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arXiv:2209.01141 · Annales Henri Poincaré

Stability of the Spectral Gap and Ground State Indistinguishability for a Decorated AKLT Model

Angelo Lucia, Alvin Moon, Amanda Young

Published Dec 27, 2023• 3 citations
arXivHTMLPDF

Abstract

Abstract We use cluster expansion methods to establish local the indistiguishability of the finite volume ground states for the AKLT model on decorated hexagonal lattices with decoration parameter at least 5. Our estimates imply that the model satisfies local topological quantum order, and so, the spectral gap above the ground state is stable against local perturbations.

DOI
10.1007/s00023-023-01398-8

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Abstract We use cluster expansion methods to establish local the indistiguishability of the finite volume ground states for the AKLT model on decorated hexagonal lattices with decoration parameter at least 5. Our estimates imply that the model satisfies local topological quantum order, and so, the spectral gap above the ground state is stable against local perturbations.
We prove that the ground state of the AKLT models on the hexagonal lattice and the Lieb lattice satisfy the local topological quantum order (LTQO) condition. This will be a consequence of proving that the finite volume ground states are indistinguishable from a unique infinite volume ground state. Concretely, we identify a sequence of increasing and absorbing finite volumes for which any finite volume ground state expectation is well approximated by the infinite volume state with error decaying at a uniform exponential rate in the distance between the support of the observable and boundary of the finite volume. As a corollary to the LTQO property, we obtain that the spectral gap above the ground state in these models is stable under general small perturbations of sufficient decay. We prove these results by a detailed analysis of the polymer representation of the ground states state derived by Kennedy, Lieb and Tasaki (1988) with the necessary modifications required for proving the strong form of ground state indistinguishability needed for LTQO.
The nonzero spectral gap of the original two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) models has remained unproven for more than three decades. Recently, Abdul-Rahman et al. (arXiv:1901.09297) provided an elegant approach and proved analytically the existence of a nonzero spectral gap for the AKLT models on the decorated honeycomb lattice (for the number nnn of spin-1 decorated sites on each original edge no less than 3). We perform calculations for the decorated square lattice and show that the corresponding AKLT models are gapped if $n\ensuremath{\ge}4$. Combining both results, we also show that a family of decorated hybrid AKLT models, whose underlying lattice is of mixed vertex degrees 3 and 4, are also gapped for $n\ensuremath{\ge}4$. We develop a numerical approach that extends beyond what was accessible previously. Our numerical results further improve the nonzero gap to $n\ensuremath{\ge}2$, including the establishment of the gap for n=2n=2n=2 in the decorated triangular and cubic lattices. The latter case is interesting, as this shows that the AKLT states on the decorated cubic lattices are not N\'eel ordered, in contrast to the state on the undecorated cubic lattice.
We consider the spectral gap question for Affleck, Kennedy, Lieb, and Tasaki models defined on decorated versions of simple, connected graphs G. This class of decorated graphs, which are defined by replacing all edges of G with a chain of n sites, in particular includes any decorated multi-dimensional lattice. Using the Tensor Network States approach from [Abdul-Rahman et al., Analytic Trends in Mathematical Physics, Contemporary Mathematics (American Mathematical Society, 2020), Vol. 741, p. 1.], we prove that if the decoration parameter is larger than a linear function of the maximal vertex degree, then the decorated model has a nonvanishing spectral gap above the ground state energy.
Abstract We study the stability with respect to a broad class of perturbations of gapped ground-state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi–Hastings–Michalakis (BHM) strategy that under a condition of local topological quantum order (LTQO), the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work, we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of an indistinguishability radius, which we introduce. Using the uniform finite-volume results, we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state.
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.
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.