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arXiv:2007.07667 · Journal of Mathematical Physics

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

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

Published Jul 1, 2022• 6 citations
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Abstract

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.

DOI
10.1063/5.0084552

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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.
We study quantum chains whose Hamiltonians are perturbations by bounded interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. We prove that, for small values of a coupling constant, the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain.
We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions. Given such a Hamiltonian H0, we prove that there exists a constant threshold ϵ>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions, the perturbed Hamiltonian H=H0+ϵV has well-defined spectral bands originating from low-lying eigenvalues of H0. These bands are separated from the rest of the spectra and from each other by a constant gap. The band originating from the smallest eigenvalue of H0 has exponentially small width (as a function of the lattice size). Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively bounded operators, and the Lieb–Robinson bound.
For a class of Hamiltonians of XXZ spin chains in a uniform external magnetic field that are small quantum perturbations of an Ising Hamiltonian, it is shown that the spectral gap above the ground-state energy remains strictly positive when the perturbation is turned on, uniformly in the length of the chain. This result is proven for rather generic perturbations of both the ferromagnetic and the antiferromagnetic Ising Hamiltonian. In the antiferromagnetic case, the external magnetic field is required to be small. For a chain of an even number of sites, the two-fold degenerate ground-state energy of the unperturbed antiferromagnetic Hamiltonian may split into two energy levels separated by a very small gap. These results are proven by using a new, quite subtle refinement of a method developed in earlier work and used to iteratively block-diagonalize Hamiltonians of systems confined to ever larger subsets of a lattice by using strictly local unitary conjugations. The new method developed in this paper provides complete control of boundary effects on the low-energy spectrum of perturbed Ising chains uniformly in their length.
In this paper we study the low-lying spectrum of the AKLT model perturbed by small, finite-range potentials and with open boundary conditions imposed at the edges of the chain. Our analysis is based on the local, iterative Lie Schwinger block-diagonalization method which allows us to control small interaction terms localized near the boundary of the chain that are responsible for the possible splitting of the ground-state energy of the AKLT Hamiltonian into energy levels separated by small gaps, for which we provide an explicit formula. This improves earlier results concerning the persistence of the so-called bulk gap in these models, besides illustrating the power of our general methods in a non-trivial application.
We consider many-body quantum systems on a finite lattice, where the Hilbert space is the tensor product of finite-dimensional Hilbert spaces associated with each site, and where the Hamiltonian of the system is a sum of local terms. We are interested in proving uniform bounds on various properties as the size of the lattice tends to infinity. An important case is when there is a spectral gap between the lowest state(s) and the rest of the spectrum which persists in this limit, corresponding to what physicists call a “phase of matter.” Here, the combination of elementary Fourier analysis with the technique of Lieb–Robinson bounds (bounds on the velocity of propagation) is surprisingly powerful. We use this to prove exponential decay of connected correlation functions, a higher-dimensional Lieb–Schultz–Mattis theorem, and a Hall conductance quantization theorem for interacting electrons with disorder.
In this paper we study the low-lying spectrum of the AKLT model perturbed by small, finite-range potentials and with open boundary conditions imposed at the edges of the chain. Our analysis is based on the \emph{local, iterative Lie Schwinger block-diagonalization method} which allows us to control small interaction terms localized near the boundary of the chain that are responsible for the possible splitting of the ground-state energy of the AKLT Hamiltonian into energy levels separated by small gaps. This improves earlier results concerning the persistence of the so called \emph{bulk} gap in these models, besides illustrating the power of our general methods in a non-trivial application.