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 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 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.
Synthetic quantum matter provides a highly tunable route to fractional quantum Hall physics beyond the constraints of conventional electronic materials. However, previous theoretical studies have mostly focused on their ground state properties. It remains unclear to what extent such platforms could reveal key excitation properties of fractional quantum Hall states. Here, we study charge-neutral collective excitations in a non-abelian lattice fractional quantum Hall state realized in the bosonic Harper-Hofstadter model at unity filling factior, realizing a Moore-Read ground state. Combining full exact diagonalization, band-projected exact diagonalization, and matrix-product-state simulations, we demonstrate the existence of a long-lived chiral graviton mode, probed by chiral 3-body correlators, for the first time on lattice non-Abelian states. The graviton signal is topological sector-independent and could be observed via geometric quenches in small open droplets directly relevant to current cold-atom experiments, while other neutral modes, such as the magnetoroton and neutral fermion, are less resolved at presently achievable volumes.
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.
In the physics of the fractional quantum Hall (FQH) effect, a zoo of Abelian topological phases can be obtained by varying the magnetic field. Aiming to reach the same phenomenology in spin like systems, we propose a family of SU(N)-symmetric models in the fundamental representation, on the square lattice with short-range interactions restricted to triangular units, a natural generalization for arbitrary N of an SU(3) model studied previously where time-reversal symmetry is broken explicitly. Guided by the recent discovery of SU(2)1 and SU(3)1 chiral spin liquids (CSL) on similar models we search for topological SU(N)1 CSL in some range of the Hamiltonian parameters via a combination of complementary numerical methods such as exact diagonalizations (ED), infinite density matrix renormalization group (iDMRG) and infinite Projected Entangled Pair State (iPEPS). Extensive ED on small (periodic and open) clusters up to N=10 and an innovative SU(N)-symmetric version of iDMRG to compute entanglement spectra on (infinitely long) cylinders in all topological sectors provide unambiguous signatures of the SU(N)1 character of the chiral liquids. An SU(4)-symmetric chiral PEPS, constructed in a manner similar to its SU(2) and SU(3) analogs, is shown to give a good variational ansatz of the N=4 ground state, with chiral edge modes originating from the PEPS holographic bulk-edge correspondence. Finally, we discuss the possible observation of such Abelian CSL in ultracold atom setups where the possibility of varying N provides a tuning parameter similar to the magnetic field in the physics of the FQH effect.
Experimental realizations of topologically ordered states of matter, such as fractional quantum Hall states, with cold atoms are now within reach. In particular, optical lattices provide a promising platform for the realization and characterization of such states, where novel detection schemes enable an unprecedented microscopic understanding. Here we show that the central charge can be directly measured in current cold atom experiments using the number entropy as a proxy for the entanglement entropy. We perform density-matrix renormalization-group simulations of Hubbard-interacting bosons on coupled chains subject to a magnetic field with $\ensuremath{\alpha}=1/4$ flux quanta per plaquette. Tuning the interchain hopping, we find a transition from a trivial quasi-one-dimensional phase to the topologically ordered Laughlin state at magnetic filling factor $\ensuremath{\nu}=1/2$ for systems of three or more chains. We resolve the transition using the central charge, on-site correlations, momentum distributions, and the many-body Chern number. Additionally, we propose a scheme to experimentally estimate the central charge from Fock basis snapshots. The model studied here is experimentally realizable with existing cold atom techniques and the proposed observables pave the way for the detection and classification of a larger class of interacting topological states of matter.
Topological states of matter, such as fractional quantum Hall states, are an active field of research due to their exotic excitations. In particular, ultracold atoms in optical lattices provide a highly controllable and adaptable platform to study such new types of quantum matter. However, finding a clear route to realize non-Abelian quantum Hall states in these systems remains challenging. Here we use the density-matrix renormalization-group (DMRG) method to study the Hofstadter-Bose-Hubbard model at filling factor $\ensuremath{\nu}=1$ and find strong indications that at $\ensuremath{\alpha}=1/6$ magnetic flux quanta per plaquette the ground state is a lattice analog of the continuum non-Abelian Pfaffian. We study the on-site correlations of the ground state, which indicate its paired nature at $\ensuremath{\nu}=1$, and find an incompressible state characterized by a charge gap in the bulk. We argue that the emergence of a charge density wave on thin cylinders and the behavior of the two- and three-particle correlation functions at short distances provide evidence for the state being closely related to the continuum Pfaffian. The signatures discussed in this letter are accessible in current cold atom experiments and we show that the Pfaffian-like state is readily realizable in few-body systems using adiabatic preparation schemes.