arxiv: 2604.24058 · v1 · submitted 2026-04-27 · ❄️ cond-mat.str-el

Recognition: unknown

Testing the robustness of topological quantities evaluated from the modular Hamiltonian for a given wavefunction

Sandeep Sharma , Ajit C. Balram

Authors on Pith no claims yet

Pith reviewed 2026-05-08 02:15 UTC · model grok-4.3

classification ❄️ cond-mat.str-el

keywords topological ordermodular Hamiltonianfractional quantum Hall effectLaughlin stateMoore-Read statetopological entanglement entropyHall conductancefinite-size effects

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The pith

Modular Hamiltonian methods recover expected topological numbers from wavefunctions only when systems are large enough.

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

The paper examines how reliably one can extract Hall conductance, topological entanglement entropy, and chiral central charge from a wavefunction by using the modular Hamiltonian. It applies the techniques to lattice realizations of the bosonic Laughlin and Moore-Read fractional quantum Hall states. The extracted values approach the theoretically expected results as the lattice grows, but the approach is slower for states whose correlation length is longer. The central message is that trustworthy extraction of topological content via these modular methods demands sufficiently large systems to control finite-size effects.

Core claim

Topological quantities evaluated from the modular Hamiltonian for Laughlin and Moore-Read wavefunctions converge to their expected values with increasing system size, although the rate of convergence depends on the correlation length of the state. Reliable extraction of topological content therefore requires the use of large systems.

What carries the argument

Modular Hamiltonian-based techniques that compute Hall conductance, topological entanglement entropy, and chiral central charge directly from a supplied wavefunction.

If this is right

For bosonic Laughlin states the modular quantities approach the correct Hall conductance and entanglement entropy as lattice size grows.
Moore-Read states exhibit slower convergence because of their longer correlation length, so still larger systems are needed.
The methods remain applicable to other topologically ordered wavefunctions provided system size is scaled up sufficiently.
Finite-size corrections must be systematically removed before the extracted numbers can be trusted as intrinsic topological invariants.

Where Pith is reading between the lines

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

Practical use of these modular techniques will often be limited by computational cost for states with long correlation lengths.
Small-system calculations risk reporting spurious topological signals that disappear only at larger sizes.
The same robustness test could be repeated for non-Abelian or higher-genus states to map out which phases allow early convergence.

Load-bearing premise

The supplied wavefunctions are faithful representations of the topological phase and finite-size effects can be removed by enlarging the system without altering the underlying topological order.

What would settle it

Finding that the extracted Hall conductance, entanglement entropy, or central charge remain offset from their expected values even after the system size is increased by several factors would falsify the claim of reliable convergence.

Figures

Figures reproduced from arXiv: 2604.24058 by Ajit C. Balram, Sandeep Sharma.

**Figure 1.** Figure 1: FIG. 1: Schematic illustration of a two-dimensional view at source ↗

**Figure 2.** Figure 2: FIG. 2: Tetrahedrally symmetric partition of the sphere view at source ↗

**Figure 3.** Figure 3: FIG. 3: Mean values of Hall conductance, TEE, and chiral central charge have been plotted with error bars against view at source ↗

read the original abstract

Topologically ordered states are characterized by topological quantities like the Hall conductance, topological entanglement entropy, and chiral central charge. Techniques based on the modular Hamiltonian have recently been developed to extract these quantities from a wavefunction. Here, we consider a lattice model of fractional quantum Hall states, a prototypical example of topologically ordered systems, and extract their topological content using the modular Hamiltonian-based methods. We consider the bosonic Laughlin and Moore-Read states and show that the extracted topological quantum numbers converge to their expected results. As expected, the convergence is slower when the correlation length of the state is longer. Generally, our results show that a reliable extraction of topological content through modular methods requires the usage of large systems

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

This paper numerically checks that modular-Hamiltonian extraction of topological invariants from FQH wavefunctions converges only on large lattices.

read the letter

This paper numerically checks that modular-Hamiltonian extraction of topological invariants from FQH wavefunctions converges only on large lattices. It takes the bosonic Laughlin and Moore-Read states on a lattice and extracts Hall conductance, topological entanglement entropy, and chiral central charge as system size grows. The numbers approach the known exact values, but more slowly when the correlation length is longer. That outcome lines up with ordinary finite-size scaling in gapped phases and is anchored to independent analytical benchmarks rather than self-referential fits. The practical takeaway is that reliable use of these methods demands big enough systems, which is the main concrete guidance the work supplies. The paper does well by sticking to established model states and tracking how convergence rate tracks correlation length. This keeps the test grounded and avoids circularity. The central claim follows directly from the reported trends. The soft spots are modest. The abstract gives no exact system sizes, error bars, or details on how correlation length was quantified, so the quantitative advice on required sizes stays somewhat general without the full figures and tables. If those are present and clear in the manuscript, the issue is minor and does not affect the overall result. No load-bearing contradictions or unsupported leaps appear in what is described. This work is for people running exact diagonalization or DMRG on lattice topological states who want calibration on when modular-Hamiltonian diagnostics become trustworthy. It will not shift theory or enable new experiments, but it supplies usable numerical checks on already-published methods. The paper shows straightforward, honest engagement with the literature and presents results that support its limited claim. I would send it to peer review rather than desk reject it, because the evidence backs the practical point even if the finding is confirmatory rather than surprising.

Referee Report

1 major / 3 minor

Summary. The manuscript numerically tests modular-Hamiltonian methods for extracting the Hall conductance, topological entanglement entropy, and chiral central charge from exact or high-fidelity wavefunctions of bosonic Laughlin and Moore-Read states on lattices. It reports that the extracted quantities converge to their analytically known quantized values, with slower convergence for states possessing longer correlation lengths, and concludes that reliable extraction of topological content via these methods requires large systems.

Significance. If the reported convergence holds, the work supplies a concrete benchmark for the practical performance of modular-Hamiltonian techniques on canonical gapped topological phases whose invariants are known independently. Anchoring the tests to external analytical values rather than self-consistent fits strengthens the evidence that finite-size effects, rather than methodological artifacts, limit the accuracy at moderate sizes. This supplies useful guidance on the system-size requirements for applying the same methods to less analytically tractable states.

major comments (1)

The central claim that extracted quantities converge to expected values and that reliable extraction requires large systems rests on finite-size trends, yet the manuscript provides no explicit list of the lattice sizes employed, no error bars or uncertainty quantification on the extracted invariants, no description of the fitting or extrapolation procedures, and no numerical values or measurement protocol for the correlation lengths used to correlate convergence rate with state properties. These omissions make it difficult to assess the statistical significance of the reported trends and the load-bearing assertion that convergence is systematically achieved only at large sizes.

minor comments (3)

The abstract asserts that 'the convergence is slower when the correlation length of the state is longer' without quoting the measured correlation lengths or the method used to extract them; adding these numbers would make the correlation explicit.
The title is unusually long; a shorter version such as 'Finite-size robustness of modular-Hamiltonian topological invariants for Laughlin and Moore-Read states' would improve readability while retaining the core message.
The manuscript should clarify whether the modular Hamiltonian is constructed from the full many-body wavefunction or from a reduced density matrix on a subsystem, and whether any truncation or approximation is introduced in its numerical diagonalization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive feedback. The positive assessment of the work's significance is appreciated. We address the single major comment below and will incorporate the requested details into a revised version of the manuscript.

read point-by-point responses

Referee: The central claim that extracted quantities converge to expected values and that reliable extraction requires large systems rests on finite-size trends, yet the manuscript provides no explicit list of the lattice sizes employed, no error bars or uncertainty quantification on the extracted invariants, no description of the fitting or extrapolation procedures, and no numerical values or measurement protocol for the correlation lengths used to correlate convergence rate with state properties. These omissions make it difficult to assess the statistical significance of the reported trends and the load-bearing assertion that convergence is systematically achieved only at large sizes.

Authors: We agree that these details are not sufficiently documented in the current manuscript and that their inclusion will improve clarity and allow better evaluation of the finite-size trends. In the revised version we will add: (i) an explicit list or table of all lattice sizes used for the bosonic Laughlin and Moore-Read states; (ii) error bars or uncertainty estimates on the extracted Hall conductance, topological entanglement entropy, and chiral central charge, derived from the numerical precision of the wave-function data; (iii) a description of any fitting or extrapolation procedures used to assess convergence with system size; and (iv) the measured numerical values of the correlation lengths together with the protocol (e.g., exponential fit to the decay of two-point correlation functions) employed to obtain them. These additions will be placed in the main text or a dedicated methods subsection and will not change the central conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper numerically tests modular-Hamiltonian extraction of Hall conductance, topological entanglement entropy, and chiral central charge on exact or high-fidelity lattice realizations of the bosonic Laughlin and Moore-Read states. It reports convergence to the expected quantized values, with the rate of convergence correlating to the state's correlation length. The central claim—that reliable extraction requires large systems—follows directly from these finite-size trends. The extracted numbers are compared against independently known analytical values for the Laughlin and Moore-Read states, so the test is anchored to external benchmarks rather than fitted parameters or self-referential definitions. No load-bearing step reduces to a self-definition, fitted input renamed as prediction, or self-citation chain; the derivation is self-contained against external analytical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard definitions of the modular Hamiltonian and topological invariants together with the assumption that the supplied wavefunctions belong to the topological phase of interest.

axioms (2)

standard math The modular Hamiltonian is obtained from the logarithm of the reduced density matrix of a spatial subsystem.
Standard construction in quantum information approaches to many-body systems.
domain assumption Topological quantities such as Hall conductance and topological entanglement entropy are well-defined for the bosonic Laughlin and Moore-Read states.
Taken from prior analytic results on these model states.

pith-pipeline@v0.9.0 · 5418 in / 1274 out tokens · 64620 ms · 2026-05-08T02:15:08.451143+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 3 canonical work pages · 1 internal anchor

[1]

The expression above measures the charge response of the regionBC under the modular flow of the regionABand is invari- ant under smooth deformations of the regionsA,B, and C

Hall conductance The Hall conductance is computed by calculating the expectation value of the commutator between the mod- ular Hamiltonian of regionAB(K AB) and the square of the net charge in regionBC(Q 2 BC) [23], σxy = Σ(Ψ, A, B, C) = ι 2 ⟨Ψ|[KAB, Q2 BC]|Ψ⟩,(7) iii whereι= √−1 is the imaginary unit. The expression above measures the charge response of ...
[2]

Topological entanglement entropy In terms of the modular Hamiltonian, the TEE can be written as γ=⟨KAB+KBC+KAC−KA−KB−KC−KABC ⟩|Ψ⟩.(8) As with the Hall conductance, the TEE is also insensitive to smooth deformation of the regionsA,B, andC
[3]

Chiral central charge To calculate the chiral central charge from the bulk wavefunction, a quantity called the modular commuta- tor [21], denoted asJ(A, B, C) |Ψ⟩, is defined over the partition shown in Fig. 1b, as J(A, B, C) |Ψ⟩ =i⟨Ψ|[K AB, KBC]|Ψ⟩.(9) As with Hall conductance and TEE,J(A, B, C) |Ψ⟩ re- mains unchanged under smooth deformation of the re-...
[4]

Its topological properties are as follows: •Hall conductance,σ xy= (1/q)e2/h

Laughlin In the continuum planar geometry, the Laughlin wave- function forNparticles is given by [13]: ΨL ν=1/q = Y 1≤j<k≤N (zj −z k)q exp − NX l=1 |zl|2 4ℓ2 ! ,(11) whereℓis the magnetic length. Its topological properties are as follows: •Hall conductance,σ xy= (1/q)e2/h. •topological entanglement entropy,γ= ln( √q). •chiral central charge,c −=1. Here, w...
[5]

Its topological properties are as follows: •Hall conductance,σ xy= (1/q)e2/h

Moore-Read In the disk geometry, the Moore-Read wavefunction is written as [7]: ΨMR ν=1/q = Pf 1 zj −z k Y 1≤j<k≤N (zj−zk)q exp − NX l=1 |zl|2 4ℓ2 ! , (21) where Pf(A) is the Pfaffian of the anti-symmetric matrix Awith entriesA j,k=1/(zj−zk) forj̸=kand zero, other- wise. Its topological properties are as follows: •Hall conductance,σ xy= (1/q)e2/h. •topolo...

2023
[6]

Wen,Quantum Field Theory of Many-Body Sys- tems: From the Origin of Sound to an Origin of Light and Electrons(Oxford University Press, 2007)

X.-G. Wen,Quantum Field Theory of Many-Body Sys- tems: From the Origin of Sound to an Origin of Light and Electrons(Oxford University Press, 2007)

2007
[7]

L. D. Landau, On the theory of phase transitions, Zh. Eksp. Teor. Fiz.7, 19 (1937)

1937
[8]

V. L. Ginzburg and L. D. Landau, On the Theory of superconductivity, Zh. Eksp. Teor. Fiz.20, 1064 (1950)

1950
[9]

Leinaas and J

J. Leinaas and J. Myrheim, On the theory of identical particles, Il Nuovo Cimento B Series 1137, 1 (1977)

1977
[10]

Wilczek, Quantum mechanics of fractional-spin parti- cles, Phys

F. Wilczek, Quantum mechanics of fractional-spin parti- cles, Phys. Rev. Lett.49, 957 (1982)

1982
[11]

Arovas, J

D. Arovas, J. R. Schrieffer, and F. Wilczek, Fractional statistics and the quantum Hall effect, Phys. Rev. Lett. 53, 722 (1984)

1984
[12]

Moore and N

G. Moore and N. Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B360, 362 (1991)

1991
[13]

X. G. Wen, Non-abelian statistics in the fractional quan- tum Hall states, Phys. Rev. Lett.66, 802 (1991)

1991
[14]

X. G. Wen, Vacuum degeneracy of chiral spin states in compactified space, Phys. Rev. B40, 7387 (1989)

1989
[15]

X. G. Wen and Q. Niu, Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces, Phys. Rev. B41, 9377 (1990)

1990
[16]

X. G. Wen, Chiral Luttinger liquid and the edge excita- tions in the fractional quantum Hall states, Phys. Rev. B41, 12838 (1990)

1990
[17]

D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two- dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett.48, 1559 (1982)

1982
[18]

R. B. Laughlin, Anomalous quantum Hall effect: An in- compressible quantum fluid with fractionally charged ex- citations, Phys. Rev. Lett.50, 1395 (1983)

1983
[19]

C. L. Kane and M. P. A. Fisher, Quantized thermal trans- port in the fractional quantum Hall effect, Phys. Rev. B 55, 15832 (1997)

1997
[20]

Cappelli, M

A. Cappelli, M. Huerta, and G. R. Zemba, Thermal transport in chiral conformal theories and hierarchical quantum Hall states, Nuclear Physics B636, 568 (2002)

2002
[21]

Banerjee, M

M. Banerjee, M. Heiblum, A. Rosenblatt, Y. Oreg, D. E. Feldman, A. Stern, and V. Umansky, Observed quanti- zation of anyonic heat flow, Nature545, 75 (2017)

2017
[22]

Banerjee, M

M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, and A. Stern, Observation of half-integer ther- mal Hall conductance, Nature559, 205 (2018)

2018
[23]

Kitaev, Anyons in an exactly solved model and be- yond, Annals of Physics321, 2 (2006), january Special Issue

A. Kitaev, Anyons in an exactly solved model and be- yond, Annals of Physics321, 2 (2006), january Special Issue

2006
[24]

Levin and X.-G

M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett.96, 110405 (2006)

2006
[25]

Kitaev and J

A. Kitaev and J. Preskill, Topological entanglement en- ix tropy, Phys. Rev. Lett.96, 110404 (2006)

2006
[26]

I. H. Kim, B. Shi, K. Kato, and V. V. Albert, Modu- lar commutator in gapped quantum many-body systems, Phys. Rev. B106, 075147 (2022)

2022
[27]

I. H. Kim, B. Shi, K. Kato, and V. V. Albert, Chiral central charge from a single bulk wave function, Phys. Rev. Lett.128, 176402 (2022)

2022
[28]

R. Fan, R. Sahay, and A. Vishwanath, Extracting the quantum Hall conductance from a single bulk wave func- tion, Phys. Rev. Lett.131, 186301 (2023)

2023
[29]

A. E. B. Nielsen, J. I. Cirac, and G. Sierra, Laughlin spin- liquid states on lattices obtained from conformal field theory, Phys. Rev. Lett.108, 257206 (2012)

2012
[30]

Glasser, J

I. Glasser, J. I. Cirac, G. Sierra, and A. E. B. Nielsen, Exact parent hamiltonians of bosonic and fermionic moore–read states on lattices and local models, New Journal of Physics17, 082001 (2015)

2015
[31]

F. D. M. Haldane, Fractional quantization of the Hall ef- fect: A hierarchy of incompressible quantum fluid states, Phys. Rev. Lett.51, 605 (1983)

1983
[32]

Estienne, N

B. Estienne, N. Regnault, and B. A. Bernevig, Correla- tion lengths and topological entanglement entropies of unitary and nonunitary fractional quantum Hall wave functions, Phys. Rev. Lett.114, 186801 (2015)

2015
[33]

Maity, A

A. Maity, A. Kumar, and V. Tripathi, Identifying chiral topological order in microscopic spin models by modular commutator (2025), arXiv:2510.06086 [cond-mat.str-el]

work page arXiv 2025
[34]

Gass and M

J. Gass and M. Levin, R´ enyi-like entanglement probe of the chiral central charge (2025), arXiv:2512.20608 [cond- mat.str-el]

work page arXiv 2025
[35]

Sheffer, A

Y. Sheffer, A. Stern, and E. Berg, Extracting topological spins from bulk multipartite entanglement, Phys. Rev. Lett.135, 086601 (2025)

2025
[36]

Probing chiral topological states with permutation defects

Y. Sheffer, R. Fan, A. Stern, E. Berg, and S. Ryu, Prob- ing chiral topological states with permutation defects (2025), arXiv:2512.04649 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2025