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Topological Vortex as a Quantum Intrinsic Invariant: Exact Diagonalization Evidence from 6×6 Lattices

GenMin Li

Published Aug 5, 2026• 0 citations
Publisher

Abstract

Topological protection against many‑body quantum fluctuations remains an open fundamental question. Using exact diagonalization on a \(6\times6\) 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 \(U/t\in[0,15]\), covering the double‑N cusp catastrophe featured in level‑spacing ratio \(\langle r\rangle\). Even deep inside chaotic parameter windows where \(\langle r\rangle\) collapses far toward the Poisson limit, the ground‑state winding number stays numerically stable at \(1.000\,0\), showing decoupling between topological invariant and many‑body quantum chaos. Three distinct magnetic‑flux values \(\Phi=\Phi_0,\Phi_0/4,\Phi_0/3\) 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.

DOI
10.5281/zenodo.21809032

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Topological protection against many‑body quantum fluctuations remains an open fundamental question. Using exact diagonalization on a (6\times6) 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\in[0,15]), covering the double‑N cusp catastrophe featured in level‑spacing ratio (\langle r\rangle). Even deep inside chaotic parameter windows where (\langle r\rangle) collapses far toward the Poisson limit, the ground‑state winding number stays numerically stable at (1.000,0), showing decoupling between topological invariant and many‑body quantum chaos. Three distinct magnetic‑flux values (\Phi=\Phi_0,\Phi_0/4,\Phi_0/3) 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.
Macroscopic Quantum Chaos from Topological Degradation We present numerical evidence that macroscopic quantum chaos can emerge without nonlinear dynamics in the evolution equation. Instead, it arises as a continuous phase transition driven solely by the topological degradation of the underlying spatial substrate. ◆ Key Features • 2D toroidal mesh: N = L × L sites (L = 20–50). • Tight-binding Hamiltonian with uncorrelated Anderson disorder (W as the primary control parameter). • Independent 10% bond-percolation channel confirming that the observed behavior is a generic consequence of connectivity loss. • Exact diagonalization on mobile ARM (Snapdragon 8 Gen 2) hardware without spectral unfolding: ~25–75 s per 50-realization ensemble, achieving ~25× speedup over conventional HPC workflows. ◆ Main Results • Perfect torus exhibits symmetry-induced spectral degeneracy, numerically close to Poisson statistics. • Moderate disorder (W ≈ 1–4) yields ⟨r⟩ = 0.530–0.534, matching GOE Wigner–Dyson statistics. • Strong disorder (W ≳ 8) restores genuine Poisson statistics (⟨r⟩ ≈ 0.39). • Critical regime (W ≈ 4–6): strongly multifractal eigenstates, monotonic decay of generalized dimensions Dq, and f(α) peaking near α ≈ 2.2–2.3. • Finite-size scaling confirms Anderson-transition-like criticality. ◆ Implications These results show that Wigner–Dyson level repulsion and algorithmic chaos are thermodynamic consequences of spatial frustration. The critical threshold (W ≈ 4–6) defines a quantitative failure boundary for distributed tensor-processing interconnects, while the unfolding-free methodology enables rapid, on-device spectral diagnostics for next-generation fault-tolerant AI accelerators.
The lifetime of a quantum state, denoted τ, is a fundamental characteristic determining the coherence time and operational reliability of quantum technologies. Conventional approaches to extend τ rely on dynamical decoupling, material purification, or engineered dissipation, yet these methods generally scale poorly with system size and are vulnerable to environmental noise. This paper introduces a novel theoretical framework whereby τ is determined by a topological invariant T of the underlying quantum system: τ = f(T). By establishing a rigorous connection between the spectrum of the Liouvillian superoperator, the topology of the system's Hilbert space, and the decay dynamics of density matrices, we demonstrate that certain topologically nontrivial phases inherently suppress relaxation channels and thus guarantee a finite lower bound on τ. We present a detailed derivation for a class of spin chains with nearest‑neighbor interactions and illustrate how the winding number of the Bloch Hamiltonian controls the lifetime of edge states. The formalism is generalized to higher‑dimensional lattice models and to systems with non‑Hermitian Hamiltonians. We discuss implications for fault‑tolerant quantum computation, robust quantum memories, and the design of synthetic quantum matter with engineered lifetimes. This work lays the foundation for a new branch of quantum stability theory that harnesses topological protection not only of energy levels but of dynamical lifetimes, potentially opening pathways toward scalable, intrinsically protected quantum devices.
We establish a quantitative bulk-boundary correspondence in interacting topological insulators by relating many-body topological invariants to characteristic degeneracy structures in the entanglement spectrum. Focusing on generalized Su-Schrieffer-Heeger chains with higher winding number, we construct a gauge-invariant many-body winding invariant based on Pancharatnam geometric phases that remains well defined in the presence of interactions. We show that this invariant uniquely determines the low-lying entanglement-spectrum degeneracy, which exhibits a universal 4ν4^ν4ν scaling with the winding number ννν, providing a concrete formulation of bulk-boundary correspondence beyond single-particle topology. Using exact diagonalization, we demonstrate the robustness of this correspondence under interactions and symmetry-preserving disorder, and identify inversion symmetry as a minimal protecting symmetry that stabilizes both the quantization of the invariant and the associated entanglement degeneracies. Our results unify geometric-phase invariants and entanglement diagnostics within a many-body framework and provide a route to identifying interacting topological phases beyond band theory.
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.
Crystalline symmetries give rise to topological invariants that can distinguish quantum phases of matter. Understanding these in strongly interacting systems is an ongoing research direction requiring non-perturbative methods. Recent developments have demonstrated that even classic models, like the Harper-Hofstadter model of free fermions on a lattice in a magnetic field, yield a host of crystalline symmetry protected topological invariants. Here we review some of these developments, focusing mainly on how to characterize, classify, and detect invariants arising from lattice translation and rotation symmetries along with charge conservation in two-dimensional systems, including integer and fractional Chern insulators.
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.