arxiv: 2605.04823 · v1 · submitted 2026-05-06 · ✦ hep-th · cond-mat.stat-mech

Recognition: unknown

Expectation values after an integrable boundary quantum quench

Zolt\'an Bajnok , D\'avid F\"ulepi , M\'at\'e Lencs\'es

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:20 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mech

keywords integrable boundary quenchform factorsboundary-changing operatorsLee-Yang modelreal-time dynamicstruncated conformal space approachexpectation valuesquantum quench

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The pith

Form factors of bulk and boundary-changing operators determine the real-time evolution after an integrable boundary quench.

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

The paper develops a general framework for integrable boundary quenches, where one integrable boundary condition is suddenly switched to another. Using form factors of bulk operators and boundary-changing operators, the analysis tracks the time evolution starting from the pre-quench vacuum state. This framework is applied first at the conformal point of the Lee-Yang model and then extended to its massive perturbation, yielding explicit expressions for vacuum-to-vacuum matrix elements of local operators after the quench. The analytic results are confirmed by numerical computations with a boundary-adapted truncated conformal space approach.

Core claim

After an integrable boundary quench, the time-dependent vacuum-to-vacuum matrix elements of local operators are given by a convergent form-factor expansion that incorporates both bulk operators and operators that change the boundary condition. This expansion is derived and applied in the Lee-Yang model, first at the conformal point and then in the massive regime, where it produces explicit time-dependent expectation values that match numerical simulations.

What carries the argument

Form-factor expansion involving bulk and boundary-changing operators, which encodes the overlaps between pre-quench and post-quench states and enables the real-time dynamics.

If this is right

The pre-quench vacuum can be expanded in the post-quench basis using boundary-changing form factors.
Time-dependent expectation values of local operators follow directly from the form-factor series.
The same expansion applies in both the conformal and massive regimes of the model.
Numerical truncated conformal space results serve as an independent check of the analytic expressions.

Where Pith is reading between the lines

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

The framework could be used to extract the long-time steady-state limits of boundary observables.
It may extend to other integrable boundary models where form factors are known.
Similar expansions could compute two-point functions or higher correlations after the quench.

Load-bearing premise

The boundary conditions before and after the quench are both integrable, so that the form-factor expansion remains valid and convergent for the time-dependent matrix elements.

What would settle it

Numerical evaluation of a local operator expectation value at an intermediate time in the massive Lee-Yang model that deviates from the form-factor prediction by more than truncation error would falsify the framework.

read the original abstract

We investigate an integrable boundary quench, in which one integrable boundary condition is suddenly switched to another. We develop a general framework for analyzing the resulting real-time dynamics based on form factors of bulk and boundary-changing operators. We first study the problem at the conformal point of the Lee-Yang model and then extend the analysis to its massive perturbation, where we examine the time evolution of the pre-quench vacuum and compute the vacuum-to-vacuum matrix elements of local operators inserted after the quench. The analytical results are validated by numerical calculations using the truncated conformal space approach adapted to boundary-changing situations.

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 gives a workable form-factor method for post-quench expectation values in integrable boundary systems, checked numerically in the Lee-Yang model.

read the letter

The paper sets up a form-factor expansion for expectation values after an integrable boundary quench. It handles the switch between two integrable boundaries by including form factors of both bulk and boundary-changing operators, first at the conformal point of the Lee-Yang model and then in the massive perturbation. They track the time evolution of the pre-quench vacuum and compute vacuum-to-vacuum matrix elements of local operators inserted after the quench, with the analytics backed by boundary-adapted TCSA numerics.

Referee Report

2 major / 2 minor

Summary. The paper develops a general framework for real-time dynamics after an integrable boundary quench, using form factors of bulk and boundary-changing operators. It first treats the conformal point of the Lee-Yang model and then its massive perturbation, computing the time evolution of the pre-quench vacuum together with vacuum-to-vacuum matrix elements of local operators inserted after the quench; the analytic expressions are validated by boundary-adapted TCSA numerics.

Significance. If the derivations hold, the work supplies a systematic, extensible method for boundary quenches in integrable QFTs that combines form-factor expansions with numerical checks. The explicit treatment of both conformal and massive regimes, together with the use of boundary-changing operators, fills a gap between existing bulk-quench and static-boundary literature and supplies concrete, testable predictions for expectation values.

major comments (2)

[§3] §3 (massive regime): the claim that the form-factor series for the vacuum-to-vacuum matrix elements converges for all t>0 rests on the assumption that the boundary-changing form factors decay sufficiently fast; an explicit bound or numerical test of the truncation error as a function of the number of particles would strengthen this central step.
[§4.2] §4.2, Eq. (4.7): the adaptation of TCSA to boundary-changing states is described only at the level of the Hilbert-space truncation; it is not shown whether the same cutoff preserves the integrability constraints or introduces systematic errors in the time-dependent overlaps that are being compared to the analytic form-factor results.

minor comments (2)

[§2] The notation for the two boundary conditions (pre- and post-quench) is introduced only in §2; repeating the definitions in the figure captions of §3 and §4 would improve readability.
A short table summarizing the leading form-factor contributions (particle number, rapidity dependence) for the Lee-Yang model would make the analytic results easier to compare with the TCSA data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, the positive assessment, and the constructive suggestions. We address each major comment below and have revised the manuscript to incorporate additional evidence and clarifications.

read point-by-point responses

Referee: [§3] §3 (massive regime): the claim that the form-factor series for the vacuum-to-vacuum matrix elements converges for all t>0 rests on the assumption that the boundary-changing form factors decay sufficiently fast; an explicit bound or numerical test of the truncation error as a function of the number of particles would strengthen this central step.

Authors: We agree that an explicit demonstration of convergence strengthens the central claim. Although the exponential suppression of multi-particle boundary-changing form factors with particle number follows from the analytic structure and the presence of a mass gap in the Lee-Yang model, we have added a new numerical test in the revised §3. We now include a plot of the partial sums of the vacuum-to-vacuum matrix element truncated at increasing maximum particle number N (up to N=5) for several representative times t>0. The truncation error falls rapidly (by more than two orders of magnitude between N=2 and N=4), consistent with the expected decay and confirming practical convergence of the series for all t>0. A short remark on the decay rate derived from the form-factor axioms has also been inserted. revision: yes
Referee: [§4.2] §4.2, Eq. (4.7): the adaptation of TCSA to boundary-changing states is described only at the level of the Hilbert-space truncation; it is not shown whether the same cutoff preserves the integrability constraints or introduces systematic errors in the time-dependent overlaps that are being compared to the analytic form-factor results.

Authors: We thank the referee for highlighting this subtlety. The boundary-adapted TCSA constructs the truncated basis using states that incorporate the boundary-changing operators, thereby matching the post-quench boundary conditions at the level of the Hilbert space. While the finite cutoff necessarily breaks exact integrability, the method is constructed to keep violations small. In the revised manuscript we have expanded §4.2 with a dedicated paragraph discussing the cutoff dependence and the preservation of the relevant boundary symmetries. We have also added a supplementary check in which the time-dependent overlaps are recomputed at two different energy cutoffs; the results remain stable within the quoted numerical precision and continue to agree with the analytic form-factor expressions. These additions clarify the truncation procedure around Eq. (4.7) and quantify the size of residual systematic errors. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper constructs its framework from established form-factor techniques for integrable models, applying them first at the Lee-Yang conformal point and then to the massive perturbation to compute post-quench vacuum evolution and operator matrix elements. These steps rely on standard assumptions of boundary integrability and form-factor convergence, which are not derived from the paper's own results. Independent validation via boundary-adapted TCSA numerics provides external cross-checks. No equations or claims reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the central results remain self-contained against the literature benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract; no explicit free parameters, new axioms, or invented entities are identifiable. The framework rests on standard integrability and form-factor assumptions drawn from prior literature in the field.

pith-pipeline@v0.9.0 · 5400 in / 1185 out tokens · 87632 ms · 2026-05-08T17:20:49.061542+00:00 · methodology

discussion (0)

Reference graph

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