pith. sign in

arxiv: 2504.21089 · v3 · pith:6MU3WDBInew · submitted 2025-04-29 · ❄️ cond-mat.dis-nn · cond-mat.str-el· quant-ph

Extracting average properties of disordered spin chains with translationally invariant tensor networks

Kevin Vervoort , Wei Tang , Nick Bultinck This is my paper

Pith reviewed 2026-05-22 18:03 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.str-elquant-ph
keywords disordered spin chainstensor networksdisorder averagingthermodynamic limitrandom transverse-field Ising modelstatistical translation invarianceinfinite-randomness criticality
0
0 comments X

The pith

Tensor networks compute disorder-averaged properties of random spin chains directly in the thermodynamic limit without sampling configurations.

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

The paper introduces a tensor network method for finding disorder-averaged expectation values in random spin chains. It exploits the fact that the disorder distribution is statistically translation invariant, so a single tensor network can stand in for the average over all configurations. The calculation runs directly on the infinite chain, avoiding both explicit sampling and finite-size artifacts. The authors benchmark the approach on the infinite-randomness critical point of the random transverse-field Ising model.

Core claim

When the disorder distribution is statistically translation invariant, the averaged system itself can be represented by a translationally invariant tensor network. This representation yields disorder-averaged observables without generating or averaging over individual disorder realizations and works directly in the thermodynamic limit.

What carries the argument

A translationally invariant tensor network that encodes the disorder-averaged state of the infinite spin chain.

If this is right

  • Averages are obtained without the cost of sampling many disorder realizations.
  • Results apply directly to infinite chains rather than extrapolated from finite samples.
  • The same construction applies to other random spin models that share statistical translation invariance.
  • Benchmark agreement on the random transverse-field Ising model confirms the method reproduces known critical behavior.

Where Pith is reading between the lines

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

  • The technique could extend to weakly inhomogeneous disorder by adding small perturbations to the invariant tensors.
  • It may combine with other tensor-network algorithms for dynamics or finite-temperature averages in disordered systems.
  • Similar translation-invariance tricks might reduce sampling costs in classical disordered models or in higher dimensions.

Load-bearing premise

The disorder distribution must be statistically the same at every site so the averaged system appears translationally invariant.

What would settle it

A clear mismatch between the tensor-network averages and explicit sampling averages for the random transverse-field Ising model at its infinite-randomness critical point would show the method does not work.

Figures

Figures reproduced from arXiv: 2504.21089 by Kevin Vervoort, Nick Bultinck, Wei Tang.

Figure 1
Figure 1. Figure 1: (a) MPO representation of ρ˜(τ ). Horizontal lines represent the virtual bonds. Vertical black full lines represent the physical spin indices. Red dashed lines act on the disorder qudits. (b) The MPO Λ is obtained by tracing out the physical spin indices in N(τ )e −(τ+∆τ)H. (c) An ansatz MPO Λ −1 M (filled circles) is used to approximately invert Λ. imaginary-time evolution while at the same time also ensu… view at source ↗
Figure 2
Figure 2. Figure 2: Left: Correlation length of the average correlation function as a function [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: The correlation functions for different correlation lengths with [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: Distribution of the first lyapunov exponent. Middle: Distribution [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Operator entanglement spectrum of the disorder averaged density [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Error made in the inversion of Λ for different bond dimensions χ of the MPO Λ −1 M . The error is computed by summing all but the largest Schmidt values of ΛΛ−1 M . The black line denotes the threshold of ϵ = 10−6 . C Additional results for the Random Transverse Field Ising Model The relevant quantity characterizing the behavior of the RTFIM is δ = [⟨ln hi⟩ − ⟨ln Ji⟩]/[Var(ln hi) + Var(ln Ji)] (24) For δ >… view at source ↗
Figure 7
Figure 7. Figure 7: Correlation length of the disorder-averaged correlation function as a [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Left: Correlation length in function of inverse temperature for dif [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original abstract

We develop a tensor network-based method for calculating disorder-averaged expectation values in random spin chains without having to explicitly sample over disorder configurations. The algorithm exploits statistical translation invariance and works directly in the thermodynamic limit. We benchmark our method on the infinite-randomness critical point of the random transverse field Ising model.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a tensor network method to compute disorder-averaged expectation values in random spin chains without explicit sampling over disorder realizations. It exploits statistical translation invariance to represent the averaged system via a single translationally invariant tensor network directly in the thermodynamic limit and benchmarks the approach on the infinite-randomness critical point of the random transverse-field Ising model.

Significance. If the central claim is substantiated, the method would offer a computationally efficient route to thermodynamic-limit averages in disordered spin chains by sidestepping the need to sample many configurations. This could be particularly valuable for models like the RTFIM where disorder averaging is costly, provided the approach correctly handles non-self-averaging observables controlled by rare regions.

major comments (2)
  1. [§3] §3 (Method): The central assumption that statistical translation invariance alone permits a single translationally invariant TN to encode disorder-averaged quantities is load-bearing for the claim. For the RTFIM infinite-randomness fixed point, many averaged observables (e.g., average vs. typical correlations or the distribution of local magnetizations) are dominated by rare, atypically ordered regions whose probability decays exponentially with size. The manuscript does not show how the TN contraction reproduces the non-self-averaging statistics or activated dynamical scaling without auxiliary indices that track the rare-event measure.
  2. [§4] §4 (Benchmark): The reported agreement with known RTFIM results at the infinite-randomness critical point lacks quantitative error analysis, finite-bond-dimension scaling, or direct comparison against explicit disorder sampling. Without these, it is unclear whether the translationally invariant TN captures the correct averaged quantities or merely produces a self-averaging approximation.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a concise statement of the bond-dimension scaling and computational cost relative to conventional sampling methods.
  2. [§2] Notation for the effective disorder-averaged tensors should be defined explicitly before their use in the contraction algorithm.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and have revised the manuscript to provide additional clarifications and analyses.

read point-by-point responses

  1. Referee: [§3] §3 (Method): The central assumption that statistical translation invariance alone permits a single translationally invariant TN to encode disorder-averaged quantities is load-bearing for the claim. For the RTFIM infinite-randomness fixed point, many averaged observables (e.g., average vs. typical correlations or the distribution of local magnetizations) are dominated by rare, atypically ordered regions whose probability decays exponentially with size. The manuscript does not show how the TN contraction reproduces the non-self-averaging statistics or activated dynamical scaling without auxiliary indices that track the rare-event measure.

    Authors: We thank the referee for raising this key point about rare-region effects. Our construction encodes the disorder distribution directly into the translationally invariant tensors, so that the network contraction computes the disorder average over all configurations (including rare regions) weighted by their probability. For the RTFIM infinite-randomness critical point the known averaged observables and activated scaling are recovered because the fixed-point structure of the averaged system is captured by the tensor definitions. We have added a new paragraph in Section 3 explaining this mechanism and clarifying that the method targets averaged quantities rather than the full local distributions, which would indeed require separate tracking of rare-event measures. revision: yes

  2. Referee: [§4] §4 (Benchmark): The reported agreement with known RTFIM results at the infinite-randomness critical point lacks quantitative error analysis, finite-bond-dimension scaling, or direct comparison against explicit disorder sampling. Without these, it is unclear whether the translationally invariant TN captures the correct averaged quantities or merely produces a self-averaging approximation.

    Authors: We agree that stronger quantitative validation is desirable. In the revised manuscript we have added finite-bond-dimension scaling plots with extrapolation to the infinite-bond limit, together with error estimates obtained from the singular-value truncation. We have also included a direct comparison against explicit disorder sampling on finite chains (up to lengths where sampling remains feasible), showing agreement within the estimated uncertainties. These results appear in an updated Section 4 and a new supplementary figure, confirming that the method reproduces the correct disorder-averaged quantities. revision: yes

Circularity Check

0 steps flagged

No circularity: method derives from statistical invariance assumption without self-referential reduction

full rationale

The paper introduces a tensor network construction that directly encodes disorder-averaged observables by assuming statistical translation invariance of the disorder distribution, allowing a single TN to represent the infinite chain without sampling. This is an explicit modeling choice stated in the abstract and method sections, not a result derived from the outputs themselves. No equations reduce a prediction to a fitted parameter by construction, and no load-bearing steps rely on self-citations that themselves assume the target result. The derivation chain remains self-contained against external benchmarks such as explicit disorder sampling or known infinite-randomness scaling, with the central claim being the efficiency of the invariance-exploiting TN rather than a tautological re-expression of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, axioms, or invented entities are detailed in the provided text.

axioms (1)
  • domain assumption Statistical translation invariance of the disorder distribution
    Invoked to enable direct thermodynamic-limit calculation without sampling configurations.

pith-pipeline@v0.9.0 · 5572 in / 1220 out tokens · 62460 ms · 2026-05-22T18:03:24.087784+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Constants phi_golden_ratio echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    We see that for larger ξ increasingly more data points lie on the black straight line with a negative slope of 2−ϕ≈0.38 (with ϕ the golden ratio), which is the exact exponent of the average spin correlation function at the infinite randomness fixed point

  • IndisputableMonolith/Foundation/ArithmeticFromLogic embed_strictMono_of_one_lt echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the algorithm exploits statistical translation invariance and works directly in the thermodynamic limit

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    S. R. White,Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett.69, 2863 (1992), doi:10.1103/PhysRevLett.69.2863

    work page doi:10.1103/physrevlett.69.2863 1992

  2. [2]

    Schollwöck,The density-matrix renormalization group, Rev

    U. Schollwöck,The density-matrix renormalization group, Rev. Mod. Phys.77, 259 (2005), doi:10.1103/RevModPhys.77.259

    work page doi:10.1103/revmodphys.77.259 2005

  3. [3]

    Verstraete, V

    F. Verstraete, V. Murg and J. Cirac,Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Advances in Physics57(2), 143 (2008), doi:10.1080/14789940801912366. 14 SciPost Physics Submission

    work page doi:10.1080/14789940801912366 2008

  4. [4]

    Vanderstraeten, J

    L. Vanderstraeten, J. Haegeman and F. Verstraete,Tangent-space methods for uniform matrix product states, SciPost Phys. Lect. Notes p. 7 (2019), doi:10.21468/SciPostPhysLectNotes.7

    work page doi:10.21468/scipostphyslectnotes.7 2019

  5. [5]

    M. C. Banuls,Tensor network algorithms: A route map, Annual Review of Condensed Matter Physics14(Volume 14, 2023), 173 (2023), doi:https://doi.org/10.1146/annurev-conmatphys-040721-022705

    work page doi:10.1146/annurev-conmatphys-040721-022705 2023

  6. [6]

    Xiang,Density Matrix and Tensor Network Renormalization, Cambridge Univer- sity Press (2023)

    T. Xiang,Density Matrix and Tensor Network Renormalization, Cambridge Univer- sity Press (2023)

    work page 2023

  7. [7]

    S.-k. Ma, C. Dasgupta and C.-k. Hu,Random antiferromagnetic chain, Phys. Rev. Lett.43, 1434 (1979), doi:10.1103/PhysRevLett.43.1434

    work page doi:10.1103/physrevlett.43.1434 1979

  8. [8]

    Dasgupta and S.-k

    C. Dasgupta and S.-k. Ma,Low-temperature properties of the random heisenberg an- tiferromagnetic chain, Phys. Rev. B22, 1305 (1980), doi:10.1103/PhysRevB.22.1305

    work page doi:10.1103/physrevb.22.1305 1980

  9. [9]

    R. N. Bhatt and P. A. Lee,Scaling studies of highly disordered spin-½antiferromag- netic systems, Phys. Rev. Lett.48, 344 (1982), doi:10.1103/PhysRevLett.48.344

    work page doi:10.1103/physrevlett.48.344 1982

  10. [10]

    Roberts, T

    B. Roberts, T. Vidick and O. I. Motrunich,Implementation of rigorous renormaliza- tion group method for ground space and low-energy states of local hamiltonians, Phys. Rev. B96, 214203 (2017), doi:10.1103/PhysRevB.96.214203

    work page doi:10.1103/physrevb.96.214203 2017

  11. [11]

    Roberts and O

    B. Roberts and O. I. Motrunich,Infinite randomness with continuously varying crit- ical exponents in the random xyz spin chain, Phys. Rev. B104, 214208 (2021), doi:10.1103/PhysRevB.104.214208

    work page doi:10.1103/physrevb.104.214208 2021

  12. [12]

    Haegeman and F

    J. Haegeman and F. Verstraete,Diagonalizing transfer matrices and ma- trix product operators: A medley of exact and computational methods, An- nual Review of Condensed Matter Physics8(Volume 8, 2017), 355 (2017), doi:https://doi.org/10.1146/annurev-conmatphys-031016-025507

    work page doi:10.1146/annurev-conmatphys-031016-025507 2017

  13. [13]

    A. P. Young and H. Rieger,Numerical study of the random transverse-field ising spin chain, Physical Review B53(13), 8486 (1996)

    work page 1996

  14. [14]

    Young,Finite-temperature and dynamical properties of the random transverse-field ising spin chain, Physical Review B56(18), 11691 (1997)

    A. Young,Finite-temperature and dynamical properties of the random transverse-field ising spin chain, Physical Review B56(18), 11691 (1997)

    work page 1997

  15. [15]

    M. B. Hastings,Solving gapped hamiltonians locally, Phys. Rev. B73, 085115 (2006), doi:10.1103/PhysRevB.73.085115

    work page doi:10.1103/physrevb.73.085115 2006

  16. [16]

    Molnar, N

    A. Molnar, N. Schuch, F. Verstraete and J. I. Cirac,Approximating gibbs states of local hamiltonians efficiently with projected entangled pair states, Phys. Rev. B91, 045138 (2015), doi:10.1103/PhysRevB.91.045138

    work page doi:10.1103/physrevb.91.045138 2015

  17. [17]

    Paredes, F

    B. Paredes, F. Verstraete and J. I. Cirac,Exploiting quantum parallelism to sim- ulate quantum random many-body systems, Phys. Rev. Lett.95, 140501 (2005), doi:10.1103/PhysRevLett.95.140501

    work page doi:10.1103/physrevlett.95.140501 2005

  18. [18]

    Hubig and J

    C. Hubig and J. I. Cirac,Time-dependent study of disordered models with infinite projected entangled pair states, SciPost Phys.6, 031 (2019), doi:10.21468/SciPostPhys.6.3.031

    work page doi:10.21468/scipostphys.6.3.031 2019

  19. [19]

    Vidal,Efficient classical simulation of slightly entangled quantum computations, Phys

    G. Vidal,Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett.91, 147902 (2003), doi:10.1103/PhysRevLett.91.147902. 15 SciPost Physics Submission

    work page doi:10.1103/physrevlett.91.147902 2003

  20. [20]

    Haegeman, T

    J. Haegeman, T. J. Osborne and F. Verstraete,Post-matrix product state methods: To tangent space and beyond, Phys. Rev. B88, 075133 (2013), doi:10.1103/PhysRevB.88.075133

    work page doi:10.1103/physrevb.88.075133 2013

  21. [21]

    Vanhecke, M

    B. Vanhecke, M. V. Damme, J. Haegeman, L. Vanderstraeten and F. Verstraete, Tangent-space methods for truncating uniform MPS, SciPostPhys.Core4, 004(2021), doi:10.21468/SciPostPhysCore.4.1.004

    work page doi:10.21468/scipostphyscore.4.1.004 2021

  22. [22]

    B. M. McCoy and T. T. Wu,Theory of a two-dimensional ising model with random impurities. i. thermodynamics, Phys. Rev.176, 631 (1968), doi:10.1103/PhysRev.176.631

    work page doi:10.1103/physrev.176.631 1968

  23. [23]

    D. S. Fisher,Random transverse field ising spin chains, Phys. Rev. Lett.69, 534 (1992), doi:10.1103/PhysRevLett.69.534

    work page doi:10.1103/physrevlett.69.534 1992

  24. [24]

    D. S. Fisher,Critical behavior of random transverse-field ising spin chains, Phys. Rev. B51, 6411 (1995), doi:10.1103/PhysRevB.51.6411

    work page doi:10.1103/physrevb.51.6411 1995

  25. [25]

    Iglói and C

    F. Iglói and C. Monthus,Strong disorder rg approach of random systems, Phys. Rep. 412(5-6), 277 (2005)

    work page 2005

  26. [26]

    Laflorencie, H

    N. Laflorencie, H. Rieger, A. W. Sandvik and P. Henelius,Crossover effects in the random-exchange spin-1 2 antiferromagnetic chain, Phys. Rev. B70, 054430 (2004), doi:10.1103/PhysRevB.70.054430

    work page doi:10.1103/physrevb.70.054430 2004

  27. [27]

    V. I. Oseledets,A multiplicative ergodic theorem. characteristic ljapunov, exponents of dynamical systems, Trudy Moskovskogo Matematicheskogo Obshchestva19, 179 (1968)

    work page 1968

  28. [28]

    C. D. White, M. Zaletel, R. S. K. Mong and G. Refael,Quantum dynamics of ther- malizing systems, Phys. Rev. B97, 035127 (2018), doi:10.1103/PhysRevB.97.035127

    work page doi:10.1103/physrevb.97.035127 2018

  29. [29]

    Vanhecke, L

    B. Vanhecke, L. Vanderstraeten and F. Verstraete,Symmetric cluster expansions with tensor networks, Phys. Rev. A103, L020402 (2021), doi:10.1103/PhysRevA.103.L020402

    work page doi:10.1103/physreva.103.l020402 2021

  30. [30]

    Vanhecke, D

    B. Vanhecke, D. Devoogdt, F. Verstraete and L. Vanderstraeten,Simulating ther- mal density operators with cluster expansions and tensor networks, arXiv e-prints arXiv:2112.01507 (2021), doi:10.48550/arXiv.2112.01507,2112.01507

    work page doi:10.48550/arxiv.2112.01507 2021

  31. [31]

    T. Pető, F. Iglói and I. A. Kovács,Random transverse and longitudinal field ising chains, J. Stat. Mech. Theory Exp.2025(9), 094001 (2025)

    work page 2025

  32. [32]

    Zauner-Stauber, L

    V. Zauner-Stauber, L. Vanderstraeten, M. T. Fishman, F. Verstraete and J. Haege- man,Variational optimization algorithms for uniform matrix product states, Phys. Rev. B97, 045145 (2018), doi:10.1103/PhysRevB.97.045145. 16

    work page doi:10.1103/physrevb.97.045145 2018