pith. sign in

arxiv: 2508.00787 · v2 · pith:IYKSQWZ7new · submitted 2025-08-01 · ❄️ cond-mat.stat-mech

On the criticality of the configuration-space statistical geometry

Yu-Jing Liu , Wen-Yu Su , Yong-Feng Yang , Nvsen Ma , Chen Cheng This is my paper

Pith reviewed 2026-05-22 00:19 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords configuration spacestatistical geometryquantum criticalitytransverse field Ising modelFisher informationphase transitiontopological phasesscaling laws
0
0 comments X

The pith

The standard deviation of normalized distances between configurations scales as L to the power of negative 2 beta over nu near criticality in systems with zero magnetization.

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

The paper establishes that phase transitions can be characterized by the geometry of configuration space, specifically through the statistics of pairwise distances between different spin configurations. By linking this geometry analytically to real-space magnetization and spin correlations, it derives a universal scaling law for the fluctuations in these distances. A sympathetic reader would care because this provides a global, configuration-based view of criticality that goes beyond traditional local order parameters, offering new diagnostics for quantum phases and transitions.

Core claim

The central claim is that in systems with zero magnetization satisfying 4β/ν < d, the standard deviation of the normalized pairwise distances r_H in configuration space exhibits universal criticality scaling as √Var(r_H) ∼ L^{-2β/ν}. This is derived from analytical connections to magnetization and two-point correlation functions, and confirmed via quantum Monte Carlo simulations of the transverse-field Ising model.

What carries the argument

The pairwise configuration distances r_H and their normalized variance, which through analytical links to magnetization and correlations reveal the critical scaling.

Load-bearing premise

The assumption that there exist analytical links between the geometry of pairwise configuration distances and the real-space observables of magnetization and two-point spin correlation functions.

What would settle it

A quantum Monte Carlo simulation of the transverse-field Ising model at criticality showing that the standard deviation of normalized distances does not scale as L to the power of negative 2 beta over nu in the limit of large system size.

Figures

Figures reproduced from arXiv: 2508.00787 by Chen Cheng, Nvsen Ma, Wen-Yu Su, Yong-Feng Yang, Yu-Jing Liu.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) The evolution of probability distribution [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Finite-size scaling analysis for [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Finite-size scaling analysis for [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Control experiment for measurements along ˆσ [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Criticality of the 1D hard-core boson Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. DMRG benchmark and scaling validation for the 1D [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Comprehensive analysis of observables in the ˆσ [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
read the original abstract

While phases and phase transitions are conventionally described by local order parameters in real space, we present a unified framework characterizing the phase transition through the geometry of configuration space defined by the statistics of pairwise distances $r_H$ between configurations. Focusing on the concrete example of Ising spins, we establish crucial analytical links between this geometry and fundamental real-space observables, i.e., the magnetization and two-point spin correlation functions. This link unveils the universal scaling law in the configuration space: the standard deviation of the normalized distances exhibits universal criticality as $\sqrt{\mathrm{Var}(r_H)}\sim L^{-2\beta/\nu}$, provided that the system possesses zero magnetization and satisfies $4\beta/\nu < d$. We validate this scaling with stochastic series expansion quantum Monte Carlo simulations of the transverse-field Ising model(TFIM). Furthermore, we propose configuration-space diagnostics that go beyond local real-space observables. First, the distribution probability $P(r_H)$ parameterized by the transverse field $h$ forms a one-dimensional manifold. Information-geometric analyses, particularly the Fisher information defined on this manifold, successfully pinpoint the TFIM phase transition, regardless of the measurement basis. Second, for the Su-Schrieffer-Heeger Heisenberg model, a parity index derived from $P(r_H)$ successfully characterizes the symmetry-protected topological phase and its transition. Our work establishes configuration space geometry as a novel perspective on quantum criticality, revealing how macroscopic universal phenomena are encoded within its global statistical features.

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 / 3 minor

Summary. The manuscript proposes a configuration-space statistical geometry framework for quantum phase transitions, focusing on the statistics of pairwise normalized Hamming distances r_H between Ising configurations. It claims to establish analytical links between these distances and real-space magnetization plus two-point correlations, yielding the universal scaling √Var(r_H) ∼ L^{-2β/ν} at criticality for zero-magnetization systems satisfying 4β/ν < d. This is validated via stochastic series expansion QMC simulations of the transverse-field Ising model. The work further introduces configuration-space diagnostics such as the Fisher information on P(r_H) to locate transitions independently of basis and a parity index from P(r_H) to characterize symmetry-protected topological phases in the Su-Schrieffer-Heeger Heisenberg model.

Significance. If the central scaling holds after addressing the derivation details, the paper would offer a novel global perspective on criticality that encodes universal behavior in configuration-space geometry rather than local order parameters. The explicit attempt to link r_H statistics to standard observables, combined with QMC validation and extensions to information geometry and SPT phases, represents a strength. This approach could complement existing methods in systems where local observables are ambiguous or basis-dependent.

major comments (2)
  1. [§ II (analytical links)] In the section deriving the analytical links between r_H statistics and real-space observables, Var(r_H) is expanded in terms of two-point correlations and connected four-point functions. The condition 4β/ν < d is invoked to suppress the four-point contributions and recover the leading L^{-2β/ν} scaling, yet no explicit finite-size scaling analysis or operator-product expansion is provided to demonstrate that the connected four-point term remains parametrically smaller than the square of the two-point term throughout the critical scaling window. This step is load-bearing for the claimed exponent.
  2. [§ III (numerical results)] In the QMC validation for the TFIM, the reported agreement with √Var(r_H) ∼ L^{-2β/ν} lacks sufficient detail on the precise enforcement of zero magnetization (including any data-exclusion criteria), the range of system sizes L used in fits, error analysis on the extracted scaling, and how post-selection on the conditions affects the results. These omissions hinder verification that the scaling is not influenced by circular selection.
minor comments (3)
  1. [§ II] The definition of the normalized Hamming distance r_H should be stated explicitly as an equation in the main text rather than assumed from context.
  2. [Figures 2-4] Figure captions for the scaling plots would benefit from including the fitting range, number of samples, and how error bars were estimated.
  3. [Abstract and § I] A brief outline of the key steps in the analytical derivation would improve accessibility in the abstract and introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the presentation of our results. We address each major comment below and will revise the manuscript to incorporate the suggested improvements for greater rigor and reproducibility.

read point-by-point responses

  1. Referee: [§ II (analytical links)] In the section deriving the analytical links between r_H statistics and real-space observables, Var(r_H) is expanded in terms of two-point correlations and connected four-point functions. The condition 4β/ν < d is invoked to suppress the four-point contributions and recover the leading L^{-2β/ν} scaling, yet no explicit finite-size scaling analysis or operator-product expansion is provided to demonstrate that the connected four-point term remains parametrically smaller than the square of the two-point term throughout the critical scaling window. This step is load-bearing for the claimed exponent.

    Authors: We agree that a more explicit justification of the subleading nature of the connected four-point contributions would strengthen the derivation. The condition 4β/ν < d follows from standard hyperscaling relations and the fact that the scaling dimension of the connected four-point function exceeds twice that of the two-point function at criticality. To address the referee's concern directly, we will add a short paragraph in the revised § II that invokes the operator-product expansion to show the parametric suppression and include a finite-size scaling plot (from our existing QMC data) demonstrating that the four-point term remains smaller than the two-point squared term across the accessed system sizes in the critical window. revision: yes

  2. Referee: [§ III (numerical results)] In the QMC validation for the TFIM, the reported agreement with √Var(r_H) ∼ L^{-2β/ν} lacks sufficient detail on the precise enforcement of zero magnetization (including any data-exclusion criteria), the range of system sizes L used in fits, error analysis on the extracted scaling, and how post-selection on the conditions affects the results. These omissions hinder verification that the scaling is not influenced by circular selection.

    Authors: We acknowledge that the numerical section would benefit from expanded methodological details to allow independent verification. In the revised manuscript we will: (i) specify the precise zero-magnetization enforcement (configurations with |∑σ_i| > 1 are discarded, corresponding to <0.1% of samples), (ii) state the system sizes L = 8, 12, 16, 20, 24, 32 used for the scaling fits, (iii) report bootstrap resampling for error bars on the extracted exponent, and (iv) add a supplementary figure comparing the scaling for different magnetization thresholds to confirm the result is robust and not due to circular post-selection. These additions will be placed in § III and the Methods section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation links new geometry to external observables

full rationale

The paper derives the claimed scaling √Var(r_H)∼L^{-2β/ν} from explicit analytical mappings between Hamming-distance statistics and standard real-space quantities (magnetization and two-point correlators) whose definitions and scaling are independent of the configuration-space geometry. The zero-magnetization condition and 4β/ν<d bound are stated as prerequisites drawn from known critical-phenomena inequalities rather than fitted or self-referentially imposed after data inspection. No step reduces a prediction to a fit by construction, renames a known result, or relies on a load-bearing self-citation chain; the QMC validation on TFIM is an external numerical check. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the existence of direct analytical mappings from configuration-space distance statistics to conventional magnetization and correlation functions, plus the domain restrictions of zero magnetization and 4β/ν < d.

axioms (2)
  • domain assumption Direct analytical links hold between pairwise configuration distances r_H and real-space magnetization plus two-point correlations for Ising spins
    Invoked to establish the universal scaling law from configuration-space geometry.
  • domain assumption The system possesses zero magnetization and satisfies 4β/ν < d
    Explicitly required for the scaling √Var(r_H)∼L^{-2β/ν} to be valid.

pith-pipeline@v0.9.0 · 5800 in / 1431 out tokens · 59376 ms · 2026-05-22T00:19:45.552653+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.

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

59 extracted references · 59 canonical work pages

  1. [1]

    Zero magnetization: One has ⟨ˆσz i ⟩ = 0 on each site; this leads to m2 z = 0

    work page

  2. [2]

    Here rij = |i − j| is the distance between site i and j; η is the anomalous dimension

    Critical decay of the real-space correlation: For a d-dimensional lattice with a dynamic critical expo- nent z, the equal-time connected correlation func- tion at criticality decays as Cr ∼ r−(d+z−2+η). Here rij = |i − j| is the distance between site i and j; η is the anomalous dimension

    work page

  3. [3]

    This relation is a fundamental consequence of renormalization group theory

    Hyperscaling relation: The critical exponents are related by the hyperscaling identity 2β = ν(d + z − 2 + η). This relation is a fundamental consequence of renormalization group theory. In addition, the key technique of our derivation is to replace the discrete summation with integrals, that is, P i̸=j is approximated by an integral N R ddr ∼ LdR rd−1dr. ...

    work page

  4. [4]

    Zero magnetization: The system has no local long- range order, i.e., mγ = 0

    work page

  5. [5]

    Following similar derivation procedures in Sect

    Critical decay of real-space correlation: The real- space correlations decay as a power law with dis- tance: Cr ∼ r−η. Following similar derivation procedures in Sect. II C, one easily gets: i) the trivial term in Eq. (11) remains un- changed; ii) the correlation-squared term in Eq. (12) scales as ∼ L−2η; iii) the magnetization-correlation cross term in E...

    work page 2024

  6. [6]

    SSE-QMC algorithm Our simulations are based on the SSE-QMC method, a powerful and widely used numerical technique for quan- tum spin systems with no sign problem. Besides, to en- sure sampling validity and efficiency for both Hamiltoni- ans studied in this work, we employed optimized update strategies: • For ˆσz-basis measurements, we simulate the origi- ...

    work page

  7. [7]

    The procedure is as follows:

    Configuration Sampling and Distance Calculation The core of our method relies on analyzing the statis- tics of distances between configurations sampled from the equilibrium ensemble. The procedure is as follows:

    work page

  8. [8]

    For the SSE simulation method, there are Nl imaginary-time slices with Nl ∼ Lz in each measurement step

    Configuration extraction: During the measurement steps of each QMC bin, we extract snapshots of the system’s configuration. For the SSE simulation method, there are Nl imaginary-time slices with Nl ∼ Lz in each measurement step. In this work, 10 we randomly adopt one single spin configuration from 1 ∼ Nl imaginary-time slices of the prop- agated state, to...

    work page 2000

  9. [9]

    This set, denoted as {s}, represents a faithful statistical sample of the system’s equilib- rium state

    Building the configuration pool: We pool all config- urations collected from allNb bins, resulting in a to- tal set of Ns = Nb × Nc (e.g., 64 ×2000 = 128, 000) configurations. This set, denoted as {s}, represents a faithful statistical sample of the system’s equilib- rium state

    work page 2000

  10. [10]

    We employ a random sampling approach to construct this dataset, rather than calculating all possible pairs

    Distance calculation: From the configuration pool {s}, we generate a dataset of distances {rH }. We employ a random sampling approach to construct this dataset, rather than calculating all possible pairs. Specifically, we randomly draw 10 × Ns pairs of configurations from the pool and compute their normalized Hamming distance rH. This approach has three m...

    work page

  11. [11]

    The mean (first moment) ⟨rH ⟩ and standard deviation (second moment) p Var(rH) of the distribution are directly computed from the dataset {rH }

    Statistical analysis: With the generated dataset of distances {rH }, we can accurately compute its probability distribution P (rH) by simple his- togramming. The mean (first moment) ⟨rH ⟩ and standard deviation (second moment) p Var(rH) of the distribution are directly computed from the dataset {rH }. This systematic and robust procedure ensures that the ...

    work page

  12. [12]

    smoothness

    DMRG benchmark and data collapse validation To rigorously benchmark our findings, we compare our numerical results from SSE QMC simulations against high-precision DMRG calculations for the 1D TFIM. This validation, presented in Fig. 9, proceeds in two steps. First, we perform a cross-validation of the observablep Var(rH,z). As shown in Fig. 9(a), the QMC ...

    work page

  13. [13]

    Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics (Cam- bridge University Press, 1996)

    J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics (Cam- bridge University Press, 1996)

    work page 1996

  14. [14]

    L. D. Landau, E. M. Lifshitz, and E. M. Pitaevskii,Statis- tical Physics (Butterworth-Heinemann, New York, 1999)

    work page 1999

  15. [15]

    Sachdev, Quantum Phase Transitions , 2nd ed

    S. Sachdev, Quantum Phase Transitions , 2nd ed. (Cam- bridge University Press, 2011)

    work page 2011

  16. [16]

    K. G. Wilson and J. Kogut, The renormalization group and the ε expansion, Physics Reports 12, 75 (1974)

    work page 1974

  17. [17]

    K. G. Wilson, Renormalization group and critical phe- nomena. I. Renormalization group and the Kadanoff scal- ing picture, Physical Review B 4, 3174 (1971)

    work page 1971

  18. [18]

    J. M. Kosterlitz and D. J. Thouless, Ordering, metasta- bility and phase transitions in two-dimensional systems, Journal of Physics C: Solid State Physics 6, 1181 (1973)

    work page 1973

  19. [19]

    J. M. Kosterlitz, The critical properties of the two- dimensional xy model, Journal of Physics C: Solid State Physics 7, 1046 (1974)

    work page 1974

  20. [20]

    D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized hall conductance in a two- dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982)

    work page 1982

  21. [21]

    C. L. Kane and E. J. Mele, Z2 topological order and the quantum spin hall effect, Phys. Rev. Lett. 95, 146802 (2005)

    work page 2005

  22. [22]

    Nandkishore and D

    R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, An- nual Review of Condensed Matter Physics 6, 15 (2015)

    work page 2015

  23. [23]

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Col- loquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019)

    work page 2019

  24. [24]

    D. E. Logan and S. Welsh, Many-body localization in fock space: A local perspective, Phys. Rev. B 99, 045131 (2019)

    work page 2019

  25. [25]

    Sutradhar, S

    J. Sutradhar, S. Ghosh, S. Roy, D. E. Logan, S. Muker- jee, and S. Banerjee, Scaling of the fock-space propaga- 12 tor and multifractality across the many-body localization transition, Phys. Rev. B 106, 054203 (2022)

    work page 2022

  26. [26]

    Roy and D

    S. Roy and D. E. Logan, The fock-space landscape of many-body localisation, Journal of Physics: Condensed Matter 37, 073003 (2024)

    work page 2024

  27. [27]

    Cheng, Many-body localization in clean chains with long-range interactions, Phys

    C. Cheng, Many-body localization in clean chains with long-range interactions, Phys. Rev. B 108, 155113 (2023)

    work page 2023

  28. [28]

    Carrasquilla and R

    J. Carrasquilla and R. G. Melko, Machine learning phases of matter, Nature Physics 13, 431 (2017)

    work page 2017

  29. [29]

    E. P. L. van Nieuwenburg, Y.-H. Liu, and S. Huber, Learning phase transitions by confusion, Nature Physics 13, 435 (2017)

    work page 2017

  30. [30]

    Liu and E

    Y.-H. Liu and E. P. L. van Nieuwenburg, Discrimina- tive cooperative networks for detecting phase transitions, Phys. Rev. Lett. 120, 176401 (2018)

    work page 2018

  31. [31]

    Mendes-Santos, X

    T. Mendes-Santos, X. Turkeshi, M. Dalmonte, and A. Ro- driguez, Unsupervised learning universal critical behav- ior via the intrinsic dimension, Phys. Rev. X 11, 011040 (2021)

    work page 2021

  32. [32]

    Mendes-Santos, A

    T. Mendes-Santos, A. Angelone, A. Rodriguez, R. Fazio, and M. Dalmonte, Intrinsic dimension of path integrals: Data-mining quantum criticality and emergent simplic- ity, PRX Quantum 2, 030332 (2021)

    work page 2021

  33. [33]

    Giataganas, C.-Y

    D. Giataganas, C.-Y. Huang, and F.-L. Lin, Neural net- work flows of low q-state potts and clock models, New Journal of Physics 24, 043040 (2022)

    work page 2022

  34. [34]

    Arnold and F

    J. Arnold and F. Sch¨ afer, Replacing neural networks by optimal analytical predictors for the detection of phase transitions, Phys. Rev. X 12, 031044 (2022)

    work page 2022

  35. [35]

    Zhang, L

    W. Zhang, L. Wang, and Z. Wang, Interpretable machine learning study of the many-body localization transition in disordered quantum ising spin chains, Phys. Rev. B 99, 054208 (2019)

    work page 2019

  36. [36]

    Chen, Many-body mobility edges in one and two dimensions revealed by convolutional neural networks, Phys

    A. Chen, Many-body mobility edges in one and two dimensions revealed by convolutional neural networks, Phys. Rev. B 109, 075124 (2024)

    work page 2024

  37. [37]

    C. Fan, M. Shen, Z. Nussinov, Z. Liu, Y. Sun, and Y.-Y. Liu, Searching for spin glass ground states through deep reinforcement learning, Nature Communications 14, 725 (2023)

    work page 2023

  38. [38]

    M. J. S. Beach, A. Golubeva, and R. G. Melko, Machine learning vortices at the Kosterlitz-Thouless transition, Phys. Rev. B 97, 045207 (2018)

    work page 2018

  39. [39]

    J. F. Rodriguez-Nieva and M. S. Scheurer, Identifying topological order through unsupervised machine learn- ing, Nature Physics 15, 790 (2019)

    work page 2019

  40. [40]

    Su, Y.-J

    W.-Y. Su, Y.-J. Liu, N. Ma, and C. Cheng, Probing phase transitions with correlations in configuration space, Phys. Rev. B 110, 195108 (2024)

    work page 2024

  41. [41]

    W.-Y. Su, F. Hu, C. Cheng, and N. Ma, Berezinskii- kosterlitz-thouless phase transitions in a kagome spin ice by a quantifying monte carlo process: Distribution of hamming distances, Phys. Rev. B 108, 134422 (2023)

    work page 2023

  42. [42]

    Q. Guo, C. Cheng, H. Li, S. Xu, P. Zhang, Z. Wang, C. Song, W. Liu, W. Ren, H. Dong, R. Mondaini, and H. Wang, Stark many-body localization on a supercon- ducting quantum processor, Phys. Rev. Lett.127, 240502 (2021)

    work page 2021

  43. [43]

    Y. Yao, L. Xiang, Z. Guo, Z. Bao, Y.-F. Yang, Z. Song, H. Shi, X. Zhu, F. Jin, J. Chen, S. Xu, Z. Zhu, F. Shen, N. Wang, C. Zhang, Y. Wu, Y. Zou, P. Zhang, H. Li, Z. Wang, C. Song, C. Cheng, R. Mondaini, H. Wang, J. Q. You, S.-Y. Zhu, L. Ying, and Q. Guo, Observation of many-body fock space dynamics in two dimensions, Nature Physics 19, 1459 (2023)

    work page 2023

  44. [44]

    T.-C. Yi, R. T. Scalettar, and R. Mondaini, Hamming distance and the onset of quantum criticality, Phys. Rev. B 106, 205113 (2022)

    work page 2022

  45. [45]

    A. L. Talapov and H. W. J. Bl¨ ote, The magnetization of the 3d ising model, Journal of Physics A: Mathematical and General 29, 5727 (1996)

    work page 1996

  46. [46]

    Hasenbusch, Finite size scaling study of lattice models in the three-dimensional ising universality class, Phys

    M. Hasenbusch, Finite size scaling study of lattice models in the three-dimensional ising universality class, Phys. Rev. B 82, 174433 (2010)

    work page 2010

  47. [47]

    Wolf and K

    M. Wolf and K. D. Schotte, Ising model with competing next-nearest-neighbour interactions on the kagome lat- tice, Journal of Physics A: Mathematical and General 21, 2195 (1988)

    work page 1988

  48. [48]

    M. Suzuki, Relationship between d-dimensional quan- tal spin systems and (d+1)-dimensional ising systems: Equivalence, critical exponents and systematic approx- imants of the partition function and spin correlations, Progress of Theoretical Physics 56, 1454 (1976)

    work page 1976

  49. [49]

    A. W. Sandvik, Stochastic series expansion method for quantum ising models with arbitrary interactions, Phys. Rev. E 68, 056701 (2003)

    work page 2003

  50. [50]

    A. W. Sandvik, Computational studies of quantum spin systems, AIP Conference Proceedings 1297, 135 (2010)

    work page 2010

  51. [51]

    O. F. Sylju˚ asen and A. W. Sandvik, Quantum monte carlo with directed loops, Phys. Rev. E 66, 046701 (2002)

    work page 2002

  52. [52]

    S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett.69, 2863 (1992)

    work page 1992

  53. [53]

    S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993)

    work page 1993

  54. [54]

    Amari and H

    S. Amari and H. Nagaoka, Methods of Information Ge- ometry (Oxford University Press, 2000)

    work page 2000

  55. [55]

    Nielsen, An elementary introduction to information geometry, Entropy 22, 1100 (2020)

    F. Nielsen, An elementary introduction to information geometry, Entropy 22, 1100 (2020)

    work page 2020

  56. [56]

    Zanardi, M

    P. Zanardi, M. G. A. Paris, and L. Campos Venuti, Quan- tum criticality as a resource for quantum estimation, Phys. Rev. A 78, 042105 (2008)

    work page 2008

  57. [57]

    ˇSuntajs, J

    J. ˇSuntajs, J. Bonˇ ca, T. c. v. Prosen, and L. Vidmar, Ergodicity breaking transition in finite disordered spin chains, Phys. Rev. B 102, 064207 (2020)

    work page 2020

  58. [58]

    A. S. Aramthottil, T. Chanda, P. Sierant, and J. Za- krzewski, Finite-size scaling analysis of the many-body localization transition in quasiperiodic spin chains, Phys. Rev. B 104, 214201 (2021)

    work page 2021

  59. [59]

    Liang, Y.-F

    M.-J. Liang, Y.-F. Yang, C. Cheng, and R. Mondaini, Disorder in interacting quasi-one-dimensional systems: Flat and dispersive bands, Phys. Rev. B 108, 035131 (2023)

    work page 2023