Bosonic self-energy functional theory

Abstract

We derive the self-energy functional theory for bosonic lattice systems with broken U(1) symmetry by parametrizing the bosonic Baym-Kadanoff effective action in terms of one-and two-point self-energies. The formalism goes beyond other approximate methods such as the pseudoparticle variational cluster approximation, the cluster composite boson mapping, and the Bogoliubov+U theory. It simplifies to bosonic dynamical-meanfield theory when constraining to local fields, whereas when neglecting kinetic contributions of noncondensed bosons, it reduces to the static mean-field approximation. To benchmark the theory, we study the Bose-Hubbard model on the two-and three-dimensional cubic lattice, comparing with exact results from path integral quantum Monte Carlo. We also study the frustrated square lattice with next-nearest-neighbor hopping, which is beyond the reach of Monte Carlo simulations. A reference system comprising a single bosonic state, corresponding to three variational parameters, is sufficient to quantitatively describe phase boundaries and thermodynamical observables, while qualitatively capturing the spectral functions, as well as the enhancement of kinetic fluctuations in the frustrated case. On the basis of these findings, we propose self-energy functional theory as the omnibus framework for treating bosonic lattice models, in particular, in cases where path integral quantum Monte Carlo methods suffer from severe sign problems (e.g., in the presence of nontrivial gauge fields or frustration). Self-energy functional theory enables the construction of diagrammatically sound approximations that are quantitatively precise and controlled in the number of optimization parameters but nevertheless remain computable by modest means.

Anisotropic Harper-Hofstadter-Mott model: Competition between condensation and magnetic fields

“…Note that Eq. (19) is consistent with the standard lattice free-energy within the single-site meanfield approximation [37]. In fact, requiring stationarity in the symmetry breaking field F X, Y ,…”

Spectral properties and phase diagram of correlated lattice bosons in an optical cavity within bosonic dynamical mean-field theory

“…Details of this method can be found in Ref. 34. This approach allows to interpret the nature of some of the bands.…”

Legendre-spectral Dyson equation solver with super-exponential convergence

“…2. This is in stark contrast to the polynomial convergence of the standard Matsubara tail approach [42][43][44] , also shown in Fig. 2.…”

“…By staying in Legendre-coefficient space the algorithm converges super exponentially with respect to the number of Legendre coefficients N L used to represent the imaginary time Green's function 47 . This is in stark contrast to the Matsubara frequency space based approach with only polynomial convergence [42][43][44] . The exponential convergence is shared with a recently presented Chebyshev polynomial based algorithm 48 , where the convolution scales as ∼ O(N 3 L ).…”