B_{V_{J_{r, i}}^{(k, q)}}$ ⁠. 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 $b, b^{'} \in B_{V_{J_{r, i}}^{(k, q)}}$ are associated with two different (ordered) sets of rectangles $R_{b}$ and $R_{b^{'}}$ ⁠.
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 $b_{1}, b_{2} \in B_{V_{J_{r, i}}^{(k, q)}}$ ⁠, are labeled by two different sets of rectangles, $R_{b_{1}}$ and $R_{b_{2}}$ ⁠, respectively.

B. Summing over the norms of branch-operator: Weights and paths, $Γ_{b}$

Our task is to estimate the norms of the potentials

V_{J_{r, 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

V_{J_{r, i}}^{(k, q)}

can be expressed as the sum

\sum_{b \in B_{V_{J_{r, 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

\sum_{b \in B_{V_{J_{r, i}}^{(k, q)}}} ‖ b ‖ .

This can be done by assigning a “weight” to every set

R_{b}

of rectangles, the weight being proportional to the product of operator norms of the potentials associated [in step (k,q)₋₁] with each rectangle J_k,q in the set

R_{b}

⁠, i.e.,

\sum_{b \in B_{V_{J_{r, i}}^{(k, q)}}} {(c \cdot t)}^{| R_{b} | - 1} ‖ V_{L_{b}} ‖ \prod_{i \in \{1, \dots, | R_{b} | - 1\}} ‖ V_{J_{k^{(i)}, q^{(i)}}}^{{(k^{(i)}, q^{(i)})}_{- 1}} ‖,

(3.41)

where

V_{L_{b}}

is the potential labeling the leaf of

b

since a factor

(c \cdot t) \cdot ‖ V_{J_{k^{(i)}, q^{(i)}}}^{{(k^{(i)}, q^{(i)})}_{- 1}} ‖

is associated with the map

A_{J_{k^{(i)}, q^{(i)}}}

⁠; here, c > 0 is a universal constant.

In order to count the sets $R_{b}$ ⁠, we shall assign a path, $Γ_{b}$ ⁠, to each $b$ ⁠, where $Γ_{b}$ has the property to visit all the rectangles in the set $R_{b}$ ⁠. Since we must estimate the “weighted” number of sets $R_{b}$ ⁠, 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 ${J_{s^{(i)}, u^{(i)}}}_{i = 1}^{n}$ ⁠, for some $n \in N$ ⁠, with the property that $J_{s^{(i)}, u^{(i)}} \neq J_{s^{(i + 1)}, u^{(i + 1)}}$ and $J_{s^{(i)}, u^{(i)}} \cap J_{s^{(i + 1)}, u^{(i + 1)}} \neq \emptyset$ 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,
$S_{Γ} ≔ \{(J_{s^{(i)}, u^{(i)}}, J_{s^{(i + 1)}, u^{(i + 1)}}) | i = 1, \dots, n - 1\},$
is called the set of steps of the path $Γ \equiv {\{J_{s^{(i)}, u^{(i)}}\}}_{i = 1}^{n}$ ⁠.
The length, l_Γ, of the path $Γ \equiv {J_{s^{(i)}, u^{(i)}}}_{i = 1}^{n}$ is defined to be l_Γ ≔ n − 1.
The support, supp(Γ), of a path $Γ \equiv {J_{s^{(i)}, u^{(i)}}}_{i = 1}^{n}$ is defined to be
$supp (Γ) ≔ \{J_{s^{(i)}, u^{(i)}}, i \in {1 \dots n}\} .$
A path $Γ \equiv {J_{s^{(i)}, u^{(i)}}}_{i = 1}^{n}$ is closed if $J_{s^{(1)}, u^{(1)}} = J_{s^{(n)}, u^{(n)}}$ ⁠.

Each rectangle $J_{k^{(i)}, q^{(i)}}$ of the set $R_{b}$ contributes to weight (3.41) of $R_{b}$ through $c \cdot t \cdot ‖ V_{J_{k^{(i)}, q^{(i)}}}^{{(k^{(i)}, q^{(i)})}_{- 1}} ‖$ [except for $J_{k^{(| R_{b} |)}, q^{(| R_{b} |)}}$ that contributes through $c \cdot ‖ V_{L_{b}} ‖$ ], which (as it will be shown) decreases with the size of the rectangle. Thus, we have to make sure that the path $Γ_{b}$ does not visit small rectangles of $R_{b}$ ⁠, which have a “big” weight, repeatedly. This motivates the requirements imposed on the paths $Γ_{b}$ considered henceforth, in particular property (C) stated in Sec. III B 2.

2. Connected components, $Z_{ρ}^{(j)}$ ⁠, of rectangles and definition of $Γ_{b}$

Since the weight of a rectangle is a function of its size, it is convenient to write the connected set

⋃_{i \in {1, \dots, | R_{b} |}} J_{k^{(i)}, q^{(i)}}

as the union

⋃_{ρ = k_{0}}^{k} (⋃_{j = 1}^{j_{ρ}} Z_{ρ}^{(j)}),

where

{Z_{ρ}^{(j)}, j = 1, \dots, j_{ρ}}

are distinct connected components of (unions of) rectangles of a given size ρ, k₀ ≤ ρ ≤ k, starting from the lowest one k₀ ≥ 1, with the following properties:

$j_{k_{0}} = 1$ (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, $Z_{ρ}^{(j)} \cap Z_{ρ}^{(j^{'})} = \emptyset$ for j ≠ j′.

We call

supp (Z_{ρ}^{(j)})

⁠, ρ = k₀, …, k, j = 1, …, j_ρ, the set of rectangles of

Z_{ρ}^{(j)}

⁠, i.e.,

supp (Z_{ρ}^{(j)}) ≔ \{J_{k^{(i)}, q^{(i)}} : J_{k^{(i)}, q^{(i)}} \subset Z_{ρ}^{(j)}, i \in \{1, \dots, | R_{b} |\}\} .

Starting from a branch

b \in B_{V_{J_{r, i}}^{(k, q)}}

⁠, we shall inductively construct a path,

Γ_{b}

⁠, of length

l_{Γ_{b}}

bounded by

l_{Γ_{b}} \leq 2 (n_{k_{0}} + \sum_{j = 1}^{j_{2}} n_{k_{0} + 1}^{(j)} + \dots + \sum_{j = 1}^{j_{k}} n_{k}^{(j)}) - 2,

where

n_{ρ}^{(j)} ≔ | supp (Z_{ρ}^{(j)}) |

⁠, with the following properties:

the support of $Γ_{b}$ is $R_{b}$ ⁠;
for each component $Z_{ρ}^{(j)}$ consisting of the union of $n_{ρ}^{(j)}$ rectangles, at most $2 n_{ρ}^{(j)} - 2$ steps are made [i.e., there are at most $2 n_{ρ}^{(j)} - 2$ steps $σ \in S_{Γ_{b}}$ for which $σ \in supp (Z_{ρ}^{(j)}) \times supp (Z_{ρ}^{(j)})$ ];
there are at most two steps connecting rectangles in $supp (Z_{ρ}^{(j)})$ with rectangles of lower size: more precisely, for every connected component $Z_{ρ}^{(j)}$ ⁠, there is at most one J_s,u in $supp (Z_{ρ}^{(j)})$ such that $(J_{s^{'}, u^{'}}, J_{s, u}) \in S_{Γ_{b}}$ with s′ < s, $J_{s^{'}, u^{'}} \in R_{b}$ and at most one $J_{s, u} \in supp (Z_{ρ}^{(j)})$ such that $(J_{s, u}, J_{s^{'}, u^{'}}) \in S_{Γ_{b}}$ with s > s′, $J_{s^{'}, u^{'}} \in R_{b}$ ⁠.

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 $Γ_{b}^{(k^{'} - 1)}$ ⁠, with k₀+1 ≤ k′ ≤ k, fulfilling (A)–(C) for the set $\cup_{ρ = k_{0}}^{k^{'} - 1} \cup_{j = 1}^{j_{ρ}} Z_{ρ}^{(j)}$ ⁠, which is connected by property (P-i). Starting from this path, we construct a new one, denoted by $Γ_{b}^{(k^{'})}$ ⁠, 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 $Γ_{b}$ ⁠, in a way that is optimal to derive suitable bounds. In fact, we will associate a weight with the steps of the paths $Γ_{b}$ 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 $R 1$ ⁠, 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

n_{ρ}^{(j)}

rectangles in the set

supp (Z_{ρ}^{(j)})

and that, for the paths

Γ_{b}

⁠, there are at most

2 n_{ρ}^{(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 = (J_{s^{(i)}, u^{(i)}}, J_{s^{(i + 1)}, u^{(i + 1)}}) \in S_{Γ_{b}}

⁠, we can assign the weight

W_{σ} \equiv W_{s^{(i)} \to s^{(i + 1)}} = {((c + 1) t)}^{\frac{1}{2}} \cdot \min \{‖ V_{J_{s^{(i)}, u^{(i)}}}^{{(s^{(i)}, u^{(i)})}_{- 1}} ‖^{\frac{1}{2}}, ‖ V_{J_{s^{(i + 1)}, u^{(i + 1)}}}^{{(s^{(i + 1)}, u^{(i + 1)})}_{- 1}} ‖^{\frac{1}{2}}\},

where t is sufficiently small such that

(c + 1) t \cdot ‖ V_{J_{k^{(i)}, q^{(i)}}}^{{(k^{(i)}, q^{(i)})}_{- 1}} ‖ < 1

⁠, and the following estimate holds:

{(c \cdot t)}^{| R_{b} | - 1} ‖ V_{L_{b}} ‖ \prod_{i = 1}^{| R_{b} | - 1} ‖ V_{J_{k^{(i)}, q^{(i)}}}^{{(k^{(i)}, q^{(i)})}_{- 1}} ‖ \leq \frac{1}{t} \prod_{σ \in S_{Γ_{b}}} w_{σ} .

(3.42)

The previous inequality is true because, if we denote by

S_{Z_{ρ}^{(j)}}

the set of at most

2 n_{ρ}^{(j)} - 2

steps between rectangles of

supp Z_{ρ}^{(j)}

and the additional at most two steps from rectangles of lower size and back, then we have

{(c \cdot t)}^{| supp (Z_{ρ}^{(j)}) |} \prod_{J_{s, u} \in supp (Z_{ρ}^{(j)})} ‖ V_{J_{s, u}}^{{(s, u)}_{- 1}} ‖ \leq \prod_{σ \in S_{Z_{ρ}^{(j)}}} w_{σ} .

Finally, we use the estimate

\sum_{b \in B_{V_{J_{r, i}}^{(k, q)}}} c^{| R_{b} | - 1} t^{| R_{b} |} ‖ V_{L_{b}} ‖ \prod_{i \in \{1, \dots, | R_{b} | - 1\}} ‖ V_{J_{k^{(i)}, q^{(i)}}}^{{(k^{(i)}, q^{(i)})}_{- 1}} ‖

(3.43)

\leq \sum_{Γ_{b}, b \in B_{V_{J_{r, i}}^{(k, q)}}} \prod_{σ \in S_{Γ_{b}}} w_{σ} \leq C_{d} \cdot r^{2 d - 1} \cdot \sum_{j = ⌊ c_{d} \cdot \frac{r}{k} ⌋}^{\infty} {(\sum_{ρ, ρ^{'} = 1}^{k} w_{ρ \to ρ^{'}} D_{ρ, ρ^{'}})}^{j},

(3.44)

where

⌊ c_{d} \cdot \frac{r}{k} ⌋

is a lower bound for

| R_{b} |

and C_d · r^2d−1 is an upper bound on the possible positions of the rectangle

J_{k^{(| R_{b} |)}, q^{(| R_{b} |)}}

of the path, where c_d, C_d are d-dependent constants; finally,

D_{s, s^{'}} ≔ C_{d} \cdot s^{d} \cdot s^{' d - 1},

(3.45)

where

C_{d}

is a d-dependent constant, is an upper bound on the number of possible directions of a path

Γ = {J_{s^{(i)}, u^{(i)}}}_{i = 1}^{n}

⁠, extended by one more step as specified here: given the path

Γ = {J_{s^{(i)}, u^{(i)}}}_{i = 1}^{n}

⁠, the number of paths

Γ^{+} = {J_{s^{' (i)}, u^{' (i)}}}_{i = 1}^{n + 1}

of length

l_{Γ^{+}} = n

⁠, whose first n elements agree with Γ (i.e.,

{J_{s^{(i)}, u^{(i)}}}_{i = 1}^{n} = {J_{s^{' (i)}, u^{' (i)}}}_{i = 1}^{n}

⁠) and for which s′⁽ⁿ⁺¹⁾ ≔ s′ and s⁽ⁿ⁾ ≔ s, is bounded from above by D_s,s′.

A minor modification of the inequality provided in (3.44) will enable us to prove the result of Theorem 5.1 concerning regime $R 1$ ⁠.

IV. THE UNITARY CONJUGATION $e^{S_{J_{k, q}}}$ AND THE SPECTRAL GAP OF $G_{J_{k, q}}$

The operator

e^{S_{J_{k, q}}}

is constructed so as to block-diagonalize the Hamiltonian

G_{J_{k, q}} + t V_{J_{k, q}}^{{(k, q)}_{- 1}}

with respect to the decomposition of the identity

1 = P_{J_{k, q}}^{(+)} + P_{J_{k, q}}^{(-)} .

(4.46)

The operator

G_{J_{k, q}} ≔ \sum_{i \subset J_{k, q}} H_{i} + t \sum_{J_{k_{(1)}^{'}, q^{'}} \subset J_{k, q}} V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} + \dots + t \sum_{J_{k_{(| k | - 1)}^{'}, q^{'}} \subset J_{k, q}} V_{J_{k_{(| k | - 1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}}

(4.47)

is already block-diagonal with respect to ().

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

E_{J_{k, q}}

⁠, which is, in fact, the ground-state energy of the operator

G_{J_{k, q}}

⁠,

E_{J_{k, q}} ≔ ⟨ ⨂_{j \in J_{k, q}} Ω_{j}, G_{J_{k, q}} ⨂_{j \in J_{k, q}} Ω_{j} ⟩,

(4.48)

i.e.,

G_{J_{k, q}} P_{J_{k, q}}^{(-)} = E_{J_{k, q}} P_{J_{k, q}}^{(-)} .

We recall that

a d A (B) ≔ [A, B],

(4.49)

where A and B are bounded operators, and for n ≥ 2,

a d^{n} A (B) ≔ [A, a d^{n - 1} A (B)] .

(4.50)

To carry out the block-diagonalization step (k, q), the operator

S_{J_{k, q}}

is defined by the series

S_{J_{k, q}} ≔ \sum_{j = 1}^{\infty} t^{j} {(S_{J_{k, q}})}_{j}

(4.51)

with

${(S_{J_{k, q}})}_{j} ≔ a d^{- 1} G_{J_{k, q}} ({(V_{J_{k, q}}^{{(k, q)}_{- 1}})}_{j}^{o d}) ≔ \frac{1}{G_{J_{k, q}} - E_{J_{k, q}}} P_{J_{k, q}}^{(+)} {(V_{J_{k, q}}^{{(k, q)}_{- 1}})}_{j} P_{J_{k, q}}^{(-)} - h . c .,$
(4.52)
where “od” means the off-diagonal with respect to the decomposition of identity (4.46);
${(V_{J_{k, q}}^{{(k, q)}_{- 1}})}_{1} ≔ V_{J_{k, q}}^{{(k, q)}_{- 1}}$ ⁠, and for j ≥ 2,
$\begin{align} {(V_{J_{k, q}}^{{(k, q)}_{- 1}})}_{j} ≔ \sum_{p \geq 2, r_{1} \geq 1 \dots, r_{p} \geq 1; r_{1} + \dots + r_{p} = j} \frac{1}{p!} ad {(S_{J_{k, q}})}_{r_{1}} (ad {(S_{J_{k, q}})}_{r_{2}} \dots (ad {(S_{J_{k, q}})}_{r_{p}} (G_{J_{k, q}})) \dots) \\ + \sum_{p \geq 1, r_{1} \geq 1 \dots, r_{p} \geq 1; r_{1} + \dots + r_{p} = j - 1} \frac{1}{p!} ad {(S_{J_{k, q}})}_{r_{1}} (ad {(S_{J_{k, q}})}_{r_{2}} \dots (ad {(S_{J_{k, q}})}_{r_{p}} (V_{J_{k, q}}^{{(k, q)}_{- 1}})) \dots) . \end{align}$
(4.53)
We recall that
$K_{Λ_{N}^{d}}^{(k, q)} ≔ e^{S_{J_{k, q}}} K_{Λ_{N}^{d}}^{{(k, q)}_{- 1}} e^{- S_{J_{k, q}}} .$
(4.54)
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.

Proof.

In the expression

\begin{align} e^{S_{J_{k, q}}} K_{Λ_{N}^{d}}^{{(k, q)}_{- 1}} e^{- S_{J_{k, q}}} & = e^{S_{J_{k, q}}} \{\sum_{i \in Λ_{N}^{(d)}} H_{i} + t \sum_{k_{(1)}^{'}, q^{'}} V_{J_{k_{(1)}^{'}, q^{'}}}^{(k, q)} + t \sum_{k_{(2)}^{'}, q^{'}} V_{J_{k_{(2)}^{'}, q^{'}}}^{(k, q)} + \dots + \\ + t \sum_{k_{(| k |)}^{'}, q^{'}} V_{J_{k_{(| k |)}^{'}, q^{'}}}^{(k, q)} + t \sum_{k_{(| k | + 1)}^{'}, q^{'}} V_{J_{k_{(| k | + 1)}^{'}}, q^{'}}^{(k, q)} + \dots + t V_{J_{N, 1}}^{(k, q)}\} e^{- S_{J_{k, q}}}, \end{align}

(4.55)

we observe that we have the following:

For all rectangles J_l,i such that J_l,i ∩ J_k,q = ∅, we have that
$e^{S_{J_{k, q}}} V_{J_{l, i}}^{{(k, q)}_{- 1}} e^{- S_{J_{k, q}}} = V_{J_{l, i}}^{{(k, q)}_{- 1}} = V_{J_{l, i}}^{(k, q)},$
(4.56)
where the last identity is due to item (a) in Definition 2.4.
Regarding the terms constituting $G_{J_{k, q}}$ [see the definition in (2.26)], we note that if we add $V_{J_{k, q}}^{{(k, q)}_{- 1}}$ ⁠, we get
$\begin{align} e^{S_{J_{k, q}}} & (G_{J_{k, q}} + t V_{J_{k, q}}^{{(k, q)}_{- 1}}) e^{- S_{J_{k, q}}} \\ = & \sum_{i \subset J_{k, q}} H_{i} + t \sum_{J_{k_{(1)}^{'}, q^{'}} \subset J_{k, q}} V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} + \dots + t \sum_{J_{k_{(| k | - 1)}^{'}}, q^{'} \subset J_{k, q}} V_{J_{k_{(| k | - 1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} + \sum_{j = 1}^{\infty} t^{j} {(V_{J_{k, q}}^{{(k, q)}_{- 1}})}_{j}^{d i a g} \\ = & \sum_{i \subset J_{k, q}} H_{i} + t \sum_{J_{k_{(1)}^{'}, q^{'}} \subset J_{k, q}} V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} + \dots + t \sum_{J_{k_{(| k | - 1)}^{'}}, q^{'} \subset J_{k, q}} V_{J_{k_{(| k | - 1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} + t V_{J_{k, q}}^{(k, q)}, \end{align}$
(4.57)
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 $V_{J_{l, i}}^{{(k, q)}_{- 1}}$ with J_l,i ∩ J_k,q ≠ ∅, but J_k,q ⊈ J_l,i and J_l,i ⊈ J_k,q, we write
$e^{S_{J_{k, q}}} V_{J_{l, i}}^{{(k, q)}_{- 1}} e^{- S_{J_{k, q}}} = V_{J_{l, i}}^{{(k, q)}_{- 1}} + \sum_{n = 1}^{\infty} \frac{1}{n!} a d^{n} S_{J_{k, q}} (V_{J_{l, i}}^{{(k, q)}_{- 1}}),$
(4.58)
where the first term on the right-hand side is $V_{J_{l, i}}^{(k, q)}$ by definition [see item (a) in Definition 2.4] and the second term contributes to the potential $V_{J_{r, j}}^{(k, q)}$ ⁠, where J_r,j ≡ [J_l,i ∪ J_k,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
$e^{S_{J_{k, q}}} V_{J_{r, j}}^{{(k, q)}_{- 1}} e^{- S_{J_{k, q}}} .$
(4.59)
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

G_{J_{k, 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

‖ V_{J_{l, i}}^{{(k, q)}_{- 1}} ‖ \leq t^{\frac{l - 1}{4}}, l = | l | ≔ l_{1} + l_{2} + \dots + l_{d}, for 0 \leq t < t_{d},

(4.60)

the Hamiltonian

G_{J_{k, q}}

has a gap

Δ_{J_{k, q}} \geq \frac{1}{2}

for all t ∈ [0, t_d), where t_d 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,

P_{J_{l, i}}^{(+)} \leq \sum_{j \in J_{l, i}} P_{Ω_{j}}^{⊥}

(4.61)

and

\sum_{i : J_{l, i} \subset J_{k, q}} P_{J_{l, i}}^{(+)} \leq \{\prod_{j = 1}^{d} (l_{j} + 1)\} \sum_{i \in J_{k, q}} P_{Ω_{i}}^{⊥}

(4.62)

\leq {(l + 1)}^{d} \sum_{i \in J_{k, q}} P_{Ω_{i}}^{⊥} .

(4.63)

Remark 4.2.

Observe that the number of shapes²⁹ of rectangles J_l,i at fixed |l| = l is bounded above by (l + 1)^d−1 = O(l^d−1). As a consequence, we have the following:

the number of rectangles J_k,q ⊂ J_r,i with fixed circumference k is bounded by (r + 1)^d(k + 1)^d−1 = $O$ (r^dk^d−1),
the number of rectangles J_k′,q′ ⊂ J_r,i is, then, bounded by ${(r + 1)}^{d} \sum_{k = 1}^{r} {(k + 1)}^{d - 1} = O (r^{2 d})$ ⁠, and
the number of rectangles in $G_{J_{r, i}}^{(k, q)}$ is bounded by $2 d {(r + 1)}^{d - 1} \sum_{k = 1}^{r} {(k + 1)}^{d - 1} = O (r^{2 d - 1})$ ⁠.³⁰

Remark 4.3.

Our block-diagonalization procedure relies on the following crucial property: If

V_{J_{l, i}}^{(k, q)}

is block-diagonal with respect to the decomposition of the identity into

1 = P_{J_{l, i}}^{(+)} + P_{J_{l, i}}^{(-)},

i.e., if

V_{J_{l, i}}^{(k, q)} = P_{J_{l, i}}^{(+)} V_{J_{l, i}}^{(k, q)} P_{J_{l, i}}^{(+)} + P_{J_{l, i}}^{(-)} V_{J_{l, i}}^{(k, q)} P_{J_{l, i}}^{(-)},

then we have that

P_{J_{l^{'}, i^{'}}}^{(+)} [P_{J_{l, i}}^{(+)} V_{J_{l, i}}^{(k, q)} P_{J_{l, i}}^{(+)} + P_{J_{l, i}}^{(-)} V_{J_{l, i}}^{(k, q)} P_{J_{l, i}}^{(-)}] P_{J_{l^{'}, i^{'}}}^{(-)} = 0

for J_l′,i′ with J_l,i ⊂ J_l′,i′. This is seen by using

P_{J_{l, i}}^{(+)} P_{J_{l^{'}, i^{'}}}^{(-)} = 0

(4.64)

in the first term and

P_{J_{l, i}}^{(-)} V_{J_{l, i}}^{(k, q)} P_{J_{l, i}}^{(-)} P_{J_{l^{'}, i^{'}}}^{(-)} = P_{J_{l^{'}, i^{'}}}^{(-)} P_{J_{l, i}}^{(-)} V_{J_{l, i}}^{(k, q)} P_{J_{l, i}}^{(-)} P_{J_{l^{'}, i^{'}}}^{(-)}

(4.65)

combined with

P_{J_{l^{'}, i^{'}}}^{(+)} P_{J_{l^{'}, i^{'}}}^{(-)} = 0

(4.66)

in the second term.

Lemma 4.4.

Assuming (4.60), the following bound on the operator

G_{J_{k, q}}

holds:

\begin{align} P_{J_{k, q}}^{(+)} G_{J_{k, q}} P_{J_{k, q}}^{(+)} & \geq (1 - 2 \cdot C_{d} \cdot t \sum_{l = 1}^{\infty} {(l + 1)}^{2 d} \cdot t^{\frac{l - 1}{4}}) P_{J_{k, q}}^{(+)} \\ + P_{J_{k, q}}^{(+)} [t \sum_{J_{k_{(1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉 + \dots + t \sum_{J_{k_{(| k | - 1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(| k | - 1)}^{'}}, q^{'}}^{{(k, q)}_{- 1}} 〉] P_{J_{k, q}}^{(+)} \end{align}

(4.67)

for t ∈ [0, t_d), with t_d independent of (k, q) and N, where C_d is the d-dependent constant implicit in the estimate of the number of shapes in Remark 4.2.

Proof.

We observe that, due to Remark 4.3, for 1 ≤ j ≤ |k| − 1, we can write

P_{J_{k, q}}^{(+)} \sum_{J_{k_{(j)}^{'}, q^{'}} \subset J_{k, q}} V_{J_{k_{(j)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} P_{J_{k, q}}^{(+)} = P_{J_{k, q}}^{(+)} \sum_{J_{k_{(j)}^{'}, q^{'}} \subset J_{k, q}} P_{J_{k_{(j)}^{'}, q^{'}}}^{(+)} V_{J_{k_{(j)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} P_{J_{k_{(j)}^{'}, q^{'}}}^{(+)} P_{J_{k, q}}^{(+)} + P_{J_{k, q}}^{(+)} \sum_{J_{k_{(j)}^{'}, q^{'}} \subset J_{k, q}} ⟨ V_{J_{k_{(j)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} ⟩ P_{J_{k_{(j)}^{'}, q^{'}}}^{(-)} P_{J_{k, q}}^{(+)}

(4.68)

with

〈 V_{J_{l, i}}^{{(k, q)}_{- 1}} 〉 ≔ ⟨(⨂ \prod_{j \in J_{l, i}} P_{Ω_{j}}), V_{J_{l, i}}^{{(k, q)}_{- 1}} (⨂ \prod_{j \in J_{l, i}} P_{Ω_{j}})⟩ .

Furthermore, we can estimate

\pm P_{J_{k, q}}^{(+)} \sum_{J_{k_{(j)}^{'}, q^{'}} \subset J_{k, q}} P_{J_{k_{(j)}^{'}, q^{'}}}^{(+)} V_{J_{k_{(j)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} P_{J_{k_{(j)}^{'}, q^{'}}}^{(+)} P_{J_{k, q}}^{(+)}

(4.69)

\leq P_{J_{k, q}}^{(+)} \sum_{J_{k_{(j)}^{'}, q^{'}} \subset J_{k, q}} ‖ V_{J_{k_{(j)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} ‖ P_{J_{k_{(j)}^{'}, q^{'}}}^{(+)} P_{J_{k, q}}^{(+)}

(4.70)

\leq P_{J_{k, q}}^{(+)} t^{\frac{j - 1}{4}} \sum_{J_{k_{(j)}^{'}, q^{'}} \subset J_{k, q}} P_{J_{k_{(j)}^{'}, q^{'}}}^{(+)} P_{J_{k, q}}^{(+)}

(4.71)

\leq P_{J_{k, q}}^{(+)} C_{d} \cdot {(j + 1)}^{2 d} \cdot t^{\frac{j - 1}{4}} \sum_{i \in J_{k, q}} P_{Ω_{i}}^{⊥} P_{J_{k, q}}^{(+)},

(4.72)

where we have used the following:

the bound in (4.60) for the step from (4.70) to (4.71) and
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 ()]

\sum_{i \subset J_{k, q}} H_{i} \geq \sum_{i \in J_{k, q}} P_{Ω_{i}}^{⊥}

(4.73)

with ()–(), and we get

P_{J_{k, q}}^{(+)} G_{J_{k, q}} P_{J_{k, q}}^{(+)}

(4.74)

\geq P_{J_{k, q}}^{(+)} [(1 - C_{d} \cdot t \sum_{l = 1}^{\infty} {(l + 1)}^{2 d} \cdot t^{\frac{l - 1}{4}}) \sum_{i \in J_{k, q}} P_{Ω_{i}}^{⊥}] P_{J_{k, q}}^{(+)}

(4.75)

+ P_{J_{k, q}}^{(+)} [t \sum_{J_{k_{(1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉 P_{J_{k_{(1)}^{'}, q^{'}}}^{(-)} + \dots + t \sum_{J_{k_{(| k | - 1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(| k | - 1)}^{'}}, q^{'}}^{{(k, q)}_{- 1}} 〉 P_{J_{k_{(| k | - 1)}^{'}, q^{'}}}^{(-)}] P_{J_{k, q}}^{(+)} .

(4.76)

Next, we use the identity

1 = P_{J_{k_{(j)}^{'}, q^{'}}}^{(-)} + P_{J_{k_{(j)}^{'}, q^{'}}}^{(+)}

in the right-hand side of (), and we get

P_{J_{k, q}}^{(+)} G_{J_{k, q}} P_{J_{k, q}}^{(+)}

(4.77)

\geq P_{J_{k, q}}^{(+)} [(1 - C_{d} \cdot t \sum_{l = 1}^{\infty} {(l + 1)}^{2 d} \cdot t^{\frac{l - 1}{4}}) \sum_{i \in J_{k, q}} P_{Ω_{i}}^{⊥}] P_{J_{k, q}}^{(+)}

(4.78)

- P_{J_{k, q}}^{(+)} [t \sum_{J_{k_{(1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉 P_{J_{k_{(1)}^{'}, q^{'}}}^{(+)} + \dots + t \sum_{J_{k_{(| k | - 1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(| k | - 1)}^{'}}, q^{'}}^{{(k, q)}_{- 1}} 〉 P_{J_{k_{(| k | - 1)}^{'}, q^{'}}}^{(+)}] P_{J_{k, q}}^{(+)}

(4.79)

+ P_{J_{k, q}}^{(+)} [t \sum_{J_{k_{(1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉 + \dots + t \sum_{J_{k_{(| k | - 1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(| k | - 1)}^{'}}, q^{'}}^{{(k, q)}_{- 1}} 〉] P_{J_{k, q}}^{(+)} .

(4.80)

By invoking the obvious bound

| 〈 V_{J_{k_{(j)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉 | \leq ‖ V_{J_{k_{(j)}^{'}}, q^{'}}^{{(k, q)}_{- 1}} ‖,

we finally get

P_{J_{k, q}}^{(+)} G_{J_{k, q}} P_{J_{k, q}}^{(+)}

(4.81)

\geq P_{J_{k, q}}^{(+)} [(1 - 2 \cdot C_{d} \cdot t \sum_{l = 1}^{\infty} {(l + 1)}^{2 d} \cdot t^{\frac{l - 1}{4}}) \sum_{i \in J_{k, q}} P_{Ω_{i}}^{⊥}] P_{J_{k, q}}^{(+)}

(4.82)

+ P_{J_{k, q}}^{(+)} [t \sum_{J_{k_{(1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉 + \dots + t \sum_{J_{k_{(| k | - 1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(| k | - 1)}^{'}}, q^{'}}^{{(k, q)}_{- 1}} 〉] P_{J_{k, q}}^{(+)}

(4.83)

\geq (1 - 2 \cdot C_{d} \cdot t \sum_{l = 1}^{\infty} {(l + 1)}^{2 d} \cdot t^{\frac{l - 1}{4}}) P_{J_{k, q}}^{(+)}

(4.84)

+ P_{J_{k, q}}^{(+)} [t \sum_{J_{k_{(1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉 + \dots + t \sum_{J_{k_{(| k | - 1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(| k | - 1)}^{'}}, q^{'}}^{{(k, q)}_{- 1}} 〉] P_{J_{k, q}}^{(+)},

(4.85)

where Lemma A.1 is used for the last inequality and t(>0) is assumed small enough such that

1 - 2 \cdot C_{d} \cdot t \sum_{l = 1}^{\infty} {(l + 1)}^{2 d} \cdot t^{\frac{l - 1}{4}} > 0 .

(4.86)

□

Lemma 4.4 implies that under the assumption in (4.60), the Hamiltonian $G_{J_{k, q}}$ has a gap that can be estimated from below by $\frac{1}{2}$ 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

G_{J_{k, q}}

has a gap

Δ_{J_{k, q}} \geq \frac{1}{2}

above the ground state energy

E_{J_{k, q}} = t \sum_{J_{k_{(1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉 + \dots + t \sum_{J_{k_{(j_{k - 1})}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(j_{k - 1})}^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉

corresponding to the ground state vector

⨂_{i \in J_{k, q}} Ω_{i}

due to the identity

\begin{matrix} P_{J_{k, q}}^{(-)} G_{J_{k, q}} P_{J_{k, q}}^{(-)} \end{matrix}

(4.87)

\begin{matrix} = & P_{J_{k, q}}^{(-)} [t \sum_{J_{k_{(1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉 P_{J_{k_{(1)}^{'}, q^{'}}}^{(-)} + \dots + t \sum_{J_{k_{(| k | - 1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(| k | - 1)}^{'}}, q^{'}}^{{(k, q)}_{- 1}} 〉 P_{J_{k_{(| k | - 1)}^{'}, q^{'}}}^{(-)}] P_{J_{k, q}}^{(-)} \\ = & P_{J_{k, q}}^{(-)} [t \sum_{J_{k_{(1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(1)}^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉 + \dots + t \sum_{J_{k_{(| k | - 1)}^{'}, q^{'}} \subset J_{k, q}} 〈 V_{J_{k_{(| k | - 1)}^{'}}, q^{'}}^{{(k, q)}_{- 1}} 〉] P_{J_{k, q}}^{(-)} . \end{matrix}

Remark 4.6.

Estimates () and ()–() 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 J_k,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,

\sum_{i \subset J_{k, q}} H_{i}

⁠, i.e., for

δ ≔ dist (z; σ (\sum_{i \subset J_{k, q}} H_{i})) > 0

⁠, the resolvent

\frac{1}{P_{J_{k, q}}^{(+)} (G_{J_{k, q}} - E_{J_{k, q}} - z) P_{J_{k, q}}^{(+)}}

(4.88)

is well defined on

P_{J_{k, q}}^{(+)} H^{(N)}

⁠, provided that 0 < t < t_δ,z, where the constant t_δ,z only depends on δ and z but is independent of J_k,q.

V. CONTROL OF $‖ V_{J_{r, i}}^{(k, q)} ‖$

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 t_d > 0 such that for 0 ≤ t < t_d, the Hamiltonians $G_{J_{k, q}}$ and $K_{N^{d}}^{(k, q)}$ are well defined, and for any rectangle J_r,i, with r = |r| ≥ 1, and for x_d ≔ 20d, we have the following:

Let (k, q)_* ≔ (k_*, q_*) be defined for some (k_*, q_*) such that $| k_{*} | = ⌊ r^{\frac{1}{4}} ⌋$ , where ⌊·⌋ is the integer part. If (k, q) ≺ (k,q)_*, then
$‖ V_{J_{r, i}}^{(k, q)} ‖ \leq \frac{t^{\frac{r - 1}{3}}}{r^{x_{d} + 2 d}} .$
(5.89)
Let (k, q)_** ≔ (k_**, q_**) be defined for some (k_**, q_**) such that $| k_{* *} | = r - ⌊ r^{\frac{1}{4}} ⌋$ . If (k, q)_** ≻ (k, q) ≥ (k, q)_*, then
$‖ V_{J_{r, i}}^{(k, q)} ‖ \leq 2 \cdot \frac{t^{\frac{r - 1}{3}}}{r^{x_{d} + 2 d}} .$
(5.90)
If (r, i) ≻ (k, q) ⪰ (k,q)_**, then
$‖\frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1}, P_{J_{r, i}}^{(#)}, V_{J_{r, i}}^{(k, q)}, P_{J_{r, i}}^{(\hat{#})}, \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1}‖ \leq 3 \cdot \frac{t^{\frac{r - 1}{3}}}{r^{x_{d} + 2 d}}, #, \hat{#} = \pm$
(5.91)
and
$‖ V_{J_{r, i}}^{(k, q)} ‖ \leq 48 \cdot \frac{t^{\frac{r - 1}{3}}}{r^{x_{d}}} .$
(5.92)
If (k, q) ⪰ (r, i), then
$‖ V_{J_{r, i}}^{(k, q)} ‖ \leq 96 \cdot \frac{t^{\frac{r - 1}{3}}}{r^{x_{d}}} .$
(5.93)
$G_{J_{{(k, q)}_{+ 1}}}$ has spectral gap $Δ_{J_{{(k, q)}_{+ 1}}} \geq \frac{1}{2}$ above its ground state energy, where $G_{J_{k, q}}$ is defined in (2.26) for |k| ≥ 2, and
$G_{J_{{(1_{j}, q)}_{+ 1}}} ≔ H_{J_{{(1_{j}, q)}_{+ 1}}}^{(0)} ≔ \sum_{i \in J_{{(1_{j}, q)}_{+ 1}}} H_{i},$
provided that ${(1_{j}, q)}_{+ 1}$ is of the form (1_j′, q′) for some j′ and q′; (1_j, q) is defined in (1.13).

Proof.

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 $G_{J_{{(k, q)}_{+ 1}}}$ ⁠, and it is not defined for (k, q) = (N − 1, 1)). That is, we assume that (S1) holds for all $V_{J_{r, i}}^{(k^{'}, q^{'})}$ with (k′, q′) ≺ (k, q) and (S2) for all (k′, q′) ≺ (k, q). Then, we show that they hold for all $V_{J_{r, i}}^{(k, q)}$ and for $G_{J_{{(k, q)}_{+ 1}}}$ ⁠. By Lemma A.3, this implies that $S_{J_{k, q}}$ and, consequently, that $K_{Λ_{N}^{d}}^{(k, q)}$ are well defined operators [see (4.54)].

For (k, q) = (0, N), (S1) can be verified by direct computation because

‖ V_{J_{1_{j}, q}}^{(0, N)} ‖ = ‖ V_{J_{1_{j}, q}} ‖ \leq 1,

and

V_{J_{r, i}}^{(0, N)} = 0

otherwise; (S2) holds trivially since, by definition, (0, N)₊₁ = (1₁, 1) and

G_{J_{1_{1}, 1}} = H_{J_{1_{1}, 1}}^{(0)}

[recall that 1_j is defined in ()].

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 t_d > 0 for which all the stages hold true for 0 ≤ t < t_d.

Warning: Throughout the proof, several positive constants are introduced. We shall use the symbols c, C for those that are universal and the symbols c_d, C_d 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 J_r,i such that i ≠ q. Then, the possible cases are described in (a); see Definition 2.4, and we have that
$‖ V_{J_{r, i}}^{(k, q)} ‖ = ‖ V_{J_{r, i}}^{{(k, q)}_{- 1}} ‖ .$
(5.94)
Let k = 1, and assume that J_r,i is equal to J_k,q. Then, we refer to case (b) and find that
$‖ V_{J_{k, q}}^{(k, q)} ‖ \leq 2 ‖ V_{J_{k, q}}^{{(k, q)}_{- 1}} ‖ \leq 2,$
(5.95)
where we have following:
- the inequality $‖ V_{J_{k, q}}^{(k, q)} ‖ \leq 2 ‖ V_{J_{k, q}}^{{(k, q)}_{- 1}} ‖$ 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 $‖ V_{J_{k, q}}^{{(k, q)}_{- 1}} ‖ = ‖ V_{J_{k, q}}^{{(k, q)}_{- 2}} ‖ = \dots = ‖ V_{J_{k, q}}^{(0, N)} ‖ \leq 1$ ⁠.
Inequality (5.91) follows trivially by using $‖ \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ‖ \leq 1$ and $‖ P_{J_{r, i}}^{(#)} V_{J_{r, i}}^{(k, q)} P_{J_{r, i}}^{(\hat{#})} ‖ \leq ‖ V_{J_{r, i}}^{(k, q)} ‖$ ⁠.
Case r = 2.
This case is not much different from the one corresponding to r = 1 with the exception that
$‖V_{J_{r, i}}^{(k^{'}, q^{'})}‖ \leq ‖V_{J_{r, i}}^{{(k^{'}, q^{'})}_{- 1}}‖ + ‖\sum_{J_{k^{″}, q^{″}} \in G_{J_{r, i}}^{(k^{'}, q^{'})}}, \sum_{n = 1}^{\infty}, \frac{1}{n!}, a, d^{n}, S_{J_{k^{'}, q^{'}}}, (V_{J_{k^{″}, q^{″}}}^{{(k^{'}, q^{'})}_{- 1}})‖$
(5.96)
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 $‖ V_{J_{r, i}}^{(k, q)} ‖$ ⁠, 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 J_l′,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 J_r,i, but if (5.91) is verified, then, consequently, (5.92) and (5.93) also hold true [in step (k, q)].

A. Regime (⁠ $R 1$ ⁠)

Here, we apply the argument explained in Sec. III B in order to show that (S1) holds for

V_{J_{r, 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 () in Lemma A.3 so as to conclude that for any (bounded) operator V,

‖ A_{J_{k^{″}, q^{″}}} (V) ‖ \leq c \cdot t \cdot ‖ V_{J_{k^{″}, q^{″}}}^{{(k^{″}, q^{″})}_{- 1}} ‖ \cdot ‖ V ‖

(5.97)

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

V_{J_{r, i}}^{(k, q)}

according to the prescriptions of Definition 3.1. Consequently, the potential can be expressed as the sum

\sum_{b \in B_{V_{J_{r, 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

R_{b}

that are associated with

V_{J_{r, 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 () repeatedly, i.e.,

‖ A_{J_{k^{(1)}, q^{(1)}}} (A_{J_{k^{(2)}, q^{(2)}}} (\dots A_{J_{k^{(| R_{b} | - 1)}, q^{(| R_{b} | - 1)}}} (V_{L_{b}}) \dots)) ‖ \leq {(c \cdot t)}^{| R_{b} | - 1} ‖ V_{L_{b}} ‖ \prod_{i = 1}^{| R_{b} | - 1} ‖ V_{J_{k^{(i)}, q^{(i)}}}^{{(k^{(i)}, q^{(i)})}_{- 1}} ‖,

(5.98)

where

V_{L_{b}}

is the potential labeling the leaf of

b

⁠, and compute the “weighted” number of sets

{J_{k^{(i)}, q^{(i)}}, i \in {1, \dots, | R_{b} |}}

⁠, weighted in the sense that each rectangle

J_{k^{(i)}, q^{(i)}}

is given the weight

c \cdot t \cdot ‖ V_{J_{k^{(i)}, q^{(i)}}}^{{(k^{(i)}, q^{(i)})}_{- 1}} ‖

except for the one associated with the leaf of the branch, that is, given the weight

‖ V_{L_{b}} ‖

⁠.

Following the scheme described in Sec. III B, we estimate the weighted sum of sets $R_{b}$ in terms of a weighted sum of paths $Γ_{b}$ ⁠. 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 ${(c \cdot t)}^{| R_{b} | - 1}$ that is small enough due to the upper bound on k, $k \leq ⌊ r^{\frac{1}{4}} ⌋$ ⁠, in regime $R 1$ ⁠. Indeed, the latter implies the lower bound $| R_{b} | \geq ⌊ c_{d} \cdot r / k ⌋$ ⁠.

In detail, concerning the powers of t, note that from the product

{(c \cdot t)}^{| R_{b} | - 1} ‖ V_{L_{b}} ‖ \prod_{i = 1}^{| R_{b} | - 1} ‖ V_{J_{k^{(i)}, q^{(i)}}}^{{(k^{(i)}, q^{(i)})}_{- 1}} ‖,

(5.99)

we get at least

t^{\frac{r - 1}{3}}

due to (1) the requirement that J_r,i is the minimal rectangle associated with

\cup_{i \in {1, \dots, | R_{b} |}} J_{k^{(i)}, q^{(i)}}

and (2) borrowing a power

t^{\frac{2}{3}}

from each factor t in

{(c \cdot t)}^{| R_{b} | - 1}

⁠. Hence, in product (), we can factor out

t^{\frac{r - 1}{3}}

and keep a power

t^{\frac{1}{3}}

for each rectangle of

R_{b}

except the one associated with the leaf of the branch. This also means that we can assign at least a factor

(c + 1) \frac{t^{1 / 6}}{ρ^{x_{d}}}

(5.100)

to each rectangle of size ρ in

R_{b}

⁠.

Consider the rectangles of the set

supp (Z_{ρ}^{(j)})

(see Sec. III B 2): there are

n_{ρ}^{(j)}

such rectangles, and for the constructed paths

Γ_{b}

⁠, there are at most

2 n_{ρ}^{(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

S_{Γ_{b}} ∋ σ = (J_{s^{(i)}, u^{(i)}}, J_{s^{(i + 1)}, u^{(i + 1)}})

we assign the weight

w_{σ} ≔ {((c + 1) \frac{t^{1 / 6}}{s_{σ}^{x_{d}}})}^{1 / 2}

with s_σ ≔ max{s⁽ⁱ⁾, s⁽ⁱ⁺¹⁾}, with w_σ < 1 for t > 0 sufficiently small.

From the considerations regarding () and (), we get the first inequality in the following formula:

{(c \cdot t)}^{| R_{b} | - 1} ‖ V_{L_{b}} ‖ \prod_{i = 1}^{| R_{b} | - 1} ‖ V_{J_{k^{(i)}, q^{(i)}}}^{{(k^{(i)}, q^{(i)})}_{- 1}} ‖ \leq t^{\frac{r - 1}{3}} \prod_{ρ = 1; j_{ρ} \neq 0}^{k} \prod_{j = 1}^{j_{ρ}} {((c + 1) \frac{t^{1 / 6}}{ρ^{x_{d}}})}^{n_{ρ}^{(j)}} \leq t^{\frac{r - 1}{3}} \cdot \prod_{σ \in S_{Γ_{b}}} w_{σ},

(5.101)

whereas for the second inequality, we use the following observation: if we denote by

S_{Z_{ρ}^{(j)}}

the set consisting of at most

2 n_{ρ}^{(j)} - 2

steps between rectangles of

supp Z_{ρ}^{(j)}

and the additional at most two steps from rectangles of lower size and back, then we have

{((c + 1) \frac{t^{\frac{1}{6}}}{ρ^{x_{d}}})}^{n_{ρ}^{(j)}} \leq \prod_{σ \in S_{Z_{ρ}^{(j)}}} w_{σ}

since w_σ,

σ \in S_{Z_{ρ}^{(j)}}

⁠, coincides with

{((c + 1) \frac{t^{\frac{1}{6}}}{ρ^{x_{d}}})}^{\frac{1}{2}} < 1

and

| S_{Z_{ρ}^{(j)}} | \leq 2 n_{ρ}^{(j)}

⁠, by construction.

Hence, the total weighted number of rectangles

\sum_{b \in B_{V_{J_{r, i}}^{(k, q)}}} {(c \cdot t)}^{| R_{b} | - 1} ‖ V_{L_{b}} ‖ \prod_{i = 1}^{| R_{b} | - 1} ‖ V_{J_{k^{(i)}, q^{(i)}}}^{{(k^{(i)}, q^{(i)})}_{- 1}} ‖ \leq \sum_{Γ_{b, b \in B_{V_{J_{r, i}}^{(k, q)}}}} t^{\frac{r - 1}{3}} \cdot \prod_{σ \in S_{Γ_{b}}} w_{σ}

(5.102)

can be bounded from above by estimating the number of weighted paths

Γ_{b}

as follows:

\sum_{Γ_{b}, b \in B_{V_{J_{r, i}}^{(k, q)}}} t^{\frac{r - 1}{3}} \cdot \prod_{σ \in S_{Γ_{b}}} w_{σ}

(5.103)

\leq C_{d} \cdot r^{2 d - 1} \cdot t^{\frac{r - 1}{3}} \cdot \sum_{j = ⌊ c_{d} \cdot r / k ⌋}^{\infty} {(\sum_{ρ, ρ^{'} = 1}^{k} {((c + 1) \frac{t^{1 / 6}}{{(\max {ρ, ρ^{'}})}^{x_{d}}})}^{1 / 2} D_{ρ, ρ^{'}})}^{j},

(5.104)

where we have the following:

$\sum_{ρ^{'} = 1}^{k} {((c + 1) \frac{t^{1 / 6}}{{(\max {ρ, ρ^{'}})}^{x_{d}}})}^{1 / 2} D_{ρ, ρ^{'}}$
accounts for all the weighted directions for a step from a rectangle of size ρ, where D_ρ,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 C_d ⋅ r^2d−1 is a bound³¹ on the number of possible initial rectangles of a fixed path $Γ_{b}$ ⁠;
the sum over j is the sum over the number of steps of $Γ_{b}$ ⁠, which by construction is bounded from below by ⌊c_d · r/k⌋.

Next, we bound

(5.104) \leq C_{d} \cdot r^{2 d - 1} \cdot t^{\frac{r - 1}{3}} \cdot \sum_{j = ⌊ c_{d} \cdot r / k ⌋}^{\infty} {({(c + 1)}^{1 / 2} \cdot t^{\frac{1}{12}} \cdot 2 \sum_{ρ = 1}^{k} \frac{ρ \cdot D_{ρ, ρ}}{ρ^{x_{d} / 2}})}^{j}

(5.105)

\begin{matrix} \leq & C_{d} \cdot r^{2 d - 1} \cdot t^{\frac{r - 1}{3}} \cdot t^{\frac{1}{24} \cdot ⌊ c_{d} \cdot r / k ⌋} \cdot \sum_{j = ⌊ c_{d} \cdot r / k ⌋}^{\infty} {((c + 1) \cdot t^{\frac{1}{24}} \cdot 2 \sum_{ρ = 1}^{k} \frac{ρ \cdot D_{ρ, ρ}}{ρ^{x_{d} / 2}})}^{j} \\ \leq & \frac{t^{\frac{r - 1}{3}}}{r^{x_{d} + 2 d}}, \end{matrix}

(5.106)

where t ≥ 0 has been chosen small enough such that (recall

k \leq ⌊ r^{\frac{1}{4}} ⌋

⁠)

C_{d} \cdot r^{4 d - 1 + x_{d}} \cdot t^{\frac{1}{24} \cdot ⌊ c_{d} \cdot r / k ⌋} \cdot \sum_{j = ⌊ c_{d} \cdot r / k ⌋}^{\infty} {({(c + 1)}^{1 / 2} \cdot t^{\frac{1}{24}} \cdot 2 \sum_{ρ = 1}^{k} \frac{ρ \cdot D_{ρ, ρ}}{ρ^{x_{d} / 2}})}^{j} < 1 .

(5.107)

B. Regime (⁠ $R 2$ ⁠)

For (k, q) in this regime, starting from the inequality

‖ V_{J_{r, i}}^{(k, q)} ‖ \leq ‖ V_{J_{r, i}}^{{(k, q)}_{- 1}} ‖ + ‖ \sum_{J_{k^{'}, q^{'}} \in G_{J_{r, i}}^{(k, q)}} \sum_{n = 1}^{\infty} \frac{1}{n!} a d^{n} S_{J_{k, q}} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}}) ‖,

(5.108)

we only keep expanding the first potential on the right-hand side. Then, using inductive hypotheses (), (), (), and (), for t ≥ 0 sufficiently small, we can estimate

‖ V_{J_{r, i}}^{(k, q)} ‖ \leq ‖ V_{J_{r, i}}^{(k_{*}, q_{*})} ‖

(5.109)

+ \sum_{s = ⌊ r^{\frac{1}{4}} ⌋}^{r - ⌊ r^{\frac{1}{4}} ⌋} \sum_{s_{1} = 0}^{s} \sum_{s_{2} = 0}^{s - s_{1}} \dots \sum_{s_{d} = 0}^{s - s_{1} - \dots - s_{d - 1}} δ_{s_{1} + s_{2} + \dots + s_{d} - s} \cdot c_{d} \cdot r^{2 d} \cdot t \cdot \frac{t^{\frac{s - 1}{3}}}{s^{x_{d}}} \cdot \frac{t^{\frac{r - s - 1}{3}}}{{(r - s)}^{x_{d}}},

(5.110)

where we have the following:

(k_*, q_*) is the greatest rectangle of regime $R 1$ with respect to the ordering ≻, and by construction, $k_{*} = ⌊ r^{\frac{1}{4}} ⌋$ ⁠;
the factor
$c_{d} \cdot r^{2 d - 1} \cdot t \cdot \frac{t^{\frac{s - 1}{3}}}{s^{x_{d}}} \cdot \frac{t^{\frac{r - s - 1}{3}}}{{(r - s)}^{x_{d}}}$
is an upper bound to the sum of the products of the type $‖ A_{J_{(k, q})_{- j}} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- j - 1}}) ‖$ for some j, where the size of the rectangle associated with (k, q)_−j is equal to s;
the multiplicative factor $O (r^{2 d - 1})$ is an upper bound estimate (see Remark 4.2) to the number of rectangles J_k′,q′ ⊂ J_r,i such that [J_k′,q′ ∪ J_k,q] = J_r,i.

Now, for any s with

⌊ r^{\frac{1}{4}} ⌋ \leq s \leq r - ⌊ r^{\frac{1}{4}} ⌋

⁠, we have

\sum_{s_{1} = 0}^{s} \sum_{s_{2} = 0}^{s - s_{1}} \dots \sum_{s_{d} = 0}^{s - s_{1} - \dots - s_{d - 1}} δ_{s_{1} + s_{2} + \dots + s_{d} - s} \cdot c_{d} \cdot r^{2 d} \cdot t \cdot \frac{t^{\frac{s - 1}{3}} \cdot}{s^{x_{d}}} \frac{t^{\frac{r - s - 1}{3}}}{{(r - s)}^{x_{d}}}

(5.111)

\leq s^{d} \cdot c_{d} \cdot r^{2 d - 1} \cdot t^{\frac{2}{3}} \cdot \frac{t^{\frac{r - 1}{3}}}{s^{x_{d}} \cdot {(r - s)}^{x_{d}}}

(5.112)

\leq r^{d} \cdot c_{d} \cdot r^{2 d - 1} \cdot t^{\frac{2}{3}} \cdot \frac{t^{\frac{r - 1}{3}}}{s^{x_{d}} \cdot {(r - s)}^{x_{d}}}

(5.113)

\leq 2^{x_{d}} \cdot c_{d} \cdot r^{2 d - 1} \cdot t^{\frac{2}{3}} \cdot \frac{t^{\frac{r - 1}{3}}}{r^{x_{d}} \cdot r^{x_{d} / 4}}

(5.114)

as

\max_{⌊ r^{\frac{1}{4}} ⌋ \leq s \leq r - ⌊ r^{\frac{1}{4}} ⌋} \frac{1}{s^{x_{d}} \cdot {(r - s)}^{x_{d}}} \leq \frac{1}{r^{x_{d} / 4} \cdot {(r - r^{\frac{1}{4}})}^{x_{d}}} \leq \frac{2^{x_{d}}}{r^{x_{d}} \cdot r^{x_{d} / 4}}

since

r - ⌊ r^{\frac{1}{4}} ⌋ \geq \frac{r}{2}

⁠. However, then, using the inductive hypothesis for

‖ V_{J_{r, i}}^{{(k, q)}_{*}} ‖

⁠,

‖ V_{J_{r, i}}^{(k, q)} ‖ \leq ‖ V_{J_{r, i}}^{{(k, q)}_{*}} ‖ + \sum_{s = ⌊ r^{\frac{1}{4}} ⌋}^{r - ⌊ r^{\frac{1}{4}} ⌋} r^{d} \cdot 2^{x_{d}} \cdot c_{d} \cdot r^{2 d - 1} \cdot t^{\frac{2}{3}} \cdot \frac{t^{\frac{r - 1}{3}}}{r^{x_{d}} \cdot r^{x_{d} / 4}}

(5.115)

\leq \frac{t^{\frac{r - 1}{3}}}{r^{x_{d} + 2 d}} + 2^{x_{d}} \cdot c_{d} \cdot t^{\frac{2}{3}} \cdot \frac{t^{\frac{r - 1}{3}}}{r^{\frac{5 x_{d}}{4} - 3 d}}

(5.116)

\leq 2 \cdot \frac{t^{\frac{r - 1}{3}}}{r^{x_{d} + 2 d}}

(5.117)

since x_d = 20d and t ≥ 0 is small enough.

C. Regime (⁠ $R 3$ ⁠)

Proof of (5.91).

For (k, q)_** ≺ (k, q) ≺ (r, i), we first consider

\frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} P_{J_{r, i}}^{(+)} V_{J_{r, i}}^{(k, q)} P_{J_{r, i}}^{(-)} \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1}

(5.118)

= \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} P_{J_{r, i}}^{(+)} V_{J_{r, i}}^{(k, q)} P_{J_{r, i}}^{(-)} .

(5.119)

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 ()–(), we get

\frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} P_{J_{r, i}}^{(+)} V_{J_{r, i}}^{(k, q)} P_{J_{r, i}}^{(-)}

(5.120)

= \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} P_{J_{r, i}}^{(+)} V_{J_{r, i}}^{{(k, q)}_{- 1}} P_{J_{r, i}}^{(-)}

(5.121)

+ \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} P_{J_{r, i}}^{(+)} \{\sum_{n = 1}^{\infty} \frac{1}{n!} a d^{n} S_{J_{k, q}} (V_{J_{r, i}}^{{(k, q)}_{- 1}})\} P_{J_{r, i}}^{(-)}

(5.122)

+ \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} P_{J_{r, i}}^{(+)} \{\sum_{J_{k^{'}, q^{'}} \in G_{J_{r, i}}^{(k, q)}} \sum_{n = 1}^{\infty} \frac{1}{n!} a d^{n} S_{J_{k, q}} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}})\} P_{J_{r, i}}^{(-)} .

(5.123)

We shall keep re-expanding the terms analogous to

V_{J_{r, i}}^{{(k, q)}_{- 1}}

in () from (k, q)₋₁ down to (k_**, q_**). The pair (k_**, q_**) represents the greatest rectangle with respect to the ordering ≻ in regime

R 2

and by construction has

k_{* *} = r - ⌊ r^{\frac{1}{4}} ⌋

⁠.

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 $V_{J_{r, i}}^{{(k, q)}_{- 1}}$ and $V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}}$ that are contained in them.

Estimate of (5.122).

Concerning (), we observe that using inductive hypotheses ()–() along with Lemma (A.3), we can bound

‖ (5.122) ‖ \leq c \cdot t \cdot \frac{t^{\frac{k - 1}{3}}}{k^{x_{d}}} \cdot \frac{t^{\frac{r - 1}{3}}}{r^{x_{d}}} .

(5.124)

At fixed k, the number of contributions of type () can be estimated from above by

O (r^{d} \cdot k^{d - 1})

⁠; see Remark 4.2. Being

k \geq r - ⌊ r^{\frac{1}{4}} ⌋

in regime

R 3

⁠, the power

t^{\frac{k - 1}{3}}

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, (), into

(5.123)

(5.125)

= \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} \{P_{J_{r, i}}^{+}, \sum_{J_{k^{'}, q^{'}} \in G_{J_{r, i}}^{(k, q)}} a d S_{J_{k, q}} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}})\} P_{J_{r, i}}^{(-)}

(5.126)

+ \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} \{P_{J_{r, i}}^{+}, \sum_{J_{k^{'}, q^{'}} \in G_{J_{r, i}}^{(k, q)}} \sum_{n = 2}^{\infty} \frac{1}{n!} a d^{n} S_{J_{k, q}} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}})\} P_{J_{r, i}}^{(-)} .

(5.127)

In (), we distinguish J_k′,q′ small and large depending on whether (k′, q′) ≺ (k, q) or (k, q) ⪯ (k′, q′), respectively, and denote by

{(G_{J_{r, i}}^{(k, q)})}_{small}

the subset formed by the small J_k′,q′ belonging to the set

G_{J_{r, i}}^{(k, q)}

⁠. We call

{(5.126)}_{small} and {(5.126)}_{large},

respectively, the corresponding contributions to (). Next, we study some commutators that enter the expression ()_small estimated below. We observe that

[S_{J_{k, q}}, V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}}]

(5.128)

= [S_{J_{k, q}}, P_{J_{k^{'}, q^{'}}}^{(+)} V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} P_{J_{k^{'}, q^{'}}}^{(+)} + P_{J_{k^{'}, q^{'}}}^{(-)} V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} P_{J_{k^{'}, q^{'}}}^{(-)}]

(5.129)

= [S_{J_{k, q}}, P_{J_{k^{'}, q^{'}}}^{(+)} V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} P_{J_{k^{'}, q^{'}}}^{(+)} + ⟨ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ⟩ P_{J_{k^{'}, q^{'}}}^{(-)}]

(5.130)

= [S_{J_{k, q}}, P_{J_{k^{'}, q^{'}}}^{(+)} V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} P_{J_{k^{'}, q^{'}}}^{(+)}] - [S_{J_{k, q}}, < V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} > P_{J_{k^{'}, q^{'}}}^{(+)}],

(5.131)

where we have exploited that

V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}}

is block-diagonalized since, by definition, small means (k′, q′) ≺ (k, q). We also observe that

P_{J_{k^{'}, q^{'}}}^{(+)} P_{J_{r, i}}^{(-)} = 0

since J_k′,q′ ⊂ J_r,i by construction; hence,

P_{J_{r, i}}^{(+)} [S_{J_{k, q}}, P_{J_{k^{'}, q^{'}}}^{(+)} V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} P_{J_{k^{'}, q^{'}}}^{(+)}] P_{J_{r, i}}^{(-)}

(5.132)

- P_{J_{r, i}}^{(+)} [S_{J_{k, q}}, < V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} > P_{J_{k^{'}, q^{'}}}^{(+)}] P_{J_{r, i}}^{(-)}

(5.133)

= - P_{J_{r, i}}^{(+)} P_{J_{k^{'}, q^{'}}}^{(+)} V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} P_{J_{k^{'}, q^{'}}}^{(+)} S_{J_{k, q}} P_{J_{r, i}}^{(-)}

(5.134)

+ P_{J_{r, i}}^{(+)} < V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} > P_{J_{k^{'}, q^{'}}}^{(+)} S_{J_{k, q}} P_{J_{r, i}}^{(-)} .

(5.135)

We recall that for j ≥ 1,

{(S_{J_{k, q}})}_{j} ≔ \frac{1}{G_{J_{k, q}} - E_{J_{k, q}}} P_{J_{k, q}}^{(+)} {(V_{J_{k, q}}^{{(k, q)}_{- 1}})}_{j} P_{J_{k, q}}^{(-)} - h . c .,

(5.136)

and, from Lemma A.3, we get

‖ \sum_{j = 2}^{\infty} t^{j} {(S_{J_{k, q}})}_{j} ‖ \leq C \cdot t^{2} \cdot ‖ {(V_{J_{k, q}}^{(k, q - 1)})}_{1} ‖^{2}

(5.137)

for t ≥ 0 sufficiently small. Hence, we split ()_small into two contributions:

the leading order term
$- \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} P_{J_{r, i}}^{(+)} \times \{\sum_{J_{k^{'}, q^{'}} \in {(G_{J_{r; i}}^{(k, q)})}_{small}} P_{J_{k^{'}, q^{'}}}^{(+)} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉) P_{J_{k^{'}, q^{'}}}^{(+)} (\frac{t}{G_{J_{k, q}} - E_{J_{k, q}}} P_{J_{k, q}}^{(+)} V_{J_{k, q}}^{{(k, q)}_{- 1}} P_{J_{k, q}}^{(-)} - h . c .)\} P_{J_{r, i}}^{(-)}$
(5.138)
$= - \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} P_{J_{r, i}}^{(+)} \times \{\sum_{J_{k^{'}, q^{'}} \in {(G_{J_{r; i}}^{(k, q)})}_{small}} P_{J_{k^{'}, q^{'}}}^{(+)} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉) P_{J_{k^{'}, q^{'}}}^{(+)} \frac{t}{G_{J_{k, q}} - E_{J_{k, q}}} P_{J_{k, q}}^{(+)} V_{J_{k, q}}^{{(k, q)}_{- 1}} P_{J_{k, q}}^{(-)}\} P_{J_{r, i}}^{(-)},$
(5.139)
where we have used $P_{J_{k, q}}^{(+)} P_{J_{r, i}}^{(-)} = 0$ ⁠;
the remainder term
$- \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} P_{J_{r, i}}^{(+)} \{\sum_{J_{k^{'}, q^{'}} \in {(G_{J_{r; i}}^{(k, q)})}_{small}} P_{J_{k^{'}, q^{'}}}^{(+)} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - ⟨ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ⟩) P_{J_{k^{'}, q^{'}}}^{(+)} \sum_{j = 2}^{\infty} t^{j} {(S_{J_{k, q}})}_{j}\} P_{J_{r, i}}^{(-)} .$
(5.140)
In order to estimate the leading order term (5.138), we make use of the inequality
$\begin{matrix} ‖\frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} \{\sum_{J_{k^{'}, q^{'}} \in {(G_{J_{r; i}}^{(k, q)})}_{small}} P_{J_{k^{'}, q^{'}}}^{(+)} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉) P_{J_{k^{'}, q^{'}}}^{(+)} \frac{t}{G_{J_{k, q}} - E_{J_{k, q}}} P_{J_{k, q}}^{(+)} V_{J_{k, q}}^{{(k, q)}_{- 1}} P_{J_{k, q}}^{(-)}\} P_{J_{r, i}}^{(-)}‖ \\ \leq & ‖ \sum_{J_{k^{'}, q^{'}} \in {(G_{J_{r; i}}^{(k, q)})}_{small}} \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} P_{J_{r, i}}^{(+)} P_{J_{k^{'}, q^{'}}}^{(+)} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉) P_{J_{k^{'}, q^{'}}}^{(+)} ‖ \end{matrix}$
(5.141)
$\times t \cdot ‖ \frac{1}{G_{J_{k, q}} - E_{J_{k, q}}} P_{J_{k, q}}^{(+)} (\sum_{j \in J_{k, q}} P_{Ω_{j}}^{⊥} + 1) ‖ \cdot ‖ \frac{1}{\sum_{j \in J_{k, q}} P_{Ω_{j}}^{⊥} + 1} P_{J_{k, q}}^{(+)} V_{J_{k, q}}^{{(k, q)}_{- 1}} P_{J_{k, q}}^{(-)} ‖ .$
(5.142)
Now, we introduce the notation
${\sum^{̄}}_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}}} ≔ \sum_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}} \in {(G_{J_{r; i}}^{(k, q)})}_{small}; J_{k^{'}, q^{'}} \cap J_{k^{″}, q^{″}} = \emptyset}$
(5.143)
and
$\sum_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}}}^{'} ≔ \sum_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}} \in {(G_{J_{r; i}}^{(k, q)})}_{small}; J_{k^{'}, q^{'}} \cap J_{k^{″}, q^{″}} \neq \emptyset} .$
(5.144)
We can write
${‖\sum_{J_{k^{'}, q^{'}} \in {(G_{J_{r; i}}^{(k, q)})}_{small}}, \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1}, P_{J_{r, i}}^{(+)}, P_{J_{k^{'}, q^{'}}}^{(+)}, (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - ⟨ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ⟩), P_{J_{k^{'}, q^{'}}}^{(+)}‖}^{2}$
(5.145)
$\leq \sup_{‖ ψ ‖ = 1} {\sum^{̄}}_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}}} ⟨\frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ,$
(5.146)
$P_{J_{k^{'}, q^{'}}}^{(+)} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - ⟨ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ⟩) P_{J_{k^{'}, q^{'}}}^{(+)} P_{J_{k^{″}, q^{″}}}^{(+)} (V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} - ⟨ V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} ⟩) P_{J_{k^{″}, q^{″}}}^{(+)} \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ⟩$
$+ \sup_{‖ ψ ‖ = 1} \sum_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}}}^{'} ⟨\frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ,$
(5.147)
$P_{J_{k^{'}, q^{'}}}^{(+)} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - ⟨ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ⟩) P_{J_{k^{'}, q^{'}}}^{(+)} P_{J_{k^{″}, q^{″}}}^{(+)} (V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} - ⟨ V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} ⟩) P_{J_{k^{″}, q^{″}}}^{(+)} \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ⟩ .$

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

We observe that for

J_{k^{'}, q^{'}} \cap J_{k^{''}, q^{''}} = \emptyset

⁠, we have

P_{J_{k^{'}, q^{'}}}^{(+)} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉) P_{J_{k^{'}, q^{'}}}^{(+)} P_{J_{k^{″}, q^{″}}}^{(+)} (V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} 〉) P_{J_{k^{″}, q^{″}}}^{(+)}

(5.148)

= P_{J_{k^{'}, q^{'}}}^{(+)} P_{J_{k^{″}, q^{″}}}^{(+)}

(5.149)

\times (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉) P_{J_{k^{'}, q^{'}}}^{(+)} P_{J_{k^{″}, q^{″}}}^{(+)} (V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} 〉)

(5.150)

\times P_{J_{k^{'}, q^{'}}}^{(+)} P_{J_{k^{″}, q^{″}}}^{(+)}

(5.151)

since

[P_{J_{k^{'}, q^{'}}}^{(+)}, V_{J_{k^{''}, q^{''}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{''}, q^{''}}}^{{(k, q)}_{- 1}} 〉] = [V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉, P_{J_{k^{''}, q^{''}}}^{(+)}] = 0 .

On the contrary, we note that

[P_{J_{k^{'}, q^{'}}}^{(+)}, P_{J_{k^{″}, q^{″}}}^{(+)}] = [P_{J_{k^{'}, q^{'}}}^{(-)}, P_{J_{k^{″}, q^{″}}}^{(-)}] = 0

(5.152)

even if

J_{k^{'}, q^{'}} \cap J_{k^{''}, q^{''}} \neq \emptyset

⁠. Indeed,

P_{J_{k^{'}, q^{'}}}^{(-)} = P_{J_{k^{'}, q^{'}} \ J_{k^{''}, q^{''}}}^{(-)} ⨂ P_{J_{k^{'}, q^{'}} \cap J_{k^{''}, q^{''}}}^{(-)}, P_{J_{k^{''}, q^{''}}}^{(-)} = P_{J_{k^{''}, q^{''}} \ J_{k^{'}, q^{'}}}^{(-)} ⨂ P_{J_{k^{'}, q^{'}} \cap J_{k^{''}, q^{''}} .}^{(-)}

Hence,

P_{J_{k^{'}, q^{'}}}^{(-)} P_{J_{k^{″}, q^{″}}}^{(-)} = (P_{J_{k^{'}, q^{'}} \ J_{k^{″}, q^{″}}}^{(-)} ⨂ P_{J_{k^{'}, q^{'}} \cap J_{k^{″}, q^{″}}}^{(-)}) (P_{J_{k^{″}, q^{″}} \ J_{k^{'}, q^{'}}}^{(-)} ⨂ P_{J_{k^{'}, q^{'}} \cap J_{k^{″}, q^{″}}}^{(-)})

(5.153)

= P_{J_{k^{'}, q^{'}} \ J_{k^{″}, q^{″}}}^{(-)} ⨂ P_{J_{k^{'}, q^{'}} \cap J_{k^{″}, q^{″}}}^{(-)} ⨂ P_{J_{k^{″}, q^{″}} \ J_{k^{'}, q^{'}}}^{(-)}

(5.154)

= (P_{J_{k^{″}, q^{″}} \ J_{k^{'}, q^{'}}}^{(-)} ⨂ P_{J_{k^{'}, q^{'}} \cap J_{k^{″}, q^{″}}}^{(-)}) (P_{J_{k^{'}, q^{'}} \ J_{k^{″}, q^{″}}}^{(-)} ⨂ P_{J_{k^{'}, q^{'}} \cap J_{k^{″}, q^{″}}}^{(-)})

(5.155)

= P_{J_{k^{″}, q^{″}}}^{(-)} P_{J_{k^{'}, q^{'}}}^{(-)} .

(5.156)

Hence, we can estimate

\sup_{‖ ψ ‖ = 1} {\sum^{̄}}_{J_{k^{'}, q^{'}, J_{k^{″}, q^{″}}}} |〈 \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ,

(5.157)

\begin{matrix} P_{J_{k^{'}, q^{'}}}^{(+)} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉) P_{J_{k^{'}, q^{'}}}^{(+)} P_{J_{k^{″}, q^{″}}}^{(+)} (V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} 〉) P_{J_{k^{″}, q^{″}}}^{(+)} \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ 〉| \\ \leq & \sup_{‖ ψ ‖ = 1} {\sum^{̄}}_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}}} {‖\frac{P_{J_{k^{'}, q^{'}}}^{(+)} P_{J_{k^{″}, q^{″}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ‖}^{2} \cdot ‖ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} 〉 ‖ \cdot ‖ V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} - 〈 V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} 〉 ‖ \\ \leq & \sup_{‖ ψ ‖ = 1} {\sum^{̄}}_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}}} 4 ‖ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ‖ \cdot ‖ V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} ‖ \cdot {‖\frac{P_{J_{k^{'}, q^{'}}}^{(+)} P_{J_{k^{″}, q^{″}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ‖}^{2} \end{matrix}

(5.158)

\leq \sup_{‖ ψ ‖ = 1} \sum_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}} \in G_{r, i}^{(k, q)}} 4 ‖ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ‖ \cdot ‖ V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} ‖ \cdot {‖\frac{P_{J_{k^{'}, q^{'}}}^{(+)} P_{J_{k^{″}, q^{″}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ‖}^{2}

(5.159)

= \sup_{‖ ψ ‖ = 1} ⟨\sum_{J_{k^{'}, q^{'}} \in G_{J_{r, i}}^{(k, q)}} 2 ‖ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ‖ \frac{P_{J_{k^{'}, q^{'}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ, \sum_{J_{k^{″}, q^{″}} \in G_{J_{r, i}}^{(k, q)}} 2 ‖ V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} ‖ \frac{P_{J_{k^{″}, q^{″}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ⟩

(5.160)

= {‖\sum_{J_{k^{'}, q^{'}} \in G_{J_{r, i}}^{(k, q)}} 2 ‖ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ‖ \frac{P_{J_{k^{'}, q^{'}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1}‖}^{2},

(5.161)

where in the step from () and () we have used

[P_{J_{k^{'}, q^{'}}}^{(+)}, P_{J_{k^{''}, q^{''}}}^{(+)}] = 0

⁠.

Now, suppose that there are 1 ≤ l ≤ d 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 l ≤ d − 1, we get

‖\sum_{J_{k^{'}, q^{'}} \in G_{J_{r, i}}^{(k, q)}} 2 ‖ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ‖ \frac{P_{J_{k^{'}, q^{'}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1}‖

(5.162)

= ‖\sum_{s : \exists u with J_{s, u} \in G_{J_{r, i}}^{(k, q)}} \sum_{u : J_{s, u} \in G_{J_{r, i}}^{(k, q)}} 2 ‖ V_{J_{s, u}}^{{(k, q)}_{- 1}} ‖ \frac{P_{J_{s, u}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1}‖

(5.163)

\leq C \cdot \{\sum_{s_{1} = r_{1} - k_{1}}^{r} \dots \sum_{s_{l} = r_{l} - k_{l}}^{r} \sum_{s_{l + 1} = 0}^{r} \dots \sum_{s_{d} = 0}^{r} \frac{t^{(\sum_{j = 1}^{d} \frac{s_{j}}{3}) - \frac{1}{3}}}{{(s_{1} + \dots + s_{d})}^{x_{d}}} \cdot [\prod_{j = l + 1}^{d} (s_{j} + 1)]\}

(5.164)

(we call s₁, …, s_d the components of k′) where in the step from () and () we use the following:

an upper bound for $‖ V_{J_{s, u}}^{{(k, q)}_{- 1}} ‖$ that is independent of u by means of inductive hypothesis (5.93),
$‖ V_{J_{s, u}}^{{(k, q)}_{- 1}} ‖ \leq 96 \cdot \frac{t^{\frac{s - 1}{3}}}{s^{x_{d}};}$
(5.165)
the fact that, for fixed k′, if k_j ≠ r_j for j = 1, …, l, then $q_{1}^{'}, \dots, q_{l}^{'}$ are uniquely³² determined by the condition [J_k′,q′ ∪ J_k,q] = J_r,i;
the estimate
$\sum_{u : u_{1}, \dots, u_{l} = fixed, J_{s, u} \in G_{J_{r, i}}^{(k, q)}} P_{J_{s, u}}^{(+)} \leq \{\prod_{j = l + 1}^{d} (s_{j} + 1)\} \sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥}$
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 $\prod_{j = l + 1}^{d} (s_{j} + 1)$ replaced by 1.

Next, for j = 1, …, l, we set

ρ_{j} ≔ s_{j} - (r_{j} - k_{j}) \Rightarrow s_{j} = ρ_{j} + (r_{j} - k_{j}),

(5.166)

and we observe that since s_j ≥ r_j − k_j for j = 1, …, l and s_j ≥ 0 for j = l + 1, …, d, we have

{(s_{1} + \dots + s_{d})}^{x_{d}} \geq {(r_{1} - k_{1} + \dots + r_{l} - k_{l})}^{x_{d}} .

(5.167)

Hence, we can estimate

\begin{matrix} (5.164) \\ \leq & C \cdot t^{- 1 / 3} \cdot t^{\sum_{j = 1}^{l} \frac{(r_{j} - k_{j})}{3}} \cdot \{\sum_{s_{l + 1} = 0}^{r} \dots \sum_{s_{d} = 0}^{r} \frac{1}{{(r_{1} - k_{1} + \dots + r_{l} - k_{l})}^{x_{d}}} \cdot [t^{\sum_{j = l + 1}^{d} \frac{s_{j}}{3}} \cdot \prod_{j = l + 1}^{d} (s_{j} + 1)] \sum_{ρ_{1} = 0}^{\infty} \dots \sum_{ρ_{l} = 0}^{\infty} t^{\sum_{j = 1}^{l} \frac{ρ_{j}}{3}}\} \\ = & C \cdot t^{- 1 / 3} \cdot t^{\sum_{j = 1}^{l} \frac{(r_{j} - k_{j})}{3}} \cdot \frac{1}{{(r_{1} - k_{1} + \dots + r_{l} - k_{l})}^{x_{d}}} \end{matrix}

(5.168)

\begin{matrix} \times \{\sum_{s_{l + 1} = 0}^{r} \dots \sum_{s_{d} = 0}^{r} [t^{\sum_{j = l + 1}^{d} \frac{s_{j}}{3}} \cdot \prod_{j = l + 1}^{d} (s_{j} + 1)] \sum_{ρ_{1} = 0}^{\infty} \dots \sum_{ρ_{l} = 0}^{\infty} t^{\sum_{j = 1}^{l} \frac{ρ_{j}}{3}}\} \\ \leq & C_{d} \cdot t^{- 1 / 3} (\frac{t^{\frac{r - 1}{3}}}{t^{\frac{k - 1}{3}}}) \cdot \frac{1}{{(r_{1} - k_{1} + \dots + r_{l} - k_{l})}^{x_{d}}}, \end{matrix}

(5.169)

where we have exploited the following:

the quantity
$\sum_{s_{l + 1} = 0}^{r} \dots \sum_{s_{d} = 0}^{r} [t^{\sum_{j = l + 1}^{d} \frac{s_{j}}{3}} \cdot \prod_{j = l + 1}^{d} (s_{j} + 1)] \sum_{ρ_{1} = 0}^{\infty} \dots \sum_{ρ_{l} = 0}^{\infty} t^{\sum_{j = 1}^{l} \frac{ρ_{j}}{3}}$
(5.170)
is bounded from above by a d-dependent constant;
for the considered k,
$t^{\sum_{j = 1}^{l} \frac{(r_{j} - k_{j})}{3}} = t^{\frac{r - k}{3}}$
(5.171)
since k_j = r_j 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

a b \leq \frac{a^{2}}{2} + \frac{b^{2}}{2}

⁠, we estimate [recall the notation ∑′ in ()]

\sup_{‖ ψ ‖ = 1} \sum_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}}}^{'} |⟨\frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ,

(5.172)

\begin{matrix} P_{J_{k^{'}, q^{'}}}^{(+)} (V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} - ⟨ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ⟩) P_{J_{k^{'}, q^{'}}}^{(+)} P_{J_{k^{″}, q^{″}}}^{(+)} (V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} - ⟨ V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} ⟩) P_{J_{k^{″}, q^{″}}}^{(+)} \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ⟩| \\ \leq & \sup_{‖ ψ ‖ = 1} \sum_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}}}^{'} 4 ‖ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ‖ \cdot ‖ V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} ‖ \cdot \{\frac{1}{2} {‖\frac{P_{J_{k^{'}, q^{'}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ‖}^{2} + \frac{1}{2} {‖\frac{P_{J_{k^{″}, q^{″}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ‖}^{2}\} . \end{matrix}

(5.173)

Since the expression in () is symmetric under the permutation of J_k′,q′ with

J_{k^{''}, q^{''}}

⁠, we can write

(5.172)

(5.174)

\begin{matrix} \leq & \sup_{‖ ψ ‖ = 1} \sum_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}}}^{'} 4 ‖ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ‖ \cdot ‖ V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} ‖ \cdot \{{‖\frac{P_{J_{k^{'}, q^{'}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ‖}^{2}\} \\ = & \sup_{‖ ψ ‖ = 1} \sum_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}}}^{'} 4 ‖ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ‖ \cdot ‖ V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} ‖ \cdot \{⟨〈 \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ, \frac{P_{J_{k^{'}, q^{'}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ 〉⟩\} \\ = & \sup_{‖ ψ ‖ = 1} \{⟨〈 \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ, \sum_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}}}^{'} 4 ‖ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ‖ \cdot ‖ V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} ‖ \frac{P_{J_{k^{'}, q^{'}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} ψ 〉⟩\} \\ \leq & ‖\sum_{J_{k^{'}, q^{'}}, J_{k^{″}, q^{″}}}^{'} 4 ‖ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ‖ \cdot ‖ V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} ‖ \frac{P_{J_{k^{'}, q^{'}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1}‖ . \end{matrix}

(5.175)

With steps similar to ()–(), assume that there are 1 ≤ l ≤ d 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 () as [warning: for l = d,

\prod_{j = l + 1}^{d} (s_{j} + 1), \prod_{j = l + 1}^{d} s_{j}

must be replaced by 1 in () and related formulas]

\begin{matrix} (5.175) & \leq & ‖\sum_{J_{k^{'}, q^{'}} \in G_{J_{r, i}}^{(k, q)}} 4 ‖ V_{J_{k^{'}, q^{'}}}^{{(k, q)}_{- 1}} ‖ \frac{P_{J_{k^{'}, q^{'}}}^{(+)}}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} \sum_{J_{k^{″}, q^{″}} \in G_{J_{r, i}}^{(k, q)} : J_{k^{'}, q^{'}} \cap J_{k^{″}, q^{″}} \neq \emptyset} ‖ V_{J_{k^{″}, q^{″}}}^{{(k, q)}_{- 1}} ‖‖ \\ \leq & C \cdot \{\sum_{s_{1} = r_{1} - k_{1}}^{r} \dots \sum_{s_{l} = r_{l} - k_{l}}^{r} \sum_{s_{l + 1} = 0}^{r} \dots \sum_{s_{d} = 0}^{r} \frac{t^{(\sum_{j = 1}^{d} \frac{s_{j}}{3}) - \frac{1}{3}}}{{(s_{1} + \dots + s_{d})}^{x_{d}}} \cdot [\prod_{j = l + 1}^{d} (s_{j} + 1)]\} \\ \times (\sum_{w = r - k}^{r} \frac{t^{\frac{w - 1}{3}}}{w^{x_{d}}} \cdot (\prod_{j = 1}^{d} s_{j}) \cdot w^{d - 1}) \end{matrix}

(5.176)

due to the estimate

\sum_{J_{k^{''}, q^{''}} \in G_{J_{r, i}}^{(k, q)} : J_{s, q^{'}} \cap J_{k^{''}, q^{''}} \neq \emptyset} ‖ V_{J_{k^{''}, q^{''}}}^{{(k, q)}_{- 1}} ‖ \leq O (\sum_{w = r - k}^{r} \frac{t^{\frac{w - 1}{3}}}{w^{x_{d}}} \cdot (\prod_{j = 1}^{d} s_{j}) \cdot w^{d - 1}),

where we have the following:

$O ((\prod_{j = 1}^{d} s_{j}) \cdot w^{d - 1})$ bounds from above the number of rectangles $J_{w, q^{''}}$ overlapping with the rectangle J_s,q′;
$O (\frac{t^{\frac{w - 1}{3}}}{w^{x_{d}}})$ is the bound to $‖ V_{J_{w, q^{''}}}^{{(k, q)}_{- 1}} ‖$ ⁠, provided by the inductive hypotheses.

Next, using the definition in () and arguments as in () and (), we write

\begin{matrix} (5.176) \\ \leq & C \cdot {(t^{- 1 / 3} \frac{t^{\frac{r - 1}{3}}}{t^{\frac{k - 1}{3}}})}^{2} \cdot \{\frac{1}{{(r_{1} - k_{1} + \dots + r_{l} - k_{l})}^{x_{d}}} \cdot \sum_{s_{l + 1} = 0}^{r} \dots \sum_{s_{d} = 0}^{r} \sum_{w = r - k}^{r} [w^{d - 1} \frac{1}{w^{x_{d}}}] \\ \times [(\prod_{j = l + 1}^{d} s_{j}) \cdot t^{\sum_{j = l + 1}^{d} s_{j} / 3} \cdot \prod_{j = l + 1}^{d} (s_{j} + 1)] \sum_{ρ_{1} = 0}^{\infty} \dots \sum_{ρ_{l} = 0}^{\infty} t^{\sum_{j = 1}^{l} \frac{ρ_{j}}{3}} \prod_{j = 1}^{l} (ρ_{j} + r_{j} - k_{j})\} . \end{matrix}

(5.177)

Now, we multiply the right-hand side of () by

\frac{{(r_{1} - k_{1} + \dots + r_{l} - k_{l})}^{d}}{{(r_{1} - k_{1} + \dots + r_{l} - k_{l})}^{l}} \geq 1,

and we get

(5.176)

(5.178)

\leq C \cdot {(t^{- 1 / 3} \frac{t^{\frac{r - 1}{3}}}{t^{\frac{k - 1}{3}}})}^{2} \cdot \{\frac{{(r_{1} - k_{1} + \dots + r_{l} - k_{l})}^{d}}{{(r_{1} - k_{1} + \dots + r_{l} - k_{l})}^{x_{d}}} \cdot \sum_{s_{l + 1} = 0}^{r} \dots \sum_{s_{d} = 0}^{r} \sum_{w = r - k}^{r} [w^{d - 1} \frac{1}{w^{x_{d}}}]

(5.179)

\begin{matrix} \times [(\prod_{j = l + 1}^{d} s_{j}) \cdot t^{\sum_{j = l + 1}^{d} s_{j} / 3} \cdot \prod_{j = l + 1}^{d} (s_{j} + 1)] \sum_{ρ_{1} = 0}^{\infty} \dots \sum_{ρ_{l} = 0}^{\infty} t^{\sum_{j = 1}^{l} \frac{ρ_{j}}{3}} \prod_{j = 1}^{l} (\frac{ρ_{j} + r_{j} - k_{j}}{(r_{1} - k_{1} + \dots + r_{l} - k_{l})})\} \\ \leq & C_{d} \cdot {(t^{- 1 / 3} \frac{t^{\frac{r - 1}{3}}}{t^{\frac{k - 1}{3}}})}^{2} \cdot {(\frac{1}{{(r_{1} - k_{1} + \dots + r_{l} - k_{l})}^{x_{d} - d}})}^{2}, \end{matrix}

(5.180)

where, in the step from () and (), we have used that x_d ≥ d + 1 and that all the following quantities are bounded from above by a d-dependent constant:

${(r_{1} - k_{1} + \dots + r_{l} - k_{l})}^{x_{d} - d} \cdot \sum_{w = r - k}^{r} w^{d - 1} \frac{1}{w^{x_{d}}}$
$\sum_{s_{l + 1} = 0}^{r} \dots \sum_{s_{d} = 0}^{r} (\prod_{j = l + 1}^{d} s_{j}) \cdot t^{\sum_{j = l + 1}^{d} s_{j} / 3} \cdot \prod_{j = l + 1}^{d} (s_{j} + 1)$
$\sum_{ρ_{1} = 0}^{\infty} \dots \sum_{ρ_{l} = 0}^{\infty} t^{\sum_{j = 1}^{l} \frac{ρ_{j}}{3}} \prod_{j = 1}^{l} (\frac{ρ_{j} + r_{j} - k_{j}}{(r_{1} - k_{1} + \dots + r_{l} - k_{l})}) .$

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

Finally, we estimate () by exploiting the inequality

‖\frac{1}{G_{J_{k, q}} - E_{J_{k, q}}} (\sum_{j \in J_{k, q}} P_{Ω_{j}}^{⊥} + 1) \frac{1}{\sum_{j \in J_{k, q}} P_{Ω_{j}}^{⊥} + 1} P_{J_{k, q}}^{(+)} V_{J_{k, q}}^{{(k, q)}_{- 1}} P_{J_{k, q}}^{(-)}‖

(5.181)

\leq ‖\frac{1}{G_{J_{k, q}} - E_{J_{k, q}}} (\sum_{j \in J_{k, q}} P_{Ω_{j}}^{⊥} + 1)‖ ‖\frac{1}{\sum_{j \in J_{k, q}} P_{Ω_{j}}^{⊥} + 1} P_{J_{k, q}}^{(+)} V_{J_{k, q}}^{{(k, q)}_{- 1}} P_{J_{k, q}}^{(-)}‖

(5.182)

\leq 3 \cdot \frac{t^{\frac{k - 1}{3}}}{k^{x_{d} + 2 d}},

(5.183)

where the first factor can be estimated to be less than 3, provided that t_d is small enough, by using the bound in () (see Lemma 4.4) that holds due to (S2) in the previous step; for the second factor, we invoke the inductive hypothesis in ().

Hence, we conclude that at fixed k and with l components different from the corresponding components of r,

‖ (5.138) ‖ \leq C_{d} \cdot t^{2 / 3} \cdot \frac{t^{\frac{r - 1}{3}}}{{(r_{1} - k_{1} + \dots + r_{l} - k_{l})}^{x_{d} - d} \cdot k^{x_{d} + 2 d}} .

(5.184)

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 J_k′,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 S_k,q or two factors $‖ {(V_{J_{k, q}}^{{(k, q)}_{- 1}})}_{1} ‖$ [see (5.137)] or J_k′,q′ is large such that (k′, q′) ≻ (k, q); thus, we get at least an extra factor $O (t^{\frac{r - 2 r^{1 / 4} - 1}{3}})$ ⁠;
the bound from above, $O (r^{2 d - 1})$ ⁠, of the number of the elements of $G_{J_{r, i}}^{(k, q)}$ (see Remark 4.2). Hence, just using inductive hypotheses (5.92) and (5.93), we can estimate
$‖ (5.140) ‖ + ‖ {(5.126)}_{large} ‖ + ‖ (5.127) ‖$
(5.185)
$\leq C_{d} \cdot t \cdot r^{4 d - 1} \cdot t^{\frac{r - 2 \cdot r^{1 / 4} - 1}{3}} \cdot \frac{t^{\frac{r - 1}{3}}}{{(r - k)}^{x_{d}} \cdot k^{x_{d}}} .$
(5.186)
At fixed k, there are at most $O (r^{d} \cdot k^{d - 1})$ contributions of type (5.185).

Complete estimate of (5.91)

Finally, by the re-expansion outlined above and due to the estimates of (), (), (), ()_large, and () that have been derived [see (), (), and ()], we can conclude that

‖ \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} P_{J_{r, i}}^{(+)} V_{J_{r, i}}^{(k, q)} P_{J_{r, i}}^{(-)} ‖

(5.187)

\leq ‖ \frac{1}{\sum_{j \in J_{r, i}} P_{Ω_{j}}^{⊥} + 1} P_{J_{r, i}}^{(+)} V_{J_{r, i}}^{{(k, q)}_{* *}} P_{J_{r, i}}^{(-)} ‖

(5.188)

+ C_{d} \cdot t \cdot \sum_{l = 1}^{d} (\binom{d}{l}) \cdot \sum_{k_{1} = 0}^{r_{1} - 1} \dots \sum_{k_{l} = 0}^{r_{l} - 1} Θ (k - r + ⌊ r^{\frac{1}{4}} ⌋)

(5.189)

\times \frac{t^{\frac{r - 1}{3}}}{{(r_{1} - k_{1} + \dots + r_{l} - k_{l})}^{x_{d} - d} \cdot k^{x_{d} + 2 d}}

(5.190)

+ \sum_{k = r - ⌊ r^{\frac{1}{4}} ⌋}^{k = r - 1} \{c_{d} \cdot r^{d} \cdot k^{d - 1} \cdot t \cdot \frac{t^{\frac{k - 1}{3}} \cdot t^{\frac{r - 1}{3}}}{k^{x_{d}} \cdot r^{x_{d}}}

(5.191)

+ C_{d} \cdot t \cdot r^{5 d - 1} \cdot k^{d - 1} \cdot t^{\frac{r - 2 r^{\frac{1}{4}} - 1}{3}} \cdot \frac{t^{\frac{r - 1}{3}}}{{(r - k)}^{x_{d}} \cdot k^{x_{d}}}\},

(5.192)

where Θ is the characteristic function of

R^{+}

⁠; indeed,

k \geq r - ⌊ r^{\frac{1}{4}} ⌋

in regime

R 3

⁠. In addition to summand () that is smaller than

2 \cdot \frac{t^{\frac{r - 1}{3}}}{r^{x_{d} + 2 d}}

by the inductive hypothesis (), 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 () works.

Remark 5.2.

We point out that we have the following:

regarding the expression in (5.189) and (5.190), the factor $\frac{1}{k^{x_{d} + 2 d}}$ [coming from the inductive hypothesis used to estimate (5.142)] provides the expected behavior since $k \geq r - ⌊ r^{\frac{1}{4}} ⌋ </$

Local iterative block-diagonalization of gapped Hamiltonians: A new tool in singular perturbation theory

I. MODELS OF GAPPED QUANTUM LATTICE SYSTEMS AND SURVEY OF RESULTS

A. A family of quantum lattice systems

B. Main result

II. OUTLINE OF THE PROOF STRATEGY

A. Minimal rectangles

B. Effective Hamiltonians

C. The algorithm and the different regimes in the growth processes of rectangles

III. TREE STRUCTURE AND PATHS OF RECTANGLES

A. Tree expansion

1. Properties of the branches b∈BVJr,i(k,q)

B. Summing over the norms of branch-operator: Weights and paths, Γb

1. Paths of connected rectangles

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

3. Weighted sums of paths

IV. THE UNITARY CONJUGATION eSJk,q AND THE SPECTRAL GAP OF GJk,q

V. CONTROL OF ‖VJr,i(k,q)‖

A. Regime (⁠R1⁠)

B. Regime (⁠R2⁠)

C. Regime (⁠R3⁠)

1. Properties of the branches $b \in B_{V_{J_{r, i}}^{(k, q)}}$

B. Summing over the norms of branch-operator: Weights and paths, $Γ_{b}$

2. Connected components, $Z_{ρ}^{(j)}$ ⁠, of rectangles and definition of $Γ_{b}$

IV. THE UNITARY CONJUGATION $e^{S_{J_{k, q}}}$ AND THE SPECTRAL GAP OF $G_{J_{k, q}}$

V. CONTROL OF $‖ V_{J_{r, i}}^{(k, q)} ‖$

A. Regime (⁠ $R 1$ ⁠)

B. Regime (⁠ $R 2$ ⁠)

C. Regime (⁠ $R 3$ ⁠)