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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Lower bounds for the ground-state degeneracies of frustrated systems on fractal lattices

Evaldo M. F. Curado, Fernando Dantas Nobre

Published Dec 1, 2000• 1 citation
Publisher

Abstract

The total number of ground states for nearest-neighbor-interaction Ising systems with frustrations, defined on hierarchical lattices, is investigated. A simple method is presented, which allows one to factorize the ground-state degeneracy, at a given hierarchy level n, in terms of contributions due to all hierarchy levels. Such a method may yield the exact ground-state degeneracy of uniformly frustrated systems, whereas it works as an approximation for randomly frustrated models. In the latter cases, it is demonstrated that such an approximation yields lower-bound estimates for the ground-state degeneracies.

DOI
10.1103/physreve.62.8814

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The total number of ground states for nearest-neighbor-interaction Ising systems with frustrations, defined on hierarchical lattices, is investigated. A simple method is presented, which allows one to factorize the ground-state degeneracy, at a given hierarchy level n, in terms of contributions due to all hierarchy levels. Such a method may yield the exact ground-state degeneracy of uniformly frustrated systems, whereas it works as an approximation for randomly frustrated models. In the latter cases, it is demonstrated that such an approximation yields lower-bound estimates for the ground-state degeneracies.
We derive an exact lower bound to a universal measure of frustration in degenerate ground states of quantum many-body systems. The bound results in the sum of two contributions: entanglement and classical correlations arising from local measurements. We show that average frustration properties are completely determined by the behavior of the maximally mixed ground state. We identify sufficient conditions for a quantum spin system to saturate the bound, and for models with twofold degeneracy we prove that average and local frustration coincide.
We discuss a general scheme for the study of frustration in quantum systems. We introduce a universal measure of frustration for arbitrary quantum systems with Hamiltonians made of sums of local two-body contributions in terms of the overlap between the unfrustrated ground state of the pair Hamiltonians and the frustrated ground state of the total Hamiltonian. We derive an exact lower bound this overlap measure of frutration in degenerate ground states of quantum many-body systems. The bound results in the sum of two contributions: entanglement and classical correlations arising from local measurements. We show that average frustration properties are completely determined by the behavior of the maximally mixed ground state. We identify sufficient conditions for a quantum spin system to saturate the bound, and for models with twofold degeneracy we prove that average and local frustration coincide. If all the (pure) ground states of a given Hamiltonian saturate the inequality, then the system is said to be inequality saturating. These conditions provide a generalization to the quantum domain of the Toulouse criteria for classical frustration-free systems. The models satisfying these conditions can be reasonably identified as geometrically unfrustrated and subject to frustration of purely quantum origin. Our results therefore establish a unified framework for studying the intertwining of geometric and quantum contributions to frustration. The mathematical scheme can be put to test by relating the overlap measure and the inequalities it obeys to global and local observables in terms of the static structure factor and transverse n-point correlation functions.
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.
The Heisenberg-Ising Hamiltonian $H=\ensuremath{-}\frac{1}{2}{\ensuremath{\alpha}({\ensuremath{\sigma}}_{x}\ensuremath{\sigma}_{x}^{}{}_{}{}^{\ensuremath{'}}+{\ensuremath{\sigma}}_{y}\ensuremath{\sigma}_{y}^{}{}_{}{}^{\ensuremath{'}})+\ensuremath{\epsilon}({\ensuremath{\sigma}}_{z}\ensuremath{\sigma}_{z}^{}{}_{}{}^{\ensuremath{'}})}$ for rectangular one-, two-, or three-dimensional lattices are considered. The sum is over nearest neighbors and $\ensuremath{\Delta}=\frac{\ensuremath{\epsilon}}{\ensuremath{\alpha}}$ measures the anisotropy of the coupling. Upper and lower bounds for the ground-state energy are established and these bounds apply equally well to lattices of one, two, or three dimensions. Furthermore, it is shown that the ground-state energy per nearest-neighbor pair is nondecreasing as the dimension of the lattice (one, two or three) increases.
A frustrated spin-1/2 model is constructed on the square lattice, the Hamiltonian of which contains five types of interactions: nearest-neighbor (NN), diagonal (D), next-nearest-neighbor (NNN), knight's-move-distance away (KM), and further-neighbor diagonal (FND). The NN bonds in a particular direction, horizontal or vertical, are of alternating coupling strengths 7J and J. The other interaction strengths are: J (for NN and D), (J), J/2 (for NNN and KM), and J/4 (for FND). We prove that the exact ground state of the Hamiltonian is the columnar-dimer state.