The Magnus expansion is attractive for periodically driven quantum systems because every finite truncation exponentiates an anti-Hermitian generator and therefore preserves unitarity. Its practical failure in interacting fermionic systems is nevertheless subtle: a low-order approximation may remain unitary while becoming progressively less accurate as additional terms are retained. We establish order-resolved breakdown signatures for a half-filled, periodically driven one-dimensional Fermi-Hubbard chain. The one-period propagator is computed numerically exactly in the fixed-particle sector and compared with Magnus approximants through fourth order. Failure is defined operationally by a successive-order error ratio (rₙ=εₙ/εₙ₋₁≥ 1), not by an unsupported claim of divergence of the infinite series. Across a raw (9×29) interaction-frequency grid ((U/J=0…,8), (ω/J=2…,16)), at least one added order loses improvement at 36.40% of the 261 sampled points. The smallest sampled frequency above which all retained orders improve throughout the remaining high-frequency tail rises from (ω/J=2.5) at (U=0) to (11.5) at (U/J=8). At (U/J=4), the highest (r₄=1) crossing occurs at (ω/J=7.719), followed by a monotone sampled tail from (ω/J=8). High-frequency error exponents approach the expected omitted-order scaling. A norm-based convergence certificate is consistently more conservative than the observed finite-order boundary. These results show that Magnus breakdown in the Hubbard model is order dependent, interaction shifted, and detectable before unitarity is lost.
The Magnus expansion is attractive for periodically driven quantum systems because every finite truncation exponentiates an anti-Hermitian generator and therefore preserves unitarity. Its practical failure in interacting fermionic systems is nevertheless subtle: a low-order approximation may remain unitary while becoming progressively less accurate as additional terms are retained. We establish order-resolved breakdown signatures for a half-filled, periodically driven one-dimensional Fermi-Hubbard chain. The one-period propagator is computed numerically exactly in the fixed-particle sector and compared with Magnus approximants through fourth order. Failure is defined operationally by a successive-order error ratio (rₙ=εₙ/εₙ₋₁≥ 1), not by an unsupported claim of divergence of the infinite series. Across a raw (9×29) interaction-frequency grid ((U/J=0…,8), (ω/J=2…,16)), at least one added order loses improvement at 36.40% of the 261 sampled points. The smallest sampled frequency above which all retained orders improve throughout the remaining high-frequency tail rises from (ω/J=2.5) at (U=0) to (11.5) at (U/J=8). At (U/J=4), the highest (r₄=1) crossing occurs at (ω/J=7.719), followed by a monotone sampled tail from (ω/J=8). High-frequency error exponents approach the expected omitted-order scaling. A norm-based convergence certificate is consistently more conservative than the observed finite-order boundary. These results show that Magnus breakdown in the Hubbard model is order dependent, interaction shifted, and detectable before unitarity is lost.
We investigate the nonequilibrium relaxation dynamics of the one-dimensional (1D) Fermi-Hubbard model subjected to abrupt, global interaction quenches. Specifically, we benchmark the convergence properties, structural accuracy, and algorithmic breakdown of the Magnus expansion against numerically exact results obtained via full Exact Diagonalization (ED) on small, periodic lattice clusters. By tracking the real-time evolution of local observables, double occupancy (doublon density), and many-body state fidelity metrics, we map out the validity bounds of the low-order Magnus series across weak, moderate, and strong interaction regimes. Our findings demonstrate that while the Magnus expansion provides an exceptionally accurate description of short-time coherent dynamics, rapid phase matching, and initial prethermalization tendencies, its convergence is fundamentally bottlenecked at longer timescales. This breakdown is driven by the rapid growth of multi-particle entanglement, non-local operator spreading via nested commutators, and the emergence of severe state-space fragmentation inherent to dense, strongly interacting many-body spectra.
The Magnus expansion provides an exponential representation of one-parameter operator families, expressed as a series expansion in their generators. This is particularly relevant in quantum mechanics for describing a unitary evolution determined by a time-dependent Hamiltonian generator of the dynamics. The solution is constructed as a series expansion in terms of increasingly complex nested commutators that rapidly become challenging to compute directly. This work establishes a universal upper bound scaling as O((A(t)/ξ)nn−3/2) on the error incurred when the Magnus expansion is truncated at an arbitrary order. The factor n−3/2 sharpens the exponential suppression governed by the integrated norm of the generator A(t) and the convergence radius ξ, and we prove that this dependence on the order n is optimal for the corresponding majorant. The main technical ingredient of the proof is the binary tree representation introduced by Iserles and Norsett from which we derive a recursion formula to delimit the magnitude of any term in the expansion. We complement our analytic bound with explicit computation of the first tree coefficients. With these findings we aim to contribute to the understanding of the accuracy and limitations of the Magnus expansion technique, and to provide a sharper bound for approximating quantum dynamics when information on the generator is not available.
We generate the perturbative expansion of the single particle Green's function and related self-energy for a half-filled single-band Hubbard model on a square lattice. We invoke algorithmic Matsubara integration to evaluate single-particle quantities for real and Matsubara frequencies and verify results through comparison to existing data on the Matsubara axis. With low-order expansions at weak coupling we observe a number of outcomes expected at higher orders: the opening of a gap, pseudogap behavior, and Fermi-surface reconstruction. Based on low-order perturbations, we consider the phase diagram that arises from truncated expansions of the self-energy and Green's function and their relation via the Dyson equation. From Matsubara axis data, we observe insulating behavior in direct expansions of the Green's function, whereas the same order of truncation of the self-energy produces metallic behavior. This observation is supported by additional calculations for real frequencies. We attribute this difference to the order in which diagrams are implicitly summed in the Dyson series. By separating the reducible and irreducible contributions at each order we show that the reducible diagrams implicitly summed in the Dyson equation lead to incorrect physics in the half-filled Hubbard model. Our observations for this particular case lead us to question the utility of the Dyson equation for any problem that shows a disparity between reducible and irreducible contributions to the expansion of the Green's function.
Strongly correlated materials exhibit some of the most complex and technologically useful phenomena in condensed matter. We use the world line and determinant quantum Monte Carlo techniques to explore the phases of the Hubbard Hamiltonian, a model describing strong interactions, in a number of different contexts. We first consider the spin, charge, and bond order correlations of the one-dimensional extended fermion Hubbard model in the presence of a coupling to the lattice. A static alternating lattice distortion leads to enhanced charge density wave correlations at the expense of antiferromagnetic order. When the lattice degrees of freedom are dynamic, we show that a similar effect occurs even though the charge asymmetry must arise spontaneously. Although the evolution of the total energy with lattice coupling is smooth, the individual components exhibit sharp crossovers at the phase boundaries. Finally, we observe a tendency for bond order in the region between the charge and spin density wave phases. Second, we examine mixtures of bosons and fermions in one-dimensional optical lattices. We evaluate the density profiles and bosonic visibility [special characters omitted] resolving the discrepancy between theory and experiment by identifying parameter regimes where [special characters omitted] is reduced and increased. We present a simple qualitative picture of the different response to the fermion admixture in terms of the superfluid and Mott-insulating domains before and after the fermions are included. Finally, we show that [special characters omitted] exhibits kinks which are tied to the domain evolution present in the pure bosonic case, and also additional structure arising from the formation of boson-fermion molecules, a prediction for future experiments. Third, we report large scale calculations of the effective bandwidth, momentum distribution, and magnetic correlations of the square lattice fermion Hubbard Hamiltonian. The sharp Fermi surface of the non-interacting limit is significantly broadened by the electronic correlations, but retains signatures of the approach to the edges of the first Brillouin zone as the density increases. Finite size scaling of simulations on large lattices allows us to extract the interaction dependence of the antiferromagnetic order parameter, exhibiting its evolution from weak-coupling to the strong-coupling Heisenberg limit. Our lattices provide improved resolution of the momentum distribution, allowing a more quantitative comparison with time-of-flight optical lattice experiments.
Abstract We develop Ω TRINITAS ENGINE v6, a mathematical and computational framework for constraint-preserving projected unitary dynamics, with particular emphasis on invariant physical subspaces, gauge-constraint leakage, local unitary evolution, and reproducible Hamiltonian lattice gauge theory. The central problem is to distinguish approximation error in discrete or product-formula time evolution from genuine violation of a prescribed physical constraint. For an orthogonal projector P defining the physical subspace and a unitary evolution U, the framework quantifies constraint leakage through ||(I-P)UP|| and establishes the exact identity ||PUx||^2 = ||x||^2 - ||(I-P)Ux||^2 for physical initial states. In finite-dimensional Hilbert spaces, exact preservation of the physical subspace is characterized by invariance under U, equivalently by vanishing leakage. For products of local unitaries, leakage admits an additive bound in terms of the individual factors, while Hamiltonian-generated evolution is controlled through commutators between the projector and the local generators. In particular, when every local Hamiltonian term preserves the Gauss-law physical subspace, finite-depth product-formula dynamics can preserve the gauge constraint exactly even when the approximation to the target continuous-time evolution remains imperfect. We investigate these results computationally using an explicitly constructed gauge-enriched successor model based on the 1+1-dimensional U(1) lattice Schwinger model with staggered fermions, periodic boundary conditions, finite electric-flux truncation, and Gauss-law reduction. Independent constructions of the physical basis and Hamiltonian agree exactly in the audited finite systems, with zero matrix and spectral discrepancies and no detected Gauss-violating hopping transitions. A symmetric local product formula exhibits approximately third-order single-step error and second-order fixed-time convergence while maintaining numerical unitarity near machine precision. Sparse low-energy calculations at fixed physical volume reproduce the massless Schwinger rest-gap benchmark: an even-power continuum extrapolation gives 0.5642377786 in units of the gauge coupling, compared with the exact continuum value 1/sqrt(pi) = 0.5641895835. In the representative electric-flux cutoff test, variations are below the reported numerical resolution. Momentum-resolved T2 sectors further reproduce the expected relativistic dispersion with strong numerical agreement at the lowest momenta and a larger, explicitly retained deviation at higher momentum. The numerical analysis constitutes finite-regulator calibration and continuum extrapolation rather than certification of a controlled continuum limit. Systematic uncertainties associated with fit family, fit range, finite physical volume, regulator ordering, and higher-momentum behavior remain distinct from the proved constraint-preservation results. This separation prevents numerical agreement with a known continuum theory from being promoted to a stronger mathematical or physical claim. The work explicitly distinguishes mathematical proof, finite computational verification, numerical observation, calibration, and physical interpretation. Agreement with the Schwinger model is treated as a nontrivial calibration of the gauge-enriched successor construction, not as a derivation of gauge theory from canonical Ω TRINITAS dynamics. The manuscript does not establish endogenous U(1) gauge emergence from canonical Ω TRINITAS v9 and does not claim correspondence between Ω TRINITAS and the microscopic dynamics of the actual universe. Negative results, failed constructions, limitations, and unresolved questions are retained as part of the epistemic record. Ω TRINITAS ENGINE v6 therefore provides a reproducible reference framework for projected quantum dynamics, invariant subspaces, Gauss-law preservation, constraint leakage, Hamiltonian lattice gauge theory, local unitary circuits, Trotter and product-formula evolution, sparse exact diagonalization, electric-flux truncation, continuum extrapolation, and momentum-sector spectroscopy. The canonical reference implementation and machine-readable numerical results are designed to preserve a strict distinction between proved statements, computationally verified finite cases, numerical evidence, open problems, and unsupported physical correspondence, providing an auditable foundation for subsequent mathematical and computational investigations.
Efficient simulation of quantum dynamics with time-dependent Hamiltonians is important not only for time-varying systems but also for time-independent Hamiltonians in the interaction picture. Such simulations are more challenging than their time-independent counterparts due to the complexity introduced by time ordering. Existing algorithms that aim to capture commutator-based scaling either exhibit polynomial cost dependence on the Hamiltonian’s time derivatives or are limited to low-order accuracy. In this work, we establish the general commutator-scaling error bounds for the truncated Magnus expansion at arbitrary order, where only Hamiltonian terms appear in the nested commutators, with no time derivatives involved. Building on this analysis, we design a high-order quantum algorithm with explicit circuit constructions. The algorithm achieves cost scaling with the commutator structure in the high-precision regime and depends only logarithmically on the Hamiltonian’s time variation, making it efficient under the standard input model for general time-dependent settings, including Hamiltonians arising in the interaction picture.