Hyperbolic Deformation on Quantum Lattice Hamiltonians

TL;DR

This paper introduces a family of hyperbolic deformed quantum lattice Hamiltonians with position-dependent interactions, analyzing their ground states and correlations, revealing dimerization and boundary effects modulated by the deformation parameter.

Contribution

It presents a novel class of hyperbolic deformed Hamiltonians related to hyperbolic space, and studies their ground state properties using DMRG, showing dimerization and boundary effects.

Findings

Ground state becomes dimerized with finite deformation parameter

Spin correlations decay exponentially

Boundary effects diminish as deformation increases

Abstract

A group of non-uniform quantum lattice Hamiltonians in one dimension is introduced, which is related to the hyperbolic $1 + 1$-dimensional space. The Hamiltonians contain only nearest neighbor interactions whose strength is proportional to $\cosh j \lambda$, where $j$ is the lattice index and where $\lambda \ge 0$ is a deformation parameter. In the limit $\lambda \to 0$ the Hamiltonians become uniform. Spacial translation of the deformed Hamiltonians is induced by the corner Hamiltonians. As a simple example, we investigate the ground state of the deformed $S = 1/2$ Heisenberg spin chain by use of the density matrix renormalization group (DMRG) method. It is shown that the ground state is dimerized when $\lambda$ is finite. Spin correlation function show exponential decay, and the boundary effect decreases with increasing $\lambda$.