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arXiv:1602.01559 · Physical review. B./Physical review. B

Superconducting circuit simulator of Bose-Hubbard model with a flat band

Xiu–Hao Deng, Chen-Yen Lai, Chih-Chun Chien

Published Feb 26, 2016• 18 citations
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Abstract

Recent advance in quantum simulations of interacting photons using superconducting circuits offers opportunities for investigating the Bose-Hubbard model in various geometries with hopping coefficients and self-interactions tuned to both signs. Here we investigate phenomena related to localized states associated with a flat band supported by the sawtooth geometry. A localization-delocalization transition emerges in the noninteracting regime as the sign of hopping coefficient is changed. In the presence of interactions, patterns of localized states approach a uniform density distribution for repulsive interactions while interesting localized density patterns can arise in a strongly attractive regime. The density patterns indicate the underlying inhomogeneity of the simulator. Two-particle correlations can further distinguish the nature of the localized states in attractive and repulsive interaction regimes. We also survey possible experimental implementations of the simulator.

DOI
10.1103/physrevb.93.054116

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Recent advance in quantum simulations of interacting photons using superconducting circuits offers opportunities for investigating the Bose-Hubbard model in various geometries with hopping coefficients and self-interactions tuned to both signs. Here we investigate phenomena related to localized states associated with a flat band supported by the sawtooth geometry. A localization-delocalization transition emerges in the noninteracting regime as the sign of hopping coefficient is changed. In the presence of interactions, patterns of localized states approach a uniform density distribution for repulsive interactions while interesting localized density patterns can arise in a strongly attractive regime. The density patterns indicate the underlying inhomogeneity of the simulator. Two-particle correlations can further distinguish the nature of the localized states in attractive and repulsive interaction regimes. We also survey possible experimental implementations of the simulator.
Lattices with dispersionless, or flat, energy bands have attracted substantial interest in part due to the strong dependence of particle dynamics on interactions. Using superconducting circuits, we experimentally study the dynamics of one and two particles in a single plaquette of a lattice whose band structure consists entirely of flat bands. We first observe strictly localized dynamics of a single particle, the hallmark of all-bands-flat physics. Upon initializing two particles on the same site, we see an interaction-enabled delocalized walk across the plaquette. We further find localization in Fock space for two particles initialized on opposite sides of the plaquette. These results mark the first experimental observation of a quantum walk that becomes delocalized due to interactions and establishes a key building block in superconducting circuits for studying flat-band dynamics with strong interactions.
Motivated by recent experiments realizing correlated phenomena and superconductivity in two-dimensional (2D) van der Waals devices, we consider the general problem of whether correlation effects may be enhanced by modifying band structure while keeping a fixed weak interaction strength. Using determinantal quantum Monte Carlo, we study the 2D Hubbard model for two different band structures: a regular nearest-neighbor tight-binding model and a partially flat-band structure containing a nondispersing region, with identical total noninteracting bandwidth Wtot{W}_{\mathrm{tot}}Wtot​. For both repulsive and attractive weak interactions ($|U|\ensuremath{\ll}{W}_{\mathrm{tot}}$), correlated phenomena are significantly stronger in the partially flat model. In the repulsive case, even with UUU being an order of magnitude smaller than Wtot{W}_{\mathrm{tot}}Wtot​, we find the presence of a Mott insulating state near half filling of the flat region in momentum space. In the attractive case, where generically the ground state is superconducting, the partially flat model exhibits significantly enhanced superconducting transition temperatures. These results suggest the possibility of engineering correlation effects in materials by tuning the noninteracting electronic dispersion.
The physics of strongly correlated quantum particles within a flat band was originally explored as a route to itinerant ferromagnetism and, indeed, a celebrated theorem by Lieb rigorously establishes that the ground state of the repulsive Hubbard model on a bipartite lattice with an unequal number of sites in each sublattice must have nonzero spin SSS at half filling. Recently, there has been interest in Lieb geometries due to the possibility of topological insulator, nematic, and Bose-Einstein condensed (BEC) phases. In this paper, we extend the understanding of the attractive Hubbard model on the Lieb lattice by using determinant quantum Monte Carlo to study real space charge and pair correlation functions not addressed by the Lieb theorems. Specifically, our results show unusual charge and charge transfer signatures within the flat band, and a reduction in pairing order at $\ensuremath{\rho}=2/3$ and $\ensuremath{\rho}=4/3$, the points at which the flat band is first occupied and then completely filled. We compare our results to the case of flat bands in the Kagome lattice and demonstrate that the behavior observed in the two cases is rather different.
Quantum many-body systems are expected to relax to a thermal state over time, with some exceptions such as systems with atypical eigenstates. In this study, we investigate the effect of the existence of spatially localized eigenstates on the relaxation dynamics of interacting bosons loaded into a one-dimensional sawtooth lattice, which exhibits a flat band in the single-particle spectrum by tuning the hopping rates. Using the time-evolving block decimation algorithm, we simulate the time evolution of the local density profile based on the Bose-Hubbard model with different initial conditions. Our results show the presence of the flat band leads to a significant slowing down of relaxation for weak interactions. Even for strong interaction, when the initial state includes an isolated localized single-particle eigenstate in the superposition, remnants of the initial bias in the density profile persist for a long time. This particular relaxation dynamics can be tested using ultracold atoms in optical lattices.
We study the effect of density-assisted hopping on different dimerized lattice geometries, such as bilayers and ladder structures. We show analytically that the density-assisted hopping induces an attractive interaction in the lower (bonding) band of the dimer structure and a repulsion in the upper (antibonding) band. Overcoming the on-site repulsion, this can lead to the appearance of superconductivity with a pairing structure more complex than s-wave pairing. Combining numerical and analytical methods such as the matrix product states ansatz, bosonization, and perturbative calculations, we map out the phase diagram of the two-leg ladder system and identify its superconducting phase. We characterize the transition from the non-density-assisted repulsive regime to the spin-gapped superconducting regime as a Berezinskii-Kosterlitz-Thouless transition.
The Bose-Hubbard model (BHM) of interacting bosons in a lattice has been a paradigm in many-body physics, and it exhibits a Mott insulator-superfluid (MI-SF) transition at integer filling. Here a quantum simulator of the BHM using a superconducting circuit is proposed. Specifically, a superconducting transmission line resonator supporting microwave photons is coupled to a charge qubit to form one site of the BHM, and adjacent sites are connected by a tunable coupler. To obtain a mapping from the superconducting circuit to the BHM, we focus on the dispersive regime where the excitations remain photonlike. Standard perturbation theory is implemented to locate the parameter range where the MI-SF transition may be simulated. This simulator allows single-site manipulations, and we illustrate this feature by considering two scenarios where a single-site manipulation can drive a MI-SF transition. The transition can be analyzed by mean-field analyses, and the exact diagonalization was implemented to provide accurate results. The variance of the photon density and the fidelity metric clearly show signatures of the transition. Experimental realizations and other possible applications of this simulator are also discussed.