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arXiv:2108.13907 · Journal of Functional Analysis

Block-diagonalization of infinite-volume lattice Hamiltonians with unbounded interactions

Simone Del Vecchio, Jürg Martin Fröhlich, Alessandro Pizzo

Published Oct 5, 2022• 2 citations
arXivHTMLPDF
DOI
10.1016/j.jfa.2022.109734

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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.

arXiv:2007.07667•Journal of Mathematical Physics•Jul 1, 2022•6 citations
We study a three-dimensional SU(3) Hamiltonian lattice partitioned into disjoint cube blocks and single bridge links. Taking the untruncated, volume-uniform finite-box estimates of G4A as input, we construct the infinite-volume vacuum for inter-block plaquette strength 0 ≤ s ≤ 1/6144 and prove a gap at least 2/9 in its GNS representation. The full sequence of open-box vacua converges locally in trace norm to a pure, locally normal state. The gap also holds on the nontrivial cyclic space generated by bounded local gauge-invariant observables. The argument separates local energy tightness, the Nachtergaele–Sims–Young spectral flow for infinite-dimensional sites with unbounded on-site terms, and a centered spectral-measure proof of gap transfer. We state these requirements as an abstract transfer theorem. Additional quantitative consequences are an exponential spatial correlation bound with rate 32/599 in a specified coarse-site metric, convergent finite-history imaginary-time correlations with time-reflection positivity, and an isometric Spectral Chaos representation with an exact depth-norm identity. A local inverse-operator estimate gives excitation probability O(s2); combined with inherited one-cube rational data, it yields variance at least 1/300 for every internal real Wilson plaquette. Exact arithmetic and finite-geometry checks are supplied separately from the analytic proofs. All results concern a fixed lattice spacing and the stated deformed Hamiltonian. They do not establish boundary-condition-independent uniqueness, the isotropic point s = 1, or a continuum Yang–Mills theory.
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.