We investigate the B1-weighted spectral gap of the quantum direction field in a open-boundary soft-core boson Harper–Hofstadter model at fractional magnetic flux , with on-site interaction ranging from 1 to 15. Exact diagonalization on the C4-symmetric subspace yields the many-body ground state for – at . The B1-weighted spectral gap is found to be anomalously low for , a feature that persists over the entire interaction range. By decomposing the gap into contributions from the direction field texture and the natural-orbital amplitude, we show that the anomaly arises solely from a reduction of the amplitude near the center of the trap, not from any topological singularity or unusual density correlation. Unweighted spectral gaps for match those of –, and the phase field remains smooth; only exhibits a vortex. Density-density correlations on the central plaquette are nearly identical for . The amplitude–phase coupling coefficient shows a slight reduction for but remains above 0.99, indicating mild decoupling due to the weight redistribution. The OBDM entropy reveals that integer flux is topologically distinct from all fractional fluxes, while – share the same thermodynamic scaling. We interpret the half-flux anomaly as a static density modulation of the condensate natural orbital, consistent with known half-vortex physics in superfluid systems. The stability of under interaction demonstrates that the direction field gap is a protected single-particle topological signature, and its insensitivity to provides a precise benchmark for finite-size cold-atom experiments.
We investigate the B1-weighted spectral gap of the quantum direction field in a open-boundary soft-core boson Harper–Hofstadter model at fractional magnetic flux , with on-site interaction ranging from 1 to 15. Exact diagonalization on the C4-symmetric subspace yields the many-body ground state for – at . The B1-weighted spectral gap is found to be anomalously low for , a feature that persists over the entire interaction range. By decomposing the gap into contributions from the direction field texture and the natural-orbital amplitude, we show that the anomaly arises solely from a reduction of the amplitude near the center of the trap, not from any topological singularity or unusual density correlation. Unweighted spectral gaps for match those of –, and the phase field remains smooth; only exhibits a vortex. Density-density correlations on the central plaquette are nearly identical for . The amplitude–phase coupling coefficient shows a slight reduction for but remains above 0.99, indicating mild decoupling due to the weight redistribution. The OBDM entropy reveals that integer flux is topologically distinct from all fractional fluxes, while – share the same thermodynamic scaling. We interpret the half-flux anomaly as a static density modulation of the condensate natural orbital, consistent with known half-vortex physics in superfluid systems. The stability of under interaction demonstrates that the direction field gap is a protected single-particle topological signature, and its insensitivity to provides a precise benchmark for finite-size cold-atom experiments.
We investigate how the topological winding number of a real-space direction field transforms under the discrete modular group (with ), using exact-diagonalization (ED) data from a open-boundary soft-core Bose–Hubbard model at fractional magnetic flux () and on-site interaction . The direction field is constructed as the normalized phase gradient of the dominant natural orbital of the one-body density matrix, and its topological charge is the plaquette winding number of the planar phase. We report two categories of results and deliberately keep them distinct. Algebraic verifications. The Minkowski-interval identity (T2) and the transformation law (T3, algebraic part) are numerical confirmations of known algebraic and topological facts: the first follows from with , and the second from the induced or reflection action on . Across 128,376 verification points these hold with zero exceptions, as required by their algebraic character. Physical inputs and non-trivial content. Four results depend on the ED ground state or on the non-compact part of and are not algebraically guaranteed: The ED ground state produces and . This is a physical input, not a group-theoretic prediction. The non-compact (Lorentz-boost) part of pushes the direction field off and produces a large scalar component –, which is the time component of the Lorentz-transformed Minkowski vector (T6). The -weighted spectral gap varies by less than as is tuned from to (T5), and its relative variation decreases monotonically with . The mirror-flip prediction translates the abstract law into a concrete lattice operation (spatial mirror combined with hopping-phase reversal) that can be realized in cold-atom experiments. We discuss the relation between the real-space winding number and the bulk Chern number of the Harper–Hofstadter model, and present three cold-atom-testable predictions. The discrete modular group appears both in the DFWH framework and in the present direction-field construction; this work does not prove the DFWH framework, but demonstrates that the same discrete group acts on a physically realizable object with a well-defined chiral transformation law.
We study the ground-state phase diagram of the strongly interacting Harper–Hofstadter–Mott model at quarter flux on a quasi-one-dimensional lattice consisting of a single magnetic flux quantum in y -direction. In addition to superfluid phases with various density patterns, the ground-state phase diagram features quasi-one-dimensional analogs of fractional quantum Hall phases at fillings ν = 1/2 and 3/2, where the latter is only found thanks to the hopping anisotropy and the quasi-one-dimensional geometry. At integer fillings—where in the full two-dimensional system the ground-state is expected to be gapless—we observe gapped non-degenerate ground-states: at ν = 1 it shows an odd 'fermionic' Hall conductance, while the Hall response at ν = 2 consists of the transverse transport of a single particle–hole pair, resulting in a net zero Hall conductance. The results are obtained by exact diagonalization and in the reciprocal mean-field approximation.
Topological protection against many‑body quantum fluctuations remains an open fundamental question. Using exact diagonalization on a 6×6 periodic lattice, we investigate the robustness of topological vortex charge across the onset of many‑body quantum chaos for soft‑core Bose‑Hubbard and spinless fermion t‑V models. We track the central‑plaquette winding number over a wide interaction range U/t∈[0,15], covering the double‑N cusp catastrophe featured in level‑spacing ratio ⟨r⟩. Even deep inside chaotic parameter windows where ⟨r⟩ collapses far toward the Poisson limit, the ground‑state winding number stays numerically stable at 1.0000, showing decoupling between topological invariant and many‑body quantum chaos. Three distinct magnetic‑flux values Φ=Φ0,Φ0/4,Φ0/3 are compared. The observed number of chaos‑collapse valleys matches the number of resolvable Hofstadter subband gaps, furnishing exact‑diagonalization numerical evidence for the transition‑hierarchy conjecture. Further, we identify a finite‑size threshold for chaos activation controlled by Hilbert‑space subspace dimension. Our results demonstrate that topological vortex invariants can persist unperturbed through strong many‑body quantum‑chaos regimes, establishing that quantum chaos does not automatically destroy ground‑state topological quantization.
We show via tensor network methods that the Harper-Hofstadter Hamiltonian for hard-core bosons on a square geometry supports a topological phase realizing the $\ensuremath{\nu}=\frac{1}{2}$ fractional quantum Hall (FQH) effect on the lattice. We address the robustness of the ground-state degeneracy and of the energy gap, measure the many-body Chern number, and characterize the system using Green functions, showing that they decay algebraically at the edges of open geometries, indicating the presence of gapless edge modes. Moreover, we estimate the topological entanglement entropy by taking a combination of lattice bipartitions that reproduces the topological structure of the original proposals by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)] and Levin and Wen [Phys. Rev. Lett. 96, 110405 (2006)]. The numerical results show that the topological contribution is compatible with the expected value $\ensuremath{\gamma}=\frac{1}{2}$. Our results provide extensive evidence that FQH states are within reach of state-of-the-art cold-atom experiments.
We investigate the ground-state properties of a bosonic Harper-Hofstadter model with local interactions on a finite cylindrical lattice with filling fraction $\ensuremath{\nu}=1/2$. We find that our system supports topologically ordered states by calculating the topological entanglement entropy, and its value is in good agreement with the theoretical value for the $\ensuremath{\nu}=1/2$ Laughlin state. By exploring the behavior of the density profiles, edge currents, and single-particle correlation functions, we find that the ground state on the cylinder shows all signatures of a fractional quantum Hall state even for large values of the magnetic flux density. Furthermore, we determine the dependence of the correlation functions and edge currents on the interaction strength. We find that depending on the magnetic flux density, the transition toward Laughlin-like behavior can be either smooth or it can happen abruptly for some critical interaction strength.
Stability of Fractional Chern Insulators in the Effective Continuum Limit of ∣C∣>1 Harper-Hofstadter Bands
We study the stability of composite fermion fractional quantum Hall states in Harper-Hofstadter bands with Chern number ∣C∣>1. We analyze the states of the composite fermion series for bosons with contact interactions and (spinless) fermions with nearest-neighbor interactions. We examine the scaling of the many-body gap as the bands are tuned to the effective continuum limit nϕ→1/∣C∣. Near these points, the Hofstadter model realises 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, we show that almost all finite-size spectra yield the ground state degeneracy predicted by composite fermion theory. We confirm that states at low ranks in the composite fermion hierarchy are the most robust, and yield a clear gap in the thermodynamic limit. For bosons 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 of fermions in ∣C∣>1 bands, which have large entanglement gaps. For cases with a clear spectral gap, we confirm that the thermodynamic limit commutes with the effective continuum limit. We analyze the nature of the correlation functions for the Abelian composite fermion states and find that they feature ∣C∣2 smooth sheets. We examine two cases associated with a bosonic integer quantum Hall effect (BIQHE): for ν=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 ν=1 in ∣C∣=2 bands, we present additional data confirming the existence of a BIQHE state.