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arxiv: 2607.06286 · v1 · pith:3REY6O2F · submitted 2026-07-07 · hep-th · cond-mat.stat-mech· hep-lat· hep-ph· quant-ph

Determination of thermodynamics from entanglement entropy in the finite-density O(N) model

Niko Jokela , Aatu Rajala , Tobias Rindlisbacher This is my paper

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classification hep-th cond-mat.stat-mechhep-lathep-phquant-ph
keywords entropydensityentanglementderivativelatticelimitmodelsubregions
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The pith

Entanglement entropy yields thermal entropy density in large slabs

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that for slab-shaped entangling regions in general quantum field theories, the derivative of entanglement entropy with respect to the slab width, divided by the cross-sectional area, equals the thermal entropy density s, provided all linear sizes exceed the longest correlation length. The argument proceeds by decomposing the replicated free energy into bulk contributions from the entangling region and its complement, showing that the ℓ-derivative isolates the difference of free energy densities ω(rβ, μ) − rω(β, μ), whose r→1 limit is exactly s. The authors verify this relation on the lattice for the three-dimensional O(4) model at finite chemical potential, using a dual-variable worm algorithm that avoids the sign problem. They confirm that the mixed derivative of the second Rényi entropy with respect to slab width and chemical potential satisfies the same Maxwell relation as the thermal entropy density, up to the point where the correlation length grows comparable to the slab width near the finite-density phase transition.

Core claim

The central result is Equation (20): in the regime where all correlation lengths are much smaller than the slab width and system size, the quantity (1/V⊥) ∂S_EE/∂ℓ is identically the thermal entropy density s. This is not an approximation or a scaling relation but an exact identity derived from the replica trick. The paper then demonstrates it nonperturbatively by showing that the r=2 Rényi analogue satisfies the corresponding Maxwell relation (Equation 87) on the lattice, with agreement persisting up to ξ_max/ℓ ≈ 0.5–1.0 depending on temperature.

What carries the argument

The derivation hinges on the observation that when a thin slice is moved from deep inside region B to deep inside region A (both regions being much larger than the correlation length), the interface between A and B can be neglected and the bulk free energy densities ω(β, μ) and ω(rβ, μ) apply independently. This reduces the ℓ-derivative of the replicated free energy to a difference of free energy densities, which upon taking the r→1 limit yields the standard thermodynamic expression for entropy. The lattice verification uses a dual-variable representation of the O(N) model that eliminates the sign problem, combined with a boundary-deformation worm algorithm to compute Rényi entropyderivativ.

If this is right

  • If the relation holds broadly, entanglement entropy measurements on the lattice could serve as an alternative route to extracting the full equation of state of interacting quantum field theories, including those where direct thermodynamic measurements are difficult.
  • The method is most cleanly applicable in confining or gapped phases where correlation lengths are finite; near critical points where correlation lengths diverge, the window of validity shrinks and the relation breaks down, as confirmed by the lattice data near μ_c ≈ 0.5.
  • The framework extends to any Rényi order r, where the derivative yields a step-scaling approximation s_r of the entropy density, converging to s as r→1; this means even r=2 Rényi data (which are far more accessible on the lattice than von Neumann entropy) carry genuine thermodynamic information.
  • The relation works in reverse: in theories where thermal entropy is easier to compute than entanglement entropy, one can use thermodynamic data to infer entanglement properties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If entanglement entropy derivatives encode the equation of state, then measuring S_EE at multiple temperatures and chemical potentials could in principle reconstruct the full thermodynamic phase diagram from purely information-theoretic observables, though the practicality depends on whether the correlation-length constraint can be satisfied across the full parameter space.
  • The breakdown near criticality suggests a natural complementarity: entanglement entropy is most useful as a thermodynamic probe away from phase transitions (where it gives bulk thermodynamics) and most useful as a diagnostic tool near phase transitions (where it detects critical behavior but not the equation of state).
  • The dual-variable worm algorithm and boundary-deformation technique developed here could be adapted to other theories admitting dual formulations, potentially including CP(N−1) models or abelian gauge-Higgs systems, extending the reach of entanglement-based thermodynamic extraction beyond O(N) models.

Load-bearing premise

The derivation assumes that when a thin slice is transferred between the two regions, the interface between them can be neglected and the bulk free energy densities apply independently. This requires the correlation length to be simultaneously much smaller than the slab width and the system size, a condition that fails near the critical point where the correlation length diverges.

What would settle it

Compute (1/V⊥) ∂S_EE/∂ℓ and the independently measured thermal entropy density s in the same theory at the same parameters; if they disagree in the regime ξ ≪ ℓ ≪ L, the identity (Equation 20) is falsified. The lattice test via the Maxwell relation (Equation 87) is an indirect version of this check.

Figures

Figures reproduced from arXiv: 2607.06286 by Aatu Rajala, Niko Jokela, Tobias Rindlisbacher.

Figure 1
Figure 1. Figure 1: Sketches demonstrating the temporal topologies of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of how the dimensionless free energy [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An illustration of the worm update in the [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A diagram of a disconnected worm move. By shift [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The figure shows how the endpoints of temporal [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A visualization of the plaquette boundary update. Worm updates constrained to a temporal plaquette are performed [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A visualization of the boundary worm update. A pair of defects introduced by the change of temporal boundary [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Mass m0 (top) and corresponding correlation length ξ0 = 1/m0 (bottom) of the three degenerate pseudo￾Goldstone modes in the three-dimensional non-linear lattice O(4) model at κ = 1.2 and µ = 0, as a function of the source j3. The data was obtained on lattices of size N 2 s Nt = 183 . heavy and effectively decouples,4 leaving three Goldstone modes, {ϕ +, ϕ−, ϕ0}, as the relevant low-energy degrees of freedo… view at source ↗
Figure 9
Figure 9. Figure 9: Mass spectrum (top) and corresponding correlation [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: In a replicated system as described by ( [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: ∂ℓH2 of the three-dimensional non-linear O(4) model, with the simulation parameters described in section 6, as a function of ℓ for Nt ∈ {5, 8} and µ ∈ {0.30, 0.60}. ∂ℓH2 quickly reaches a Nt and µ dependent plateau value marked in the plot with dashed lines. As argued in section 3, this value corresponds, up to a factor of 1/V⊥, to the r = 2 step scaling approximation (28) of the thermal entropy density, … view at source ↗
Figure 12
Figure 12. Figure 12 [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of results for − ∂ 2(log Z˜) ∂µ∂ℓ as a function of µ in the three-dimensional non-linear O(4) model as com￾puted from ∂µ∂ℓH2 and −2NtV ∂ℓn˜ for two Nt values. Both cases show a clear agreement between the two evaluations. where the asterisk (∗) refers to the requirement ξmax ≪ ℓ, Nx/2, Ns for the equality to hold. Both sides of the relation (87) can be computed from lattice simulations. The lef… view at source ↗
Figure 16
Figure 16. Figure 16: Approximation of the rescaled thermal entropy [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The thermal entropy density of the non-linear [PITH_FULL_IMAGE:figures/full_fig_p019_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: n/T 2 as a function of µ/m0 for the three￾dimensional non-linear O(4) model. The results have been obtained with (83) from simulations of unreplicated systems. respect to the sub-region volume approaches a step scal￾ing approximation sr with scaling factor r of the thermal entropy density, as defined in (28). These relations are expected to hold for general QFTs. We tested this idea with lattice simulatio… view at source ↗
Figure 19
Figure 19. Figure 19: A visualization of the plaquette update chain. The upper graph is for the whole update chain and the lower one is [PITH_FULL_IMAGE:figures/full_fig_p022_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: A visualization of the boundary worm update chain corresponding to adding a site to region [PITH_FULL_IMAGE:figures/full_fig_p023_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: ∂ℓH2 as a function of µ obtained with both the plaquette and the boundary worm updates. The results for the two updates show strong agreement. To investigate the efficiency of both boundary update algorithms, we have measured their acceptance rates and update times with the simulation parameters used in this work. The acceptance rates are shown in table II. We see that the acceptance rates for both update… view at source ↗
read the original abstract

We nonperturbatively compute R\'enyi 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 6 minor

Summary. This paper establishes a relation between the derivative of entanglement entropy (and Rényi entropies) with respect to the entangling region width and the thermal entropy density for slab-shaped subregions in general QFTs, in the limit where all linear sizes exceed the longest correlation length. The authors then test this relation nonperturbatively in the (2+1)-dimensional O(4) model at finite density using a dual-variable worm algorithm that avoids the sign problem. The lattice verification proceeds by checking a Maxwell relation (Eq. 87) rather than a direct comparison of the entropy density with the entanglement derivative. The theoretical derivation (Section 3) is clean and parameter-free; the lattice methodology is technically sound with two independent boundary-update algorithms showing consistent results.

Significance. The paper makes a worthwhile contribution on two fronts. First, the theoretical relation (Eq. 20/27) is derived from first principles (replica trick, thermodynamic identities, bulk free energy density assumption) with no fitted parameters, and applies to general QFTs. Second, the lattice implementation is a genuine technical achievement: the dual-variable worm algorithm with boundary deformation in the replicated geometry is non-trivial, the sign problem is circumvented, and the authors provide two independent update schemes (plaquette and boundary worm) with fully consistent results (Fig. 21, Table II). The observation that the phase transition at μc is visible in ∂ℓH₂ (Fig. 13) is a nice physical result. The framework offers a novel route to extracting thermodynamic equations of state from entanglement data, which is of broad interest.

major comments (2)
  1. §7, Eq. (87) and Fig. 15: The central theoretical claim is (1/V⊥)∂ℓH_r = s_r(T,μ) (Eq. 27), but the lattice test verifies only its μ-derivative: ∂μ[(1/V⊥)∂ℓH₂] = ∂μ[s₂(T,μ)] (Eq. 87). A μ-independent offset C(T) at finite ℓ — for instance from subleading boundary/interface contributions that are O(ξ/ℓ) but not exactly zero — would be invisible to this test. The authors acknowledge this gap explicitly in §8 ('we tested the proposed framework here through the Maxwell relation rather than by directly comparing s with (1/V⊥)∂ℓSEE'), so it is not a hidden flaw. However, the abstract and conclusion could be read as claiming direct verification. I recommend the authors strengthen the presentation by: (a) stating more prominently in the abstract that the Maxwell relation (not the direct equality) is what is verified, and (b) providing a quantitative bound on the plateau value of ∂ℓH₂/V⊥ against
  2. §3, Eq. (18): The derivation assumes that when a thin slice is moved from deep inside region B to deep inside region A, the interface between A and B can be neglected and the bulk free energy densities ω(β,μ) and ω(rβ,μ) apply independently. This requires ξ ≪ ℓ ≪ L simultaneously. Near the critical point μc ≈ 0.5, ξ_max diverges (Eq. 81), shrinking the validity window. The lattice data in Fig. 15 confirms agreement only up to ξ_max/ℓ ≈ 0.5, and the authors restrict entropy extraction to μ ≤ 0.4. This is handled reasonably, but the manuscript would benefit from a brief quantitative discussion of the expected scaling of finite-ℓ corrections — e.g., whether the residual at ξ/ℓ ≈ 0.5 is consistent with O(ξ/ℓ) or O((ξ/ℓ)²) — to justify the choice of cutoff.
minor comments (6)
  1. §3, Eq. (18): The notation lim_{ℓ,L→∞, ℓ≪L} is slightly ambiguous about the order of limits. Clarifying whether the limit is ℓ→∞ at fixed L→∞ with ℓ/L→0, or a double limit with the constraint maintained, would improve precision.
  2. §6: The simulation parameters (κ=1.2, j₃=0.2, Ns=12, Nx=36) are stated, but no continuum extrapolation is attempted. The authors note this is intentional. A brief remark on the expected size of lattice artifacts at these parameters would be welcome.
  3. Fig. 15: The horizontal axis is ξ_max(μ)/ℓ, but the different temperature curves correspond to different μ ranges. It would help the reader if the μ values corresponding to the plotted ξ_max/ℓ range were indicated, perhaps as a secondary axis.
  4. §7, Figs. 16–17: The entropy density extracted from ∂ℓH₂ is labeled s₂ (the r=2 step-scaling approximation), but the figure captions and axis labels sometimes refer to 'thermal entropy density s' without the subscript. Consistency in notation would avoid confusion about whether s or s₂ is being shown.
  5. §4.2, Eq. (65): The partition function in dual variables is lengthy. A compact summary table of the dual variables and their physical meanings (which is currently spread across the text) would improve readability.
  6. Reference [57] is cited as a companion letter. If it contains overlapping results, a brief statement of what is new in this paper versus [57] would help assess novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful and constructive report. Both major comments are well-taken and will be addressed in a revised manuscript. The first concerns the distinction between direct verification of Eq. (27) and verification of its μ-derivative (the Maxwell relation, Eq. (87)); we agree the abstract and conclusion should state this more prominently and will add a quantitative bound on the plateau value. The second concerns finite-ℓ corrections near the critical point and the expected scaling; we will add a quantitative discussion of the residual at ξ/ℓ ≈ 0.5.

read point-by-point responses

  1. Referee: §7, Eq. (87) and Fig. 15: The central theoretical claim is (1/V⊥)∂ℓH_r = s_r(T,μ) (Eq. 27), but the lattice test verifies only its μ-derivative. A μ-independent offset C(T) at finite ℓ would be invisible to this test. The authors acknowledge this gap in §8, but the abstract and conclusion could be read as claiming direct verification. Recommendations: (a) state more prominently in the abstract that the Maxwell relation (not the direct equality) is what is verified, and (b) provide a quantitative bound on the plateau value of ∂ℓH₂/V⊥ against s₂.

    Authors: We fully agree with this assessment. The referee correctly identifies that our lattice test verifies the Maxwell relation (Eq. 87), i.e., the μ-derivative of Eq. (27), rather than the direct equality (1/V⊥)∂ℓH₂ = s₂(T,μ) itself. A μ-independent offset C(T) from subleading boundary/interface contributions would indeed be invisible to our test. We acknowledge this explicitly in §8 but agree that the abstract and conclusion could be misread as claiming direct verification. We will implement both recommendations: (a) We will revise the abstract to state more prominently that the Maxwell relation—not the direct equality—is what is verified on the lattice. The current abstract already mentions the Maxwell relation in its final sentence, but we will make the distinction sharper and more upfront. (b) We will provide a quantitative bound on the plateau value of ∂ℓH₂/V⊥ against s₂(T,μ). Concretely, from the data in Fig. 11, the plateau in ∂ℓH₂ is reached for ℓ ≥ 5 at the parameter values shown. At ℓ = ℓ* = 17.5, the residual ℓ-dependence is consistent with zero within errors for μ ≤ 0.4 (where ξ_max/ℓ ≲ 0.5). We will extract a numerical bound on |∂ℓH₂/V⊥ − s₂| at fixed T and μ from the plateau region and include it in the revised §7, along with a discussion of the expected O(ξ/ℓ) scaling of any residual offset. We note that a fully direct comparison of ∂ℓH₂/V⊥ with an independently computed s₂ requires a separate lattice determination of the thermal entropy density in the O(4) model, which to our knowledge does not yet exist in the literature. We state this explicitly in §8 and identify it as an important next step. revision: yes

  2. Referee: §3, Eq. (18): The derivation assumes ξ ≪ ℓ ≪ L simultaneously. Near μc ≈ 0.5, ξ_max diverges, shrinking the validity window. The lattice data in Fig. 15 confirms agreement only up to ξ_max/ℓ ≈ 0.5. The manuscript would benefit from a brief quantitative discussion of the expected scaling of finite-ℓ corrections — e.g., whether the residual at ξ/ℓ ≈ 0.5 is consistent with O(ξ/ℓ) or O((ξ/ℓ)²) — to justify the choice of cutoff.

    Authors: This is a fair point. The derivation of Eq. (18) requires ξ ≪ ℓ ≪ L, and near μc the divergence of ξ_max (Eq. 81) shrinks the validity window. We will add a quantitative discussion of the expected finite-ℓ corrections. The interface contributions that are neglected in going from Eq. (17) to Eq. (18) arise from the boundary between regions A and B. For a slab geometry, these are surface terms scaling as the area of the entangling surface, i.e., O(ξ^d−2) relative to the bulk O(ℓ · ξ^d−2) contribution, giving corrections of order O(ξ/ℓ). The data in Fig. 15 shows agreement up to ξ_max/ℓ ≈ 0.5 at the lowest temperature and up to ξ_max/ℓ ≈ 1.0 at the highest temperature (where thermal effects shorten the effective correlation length below ξ_max). We will extract the residual deviation as a function of ξ_max/ℓ from the data underlying Fig. 15 and discuss whether it is consistent with linear O(ξ/ℓ) scaling or whether the data favor a faster, e.g., O((ξ/ℓ)²), falloff. This analysis will be included in the revised §7 to justify the choice of μ ≤ 0.4 as the cutoff for entropy extraction. We note that the current data, while clearly showing the onset of deviation as ξ_max/ℓ increases, may not have sufficient resolution to cleanly distinguish O(ξ/ℓ) from O((ξ/ℓ)²) across the full range; we will be transparent about this limitation. revision: yes

Circularity Check

1 steps flagged

No significant circularity; one minor self-citation for the boundary deformation method, but the central derivation is self-contained and the lattice test compares independently computed quantities.

specific steps
  1. self citation load bearing [Section 3, Eq. (17) and surrounding text]
    "Next, following the argument from [7], we write the ℓ-derivative in (16) as ∂ ˜Ω(ℓ, β, V, µ, r)/∂ℓ = [˜Ω(ℓ + δℓ, β, V, µ, r) − ˜Ω(ℓ, β, V, µ, r)] / δℓ ."

    Reference [7] (Jokela, Rummukainen, Salami, Pönni, Rindlisbacher) shares three authors with the present paper. The cited work introduces the boundary deformation method for computing ℓ-derivatives of free energies. However, this citation is methodological (how to compute the finite difference on the lattice), not load-bearing for the theoretical derivation itself. The central result (Eq. 20) follows from the replica trick (Eq. 3), the thermodynamic identity s = β∂ω/∂β − ω (Eq. 12), and the physical assumption that bulk free energy densities apply when ξ ≪ ℓ ≪ L (Eq. 18). No step in the chain from Eq. (3) to Eq. (20) depends on [7] for its mathematical content. The self-citation provides a computational technique, not a premise.

full rationale

The paper's central claim (Eq. 20) is derived from first principles: the replica trick definition of EE (Eq. 3), the standard thermodynamic identity s = β∂ω/∂β − ω (Eq. 12), and the physical assumption that bulk free energy densities ω(β,µ) and ω(rβ,µ) apply independently in regions A and B when ξ ≪ ℓ ≪ L (Eq. 18). No fitted parameters enter the derivation. The lattice test (Eq. 87, Fig. 15) compares two independently computed quantities: ∂µ∂ℓH₂ from replicated simulations and −2Nt(n(2Nt)−n(Nt)) from unreplicated simulations. These are measured on different ensembles with different algorithms, so the agreement is a genuine cross-check, not a tautology. The skeptic's concern that the Maxwell relation test only verifies the µ-derivative (not the absolute relation) is a valid limitation of the evidence, but it is not circularity: the test does not assume its own conclusion. The authors explicitly acknowledge this gap ('we tested the proposed framework here through the Maxwell relation rather than by directly comparing s with (1/V⊥)∂ℓSEE'). The one self-citation to [7] for the boundary deformation method is minor and methodological, not load-bearing for the theoretical result. Score 2 reflects this minor self-citation with no impact on the central derivation.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

No new physical entities are postulated. The dual variables (k, l, χ, p, q, n) are reformulations of the original O(N) fields, not new physics. The step-scaling derivative ∆_T^r (Eq. 28) is a new definition but not a new entity. All free parameters are simulation choices, not fitted to the thermodynamic relation.

free parameters (4)
  • κ (hopping parameter) = 1.2
    Set to place the system in the spontaneously broken phase; not fitted to the target result.
  • j₃ (source) = 0.2
    Introduced to give Goldstone modes a mass and ensure finite correlation length; not fitted to the thermodynamic relation.
  • λ (nonlinear limit) = ∞
    Set to infinity to select the nonlinear O(4) sigma model; a model choice, not a fit.
  • p_b (boundary update probability) = 0.25
    Algorithmic parameter controlling ratio of boundary to bulk updates; not a physics parameter.
axioms (4)
  • standard math Replica trick: tr(ρ_A^r) = Z̃(ℓ,r)/Z^r (Eq. 2)
    Standard replica method from Calabrese-Cardy [2,58], used as the starting point for EE computation.
  • domain assumption Bulk free energy densities apply deep inside regions A and B when ξ ≪ ℓ ≪ L (argument leading to Eq. 18)
    The central physical assumption: interface effects between A and B are negligible when both regions are much larger than the correlation length. This is the load-bearing premise.
  • standard math The r→1 limit commutes with the ℓ-derivative (Eq. 16, first equality)
    Assumed interchangeability of limit and derivative; standard in well-behaved QFTs but not proven here.
  • domain assumption Dual variable representation of the O(N) partition function is sign-problem-free (Section 4.2)
    The cancellation ∏_x e^{iφ_j p_x} = 1 follows from ∑_x p_x = 0, which is shown in the text. This is a property of the formalism, not an assumption.

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Reference graph

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    SIMULA TION ALGORITHMS FOR ENT ANGLEMENT ENTROPY IN LA TTICE O(N) MODELS To compute the derivative of entanglement entropy (16) on the lattice, both the ℓ and the r derivative must be approximated as finite differences. By approximating ∂r with a discrete forward derivative, followed by set- ting r → 1, one approximates SEE by the second R´ enyi entropy (...

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    ON SELECTION OF SIMULA TION P ARAMETERS The purpose of the simulations carried out in this work is to verify the relations between entanglement and R´ enyi entropies, and thermodynamic quantities, as discussed in Sec. 3. These relations were derived without referring to a specific type of system and should therefore hold for any sufficiently well behaved ...

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    RESUL TS Figure 10 shows the charged two-point correlation function ⟨ϕ−(x0)ϕ+(x)⟩ for the non-linear, (2 + 1)- dimensional lattice O(4) model, simulated in a r = 2- replica system of size Nx × Ny × r · Nt = 18 × 12 × 2 · 12, using the simulation parameter values κ = 1.2, j3 = 0.5, and µ = 0.3, and entangling region width ℓ = 4. Since translation invarianc...

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