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This paper claims that the transient, pre-ground-state stage of imaginary-time evolution in an iPEPS simulation of the 2D transverse-field Ising model already contains universal quantum critical scaling, from which both the critical point a

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2026-08-03 22:18 UTC pith:DVEW57RO

load-bearing objection Useful demonstration that iPEPS can extract h_c and β/(νz) from imaginary-time relaxation; the claimed θ extraction is not backed up and the 'good agreement' with QMC is overstated. the 3 major comments →

arxiv 2511.10934 v2 pith:DVEW57RO submitted 2025-11-14 cond-mat.str-el quant-ph

Probing universal imaginary-time relaxation critical dynamics with infinite projected entangled pair states

He-Yu Lin , Shuai Yin , Z. Y. Xie , Zhong-Yi Lu This is my paper
classification cond-mat.str-el quant-ph
keywords imaginary-time evolutioninfinite projected entangled pair statestransverse-field Ising modelcritical dynamicsinitial-slip exponentdynamical critical exponentsquantum phase transitiontensor network
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 tries to establish that universal quantum critical dynamics can be read off from the transient, pre-equilibrium stage of imaginary-time evolution, using a tensor network that works directly in the thermodynamic limit. Simulating the square-lattice transverse-field Ising model from two types of initial states, it finds clean power laws for the magnetization: a decay M ~ tau^(-beta/(nu z)) from a fully polarized start and a growth M ~ tau^theta from a nearly paramagnetic start. The extracted critical field and exponents agree with established values, with the initial-slip exponent theta extrapolating to 0.1958, close to the quantum Monte Carlo estimate 0.209(4). If correct, this means two-dimensional quantum criticality can be probed without ground-state convergence, free of the sign problem, and at modest computational cost, opening frustrated and fermionic models to dynamical criticality studies.

Core claim

Starting from a fully polarized state, the imaginary-time evolved magnetization obeys M(tau) proportional to tau^(-beta/(nu z)). For each bond dimension D, the transverse field h* that gives the straightest log-log curve is selected: h* = 3.064, 3.055, 3.051 for D = 3, 4, 5, with fitted slopes -0.52339, -0.52204, -0.51983, all close to the universal -0.518, and an extrapolated critical field h_c = 3.0445. Starting from a product state with tiny magnetization M0 = 0.005, the same protocol gives initial-slip growth M proportional to tau^theta with per-D slopes 0.12586, 0.17889, 0.1902 and a large-D extrapolated theta = 0.19584, consistent with quantum Monte Carlo theta = 0.209(4). The paper co

What carries the argument

The scaling theory of imaginary-time relaxation near a quantum critical point: the order parameter obeys M(tau, g, M0) = b^(-beta/nu) M(tau b^(-z), g b^(1/nu), U(M0, b)), which at criticality reduces to a power-law decay for saturated initial states and to the initial-slip law M ~ M0 tau^theta for tiny M0, with theta = (x0 - beta/nu)/z. The numerical engine is the infinite projected entangled pair state (iPEPS), a tensor-network ansatz with a 2x2 unit cell, bond dimension D, and environment bond dimension chi, evolved by the full-update strategy with Trotter-Suzuki slicing. This machinery converts the universal scaling prediction into a concrete observable: the log-log slope of M(tau) at the

Load-bearing premise

The whole extraction rests on treating the slope of the magnetization curve at the field that happens to look most like a straight line as the true universal exponent, even though that field is not exactly the critical field and the fitting range changes with the numerical accuracy.

What would settle it

Run the D=5 simulation at the known critical field h_c = 3.044 rather than at h* = 3.051, with a fixed fitting window such as 0.5 < tau < 0.84, and check whether the extracted theta still falls within the quoted uncertainty or whether the log-log curve loses its linearity. Equivalently, a D=6 calculation whose extrapolated theta moves away from 0.209 would contradict the claimed convergence toward the universal value.

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

If this is right

  • The critical point of the 2D transverse-field Ising model can be estimated as h_c ≈ 3.0445 by extrapolating h*(D) to infinite bond dimension, matching established high-precision values.
  • The decay exponent -beta/(nu z) remains stable across bond dimensions (around -0.52) and close to the universal -0.518, so the exponent can be extracted without first pinpointing the exact critical field.
  • The initial-slip exponent theta extrapolates to 0.1958, in line with the quantum Monte Carlo value 0.209(4), and improves systematically as the bond dimension grows.
  • Universal scaling appears in the short-time stage, so expensive ground-state convergence is unnecessary for extracting critical properties.
  • Because the method is sign-problem-free and operates in the thermodynamic limit, it can be extended to frustrated magnets and interacting fermionic systems where quantum Monte Carlo fails.

Where Pith is reading between the lines

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

  • A direct consistency test would be to fix the field at the independently known h_c ≈ 3.044 rather than at the bond-dimension-dependent h*, and check whether M(tau) remains a clean power law; if it does, the h*-fitting procedure is not essential to the exponent extraction.
  • The roughly 6% offset between the extrapolated theta = 0.1958 and the quantum Monte Carlo theta = 0.209(4) may stem from per-D fitting-window choices; applying one fixed window across all bond dimensions would reveal whether the residual drift is systematic.
  • The same imaginary-time relaxation dynamics could be mirrored in real-time evolution or on quantum simulators, where discrete Trotter steps play the role of imaginary-time slices, offering a blueprint for extracting critical exponents from pre-thermal plateaus.
  • If the method transfers to models without a quantum Monte Carlo benchmark, it would provide estimates of z and beta/nu in frustrated or doped systems where entanglement growth currently limits tensor-network ground-state studies.

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

3 major / 5 minor

Summary. The paper studies imaginary-time relaxation critical dynamics of the two-dimensional transverse-field Ising model using infinite projected entangled pair states (iPEPS) with the full-update strategy, working directly in the thermodynamic limit. For a fully polarized initial state, the magnetization is claimed to decay as M(τ)∼τ^{-β/(νz)}, and the field h* giving the best power law is used to estimate the critical point and the exponent. For a nearly paramagnetic product state with small M0, the short-time growth M(τ)∼τ^θ is analyzed, and a 1/D extrapolation of the fitted slopes yields θ≈0.19584, which the authors state is in good agreement with the quantum Monte Carlo result θ=0.209(4). The paper concludes that iPEPS is a robust and scalable method for probing dynamical critical phenomena in two dimensions.

Significance. If the main claims were quantitatively established, the work would be a useful methodological contribution: it demonstrates that iPEPS imaginary-time evolution can access universal short-time dynamics in the thermodynamic limit, avoiding finite-size and sign-problem limitations. The extracted critical field h_c≈3.0445 from the fully polarized sector agrees very well with the known value, and the saturated-state slopes (−0.523 to −0.520) are close to the expected −0.518. These are genuine positive results. However, the central demonstration of the method's quantitative power is the initial-slip exponent θ, and the reported value 0.19584 lies outside the quoted QMC error bar 0.209(4). The extraction procedure has several uncontrolled elements, so the claimed agreement and the broad conclusion that iPEPS reliably captures universal early-time dynamics are not yet established.

major comments (3)
  1. [Sec. IV.B, Fig. 4] The headline initial-slip exponent θ≈0.19584 disagrees with the cited QMC value θ=0.209(4) (Ref. [10]) by about 3.3σ (difference 0.0132, error 0.004). The text states this 'agrees well' and is 'consistent with the QMC estimation,' but a value outside the quoted error bar does not support that statement. Since the initial-slip exponent is exactly the quantity that would demonstrate the method's ability to capture universal short-time dynamics, this discrepancy is load-bearing and must be addressed quantitatively, e.g., by reporting uncertainty on the extrapolation and showing that the discrepancy is within the expected finite-entanglement error.
  2. [Sec. IV.B and Fig. 3] The extraction procedure is circular in a way that can bias the exponent: h* is selected as the field giving the 'best power law' in the log-log plot, and the slope is then fitted at that same h*. For D=3, h*=3.063, which is 0.019 away from the known h_c≈3.044, so the fit is performed off-critical. The fitting windows are also chosen per D by eye (0.2<τ<0.5 for D=3; 0.5<τ<0.84 for D=4 and D=5) with no stated criterion. Off-critical corrections to scaling and window choice can systematically bend the fitted slope. The authors should show robustness, for example by fitting at the fixed known h_c=3.044, by varying the fitting window, and by estimating the resulting spread in k*.
  3. [Sec. IV.B, Fig. 4] The 1/D extrapolation uses only three points (D=3,4,5) with a 'simple power-law fit' whose functional form is not specified, and no error bars are assigned to the slopes or to the extrapolated θ. Given that the D=3 point (0.12586) is far from the D=4/5 values (0.17889, 0.1902), the extrapolated θ depends sensitively on the assumed form. The authors should specify the fitting function, include uncertainties, and test sensitivity (e.g., linear in 1/D, quadratic, or excluding D=3). Without this, the claimed value θ=0.19584 is not a controlled estimate.
minor comments (5)
  1. [Sec. II] Typo: 'Hamailtonian' should be 'Hamiltonian'.
  2. [Sec. IV.B] Typo: 'preform' should be 'perform'.
  3. [Fig. 3] The y-axis is M(τ)/M0, whereas Eq. (4) writes M(τ)∼M0 τ^θ; the caption should state clearly that the plotted quantity is normalized by M0.
  4. [Sec. IV.A] The statement that the fitted slopes are 'remarkably consistent' would be stronger with error bars or a table reporting the fitting uncertainty for each k*; currently only the raw slopes are given.
  5. [Sec. IV.B, Fig. 5] The exponential fit in M0 is described only briefly. It would be helpful to state its functional form and show the fitted curve parameters, since the M0→0 extrapolation is used to support the universal value.

Circularity Check

0 steps flagged

No significant circularity: the iPEPS simulations produce new data; the cited scaling theory and QMC benchmark are external inputs, not redefinitions of the results.

full rationale

The paper's derivation chain is: adopt the scaling theory from Refs. [8,9] for imaginary-time relaxation, simulate the 2D TFIM with iPEPS for two initial states, extract slopes at estimated critical fields, and compare with previous QMC results. No step reduces to its own input by construction. The critical field h* is defined as the field giving the 'best power behavior' of M(τ), and the exponent is then obtained by fitting the slope at that same h*. This is a standard estimation procedure, not a self-definitional one: the fitted exponent is not used to define h*, and h* is not defined in terms of the target exponent. The extracted exponents (e.g., −0.52339, −0.52204, −0.51983 for -β/(νz), and k* = 0.12586, 0.17889, 0.1902 for θ) are new iPEPS data. The comparison values (β/(νz) ≈ 0.518, θ ≈ 0.209(4)) come from QMC [10], an independent numerical method with different systematic errors; the overlap of one author (S. Yin) does not make the QMC result fitted to the iPEPS data, and it is externally falsifiable. The scaling theory in Refs. [8,9] is a published RG framework, also anchored in classical short-time dynamics [12-15], and is used as a prior, not as a conclusion derived from the present data. The 1/D extrapolation is a fit of three points, but this is an extrapolation of the paper's own results, not a renaming or a circular prediction. Concerns about the off-critical fit at h*, hand-selected fitting windows, and the discrepancy θ=0.19584 vs 0.209(4) are accuracy/statistical issues, not circularity. The paper is self-contained in that its central numerical quantities are newly computed and are not equivalent to the cited inputs by construction. Therefore no circular step is present.

Axiom & Free-Parameter Ledger

11 free parameters · 6 axioms · 0 invented entities

The central quantitative claims rest on hand-selected fitting fields and windows, on a three-point extrapolation in 1/D, and on scaling forms taken from same-group prior work. No new physical entities are introduced.

free parameters (11)
  • h* (fully polarized, D=3) = 3.064
    Chosen as the transverse field giving the most linear log-log M(τ); used as estimator for h_c.
  • h* (fully polarized, D=4) = 3.055
    Same best-power-law selection; D=4 estimate.
  • h* (fully polarized, D=5) = 3.051
    Same best-power-law selection; D=5 estimate.
  • h* (small M0, D=3) = 3.063
    Best-power-law selection for the initial-slip curves.
  • h* (small M0, D=4) = 3.044
    Best-power-law selection for the initial-slip curves.
  • h* (small M0, D=5) = 3.044
    Best-power-law selection for the initial-slip curves.
  • slopes k* (fully polarized, D=3,4,5) = -0.52339, -0.52204, -0.51983
    Power-law fits to M(τ) at each h*; estimates of -β/(νz).
  • slopes k* (small M0, D=3,4,5) = 0.12586, 0.17889, 0.1902
    Power-law fits to M(τ) at each h*; estimates of θ.
  • extrapolated θ = 0.19584
    Large-D limit from a 'simple power-law fit' to the three k*(1/D) points; functional form not specified.
  • extrapolated h_c = 3.0445
    Large-D limit from a power-law fit to the three h*(1/D) points (D=3,4,5).
  • fitting windows for small-M0 slopes = 0.2<τ<0.5 (D=3); 0.5<τ<0.84 (D=4,5)
    Hand-selected τ ranges where M(τ) appears most power-law; affects the extracted slope.
axioms (6)
  • domain assumption Scaling form Eq. (2): M(τ,g,M0)=b^{-β/ν}M(τb^{-z}, gb^{1/ν}, U(M0,b))
    Taken from Refs [8,9]; assumed to describe imaginary-time relaxation near quantum critical points.
  • standard math Taylor expansion of f_M to first order in M0 τ^{x0/z}, giving M~M0 τ^θ
    Linearization of the universal scaling function; standard short-time initial-slip analysis (Refs [12-14]).
  • domain assumption 2D TFIM belongs to the 3D classical Ising universality class with z=1, β≈0.3264, ν≈0.62997
    Standard quantum-to-classical mapping, used to compare fitted exponents.
  • ad hoc to paper h* chosen as the best power-law field is an unbiased estimator for h_c
    No formal justification is given; finite-D shifts h* away from h_c (e.g., 3.064 vs 3.044).
  • domain assumption Finite bond dimension D and environment dimension χ provide controlled approximations that improve systematically as D grows
    Standard iPEPS assumption; no error bound is given, and the θ extrapolation using only three D values is not robust.
  • domain assumption Trotter-Suzuki decomposition with a small time step δ is accurate
    δ is never specified in the paper; accuracy in δ is assumed.

pith-pipeline@v1.3.0-alltime-deepseek · 9475 in / 13708 out tokens · 114132 ms · 2026-08-03T22:18:45.835549+00:00 · methodology

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read the original abstract

We investigate the imaginary-time relaxation critical dynamics of the two-dimensional transverse-field Ising model using infinite projected entangled pair states (iPEPS) with the full-update strategy. Simulating directly in the thermodynamic limit, we explore the relaxation process near the critical point with two types of initial states: a fully polarized state and a product state with a small magnetization. For the fully polarized state, the magnetization shows a power law scaling $M\propto \tau^{-\beta/(\nu z)}$ in the imaginary-time evolution, from which both the critical point and critical exponent can be determined with high accuracy. For the nearly paramagnetic state, the relaxation process exhibits a behavior of $M\propto \tau^\theta$ with $\theta=0.1958$ being the critical initial-slip exponent, which is in good agreement with that obtained from the dynamic scaling of the self-correlation in quantum Monte Carlo method. These universal features emerge well before the system converges to the ground state, demonstrating the efficiency of imaginary-time evolution for probing quantum criticality. Our results demonstrate that iPEPS can serve as a robust and scalable method for studying dynamical critical phenomena in two-dimensional quantum many-body systems.

Figures

Figures reproduced from arXiv: 2511.10934 by He-Yu Lin, Shuai Yin, Zhong-Yi Lu, Z. Y. Xie.

Figure 1
Figure 1. Figure 1: FIG. 1: Imaginary-time relaxation dynamics for a fully polarized initial state, plotted in log-log scale. Magnetization [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Extrapolation of the estimated critical field [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Imaginary-time relaxation dynamics for a product initial state with a tiny magnetization [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The fitting power at [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Estimated critical initial-slip exponent [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

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