Detecting the Largest Correlations using the Correlation Density Matrix: a Quantum Monte Carlo Approach

Aditya Chincholi; Fabien Alet; Sylvain Capponi

arxiv: 2507.21697 · v2 · pith:XCR2WDLWnew · submitted 2025-07-29 · ❄️ cond-mat.str-el

Detecting the Largest Correlations using the Correlation Density Matrix: a Quantum Monte Carlo Approach

Aditya Chincholi , Sylvain Capponi , Fabien Alet This is my paper

Pith reviewed 2026-05-25 08:15 UTC · model grok-4.3

classification ❄️ cond-mat.str-el

keywords quantummonteapproachcarlocorrelationcorrelationsdensitylarge

0 comments

The pith

The correlation density matrix between two small subsystems reveals the dominant correlations in large many-body systems.

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

This paper presents a quantum Monte Carlo method to detect the strongest correlations in many-body quantum systems by examining the correlation density matrix between two small embedded subsystems. The approach requires no prior knowledge of possible order parameters and ranks correlations by their strength. It is demonstrated on zero-temperature phase transitions in the one- and two-dimensional quantum Ising models and the two-dimensional bilayer Heisenberg antiferromagnet. A sympathetic reader would care because it offers a way to explore unknown phases where conventional identification of order is difficult.

Core claim

The correlation density matrix between two small subsystems embedded in the full sample allows detection and computation of the most dominant correlations for many-body systems without prior knowledge, as shown through benchmarks on quantum Ising and Heisenberg models.

What carries the argument

The correlation density matrix measured between two small subsystems, which captures and ranks the dominant correlations of the large system.

Load-bearing premise

That the correlation density matrix between two small subsystems captures the dominant correlations of the entire large system.

What would settle it

Observing that in a system with known dominant long-range order, the correlation density matrix from small subsystems fails to identify it as the largest correlation.

Figures

Figures reproduced from arXiv: 2507.21697 by Aditya Chincholi, Fabien Alet, Sylvain Capponi.

**Figure 2.** Figure 2: FIG. 2: Schemes of the partition function sampling ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: 1D TFIM: ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: 1D TFIM: Singular values vs field ( [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Ising 1D: Physical correlations vs length for [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8: Data collapse for the 2D TFIM: (left) Finite [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9: 2D TFIM: 2-site and 4-site singular values at [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10: 2D TFIM: Log-Log plot of [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11: Heisenberg bilayer: Layout of clusters [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12: Heisenberg bilayer: Singular values from 2-site [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13: Heisenberg bilayer: Highest singular value [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗

**Figure 14.** Figure 14: FIG. 14: Data collapse for the Heisenberg bilayer: (a) [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗

read the original abstract

We present a quantum Monte Carlo-based approach to detect and compute the most dominant correlations for many-body systems without prior knowledge. It is based on the measurement and analysis of the correlation density matrix between two (small) subsystems embedded in the full (large) sample. In order to benchmark this procedure, we investigate zero-temperature quantum phase transitions in one- and two-dimensional quantum Ising model as well as the two-dimensional bilayer Heisenberg antiferromagnet. The method paves the way for a systematic identification of unknown or exotic order parameters in unexplored phases on large systems accessible to quantum Monte Carlo methods.

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.

Desk Editor's Note private letter to a colleague

The correlation density matrix method recovers known orders in QMC benchmarks but its claim to work without prior knowledge for arbitrary phases rests on an assumption about locality that is not generally validated.

read the letter

The paper shows how to measure a correlation density matrix between two small embedded subsystems in a QMC simulation and use its dominant eigenvectors to identify the strongest correlations present. They apply this to the 1D and 2D transverse-field Ising models and the bilayer Heisenberg antiferromagnet, recovering the expected order parameters at the known transitions without having to guess the operators in advance. The implementation details and the numerical results on these standard cases are the concrete advance. The benchmarks are clean and the procedure is straightforward to code once the idea is stated. The limitation is that the method assumes the dominant correlations can be captured by correlations between two small local subsystems. This works for the Ising and Heisenberg cases because the order parameters produce clear two-site signatures, but the paper provides no test or argument for orders whose leading correlations are multi-point, involve longer-range structure, or are not reducible to a two-subsystem matrix. Without that check, the claim that the approach works for unknown or exotic phases remains an extrapolation. The work is aimed at QMC practitioners who already simulate large systems and want a systematic scan for order rather than a hand-picked list of observables. It is worth sending to referees because the technique is new in this setting and the existing benchmarks are reproducible, even though the generality question will need more evidence in revision.

Referee Report

1 major / 0 minor

Summary. The manuscript presents a quantum Monte Carlo method to detect and rank the dominant correlations in many-body systems without prior knowledge of the order parameter. The approach relies on constructing and analyzing the correlation density matrix between two small subsystems embedded in a larger sample, with benchmarks on zero-temperature quantum phase transitions in the 1D and 2D quantum Ising models as well as the 2D bilayer Heisenberg antiferromagnet.

Significance. If the local two-subsystem correlation density matrix measurement reliably identifies the globally dominant correlations, the method would enable systematic exploration of unknown or exotic ordered phases in large systems accessible to QMC, providing a practical tool beyond conventional order-parameter searches.

major comments (1)

[Abstract] Abstract and benchmarking description: the claim that measurements on any two small embedded subsystems suffice to rank the largest correlations present anywhere in the system is supported only by benchmarks on models whose order parameters produce clear local signatures (Ising and bilayer Heisenberg); no demonstration or counter-example analysis is given for cases where the dominant correlations are non-local, multi-point, or otherwise not reducible to two-subsystem operator correlations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract and benchmarking description: the claim that measurements on any two small embedded subsystems suffice to rank the largest correlations present anywhere in the system is supported only by benchmarks on models whose order parameters produce clear local signatures (Ising and bilayer Heisenberg); no demonstration or counter-example analysis is given for cases where the dominant correlations are non-local, multi-point, or otherwise not reducible to two-subsystem operator correlations.

Authors: We agree with the referee that our benchmarks are restricted to models (1D/2D quantum Ising and bilayer Heisenberg) whose dominant correlations have clear local two-point character that is directly captured by the correlation density matrix (CDM) between two small subsystems. The method as formulated extracts the dominant correlation operators from the spectrum of this two-subsystem CDM; it therefore identifies the largest correlations that can be expressed as operators acting on the chosen pair of subsystems. We have not provided explicit demonstrations or counter-examples for cases in which the globally dominant correlations are inherently non-local, require multi-point operators, or cannot be reduced to two-subsystem correlations. We will revise the abstract to clarify the scope of the claim and add a dedicated paragraph in the discussion section that states the current limitations and the conditions under which the two-subsystem CDM is expected to recover the leading correlations. revision: yes

Circularity Check

0 steps flagged

No circularity: method is a direct QMC measurement procedure validated on benchmarks

full rationale

The paper introduces a computational procedure to measure the correlation density matrix between two small embedded subsystems and use it to rank dominant correlations. This is framed as an empirical detection method benchmarked on standard models (1D/2D Ising, bilayer Heisenberg) with known order parameters. No derivation chain reduces a claimed prediction or result to a fitted input or self-citation by construction; the approach relies on standard QMC sampling rather than any self-referential definition or ansatz smuggling. The central claim about sufficiency of local subsystem measurements is presented as a testable hypothesis, not a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities; ledger left empty pending full text.

pith-pipeline@v0.9.0 · 5626 in / 888 out tokens · 19268 ms · 2026-05-25T08:15:07.598883+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Decomposition of the correlation density matrix We reshape the correlation density matrix to separate the indices on A and B: (ρc AB)i,j = (ρc AB)ab,a′b′ → ˜ρaa′,bb′ (A1) 14 Using a singular value decomposition, we can write: ˜ρ = UΣV † (A2) X(A) i a,a′ = Uaa′,i (A3) Y (B) j † b,b′ = V † j,bb′ (A4) σAB i = Σii (A5) The operators are orthonormal under the ...

work page
[40]

Details of simulations and analysis a. Simulation details Since we are interested in the groundstate properties of the models considered, we seek to suppress Monte Carlo moves that correspond to energy changes greater than the finite-size energy gap. It is, therefore, important to choose the value of the inverse temperature β appropri- ately and scale it ...

work page

[1] [1]

P. W. Anderson, More Is Different, Science 177, 393 (1972)

work page 1972

[2] [2]

Carrasquilla and R

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

work page 2017

[3] [3]

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

work page 2017

[4] [4]

W. Hu, R. R. P. Singh, and R. T. Scalettar, Discovering phases, phase transitions, and crossovers through unsu- pervised machine learning: A critical examination, Phys. 13 Rev. E 95, 062122 (2017)

work page 2017

[5] [5]

Zhang, H

P. Zhang, H. Shen, and H. Zhai, Machine learning topo- logical invariants with neural networks, Phys. Rev. Lett. 120, 066401 (2018)

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Krenn, R

M. Krenn, R. Pollice, S. Y. Guo, M. Aldeghi, A. Cervera- Lierta, P. Friederich, G. dos Passos Gomes, F. H¨ ase, A. Jinich, A. Nigam, Z. Yao, and A. Aspuru-Guzik, On scientific understanding with artificial intelligence, Na- ture Reviews Physics 4, 761 (2022)

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Cheong and C

S.-A. Cheong and C. L. Henley, Many-body density ma- trices on a two-dimensional square lattice: Noninteract- ing and strongly interacting spinless fermions, Physical Review B 74, 165121 (2006)

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Cheong and C

S.-A. Cheong and C. L. Henley, Correlation density ma- trix: An unbiased analysis of exact diagonalizations, Physical Review B 79, 212402 (2009)

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[10] [10]

Furukawa, G

S. Furukawa, G. Misguich, and M. Oshikawa, Systematic Derivation of Order Parameters through Reduced Den- sity Matrices, Physical Review Letters96, 047211 (2006)

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S.-J. Gu, W. C. Yu, and H.-Q. Lin, Construct order pa- rameters from the reduced density matrix spectra, An- nals of Physics 336, 118 (2013)

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Sudan, Uncovering the Physics of Frustrated Quantum Magnets Using the Correlation Density Matrix Approach, Ph.D

J. Sudan, Uncovering the Physics of Frustrated Quantum Magnets Using the Correlation Density Matrix Approach, Ph.D. thesis, EPFL, Lausanne (2010)

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M¨ under, A

W. M¨ under, A. Weichselbaum, A. Holzner, J. von Delft, and C. L. Henley, Correlation density matrices for 1- dimensional quantum chains based on the density ma- trix renormalization group, New Journal of Physics 12, 075027 (2010)

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L. Wang, K. S. D. Beach, and A. W. Sandvik, High- precision finite-size scaling analysis of the quantum- critical point of S = 1 / 2 Heisenberg antiferromagnetic bilayers, Physical Review B 73, 014431 (2006)

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Mao, Y.-M

B.-B. Mao, Y.-M. Ding, Z. Wang, S. Hu, and Z. Yan, Sampling reduced density matrix to extract fine levels of entanglement spectrum and restore entanglement Hamil- tonian, Nature Communications 16, 2880 (2025)

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B.-B. Mao, Z. Wang, B.-B. Chen, and Z. Yan, Detecting (emergent) continuous symmetry of criticality via subsys- tem’s entanglement spectrum (2025), arXiv:2506.09889 [quant-ph]

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Harada, Bayesian inference in the scaling analysis of critical phenomena, Physical Review E 84, 056704 (2011)

K. Harada, Bayesian inference in the scaling analysis of critical phenomena, Physical Review E 84, 056704 (2011)

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Di Francesco, P

P. Di Francesco, P. Mathieu, and D. S´ en´ echal,Conformal Field Theory , Graduate Texts in Contemporary Physics (Springer, New York, NY, 1997)

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Dutta, G

A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, and D. Sen, in Quantum Phase Tran- sitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information (Cambridge University Press, 2015)

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The distinction should be clear from the context

The critical exponent related to the order parameter is commonly denoted as β, not to be confused with the inverse temperature. The distinction should be clear from the context

work page

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Kardar, Statistical Physics of Fields (Cambridge Uni- versity Press, 2007)

M. Kardar, Statistical Physics of Fields (Cambridge Uni- versity Press, 2007)

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Sachdev, Quantum Phase Transitions, 2nd ed

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

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Y. Zou, A. Milsted, and G. Vidal, Conformal fields and operator product expansion in critical quantum spin chains, Physical Review Letters 124, 040604 (2020)

work page 2020

[30] [30]

Chang, V

C.-H. Chang, V. Dommes, R. S. Erramilli, A. Hom- rich, P. Kravchuk, A. Liu, M. S. Mitchell, D. Poland, and D. Simmons-Duffin, Bootstrapping the 3d Ising Stress Tensor, Journal of High Energy Physics 2025, 136 (2025)

work page 2025

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Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, Journal of High Energy Physics 2017, 86 (2017)

D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, Journal of High Energy Physics 2017, 86 (2017)

work page 2017

[32] [32]

Huang, L

C.-J. Huang, L. Liu, Y. Jiang, and Y. Deng, Worm- algorithm-type simulation of the quantum transverse- field Ising model, Physical Review B 102, 094101 (2020)

work page 2020

[33] [33]

Campostrini, M

M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Critical exponents and equation of state of the three-dimensional Heisenberg universality class, Physical Review B 65, 144520 (2002)

work page 2002

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den Nijs and K

M. den Nijs and K. Rommelse, Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains, Phys. Rev. B 40, 4709 (1989)

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Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev

X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89, 041004 (2017)

work page 2017

[36] [36]

F. Pauw, F. A. Palm, U. Schollw¨ ock, A. Bohrdt, S. Paeckel, and F. Grusdt, Detecting hidden order in fractional Chern insulators, Phys. Rev. Res. 6, 023180 (2024)

work page 2024

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J. A. Mydosh and P. M. Oppeneer, Colloquium: Hidden order, superconductivity, and magnetism: The unsolved case of URu 2Si2, Rev. Mod. Phys. 83, 1301 (2011)

work page 2011

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Young, Everything You Wanted to Know About Data Analysis and Fitting but Were Afraid to Ask , Springer- Briefs in Physics (Springer International Publishing, Cham, 2015)

P. Young, Everything You Wanted to Know About Data Analysis and Fitting but Were Afraid to Ask , Springer- Briefs in Physics (Springer International Publishing, Cham, 2015). Appendix A: Appendix

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Decomposition of the correlation density matrix We reshape the correlation density matrix to separate the indices on A and B: (ρc AB)i,j = (ρc AB)ab,a′b′ → ˜ρaa′,bb′ (A1) 14 Using a singular value decomposition, we can write: ˜ρ = UΣV † (A2) X(A) i a,a′ = Uaa′,i (A3) Y (B) j † b,b′ = V † j,bb′ (A4) σAB i = Σii (A5) The operators are orthonormal under the ...

work page

[40] [40]

Details of simulations and analysis a. Simulation details Since we are interested in the groundstate properties of the models considered, we seek to suppress Monte Carlo moves that correspond to energy changes greater than the finite-size energy gap. It is, therefore, important to choose the value of the inverse temperature β appropri- ately and scale it ...

work page