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The anisotropic Harper-Hofstadter-Mott model: supersolid, striped superfluid, and symmetry protected topological groundstates

Dario Hügel, Hugo U. R. Strand, Philipp Werner, Lode Pollet

Published Sep 13, 2016• 0 citations
OpenAlex

Abstract

Since the discovery of the quantum Hall effect, the lattice geometry's influence on charged particles in magnetic fields has been the subject of extensive research. Prototypical models such as the non-interacting Harper-Hofstadter model (HHm) exhibit fractionalization of the Bloch bands with non-trivial topology, manifesting in quantum (spin) Hall phases. Ultracold atomic gases with artificial magnetic fields enabled the experimental study of the non-interacting model, while the effect of strong interactions on the band properties remains an open problem. In this work we study the strongly-interacting bosonic Harper-Hofstadter-Mott model (HHMm) using a reciprocal cluster mean-field (RCMF) method. We obtain a rich groundstate phase diagram featuring striped superfluid, supersolid, and symmetry protected topological (SPT) phases. At filling $\nu=2$ (in contrast to fillings $\nu=1, 3$) the SPT phase has no fermionic counterpart. Incompressible metastable states at fractional filling are also observed. The SPT phases in the HHMm are promising candidates for realizing strongly-correlated topological phases using cold atoms.

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

Dario Hügel, Hugo U. R. Strand, Philipp Werner +1 more

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 $\mathcal{C}\mathcal{T}$-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.

arXiv:1609.03760•Physical review. B./Physical review. B•Aug 22, 2017•26 citations

Quasi-one-dimensional Hall physics in the Harper–Hofstadter–Mott model

Filip Kozarski, Dario Hügel, Lode Pollet

We study the ground-state phase diagram of the strongly interacting Harper–Hofstadter–Mott model at quarter flux on a quasi-one-dimensional lattice consisting of a single magnetic flux quantum in y -direction. In addition to superfluid phases with various density patterns, the ground-state phase diagram features quasi-one-dimensional analogs of fractional quantum Hall phases at fillings ν = 1/2 and 3/2, where the latter is only found thanks to the hopping anisotropy and the quasi-one-dimensional geometry. At integer fillings—where in the full two-dimensional system the ground-state is expected to be gapless—we observe gapped non-degenerate ground-states: at ν = 1 it shows an odd 'fermionic' Hall conductance, while the Hall response at ν = 2 consists of the transverse transport of a single particle–hole pair, resulting in a net zero Hall conductance. The results are obtained by exact diagonalization and in the reciprocal mean-field approximation.

arXiv:1710.08931•New Journal of Physics•Feb 19, 2018•2 citations

Competing ground states of strongly correlated bosons in the Harper-Hofstadter-Mott model

Stefan S. Natu, Erich J. Mueller, S. Das Sarma

Using an efficient cluster approach, we study the physics of two-dimensional lattice bosons in a strong magnetic field in the regime where the tunneling is much weaker than the on-site interaction strength. We study both the dilute, hard-core bosons at filling factors much smaller than unity occupation per site and the physics in the vicinity of the superfluid-Mott lobes as the density is tuned away from unity. For hard-core bosons, we carry out extensive numerics for a fixed flux per plaquette $\ensuremath{\phi}=1/5$ and $\ensuremath{\phi}=1/3$. At large flux, the lowest-energy state is a strongly correlated superfluid, analogous to He-4, in which the order parameter is dramatically suppressed, but nonzero. At filling factors $\ensuremath{\nu}=1/2,1$, we find competing incompressible states which are metastable. These appear to be commensurate density wave states. For small flux, the situation is reversed and the ground state at $\ensuremath{\nu}=1/2$ is an incompressible density wave solid. Here, we find a metastable lattice supersolid phase, where superfluidity and density wave order coexist. We then perform careful numerical studies of the physics near the vicinity of the Mott lobes for $\ensuremath{\phi}=1/2$ and $\ensuremath{\phi}=1/4$. At $\ensuremath{\phi}=1/2$, the superfluid ground state has commensurate density wave order. At $\ensuremath{\phi}=1/4$, incompressible phases appear outside the Mott lobes at densities $n=1.125$ and $n=1.25$, corresponding to filling fractions $\ensuremath{\nu}=1/2$ and 1, respectively. These phases, which are absent in single-site mean-field theory, are metastable and have slightly higher energy than the superfluid, but the energy difference between them shrinks rapidly with increasing cluster size, suggestive of an incompressible ground state. We thus explore the interplay between Mott physics, magnetic Landau levels, and superfluidity, finding a rich phase diagram of competing compressible and incompressible states.

arXiv:1601.06172•Physical Review A•Jun 10, 2016•19 citations

Fractional quantum Hall effect in optical lattices

Deepak Gaur, Hrushikesh Sable, D. Angom

In this research, we study the bosonic fractional quantum Hall (FQH) states in a system of ultracold bosons in a two-dimensional optical lattice in the presence of a synthetic magnetic field, described by the bosonic Harper–Hofstadter Hamiltonian. We use the cluster Gutzwiller mean-field and exact diagonalization techniques in our work. We obtain incompressible states as ground states at various filling factors similar to those of the FQH states. We focus in particular on the ν = 1/2 FQH state, and it is characterized by the two-point correlation function and the many-body Chern number. We further investigate the effect of dipolar interaction on the ν = 1/2 FQH state. We find that the dipolar interaction stabilizes the FQH state against the competing superfluid state.

Frontiers in Physics•Jan 16, 2023•2 citations

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

Dario Hügel

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.

•Jan 1, 2018•0 citations

Symmetry-protected topological phases in spinful bosons with a flat band

Hong Yang, Hayate Nakano, Hosho Katsura

We theoretically demonstrate that interacting symmetry-protected topological (SPT) phases can be realized with ultracold spinful bosonic atoms loaded on the lattices which have a flat band at the bottom of the band structure.Ground states of such systems are not conventional Mott insulators in the sense that the ground states possess not only spin fluctuations but also non-negligible charge fluctuations.The SPT phases in such systems are determined by both spin and charge fluctuations at zero temperature.We find that the many-body ground states of such systems can be exactly obtained in some special cases, and these exact ground states turn out to serve as representative states of the SPT phases.As a concrete example, we demonstrate that spin-1 bosons on a sawtooth chain can be in an SPT phase protected by Z 2 × Z 2 spin rotation symmetry or time-reversal symmetry, and this SPT phase is a result of spin fluctuations.We also show that spin-3 bosons on a kagome lattice can be in an SPT phase protected by D 2 point group symmetry, but this SPT phase is, however, a result of charge fluctuations.

arXiv:2003.01705•Physical Review Research•Jun 14, 2021•11 citations
Since the discovery of the quantum Hall effect, the lattice geometry's influence on charged particles in magnetic fields has been the subject of extensive research. Prototypical models such as the non-interacting Harper-Hofstadter model (HHm) exhibit fractionalization of the Bloch bands with non-trivial topology, manifesting in quantum (spin) Hall phases. Ultracold atomic gases with artificial magnetic fields enabled the experimental study of the non-interacting model, while the effect of strong interactions on the band properties remains an open problem. In this work we study the strongly-interacting bosonic Harper-Hofstadter-Mott model (HHMm) using a reciprocal cluster mean-field (RCMF) method. We obtain a rich groundstate phase diagram featuring striped superfluid, supersolid, and symmetry protected topological (SPT) phases. At filling ν=2\nu=2ν=2 (in contrast to fillings ν=1,3\nu=1, 3ν=1,3) the SPT phase has no fermionic counterpart. Incompressible metastable states at fractional filling are also observed. The SPT phases in the HHMm are promising candidates for realizing strongly-correlated topological phases using cold atoms.
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.
We study the ground-state phase diagram of the strongly interacting Harper–Hofstadter–Mott model at quarter flux on a quasi-one-dimensional lattice consisting of a single magnetic flux quantum in y -direction. In addition to superfluid phases with various density patterns, the ground-state phase diagram features quasi-one-dimensional analogs of fractional quantum Hall phases at fillings ν = 1/2 and 3/2, where the latter is only found thanks to the hopping anisotropy and the quasi-one-dimensional geometry. At integer fillings—where in the full two-dimensional system the ground-state is expected to be gapless—we observe gapped non-degenerate ground-states: at ν = 1 it shows an odd 'fermionic' Hall conductance, while the Hall response at ν = 2 consists of the transverse transport of a single particle–hole pair, resulting in a net zero Hall conductance. The results are obtained by exact diagonalization and in the reciprocal mean-field approximation.
Using an efficient cluster approach, we study the physics of two-dimensional lattice bosons in a strong magnetic field in the regime where the tunneling is much weaker than the on-site interaction strength. We study both the dilute, hard-core bosons at filling factors much smaller than unity occupation per site and the physics in the vicinity of the superfluid-Mott lobes as the density is tuned away from unity. For hard-core bosons, we carry out extensive numerics for a fixed flux per plaquette $\ensuremath{\phi}=1/5$ and $\ensuremath{\phi}=1/3$. At large flux, the lowest-energy state is a strongly correlated superfluid, analogous to He-4, in which the order parameter is dramatically suppressed, but nonzero. At filling factors $\ensuremath{\nu}=1/2,1$, we find competing incompressible states which are metastable. These appear to be commensurate density wave states. For small flux, the situation is reversed and the ground state at $\ensuremath{\nu}=1/2$ is an incompressible density wave solid. Here, we find a metastable lattice supersolid phase, where superfluidity and density wave order coexist. We then perform careful numerical studies of the physics near the vicinity of the Mott lobes for $\ensuremath{\phi}=1/2$ and $\ensuremath{\phi}=1/4$. At $\ensuremath{\phi}=1/2$, the superfluid ground state has commensurate density wave order. At $\ensuremath{\phi}=1/4$, incompressible phases appear outside the Mott lobes at densities n=1.125n=1.125n=1.125 and n=1.25n=1.25n=1.25, corresponding to filling fractions $\ensuremath{\nu}=1/2$ and 1, respectively. These phases, which are absent in single-site mean-field theory, are metastable and have slightly higher energy than the superfluid, but the energy difference between them shrinks rapidly with increasing cluster size, suggestive of an incompressible ground state. We thus explore the interplay between Mott physics, magnetic Landau levels, and superfluidity, finding a rich phase diagram of competing compressible and incompressible states.
In this research, we study the bosonic fractional quantum Hall (FQH) states in a system of ultracold bosons in a two-dimensional optical lattice in the presence of a synthetic magnetic field, described by the bosonic Harper–Hofstadter Hamiltonian. We use the cluster Gutzwiller mean-field and exact diagonalization techniques in our work. We obtain incompressible states as ground states at various filling factors similar to those of the FQH states. We focus in particular on the ν = 1/2 FQH state, and it is characterized by the two-point correlation function and the many-body Chern number. We further investigate the effect of dipolar interaction on the ν = 1/2 FQH state. We find that the dipolar interaction stabilizes the FQH state against the competing superfluid state.
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 theoretically demonstrate that interacting symmetry-protected topological (SPT) phases can be realized with ultracold spinful bosonic atoms loaded on the lattices which have a flat band at the bottom of the band structure.Ground states of such systems are not conventional Mott insulators in the sense that the ground states possess not only spin fluctuations but also non-negligible charge fluctuations.The SPT phases in such systems are determined by both spin and charge fluctuations at zero temperature.We find that the many-body ground states of such systems can be exactly obtained in some special cases, and these exact ground states turn out to serve as representative states of the SPT phases.As a concrete example, we demonstrate that spin-1 bosons on a sawtooth chain can be in an SPT phase protected by Z 2 × Z 2 spin rotation symmetry or time-reversal symmetry, and this SPT phase is a result of spin fluctuations.We also show that spin-3 bosons on a kagome lattice can be in an SPT phase protected by D 2 point group symmetry, but this SPT phase is, however, a result of charge fluctuations.