Two-dimensional Bose-Hubbard Model Research Articles

Resistivity of the two-dimensional Bose-Hubbard model at weak coupling

Resistivity of the two-dimensional Bose-Hubbard model at weak coupling

Tensor network simulation of the quantum Kibble-Zurek quench from the Mott to the superfluid phase in the two-dimensional Bose-Hubbard model

Quantum simulations of the Bose-Hubbard model (BHM) at commensurate filling can follow spreading of correlations after a sudden quench for times long enough to estimate their propagation velocities. In this work we perform tensor network simulation of the quantum Kibble-Zurek (KZ) ramp from the Mott towards the superfluid phase in the square lattice BHM and demonstrate that even relatively short ramp/quench times allow one to test the power laws predicted by the KZ mechanism (KZM). They can be verified for the correlation length and the excitation energy but the most reliable test is based on the KZM scaling hypothesis for the single-particle correlation function: scaled correlation functions for different quench times evaluated at the same scaled time collapse to the same scaling function of the scaled distance. The scaling of the space and time variables is done according to the KZ power laws.

Quantum extraordinary-log universality of boundary critical behavior

The recent discovery of extraordinary-log universality has generated intense interest in classical and quantum boundary critical phenomena. Despite tremendous efforts, the existence of quantum extraordinary-log universality remains extremely controversial. Here, by utilizing quantum Monte Carlo simulations, we study the quantum edge criticality of a two-dimensional Bose-Hubbard model featuring emergent bulk criticality. On top of an insulating bulk, the open edges experience a Kosterlitz-Thouless-like transition into the superfluid phase when the hopping strength is sufficiently enhanced on edges. At the bulk critical point, the open edges exhibit the special, ordinary, and extraordinary critical phases. In the extraordinary phase, logarithms are involved in the finite-size scaling of two-point correlation and superfluid stiffness, which admit a classical-quantum correspondence for the extraordinary-log universality. Thanks to modern quantum emulators for interacting bosons in lattices, the edge critical phases might be realized in experiments.

Tensor-network study of correlation-spreading dynamics in the two-dimensional Bose-Hubbard model

Recent developments in analog quantum simulators based on cold atoms and trapped ions call for cross-validating the accuracy of quantum-simulation experiments with use of quantitative numerical methods; however, it is particularly challenging for dynamics of systems with more than one spatial dimension. Here we demonstrate that a tensor-network method running on classical computers is useful for this purpose. We specifically analyze real-time dynamics of the two-dimensional Bose-Hubbard model after a sudden quench starting from the Mott insulator by means of the tensor-network method based on infinite projected entangled pair states. Calculated single-particle correlation functions are found to be in good agreement with a recent experiment. By estimating the phase and group velocities from the single-particle and density-density correlation functions, we predict how these velocities vary in the moderate interaction region, which serves as a quantitative benchmark for future experiments and numerical simulations.

Spin-orbit-coupling-driven superfluid states in optical lattices at zero and finite temperatures

We investigate the quantum phase transitions of a two-dimensional Bose-Hubbard model in the presence of a Rashba spin-orbit coupling with and without thermal fluctuations. The interplay of single-particle hopping, strength of spin-orbit coupling, and interspin interaction leads to superfluid phases with distinct properties. With interspin interactions weaker than intraspin interactions, the spin-orbit coupling induces two finite-momentum superfluid phases. One of them is a phase-twisted superfluid that exists at low hopping strengths and reduces the domain of insulating phases. At comparatively higher hopping strengths, there is a transition from the phase-twisted to a finite momenta stripe superfluid. With interspin interactions stronger than the intraspin interactions, the system exhibits phase-twisted to ferromagnetic phase transition. At finite temperatures, the thermal fluctuations destroy the phase-twisted superfluidity and lead to a wide region of normal-fluid states. These findings can be observed in recent quantum gas experiments with spin-orbit coupling in optical lattices.

Two-dimensional Bose-Hubbard model for helium on graphene

An exciting development in the field of correlated systems is the possibility of realizing two-dimensional (2D) phases of quantum matter. For a systems of bosons, an example of strong correlations manifesting themselves in a 2D environment is provided by helium adsorbed on graphene. We construct the effective Bose-Hubbard model for this system which involves hard-core bosons $(U\approx\infty)$, repulsive nearest-neighbor $(V>0)$ and small attractive $(V'<0)$ next-nearest neighbor interactions. The mapping onto the Bose-Hubbard model is accomplished by a variety of many-body techniques which take into account the strong He-He correlations on the scale of the graphene lattice spacing. Unlike the case of dilute ultracold atoms where interactions are effectively point-like, the detailed microscopic form of the short range electrostatic and long range dispersion interactions in the helium-graphene system are crucial for the emergent Bose-Hubbard description. The result places the ground state of the first layer of $^4$He adsorbed on graphene deep in the commensurate solid phase with $1/3$ of the sites on the dual triangular lattice occupied. Because the parameters of the effective Bose-Hubbard model are very sensitive to the exact lattice structure, this opens up an avenue to tune quantum phase transitions in this solid-state system.

Numerical Evidence for Many-Body Localization in Two and Three Dimensions.

Disorder and interactions can lead to the breakdown of statistical mechanics in certain quantum systems, a phenomenon known as many-body localization (MBL). Much of the phenomenology of MBL emerges from the existence of ℓ bits, a set of conserved quantities that are quasilocal and binary (i.e., possess only ±1 eigenvalues). While MBL and ℓ bits are known to exist in one-dimensional systems, their existence in dimensions greater than one is a key open question. To tackle this question, we develop an algorithm that can find approximate binary ℓ bits in arbitrary dimensions by adaptively generating a basis of operators in which to represent the ℓ bit. We use the algorithm to study four models: the one-, two-, and three-dimensional disordered Heisenberg models and the two-dimensional disordered hard-core Bose-Hubbard model. For all four of the models studied, our algorithm finds high-quality ℓ bits at large disorder strength and rapid qualitative changes in the distributions of ℓ bits in particular ranges of disorder strengths, suggesting the existence of MBL transitions. These transitions in the one-dimensional Heisenberg model and two-dimensional Bose-Hubbard model coincide well with past estimates of the critical disorder strengths in these models, which further validates the evidence of MBL phenomenology in the other two- and three-dimensional models we examine. In addition to finding MBL behavior in higher dimensions, our algorithm can be used to probe MBL in various geometries and dimensionality.

Finite-size scaling analysis of localization transitions in the disordered two-dimensional Bose-Hubbard model within the fluctuation operator expansion method

The disordered Bose-Hubbard model in two dimensions at non-integer filling admits a superfluid to Bose-glass transition at weak disorder. Less understood are the properties of this system at strong disorder and energy densities corresponding to excited states. In this work we study the Bose-glass transition of the ground state and the related finite energy localization transition, the mobility edge of the quasiparticle spectrum, a critical energy separating extended from localized quasiparticle excitations. To study these the fluctuation operator expansion is used. The level spacing statistics of the quasiparticle excitations, the fractal dimension and decay of the corresponding wave functions are consistent with a many-body mobility edge. The finite-size scaling of the lowest gaps yields a correction to the mean-field prediction of the superfluid to Bose-glass transition. In its vicinity we discuss spectral properties of the ground state in terms of the dynamic structure factor and the spectral function which also shows distinct behavior above and below the mobility edge.

Mobility edge of the two-dimensional Bose-Hubbard model

We analyze the disorder driven localization of the two dimensional Bose-Hubbard model by evaluating the full low energy quasiparticle spectrum via a recently developed fluctuation operator expansion method. For any strength of the local interaction we find a mobility edge that displays an approximately exponential decay with increasing disorder strength. We determine the finite-size scaling collapse and exponents at this critical line finding that the localization of excitations is characterized by weak multi-fractality and a thermal-like critical gap ratio. A direct comparison to a recent experiment yields an excellent match of the predicted finite-size transition point and scaling of single particle correlations.

Entanglement measures and nonequilibrium dynamics of quantum many-body systems: A path integral approach

We present a path integral formalism for expressing matrix elements of the density matrix of a quantum many-body system between any two coherent states in terms of standard Matsubara action with periodic(anti-periodic) boundary conditions on bosonic(fermionic) fields. We show that this enables us to express several entanglement measures for bosonic/fermionic many-body systems described by a Gaussian action in terms of the Matsubara Green function. We apply this formalism to compute various entanglement measures for the two-dimensional Bose-Hubbard model in the strong-coupling regime, both in the presence and absence of Abelian and non-Abelian synthetic gauge fields, within a strong coupling mean-field theory. In addition, our method provides an alternative formalism for addressing time evolution of quantum-many body systems, with Gaussian actions, driven out of equilibrium without the use of Keldysh technique. We demonstrate this by deriving analytical expressions of the return probability and the counting statistics of several operators for a class of integrable models represented by free Dirac fermions subjected to a periodic drive in terms of the elements of their Floquet Hamiltonians. We provide a detailed comparison of our method with the earlier, related, techniques used for similar computations, discuss the significance of our results, and chart out other systems where our formalism can be used.

Phase diagrams of the disordered Bose-Hubbard model with cavity-mediated long-range and nearest-neighbor interactions

Recent experiments with ultracold atoms in an optical lattice have realized cavity-mediated long-range interaction and observed the emergence of a supersolid phase and a density wave phase in addition to Mott insulator and superfluid phases. Here we consider theoretically the effect of uncorrelated disorder on the phase diagram of this system and study the two-dimensional Bose-Hubbard model with cavity-mediated long-range interactions and uncorrelated diagonal disorder. We also study the phase diagram of the extended Bose-Hubbard model with nearest-neighbor interactions in the presence of uncorrelated diagonal disorder. The extended Bose-Hubbard model with nearest-neighbor interactions has been realized in the experiment using dipolar interaction recently. With the help of quantum Monte Carlo simulations using the worm algorithm, we determine the phase diagram of those two models. We compare the phase diagrams of cavity-mediated long-range interactions with nearest-neighbor interactions. We show that two kinds of Bose glass phases exist: one with and one without density wave order. We also find that weak disorder enhances the supersolid phase.Graphical abstract

The Effect of Disorder on the Phase Diagrams of Hard-Core Lattice Bosons With Cavity-Mediated Long-Range and Nearest-Neighbor Interactions

We use quantum Monte Carlo simulations with the worm algorithm to study the phase diagram of a two-dimensional Bose-Hubbard model with cavity-mediated long-range interactions and uncorrelated disorder in the hard-core limit. Our study shows the system is in a supersolid phase at weak disorder and a disordered solid phase at stronger disorder. Due to long-range interactions, a large region of metastable states exists in both clean and disordered systems. By comparing the phase diagrams for both clean and disordered systems, we find that disorder suppresses metastable states and superfluidity. We compare these results with the phase diagram of the extended Bose-Hubbard model with nearest-neighbor interactions. Here, the supersolid phase does not exist even at weak disorder. We identify two kinds of glassy phases: a Bose glass phase and a disordered solid phase. The glassy phases intervene between the density-wave and superfluid phases as the Griffiths phase of the Bose-Hubbard model. The disordered solid phase intervenes between the density-wave and Bose glass phases since both have a finite structure factor.

Variational Monte-Carlo study of the extended Bose-Hubbard model with short- and infinite-range interactions

In this paper, we study the two-dimensional Bose-Hubbard model with short- and long-range interactions in the canonical ensemble. Using a Variational Monte-Carlo method, we obtain the phase diagrams containing four different phases: Mott insulator, density wave, superfluid and supersolid. The transition lines are determined using the structure factor and the superfluid density. We observe that the phase diagrams for short and long-range interactions are very similar quantitatively and qualitatively, but show also some significant differences. To evaluate the quality of our Variational approach, we compare our results with results published by other groups, obtained with various methods, such as an exact quantum Monte-Carlo algorithm and Gutzwiller Ansatz.

Anomalous pairing of bosons: Effect of multibody interactions in an optical lattice

An interesting first order type phase transition between Mott lobes has been reported in Phys. Rev. Lett. 109, 135302 (2012) for a two-dimensional Bose-Hubbard model in the presence of attractive three-body interaction. We re-visit the scenario in a system of ultracold bosons in a one-dimensional optical lattice using the density matrix renormalization group method and show that an unconventional pairing of particles occurs due to the competing two-body repulsive and three-body attractive interactions. This leads to a pair superfluid phase sandwiched between the Mott insulator lobes corresponding to densities $\rho=1$ and $\rho=3$ in the strongly interacting regime. We further extend our analysis to a two dimensional Bose-Hubbard model using the self consistent cluster-mean-field theory approach and confirm that the unconventional pair superfluid phase stabilizes in the region between the Mott lobes in contrast to the direct first order jump as predicted before. In the end we establish connection to the most general Bose-Hubbard model and analyse the fate of the pair superfluid phase in presence of an external trapping potential.

Slater determinant and exact eigenstates of the two-dimensional Fermi–Hubbard model**roject supported by the National Natural Science Foundation of China (Grant No. 11674059) and Natural Science Foundation of Fujian Province, China (Grant Nos. 2016J01008 and 2016J01009).

We consider the construction of exact eigenstates of the two-dimensional Fermi–Hubbard model defined on an L × L lattice with a periodic condition. Based on the characteristics of Slater determinants, several methods are introduced to construct exact eigenstates of the model. The eigenstates constructed are independent of the on-site electron interaction and some of them can also represent exact eigenstates of the two-dimensional Bose–Hubbard model.

Hypergeometric continuation of divergent perturbation series: II. Comparison with Shanks transformation and Padé approximation

We explore in detail how analytic continuation of divergent perturbation series by generalized hypergeometric functions is achieved in practice. Using the example of strong-coupling perturbation series provided by the two-dimensional Bose–Hubbard model, we compare hypergeometric continuation to Shanks and Padé techniques, and demonstrate that the former yields a powerful, efficient and reliable alternative for computing the phase diagram of the Mott insulator-to-superfluid transition. In contrast to Shanks transformations and Padé approximations, hypergeometric continuation also allows us to determine the exponents which characterize the divergence of correlation functions at the transition points. Therefore, hypergeometric continuation constitutes a promising tool for the study of quantum phase transitions.

Hypergeometric continuation of divergent perturbation series: I. Critical exponents of the Bose–Hubbard model

We study the connection between the exponent of the order parameter of the Mott insulator-to-superfluid transition occurring in the two-dimensional Bose–Hubbard model, and the divergence exponents of its one- and two-particle correlation functions. We find that at the multicritical points all divergence exponents are related to each other, allowing us to express the critical exponent in terms of one single divergence exponent. This approach correctly reproduces the critical exponent of the three-dimensional XY universality class. Because divergence exponents can be computed in an efficient manner by hypergeometric analytic continuation, our strategy is applicable to a wide class of systems.

Phase diagram of bosons in a two-dimensional optical lattice with infinite-range cavity-mediated interactions

A high-finesse optical cavity allows the establishment of long-range interactions between bosons in an optical lattice when most cold-atom experiments are restricted to short-range interactions. Supersolid phases have recently been experimentally observed in such systems. Using both exact quantum Monte Carlo simulations and the Gutzwiller approximation, we study the ground-state phase diagrams of a two-dimensional Bose-Hubbard model with infinite-range interactions which describes such experiments. In addition to superfluid and insulating Mott phases, the infinite-range checkerboard interactions introduce charge-density waves and supersolid phases. We study here the system at various particle densities, elucidate the nature of the phases and quantum phase transitions, and discuss the stability of the phases with respect to phase separation. In particular we confirm the existence and stability of a supersolid phase detected experimentally.

Locating the quantum critical point of the Bose-Hubbard model through singularities of simple observables.

We show that the critical point of the two-dimensional Bose-Hubbard model can be easily found through studies of either on-site atom number fluctuations or the nearest-neighbor two-point correlation function (the expectation value of the tunnelling operator). Our strategy to locate the critical point is based on the observation that the derivatives of these observables with respect to the parameter that drives the superfluid-Mott insulator transition are singular at the critical point in the thermodynamic limit. Performing the quantum Monte Carlo simulations of the two-dimensional Bose-Hubbard model, we show that this technique leads to the accurate determination of the position of its critical point. Our results can be easily extended to the three-dimensional Bose-Hubbard model and different Hubbard-like models. They provide a simple experimentally-relevant way of locating critical points in various cold atomic lattice systems.

Prethermal Floquet Steady States and Instabilities in the Periodically Driven, Weakly Interacting Bose-Hubbard Model

We explore prethermal Floquet steady states and instabilities of the weakly interacting two-dimensional Bose-Hubbard model subject to periodic driving. We develop a description of the nonequilibrium dynamics, at arbitrary drive strength and frequency, using a weak-coupling conserving approximation. We establish the regimes in which conventional (zero-momentum) and unconventional [(π,π)-momentum] condensates are stable on intermediate time scales. We find that condensate stability is enhanced by increasing the drive strength, because this decreases the bandwidth of quasiparticle excitations and thus impedes resonant absorption and heating. Our results are directly relevant to a number of current experiments with ultracold bosons.