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Self-consistent methods for interacting lattice bosons with U(1)-symmetry-breaking

Dario Hügel

Published Jan 1, 2018• 0 citations
Publisher

Abstract

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.

DOI
10.5282/edoc.21897

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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.
We extend the self-energy functional theory to the case of interacting lattice bosons in the presence of symmetry breaking and quenched disorder. The self-energy functional we derive depends only on the self-energies of the disorder-averaged propagators, allowing for the construction of general non-perturbative approximations. Using a simple single-site reference system with only three variational parameters, we are able to reproduce numerically exact quantum Monte Carlo (QMC) results on local observables of the Bose–Hubbard model with box disorder with high accuracy. At strong interactions, the phase boundaries are reproduced qualitatively but shifted with respect to the ones observed with QMC due to the extremely low condensate fraction in the superfluid phase. Deep in the strongly-disordered weakly-interacting regime, the simple reference system employed is insufficient and no stationary solutions can be found within its restricted variational subspace. By systematically analyzing thermodynamical observables and the spectral function, we find that the strongly interacting Bose glass is characterized by different regimes, depending on which local occupations are activated as a function of the disorder strength. We find that the particles delocalize into isolated superfluid lakes over a strongly localized background around maximally-occupied sites whenever these sites are particularly rare. Our results indicate that the transition from the Bose glass to the superfluid phase around unit filling at strong interactions is driven by the percolation of superfluid lakes which form around doubly occupied sites.
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-mean-field 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.
We derive an approximate analytical solution of the self-consistency equations of the bosonic dynamical mean-field theory (B-DMFT) in the strong-coupling limit. The approach is based on a linked-cluster expansion in the hybridization function of normal bosons around the atomic limit. The solution is used to compute the phase diagram of the bosonic Hubbard model for different lattices. We compare our results with numerical solutions of the B-DMFT equations and numerically exact methods, respectively. The very good agreement with those numerical results demonstrates that our approach captures the essential physics of correlated bosons both in the Mott insulator and in the superfluid phase. Close to the transition into the superfluid phase the momentum distribution function at zero momentum is found to be strongly enhanced already in the normal phase. The linked-cluster expansion also allows us to compute dynamical properties such as the spectral function of bosons. The evolution of the spectral function across the transition from the normal to the superfluid phase is seen to be characteristically different for the interaction driven and density driven transition, respectively.
A versatile and numerically inexpensive method is presented allowing the accurate calculation of phase diagrams for bosonic lattice models. By treating clusters within the Gutzwiller theory, a surprisingly good description of quantum fluctuations beyond the mean-field theory is achieved approaching quantum Monte Carlo predictions for large clusters. Applying this powerful method to the Bose-Hubbard model, we demonstrate that it yields precise results for the superfluid to Mott-insulator transition in square, honeycomb, and cubic lattices. Due to the exact treatment within a cluster, the method can be effortlessly adapted to more complicated Hamiltonians in the fast progressing field of optical lattice experiments. This includes state- and site-dependent superlattices, large confined atomic systems, and disordered potentials, as well as various types of extended Hubbard models. Furthermore, the approach allows an excellent treatment of systems with arbitrary filling factors. We discuss the perspectives that allow for the computation of large, spatially varying lattices, low-lying excitations, and time evolution.
We derive the reciprocal cluster mean-field method to study the strongly interacting bosonic Harper-Hofstadter-Mott model. The system exhibits a rich phase diagram featuring band insulating, striped superfluid, and supersolid phases. Furthermore, for finite hopping anisotropy, we observe gapless uncondensed liquid phases at integer fillings, which are analyzed by exact diagonalization. The liquid phases at fillings $\ensuremath{\nu}=1,3$ exhibit the same band fillings as the fermionic integer quantum Hall effect, while the phase at $\ensuremath{\nu}=2$ is CT\mathcal{C}\mathcal{T}CT-symmetric with zero charge response. We discuss how these phases become gapped on a quasi-one-dimensional cylinder, leading to a quantized Hall response, which we characterize by introducing a suitable measure for nontrivial many-body topological properties. Incompressible metastable states at fractional filling are also observed, indicating competing fractional quantum Hall phases. The combination of reciprocal cluster mean-field and exact diagonalization yields a promising method to analyze the properties of bosonic lattice systems with nontrivial unit cells in the thermodynamic limit.
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 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 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.