Local temperatures and local terms in modular Hamiltonians

TL;DR

This work introduces local temperatures for quantum field theories by leveraging relative entropy between vacuum and localized excitations, generalizing the Unruh temperature to arbitrary regions. It shows that, while the Rindler modular Hamiltonian is local and stress-tensor–driven, general regions in dimensions $d\ge 3$ require nonlocal terms, whereas in $d=2$ the leading null-direction temperatures are universal and governed by the region's geometry, independent of mass or field content for free fields. The authors derive and test analytic structure for local terms, including a universal null-temperature form in $d=2$ and a bilinear local term for free scalars, with extensive lattice confirmations for one- and two-interval regions. These results reveal a deep link between geometry, vacuum entanglement, and high-energy tails of reduced density matrices, suggesting a universal, geometry-driven imprint on local excitations across QFTs. The findings have potential implications for entanglement-based constraints, holography, and the study of locality in modular Hamiltonians across different spacetime dimensions and states.

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.

Figures (12)

• Figure 1: We are testing the modular Hamiltonian of a region (here the Rindler wedge $A$) by unitary operator well localized around a point $a$ on Cauchy surface (horizontal line in the figure). The state produced from this unitary acting on the vacuum spreads out in the past and future of the Cauchy surface.
• Figure 2: The causal region $A$ (black) is included in the Rindler wedge and includes a double cone elongated along a null line (shown with dashed lines). The contributions to $\Delta K$ for these three regions from a unitary excitation localized near a point (marked with a red circle) are ordered according to the same relations.
• Figure 3: The modular Hamiltonian written in two different Cauchy surfaces must have the same expectation value when vacuum is perturbed by an energetic ray which follows an approximately null trajectory.
• Figure 4: Scalar in a single interval. Kernel $N$ for a scalar field.
• Figure 5: Scalar in a single interval. Kernel $M$ for a scalar field of mass $mL=1$.