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arXiv:2001.10763

Computing local properties in the trivial phase

Yichen Huang

Published Jan 29, 2020• 1 citation
arXivHTMLPDF

Abstract

A translation-invariant gapped local Hamiltonian is in the trivial phase if it can be connected to a completely decoupled Hamiltonian with a smooth path of translation-invariant gapped local Hamiltonians. For the ground state of such a Hamiltonian, we show that the expectation value of a local observable can be computed in time $\text{poly}(1/δ)$ in one spatial dimension and $e^{\text{poly}\log(1/δ)}$ in two and higher dimensions, where $δ$ is the desired (additive) accuracy. The algorithm applies to systems of finite size and in the thermodynamic limit. It only assumes the existence but not any knowledge of the path.

DOI
10.48550/arxiv.2001.10763

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A translation-invariant gapped local Hamiltonian is in the trivial phase if it can be connected to a completely decoupled Hamiltonian with a smooth path of translation-invariant gapped local Hamiltonians. For the ground state of such a Hamiltonian, we show that the expectation value of a local observable can be computed in time poly(1/δ)\text{poly}(1/δ)poly(1/δ) in one spatial dimension and epolylog⁡(1/δ)e^{\text{poly}\log(1/δ)}epolylog(1/δ) in two and higher dimensions, where δδδ is the desired (additive) accuracy. The algorithm applies to systems of finite size and in the thermodynamic limit. It only assumes the existence but not any knowledge of the path.
We prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a continuous density of states above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. The type of interactions allowed for include translational invariance in a disorder (i.e., probabilistic) sense with some assumptions on the local distributions. Examples include many-body localization and random spin models. We calculate the scaling of the gap with the system's size when the local terms are distributed according to a Gaussian β orthogonal random matrix ensemble. As a corollary, there exist finite size partitions with respect to which the ground state is arbitrarily close to a product state. When the local eigenvalue distribution is discrete, in addition to the lack of an energy gap in the limit, we prove that the ground state has finite size degeneracies. The proofs are simple and constructive. This work excludes the important class of truly translationally invariant Hamiltonians where the local terms are all equal.
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Understanding the asymptotic behavior of physical quantities in the thermodynamic limit is a fundamental problem in statistical mechanics. In this paper, we study how fast the eigenstate expectation values of a local operator converge to a smooth function of energy density as the system size diverges. In translation-invariant quantum lattice systems in any spatial dimension, we prove that for all but a measure zero set of local operators, the deviations of finite-size eigenstate expectation values from the aforementioned smooth function are lower bounded by 1/O(N)1/O(N)1/O(N), where NNN is the system size. The lower bound holds regardless of the integrability or chaoticity of the model, and is saturated in systems satisfying the eigenstate thermalization hypothesis.
We show that the 2D local Hamiltonian problem with the constraint that the ground state obeys area laws is QMA-complete. We also prove similar results in 2D translation- invariant systems and for the 3D Heisenberg and Hubbard models with local magnetic fields. Consequently, unless MA = QMA, not all ground states of 2D local Hamiltonians with area laws have efficient classical representations that support efficient computation of local expectation values. In the future, even if area laws are proved for ground states of 2D gapped systems, the computational complexity of these systems remains unclear.