Signatures of Lattice Excitations in Quantum Channels: Limit of Parent Hamiltonians

TL;DR

This paper demonstrates that the low energy spectrum of parent Hamiltonians for injective Matrix Product States depends only on the quantum channel, revealing a connection between lattice excitations and quantum channels, with implications for localization.

Contribution

It introduces a framework linking the low energy spectrum of parent Hamiltonians to quantum channels, providing a new perspective on lattice excitations and localization phenomena.

Findings

Spectrum depends solely on the quantum channel properties.

Normal quantum channels simplify the spectral expression.

Framework offers insights into many-body localization.

Abstract

We prove that every injective Matrix Product State is the unique ground state of a simple hopping theory. We start by studying the low energy spectrum of parent Hamiltonians of injective Matrix Product States in a particular long range and system size limit under the validity of an asymptotic regime with low particle density. We show that in this limit a natural first quantization arises. This allows us to compute a specific type of low energy spectrum. This spectrum depends solely on the properties of a quantum channel, i.e. transfer matrix of the ground state, and not on any other details of the ground-state. We also review normal quantum channels for which the expression is more simplified. The construction possibly has some interesting uses for the study of quantum and classical Markov processes which we briefly expose. As an application, we revisit the notion of (many-body)-localization with our framework. Our calculations revealed that translational invariant Matrix Product States can be interpreted as a stationary sea of particles. As a next step rather than starting from some local Hamiltonian with random potentials, we consider fluctuations of the local tensors of a continuous one-parameter family of Matrix Product States. Localization in 1-dimension, is then understood from a simple study of spectral and mixing properties of finite dimensional quantum channels.