Local temperatures and local terms in modular Hamiltonians

TL;DR

This paper explores the concept of local temperatures in quantum field theories, revealing their universal features, especially in two dimensions, and analyzing the structure of modular Hamiltonians across different dimensions.

Contribution

It introduces a framework for defining local temperatures via relative entropy, investigates their universality, and characterizes local terms in modular Hamiltonians for various theories and dimensions.

Findings

Local temperatures can be defined for any quantum field theory and region.

In 2D, local temperatures may originate from stress tensor terms in the modular Hamiltonian.

In dimensions ≥3, local terms cannot be solely from the stress tensor, and their structure varies.

Abstract

We show there are analogues to the Unruh temperature that can be defined for any quantum field theory and region of the space. These local temperatures are defined using relative entropy with localized excitations. We show important restrictions arise from relative entropy inequalities and causal propagation between Cauchy surfaces. These suggest a large amount of universality for local temperatures, specially the ones affecting null directions. For regions with any number of intervals in two space-time dimensions the local temperatures might arise from a term in the modular Hamiltonian proportional to the stress tensor. We argue this term might be universal, with a coefficient that is the same for any theory, and check analytically and numerically this is the case for free massive scalar and Dirac fields. In dimensions $d\ge 3$ the local terms in the modular Hamiltonian producing these local temperatures cannot be formed exclusively from the stress tensor. For a free scalar field we classify the structure of the local terms.