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Quench to criticality, read off the conformal spectrum

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2026-07-10 03:37 UTC pith:I6W2ZJXM

load-bearing objection Solid CFT extraction protocol with a real generalization gap — tested only on integrable Ising at short times the 2 major comments →

arxiv 2607.08649 v1 pith:I6W2ZJXM submitted 2026-07-09 quant-ph cond-mat.stat-mechcond-mat.str-elhep-th

Extracting conformal data from Loschmidt echoes after critical quenches

Aleix Bou-Comas , Stefano Carignano , Sergio Cerezo-Roquebr\'un , Esperanza Lopez , Luca Tagliacozzo This is my paper
classification quant-ph cond-mat.stat-mechcond-mat.str-elhep-th PACS 05.70.Jk05.30.-d03.67.Ac
keywords criticalloschmidtscalingboundaryconformalamplitudedataechoes
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.

This paper claims that the universal data of a conformal field theory — the central charge, the boundary operator spectrum, and the scaling dimensions — can be extracted from the real-time dynamics of a quantum system after a sudden quench to a critical point, without ever preparing the ground state. The mechanism is a geometric duality: after analytic continuation, the Loschmidt amplitude (the overlap between the initial product state and the time-evolved state) is reinterpreted as a partition function on a strip, and the transfer matrix running across the strip has eigenvalues whose gaps are exactly the boundary scaling dimensions of the CFT. The central charge appears as a universal phase correction to the leading eigenvalue. The authors show that localized perturbations during the evolution probe one- and two-point correlation functions on the strip, recovering the scaling dimension of the spin field to within 1% in the critical Ising chain. They then show that the central charge can be extracted either by reconstructing the phase of the Loschmidt amplitude from modulus measurements, or from the logarithmic scaling of generalized temporal Rényi entropies, provided one includes the leading finite-time correction predicted by CFT. Finally, they demonstrate that for a finite chain the Loschmidt echo, as a function of system size, decomposes into damped oscillations whose frequencies are proportional to differences of boundary scaling dimensions and whose decay-rate ratios eliminate the non-universal extrapolation length. Using harmonic-inversion techniques on chains of 3–40 sites, they reconstruct the low-lying boundary spectrum for the Ising CFT with agreement to exact predictions within a few percent, and show that even chains of ~10 sites can resolve the first boundary gap.

Core claim

The boundary-CFT spectrum of a critical system is encoded in the system-size dependence of finite-chain Loschmidt echoes: each transverse eigenmode contributes a damped oscillation whose frequency gives a difference of boundary scaling dimensions and whose decay-rate ratio gives a universal combination of the same dimensions, independent of the non-universal extrapolation length. This means the conformal data can be spectroscopically extracted from return probabilities alone, without ground-state preparation or direct access to the transfer matrix.

What carries the argument

The transverse transfer matrix of the analytically continued Loschmidt amplitude, whose eigenvalues take the form t_i = exp[(i/vT)(−πc/24 + πx_i) + ...], so that eigenvalue gaps encode boundary scaling dimensions x_i and the leading phase encodes the central charge c. For finite systems, the Loschmidt echo becomes a sum of damped oscillations L(T,N) = Σ |c_i|² e^{−NΔ_i} + 2Σ|c_i c_j| e^{−NΔ_{ij}} cos(ω_{ij}N + φ_{ij}), where ω_{ij} = π(x_i − x_j)/(vT) and Γ_{ij} = Δ_{ij}/Δ_{10} = (x_i + x_j − 2x_0)/(x_1 − x_0), allowing reconstruction of the spectrum from measurable quantities.

Load-bearing premise

The analytic continuation from imaginary to real time preserves the spectral structure of the transverse transfer matrix so that the leading CFT-predicted eigenvalue gaps dominate the finite-size signal. This is most strained at the short evolution times (T ~ 1–2) where mode separation is best but finite-time corrections are largest, creating a tension the paper acknowledges but does not fully resolve.

What would settle it

If, at the short evolution times required for finite-size spectroscopy, the subleading non-universal corrections to the transverse eigenvalues are large enough to scramble the separation between the first two or three modes, then the harmonic-inversion reconstruction would extract spurious frequencies and decay rates, and the recovered scaling dimensions would not match the CFT predictions. The paper's own data for free–free boundary conditions at T ≈ 2 already shows significant deviation, suggesting this regime is near the edge of validity.

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

If this is right

  • Quantum simulators with ~10–40 qubits could extract the lowest boundary scaling dimensions and the central charge of a critical model without adiabatic ground-state preparation, bypassing a major bottleneck in current experiments.
  • The finite-size spectroscopy protocol could be applied to non-integrable critical models where the CFT data is not known analytically, turning Loschmidt echo measurements into a diagnostic tool for identifying the universality class.
  • The phase-reconstruction route to the central charge via complex-time Cauchy–Riemann relations could be combined with randomized measurement protocols already demonstrated on intermediate-scale devices.
  • The framework extends naturally to other rational CFTs (e.g., the three-state Potts model), where the highly constrained fusion rules would make even a few reconstructed scaling dimensions sufficient to identify the full operator content.

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 / 6 minor

Summary. This manuscript develops a framework for extracting conformal field theory (CFT) data—boundary scaling dimensions and the central charge—from real-time Loschmidt echoes following quenches to criticality. The theoretical foundation, drawn from Ref. [38], maps the analytically continued Loschmidt amplitude to a boundary-CFT strip partition function whose transverse transfer-matrix spectrum encodes boundary scaling dimensions, while the leading phase correction encodes the central charge. The paper presents four concrete extraction routes: (1) direct diagonalization of the transverse transfer matrix in tensor-network simulations, yielding gap ratios matching BCFT predictions to <2% in the critical Ising chain; (2) localized perturbations probing strip one- and two-point correlation functions, recovering the spin scaling dimension h=1/8; (3) two complementary approaches to the central charge—phase-sensitive reconstruction and generalized temporal Rényi entropies, the latter giving c≈0.518 vs. exact 0.5; and (4) a finite-size protocol reconstructing the low-lying boundary spectrum from the system-size dependence of Loschmidt echoes via harmonic inversion, demonstrated for chains of 3–40 sites. The finite-size protocol is the most novel and experimentally oriented contribution.

Significance. The paper addresses a practically important problem: extracting universal CFT data without preparing low-energy critical states, which is a known bottleneck in quantum simulation experiments. The framework is internally consistent and the numerical benchmarks in the integrable Ising model are convincing for the infinite-system quantities (transfer-matrix gap ratios, correlation-function exponents, central charge from temporal entropies). The finite-size reconstruction protocol is a genuinely new contribution with clear experimental motivation. The authors are explicit about non-universal parameters (β₀, γ, amplitudes A₀, Aₙ) and design ratios that cancel them (Eqs. 15, 71), which is a strength. The proposal for measuring generalized temporal purities via replicated systems (Ref. [55]) gives the work concrete experimental traction.

major comments (2)
  1. Section IV, Eqs. (9) and (65)–(74): The finite-size reconstruction protocol operates at T = 1.0–2.5 (Figs. 7–8), where the expansion parameter 1/T is O(0.4–1.0). The CFT prediction for the transverse eigenvalues (Eq. 9) is a large-T expansion with O(T⁻²) and O(T⁻³) corrections that are not controlled at these times. Appendix B establishes 1/T² convergence for the infinite-system transfer-matrix gap ratios only at T = 5–7, which is well outside the regime used for finite-size spectroscopy. The paper does not provide a quantitative analysis of how finite-time corrections propagate through the matrix-pencil inversion to bias the extracted scaling dimensions. The 'intermediate temporal window' identified in Fig. 8 could be specific to the integrable Ising model rather than a generic feature of the CFT expansion. The authors should either (a) provide a systematic error analysis showing that O
  2. Section IV.B and Abstract: The claim that the protocol 'can extract these quantities from simulations or experiments on state-of-the-art quantum platforms' is supported only by a benchmark in the integrable Ising model. Integrability ensures particularly simple Cardy boundary states and may accelerate CFT convergence relative to generic interacting critical theories. Without at least one non-integrable benchmark (e.g., the 3-state Potts model or a critical non-integrable spin chain), the generality of the finite-size protocol remains untested. The authors should either add such a benchmark or temper the experimental-applicability claims to reflect that only the integrable case has been demonstrated.
minor comments (6)
  1. Eq. (23): The amplitude A_{↑,+}(T) = ⟨↑|U(T)|↑⟩ is labeled as free-fixed boundary conditions, but the bra and ket states are identical (both |↑⟩), which would correspond to free-free conditions. This appears to be a typo; the bra should likely be ⟨+|.
  2. Appendix D: The (+,−) boundary spectrum is listed as x_i ∈ {0, 1/2, 3/2, 2, 5/2, ...}, which differs from the main text Eq. (20) where x_i ∈ {1/2, 3/2, 5/2, 7/2, ...}. Please reconcile.
  3. Table I: The row label 'aaaaa' appears to be a formatting artifact. Please correct.
  4. Section III.A, Eq. (43): The phase is written as ϕ(T) = avT − κ/(vT), but the fit form in Eq. (45) uses ϕ(T)/T = A + B/T − C/T². The relationship between the non-universal constant 'a' in Eq. (43) and the fitted parameters A, B, C could be stated more explicitly to help the reader understand what is universal versus non-universal in the phase reconstruction protocol.
  5. Fig. 2(a): The caption mentions 'overlaid fits whose slope is consistent with the analytical prediction −h = −1/8,' but it is unclear whether the fits are shown as solid lines or whether only the data points are plotted. Clarifying which curves are fits would help.
  6. The notation transitions between ℓ_β and ℓ_T for the strip width (Eqs. 3, 8, 48). While defined, consistently using one symbol or explicitly stating the replacement would improve readability.

Circularity Check

0 steps flagged

No significant circularity; framework self-citation is load-bearing but not circular

full rationale

The paper's derivation chain proceeds as follows: (1) Standard boundary-CFT results (Cardy [7,8,11], Affleck [9]) give the transverse transfer matrix eigenvalue structure in Eq. (4), where t_i = exp[(κ - πx_i)/(vℓ_β) + ...] with κ = πc/24. These are textbook BCFT results, not authored by the present paper's authors. (2) The analytic continuation from Euclidean to real time (Eqs. 7-9) is drawn from Ref. [38] (Carignano & Tagliacozzo, two of five authors). This self-citation is load-bearing for the framework but is a theoretical construction, not a fitted result or uniqueness theorem. (3) The key predictions—ratio of gaps ∆λ_i/∆λ_j = (x_i - x_0)/(x_j - x_0) in Eq. (15), and damped oscillation frequencies ω_ij = π(x_i - x_j)/(vT) in Eq. (69)—are parameter-free consequences of BCFT. They are tested against independently known exact BCFT values for the Ising model (e.g., x_i ∈ {0, 2, 3, 4, ...} for fixed boundary conditions, from Cardy's work). (4) The finite-size reconstruction protocol (Eqs. 65-74) inverts the theoretical predictions to extract scaling dimensions from measurable quantities; this is a genuine inversion, not a fit renamed as prediction. No step reduces to its own inputs by construction. The self-citation to Ref. [38] provides the theoretical scaffolding but does not make the tested predictions tautological, since the predictions could have failed numerically (and the paper honestly reports where they do, e.g., free-free boundary conditions at T≈2 in Fig. 8). The concerns about finite-time corrections at short T are correctness/applicability issues, not circularity.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities, particles, or forces. All objects (transverse transfer matrix, boundary states, temporal Rényi entropies) are standard in BCFT and tensor-network literature. The free parameters are all non-universal quantities that are either eliminated in universal ratios or fitted to numerical data.

free parameters (6)
  • β0 (extrapolation length) = not reported; eliminated in ratios
    Non-universal imaginary-time regulator associated with the initial state. Enters Eqs. (4), (9), (12). Eliminated in gap ratios (Eq. 15) and decay-rate ratios (Eq. 71), but appears in absolute decay rates.
  • γ (leading non-universal correction) = not reported; eliminated in ratios
    Non-universal correction at order ℓβ^{-2}. Enters Eqs. (4), (9). Eliminated in gap ratios but affects absolute eigenvalues.
  • A0 / A1 (non-universal amplitudes for one-point function) = fitted by hand in Fig. 2
    Multiplicative constant in the CFT one-point function (Eq. 29). Absorbed into A1 in the logarithmic form (Eq. 30).
  • An (non-universal amplitude for Rényi entropy correction) = fitted in Fig. 5 and Fig. 6
    Complex amplitude for the leading finite-time correction to temporal Rényi entropies (Eq. 56). Fitted to numerical data.
  • a (non-universal extensive phase constant) = fitted in Fig. 4
    Non-universal constant in the Loschmidt amplitude phase (Eq. 43), related to Hamiltonian normalization. Fitted as parameter A in Eq. (45).
  • B (integration-dependent phase offset) = fitted in Fig. 4
    Offset arising from the phase reconstruction formula (Eq. 42). Fitted as parameter B in Eq. (45).
axioms (5)
  • domain assumption The microscopic initial product state is replaced by a conformal boundary state |b⟩ evolved for a short imaginary time β0, which regularizes the theory.
    Standard BCFT quench setup from Calabrese-Cardy (Refs. [21, 22]). Invoked in Section II, paragraph after Eq. (2). This is the foundational assumption enabling the strip geometry.
  • domain assumption Analytic continuation of only the physical Euclidean time β→iT, while keeping β0 real, preserves the CFT spectral structure.
    Invoked at Eq. (7)-(8). This is the standard analytic continuation used in CFT quench calculations. Its validity for the transverse transfer matrix spectrum is the structural basis for all real-time predictions.
  • domain assumption The transverse transfer matrix is diagonalizable with a discrete spectrum.
    Invoked at Eq. (62). Required for the spectral decomposition of the finite-system Loschmidt amplitude. Standard for transfer matrices of lattice models but not formally proven for the real-time case.
  • domain assumption The leading finite-time corrections to the temporal Rényi entropy are governed by the most relevant even operator (energy operator, x=1 for Ising).
    Invoked in Eq. (46) and Appendix C. Standard CFT result from Calabrese-Cardy (Refs. [22, 66-68]). The accuracy of the central-charge extraction depends on this being the correct leading correction.
  • standard math The matrix-pencil method can reliably extract complex exponents from a finite sum of damped oscillations when the number of modes is small.
    Standard signal processing result (Refs. [71, 72]). Invoked in Section IV. The practical limitation to a small number of modes is acknowledged.

pith-pipeline@v1.1.0-glm · 29438 in / 3729 out tokens · 510213 ms · 2026-07-10T03:37:53.311915+00:00 · methodology

0 comments
read the original abstract

Conformal field theory provides universal predictions for Loschmidt amplitudes following quenches from product states to critical Hamiltonians. Building on this observation, we develop a route to extracting conformal data from real-time dynamics without preparing critical low-energy states. After analytic continuation, the Loschmidt amplitude is described by a boundary-CFT partition function on a strip, whose transverse transfer matrix encodes both the boundary operator spectrum and the central charge. Local space-time perturbations of the amplitude are governed by equilibrium correlation functions, and therefore provide access to critical exponents. In parallel, generalized temporal entropies exhibit scaling with time analogous to the equilibrium scaling of spatial entanglement entropy. We show that the low-lying boundary spectrum can be reconstructed from the system-size dependence of finite-chain Loschmidt echoes, whose damped oscillations encode differences of boundary scaling dimensions. Finally, we propose a finite-size scaling protocol that can extract these quantities from simulations or experiments on state-of-the-art quantum platforms.

Figures

Figures reproduced from arXiv: 2607.08649 by Aleix Bou-Comas, Esperanza Lopez, Luca Tagliacozzo, Sergio Cerezo-Roquebr\'un, Stefano Carignano.

Figure 1
Figure 1. Figure 1: FIG. 1. Ratios of universal dynamical exponents obtained from the transverse transfer matrix of Loschmidt amplitudes [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Panel (a) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Phase of the Loschmidt echo estimated through the phase-sensitive reconstruction protocol. In all simulations, [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Deviation of the real part (a) and imaginary part (b) of the generalized temporal R´enyi entropy from the [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Generalized temporal R´enyi-2 entropy [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Loschmidt return probability [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Estimate of the first boundary gap [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Tensor-network representation of the Loschmidt amplitude [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Definition of the left and right temporal boundary states. For any spatial cut [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Plot of the quotient ∆ [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Boundary scaling dimensions found by using the matrix-pencil method and Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗

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