Sub-Landau levels in two-dimensional electron system in magnetic field

Guo-Qiang Hai

arxiv: 2406.06212 · v2 · submitted 2024-06-10 · ❄️ cond-mat.str-el · cond-mat.mes-hall

Sub-Landau levels in two-dimensional electron system in magnetic field

Guo-Qiang Hai This is my paper

Pith reviewed 2026-05-24 00:05 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hall

keywords sub-Landau levelsrelative angular momentumlowest Landau levelquantum Hall effectcorrelated pairstrial wavefunctionstwo-electron solutionscenter-of-mass degeneracy

0 comments

The pith

Exact two-electron solutions in a magnetic field organize into sub-Landau levels indexed by relative angular momentum m.

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

The paper demonstrates that the exact solutions for two interacting electrons in a strong magnetic field fall into distinct groups labeled by their relative angular momentum m. Each group forms a subspace inside the usual Landau level that keeps the full degeneracy coming from the center-of-mass motion but restricts the states available for relative motion. This partitioning reduces the effective number of states per correlation channel and suggests a fractional occupancy picture. The authors then use these pair states to build trial wave functions for larger numbers of electrons, in which every pair has the same m and therefore vanishes at short range. A sympathetic reader would care because the construction offers a direct microscopic link between solvable two-body problems and the correlated many-body states that appear in the quantum Hall effect.

Core claim

Numerically exact solutions for two interacting electrons organize into a set of sub-Landau levels characterized by relative angular momentum quantum number m. These sub-levels define correlation-resolved subspaces of the Landau-level Hilbert space, while retaining the full degeneracy associated with center-of-mass motion. Within this structure, the accessible states in each correlation channel are effectively reduced, leading to a natural organization of guiding-center states consistent with a fractional occupancy. Building on the two-electron states, a class of many-electron trial wavefunctions based on correlated electron pairs with fixed m is constructed, which encode short-range correl

What carries the argument

The sub-Landau levels indexed by the relative angular momentum quantum number m; they organize the Landau-level states into correlation channels that preserve center-of-mass degeneracy but limit the relative-motion Hilbert space.

If this is right

The Landau level Hilbert space factors into independent m-subspaces.
Trial many-electron states can be assembled from fixed-m pairs without mixing correlation channels.
Short-range correlations arise automatically from the pair wave function vanishing at small distances.
Spin-polarized pair states become favored when Zeeman energy and disorder are included.
The structure supplies a microscopic basis for correlated phases at fractional fillings in the lowest Landau level.

Where Pith is reading between the lines

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

If the m-subspaces survive in larger systems, they could provide an alternative counting of states at fractional quantum Hall fillings.
One could check the robustness of the sub-level structure by performing exact calculations for three or four electrons under the same interaction.
The pair-based construction might be adapted to study pairing instabilities or bilayer quantum Hall systems.

Load-bearing premise

The numerically exact two-electron eigenstates obtained for finite systems or specific interactions remain representative of the organization of the full Landau-level Hilbert space in the thermodynamic limit.

What would settle it

A calculation of the exact eigenstates for three electrons showing that the wave functions or energies mix states from different m values in a way that cannot be explained by the two-body sub-level structure.

Figures

Figures reproduced from arXiv: 2406.06212 by Guo-Qiang Hai.

**Figure 2.** Figure 2: (a) shows the sub-Landau levels E pair NM,nm for N = 0 (with Ecm 0M = ~ωc/2) and m = +3, +1, −1, −3, −5, −7, and −9 (i.e., only the odd m for triplet states are plotted). The energy in the figure is given by E pair NM,nm divided by 2 (i.e., measured by energy per electron) in order to compare with the Landau levels indicated by ns. Notice that, the energy levels with m ≤ −10 are not shown in the figure.… view at source ↗

**Figure 3.** Figure 3: FIG. 3. Diagram of the correlated rotating electron pair wit [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

read the original abstract

We study two interacting electrons in a two-dimensional system under a strong magnetic field and show that their numerically exact solutions organize into a set of {\em sub-Landau levels} characterized by relative angular momentum quantum number $m$. These sub-levels define correlation-resolved subspaces of the Landau-level Hilbert space, while retaining the full degeneracy associated with center-of-mass motion. Within this structure, the accessible states in each correlation channel are effectively reduced, leading to a natural organization of guiding-center states consistent with a fractional occupancy. We further analyze the role of electron correlation, Zeeman splitting, and disorder in stabilizing spin-polarized electron-pair states. Building on the two-electron states, we construct a class of many-electron trial wavefunctions based on correlated electron pairs with fixed $m$, which encode short-range correlations through the vanishing of the pair wavefunction at small separation. Our results establish a direct connection between exact two-body physics and the organization of correlated many-electron states in the lowest Landau level, providing a microscopic perspective on how relative angular momentum structures can underpin the emergence of correlated phases in quantum Hall 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

Two-electron exact solutions are grouped by relative angular momentum m into sub-Landau levels that keep center-of-mass degeneracy, and the paper builds many-body trial states from fixed-m pairs.

read the letter

The core observation is that numerically exact two-electron wavefunctions in the lowest Landau level fall into distinct channels labeled by relative angular momentum m. Each channel forms a reduced subspace while the center-of-mass degeneracy stays intact, and the authors use these channels to write trial many-electron states built from correlated pairs that vanish at small separation. That framing is the main new element; it supplies a direct two-body route to short-range correlation structure inside the LLL without invoking composite fermions or pseudopotentials up front. The construction is straightforward and the pair-wavefunction vanishing is a clean way to encode repulsion. The numerics are presented as exact for two particles, which is a solid starting point if the basis and interaction details hold up in the full text. The soft spot is the jump from two-body spectra to many-body trial functions. The paper assumes the m-channel organization extracted from finite two-particle systems carries over unchanged when many electrons are present, but no three-body spectra, continuum extrapolation, or direct energy comparisons to established FQHE states are mentioned to test that step. Screening or Landau-level mixing could mix the channels in the thermodynamic limit, and without those checks the claim that the subspaces organize the full Hilbert space remains provisional. The work is aimed at theorists who build microscopic wavefunctions for the quantum Hall regime. It is coherent on its own terms and engages the existing literature on pair correlations, so it is worth sending to a serious referee even though the many-body evidence is still thin. I would ask the authors for explicit checks on larger systems before accepting the organizing principle as general.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that numerically exact solutions for two interacting electrons in a strong magnetic field organize into sub-Landau levels labeled by relative angular momentum quantum number m. These sub-levels define correlation-resolved subspaces of the Landau-level Hilbert space while retaining full center-of-mass degeneracy, leading to a natural organization consistent with fractional occupancy. The work analyzes the roles of electron correlation, Zeeman splitting, and disorder in stabilizing spin-polarized pair states, and constructs many-electron trial wavefunctions based on correlated pairs with fixed m that encode short-range correlations through vanishing of the pair wavefunction at small separation. This establishes a direct connection between exact two-body physics and the organization of correlated many-electron states in the lowest Landau level.

Significance. If the sub-Landau level organization and its extension to many-body states hold, the work supplies a microscopic perspective linking two-body relative angular momentum channels to the emergence of correlated phases in quantum Hall systems. The use of numerically exact two-electron solutions is a positive feature, as it avoids ad-hoc fitting and provides a parameter-free starting point for trial states.

major comments (1)

[Abstract] The central construction of correlation-resolved subspaces and many-electron trial wavefunctions (abstract, paragraph on two-electron solutions and subsequent many-body construction) rests on the assumption that finite-system two-electron eigenstates labeled by m remain representative when embedded in the infinite-system or thermodynamic-limit Landau-level Hilbert space. Potential mixing from screening, higher-Landau-level effects, or collective modes could alter the claimed retention of center-of-mass degeneracy and the subspace definition; the manuscript should supply explicit convergence checks, three-body spectra, or continuum extrapolation to secure this step.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its potential significance. We address the major comment below.

read point-by-point responses

Referee: [Abstract] The central construction of correlation-resolved subspaces and many-electron trial wavefunctions (abstract, paragraph on two-electron solutions and subsequent many-body construction) rests on the assumption that finite-system two-electron eigenstates labeled by m remain representative when embedded in the infinite-system or thermodynamic-limit Landau-level Hilbert space. Potential mixing from screening, higher-Landau-level effects, or collective modes could alter the claimed retention of center-of-mass degeneracy and the subspace definition; the manuscript should supply explicit convergence checks, three-body spectra, or continuum extrapolation to secure this step.

Authors: We appreciate the referee's concern regarding the extrapolation of our finite-system two-electron results. Our two-electron eigenstates are obtained by exact diagonalization strictly within the lowest Landau level (LLL) on a finite torus, where magnetic translation symmetry rigorously enforces center-of-mass degeneracy for any system size. The organization into sub-Landau levels follows directly from the conservation of relative angular momentum m, which is a good quantum number due to rotational invariance in the relative coordinate; this symmetry persists in the LLL-projected many-body Hilbert space. Our pair-based trial wavefunctions are constructed to lie entirely within the LLL and are therefore variational upper bounds that encode the exact short-distance correlations of the two-body problem. While we acknowledge that higher-Landau-level mixing, screening, and collective modes lie outside the strict LLL projection (standard in the quantum Hall literature), these effects do not lift the CM degeneracy within the projected theory. In the revised manuscript we will add an explicit discussion of these assumptions and limitations in the sections describing the two-body solutions and the many-body construction. We note, however, that three-body spectra or full continuum extrapolations lie beyond the computational scope of the present work. revision: partial

Circularity Check

0 steps flagged

No circularity detected; two-body numerics feed one-way into many-body construction

full rationale

The derivation begins with numerically exact two-electron solutions in a magnetic field, which are organized by relative angular momentum m into sub-Landau levels that define correlation subspaces while preserving center-of-mass degeneracy. These subspaces are then used to construct many-electron trial wavefunctions based on fixed-m correlated pairs. This is a unidirectional construction from independent numerical two-body eigenstates to a many-body ansatz; no parameter is fitted inside the two-body step and then relabeled as a prediction, no self-definition equates the output to the input, and no load-bearing self-citation or uniqueness theorem is invoked. The representativeness of finite two-body results for the thermodynamic limit is an applicability assumption, not a circular reduction. The paper is therefore self-contained against external benchmarks with no reduction of its central claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The numerical exactness of the two-electron solutions is treated as an unexamined background assumption.

pith-pipeline@v0.9.0 · 5718 in / 1287 out tokens · 20008 ms · 2026-05-24T00:05:30.012531+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages · 1 internal anchor

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The degeneracy (i.e., the number of single-electron states per unit area) of the lowest Landau level is given by nφ0 = 1/ (2πl 2 B) = B/φ 0, where φ 0 = 2π ℏc/e is the quantum of ﬂux. When there are Ne electrons conﬁned in the 2D plane in the magnetic ﬁeld, the Hamiltonian of the many- electron system can be written as, ˆH = Ne∑ i=1 ˆHs(ri) + Ne∑ i=1 ∑ j<...

work page
[2]

For γB > 0, the relative motion maintains the angular symmetry

and ( 4). For γB > 0, the relative motion maintains the angular symmetry. The or- bital angular momentum ( ˆLrel = −iℏ∂/∂θ ) is conserved. The eigenfunction has the following form, ψ rel nm(r,θ ) = eimθ √ 2πRrel nm(r), (13) for m = 0, ± 1, ± 2, · · ·. The angular momentum of the relative motion is given by Lrel m = mℏ. And the radial wavefunction can be w...

work page
[3]

(17) Here σs = 0 corresponds to the spin-singlet state and σs = 1 to spin-triplet state

must have the following property, ψ rel nm(r,θ +π ) = eiσsπψ rel nm(r,θ ), for σs = 0 or 1. (17) Here σs = 0 corresponds to the spin-singlet state and σs = 1 to spin-triplet state. The condition σs = 0 (σs = 1) is satisﬁed when taking m as an even (odd) number in Eq. ( 13). Therefore, we obtain the wave- functions with m = 0, ± 2, ± 4, · · ·for singlet an...

work page
[4]

is obtained from the numerical calculations. At speciﬁc γB where there exists /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48 /s49/s46 /s53 /s50/s46 /s48 /s50/s46 /s53 /s51/s46 /s48 /s51/s46 /s53 /s52/s46 /s48 /s48/s46 /s53 /s49/s46 /s48 /s49/s46 /s53 /s50/s46 /s48 /s50/s46 /s53 /s51/s46 /s48 /s51/s46 /s53 /s52/s46 /s48 /s40 /s48 /s44 /s50 /s41 /s40 /s49 /s44 /...

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states for single electrons per unit area

and (9) of the same Hamiltonian of the two-electron system in diﬀerent coordinates. The above degeneracy is equiv- alent to two non-interacting electron systems of each one /s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s40/s97/s41 /s40/s49/s44/s43/s51/s41 ...

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We may denote an electron pair α at ξα = (rα,1, rα,2) = ( Rα, rα) in the state µ = ( NM,nm ) with wavefunction ψ µ(ξα) = Ψ N M,nm(rα,1, rα,2) given by Eq

becomes, ˆH = Np∑ α=1 ˆHp(rα,1, rα,2) + Np∑ β <α 2∑ i,j=1 γB |rα,i − rβ,j |, (22) where rα,i represents the pair α with two electrons i=1 and 2. We may denote an electron pair α at ξα = (rα,1, rα,2) = ( Rα, rα) in the state µ = ( NM,nm ) with wavefunction ψ µ(ξα) = Ψ N M,nm(rα,1, rα,2) given by Eq. ( 19). There are the following relations∫ dξψ ∗ µ(ξ)ψ µ′(...

work page
[7]

The inter-pair interaction has to be considered through the pair-pair interaction potential in Eq

includes only the intra-pair e-e interaction. The inter-pair interaction has to be considered through the pair-pair interaction potential in Eq. (

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for α ̸=β . The inter-pair correlation energy should be small because the overlap between diﬀerent pairs is small and the intra-pair and inter-pair correlations are not entangled.[17] On the other hand, our wavefunction is presented for both the IQHE (m = − 1) and FQHE (m ≤ − 3) states. Each state has well deﬁned energy level and degeneracy nφ0/ |m|. Ther...

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[1] [1]

The degeneracy (i.e., the number of single-electron states per unit area) of the lowest Landau level is given by nφ0 = 1/ (2πl 2 B) = B/φ 0, where φ 0 = 2π ℏc/e is the quantum of ﬂux. When there are Ne electrons conﬁned in the 2D plane in the magnetic ﬁeld, the Hamiltonian of the many- electron system can be written as, ˆH = Ne∑ i=1 ˆHs(ri) + Ne∑ i=1 ∑ j<...

work page

[2] [2]

For γB > 0, the relative motion maintains the angular symmetry

and ( 4). For γB > 0, the relative motion maintains the angular symmetry. The or- bital angular momentum ( ˆLrel = −iℏ∂/∂θ ) is conserved. The eigenfunction has the following form, ψ rel nm(r,θ ) = eimθ √ 2πRrel nm(r), (13) for m = 0, ± 1, ± 2, · · ·. The angular momentum of the relative motion is given by Lrel m = mℏ. And the radial wavefunction can be w...

work page

[3] [3]

(17) Here σs = 0 corresponds to the spin-singlet state and σs = 1 to spin-triplet state

must have the following property, ψ rel nm(r,θ +π ) = eiσsπψ rel nm(r,θ ), for σs = 0 or 1. (17) Here σs = 0 corresponds to the spin-singlet state and σs = 1 to spin-triplet state. The condition σs = 0 (σs = 1) is satisﬁed when taking m as an even (odd) number in Eq. ( 13). Therefore, we obtain the wave- functions with m = 0, ± 2, ± 4, · · ·for singlet an...

work page

[4] [4]

is obtained from the numerical calculations. At speciﬁc γB where there exists /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48 /s49/s46 /s53 /s50/s46 /s48 /s50/s46 /s53 /s51/s46 /s48 /s51/s46 /s53 /s52/s46 /s48 /s48/s46 /s53 /s49/s46 /s48 /s49/s46 /s53 /s50/s46 /s48 /s50/s46 /s53 /s51/s46 /s48 /s51/s46 /s53 /s52/s46 /s48 /s40 /s48 /s44 /s50 /s41 /s40 /s49 /s44 /...

work page

[5] [5]

states for single electrons per unit area

and (9) of the same Hamiltonian of the two-electron system in diﬀerent coordinates. The above degeneracy is equiv- alent to two non-interacting electron systems of each one /s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s40/s97/s41 /s40/s49/s44/s43/s51/s41 ...

work page

[6] [6]

We may denote an electron pair α at ξα = (rα,1, rα,2) = ( Rα, rα) in the state µ = ( NM,nm ) with wavefunction ψ µ(ξα) = Ψ N M,nm(rα,1, rα,2) given by Eq

becomes, ˆH = Np∑ α=1 ˆHp(rα,1, rα,2) + Np∑ β <α 2∑ i,j=1 γB |rα,i − rβ,j |, (22) where rα,i represents the pair α with two electrons i=1 and 2. We may denote an electron pair α at ξα = (rα,1, rα,2) = ( Rα, rα) in the state µ = ( NM,nm ) with wavefunction ψ µ(ξα) = Ψ N M,nm(rα,1, rα,2) given by Eq. ( 19). There are the following relations∫ dξψ ∗ µ(ξ)ψ µ′(...

work page

[7] [7]

The inter-pair interaction has to be considered through the pair-pair interaction potential in Eq

includes only the intra-pair e-e interaction. The inter-pair interaction has to be considered through the pair-pair interaction potential in Eq. (

work page

[8] [8]

for α ̸=β . The inter-pair correlation energy should be small because the overlap between diﬀerent pairs is small and the intra-pair and inter-pair correlations are not entangled.[17] On the other hand, our wavefunction is presented for both the IQHE (m = − 1) and FQHE (m ≤ − 3) states. Each state has well deﬁned energy level and degeneracy nφ0/ |m|. Ther...

work page 2024

[9] [9]

Lectures on the Quantum Hall Effect

D. Tong, Lectures on the Quantum Hall Eﬀect, arXiv:1606.06687 [hep-th] (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016

[10] [10]

von Klitzing, G

K. von Klitzing, G. Dorda, and M. Pepper, New Method for High-Accuracy Determination of the Fine Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett. 45, 494 (1980)

work page 1980

[11] [11]

D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two- Dimensional Magnetotransport in the Extreme Quantum Limit, Phys. Rev. Lett. 48, 1559 (1982)

work page 1982

[12] [12]

R. B. Laughlin, The Anomalous Quantum Hall Eﬀect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Phys. Rev. Lett. 50, 1395 (1983)

work page 1983

[13] [13]

J. K. Jain, Composite-Fermion Approach for the Frac- tional Quantum Hall Eﬀect, Phys. Rev. Lett. 63, 199 (1989)

work page 1989

[14] [14]

H. K. Choi, I. Sivan, A. Rosenblatt, M. Heiblum, V. Umansky, and D. Mahalu, Robust electron pairing in the integer quantum hall eﬀect regime, Nat. Commun. 6:7435 doi: 10.1038/ncomms8435 (2015)

work page doi:10.1038/ncomms8435 2015

[15] [15]

Biswas, H

S. Biswas, H. K. Kundu, V. Umansky, and M. Heiblum, Electron Pairing of Interfering Interface-Based Edge Modes, Phys. Rev. Lett. 131, 096302 (2023)

work page 2023

[16] [16]

Demir, N

A. Demir, N. Staley, S. Aronson, S. Tomarken, K. West, K. Baldwin, L. Pfeiﬀer, and R. Ashoori, Corre- lated double-electron additions at the edge of a two- dimensional electronic system, Phys. Rev. Lett. 126, 256802 (2021)

work page 2021

[17] [17]

Curilef and F

S. Curilef and F. Claro, Dynamics of two interacting par- ticles in a magnetic ﬁeld in two dimensions, Am. J. Phys. 65, 244 (1997)

work page 1997

[18] [18]

R. B. Laughlin, Quantized motion of three two- dimensional electrons in a strong magnetic ﬁeld, Phys. Rev. B 27, 3383 (1983)

work page 1983

[19] [19]

Ver¸ cin, Two anyons in a static, uniform magnetic ﬁeld

A. Ver¸ cin, Two anyons in a static, uniform magnetic ﬁeld. Exact solution, Phys. Lett. B 260, 120 (1991)

work page 1991

[20] [20]

A. V. Turbiner, Quasi-Exactly-Solvable Problems and sl(2) Algebra, Commun. Math. Phys. 118, 467 (1988)

work page 1988

[21] [21]

Taut, Two electrons in a homogeneous magnetic ﬁeld: particular analytical solutions, J

M. Taut, Two electrons in a homogeneous magnetic ﬁeld: particular analytical solutions, J. Phys. A: Math. Gen. 27, 1045 (1994) [ 27, 4723 (1994) erratum]

work page 1994

[22] [22]

T. T. Truong and D. Bazzali, Exact low-lying states of two interacting equally charged particles in a magnetic ﬁeld, Phys. Lett. A 269, 186 (2000)

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