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arXiv:2603.07635

Thermal and chemical response from entanglement entropy

Niko Jokela, Aatu Rajala, Tobias Rindlisbacher

hep-thcond-mat.stat-mechhep-lathep-phquant-phSubmitted Mar 8, 2026• Announced Mar 10, 2026• 0 citations
v1 · Mar 8, 2026v2 · Jul 20, 2026
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Abstract

We study entanglement entropy (EE) in interacting quantum field theories (QFTs) at finite density. We argue that, in the limit of large subregions, the derivative of EE with respect to the size of the entangling region approaches the thermal entropy density, independently of microscopic details. We make this relation explicit using slab-shaped subregions, where the limiting behavior can be directly identified. At finite chemical potential, we show that EE satisfies thermodynamic response relations, including a generalized Maxwell relation linking chemical potential and charge density. We provide strong nonperturbative evidence for these statements in the three-dimensional $\operatorname{O}\left(4\right)$ model, and conjecture that they are generic features of continuum QFTs, establishing a two-way link between entanglement and thermodynamics that opens a route toward extracting the equation-of-state information from entanglement data.

Comments
7 pages, 3 figures

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We study entanglement entropy (EE) in interacting quantum field theories (QFTs) at finite density. We argue that, in the limit of large subregions, the derivative of EE with respect to the size of the entangling region approaches the thermal entropy density, independently of microscopic details. We make this relation explicit using slab-shaped subregions, where the limiting behavior can be directly identified. At finite chemical potential, we show that EE satisfies thermodynamic response relations, including a generalized Maxwell relation linking chemical potential and charge density. We provide strong nonperturbative evidence for these statements in the three-dimensional O⁡(4)\operatorname{O}\left(4\right)O(4) model, and conjecture that they are generic features of continuum QFTs, establishing a two-way link between entanglement and thermodynamics that opens a route toward extracting the equation-of-state information from entanglement data.
We nonperturbatively compute Rényi entropies for strip-shaped subregions in the three-dimensional O(4) model at finite density on the lattice. By using a dual variable representation and a tailored worm algorithm, we circumvent the sign problem when sampling the grand canonical ensemble. In the limit of large subregions, we also establish a direct, quantitative relationship between the derivative of entanglement entropy with respect to the size of the entangling region and the thermal entropy density for general quantum field theories, providing a new way to study their thermodynamics. We corroborate this argument with our lattice results by demonstrating that, in the appropriate limit, the derivative of entanglement entropy satisfies the same Maxwell relation as the thermal entropy density.
Entropy is a quantity for counting physical degrees of freedom in a system. At a finite temperature, one can use thermal entropy to study thermodynamical properties. At zero temperature, entanglement entropy is expected to provide a suitable order parameter of a phase structure. Especially, the entanglement entropy exhibits an interesting codimension two area law in a strongly coupled conformal field theory. We compute thermal entropy in a non-relativistic model with an infinite fermion mass limit from an exact effective potential to obtain thermal entropy at an infinite strong coupling limit. The computational result provides vanishing thermal entropy at an infinite strong coupling limit with a finite lattice spacing. The non-trivial topological term can be included in the strongly coupled lattice system to obtain the non-trivial entropy and the topology can be marked from the entropy. We first compute the thermal entropy in a two dimensional lattice topological quantum field theory to study the lattice artifact and also argue that a theory possibly has translational invariance if a system does not have a volume law in the entanglement entropy. Finally, we show that a coefficient of a universal term of the entanglement entropy should not be affected by a choice of an entangling surface in two dimensional conformal field theory for one interval case. We also discuss a choice of the entangling surface in the entanglement entropy in two dimensional CPN−1\mathrm{CP}^{N-1}CPN−1 model at the large NNN limit.
Abstract Entropy is a quantity for counting physical degrees of freedom in a system. At a finite temperature, one can use thermal entropy to study thermodynamical properties. At zero temperature, entanglement entropy is expected to provide a suitable order parameter of a phase structure. Especially, the entanglement entropy exhibits an interesting codimension two area law in a strongly coupled conformal field theory. We compute thermal entropy in a non-relativistic model with an infinite fermion mass limit from an exact effective potential to obtain thermal entropy at an infinite strong coupling limit. The computational result provides vanishing thermal entropy at an infinite strong coupling limit with a finite lattice spacing. The non-trivial topological term can be included in the strongly coupled lattice system to obtain the non-trivial entropy and the topology can be marked from the entropy. We first compute the thermal entropy in a two-dimensional (2D) lattice topological quantum field theory to study the lattice artifact and also argue that a theory possibly has translational invariance if a system does not have a volume law in the entanglement entropy. Finally, we show that a coefficient of a universal term of the entanglement entropy should not be affected by a choice of an entangling surface in 2D conformal field theory for one interval case. We also discuss a choice of the entangling surface in the entanglement entropy in 2D model at the large N limit.
Abstract We study different aspects of quantum field theory at finite density using methods from quantum information theory. For simplicity we focus on massive Dirac fermions with nonzero chemical potential, and work in 1 + 1 space-time dimensions. Using the entanglement entropy on an interval, we construct an entropic c-function that is finite. Unlike what happens in Lorentz-invariant theories, this c-function exhibits a strong violation of monotonicity; it also encodes the creation of long-range entanglement from the Fermi surface. Motivated by previous works on lattice models, we next calculate numerically the Renyi entropies and find Friedel-type oscillations; these are understood in terms of a defect operator product expansion. Furthermore, we consider the mutual information as a measure of correlation functions between different regions. Using a long-distance expansion previously developed by Cardy, we argue that the mutual information detects Fermi surface correlations already at leading order in the expansion. We also analyze the relative entropy and its Renyi generalizations in order to distinguish states with different charge and/or mass. In particular, we show that states in different superselection sectors give rise to a super-extensive behavior in the relative entropy. Finally, we discuss possible extensions to interacting theories, and argue for the relevance of some of these measures for probing non-Fermi liquids.
From black hole thermodynamics, the Bekenstein bound has been proposed as a universal thermal entropy bound. It has been further generalized to an entanglement entropy bound which is valid even in a quantum system. In a quantumly entangled system, the non-negativity of the relative entropy leads to the entanglement entropy bound. When the entanglement entropy bound is saturated, a quantum system satisfies the thermodynamicslike law with an appropriately defined entanglement temperature. We show that the saturation of the entanglement entropy bound accounts for a universal feature of the entanglement temperature proportional to the inverse of the system size. In addition, we show that the deformed modular Hamiltonian under a global quench also satisfies the generalized entanglement entropy boundary after introducing a new quantity called the entanglement chemical potential.
We study aspects of quantum field theory at finite density using techniques and concepts from quantum information theory. We focus on massive Dirac fermions with chemical potential in 1+1 space-time dimensions. Using the entanglement entropy on an interval, we construct an entropic c-function that is finite. This c-function is not monotonous,and incorporates the long-range entanglement from the Fermi surface. Motivated by previous works on lattice models,we next compute the Renyi entropies numerically, and find Friedel-type oscillations. Next, we analyze the mutual in-formation as a measure of correlation functions between different regions. Using a long-distance expansion developed by Cardy, we show how the mutual information detects the Fermi surface correlations already at leading order in the expansion. Finally, we analyze the relative entropy and its Renyi generalizations in order to distinguish states with different charge. We find that states in different superselection sectors give rise to a super-extensive behavior in the relative entropy.