We study the quantum Hall phases that appear in the fast-rotation limit for Bose-Einstein condensates of spinless bosonic atoms. We use exact diagonalization in a spherical geometry to obtain low-lying states of a small number of bosons as a function of the angular momentum. This allows us to understand or guess the physics at a given filling fraction $\ensuremath{\nu}$, the ratio of the number of bosons to the number of vortices. This is also the filling factor of the lowest Landau level. In addition to the well-known Bose-Laughlin state at $\ensuremath{\nu}=1∕2$, we give evidence for the Jain principal sequence of incompressible states at $\ensuremath{\nu}=p∕(p\ifmmode\pm\else\textpm\fi{}1)$ for a few values of
p. There is a collective mode in these states whose phenomenology is in agreement with standard arguments coming, e.g., from the composite-fermion picture. At filling factor 1, the potential Fermi sea of composite fermions is replaced by a paired state, the Moore-Read state. This is most clearly seen from the half-flux nature of elementary excitations. We find that the hierarchy picture does not extend up to the point of transition towards a vortex lattice. While we cannot come to any definite conclusions, we investigate the clustered Read-Rezayi states and show evidence for incompressible states at the expected ratio of flux versus number of Bose particles.
In this paper, the Bose-Hubbard model (BHM) with the nearest-neighbor (NN) repulsions is studied from the viewpoint of possible bosonic analogs of the fractional quantum Hall (FQH) state in the vicinity of the Mott insulator (MI). First, by means of the Gutzwiller approximation, we obtain the phase diagram of the BHM in a magnetic field. Then, we introduce an effective Hamiltonian describing excess particles on a MI and calculate the vortex density, momentum distribution, and the energy gap. These calculations indicate that the vortex solid forms for small NN repulsions, but a homogeneous featureless ``Bose metal'' takes the place of it as the NN repulsion increases. We consider particular filling factors at which the bosonic FQH state is expected to form. Chern-Simons (CS) gauge theory to the excess particle is introduced, and a modified Gutzwiller wave function, which describes bosons with attached flux quanta, is introduced. The energy of the excess particles in the bosonic FQH state is calculated using that wave function, and it is compared with the energy of the vortex solid and Bose metal. We found that the energy of the bosonic FQH state is lower than that of the Bose metal and comparable with the vortex solid. Finally, we clarify the condition that the composite fermion appears by using CS theory on the lattice that we previously proposed for studying the electron FQH effect.
We study the stability of composite fermion fractional quantum Hall states in Harper-Hofstadter bands with Chern number
∣C∣>1. From composite fermion theory, states are predicted to be found at filling factors $\ensuremath{\nu}=r/(kr|C|+1),r\ensuremath{\in}\mathbb{Z}$, with
k=1 for bosons and
k=2 for fermions. Here, we closely analyze these series in both cases, with contact interactions for bosons and nearest-neighbor interactions for (spinless) fermions. In particular, we analyze how the many-body gap scales as the bands are tuned to the effective continuum limit of Chern number
∣C∣ bands, realized near flux density ${n}_{\ensuremath{\phi}}=1/|C|$. Near these points, the Hofstadter model requires large magnetic unit cells that yield bands with perfectly flat dispersion and Berry curvature. We exploit the known scaling of energies in the effective continuum limit in order to maintain a fixed square aspect ratio in finite-size calculations. Based on exact diagonalization calculations of the band-projected Hamiltonian for these lattice geometries, we show that for both bosons and fermions, the vast majority of finite-size spectra yield the ground-state degeneracy predicted by composite fermion theory. For the chosen interactions, we confirm that states with filling factor $\ensuremath{\nu}=1/(k|C|+1)$ are the most robust and yield a clear gap in the thermodynamic limit. For bosons with contact interactions in
∣C∣=2 and
∣C∣=3 bands, our data for the composite fermion states are compatible with a finite gap in the thermodynamic limit. We also report new evidence for gapped incompressible states stabilized for fermions with nearest-neighbor interactions in
∣C∣>1 bands. For cases with a clear gap, we confirm that the thermodynamic limit commutes with the effective continuum limit within finite-size error bounds. We analyze the nature of the correlation functions for the Abelian composite fermion states and find that the correlation functions for
∣C∣>1 states are smooth functions for positions separated by
∣C∣ sites along both axes, giving rise to
∣C∣2 sheets; some of which can be related by inversion symmetry. We also comment on two cases which are associated with a bosonic integer quantum Hall effect (BIQHE): For $\ensuremath{\nu}=2$ in
∣C∣=1 bands, we find a strong competing state with a higher ground-state degeneracy, so no clear BIQHE is found in the band-projected Hofstadter model; for $\ensuremath{\nu}=1$ in
∣C∣=2 bands, we present additional data confirming the existence of a BIQHE state.
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=1/3 Laughlin phase described by the
V1 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=1/3 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.
The Monte Carlo method is very useful for studying various model states proposed for the fractional quantum Hall systems. In this paper, we introduce a lattice Monte Carlo method based on an exact lattice formalism for quantum Hall problems defined on a torus. This ``lattice representation'' is applied to study the many body Berry phase of the composite-Fermi liquid phase in a half-filled Landau level. The Monte Carlo result agrees with the exact numerical result found on small systems sizes and is consistent with the prediction from the Dirac fermion low-energy effective theory. Many other aspects of other model states, including the structure factor, Coulomb energy and particle-hole symmetry, are studied and discussed.
We investigate the feasibility of many candidate quantum Hall states for two-component bosons in the lowest Landau level. We identify interactions for which spin-singlet incompressible states occur at filling factors $\ensuremath{\nu}=2/3$,
4/5, and
4/3, and partially spin-polarized states at filling factors
3/4 and
3/2, where ``spin'' serves as a generic label for the two components. We study ground states, excitations, edge states, and entanglement spectrum for systems with up to 16 bosons and construct explicit trial wave functions to clarify the underlying physics. The composite fermion theory very accurately describes the ground states as well as excitations at $\ensuremath{\nu}=2/3$,
4/5, and
3/4, although it is less satisfactory for the $\ensuremath{\nu}=3/2$ state. For $\ensuremath{\nu}=4/3$ a ``non-Abelian spin-singlet'' state, which is the exact ground state of a three-body contact interaction, has been proposed to occur even for a two-body contact interaction; our trial wave functions are very accurate for the excitations of the three-body interaction, but they do not describe the excitations of the two-body interaction very well. Instead, we find that the $\ensuremath{\nu}=4/3$ state is more likely to be a spin-singlet state of reverse-flux-attached composite fermions at filling ${\ensuremath{\nu}}^{*}=4$. We also consider incompressible states at integral filling factors $\ensuremath{\nu}=1$ and 2. The incompressible state at $\ensuremath{\nu}=1$ is shown to be well described by the parton-based Jain spin-singlet wave function, and the incompressible state at $\ensuremath{\nu}=2$ as the spin-singlet state of reverse-flux-attached composite fermions at ${\ensuremath{\nu}}^{*}=2$, which provides an example of the bosonic integer topological phase.