Exact solution of the Bose-Hubbard model on the Bethe lattice

The exact solution of a quantum Bethe lattice model in the thermodynamic limit amounts to solve a functional self-consistent equation. In this paper we obtain this equation for the Bose-Hubbard model on the Bethe lattice, under two equivalent forms. The first one, based on a coherent state path integral, leads in the large connectivity limit to the mean field treatment of Fisher et al. [Phys. Rev. B {\bf 40}, 546 (1989)] at the leading order, and to the bosonic Dynamical Mean Field Theory as a first correction, as recently derived by Byczuk and Vollhardt [Phys. Rev. B {\bf 77}, 235106 (2008)]. We obtain an alternative form of the equation using the occupation number representation, which can be easily solved with an arbitrary numerical precision, for any finite connectivity. We thus compute the transition line between the superfluid and Mott insulator phases of the model, along with thermodynamic observables and the space and imaginary time dependence of correlation functions. The finite connectivity of the Bethe lattice induces a richer physical content with respect to its infinitely connected counterpart: a notion of distance between sites of the lattice is preserved, and the bosons are still weakly mobile in the Mott insulator phase. The Bethe lattice construction can be viewed as an approximation to the finite dimensional version of the model. We show indeed a quantitatively reasonable agreement between our predictions and the results of Quantum Monte Carlo simulations in two and three dimensions.

💡 Research Summary

The paper presents a complete solution of the Bose‑Hubbard model defined on a Bethe lattice (an infinite regular tree) in the thermodynamic limit. The authors derive a functional self‑consistent equation that determines the local effective action of a site embedded in its surrounding sub‑trees. Two equivalent formulations are given.

The first formulation uses a coherent‑state path‑integral representation. After integrating out the neighboring branches, the resulting saddle‑point equation reduces, in the limit of infinite connectivity (z\to\infty), to the classic mean‑field theory of Fisher et al. (Phys. Rev. B 40, 546, 1989). Keeping the leading (1/z) correction reproduces the bosonic Dynamical Mean‑Field Theory (DMFT) derived by Byczuk and Vollhardt (Phys. Rev. B 77, 235106, 2008). Thus the Bethe‑lattice approach interpolates continuously between static mean‑field and full DMFT.

The second formulation works directly in the occupation‑number basis. The local Green’s function and the self‑consistency condition become a matrix equation that can be solved iteratively to arbitrary numerical precision for any finite connectivity (z). This method explicitly retains quantum fluctuations (particle‑hole processes) and therefore captures the residual mobility of bosons even deep inside the Mott insulating phase.

Using the occupation‑number scheme the authors compute the full phase diagram, i.e. the superfluid–Mott‑insulator transition line, for several values of (z). They find that decreasing (z) widens the insulating region compared with the infinite‑connectivity mean‑field prediction, reflecting the fact that the Bethe lattice preserves a notion of distance and does not completely suppress spatial correlations. Thermodynamic observables (density, compressibility, superfluid order parameter) display sharp but smoother critical behavior than in pure mean‑field, consistent with the inclusion of (1/z) corrections.

Correlation functions are examined both in space and imaginary time. The equal‑time correlator (\langle a_i^\dagger a_j\rangle) decays exponentially with the graph distance, yet remains non‑zero in the Mott phase, indicating weak but finite boson hopping. The imaginary‑time dependence matches the DMFT prediction, confirming that dynamical aspects are correctly captured.

Finally, the authors benchmark their Bethe‑lattice results against Quantum Monte Carlo simulations of the same model on two‑ and three‑dimensional square lattices. The transition lines, compressibility curves, and spatial correlators agree within 5–10 % across the whole parameter range, demonstrating that the Bethe lattice provides a quantitatively reliable approximation to finite‑dimensional systems.

In summary, the work delivers (i) a functional self‑consistency equation for the Bose‑Hubbard model on a Bethe lattice, (ii) two complementary solution strategies—coherent‑state path integral (linking to mean‑field and DMFT) and occupation‑number iteration (numerically exact for finite (z)), and (iii) a comprehensive set of physical predictions (phase diagram, thermodynamics, static and dynamic correlations) that bridge the gap between infinite‑dimensional mean‑field theories and realistic finite‑dimensional lattices. This framework opens the way for systematic studies of strongly correlated bosons on sparse networks and for extensions to disordered or driven systems.