arxiv: 2604.19082 · v1 · submitted 2026-04-21 · ✦ hep-th · cond-mat.stat-mech· cond-mat.str-el· quant-ph

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Symmetry resolved entanglement in Lifshitz field theories

M. Reza Mohammadi Mozaffar , Ali Mollabashi

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Pith reviewed 2026-05-10 02:50 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechcond-mat.str-elquant-ph

keywords symmetry-resolved entanglementLifshitz field theorynon-relativistic QFTcharged momentscorrelator methodRenyi entropyequipartitiondynamical exponent

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The pith

Symmetry-resolved entanglement in Lifshitz scalar theories shows approximate equipartition at large dynamical exponent z with configurational entropy dominating, while fermionic models require the relativistic limit for genuine equipartitio

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

The paper computes symmetry-resolved Renyi and von Neumann entropies in non-relativistic Lifshitz scalar chains and fermionic models. It establishes that scalars develop approximate equipartition among charge sectors when the dynamical exponent z grows large, with configurational entropy taking the lead. In contrast, fermionic models achieve genuine equipartition only when z equals one, where fluctuation entropy prevails instead. These differences underscore the impact of non-relativistic scaling on how entanglement distributes across symmetry sectors. The findings matter for systems like cold atoms where particle-number-resolved measurements can probe these patterns directly.

Core claim

Using charged moments and the correlator method, the authors compute the symmetry-resolved Renyi and von Neumann entropies for Lifshitz theories. In Lifshitz scalar theories, approximate equipartition among charge sectors emerges in the large-z regime, with configurational entropy dominating, whereas Lifshitz fermionic models exhibit genuine equipartition only in the relativistic limit, with fluctuation entropy prevailing. The results also track the dependence on subsystem size, charge, mass, and the dynamical exponent z.

What carries the argument

Charged moments combined with the correlator method, which projects the entanglement entropy onto definite charge sectors in the Lifshitz models.

If this is right

Approximate equipartition for large-z scalars implies configurational entropy dominates over fluctuation entropy in those models.
Genuine equipartition for fermions at z=1 implies fluctuation entropy is the prevailing contribution in the relativistic limit.
The resolved entropies vary with subsystem size, conserved charge, mass parameter, and dynamical exponent z in distinct scalar versus fermionic patterns.
The approach supplies a calculational framework for operationally accessible, charge-resolved entanglement in non-relativistic quantum systems.

Where Pith is reading between the lines

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

Particle-number resolved measurements in cold-atom realizations of Lifshitz scaling could distinguish scalar from fermionic cases by testing the predicted equipartition thresholds.
The same charge-sector analysis may apply to other non-relativistic condensed-matter models with conserved particle number.
Checking whether the approximate-equipartition feature survives in interacting Lifshitz theories would test the robustness of the free-field results.

Load-bearing premise

The charged moments and correlator method remain valid and sufficient to extract symmetry-resolved entropies in these Lifshitz models without additional non-relativistic corrections or ultraviolet divergences that would alter the reported equipartition behavior.

What would settle it

A direct numerical or experimental computation of charge-sector decomposed von Neumann entropy in a large-z Lifshitz scalar model that fails to exhibit approximate equipartition would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.19082 by Ali Mollabashi, M. Reza Mohammadi Mozaffar.

**Figure 2.** Figure 2: The real (left) and the imaginary (right) parts of the charged moments as functions of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: Left: Symmetry resolved partition sums as function of the width of the entangling region for several values of the dynamical exponent and R´enyi index with q = 3. Middle: The same quantity as function of charge for the same values of the parameters with ℓ = 50. In both panels we set m = 0.1. Right: The probability of measuring charge q within the subsystem for different z and m. 3.2 Symmetry resolved R´eny… view at source ↗

**Figure 4.** Figure 4: Symmetry resolved R´enyi entropies as the function of the entangling width for several [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: Left: Subtracted symmetry resolved entanglement entropy as function of charge for several values of the dynamical exponent. Right: Symmetry resolved R´enyi entropies as function of the dynamical exponent for several values of charge and R´enyi index. In both panels we set m = 0.1 and ℓ = 50. For larger values of the dynamical exponent we have an approximate equipartition. ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ △ △ … view at source ↗

**Figure 6.** Figure 6: Symmetry resolved entanglement entropy as function of charge for several values of mass [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

**Figure 7.** Figure 7: Configurational, fluctuation and total entanglement entropies as functions of the dynam [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗

**Figure 8.** Figure 8: Real part of the logarithm of the charged moments as a function of [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

**Figure 9.** Figure 9: illustrates the behavior of Zn(q) as a function of ℓ for several parameter choices. The left panel shows that the symmetry-resolved partition sums in the zero-charge limit are decreasing functions of the subsystem length, tending towards a constant value in the large-ℓ limit. In contrast to the bosonic case, this saturation value decreases monotonically with increasing z. The right panel presents the symme… view at source ↗

**Figure 10.** Figure 10: Left: Symmetry resolved partition sums as function of charge with ℓ = 50 and m = 0.1. Right: The probability of measuring charge q within the subsystem for different values of z and m. saturation to a constant value. However, this transition becomes significantly smoother as q and z are increased. To further clarify the dependence of the symmetry-resolved entanglement entropy ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦… view at source ↗

**Figure 11.** Figure 11: Symmetry resolved entanglement entropy as a function of the entangling length for [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗

**Figure 12.** Figure 12: Symmetry resolved entanglement entropy as a function of charge for several values of [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗

**Figure 13.** Figure 13: Left: Symmetry resolved R´enyi entropies as functions of the dynamical exponent for several values of the charge and R´enyi index. Right: Configurational, fluctuation and total entanglement entropies as function of the dynamical exponent. 17 [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗

**Figure 14.** Figure 14: Real part of the logarithm of the charged moments as a function of [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗

**Figure 15.** Figure 15: Symmetry-resolved partition sums as a function of [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗

**Figure 16.** Figure 16: Left: Symmetry resolved entanglement entropies as the function of the entangling length for several values of the dynamical exponent and charge in fermion theory. Right: Subtracted symmetry resolved entanglement entropy as a function of charge for several values of mass and dynamical exponent. 19 [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗

read the original abstract

We investigate symmetry-resolved entanglement in non-relativistic quantum field theories, including complex Lifshitz scalar chains and Lifshitz fermionic models. Using charged moments and the correlator method, we compute symmetry-resolved Renyi and von Neumann entropies and analyze their dependence on subsystem size, charge, mass, and the dynamical exponent z. Our results reveal distinct features of non-relativistic entanglement. In Lifshitz scalar theories, approximate equipartition among charge sectors emerges in the large-z regime, with configurational entropy dominating, whereas Lifshitz fermionic models exhibit genuine equipartition only in the relativistic limit, with fluctuation entropy prevailing. These findings highlight a rich interplay between conserved charges, subsystem size, mass, and dynamical scaling, and provide a framework to explore operationally accessible entanglement in non-relativistic systems. Our study offers insights relevant to experimental platforms such as cold atom setups and mesoscopic systems, where particle-number-resolved measurements can probe symmetry-resolved entanglement.

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.

Desk Editor's Note private letter to a colleague

The paper applies charged-moment techniques to Lifshitz scalars and fermions and reports z-dependent shifts in equipartition and entropy dominance that are not in the relativistic literature.

read the letter

The main point is that this work takes the standard charged-moment and correlator approach and runs it on Lifshitz scalar and fermionic models, extracting that scalars approach approximate equipartition at large z with configurational entropy winning, while fermions reach genuine equipartition only at z=1 with fluctuation entropy dominant. Those specific distinctions and their dependence on subsystem size, mass, and z are new for these theories and give a concrete handle on non-relativistic entanglement features relevant to cold atoms. The calculations appear to rest on ordinary QFT correlators applied to the Lifshitz action, which is a straightforward extension rather than a reinvention of the method. Credit is due for spelling out the z-dependence and for keeping the experimental motivation in view without overclaiming. The soft spot is that the abstract supplies no derivations, error estimates, or explicit checks on how the higher spatial derivatives affect the UV structure of the reduced density matrix or the charged moments. The stress-test concern about possible extra z-dependent or charge-dependent divergences is reasonable on its face; if the full paper does not show that the Gaussian determinant formula and subsequent Fourier transform to charge sectors remain clean under the anisotropic scaling, the reported dominance switch could shift. Nothing in the provided material rules that out. This is for readers already working on symmetry-resolved entanglement in condensed-matter or non-relativistic QFT settings. They will find usable numbers and a clear contrast between the two models, but the paper does not reorganize the broader subject. It deserves a serious referee because the topic is timely, the method is standard, and the claimed distinctions are falsifiable once the derivations are on the table.

Referee Report

2 major / 2 minor

Summary. The paper investigates symmetry-resolved Rényi and von Neumann entropies in Lifshitz scalar chains and fermionic models using charged moments and the correlator method. It reports approximate equipartition among charge sectors for scalars in the large-z regime (configurational entropy dominant) and genuine equipartition for fermions only in the relativistic limit z=1 (fluctuation entropy dominant), with dependence on subsystem size, charge, and mass.

Significance. If the central calculations are robust, the work identifies distinct non-relativistic signatures in symmetry-resolved entanglement and supplies a framework for probing operationally accessible entanglement in systems such as cold atoms, where particle-number resolution is feasible.

major comments (2)

[§3] §3 (Charged moments for Lifshitz scalars): the derivation of Z_n(α) via the correlator method assumes the standard Gaussian form without explicit regularization of the anisotropic UV divergences arising from the (∂^z ϕ)^2 term; this is load-bearing because any z-dependent charge-sector corrections would modify the Fourier transform to p(Q) and the reported dominance of configurational entropy at large z.
[§5] §5 (Fermionic Lifshitz results): the claim of genuine equipartition only at z=1 rests on the two-point function scaling |x-y|^{-1/z} yielding no extra charge-dependent terms in the charged moments for z≠1, yet no explicit check or counter-term analysis is provided to confirm the absence of such corrections when the Hamiltonian contains higher spatial derivatives.

minor comments (2)

[Abstract] The abstract states the main results without referencing the lattice discretization or continuum extrapolation procedure employed for the Lifshitz chains.
[Figures] Figure captions for the entropy plots should include the numerical precision or fitting range used in the correlator evaluations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive feedback. We address each of the major comments in detail below. We agree that additional clarifications on regularization and counter-term analysis will strengthen the presentation, and we will incorporate these in the revised version.

read point-by-point responses

Referee: [§3] §3 (Charged moments for Lifshitz scalars): the derivation of Z_n(α) via the correlator method assumes the standard Gaussian form without explicit regularization of the anisotropic UV divergences arising from the (∂^z ϕ)^2 term; this is load-bearing because any z-dependent charge-sector corrections would modify the Fourier transform to p(Q) and the reported dominance of configurational entropy at large z.

Authors: We thank the referee for this important remark. Our derivation employs the correlator method for the Gaussian state of the Lifshitz scalar on a discrete chain, where the lattice spacing serves as the UV regulator. The anisotropic divergences associated with the higher-derivative term are present but, crucially, they appear as overall multiplicative factors in the expression for the charged moments Z_n(α) that are independent of the charge twist parameter α. This is because α modifies the boundary conditions or the phase in the correlation functions within the subsystem, but does not alter the UV behavior of the bulk propagator. As a result, these factors cancel in the normalized probabilities p(Q) obtained via Fourier transform, leaving the charge dependence unaffected. We will revise §3 to include an explicit demonstration of this α-independence through a brief regularization analysis, thereby confirming that the reported dominance of configurational entropy at large z remains valid. revision: yes
Referee: [§5] §5 (Fermionic Lifshitz results): the claim of genuine equipartition only at z=1 rests on the two-point function scaling |x-y|^{-1/z} yielding no extra charge-dependent terms in the charged moments for z≠1, yet no explicit check or counter-term analysis is provided to confirm the absence of such corrections when the Hamiltonian contains higher spatial derivatives.

Authors: We appreciate the referee pointing out the lack of explicit verification for the fermionic model. In the free fermionic Lifshitz theory, the charged moments are computed from the two-point correlation functions with the insertion of the U(1) phase factor e^{iα Q} for the subsystem. The scaling |x-y|^{-1/z} arises from the dispersion relation, and the higher spatial derivatives in the Hamiltonian modify the form of the correlator but maintain the Gaussian (bilinear) structure of the action. Consequently, no additional charge-dependent UV divergences or counterterms are introduced beyond those already accounted for in the standard treatment. To rigorously substantiate this, we will add an explicit counter-term analysis in the revised §5 (or a new appendix), computing the possible corrections for z ≠ 1 and showing that they do not generate extra α-dependent contributions that would spoil the equipartition at z=1. This will also clarify why fluctuation entropy dominates only in the relativistic limit. revision: yes

Circularity Check

0 steps flagged

No circularity: standard charged-moment computations applied to Lifshitz action

full rationale

The paper computes symmetry-resolved Rényi and von Neumann entropies via the charged moments Z_n(α) and the correlator method applied directly to the Lifshitz scalar and fermionic actions. These are standard QFT techniques whose validity for Gaussian states is independent of the target equipartition results; the reported large-z scalar behavior and z=1 fermionic behavior follow from the explicit two-point functions and Fourier transforms rather than from any redefinition, fitted parameter renamed as prediction, or self-citation chain. No equation reduces the final equipartition statements to the inputs by construction, and the derivation remains self-contained against external benchmarks such as the usual replica-trick formulas.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the work relies on standard definitions of Lifshitz scaling and charged moments from prior literature.

pith-pipeline@v0.9.0 · 5475 in / 1286 out tokens · 44613 ms · 2026-05-10T02:50:59.883198+00:00 · methodology

discussion (0)

Reference graph

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Entanglement entropy and mutual information of circular entangling surfaces in the 2 + 1-dimensional quantum Lifshitz model,

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Entanglement in Lifshitz-type quantum field theories,

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Entanglement entropy in Lifshitz theories,

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Entanglement Evolution in Lifshitz-type Scalar Theories,

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Universal Scaling in Fast Quenches Near Lifshitz-Like Fixed Points,

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A field theory study of entanglement wedge cross section: Odd entropy,

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Time scaling of entanglement in in- tegrable scale-invariant theories,

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Entanglement and separability in con- tinuum Rokhsar-Kivelson states,

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Entanglement in Lifshitz fermion theories,

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S. Khoshdooni, K. Babaei Velni and M. R. Mohammadi Mozaffar, “Capacity of entangle- ment in Lifshitz theories,” Phys. Rev. D112, no.2, 026027 (2025) doi:10.1103/7cg6-m7dn [arXiv:2505.08297 [hep-th]]

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Observing the Quantum Mpemba Effect in Quantum Simulations,

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M. R. Mohammadi Mozaffar and A. Mollabashi, work in progress. 28