KMS states on Quantum Grammars

TL;DR

This paper studies quantum lattice systems with evolving lattices, proving the existence of dynamics and KMS states, and introduces a renormalization approach for high-temperature regimes to relate evolving lattices to fixed lattice models.

Contribution

It provides exact mathematical results for quantum lattice models with evolving lattices, including the construction of dynamics and KMS states, and introduces a renormalization method for high-temperature regimes.

Findings

Existence of dynamics in Schrödinger and Heisenberg pictures.

Construction of KMS states on appropriate C*-algebras.

High-temperature scaling leading to fixed lattice quantum spin systems.

Abstract

We consider quantum (unitary) continuous time evolution of spins on a lattice together with quantum evolution of the lattice itself. In physics such evolution was discussed in connection with quantum gravity. It is also related to what is called quantum circuits, one of the incarnations of a quantum computer. We consider simpler models for which one can obtain exact mathematical results. We prove existence of the dynamics in both Schroedinger and Heisenberg pictures, construct KMS states on appropriate C*-algebras. We show (for high temperatures) that for each system where the lattice undergoes quantum evolution, there is a natural scaling leading to a quantum spin system on a fixed lattice, defined by a renormalized Hamiltonian.