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arxiv: 2509.18265 · v1 · submitted 2025-09-22 · ❄️ cond-mat.str-el · cond-mat.stat-mech

Instability of Laughlin FQH liquids into gapless power-law correlated states with continuous exponents in ideal Chern bands: rigorous results from plasma mapping

Saranyo Moitra , Inti Sodemann Villadiego This is my paper

Pith reviewed 2026-05-18 14:01 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.stat-mech
keywords Laughlin wavefunctionfractional quantum HallChern bandsplasma mappingpower-law correlationsinhomogeneous magnetic fieldsdielectric phaseBerezinskii-Kosterlitz-Thouless transition
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The pith

Laughlin wavefunctions in ideal Chern bands with inhomogeneous fields transition to gapless power-law dielectric states at fixed filling.

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

The paper establishes that the ideal Laughlin state at filling 1/m undergoes a phase transition as the magnetic field becomes more spatially inhomogeneous. Even with fixed particle density, the state loses its topological gap and enters a dielectric phase whose density correlations decay as a power law. The governing exponent varies continuously with the degree of field inhomogeneity, taking values between 4 near the transition and 2m in the limit of highly localized fields. This result follows from an exact mapping of the wavefunction to a classical one-component plasma in a nonuniform neutralizing background, combined with known results on plasma phases.

Core claim

The ideal Laughlin wave-function undergoes a phase transition from its well-known fully gapped topologically ordered plasma state into a power-law correlated dielectric state even for the fixed filling of 1/3, as the magnetic field becomes increasingly more inhomogeneous. This dielectric state is gapless even though it does not spontaneously break translational symmetry. For a fixed filling ν=1/m, the exponent governing density correlations in this state changes continuously as a function of the degree of spatial inhomogeneity of the magnetic field, and can range from 4 near a Berezinskii-Kosterlitz-Thouless transition to the plasma state, up to 2m in the limit of fields generated by point-s

What carries the argument

Exact mapping of the Laughlin wavefunction in generalized zero Landau levels with spatially dependent magnetic fields onto a classical one-component plasma in a nonuniform neutralizing background.

If this is right

  • The resulting dielectric phase remains gapless while preserving translational symmetry.
  • Density correlations obey a tunable power-law exponent set by the magnetic-field inhomogeneity.
  • The transition out of the gapped Laughlin plasma occurs via a Berezinskii-Kosterlitz-Thouless-like mechanism.
  • In the limit of point-solenoid fields the exponent saturates at 2m.

Where Pith is reading between the lines

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

  • Similar instabilities may appear in fractional Chern insulator candidates realized in moiré heterostructures whenever the effective magnetic field is modulated by the lattice.
  • The continuous exponent could be probed by measuring the structure factor in momentum-resolved spectroscopy.
  • The dielectric phase may host new transport signatures distinct from both gapped fractional quantum Hall liquids and conventional metals.

Load-bearing premise

The plasma mapping remains exact without extra quantum corrections arising from the geometry of the Chern band.

What would settle it

Direct numerical computation of the two-point density correlation function in a model Chern band whose magnetic field strength varies continuously from uniform to highly localized, checking whether the decay exponent changes continuously between 4 and 2m.

Figures

Figures reproduced from arXiv: 2509.18265 by Inti Sodemann Villadiego, Saranyo Moitra.

Figure 1
Figure 1. Figure 1: (a) Depiction of effective magnetic field in moiré Chern bands with one flux per unit cell, and (b) in ICB of solenoids with m-fluxes per unit cell. (c) Phase diagram of the Coulomb gas with neutralizing background of the ions from panel (b), adapted from Ref. [29]. The BKT transition is expected to occur for small radius and at T ϵ = 1/4. (see [36] for details), as follows: ϵ −1 (r, r ′ ) = δ(r−r ′ )+2m Z… view at source ↗
read the original abstract

We investigate the fate of Laughlin's wave-function in ideal Chern bands which can be mapped to generalized zero Landau levels in spatially dependent magnetic fields. By exploiting its exact mapping onto a classical Coulomb gas and leveraging previous results of one-component plasmas in nonuniform neutralizing backgrounds, we demonstrate that the ideal Laughlin wave-function undergoes a phase transition from its well-known fully gapped topologically ordered plasma state into a power-law correlated dielectric state even for the fixed filling of $1/3$, as the magnetic field becomes increasingly more inhomogeneous. This dielectric state is gapless even though it does not spontaneously break translational symmetry. Remarkably, for a fixed filling $\nu=1/m$, the exponent governing density correlations in this state changes continuously as a function of the degree of spatial inhomogeneity of the magnetic field, and can range from $4$ near a Berezinskii-Kosterlitz-Thouless transition to the plasma state, up to $2 m$ in the limit of fields generated by point solenoids.

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 claims that the Laughlin wavefunction at filling 1/m in ideal Chern bands (equivalently, generalized zero Landau levels with spatially varying magnetic fields) maps exactly onto a classical one-component plasma in a nonuniform neutralizing background. Leveraging prior mathematical results on such plasmas, it concludes that increasing magnetic-field inhomogeneity drives a BKT-type transition out of the gapped, topologically ordered plasma phase into a gapless dielectric phase whose density-density correlations decay as a power law with an exponent that varies continuously between 4 (near the transition) and 2m (point-solenoid limit), all at fixed filling and without spontaneous translational symmetry breaking.

Significance. If the exactness of the plasma mapping without residual Chern-band or position-dependent magnetic-length corrections can be established, the result would be notable: it supplies a concrete, tunable example of a gapless, power-law correlated state emerging from a Laughlin wavefunction purely through inhomogeneity, with continuously variable exponents and no symmetry breaking. The use of previously established plasma theorems and the absence of free parameters in the exponent tuning are strengths that would make the work of interest to the fractional quantum Hall and Chern-band communities.

major comments (2)
  1. [§2] §2 (Plasma mapping): the central claim that the norm of the Laughlin wavefunction is exactly the Boltzmann weight of a classical 2D one-component plasma in a nonuniform background, with no additive quantum corrections arising from the Chern-band geometry, Berry curvature, or spatially varying magnetic length, is load-bearing for the entire phase-transition and exponent-tuning argument. An explicit derivation showing that all such corrections vanish identically (or are irrelevant) is required; without it the invocation of prior nonuniform-plasma results does not yet rigorously imply the stated BKT transition or continuous exponent range.
  2. [§4] §4 (Dielectric phase and exponent): the assertion that the dielectric phase remains gapless while preserving translational symmetry, and that the density-correlation exponent interpolates continuously from 4 to 2m, rests on the applicability of the classical plasma analysis. A concrete check (e.g., via the pair-correlation function or the effective potential) that no geometry-induced relevant operators appear at the transition is needed to confirm the exponent is not pinned or discretized by band effects.
minor comments (2)
  1. [Abstract] The abstract and introduction should clarify whether the mapping is claimed to be exact for any ideal Chern band or only for a specific class of generalized Landau levels; the current wording leaves this ambiguous.
  2. [§2] Notation for the local magnetic length and the neutralizing background charge density should be introduced once and used consistently; occasional redefinitions in later sections hinder readability.

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 the two major points below, clarifying the exactness of the plasma mapping and the implications for the dielectric phase. We will incorporate expanded derivations and explicit checks in the revised version.

read point-by-point responses

  1. Referee: §2 (Plasma mapping): the central claim that the norm of the Laughlin wavefunction is exactly the Boltzmann weight of a classical 2D one-component plasma in a nonuniform background, with no additive quantum corrections arising from the Chern-band geometry, Berry curvature, or spatially varying magnetic length, is load-bearing for the entire phase-transition and exponent-tuning argument. An explicit derivation showing that all such corrections vanish identically (or are irrelevant) is required; without it the invocation of prior nonuniform-plasma results does not yet rigorously imply the stated BKT transition or continuous exponent range.

    Authors: We appreciate the referee drawing attention to this foundational step. In the ideal Chern band, the single-particle states are holomorphic with respect to the local complex structure defined by the position-dependent vector potential, so that the Laughlin wavefunction takes the form Ψ = ∏_{i<j} (z_i − z_j)^m exp(−½ ∫^r B(r′) |z − r′|^2 d²r′) (up to normalization). Its norm squared is then obtained by direct integration over the coordinates, yielding exactly the Boltzmann weight of the one-component plasma with background density ρ_b(r) ∝ B(r). The uniform Berry curvature of the ideal band enters the definition of the holomorphic coordinates but cancels identically in |Ψ|²; likewise, the spatially varying magnetic length is fully absorbed into the Gaussian factor and produces no additional position-dependent operators in the probability measure. This reduction is shown in Section 2 by explicit comparison with the uniform Landau-level case. We will add an appendix containing the full intermediate steps of the norm calculation to render every cancellation manifest. revision: yes

  2. Referee: §4 (Dielectric phase and exponent): the assertion that the dielectric phase remains gapless while preserving translational symmetry, and that the density-correlation exponent interpolates continuously from 4 to 2m, rests on the applicability of the classical plasma analysis. A concrete check (e.g., via the pair-correlation function or the effective potential) that no geometry-induced relevant operators appear at the transition is needed to confirm the exponent is not pinned or discretized by band effects.

    Authors: Because the mapping of the quantum norm to the classical plasma measure is exact, every correlation function of the many-body state—including the density-density correlator—is identical to that of the classical plasma. The mathematical results on nonuniform one-component plasmas (cited in the manuscript) establish that the dielectric phase is gapless, translationally invariant, and exhibits power-law density correlations whose exponent varies continuously with background inhomogeneity, ranging from 4 at the BKT point to 2m in the point-solenoid limit. No additional relevant operators arise from the band geometry or varying magnetic length, as these are already encoded in the nonuniform background charge and do not generate new long-wavelength terms beyond those of the classical model. In the revised manuscript we will include an explicit expression for the pair-correlation function g(r) obtained from the plasma analogy, together with a short discussion confirming the absence of geometry-induced pinning of the exponent. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation applies independent plasma results after exact mapping

full rationale

The paper claims an exact mapping of the Laughlin wavefunction in ideal Chern bands (generalized zero Landau levels with position-dependent B) onto the energy of a classical 2D one-component plasma in a nonuniform neutralizing background. It then directly invokes prior independent mathematical results on the phases of such plasmas to establish the BKT-type transition to a gapless dielectric state and the continuous tuning of the density-correlation exponent (from 4 to 2m) with inhomogeneity strength. No quoted step reduces a claimed prediction or exponent to a parameter fitted inside the paper, nor does any load-bearing premise collapse to a self-citation whose validity is internal to this work. The central results are therefore obtained by applying external plasma benchmarks to the mapped system rather than by construction or self-referential fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the exact plasma mapping for generalized zero Landau levels with spatially dependent fields and on the direct applicability of prior mathematical results for one-component plasmas in nonuniform neutralizing backgrounds. No free parameters or new postulated entities are introduced.

axioms (2)
  • domain assumption The Laughlin wave-function in ideal Chern bands with spatially dependent magnetic fields admits an exact mapping onto a classical one-component Coulomb plasma in a nonuniform neutralizing background.
    This mapping is the foundational step that allows use of classical plasma results.
  • domain assumption Established mathematical results on the phase diagram of one-component plasmas in nonuniform backgrounds apply without modification to the quantum Hall setting.
    The paper explicitly leverages these prior results to identify the transition and the range of exponents.

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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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    the probability density of these generalized Laughlin states is identical to the finite temperature probability density of a gas of classical particles repelling via 2D Coulomb forces... |Ψ|^2 = exp[-β U] with U = -∑_{i<j} ln|ri-rj| + ∑ ϕ(ri)/m

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
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unclear
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Reference graph

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