We extend the variational cluster approach to deal with strongly correlated lattice bosons in the superfluid phase. To this end, we reformulate the approach within a pseudoparticle formalism, whereby cluster excitations are described by particlelike excitations. The approximation amounts to solving a multicomponent noninteracting bosonic system by means of a multimode Bogoliubov approximation. A source-and-drain term is introduced in order to break U(1) symmetry at the cluster level. We provide an expression for the grand potential, the single-particle normal and anomalous Green's functions, the condensate density, and other static quantities. As a first nontrivial application of the method we choose the two-dimensional Bose-Hubbard model and evaluate results in both the Mott and the superfluid phases. Our results show an excellent agreement with quantum Monte Carlo calculations.
We extend the variational cluster approach to deal with strongly correlated lattice bosons in the superfluid phase. To this end, we reformulate the approach within a pseudoparticle formalism, whereby cluster excitations are described by particlelike excitations. The approximation amounts to solving a multicomponent noninteracting bosonic system by means of a multimode Bogoliubov approximation. A source-and-drain term is introduced in order to break U(1) symmetry at the cluster level. We provide an expression for the grand potential, the single-particle normal and anomalous Green's functions, the condensate density, and other static quantities. As a first nontrivial application of the method we choose the two-dimensional Bose-Hubbard model and evaluate results in both the Mott and the superfluid phases. Our results show an excellent agreement with quantum Monte Carlo calculations.
Among the various numerical techniques to study the physics of strongly correlated quantum many-body systems, the self-energy functional approach (SFA) has become increasingly important. In its previous form, however, SFA is not applicable to Bose-Einstein condensation or superfluidity. In this paper, we show how to overcome this shortcoming. To this end, we identify an appropriate quantity, which we term D, that represents the correlation correction of the condensate order parameter, as it does the self-energy for Green's function. An appropriate functional is derived, which is stationary at the exact physical realization of D and of the self-energy. Its derivation is based on a functional-integral representation of the grand potential followed by an appropriate sequence of Legendre transformations. The approach is not perturbative and, therefore, applicable to a wide range of models with local interactions. We show that the variational cluster approach based on the extended self-energy functional is equivalent to the ``pseudoparticle'' approach proposed in Phys. Rev. B 83, 134507 (2011). We present results for the superfluid density in the two-dimensional Bose-Hubbard model, which shows a remarkable agreement with those of quantum-Monte-Carlo calculations.
Convergence properties of the variational cluster approach with respect to the variational parameter space, cluster size, and boundary conditions of the reference system are investigated and discussed for bosonic many-body systems. Specifically, the variational cluster approach is applied to the one-dimensional Bose-Hubbard model, which exhibits a quantum phase transition from Mott to superfluid phase. In order to benchmark the variational cluster approach, results for the phase boundary delimiting the first Mott lobe are compared with essentially exact density matrix renormalization group data. Furthermore, static quantities, such as the ground state energy and the one-particle density matrix are compared with high-order strong coupling perturbation theory results. For reference systems with open boundary conditions the variational parameter space is extended by an additional variational parameter which allows for a more uniform particle density on the reference system and thus drastically improves the results. It turns out that the variational cluster approach yields accurate results with relatively low-computational effort for both the phase boundary as well as the static properties of the one-dimensional Bose-Hubbard model, even at the tip of the first Mott lobe where correlation effects are most pronounced.
We discuss the application of the variational cluster perturbation theory (VCPT) to the Mott-insulator-to-superfluid transition in the Bose–Hubbard model. We show how the VCPT can be formulated in such a way that it gives a translation invariant excitation spectrum—free of spurious gaps—despite the fact that it formally breaks translation invariance. The phase diagram and the single-particle Green function in the insulating phase are obtained for one-dimensional systems. When the chemical potential of the cluster is taken as a variational parameter, the VCPT reproduces the dimensional dependence of the phase diagram even for one-site clusters. We find a good quantitative agreement with the results of the density-matrix renormalization group when the number of sites in the cluster becomes of order 10. The extension of the method to the superfluid phase is discussed.
Bosonic lattice systems with nontrivial interactions represent an intriguing platform to study exotic phases of matter. Here, we study the effects of extended correlated hopping processes in a system of bosons trapped in a lattice geometry. The interplay between single particle tunneling terms, correlated hopping processes, and onsite repulsion is studied by means of a combination of exact diagonalization, strong coupling expansion, and cluster mean field theory. We identify a rich ground state phase diagram where, apart from the usual Mott and superfluid states, superfluid phases with interesting clustering properties occur.
Spectral properties of the two-dimensional Bose-Hubbard model, which emulates ultracold gases of atoms confined in optical lattices, are investigated by means of the variational cluster approach. The phase boundary of the quantum phase transition from Mott phase to superfluid phase is calculated and compared to recent work. Moreover the single-particle spectral functions in both the first and the second Mott lobe are presented and the corresponding densities of states and momentum distributions are evaluated. A qualitatively similar intensity distribution of the spectral weight can be observed for spectral functions in the first and the second Mott lobe.
This thesis is dedicated to the derivation and benchmarking of self-consistent numerical methods that can be applied to interacting bosonic lattice models. The central goal is to derive methods with low numerical complexity but high accuracy, to be applied to complex systems which are out-of reach for established methods such as path integral quantum Monte Carlo (QMC) or the density matrix renormalization group. In the first part we derive the self-energy functional theory (SFT) for bosons. Building upon previous works on lattice systems without U(1)-symmetry-breaking, we systematically extend SFT to the possibility of a broken U(1)-symmetry and the presence of disorder. SFT incorporates bosonic dynamical mean-field theory as a certain limit, and represents a general non-perturbative framework, enabling the construction of diagrammatically sound approximations in the thermodynamical limit that are controlled in the number of optimization parameters. Using just three variational parameters, we are able to study the Bose-Hubbard model both in its clean version and in the presence of local disorder, showing excellent agreement with numerically exact QMC results. We systematically analyze the corresponding spectral functions, which cannot be fully captured by QMC. In particular, we find that in the presence of disorder the phase transition from the Bose glass to the superfluid phase at strong interactions is driven by the percolation of superfluid lakes which form around doubly occupied sites, leading to a small condensate fraction over a strongly-localized background. The second part is dedicated to the derivation of reciprocal cluster mean-field theory (RCMF) and its application to the strongly-interacting Harper-Hofstadter-Mott model (HHMm). In RCMF the full lattice in the thermodynamical limit is projected onto finite-size clusters, which are decoupled in reciprocal space through a mean-field decoupling approximation, crucially preserving the symmetries of the non-interacting dispersion. The resulting groundstate phase diagram of the HHMm exhibits band insulating, striped superfluid, and supersolid phases. Furthermore, we observe gapless uncondensed liquid phases at integer fillings, and a metastable competing fractional quantum Hall (fQH) phase. The fQH phase, predicted as the groundstate by other methods, is most likely underestimated by RCMF. We then show how a quasi-one-dimensional geometry stabilizes gapped topologically non-trivial groundstates in the HHMm. We observe quasi-one-dimensional analogues of fQH phases at fillings 1/2 and 3/2, and unconventional gapped non-degenerate groundstates at integer filling with quantized Hall responses. By systematically comparing results computed with RCMF and exact diagonalization (ED), we are able to give conclusive quantitative answers on the phase boundaries of the system, as the two methods approach the thermodynamical limit from opposite sides, since RCMF favours gapless and ED gapped phases.