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arxiv: 2606.00214 · v1 · pith:CR3PJZXAnew · submitted 2026-05-29 · ❄️ cond-mat.str-el · hep-th· quant-ph

Extracting central charge from ground-state overlaps of spatially deformed Hamiltonians

Chen Bai , Xinyu Sun , Liang-Hong Mo , Hong-Hao Tu This is my paper

Pith reviewed 2026-06-28 20:40 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-thquant-ph
keywords central chargeconformal anomalyground-state overlapq-Möbius deformationcritical quantum chainsChern insulator edge modesentanglement spectrum
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The pith

The central charge of a (1+1)D CFT is encoded in the overlap between undeformed and q-Möbius deformed ground states.

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

The paper shows that in (1+1)-dimensional conformal field theories an exact formula relates the overlap of the undeformed ground state with the ground state of a spatially deformed Hamiltonian to the central charge alone. The deformation considered is the q-Möbius map, and the logarithm of the overlap is proportional to c times a function of q that can be computed independently of the model. This relation is turned into a lattice estimator that uses only ground-state wavefunction overlaps, and the estimator is tested on critical quantum spin chains and on the gapless edge of a two-dimensional Chern insulator. The same deformed states are shown to preserve universal features in their entanglement spectra and entropies.

Core claim

We show that the conformal anomaly of a (1+1)-dimensional conformal field theory can be extracted directly from a ground-state wave-function overlap associated with a spatial conformal deformation. Focusing on the q-Möbius deformation, we derive an exact overlap formula between the deformed and undeformed ground states, whose exponent depends only on the central charge. Motivated by this result, we construct a lattice estimator based solely on ground-state overlaps and apply it to representative critical quantum chains and the gapless edge modes of a two-dimensional Chern insulator.

What carries the argument

The q-Möbius deformation, a one-parameter family of spatial conformal maps on the circle that produces a deformed Hamiltonian whose ground-state overlap with the original state has an exponent fixed solely by the central charge.

If this is right

  • Central charge can be obtained from ground-state wavefunctions alone without computing energies or correlation functions.
  • The same overlap construction applies to gapless edge modes of two-dimensional topological insulators.
  • Deformed ground states retain universal geometric structures in their entanglement spectra and entanglement entropies.
  • The lattice estimator is simple to implement and numerically robust for a range of critical spin chains.

Where Pith is reading between the lines

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

  • The method may allow extraction of central charge in models whose continuum limit is not known in advance.
  • Other families of conformal deformations could be examined to see whether they also isolate the central charge in an overlap formula.
  • The preservation of universal entanglement features under deformation hints that wavefunction overlaps may capture additional CFT data beyond the central charge.

Load-bearing premise

The physical system must be described by a continuum (1+1)D conformal field theory whose lattice regularization preserves the conformal properties needed for the exact overlap formula to hold.

What would settle it

For the critical transverse-field Ising chain (known c=1/2), compute the overlap for a chosen q-Möbius deformation and check whether the exponent of the overlap matches the value predicted by the formula; a statistically significant mismatch would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.00214 by Chen Bai, Hong-Hao Tu, Liang-Hong Mo, Xinyu Sun.

Figure 1
Figure 1. Figure 1: FIG. 1. Finite-size extrapolation of the effective central [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Entanglement spectrum and entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Edge-state central charge and entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We show that the conformal anomaly of a $(1+1)$-dimensional conformal field theory can be extracted directly from a ground-state wave-function overlap associated with a spatial conformal deformation. Focusing on the $q$-M\"obius deformation, we derive an exact overlap formula between the deformed and undeformed ground states, whose exponent depends only on the central charge. Motivated by this result, we construct a lattice estimator based solely on ground-state overlaps and apply it to representative critical quantum chains and the gapless edge modes of a two-dimensional Chern insulator. Numerical results demonstrate that the resulting overlaps provide a simple and robust probe of the central charge in microscopic models. We further demonstrate that the deformed ground states retain universal geometric structures in their entanglement spectra and entanglement entropies. These results provide a simple wave-function-based route to probing conformal data in critical systems and topological edge modes.

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 / 1 minor

Summary. The manuscript claims that the conformal anomaly (central charge c) of (1+1)D CFTs can be extracted from ground-state wave-function overlaps under a q-Möbius spatial deformation. An exact overlap formula is derived in the continuum whose exponent depends only on c; a lattice estimator is then constructed from this relation and applied to critical quantum chains and the gapless edge modes of a 2D Chern insulator. Numerical results are presented to demonstrate robustness, and the deformed states are shown to retain universal geometric features in their entanglement spectra and entropies.

Significance. If the continuum derivation is rigorous and lattice corrections are demonstrably controlled, the approach supplies a direct wave-function probe of c that requires only ground-state overlaps. This would be useful for tensor-network or DMRG studies of critical chains and topological edge modes, where overlaps are often accessible. The numerical tests on representative models provide practical evidence of utility.

major comments (2)
  1. [Derivation of the overlap formula (continuum CFT section)] The abstract and introduction assert an exact overlap formula whose logarithm is strictly proportional to c after a CFT derivation, yet the provided text contains no derivation steps, intermediate Ward-identity manipulations, or explicit verification that no other CFT data (e.g., operator dimensions) enter the exponent. This is load-bearing for the central claim that the estimator depends only on c.
  2. [Lattice estimator and numerical results] The lattice implementation of the q-Möbius deformation on finite chains and edge modes is presented without quantitative bounds on corrections from irrelevant operators or broken conformal Ward identities at finite spacing. The numerical success on representative models does not yet rule out non-universal contributions to the extracted exponent; an explicit finite-size or continuum-limit analysis is required to confirm the claimed parameter-free dependence on c.
minor comments (1)
  1. The abstract refers to 'representative critical quantum chains' without naming the specific Hamiltonians, system sizes, or bond dimensions used; these details are needed for reproducibility even if the central claim holds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting two key areas where the manuscript can be strengthened. We agree that both the continuum derivation and the lattice analysis require additional explicit detail to fully support the central claims. Below we respond point by point and outline the revisions we will make.

read point-by-point responses

  1. Referee: The abstract and introduction assert an exact overlap formula whose logarithm is strictly proportional to c after a CFT derivation, yet the provided text contains no derivation steps, intermediate Ward-identity manipulations, or explicit verification that no other CFT data (e.g., operator dimensions) enter the exponent. This is load-bearing for the central claim that the estimator depends only on c.

    Authors: We acknowledge that the submitted manuscript presents the final overlap formula without the intermediate steps of the derivation. In the revised version we will expand the continuum CFT section to include the full derivation: starting from the q-Möbius transformation, applying the appropriate Ward identities for the stress-tensor insertions, and explicitly showing that all contributions involving operator dimensions cancel, leaving an exponent that depends only on the central charge c. This will make the load-bearing step transparent and verifiable. revision: yes

  2. Referee: The lattice implementation of the q-Möbius deformation on finite chains and edge modes is presented without quantitative bounds on corrections from irrelevant operators or broken conformal Ward identities at finite spacing. The numerical success on representative models does not yet rule out non-universal contributions to the extracted exponent; an explicit finite-size or continuum-limit analysis is required to confirm the claimed parameter-free dependence on c.

    Authors: We agree that the current numerical section lacks a systematic finite-size or continuum-limit study. In the revision we will add (i) data for multiple system sizes with explicit extrapolation of the extracted exponent to the thermodynamic limit, (ii) a quantitative estimate of the leading irrelevant-operator corrections based on the known scaling dimensions of the models studied, and (iii) a direct comparison of the lattice overlaps against the continuum formula at the largest accessible sizes. These additions will provide the requested bounds and confirm that non-universal contributions are under control. revision: yes

Circularity Check

0 steps flagged

No circularity: CFT derivation supplies independent input; lattice tests are external validation

full rationale

The central result is an exact overlap formula derived from continuum (1+1)D CFT under the q-Möbius deformation, with the exponent fixed solely by the conformal anomaly (standard Ward identities). This supplies the functional dependence on c as an external theoretical input rather than a fit to the lattice data. The subsequent lattice estimator applies this pre-derived relation to numerical ground-state overlaps on concrete models (critical chains, Chern edge modes); the numerical agreement tests the applicability of the continuum formula but does not redefine or force the exponent. No self-citation chain, self-definitional loop, or fitted-input-renamed-as-prediction appears in the derivation. The continuum-to-lattice step is an assumption about irrelevant-operator corrections, not a circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the continuum CFT derivation for the overlap formula and on the assumption that lattice models realize the same universal overlap behavior in the scaling limit.

axioms (1)
  • domain assumption The system is described by a (1+1)D conformal field theory in the continuum limit
    The exact overlap formula is derived using CFT techniques that require conformal invariance.

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Reference graph

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