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arXiv:1402.0284 · Journal of the Physical Society of Japan

Superfluid, Supersolid, and Checkerboard Solid in Two-Component Bosons in an Optical Lattice: Study by Means of Gross–Pitaevskii Theory and Monte-Carlo Simulations

Yoshihito Kuno, Keita Suzuki, Ikuo Ichinose

Published Jun 13, 2014• 3 citations
arXivHTMLPDF

Abstract

The bosonic t-J model is a strong-on-site repulsion limit of the two-component Bose-Hubbard model and is expected to be realized by experiments of cold atoms in an optical lattice. In previous papers, we studied the bosonic t-J model by both analytical methods and numerical Monte - Carlo (MC) simulations. However, in the case of finite $J_z$, where $J_z$ is the $z$-component coupling constant of the pseudospin interaction, the phase diagram of the model was investigated by assuming the checkerboard type of boson densities. In this study, we shall continue our previous study of the bosonic t-J model using both the Gross-Pitaevskii (GP) theory and MC simulations without assuming any pattern of boson densities. These two methods complement each other and give reliable results. We show that as $J_z$ is increased, the superfluid state evolves into a supersolid (SS), and furthermore into a genuine solid with the checkerboard symmetry. In the present study, we propose a method identifying quantum phase transitions in the GP theory. We also study finite-temperature phase transitions of the superfluidity and the diagonal solid order of the SS by MC simulations.

DOI
10.7566/jpsj.83.074501

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The bosonic t-J model is a strong-on-site repulsion limit of the two-component Bose-Hubbard model and is expected to be realized by experiments of cold atoms in an optical lattice. In previous papers, we studied the bosonic t-J model by both analytical methods and numerical Monte - Carlo (MC) simulations. However, in the case of finite JzJ_zJz​, where JzJ_zJz​ is the zzz-component coupling constant of the pseudospin interaction, the phase diagram of the model was investigated by assuming the checkerboard type of boson densities. In this study, we shall continue our previous study of the bosonic t-J model using both the Gross-Pitaevskii (GP) theory and MC simulations without assuming any pattern of boson densities. These two methods complement each other and give reliable results. We show that as JzJ_zJz​ is increased, the superfluid state evolves into a supersolid (SS), and furthermore into a genuine solid with the checkerboard symmetry. In the present study, we propose a method identifying quantum phase transitions in the GP theory. We also study finite-temperature phase transitions of the superfluidity and the diagonal solid order of the SS by MC simulations.
In this paper, we study an extended bosonic t-J model in an optical lattice, which describes two-component hard-core bosons with nearest-neighbor pseudospin interactions and, also, inter- and intraspecies dipole-dipole interactions. In particular, we focus on the case in which two-component hard-core bosons have antiparallel polarized dipoles with each other. The global phase diagram is studied by means of the Gutzwiller variational method and also quantum Monte Carlo (QMC) simulations. Both calculations show that a striped solid order, besides a checkerboard one, appears as a result of the dipole-dipole interactions. By QMC, we find that two kinds of supersolids (SSs) form, i.e., checkerboard SS and striped SS, and we also verify the existence of an exotic phase between the striped solid and the checkerboard SS. Finally, by QMC, we study the t-J-like model, which was recently realized experimentally by A. de Paz et al. [Phys. Rev. Lett. 111, 185305 (2013)].
In the present paper, we study finite-temperature phase structure of two-component hard-core bosons in a cubic optical lattice. The system that we study in the present paper is an effective model for the Bose-Hubbard model with strong on-site repulsions and is called bosonic t-J model. This model is a bosonic counterpart of the t-J model for the strongly-correlated electron systems like the high-temperature superconducting materials. We study the model by means of path-integral methods and Monte-Carlo simulations. We found that this system has a very rich phase structure including checkerboard-type "insulating" state, superfluid, phase-separated state, inhomogeneous cloudlet state, etc. We are also interested in the possible supersolid phase with both the checkerboard order and superfluidity and found that additional nearest-neighbor inter-species attractive force induces the supersolid state. In the supersolid state, paired superfluid appears in addition to the superfluid of single atom. This result gives important insight into mechanism of the high-temperature superconductivity of the cuprate.
We study the three-dimensional bosonic ttt-JJJ model, i.e., the ttt-JJJ model of ``bosonic electrons,'' at finite temperatures. This model describes the s=12s=\frac{1}{2}s=21​ Heisenberg spin model with the anisotropic exchange coupling ${J}_{\ensuremath{\perp}}=\ensuremath{\alpha}{J}_{z}$ and doped bosonic holes, which is an effective system of the Bose-Hubbard model with strong repulsions. The bosonic ``electron'' operator ${B}_{r\ensuremath{\sigma}}$ at the site rrr with a two-component (pseudo)spin $\ensuremath{\sigma}(=1,2)$ is treated as a hard-core boson operator. By means of Monte Carlo simulations, we study its finite-temperature phase structure including the $\ensuremath{\alpha}$ dependence, the possible phenomena-like appearance of checkerboard long-range order, supercounterflow, superfluidity, phase separation, etc. The obtained results, which clarify the relation between various phases, may be taken as predictions about experiments of two-component cold bosonic atoms in a cubic optical lattice.
This thesis is divided into two parts. In the first part we study quantum phases in bosonic systems in an optical lattice by using numerical unbiased Quantum Monte Carlo (QMC) method. Ultracold atoms experiments in optical lattices provide a possibility to simulate the materials of condensed matter in a clean and well-controled way. We focus on two components bosonic systems where exotic phases such as pair-superfluid and pair-supersolid appear. In the second part we provide numerical methods based on QMC simulations to study the entanglement properties of strong correlated systems. By employing the replica trick, we reconstruct the entanglement spectrum from the Renyi entropies. Furthermore, we study the trace of the power of the partial transposed reduced density matrix, which is related to the entanglement measurement negativity for mixed states. In the following we briefly summarize each chapter. In Chapter 1 we very briefly introduce the cold atom systems and the connection with QMC simulations. In Chapter 2 we review the QMC method based on the path integral (world line) representation. We focus on the directed worm algorithm which is an efficient algorithm with global updtae for bosonic and spin systems. All the following methods in this thesis – the method of two component bosonic systems, the reconstruction of entanglement spectrum and the measuring of partial transposed quantities, are based on directed worm algorithm introduced in this chapter. In Chapter 3 we discuss two component bosons in a square lattice. We show that the interspecies attraction and nearest-neighbor intraspecies repulsion result in the pair-supersolid phase, where a diagonal solid order coexists with an off-diagonal pair-superfluid order. The quantum and thermal transitions out of the pair-supersolid phase are characterized. It is found that there is a direct first-order transition from the pair-supersolid phase to the double-superfluid phase without an intermediate region. Furthermore, the melting of the pair-supersolid occurs in two steps. Upon heating, first the pair-superfluid is destroyed via a Kosterlitz-Thouless transition, then the solid order melts via an Ising transition. In Chapter 4 we represent a new method to reconstruct a subset of the entanglement spectrum of quantum many body systems by QMC, where the method can in principle be applied to two or higher dimension. The approach builds on the replica trick to evaluate particle number resolved traces of the first n of powers of a reduced density matrix. From this information we reconstruct first n entanglement spectrum levels using a polynomial root solver. We illustrate the power and limitations of the method by an application to the extended Bose-Hubbard model in one dimension where we are able to resolve the quasidegeneracy of the entanglement spectrum in the Haldane-insulator phase. In general, the method is able to reconstruct the largest few eigenvalues in each symmetry sector. In Chapter 5 we devise a Quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two intervals embedded in a chain. In particular, we study several scale invariant combinations of the moments for the 1D hard-core boson model. For two adjacent intervals unusual finite size corrections are present, showing parity effects that oscillate with a filling dependent period. For large chains we find perfect agreement with conformal field theory (CFT) calculations. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. The finite-size effects are smaller and QMC data are already in impressive agreement with CFT at moderate large sizes.
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.
We study the extended Bose-Hubbard model on a two-dimensional honeycomb lattice by using large-scale quantum Monte Carlo simulations. We present the ground-state phase diagrams for both the hard-core and the soft-core bosons. For the hard-core case, the transition between the $\ensuremath{\rho}=1∕2$ solid and the superfluid is first order, and the supersolid state is unstable toward phase separation. For the soft-core case, due to the presence of multiple occupation, a stable particle-induced supersolid (SS-p) phase emerges when $1∕2<\ensuremath{\rho}<1$. The transition from the solid at $\ensuremath{\rho}=1∕2$ to the SS-p phase is second order with the superfluid density scaling as ${\ensuremath{\rho}}_{s}\ensuremath{\sim}\ensuremath{\rho}\ensuremath{-}1∕2$. The SS-p phase has the same diagonal order as the solid at $\ensuremath{\rho}=1∕2$. As the chemical potential increases further, the SS-p phase turns into a solid where two bosons occupy each site of one sublattice through a first-order transition. We also calculate the critical exponents of the transition between the $\ensuremath{\rho}=1∕2$ solid and superfluid at the Heisenberg point for the hard-core case. We find the dynamical critical exponent z=0.15z=0.15z=0.15, which is smaller than results obtained on smaller lattices. This indicates that zzz approaches zero in the thermodynamic limit, and thus the transition is also first order even at the Heisenberg point.