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arXiv:2603.25639 · Low Temperature Physics

Symplectic Split-Operator Propagators from Tridiagonalized Multi-Mode Bosonic Hilbert Spaces for Bose-Hubbard Hamiltonians

Denys I. Bondar, Ole Steuernagel

quant-phSubmitted Mar 26, 2026• Announced Mar 27, 2026• 0 citations
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Abstract

In this methods paper, we show how to tridia\-go\-nalize two families of bosonic multimode systems: optomechanical and Bose-Hubbard hamiltonians. Using tools from number theory, we devise a rendering of these systems in the form of exact $D \times D$ tridiagonal symmetric matrices with real-valued entries. Such matrices can subsequently be exactly diagonalized using specialized sparse-matrix algorithms that need on the order of $D \ln(D)$ steps. This makes it possible to describe systems with much larger numbers of basis states than available to date. It also allows for efficient diagonal representation of large, accurate, symplectic split-operator propagators for which we moreover show that the required basis changes can be implemented by simple re-indexing, at marginal computational cost.

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8 pages and 2 figures

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In this methods paper, we show how to tridia-go-nalize two families of bosonic multimode systems: optomechanical and Bose-Hubbard hamiltonians. Using tools from number theory, we devise a rendering of these systems in the form of exact D×DD \times DD×D tridiagonal symmetric matrices with real-valued entries. Such matrices can subsequently be exactly diagonalized using specialized sparse-matrix algorithms that need on the order of Dln⁡(D)D \ln(D)Dln(D) steps. This makes it possible to describe systems with much larger numbers of basis states than available to date. It also allows for efficient diagonal representation of large, accurate, symplectic split-operator propagators for which we moreover show that the required basis changes can be implemented by simple re-indexing, at marginal computational cost.
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We present a fast, memory-efficient, unitarity-preserving numerical method beyond the rotating-wave approximation for the closed Tavis-Cummings model in which a multilevel spin system interacts with a cavity mode. This model can describe the interaction of an ensemble of spins with a cavity mode in which the spin frequency and other parameters are time-dependent. The method exploits the fact that, while the Tavis-Cummings model is not tri-diagonal, it can be brought into tri-diagonal form by a change of basis that can be implemented purely by re-indexing (permuting basis elements), which is a fast operation. By truncating the Fock basis of the cavity mode, the computational complexity of the method is linear in the total dimension of the coupled system, both in time and memory. The method can be employed to simulate any closed quantum system whose Hamiltonian terms can be brought into tri-diagonal form.
Computing bosonic Gaussian ground states via variational optimization is challenging because the covariance matrices must satisfy the uncertainty principle, rendering constrained or Riemannian optimization costly, delicate, and thus difficult to scale, particularly in large and inhomogeneous systems. We introduce a symplectic optimization framework that addresses this challenge by parameterizing covariance matrices directly as positive-definite symplectic matrices using unit-triangular factorizations. This approach enforces all physical constraints exactly, yielding a globally unconstrained variational formulation of the bosonic ground-state problem. The unconstrained structure also naturally supports solution reuse across nearby Hamiltonians: warm-starting from previously optimized covariance matrices substantially reduces the number of optimization steps required for convergence in families of related configurations, as encountered in crystal lattices, molecular systems, and fluids. We demonstrate the method on weakly dipole-coupled lattices, recovering ground-state energies, covariance matrices, and spectral gaps accurately. The framework further provides a foundation for large-scale approximate treatments of weakly non-quadratic interactions and offers potential scaling advantages through tensor-network enhancements.
We uncover a previously unrecognized hierarchical algebraic structure in the Bose-Hubbard model that enables a systematic construction of its spectrum without relying on integrability. We develop a recursive framework in which, within each fixed particle-number sector, the eigenproblem of a p -site system is reduced to a closed set of algebraic relations determined entirely by the eigenpairs and single-particle transfer amplitudes of the ( p − 1 ) -site subsystem. This reduction yields two complementary sets of Bethe-type equations whose solutions exhaust the spectrum. The underlying structure originates from the locality of the Hamiltonian, which constrains the coupling between subsystems and gives rise to an emergent sparse, near-tridiagonal representation of the many-body problem. As an explicit demonstration, we construct the complete spectrum of the three-site Bose-Hubbard model from that of the two-site system for particle numbers n ≤ 3 , obtaining exact agreement with direct diagonalization. The resulting formulation substantially reduces computational cost and reveals a form of hierarchical solvability in a nonintegrable quantum many-body system. More broadly, our results suggest that recursive algebraic constructions may provide a general framework for analyzing lattice models beyond the conventional integrable paradigm.
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We take the Bose-Hubbard model to illustrate exact diagonalization techniques in a pedagogical way. We follow the road of first generating all the basis vectors, then setting up the Hamiltonian matrix with respect to this basis, and finally using the Lanczos algorithm to solve low lying eigenstates and eigenvalues. Emphasis is placed on how to enumerate all the basis vectors and how to use the hashing trick to set up the Hamiltonian matrix or matrices corresponding to other quantities. Although our route is not necessarily the most efficient one in practice, the techniques and ideas introduced are quite general and may find use in many other problems.