Thermal and chemical response from entanglement entropy

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.

• The paper establishes nonperturbatively that the derivative of entanglement entropy with respect to a large slab’s width approaches the thermal entropy density, provided the region exceeds the correlation length.
• The paper derives generalized Maxwell relations linking mixed chemical-potential and size derivatives of entanglement or Rényi entropy to charge-density responses, extending thermodynamic identities to entanglement observables.
• The paper’s sign-problem-free 3D O(4) lattice simulations validate the Rényi-2 relation near finite-density transitions, with agreement extending to correlation-length-to-region-size ratios of roughly 0.5–1.0.

Overview

This Letter by Jokela, Rajala, and Rindlisbacher (2603.07635) establishes a direct, nonperturbative connection between entanglement entropy (EE) and thermodynamics in interacting quantum field theories at finite density. The central claim is that for sufficiently large entangling regions, the derivative of EE with respect to the size of the region approaches the thermal entropy density $s(T,\mu)$, independent of microscopic details. At finite chemical potential $\mu$, this statement extends to full thermodynamic response relations, including a generalized Maxwell relation linking chemical potential and charge density. The authors provide nonperturbative lattice evidence in the three-dimensional $O(4)$ model and conjecture that these relations are generic features of continuum QFTs.

The result is distinct from the first law of entanglement (Bhattacharya et al., 2012, Blanco et al., 2013), which relates variations of EE to expectation values of the modular Hamiltonian at fixed region geometry. Here, the relevant variation is with respect to the size of the entangling region itself — an infinitesimal rigid displacement of the entangling surface — which probes bulk thermodynamic structure directly.

Derivation of the entropy density relation

The argument proceeds from the replica construction. For a slab-shaped region $A$ of width $\ell$, the replica trick gives

$$\text{tr}(\rho_A^r) = \frac{\tilde Z(A,\beta,V,\mu,r)}{Z(\beta,V,\mu)^r},$$

where $\tilde Z$ is the path integral on a geometry with Euclidean time $r\beta$-periodic over $A$. Assuming that the limit $r\to 1$ commutes with the $\mu$0-derivative, the key input is a large-region approximation: when both $\mu$1 and its complement have linear sizes much larger than the longest correlation length $\mu$2 (i.e., $\mu$3), the $\mu$4-derivative of $\mu$5 per transverse volume $\mu$6 reduces to a difference of free energy densities,

$\mu$7

Combining this with the standard extraction of entropy density from the dimensionless free energy yields the central identity:

$\mu$8

Two points deserve emphasis. First, the derivation is nonperturbative and does not rely on the replica formalism beyond being an intermediate computational device. Second, the relation is most sharply realized in phases with finite correlation length, which is why the authors introduce an explicit symmetry-breaking source in their numerics. The authors also note that analogous saturation behavior was previously observed in deconfining Yang–Mills phases (Nakagawa et al., 2011, Jokela et al., 2023) and follows by construction from the Ryu–Takayanagi prescription in holography [hep-th/0603001], but a precise quantitative comparison to thermal entropy had not been established there.

Generalized Maxwell relations and Rényi entropies

Replacing $\mu$9 in the ordinary Maxwell relation $O(4)$0 by $O(4)$1 gives a generalized Maxwell relation:

$O(4)$2

Analogous relations hold for integer-order Rényi entropies $O(4)$3: their $O(4)$4-derivative approaches a step-scaling entropy density $O(4)$5, a discrete approximation to the temperature derivative with scaling factor $O(4)$6, which reduces to $O(4)$7 as $O(4)$8. Correspondingly, the mixed derivative satisfies

$O(4)$9

again reducing to the Maxwell relation in the $A$0 limit. This step-scaling structure is practically important because numerical computations are restricted to integer $A$1.

Nonperturbative test in the 3d $A$2 model

The numerical platform is the lattice $A$3 model with a conserved $A$4 charge coupled to $A$5. At nonzero $A$6 the action becomes complex, but a dual reformulation in terms of integer-valued flux variables renders the partition function sign-problem-free and amenable to worm-algorithm sampling [prokof2001worm; (Bruckmann et al., 2015)]. This is a decisive practical advantage over QCD, where finite-density simulations require sign-problem circumvention techniques.

Since $A$7 is inaccessible numerically, EE is estimated via the second Rényi entropy $A$8, and the $A$9-derivative is evaluated as a finite difference between $\ell$0 and $\ell$1. A severe overlap problem arises because the two ensembles differ on $\ell$2 sites; the authors overcome it with the boundary-deformation method introduced for $\ell$3 gauge theories (Rindlisbacher et al., 2022, Jokela et al., 2023), supplemented by worm updates that repair or preempt constraint defects caused by changing temporal boundary conditions. Technical details are deferred to a companion paper.

The simulations use $\ell$4 (inside the $\ell$5 broken-symmetry phase), a source $\ell$6 giving the Goldstone modes a mass $\ell$7, lattices with $\ell$8, $\ell$9, $$\text{tr}(\rho_A^r) = \frac{\tilde Z(A,\beta,V,\mu,r)}{Z(\beta,V,\mu)^r},$$0, and $$\text{tr}(\rho_A^r) = \frac{\tilde Z(A,\beta,V,\mu,r)}{Z(\beta,V,\mu)^r},$$1. The lightest mode mass behaves as $$\text{tr}(\rho_A^r) = \frac{\tilde Z(A,\beta,V,\mu,r)}{Z(\beta,V,\mu)^r},$$2 below the critical value, so the longest correlation length diverges as $$\text{tr}(\rho_A^r) = \frac{\tilde Z(A,\beta,V,\mu,r)}{Z(\beta,V,\mu)^r},$$3.

Two consistency checks precede the main result. First, the mixed derivative $$\text{tr}(\rho_A^r) = \frac{\tilde Z(A,\beta,V,\mu,r)}{Z(\beta,V,\mu)^r},$$4 computed two independent ways — from $$\text{tr}(\rho_A^r) = \frac{\tilde Z(A,\beta,V,\mu,r)}{Z(\beta,V,\mu)^r},$$5 at neighboring $$\text{tr}(\rho_A^r) = \frac{\tilde Z(A,\beta,V,\mu,r)}{Z(\beta,V,\mu)^r},$$6, and from $$\text{tr}(\rho_A^r) = \frac{\tilde Z(A,\beta,V,\mu,r)}{Z(\beta,V,\mu)^r},$$7 at neighboring $$\text{tr}(\rho_A^r) = \frac{\tilde Z(A,\beta,V,\mu,r)}{Z(\beta,V,\mu)^r},$$8 — agree well across $$\text{tr}(\rho_A^r) = \frac{\tilde Z(A,\beta,V,\mu,r)}{Z(\beta,V,\mu)^r},$$9, confirming internal algorithmic consistency. Notably, the entanglement-based observable clearly resolves the finite-density phase transition at $\tilde Z$0, demonstrating that $\tilde Z$1 and related quantities are sensitive probes of phase structure.

Second, the generalized Maxwell relation at $\tilde Z$2,

$\tilde Z$3

is tested against independent charge-density measurements. Agreement holds up to $\tilde Z$4 at the lowest temperature and almost up to $\tilde Z$5 at the highest temperature. The authors attribute the wider validity window at higher temperature to thermal truncation of the effective correlation length, since $\tilde Z$6 is defined at zero temperature. This constitutes strong nonperturbative evidence that entanglement variations encode equation-of-state information.

Limitations and open questions

Several qualifications bear directly on the strength of the results. The equality in the Maxwell relation carries an asterisk: it requires $\tilde Z$7, i.e., all linear sizes large compared to the correlation length, and the empirical agreement degrades precisely as $\tilde Z$8 approaches unity. The derivation also assumes that the $\tilde Z$9 limit commutes with the $r\beta$0-derivative, an interchange whose justification is not proven here. Numerically, EE is approximated by $r\beta$1 with a single unit-lattice-spacing finite difference in $r\beta$2, so the reported agreement validates the $r\beta$3 step-scaling version rather than the $r\beta$4 relation itself. The test system is a gapped, symmetry-broken $r\beta$5 model with an explicitly introduced source; whether the relations hold quantitatively in gapless phases, near criticality where $r\beta$6 diverges, or in gauge theories with sign problems remains untested. Finally, the behavior in the deconfining phase of Yang–Mills theories, where saturation was observed but not compared quantitatively to thermal entropy, is left as an open question requiring precise measurement.

Conclusion

The paper establishes, both analytically and through nonperturbative lattice simulation, that size derivatives of entanglement entropy in large subregions reproduce thermal entropy densities and satisfy generalized Maxwell relations at finite chemical potential. The demonstration in the sign-problem-free 3d $r\beta$7 model — with agreement between entanglement-derived and independently measured charge response up to $r\beta$8–$r\beta$9 — supports the conjecture that these relations are generic in continuum QFTs. The practical consequence is that entanglement data can, in principle, be used to extract equation-of-state information, positioning entanglement variations as thermodynamic response functions rather than purely information-theoretic diagnostics.