Entanglement Hamiltonian after a local quench

Riccarda Bonsignori; Viktor Eisler

Entanglement Hamiltonian after a local quench

T0 review · 0 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read After a joining quench, the entanglement Hamiltonian stays local in the continuum, with right- and left-movers carrying different weights.

desk verdict Solid paper: finite-L CFT weights for joining-quench EH, plus a lattice continuum limit that nails the prediction; worth refereeing. read the letter →

arxiv 2508.19406 v1 pith:CBJJUJEJ submitted 2025-08-26 hep-th cond-mat.stat-mechquant-ph

Riccarda Bonsignori , Viktor Eisler This is my paper · ORCID

classification hep-thcond-mat.stat-mechquant-ph

keywords entanglementHamiltonianlocalquenchconformalfieldtheoryfreefermionscontinuumlimitmodularentropyjoining

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Two independent half-chains of free fermions, each in its ground state, are suddenly joined at their ends. This paper shows that, in the continuum limit described by conformal field theory, the entanglement Hamiltonian of a subsystem is a local operator: an integral of the energy density with two distinct weight functions, one for right-moving and one for left-moving modes. The weights have zeros at the entangling point and at the ballistic front created by the quench, and their structure explains why right-movers contribute three times as much to the entanglement entropy as left-movers. Exact numerics on a hopping chain, after summing away the lattice's long-range hopping tails, reproduce these weights. The result contrasts with the global quench, where the lattice entanglement Hamiltonian stays genuinely long-range.

What carries the argument

The conformal map from the 'double-pants' geometry of the quench path integral to an annulus, ξ(z) = i√[sin(π(iλ+z)/2L)/sin(π(iλ−z)/2L)], followed by the logarithmic map w = log ζ. In the strip geometry the entanglement Hamiltonian is a translation generator, and the inverse-temperature weights follow from β = 2π/w′(z); analytic continuation τ→it turns them into the piecewise functions in (27) and (29). On the lattice side, the carrying mechanism is the continuum limit of the EH hopping matrix: substituting c_j → √a(e^{ik_F x}ψ + e^{−ik_F x}ψ̄) and summing the diagonals t_r, s_r gives β0 and β1 via (41), which at half filling decouples into purely real and imaginary diagonals and kills the p

What would settle it

Compute the continuum-limit weights from (41) for a subsystem displaced from the quench (x0 > 0) at times just after the front arrives (t slightly larger than x0): the prediction (29) has β and β̄ given by different piecewise formulas on the two sides of x = t, so the ratio β1/β0 should jump exactly at j = t. If the numerical ratio crosses smoothly or at the wrong location, the branch-cut choice in Appendix A is wrong.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is an explicit local CFT prediction for the entanglement Hamiltonian after a joining quench on a finite chain of 2L sites: H = ∫A dx [β(x,t) T(x−t) + β̄(x,t) T̄(x+t)], with piecewise trigonometric weights. For a subsystem starting at the junction (x0 = 0) the weights are β = 4L |sin(πx/2L) sin(π(x−t)/2L) / sin(πt/2L)| and β̄ with x+t, valid for λ→0, t≫λ; for an arbitrary partition x0 > 0, the branch cuts of the conformal map force three different regimes, giving the piecewise expression in (29). The lattice calculation shows that the EH matrix contains long-range real and imaginary hopping terms, but the properly defined continuum limit—summing

Load-bearing premise

The derivation depends on fixing the sign of a square root in the conformal map so that the relevant image points lie in the upper half-plane after continuing τ→it; a different branch choice changes the piecewise weights and the entropy.

Editorial extensions

If this is right

The EH is local in the continuum: a weighted integral of T00 and T01, so a local-temperature interpretation is valid for this out-of-equilibrium protocol.
The right-moving front acts as a moving entangling point inside the subsystem, which is why right-movers contribute three times the left-movers to the entropy; the total entropy matches the known CFT result S = (c/3) log[(2L/π)√(λϵ) sin(πt/2L)].
After the front reflects from the boundary at t = L, the weights satisfy β(x, 2L − t) = β̄(x, t), so the late-time EH is obtained by exchanging chiralities.
On the lattice, convergence to the CFT result requires the full sum over diagonals of the EH matrix; the first diagonal alone is not enough, especially for the imaginary (current) part.
The weight behind the front is time-independent (β0 = 2 sin πx for the half-chain), and at t = L the momentum-density weight β1 vanishes; both features are visible in the L = 100 numerics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One natural next test is away from half filling: the CFT weights should simply rescale time by the Fermi velocity, while the lattice sums in (40) lose the even/odd diagonal separation; comparing the two would isolate whether the continuum limit still converges.
The local form suggests an experimental target the paper leaves implicit: in a cold-atom or ion realization of the joining quench, one could reconstruct ρA and look for the two inverse-temperature profiles, with the zero of β0 at the moving front as a direct signature.
The strong lattice oscillations seen for small t near the unvisited region raise a testable question: do they vanish in the L→∞ limit at fixed t, or does the continuum limit require t large enough that the front has spread? Numerics at larger L would settle it.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read the Bonsignori–Eisler paper. Bottom line: the main claim holds. They generalize Cardy–Tonni to the finite-L joining quench, get explicit chiral weights beta and betabar (eqs 27,29), and then show the lattice EH for free fermions, after the appropriate continuum limit, reproduces those weights. The continuum limit involves imaginary hopping terms, which is genuinely new for this quench, and the matching is convincing, not tuned. The L→∞ limit reduces to [7] and the entanglement entropy reproduces [46]; both are good consistency checks.

Credit where due: the derivation follows the established conformal mapping, with real extra work in the branch-choice appendix. The lattice numerics are exact, and the sums in (41) contain no input from the CFT branch choice, so the agreement in Figs. 4 and 5 is real evidence, not a fit. The parity structure t_{2p}=0, s_{2p+1}=0 is a nice simplification that removes the chemical potential and current terms.

Soft spots, in proportion: the oscillations for j>t are visible, stronger at early times, and the paper calls the agreement 'remarkable' without quantifying them. That is a minor complaint; the overall shape is clearly right. The convergence of the diagonal sums in (41) is empirical—they say you essentially need the complete sum, especially for s_{2p}—but they do not provide a convergence test or error estimates. A referee should ask for that. The branch-choice step in Appendix A is the delicate one: they fix signs so that xi and xi0 sit in the UHP after analytic continuation. This is imposed, not derived from an explicit continuation path. The stress-test concern about it does not land, because the lattice sums are independent of that choice and reproduce the piecewise beta, but a referee should push for a cleaner justification or a direct numeric check of the continuation.

Who this is for: people working on entanglement Hamiltonians in CFT and lattice models, local quench dynamics, modular Hamiltonians. It does not claim broader technology, and it does not need to. It deserves a serious referee; I would send it to review. If I were editing, I would not desk-reject it.

Referee Report

0 major / 5 minor

Summary. This paper investigates the entanglement Hamiltonian (EH) after a local joining quench in a finite chain of free fermions. The authors first generalize the Cardy–Tonni CFT approach to a finite geometry with open boundaries. They map the double-pants geometry to an annulus and derive the EH as a weighted integral of the right- and left-moving stress-tensor components. For the half-chain partition (x0=0) they obtain explicit weights β(x,t) and β̄(x,t) in Eq. (27); for a decentered subsystem (x0>0) they obtain piecewise expressions in Eq. (29). They also compute the entanglement entropy from these weights and recover the known result of Stephan and Dubail. On the lattice side, they compute the exact EH matrix via the correlation-matrix method and develop a continuum limit of the lattice EH in terms of sums over hopping amplitudes (Eqs. 40–41). For the half-filling case, these sums simplify and yield the weights β0 and β1, which are compared to the CFT prediction through Eq. (42). Exact numerical data for L=100 show good agreement for various times and both subsystem geometries, with only small oscillations near the front and boundary.

Significance. This is a significant result in the study of entanglement Hamiltonians in non-equilibrium systems. It provides, to my knowledge, the first explicit CFT prediction for the local EH after a local quench in a finite system and its verification by exact lattice numerics. The CFT derivation is parameter-free, and the lattice calculation is independent, so the agreement is highly nontrivial. The result contrasts with the global quench case, where the lattice EH is genuinely long-range, and shows that for a local low-energy quench the continuum EH remains local. The paper is well organized, with detailed appendices that make the calculations reproducible. The numerical evidence is strong, and the limitations (oscillations, slow convergence of the sums) are honestly acknowledged.

minor comments (5)

[Appendix A / Eq. (29)] The branch-choice prescription (requiring ξ(z) and ξ0 to lie in the UHP after analytic continuation) is stated but not derived. Since the piecewise form of β(x,t) and β̄(x,t) is a central result, a brief justification of why this prescription follows from a consistent continuation path—or an explicit statement that it is a working assumption supported by the lattice checks—would remove residual ambiguity.
[Section IV / Eq. (41)] The convergence of the sums in the continuum limit is only empirical. The paper notes that the decay in p is slow, especially for s_{2p}(x). Adding a plot for different lattice sizes (e.g., L=50, 100, 200) or a short discussion of the asymptotic behavior would strengthen the continuum-limit claim.
[Fig. 4] The oscillations for j>t are mentioned but not quantified. A comment on their scaling with L or their relation to finite-size effects (e.g., the front width) would be useful for the reader to judge the discrepancy.
[Section III.C / Eq. (31)] The regularization A_{ε,λ} assumes t < L−λ; for t close to L the second interval [t+λ, L] is empty and Eq. (31) should be adjusted or the regime of validity stated. This is a minor technical point, but it would improve the rigor of the entropy check.
[References / Eq. (28)] The L→∞ limit is said to reproduce the result of [7], which is a general review. Consider also citing the original local-quench entropy papers (e.g., [45,46]) at this point for historical accuracy.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the CFT weights and the lattice continuum-limit weights are derived independently and compared without fitting.

full rationale

The central claim is a comparison between two independently constructed objects. The CFT weights β(x,t) and β̄(x,t) are obtained from the conformal mapping (14)–(20) plus analytic continuation and the branch choices in Appendix A; no lattice data enter this derivation. The lattice weights β0 and β1 are obtained from the exact correlation matrix via H=ln((1−C)/C), expanded in the continuum limit in Appendix B, and evaluated with the lattice sums (41); the only model input is the Fermi momentum q_F a=π/2. Equation (42) merely rewrites the CFT chiral weights into T00/T01 components for comparison. No parameter is fitted to force agreement, and the paper explicitly notes that the first diagonal alone deviates significantly from the CFT result, so the agreement is a nontrivial check. The branch-choice prescription in Appendix A is delicate, but it is not circular: an inconsistent choice would change the prediction, while the lattice sums (41) are derived without using that choice and reproduce the CFT curves. The self-citations ([26], [29], [30], [60]) are methodological or contextual; in particular, the continuum-limit identities borrowed from [26] are rederived in Appendix B, so no load-bearing argument reduces to an unverified self-citation. The infinite-chain limit (28) and the entropy (33) reproduce previously known results, serving as consistency checks rather than inputs. Overall, no step in the derivation chain is equivalent to its own input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters; λ and ε are regulators that cancel. All inputs are standard (half-filling, chain length L). The axioms are the CFT description and the path-integral representation, which are standard domain assumptions, plus the branch-choice procedure specific to this paper.

assumptions (5)

domain assumption The half-filled hopping chain's low-energy sector is described by a 1+1D CFT with central charge c=1.
Used throughout Section III to derive the EH via conformal mapping; standard for free fermions, but unproved in this paper.
domain assumption The path integral representation (12) with damping e^{-λH} and the double-pants geometry faithfully represent the time-evolved density matrix and its reduction.
Basis of the CFT derivation, following [46]; the regularization λ is taken to 0 at the end.
standard math In the strip geometry, the modular Hamiltonian is given by (17), the integral of the translation generator along the image of the cut.
Known modular Hamiltonian result in CFT, from Cardy-Tonni [14].
ad hoc to paper The analytic continuation τ→it and the branch choices in Appendix A give the physical β and β̄.
The key technical step for the finite-L result; the authors justify it by requiring UHP/LHP placement of ξ values, but it is not derived from first principles.
domain assumption The continuum limit substitution (36) and the expansion to order a are valid for the non-equilibrium EH.
Standard approach in [26] extended here to imaginary hoppings; assumed to hold for the time-dependent case.

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Cite this review

Pith. "Pith review of Entanglement Hamiltonian after a local quench." pith.science (2026). https://pith.science/paper/CBJJUJEJ

@misc{pith2026250819406,
  author       = {Pith},
  title        = {Pith review of: Entanglement Hamiltonian after a local quench},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CBJJUJEJ}},
  note         = {Machine review of arXiv:2508.19406}
}

read the original abstract

We investigate the dynamics of the entanglement Hamiltonian in a system of one-dimensional free fermions, following a local joining quench of two initially disconnected half-chains in their ground states. Applying techniques of conformal field theory, we obtain a local expression where the left- and right-moving components of the energy density are associated with different weight functions. The results are then compared to numerical calculations for the hopping chain, which requires to consider a proper continuum limit of the lattice entanglement Hamiltonian, obtaining a good agreement with the field-theory prediction.

Figures

Figures reproduced from arXiv: 2508.19406 by the authors.

**Figure 2.** FIG. 2. Mapping from the double-pants geometry to the logarithmic image of the annulus. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Inverse temperatures [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Spatial profiles of the weights [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5. Spatial profile of the weights [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Boundary quenches in (1+1)-dimensional conformal field theory
cond-mat.stat-mech 2026-07 accept novelty 7.0 of 10

A boundary quench in a (1+1)-d CFT makes one-point functions switch from old to new ground state across a light cone and makes adjacent-region entanglement jump by log(g_b/g_a).

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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