Many-Body Gap-Labeling in Quantum Hall Phases

• The many-body gap-labeling theorem defines conditions under which non-interacting Hofstadter gaps yield quantized Hall conductivity labels in interacting systems.
• It employs quasi-adiabatic continuation and denominator-independent magnetic estimates to ensure a uniformly gapped phase in the three-parameter space.
• The theorem bridges the non-interacting colored Hofstadter butterfly with interacting quantum Hall phases, covering both rational and irrational flux cases.

Searching arXiv for the specified paper and closely related background papers on many-body Hall conductivity and quasi-adiabatic continuation. The many-body gap-labeling theorem identifies a regime in which the integer labels of open gaps in a Hofstadter-like one-particle spectrum persist as quantized Hall-conductivity labels for weakly interacting lattice fermion systems. In the formulation proved in "The Colored Hofstadter Butterfly as a Many-Body Quantum Hall Phase Diagram" (Marcelli et al., 24 Jun 2026), one starts from a spectral gap of a short-range magnetic one-particle Hamiltonian at arbitrary magnetic flux $b$ and constructs an open region in the three-dimensional parameter space $(b,\mu,\lambda)$ of magnetic field, chemical potential, and interaction strength on which the infinite-volume interacting system has locally unique gapped ground states. On connected uniformly gapped regions meeting the non-interacting plane $\lambda=0$, the Hall conductivity entering the macroscopic Ohm's law is constant and quantized, with $2\pi \sigma^{\mathrm H}\in \mathbb Z$ (Marcelli et al., 24 Jun 2026).

1. Statement of the theorem

The theorem is formulated for a family of many-body Hamiltonians

$$H_{(b,\mu,\lambda)}=\mathrm d\Gamma(\mathfrak h^b-\mu)+\lambda V^b,$$

where $\mathfrak h^b$ is a short-range magnetic one-particle Hamiltonian on $\ell^2(\mathbb Z^2,\mathbb C^r)$ and $V^b$ is a gauge-invariant short-range interaction. The relevant parameter space is $\mathbb R^3_{(b,\mu,\lambda)}$, with $b\in\mathbb R$ the magnetic flux, $(b,\mu,\lambda)$0 the chemical potential, and $(b,\mu,\lambda)$1 the interaction strength (Marcelli et al., 24 Jun 2026).

A central notion is that of a uniformly gapped region. This consists of an open connected set

$(b,\mu,\lambda)$2

together with states $(b,\mu,\lambda)$3 such that there exists $(b,\mu,\lambda)$4 with each $(b,\mu,\lambda)$5 a locally unique gapped ground state of $(b,\mu,\lambda)$6 with gap at least $(b,\mu,\lambda)$7, and such that $(b,\mu,\lambda)$8 is continuous for all $(b,\mu,\lambda)$9 in the CAR-algebra $\lambda=0$0 (Marcelli et al., 24 Jun 2026).

The many-body gap-labeling theorem is then stated for an open connected nonempty set

$\lambda=0$1

described as a non-interacting gapped phase. If the family $\lambda=0$2 admits a uniformly gapped region

$\lambda=0$3

with gap $\lambda=0$4, containing $\lambda=0$5 and all straight segments $\lambda=0$6, then the map

$\lambda=0$7

is constant on $\lambda=0$8 and takes values in the discrete set $\lambda=0$9. In particular,

$2\pi \sigma^{\mathrm H}\in \mathbb Z$0

This is the precise sense in which the integer colors of the non-interacting Hofstadter butterfly persist in the interacting problem (Marcelli et al., 24 Jun 2026).

2. One-particle input and many-body setting

The one-particle Hamiltonian is built from a short-range hopping matrix

$2\pi \sigma^{\mathrm H}\in \mathbb Z$1

For magnetic flux $2\pi \sigma^{\mathrm H}\in \mathbb Z$2, the corresponding Peierls-twisted operator is

$2\pi \sigma^{\mathrm H}\in \mathbb Z$3

and this operator is invariant under the projective magnetic translations $2\pi \sigma^{\mathrm H}\in \mathbb Z$4 (Marcelli et al., 24 Jun 2026).

The spectral assumption is that at some base point $2\pi \sigma^{\mathrm H}\in \mathbb Z$5 one has $2\pi \sigma^{\mathrm H}\in \mathbb Z$6, so that a spectral gap of size $2\pi \sigma^{\mathrm H}\in \mathbb Z$7 persists in a neighborhood. This one-particle gap provides the starting point for the many-body construction. The corresponding interacting Hamiltonian

$2\pi \sigma^{\mathrm H}\in \mathbb Z$8

is defined on the CAR-algebra, with $2\pi \sigma^{\mathrm H}\in \mathbb Z$9 any short-range, gauge-invariant interaction continuous in $$H_{(b,\mu,\lambda)}=\mathrm d\Gamma(\mathfrak h^b-\mu)+\lambda V^b,$$0 (Marcelli et al., 24 Jun 2026).

The formulation is explicitly designed to cover both commensurate and incommensurate magnetic flux. For rational flux $$H_{(b,\mu,\lambda)}=\mathrm d\Gamma(\mathfrak h^b-\mu)+\lambda V^b,$$1, finite magnetic supercells of size $$H_{(b,\mu,\lambda)}=\mathrm d\Gamma(\mathfrak h^b-\mu)+\lambda V^b,$$2 are available. For irrational flux, no finite magnetic unit cell exists. The theorem is significant because it does not restrict the analysis to rational fluxes; instead, it treats arbitrary $$H_{(b,\mu,\lambda)}=\mathrm d\Gamma(\mathfrak h^b-\mu)+\lambda V^b,$$3 through denominator-independent magnetic perturbation estimates and continuity in $$H_{(b,\mu,\lambda)}=\mathrm d\Gamma(\mathfrak h^b-\mu)+\lambda V^b,$$4 (Marcelli et al., 24 Jun 2026).

A common misconception is that quantized Hall labels in Hofstadter-type problems are inherently tied to finite periodicity or Bloch theory. The theorem shows that this is not the relevant structural requirement in the weakly interacting setting under consideration. The essential input is a uniformly gapped region connected to a non-interacting gapped phase, not the existence of a finite magnetic unit cell (Marcelli et al., 24 Jun 2026).

3. Ground states and uniformly gapped regions

At $$H_{(b,\mu,\lambda)}=\mathrm d\Gamma(\mathfrak h^b-\mu)+\lambda V^b,$$5, the construction is explicit. Let

$$H_{(b,\mu,\lambda)}=\mathrm d\Gamma(\mathfrak h^b-\mu)+\lambda V^b,$$6

The infinite-volume quasi-free state

$$H_{(b,\mu,\lambda)}=\mathrm d\Gamma(\mathfrak h^b-\mu)+\lambda V^b,$$7

is the unique gapped ground state of $$H_{(b,\mu,\lambda)}=\mathrm d\Gamma(\mathfrak h^b-\mu)+\lambda V^b,$$8 (Marcelli et al., 24 Jun 2026).

For nonzero interaction strength, the states are obtained by quasi-adiabatic continuation in $$H_{(b,\mu,\lambda)}=\mathrm d\Gamma(\mathfrak h^b-\mu)+\lambda V^b,$$9. Fix an off-diagonal filter $\mathfrak h^b$0, described as rapidly decaying and satisfying

$\mathfrak h^b$1

The local generator is defined by

$\mathfrak h^b$2

and extended to a short-range interaction $\mathfrak h^b$3. The cocycle of automorphisms

$\mathfrak h^b$4

is well-defined on the CAR-algebra, and the interacting state is set to

$\mathfrak h^b$5

According to the paper, quasi-adiabatic stability results together with denominator-independent magnetic estimates imply that for $\mathfrak h^b$6 in a small cylinder $\mathfrak h^b$7, each $\mathfrak h^b$8 remains a locally unique gapped ground state with gap at least $\mathfrak h^b$9 and depends continuously on $\ell^2(\mathbb Z^2,\mathbb C^r)$0 (Marcelli et al., 24 Jun 2026).

The proof proceeds through finite-volume approximations at rational fluxes, extension to irrational flux by rational approximants $\ell^2(\mathbb Z^2,\mathbb C^r)$1, and a lifting step from finite to infinite volume. For rational fluxes, one restricts to boxes commensurate with the sublattice $\ell^2(\mathbb Z^2,\mathbb C^r)$2, constructs finite-volume gapped ground states, and proves convergence in norm and in gap bounds uniformly in $\ell^2(\mathbb Z^2,\mathbb C^r)$3. Irrational flux is then handled by continuity of the cocycles in $\ell^2(\mathbb Z^2,\mathbb C^r)$4 and by gauge-covariant resolvent estimates. This suggests that the theorem is not merely a perturbative statement at fixed rational denominator, but a framework uniform across flux arithmetic (Marcelli et al., 24 Jun 2026).

4. Hall conductivity and the gap label

The Hall conductivity of a gapped state $\ell^2(\mathbb Z^2,\mathbb C^r)$5 is defined through a double-commutator formula associated with the macroscopic Ohm's law: $\ell^2(\mathbb Z^2,\mathbb C^r)$6 Here the off-diagonal parts of the position operators are

$\ell^2(\mathbb Z^2,\mathbb C^r)$7

with a rapidly decaying kernel $\ell^2(\mathbb Z^2,\mathbb C^r)$8 (Marcelli et al., 24 Jun 2026).

In the non-interacting limit, the paper identifies this many-body expression with the standard one-body trace index. More precisely, Proposition 3.9 states that

$\ell^2(\mathbb Z^2,\mathbb C^r)$9

where

$V^b$0

In the commensurate case, this is the TKNN-Chern number; in the incommensurate or disordered case, it is the Bellissard noncommutative gap-label (Marcelli et al., 24 Jun 2026).

The theorem therefore does not introduce a new discrete invariant unrelated to established Hall quantization formulas. Instead, it shows that the Hall conductivity defined for the interacting gapped state coincides at $V^b$1 with the known one-body index, and then remains fixed throughout the connected uniformly gapped region. The gap label is thus physically the Hall conductivity and mathematically a constant element of $V^b$2 (Marcelli et al., 24 Jun 2026).

5. Mechanism of constancy and quantization

The proof separates constancy in $V^b$3 from constancy in $V^b$4. Along the interaction direction, the states satisfy

$V^b$5

By automorphic invariance of the Hall conductivity, described in the paper through the Chern-Simons formula, $V^b$6 is independent of $V^b$7 along any connected gapped path (Marcelli et al., 24 Jun 2026).

At $V^b$8, the conductivity equals the one-body index described above and therefore lies in $V^b$9. Since that one-body trace index is locally constant on the open connected non-interacting gap region $\mathbb R^3_{(b,\mu,\lambda)}$0 and takes values in a discrete set, it is constant on all of $\mathbb R^3_{(b,\mu,\lambda)}$1. Combining these two facts yields

$\mathbb R^3_{(b,\mu,\lambda)}$2

The logic is therefore topological in the sense of local constancy on connected sets, but it is implemented through explicit control of interacting ground states and their automorphic transport (Marcelli et al., 24 Jun 2026).

A plausible implication is that the theorem should be read as a persistence statement rather than a classification of all possible interacting phases. The result applies to uniformly gapped regions meeting the non-interacting plane and containing the straight interaction segments from $\mathbb R^3_{(b,\mu,\lambda)}$3. The quantized label is thus established for interacting phases continuously connected to non-interacting Hofstadter gaps under the stated assumptions (Marcelli et al., 24 Jun 2026).

6. Relation to the colored Hofstadter butterfly

The non-interacting colored Hofstadter butterfly is the graph of the Hofstadter spectrum $\mathbb R^3_{(b,\mu,\lambda)}$4 as a function of $\mathbb R^3_{(b,\mu,\lambda)}$5, with open gaps carrying integer labels identified as Chern numbers. The theorem gives this colored butterfly a many-body interpretation by promoting each non-interacting gap to an interacting quantum Hall phase in the three-parameter space $\mathbb R^3_{(b,\mu,\lambda)}$6, provided one remains within a uniformly gapped region (Marcelli et al., 24 Jun 2026).

The paper states that each non-interacting gap $\mathbb R^3_{(b,\mu,\lambda)}$7 continues to a genuine many-body gapped phase in $\mathbb R^3_{(b,\mu,\lambda)}$8, even when $\mathbb R^3_{(b,\mu,\lambda)}$9 is irrational and no finite unit cell exists. In that continuation, the labels $b\in\mathbb R$0 survive as Hall-conductivity invariants of the interacting phases. This is the precise sense in which the integer colors of the butterfly become many-body phase labels (Marcelli et al., 24 Jun 2026).

The distinction between commensurate and incommensurate flux remains methodologically important. For rational fluxes, finite magnetic supercells and Bloch-Floquet arguments enter the construction. For irrational fluxes, the proof relies instead on denominator-independent magnetic perturbation estimates and continuity in $b\in\mathbb R$1 to pass to infinite volume without periodic boundary conditions. The resulting phase diagram is therefore not a periodic-band picture extended heuristically to weak interaction, but an interacting infinite-volume construction valid at arbitrary flux (Marcelli et al., 24 Jun 2026).

7. Scope, interpretation, and limitations

The theorem establishes quantization and constancy of the Hall conductivity on connected uniformly gapped regions that meet a non-interacting gapped phase. It does not assert that every interacting gapped phase in a magnetic lattice system must arise in this way, nor does it remove the need to prove the existence of the uniformly gapped region $b\in\mathbb R$2 for the family under consideration. The existence statement proved in the paper is local, obtained in a small cylinder around a base point $b\in\mathbb R$3 through quasi-adiabatic continuation and magnetic perturbation theory (Marcelli et al., 24 Jun 2026).

Another potential misunderstanding is to treat the result as purely a rational-flux theorem because of the use of finite-volume approximations at $b\in\mathbb R$4. The paper explicitly avoids such a restriction. Rational flux serves as an approximation and technical foothold, while irrational flux is incorporated by rational approximants together with continuity of the cocycle in $b\in\mathbb R$5 and denominator-independent estimates. The absence of a finite magnetic unit cell at irrational flux is therefore not an obstruction within the stated weak-interaction regime (Marcelli et al., 24 Jun 2026).

Within those limits, the theorem gives a complete many-body interpretation of the colored Hofstadter butterfly: open spectral gaps at $b\in\mathbb R$6 persist as weakly interacting gapped quantum Hall phases, and their integer colors become quantized Hall-conductivity labels throughout the corresponding regions of the full $b\in\mathbb R$7 phase diagram (Marcelli et al., 24 Jun 2026).