Pith. sign in

REVIEW 2 major objections 5 minor 68 references

After a boundary quench in a (1+1)-dimensional conformal field theory, every scalar-primary one-point function becomes a step function in x−t, and the entanglement entropy of a subsystem adjacent to the boundary jumps by log(g_b/g_a) when t

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 13:14 UTC pith:SAGX6JTB

load-bearing objection A clean, correct derivation of a new universal entropy jump after a boundary-condition quench; the main proof-gap worry raised in the stress test is unfounded. the 2 major comments →

arxiv 2607.19166 v1 pith:SAGX6JTB submitted 2026-07-21 cond-mat.stat-mech cond-mat.str-elhep-thquant-ph

Boundary quenches in (1+1)-dimensional conformal field theory

Michele Fossati , Colin Rylands , Eytan Grosfeld , Eran Sela , Pasquale Calabrese This is my paper
classification cond-mat.stat-mech cond-mat.str-elhep-thquant-ph MSC 81T4081P42
keywords boundary quenchconformal field theoryentanglement entropyboundary g-factorone-point functionlight coneIsing CFTRényi entropy
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks what a conformal field theory does when the boundary condition on a half-line is suddenly changed from a to b at time zero. It proves that every scalar-primary one-point function becomes a step function in x−t: for x>t (space-like) it equals the ground-state expectation with boundary condition a, and for x

Core claim

The central claim is that a boundary quench from condition a to condition b can be encoded in the conformal map f(z)=(z+iϵ)/(z−iϵ), which sends the cut geometry with boundary-changing fields at ±iϵ to the upper half-plane with two boundary conditions. In the combined limits ϵ→0 and τ=it, the angular variable cosθ in the upper-half-plane one-point function becomes sgn(x−t), so the one-point function of any scalar primary collapses to the a-ground-state value for x>t and the b-ground-state value for x<t (Eq. 2.13). Repeating the same argument in the n-replicated theory with a twist field at the entangling surface gives the Rényi entropy shift: 0 for t<x and log(g_b/g_a) for t>x, for every n, a

What carries the argument

The conformal map f(z)=(z+iϵ)/(z−iϵ) from the cut half-plane (with boundary-changing fields at ±iϵ) to the upper half-plane, combined with the general boundary one-point function ⟨O(w)⟩_H = F_ab(cosθ)(2 Im w)^{−ΔO}. The light-cone step emerges because analytic continuation of cosθ to real time yields sgn(x−t). In the replica calculation, the same map is applied to the n-replicated theory with a twist field at the entangling surface; the factor g_d^{1−n} from the entangling boundary cancels between numerator and denominator, leaving the jump log(g_b/g_a) independent of d.

Load-bearing premise

The derivation rests on the assumption that a sudden boundary switch is correctly represented by a primary boundary-condition-changing operator with the e^{−ϵH_b} regularization, and that the boundary conditions involved are 'simple'; if the quench generates a non-primary or nonlocal boundary change, the light-cone step (2.13) and the entropy jump (2.21) need not hold.

What would settle it

Perform a boundary quench in a conformal field theory where the boundary-changing operator is a descendant or a nonlocal operator and check whether the one-point function still takes the step form (2.13); alternatively, measure the Rényi entropy jump for the (f,+) → (+,+) Ising quench on much larger chains and test whether the jump approaches log(1/g_f) = −log√2 uniformly in x, or whether residual x-dependence indicates the universality claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • One-point functions of scalar primaries follow a strict light-cone step: they equal the initial ground-state value for x>t and the final ground-state value for x<t.
  • The Rényi entropy of the subsystem [0,x] jumps by log(g_b/g_a) at t=x for all n, and the same holds for the von Neumann entropy; the jump is independent of the entangling-surface boundary condition.
  • The jump is universal for simple boundary conditions and primary boundary-changing operators; it depends only on the boundary g-factors, not on the quench details or the operator content.
  • In the critical Ising chain, the magnetization inside the light cone vanishes for the free boundary condition (since A_f=0) and takes the fixed-boundary value otherwise; the sign-change periodicity at t=2L for (f,f) is explained by the Z2 spin-flip action of U(2L).
  • For quenches between boundary conditions with equal g-factors (e.g., +→−), the entropy shows no jump even though the magnetization changes sign across the light cone.

Where Pith is reading between the lines

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

  • Extension: a boundary quench acts like a local operator insertion of a boundary-condition-changing field, suggesting that a quasiparticle picture for local quenches may exist where a single boundary-emitted excitation carries the entropy jump; the paper leaves this open.
  • Extension: measuring the entropy jump after a sudden boundary switch in a cold-atom or Kondo-type setup could provide a dynamical measurement of the ratio of boundary g-factors, complementing the equilibrium boundary entropy.
  • Extension: the same conformal-map argument applied to higher-point functions would predict that multipoint correlators also switch abruptly to the b ground-state values inside the light cone; this is testable in the Ising chain.
  • Extension: the finite-interval analysis hints that at special times such as t=L/v the time-evolved state may coincide with a different ground state (the numerics suggest (f,+) at t=L/v behaves like the (f,+) ground state); if confirmed analytically, this would be a new 'return-to-ground-state' effect for boundary quenches.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper investigates a local quench in a (1+1)-dimensional CFT on the half-line where the conformal boundary condition is suddenly changed from a to b. The initial state is prepared via a boundary-changing operator smeared over a short Euclidean time. Using the conformal map f(z)=(z+i eps)/(z-i eps), the authors derive Eq. (2.13): scalar primary one-point functions equal the a-ground-state value for x>t and the b-ground-state value for x<t. Applying the same argument to the replica twist field gives Eq. (2.21): the Renyi/entanglement entropy of the subsystem adjacent to the boundary jumps by log(g_b/g_a) once the subsystem is fully time-like separated from the quench. The predictions are benchmarked against MPS simulations of the critical Ising chain for several boundary conditions. The paper also studies the Z_2 entanglement asymmetry and explains a magnetization sign change at t=2L/v using Virasoro module periodicity.

Significance. If correct, the central result is a clean, parameter-free universal prediction: a sharp entanglement entropy jump determined solely by the ratio of Affleck-Ludwig g-factors. The derivation is transparent, no fitted parameters are used, and the Ising MPS results independently confirm the light-cone step in one-point functions and entropy. The paper connects boundary critical phenomena with nonequilibrium entanglement dynamics and should interest a wide audience. Its strengths are the explicit universal formulas, the direct numerical benchmark, and the algebraic finite-size argument. The main open issues are a formal gap in the analytic continuation and the unproved but plausible primariness assumption; both are fixable and do not undermine the numerical evidence.

major comments (2)
  1. [Section 2, Eqs. (2.10)-(2.11)] The step 'after setting tau=it and sending epsilon to 0' is load-bearing but not justified as written. The right-hand side of Eq. (2.10) is introduced as Re(w)/|w|, which is not holomorphic in tau; a naive substitution tau=it in the non-holomorphic expression would give a different sign pattern. The correct procedure is to treat the right-hand side as a real-analytic function of Euclidean tau, continue that real function to tau=it, and state the branch of the square roots that yields sgn(x-t) as epsilon goes to zero. Since Eqs. (2.13) and (2.21) depend on this step, the paper should supply this clarification. This is a proof gap, not a demonstrated error: the Ising numerics support the final step.
  2. [Section 2, Eq. (2.1)] The initial state assumes a primary boundary-changing operator phi-hat : H_a -> H_b, and the text says 'which we assume to be a primary', but no justification or precise condition is provided. The conformal mapping and replica arguments rely on this primariness. For the Ising examples the relevant operators are known to be primary, but the paper claims universality for simple boundary conditions. The authors should either prove or cite a result that such primary boundary-changing operators exist for the class of quenches considered, or explicitly restrict Eq. (2.13) and Eq. (2.21) to quenches where this holds.
minor comments (5)
  1. [Section 2.1] The cutoff epsilon-prime at the entangling surface is introduced without a definition; it should be distinguished from the epsilon used in Eq. (2.1).
  2. [Section 3, after Eq. (3.3)] The notation for the boundary fields is slightly ambiguous; specifying the sign convention would help.
  3. [Figures 4-6] The color-scale plots are useful, but explicit contour lines or cross-section insets would make the light-cone step and entropy jump easier to read.
  4. [Throughout] The accent in 'Renyi' is typeset inconsistently.
  5. [Section 1, last paragraph] The comparison with Ref. [52] would benefit from one sentence stating which results are non-overlapping rather than only 'to the best of our knowledge'.

Circularity Check

0 steps flagged

No circularity: the light-cone result and entropy jump follow from a self-contained conformal mapping argument; benchmark constants come from independent prior literature.

full rationale

The central derivation (Eqs. 2.5-2.13) is self-contained: it uses the explicit conformal map f(z)=(z+iε)/(z-iε), the general form of the one-point function on the upper half-plane (2.7), and the boundary limits F_ab(±1)=A^a/b_{O,I}. These are standard, independently established CFT facts, not outputs of this paper. The only non-trivial input is the assumption that the boundary-condition-changing operator φ̂ is primary, explicitly stated as an assumption ('which we assume to be a primary'), not a fitted or imported result. The entropy jump (2.21) is obtained by applying the same light-cone one-point result to the replica twist field T_n, a legitimate secondary use rather than a restatement of the input; the g-factors used in the Ising benchmark are taken from Affleck-Ludwig [42] and the bulk-boundary constants from Runkel [55], both independent of the present work. The MPS simulations are genuine external benchmarks and do not feed back into the analytic derivation. Self-citations to Calabrese-Cardy quench and entanglement-entropy literature are contextual background, not load-bearing. The only concern raised in a skeptical reading is the branch choice in the analytic continuation from (2.10) to (2.11): cosθ is obtained from non-holomorphic Re(w)/|w|, but the correct limiting sign can be obtained directly by taking ε→0 in the explicit real expression of (2.10) at τ=it, yielding (x²-t²-ε²)/[(x²-t²-ε²)²+4ε²t²]^{1/2} → sgn(x²-t²). This is at most a proof-clarity issue, not circularity, and it does not involve fitting parameters or self-referential definitions. Thus the paper's predictions are not equivalent to their inputs by construction, and no circular step is present.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The analytic results rely on standard boundary-CFT inputs and the primary boundary-changing operator assumption. No new particles, forces, or conserved quantities are introduced, and no numbers are fitted to data in the derivation of the central formulas.

axioms (4)
  • domain assumption A primary boundary-changing operator φ̂ : H_a^{R+} → H_b^{R+} exists and the initial state is e^{-ϵH_b}φ̂|gs,a⟩ (Eq. (2.1)).
    The entire calculation is anchored to this regularization. The paper states "which we assume to be a primary" and cites "simple" boundary conditions, but does not prove such an operator exists for arbitrary boundary-condition quenches.
  • domain assumption The one-point function on the upper half-plane with mixed boundary conditions has the form F_ab(cos θ)(2 Im w)^{-Δ}, with F_ab(±1) equal to bulk-boundary OPE constants (Eq. (2.7)).
    This is standard boundary-CFT structure, cited to [54], but it is an input assumption for the derivation of the light-cone step.
  • standard math Analytic continuation τ=it and the ϵ→0 limit can be interchanged, and F_ab(cos θ) can be evaluated at cosθ→±1 after continuation (Eqs. (2.10)–(2.13)).
    The result depends on the limit lim_{ϵ→0} cosθ|_{τ=it} = sgn(x−t). No rigorous justification is given for the interchange of limits, though it is standard in CFT quench calculations.
  • domain assumption The n-th Rényi entropy is computed by the replica construction where the entangling surface is assigned a boundary condition d and small-disk partition functions factor into powers of g_d and twist fields (Eqs. (2.16)–(2.19)).
    This follows the entangling-surface treatment of [57] and is essential for obtaining the g-factor jump; no independent derivation is provided in this paper.

pith-pipeline@v1.3.0-alltime-deepseek · 13427 in / 14426 out tokens · 161836 ms · 2026-08-01T13:14:53.049459+00:00 · methodology

0 comments
read the original abstract

We investigate a class of local quantum quenches in which the conformal boundary condition of a (1+1)-dimensional conformal field theory is abruptly changed. We derive a remarkably simple and universal expression for the time evolution of one-point functions on the half-line. This result provides a direct description of the propagation of the disturbance generated by the quench and, in turn, allows us to determine the dynamics of bipartite entanglement for subsystems adjacent to the boundary. We show that, once the subsystem becomes fully causally connected to the quench event, the entanglement entropy undergoes a sharp finite jump whose magnitude is universally given by the logarithm of the ratio of the boundary g-factors associated with the initial and final boundary conditions. We benchmark these analytical predictions against Matrix Product State simulations of the critical Ising spin chain, finding excellent agreement. The numerical analysis further allows us to investigate the time evolution of the spin-flip entanglement asymmetry, revealing how the symmetry-breaking perturbation emitted from the boundary propagates through the system. Our results uncover universal dynamical signatures of boundary quenches and establish a direct connection between nonequilibrium entanglement dynamics and boundary critical phenomena.

Figures

Figures reproduced from arXiv: 2607.19166 by Colin Rylands, Eran Sela, Eytan Grosfeld, Michele Fossati, Pasquale Calabrese.

Figure 1
Figure 1. Figure 1: Euclidean path-integral preparation of the ground state with boundary condition a (on the left), and path integral preparation of the initial state |ψi⟩ of (2.1) (on the right). breaking at the level of the subsystem. It has been shown to be an ideal proxy for the quantum Mpemba effect [48, 49] and it is experimentally measurable [50, 51]. We evaluate the second R´enyi entanglement asymmetry for subsystems… view at source ↗
Figure 2
Figure 2. Figure 2: Left: The geometry M, is the right half of the complex plane, with fields that change boundary condition from a to b and vice-versa inserted at ±iϵ. Points on M are denoted by z = x + iτ . Right: the geometry H is the upper half-plane, with boundary condition a on the positive real axis, and boundary condition b on the negative real axis. Boundary changing fields are at 0 and at ∞. Points on H are denoted … view at source ↗
Figure 3
Figure 3. Figure 3: The analytic continuation to euclidean time of (2.16), expressed in terms of partition functions of the n-replicated CFT. ∆Tn = c 12 (n − 1/n) is the scaling dimension of Tn [59]). This is analogous to the procedure explained in [57], with the difference that here we replicate the theory (taking n stacks of it), instead of putting the theory on a replicated manifold (n-sheeted Riemann surface). We then get… view at source ↗
Figure 4
Figure 4. Figure 4: Top: Space-time plot of the quench (+, b) → (f, b). Every row is a different quench, where b is fixed respectively to f, +, −. Columns are respectively: magnetization, jump in the entanglement entropy, second R´enyi asymmetry on A = [0, x]. The predicted light-cone behavior is observed in the magnetization and in the entanglement entropy. Bottom: The same observables, on a fixed site at x/L = 1/5 (dashed l… view at source ↗
Figure 5
Figure 5. Figure 5: Top: Space-time plots of the quench (f, b) → (+, b). Every row is a different quench, where b is fixed respectively to f, +, −. Columns are respectively: magnetization, jump in the entanglement entropy, second R´enyi asymmetry on the right subsystem B = [x, L]. Bottom: The same observables, on a fixed site at x/L = 1/5 (dashed line in top figures), as a function of time. 14 [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 6
Figure 6. Figure 6: Top: Spacetime plot for the (+, b) → (−, b) quench. Every row is a different quench, where b is fixed respectively to f, +, −. Columns display magnetization and variation in the entanglement entropy. Bottom: The same observables, on a fixed site at x/L = 1/5 (dashed line in top figures), as a function of time . 15 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

68 extracted references · 57 linked inside Pith

  1. [1]

    Calabrese and J

    P. Calabrese and J. Cardy,Time-dependence of correlation functions following a quantum quench,Phys. Rev. Lett.96(2006) 136801,arXiv:cond-mat/0601225

    Pith/arXiv arXiv 2006

  2. [2]

    Calabrese and J

    P. Calabrese and J. Cardy,Quantum Quenches in Extended Systems,J. Stat. Mech.(2007) P06008,arXiv:0704.1880

    Pith/arXiv arXiv 2007

  3. [3]

    Kinoshita, T

    T. Kinoshita, T. Wenger, and D. S. Weiss,A quantum Newton’s cradle,Nature440(2006) 900

    2006

  4. [4]

    Hofferberth, I

    S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer,Non- equilibrium coherence dynamics in one-dimensional Bose gases,Nature449(2007) 324, arXiv:0706.2259. 16

    Pith/arXiv arXiv 2007

  5. [5]

    Trotzky, Y.-A

    S. Trotzky, Y.-A. Chen, A. Flesch, I. P. McCulloch, U. Schollw¨ ock, J. Eisert, and I. Bloch,Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas,Nature Phys.8(2012) 325,arXiv:1101.2659

    Pith/arXiv arXiv 2012

  6. [6]

    Gring, M

    M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer,Relaxation and Prethermalization in an Isolated Quantum System,Science337(2012) 1224953,arXiv:1112.0013

    Pith/arXiv arXiv 2012

  7. [7]

    Cheneau, P

    M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauss, T. Fukuhara, C. Gross, I. Bloch, C. Kollath, and S. Kuhr,Light-cone-like spreading of correlations in a quantum many-body system,Nature481(2012) 484,arXiv:1111.0776

    Pith/arXiv arXiv 2012

  8. [8]

    Calabrese and J

    P. Calabrese and J. Cardy,Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech.(2005) P04010,arXiv:cond-mat/0503393

    Pith/arXiv arXiv 2005

  9. [9]

    Alba and P

    V. Alba and P. Calabrese,Entanglement and thermodynamics after a quantum quench in integrable systems,Proc. Nat. Acad. Sci.114(2017) 7947,arXiv:1608.00614

    Pith/arXiv arXiv 2017

  10. [10]

    Calabrese and J

    P. Calabrese and J. Cardy,Entanglement and correlation functions following a local quench: a conformal field theory approach,J. Stat. Mech.(2007) P10004,arXiv:0708.3750

    Pith/arXiv arXiv 2007

  11. [11]

    Eisler and I

    V. Eisler and I. Peschel,Evolution of entanglement after a local quench,J. Stat. Mech. (2007) P06005,arXiv:0703379

    2007

  12. [12]

    Eisler, D

    V. Eisler, D. Karevski, T. Platini, and I. Peschel,Entanglement evolution after connecting finite to infinite quantum chains,J. Stat. Mech.(2008) P01023,arXiv:0711.0289

    Pith/arXiv arXiv 2008

  13. [13]

    Kleine, C

    A. Kleine, C. Kollath, I. P. McCulloch, T. Giamarchi, and U. Schollw¨ ock,Excitations in two-component bose gases,New J. Phys.10(2008) 045025,arXiv:0712.1448

    Pith/arXiv arXiv 2008

  14. [14]

    Igl´ oi, Z

    F. Igl´ oi, Z. Szatm´ ari, and Y.-C. Lin,Entanglement entropy with localized and extended interface defects,Phys. Rev. B80(2009) 024405,arXiv:0903.3740

    Pith/arXiv arXiv 2009

  15. [15]

    B. Hsu, E. Grosfeld, and E. Fradkin,Quantum noise and entanglement generated by a local quantum quench,Phys. Rev. B80(2009) 235412,arXiv:0908.2622

    Pith/arXiv arXiv 2009

  16. [16]

    Polkovnikov, K

    A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore,Nonequilibrium dynamics of closed interacting quantum systems,Rev. Mod. Phys.83(2011) 863,arXiv:1007.5331

    Pith/arXiv arXiv 2011

  17. [17]

    Cardy,Measuring Entanglement Using Quantum Quenches,Phys

    J. Cardy,Measuring Entanglement Using Quantum Quenches,Phys. Rev. Lett.106(2011) 150404,arXiv:1012.5116

    Pith/arXiv arXiv 2011

  18. [18]

    Eisler and I

    V. Eisler and I. Peschel,On entanglement evolution across defects in critical chains, Europhys. Lett.99(2012) 20001,arXiv:1205.4331

    Pith/arXiv arXiv 2012

  19. [19]

    Collura and P

    M. Collura and P. Calabrese,Entanglement evolution across defects in critical anisotropic Heisenberg chains,J. Phys. A46(2013) 175001,arXiv:1302.4274

    Pith/arXiv arXiv 2013

  20. [20]

    C. T. Asplund and A. Bernamonti,Mutual information after a local quench in conformal field theory,Phys. Rev. D89(2014) 066015,arXiv:1311.4173

    Pith/arXiv arXiv 2014

  21. [21]

    Wen, P.-Y

    X. Wen, P.-Y. Chang, and S. Ryu,Entanglement negativity after a local quantum quench in conformal field theories,Phys. Rev. B92(2015) 075109,arXiv:1501.00568. 17

    Pith/arXiv arXiv 2015

  22. [22]

    Dahan, M

    D. Dahan, M. T. Ahari, G. Ortiz, B. Seradjeh, and E. Grosfeld,Non-abelian fermion parity interferometry of majorana bound states in a fermi sea,Physical Review B95(2017) 201114,arXiv:1609.04627

    Pith/arXiv arXiv 2017

  23. [23]

    Feldman and M

    N. Feldman and M. Goldstein,Dynamics of Charge-Resolved Entanglement after a Local Quench,Phys. Rev. B100(2019) 235146,arXiv:1905.10749

    Pith/arXiv arXiv 2019

  24. [24]

    Dahan, E

    D. Dahan, E. Grosfeld, and B. Seradjeh,Dynamically Induced Topology and Quantum Monodromies in a Proximity Quenched Gapless Wire,Phys. Rev. B102(2020) 125142, arXiv:1908.06111

    Pith/arXiv arXiv 2020

  25. [25]

    Di Giulio and E

    G. Di Giulio and E. Tonni,Subsystem complexity after a local quantum quench,JHEP 08(2021) 135,arXiv:2106.08282

    Pith/arXiv arXiv 2021

  26. [26]

    Kudler-Flam, Y

    J. Kudler-Flam, Y. Kusuki, and S. Ryu,The quasi-particle picture and its breakdown after local quenches: mutual information, negativity, and reflected entropy,JHEP03(2021) 146,arXiv:2008.11266

    Pith/arXiv arXiv 2021

  27. [27]

    Capizzi, S

    L. Capizzi, S. Scopa, F. Rottoli, and P. Calabrese,Domain wall melting across a defect, EPL141(2023) 31002,arXiv:2210.02162

    Pith/arXiv arXiv 2023

  28. [28]

    Rylands and P

    C. Rylands and P. Calabrese,Transport and Entanglement across Integrable Impurities from Generalized Hydrodynamics,Phys. Rev. Lett.131(2023) 156303, arXiv:2303.01779

    Pith/arXiv arXiv 2023

  29. [29]

    Kudler-Flam, M

    J. Kudler-Flam, M. Nozaki, T. Numasawa, S. Ryu, and M. T. Tan,Bridging two quantum quench problems — local joining quantum quench and M¨ obius quench — and their holographic dual descriptions,JHEP08(2024) 213,arXiv:2309.04665

    Pith/arXiv arXiv 2024

  30. [30]

    Caputa and G

    P. Caputa and G. Di Giulio,Local quenches from a Krylov perspective,JHEP07(2025) 164,arXiv:2502.19485

    arXiv 2025

  31. [31]

    Bonsignori and V

    R. Bonsignori and V. Eisler,Entanglement hamiltonian after a local quench,JHEP02 (2026) 232,arXiv:2508.19406

    Pith/arXiv arXiv 2026

  32. [32]

    Calabrese, C

    P. Calabrese, C. Hagendorf, and P. L. Doussal,Time evolution of one-dimensional gapless models from a domain wall initial state: stochastic Loewner evolution continued?,J. Stat. Mech.(2008) P07013,arXiv:0804.2431

    Pith/arXiv arXiv 2008

  33. [33]

    St´ ephan and J

    J.-M. St´ ephan and J. Dubail,Local quantum quenches in critical one-dimensional systems: entanglement, the Loschmidt echo, and light-cone effects,J. Stat. Mech.(2011) P08019, arXiv:1105.4846

    Pith/arXiv arXiv 2011

  34. [34]

    Calabrese and J

    P. Calabrese and J. Cardy,Quantum quenches in 1 + 1 dimensional conformal field theories,J. Stat. Mech.(2016) 064003,arXiv:1603.02889

    Pith/arXiv arXiv 2016

  35. [35]

    Bertini, M

    B. Bertini, M. Collura, J. De Nardis, and M. Fagotti,Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents,Phys. Rev. Lett.117(2016) 207201, arXiv:1605.09790

    Pith/arXiv arXiv 2016

  36. [36]

    F. K. Popov and Y. Wang,Factorizing Defects from Generalized Pinning Fields,Phys. Rev. Lett.135(2025) 201601,arXiv:2504.06203

    Pith/arXiv arXiv 2025

  37. [37]

    S. He, T. Numasawa, T. Takayanagi, and K. Watanabe,Quantum dimension as entangle- ment entropy in two dimensional conformal field theories,Phys. Rev. D90(2014) 041701, arXiv:1403.0702. 18

    Pith/arXiv arXiv 2014

  38. [38]

    Zhang and P

    J. Zhang and P. Calabrese,Subsystem distance after a local operator quench,JHEP02 (2020) 056,arXiv:1911.04797

    Pith/arXiv arXiv 2020

  39. [39]

    W.-Z. Guo, S. He, and Y.-X. Zhang,On the real-time evolution of pseudo-entropy in 2d CFTs,JHEP09(2022) 094,arXiv:2206.11818

    Pith/arXiv arXiv 2022

  40. [40]

    S. He, J. Yang, Y.-X. Zhang, and Z.-X. Zhao,Pseudoentropy for descendant oper- ators in two-dimensional conformal field theories,Phys. Rev. D109(2024) 025014, arXiv:2301.04891

    Pith/arXiv arXiv 2024

  41. [41]

    Krasznai and G

    A. Krasznai and G. Tak´ acs,Escaping fronts in local quenches of a confining spin chain, SciPost Phys.16(2024) 138,arXiv:2401.04193

    Pith/arXiv arXiv 2024

  42. [42]

    Affleck and A

    I. Affleck and A. W. W. Ludwig,Universal noninteger ’ground state degeneracy’ in critical quantum systems,Phys. Rev. Lett.67(1991) 161

    1991

  43. [43]

    F. Ares, S. Murciano, and P. Calabrese,Entanglement asymmetry as a probe of symmetry breaking,Nature Commun.14(2023) 2036,arXiv:2207.14693

    Pith/arXiv arXiv 2023

  44. [44]

    Casini, M

    H. Casini, M. Huerta, J. M. Mag´ an, and D. Pontello,Entanglement entropy and superse- lection sectors. Part I. Global symmetries,JHEP02(2020) 014,arXiv:1905.10487

    Pith/arXiv arXiv 2020

  45. [45]

    Casini, M

    H. Casini, M. Huerta, J. M. Mag´ an, and D. Pontello,Entropic order parameters for the phases of QFT,JHEP04(2021) 277,arXiv:2008.11748

    Pith/arXiv arXiv 2021

  46. [46]

    J. A. Vaccaro, F. Anselmi, H. M. Wiseman, and K. Jacobs,Tradeoff between extractable mechanical work, accessible entanglement, and ability to act as a reference system, under arbitrary superselection rules,Phys. Rev. A77(2008) 032114, arXiv:quant-ph/0501121

    Pith/arXiv arXiv 2008

  47. [47]

    G. Gour, I. Marvian, and R. W. Spekkens,Measuring the quality of a quantum ref- erence frame: The relative entropy of frameness,Phys. Rev. A80(2009) 012307, arXiv:0901.0943

    Pith/arXiv arXiv 2009

  48. [48]

    F. Ares, P. Calabrese, and S. Murciano,The quantum Mpemba effects,Nature Reviews Physics7(2025) 451,arXiv:2502.08087

    Pith/arXiv arXiv 2025

  49. [49]

    Calabrese,The quantum Mpemba effect in closed systems: from theory to experiment, J

    P. Calabrese,The quantum Mpemba effect in closed systems: from theory to experiment, J. Stat. Mech.(2026) 034002

    2026

  50. [50]

    L. K. Joshiet al.,Observing the quantum Mpemba effect in quantum simulations,Phys. Rev. Lett.133(2024) 010402,arXiv:2401.04270

    Pith/arXiv arXiv 2024

  51. [51]

    J.-N. Yang, L. K. Joshi, F. Ares, Y. Han, P. Zhang, and P. Calabrese,Probing entan- glement and symmetries in random states using a superconducting quantum processor, arXiv:2601.22224

    arXiv

  52. [52]

    Bajnok, D

    Z. Bajnok, D. F¨ ulepi, and M. Lencs´ es,Expectation values after an integrable boundary quantum quench,arXiv:2605.04823

    Pith/arXiv arXiv

  53. [53]

    Y. Choi, B. C. Rayhaun, Y. Sanghavi, and S.-H. Shao,Remarks on boundaries, anomalies, and noninvertible symmetries,Phys. Rev. D108(2023) 125005,arXiv:2305.09713

    Pith/arXiv arXiv 2023

  54. [54]

    T. W. Burkhardt and T. Xue,Conformal invariance and critical systems with mixed boundary conditions,Nucl. Phys. B354(1991) 653. 19

    1991

  55. [55]

    Runkel,Boundary structure constants for the A series Virasoro minimal models,Nucl

    I. Runkel,Boundary structure constants for the A series Virasoro minimal models,Nucl. Phys. B549(1999) 563,arXiv:hep-th/9811178

    Pith/arXiv arXiv 1999

  56. [56]

    Runkel,Structure constants for the D series Virasoro minimal models,Nucl

    I. Runkel,Structure constants for the D series Virasoro minimal models,Nucl. Phys. B 579(2000) 561,arXiv:hep-th/9908046

    Pith/arXiv arXiv 2000

  57. [57]

    Ohmori and Y

    K. Ohmori and Y. Tachikawa,Physics at the entangling surface,J. Stat. Mech.(2015) P04010,arXiv:1406.4167

    Pith/arXiv arXiv 2015

  58. [58]

    Calabrese and J

    P. Calabrese and J. Cardy,Entanglement entropy and quantum field theory,J. Stat. Mech. (2004) P06002,arXiv:hep-th/0405152

    Pith/arXiv arXiv 2004

  59. [59]

    Calabrese and J

    P. Calabrese and J. Cardy,Entanglement entropy and conformal field theory,J. Phys. A 42(2009) 504005,arXiv:0905.4013

    Pith/arXiv arXiv 2009

  60. [60]

    Cardy and E

    J. Cardy and E. Tonni,Entanglement hamiltonians in two-dimensional conformal field theory,J. Stat. Mech.(2016) 123103,arXiv:1608.01283

    Pith/arXiv arXiv 2016

  61. [61]

    Cardy,Effect of Boundary Conditions on the Operator Content of Two-Dimensional Conformally Invariant Theories,Nucl

    J. Cardy,Effect of Boundary Conditions on the Operator Content of Two-Dimensional Conformally Invariant Theories,Nucl. Phys. B275(1986) 200

    1986

  62. [62]

    Ferro, F

    F. Ferro, F. Ares, and P. Calabrese,Non-equilibrium entanglement asymmetry for discrete groups: the example of the XY spin chain,J. Stat. Mech.(2024) 023101, arXiv:2307.06902

    Pith/arXiv arXiv 2024

  63. [63]

    Cardy,Boundary Conditions, Fusion Rules and the Verlinde Formula,Nucl

    J. Cardy,Boundary Conditions, Fusion Rules and the Verlinde Formula,Nucl. Phys. B 324(1989) 581

    1989

  64. [64]

    Hauschild and F

    J. Hauschild and F. Pollmann,Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy),SciPost Phys. Lect. Notes(2018) 5, arXiv:1805.00055. Code available fromhttps://github.com/tenpy/tenpy

    Pith/arXiv arXiv 2018

  65. [65]

    Fossati, C

    M. Fossati, C. Rylands, and P. Calabrese,Entanglement asymmetry in CFT with boundary symmetry breaking,JHEP06(2025) 089,arXiv:2411.10244

    Pith/arXiv arXiv 2025

  66. [66]

    Affleck and A

    I. Affleck and A. W. W. Ludwig,Exact conformal-field-theory results on the multichannel Kondo effect: Single-fermion Green’s function, self-energy, and resistivity,Phys. Rev. B 48(1993) 7297

    1993

  67. [67]

    Z. Ma, A. K. Mitchell, and E. Sela,Quantum Work Statistics across a Critical Point: Full Crossover from Sudden Quench to the Adiabatic Limit,Phys. Rev. Lett.135(2025) 130402,arXiv:2502.01601

    arXiv 2025

  68. [68]

    Calabrese,Entanglement spreading in non-equilibrium integrable systems,SciPost Phys

    P. Calabrese,Entanglement spreading in non-equilibrium integrable systems,SciPost Phys. Lect. Notes(2020) 20,arXiv:2008.11080. 20

    Pith/arXiv arXiv 2020