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In current studies of mean-field quantum spin systems, much attention isplaced on the calculation of the ground-state energy and the excitation gap,especially the latter which plays an important role in quantum annealing. Inpure systems, the finite gap can be obtained by various existing methods suchas the Holstein-Primakoff transform, while the tunneling splitting atfirst-order phase transitions has also been studied in detail using instantonsin many previous works. In disordered systems, however, it remains challengingto compute the gap of large-size systems with specific realization of disorder.Hitherto, only quantum Monte Carlo techniques are practical for such studies.Recently, Knysh [Nature Comm. \textbf{7}, 12370 (2016)] proposed a method wherethe exponentially large dimensionality of such systems is condensed onto arandom potential of much lower dimension, enabling efficient study of suchsystems. Here we propose a slightly different approach, building upon themethod of static approximation of the partition function widely used foranalyzing mean-field models. Quantum effects giving rise to the excitation gapand non-extensive corrections to the free energy are accounted for byincorporating dynamical paths into the path integral. The time-dependence ofthe trace of the time-ordered exponential of the effective Hamiltonian iscalculated by solving a differential equation perturbatively, yielding afinite-size series expansion of the path integral. Formulae for the firstexcited-state energy are proposed to aid in computing the gap. We illustrateour approach using the infinite-range ferromagnetic Ising model and theHopfield model, both in the presence of a transverse field.

Effects of dynamical paths on the energy gap and the corrections to free energy in path integrals of mean-field quantum spin systems