Lattice studies of entanglement entropy in $O(N)$ models at finite densities

Abstract: As a characteristic property of all quantum systems, entanglement participates in many important quantum phenomena. In this proceeding, we employ it in the study of quantum field theories at finite density. We incorporate evaluations of entanglement entropy using the replica trick into MC simulations of $O(N)$ models at finite density with the worm algorithm and present some initial results for the nonlinear $O(4)$ model in 3 dimensions.

• The paper develops a boundary-deformation Monte Carlo method combined with dual flux variables and worm updates to calculate second Rényi entropy derivatives without the finite-density sign problem.
• In three-dimensional nonlinear O(4) simulations, the spatial derivative of the second Rényi entropy decreases rapidly with slab width, falls with temperature, and peaks near the critical chemical potential.
• The paper validates the algorithm through mixed-derivative agreement between entanglement and charge-density measurements, supporting entanglement entropy as a probe of finite-density phase transitions.

Overview

This proceeding, presented at LATTICE2025, develops a Monte Carlo framework for computing derivatives of the entanglement entropy (EE) in lattice $O(N)$ models at finite chemical potential $\mu$. The central technical contribution is an adaptation of the boundary deformation method—previously formulated for $SU(N)$ gauge theories—to $O(N)$ models simulated with a worm algorithm in their integer-valued dual flux representation. The authors present initial results for the nonlinear $O(4)$ model in three dimensions and validate their algorithm through a mixed-derivative consistency check.

The model and its dual formulation

The starting point is a general $O(N)$ lattice action with hopping parameter $\kappa$, quartic coupling $\lambda$, source fields $j$, and chemical potential $\mu$ coupled to the first two field components via the antisymmetric matrix $\mu$0. For non-zero $\mu$1 the action becomes complex, so importance sampling over the $\mu$2 fields fails due to the sign problem. The remedy is a reformulation in terms of integer-valued dual variables: charged net fluxes $\mu$3 on links, neutral-pair fluxes $\mu$4, neutral particle fluxes $\mu$5 for components $\mu$6, and monomer numbers $\mu$7, $\mu$8, $\mu$9 on sites. The partition function factorizes into link weights $SU(N)$0 and site weights $SU(N)$1 subject to two classes of constraints: local charge conservation (a discrete delta function) and evenness constraints ($SU(N)$2) tying neutral flux to monomer parity.

These constraints make local Metropolis updates inefficient, but they are naturally handled by the worm algorithm of Prokof'ev and Svistunov. A worm update inserts a source-sink pair (e.g., a $SU(N)$3 pair in the charged sector), moves the head through the lattice while shifting the corresponding flux variables by $SU(N)$4 along the path so that all constraints remain satisfied, and removes the pair when head and tail recombine. The neutral sectors work analogously with the evenness constraints.

Computing $SU(N)$5 via boundary deformation

For a slab-shaped region $SU(N)$6 of width $SU(N)$7, the replica trick gives $SU(N)$8, where in $SU(N)$9 the fields are $O(N)$0-periodic inside $O(N)$1 and $O(N)$2-periodic elsewhere. Because EE is UV-divergent while its derivative with respect to $O(N)$3 is finite, the authors target

$O(N)$4

i.e., a finite-difference approximation using the second Rényi entropy. The difficulty is that enlarging $O(N)$5 by one lattice unit is a non-local change: the configuration distributions contributing to $O(N)$6 and $O(N)$7 barely overlap, defeating standard importance sampling.

The boundary deformation method resolves this by changing the temporal boundary conditions site by site, connecting $O(N)$8 and $O(N)$9 through an ordered sequence of local deformations. The simulation walks back and forth along this sequence, collects histograms of visits to each boundary state, and obtains $O(4)$0 from differences of log-histograms, with jack-knife error estimation.

The new complication specific to $O(4)$1 models is that swapping endpoints of temporal links during a boundary change alters incoming fluxes, producing constraint violations ("defects") unless the flux configurations on the swapped links are compatible. The authors propose two remedies:

• Plaquette worms: worm updates constrained to move through a temporal plaquette containing a problematic link, adjusting $O(4)$2 and $O(4)$3 until $O(4)$4 and $O(4)$5, allowing a defect-free boundary change; the plaquette worms are then reversed to guarantee detailed balance.
• Defect-annihilation worms: since defects always appear as defect–anti-defect pairs on distinct sites, a restricted worm can transport the defect to its partner's location, where both are removed. The restriction that these worms not touch the two links whose endpoints are swapped ensures equal selection probability for the reverse move and detailed balance.

Results and validation

Simulations were performed on lattices of size $O(4)$6 with $O(4)$7, $O(4)$8, $O(4)$9, $O(N)$0, at $O(N)$1 and $O(N)$2. Three qualitative findings emerge:

Observation	Behavior
$O(N)$3 vs. $O(N)$4	Rapid initial decrease, plateau for $O(N)$5
$O(N)$6 vs. $O(N)$7	Monotonically decreasing
$O(N)$8 vs. $O(N)$9	Increases up to the critical $\kappa$0, then decreases

The non-monotonic dependence of $\kappa$1 on $\kappa$2, peaking near the critical chemical potential, supports the paper's claim that EE is a useful diagnostic of finite-density phase transitions.

The key validation exploits the identity $\kappa$3. Taking a numerical $\kappa$4-derivative yields $\kappa$5. Independently, the charge density satisfies $\kappa$6, so that

$\kappa$7

By measuring $\kappa$8 in every intermediate boundary-condition state, both sides of this relation are computed from the same simulations. The two evaluations agree across the tested parameters ($\kappa$9, $\lambda$0, multiple $\lambda$1), and the authors report that the agreement has held in all additional tests performed. This constitutes strong evidence that the modified boundary-deformation algorithm samples the correct ensemble.

Limitations and open questions

The results presented are explicitly preliminary: only the nonlinear $\lambda$2 model in $\lambda$3 is studied, at a single point $\lambda$4 in parameter space, with the second Rényi entropy used as a proxy rather than the von Neumann entropy itself. The equivalence between the large-$\lambda$5 value of $\lambda$6 and the thermal entropy density has been argued for only in certain limits and remains untested here. Whether the defect-handling procedures remain efficient near criticality—where correlation lengths grow and worm trajectories lengthen—is not addressed. Extracting critical exponents from EE, the stated goal, is left to future work.

Conclusion

The paper extends the boundary deformation method for entanglement entropy to sign-problem-free dual formulations of $\lambda$7 models at finite density, introducing two detailed-balance-preserving mechanisms for handling constraint defects during boundary updates. The demonstrated consistency between $\lambda$8 and the $\lambda$9-derivative of the charge density validates the approach, and the observed peak of $j$0 near the critical chemical potential indicates that entanglement measures can serve as probes of finite-density phase structure. Planned extensions include extracting critical exponents, comparison with $j$1 analytic results, canonical-ensemble simulations for symmetry-resolved entanglement, and systematic tests of the connection between large-$j$2 EE derivatives and thermal entropy density.