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arXiv:2610.21317

Solving fractional quantum Hall problems with thermodynamic-limit bootstrap

Qiang Gao, Taige Wang

Published Oct 15, 2026• 0 citations
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Abstract

Determining correlated ground states directly in the thermodynamic limit remains a central challenge in quantum many-body physics. Here we develop a quantum many-body bootstrap that yields a rigorous and tight lower bound on the thermodynamic ground-state energy of a fractional quantum Hall system together with an optimized reduced density matrix $ρ_W^\star$ on a spatial window. For sufficiently tight bounds, we find that $ρ_W^\star$ accurately reproduces the ground-state reduced density matrix, as directly tested by its fidelity with known model states. For exactly solvable Laughlin and Moore--Read parent Hamiltonians, the bootstrap also reproduces pair correlations and orbital entanglement spectra (OES). For Landau-level-projected Coulomb interactions across multiple fillings, the energy bounds closely track established thermodynamic estimates, while the window density matrix recovers the familiar $ν=1/3$ Laughlin and $ν=5/2$ Moore--Read states and the composite Fermi liquid at $ν=1/2$. More importantly, the same framework also resolves unsettled regimes. At $ν=1/4$, the OES and real-space correlations identify a composite Fermi liquid, while at $ν=12/5$ the combined diagnostics suggest $\overline{\mathrm{RR}}_3$ as the ground-state order. Our bootstrap therefore provides direct access to both ground-state energies and many-body structure in the thermodynamic limit.

DOI
10.48550/arxiv.2610.21317

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The bootstrap method aims to solve problems by imposing constraints on the space of physical observables, which often follow from physical assumptions such as positivity and symmetry. Here, we employ a bootstrap approach to study interacting electrons in the lowest Landau level by minimizing the energy as a function of the static structure factor subject to a set of constraints, bypassing the need to construct the full many-body wave function. This approach rigorously lower bounds the ground state energy, making it complementary to conventional variational upper bounds. We show that the lower bound we obtain is relatively tight, within at most a few percent from the ground state energy computed with exact diagonalization at small system sizes, and generally gets tighter as we include more constraints. We also show that, by combining the bootstrap lower bounds with variational Monte Carlo calculations for various quantum Hall states, we can obtain two-sided bounds on the ground state energy that rigorously bounds the error to below a few percentage for large system sizes where exact diagonalization is not accessible. Beyond the energetics, our results reproduce the correct power law dependence of the pair correlation function at short distances and the existence of a large entanglement gap in the two-particle entanglement spectra for the Laughlin states at ν = 1 / 3 . We further identify signatures of the composite Fermi liquid state close to half filling. This shows that the bootstrap approach is capable, in principle, of describing nontrivial gapped topologically ordered, as well as gapless, phases. We further study the evolution of the ground state energy and correlation functions as we interpolate between the lowest and first Landau levels at half filling and find indications of a phase transition from a composite Fermi liquid to a non-Abelian state. We also discuss more generally how the geometry of the allowed set of structure factors can be used to diagnose phase transitions. At the end, we discuss possible extensions and current limitations of this approach and how they can be overcome in future studies. Our work establishes numerical bootstrap as a promising method to study many-body phases in topological bands, paving the way to its application in moiré platforms where the energetic competition between fractional quantum anomalous Hall, symmetry broken, and gapless states remains poorly understood.
The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.
The calculation of pair correlations and density profiles of quasiholes are routine steps in the study of proposed fractional quantum Hall states. Nevertheless, the field has not adopted a standard way to present the results of such calculations in an easily reproducible form. We develop a polynomial expansion that allows for easy quantitative comparison between different candidate wavefunctions, as well as reliable scaling of correlation and quasihole profiles to the thermodynamic limit. We start from the well-known expansion introduced by Girvin [PRB, 30 (1984)] (see also [Girvin, MacDonald and Platzman, PRB, 33 (1986)]), which is physically appealing but, as we demonstrate, numerically unstable. We orthogonalize their basis set to obtain a new basis of modified Jacobi polynomials, whose coefficients can be stably calculated. We then apply our expansion to extract pair correlation expansion coefficients and quasihole profiles in the thermodynamic limit for a wide range of fractional quantum Hall wavefunctions. These include the Laughlin series, composite fermion states with both reverse and direct flux attachment, the Moore-Read Pfaffian state, and BS hierarchy states. The expansion procedure works for both abelian and non-abelian quasiholes, even when the density at the core is not zero. We find that the expansion coefficients for all quantum Hall states considered can be fit remarkably well using a cosine oscillation with exponentially decaying amplitude. The frequency and the decay length are related in an intuitive, but not elementary way to the filling fraction. Different states at the same filling fraction can have distinct values for these parameters. Finally, we also use our scaled correlation functions to calculate estimates for the magneto-roton gaps of the various states.
Recent advances in engineered quantum platforms have enabled the realization of bosonic Laughlin states at ν=1/2ν=1/2ν=1/2 and brought non-Abelian topological phases within experimental reach. A central theoretical challenge is to develop a unified framework connecting lattice fractional quantum Hall (FQH) model states, exact parent Hamiltonians, and experimentally accessible interactions. We systematically construct Hermitian parent Hamiltonians for which continuum lowest Landau level (LLL) model states sampled on lattice sites are exact zero modes. Using these model manifolds as quantitative references, we find that, within the projected lattice LLL at the flux densities studied, short-range density interactions stabilize Laughlin ground-state manifolds at ν=1/3ν=1/3ν=1/3 and 1/41/41/4, while two-body onsite repulsion supports a Moore--Read triplet at ν=1ν=1ν=1. Full multiband calculations reveal a sharp contrast: the Laughlin manifolds remain robust against interband mixing on the studied tori, whereas the Moore--Read triplet becomes less spectrally isolated and eventually undergoes a level crossing with competing states near the interband scale. Our framework provides a theoretical foundation for using lattice model states to guide the search for experimentally accessible Abelian and non-Abelian FQH states.
Nearly fifty years after its experimental debut, the quantum hall effect has produced perhaps more unique descriptions than any other subject of condensed matter physics. From topological band theory to non-commutative geometry, topological and conformal field theories, random matrix models and more, each approach captures a subset of the essential features, and compliments the others to produce a beautiful mosaic description. In this thesis, we revisit the fractional quantum hall (FQH) effect from its oldest vantage point, that of the trial wavefunctions and the parent Hamiltonians which stabilize them. Laughlin's observation of the importance of the theory of holomorphic functions to describing incompressible states in a partially filled Landau level has produced a coherent paradigm for realizing FQH parent Hamiltonians as frustration-free exactly-solvable one-dimensional models through dimensional reduction. Trial wavefunctions are then classified in terms of a small number of product states which generate all other states in the orbital-occupation expansion of the full state by a sequence of angular-momentum-preserving squeezing operations. These unentangled states are called the DNA of the FQH state, and they can be described in terms of simple local rules called generalized Pauli principles (GPP) of the form "at most r electrons can occupy k consecutive orbitals". It was only recently shown that an analysis of this form can be carried out in second-quantization for states with Landau level mixing, such as the Jain composite-fermion and parton series. In this case, the GPPs are generalized once further to entangled Pauli principles (EPP) for which the DNA may be entangled with respect to the extra Landau level index. In the first part of this thesis, we prove a theorem that allows for a rigorous computation of EPPs given a frustration free k-body parent Hamiltonian by examining only k-particle zero-modes, and give an example of its utility by demonstrating the exact-solvability of a correlated pair-hopping model that can be derived as a special limit of a FQH parent Hamiltonian. In the second part, taking advantage of the second quantized EPP analysis from the continuum, we derive a class of bosonic lattice models for states in multiple Landau levels which, in the presence of a contact interaction, are shown to stabilize non-Abelian ground states. Furthermore, we show how the EPP can be used to compute the degeneracies of quasihole excitations, and therefore the topological order through the bulk-boundary correspondence, which we verify with simple numerics. To the best of our knowledge, this work represents the first exponentially-local exactly-solvable fractional Chern insulator model for non-Abelian states with only two-body interactions.
We investigate the use of variational quantum algorithms to prepare and characterize fractional quantum Hall states on near-term quantum processors. Focusing on the ν=1/3ν=1/3ν=1/3 Laughlin phase described by the V1V_1V1​ Haldane pseudopotential, we formulate the lowest-Landau-level problem in second quantization, and implement particle-number-preserving variational circuits combined with the variational quantum eigensolver (VQE) and variational quantum deflation (VQD). We benchmark the approach in two complementary geometries: Haldane sphere and torus shape. On the Haldane sphere, the target state is a unique zero-energy Laughlin ground state, providing a controlled test of the variational workflow and of excited-state reconstruction. On the torus, the problem retains the genuinely two-dimensional periodic character of the quantum Hall liquid and exhibits the threefold topological ground-state degeneracy expected for the ν=1/3ν=1/3ν=1/3 fractional filling factor. This feature makes the torus a more demanding benchmark than the quasi-one-dimensional cylinder or thin-torus limits commonly exploited in state-preparation quantum protocols. We benchmark the hardware-optimized variational states against exact diagonalization using energy estimates, error-mitigated observables, and subspace-containment diagnostics. Our results show that hybrid quantum algorithms can approximately reconstruct the low-energy structure of small fractional quantum Hall systems, including the topological ground-state manifold on the torus. Beyond serving as a benchmark for quantum hardware, this geometry-resolved approach provides a route toward quantum simulations of fractional Chern insulators and strongly correlated topological phases in realistic two-dimensional materials.