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arXiv:2003.02411 · Physical Review A

Simulating an exact one-dimensional transverse Ising model in an optical lattice

Ren Liao, Fangyu Xiong, Xuzong Chen

Published Apr 8, 2021• 15 citations
arXivHTMLPDF

Abstract

A spinless Bose-Hubbard model in a one-dimensional (1D) double-chain tilted lattice is numerically studied at unit filling per cell. When each atom is localized in a two-site cell, the low-energy effective model gives rise to an exact textbook model of the 1D transverse Ising model via superexchange interaction. To validate the effective transverse Ising model, we calculate the energy spectrum and the nearest-neighbor correlation functions of the states in a subspace of the Bose-Hubbard model which is equivalent to the Hilbert space of a spin-1/2 magnetic model. The results show good consistency with the effective transverse Ising model, and we show that it is possible to simulate the dynamical quantum phase transition of the 1D transverse Ising model exactly with such a double-chain Bose-Hubbard model. Our results may provide some inspirations for realizing and exploring an exact 1D transverse Ising model in ultracold neutral atom systems.

DOI
10.1103/physreva.103.043312

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A spinless Bose-Hubbard model in a one-dimensional (1D) double-chain tilted lattice is numerically studied at unit filling per cell. When each atom is localized in a two-site cell, the low-energy effective model gives rise to an exact textbook model of the 1D transverse Ising model via superexchange interaction. To validate the effective transverse Ising model, we calculate the energy spectrum and the nearest-neighbor correlation functions of the states in a subspace of the Bose-Hubbard model which is equivalent to the Hilbert space of a spin-1/2 magnetic model. The results show good consistency with the effective transverse Ising model, and we show that it is possible to simulate the dynamical quantum phase transition of the 1D transverse Ising model exactly with such a double-chain Bose-Hubbard model. Our results may provide some inspirations for realizing and exploring an exact 1D transverse Ising model in ultracold neutral atom systems.
Quantum simulation of the XXZXXZXXZ model with a two-component Bose- or Fermi-Hubbard model based on a Mott insulator background has been widely used in the investigations of quantum magnetism with ultracold neutral atoms. In most cases, the diagonal spin-spin interaction is always accompanied by a large spin-exchange interaction which hinders the formation of long-range magnetic order at low temperature. Here, we show that the spin-exchange interaction can be strongly reduced in a Mott insulator of paired atoms, while the diagonal spin-spin interaction remains unaffected. Thus, the effective magnetic models become the exact quantum Ising models in a one-dimensional (1D) or two-dimensional (2D) lattice. Meanwhile, we analyzed an experimentally achievable three-component Fermi-Hubbard model of 6Li^{6}\mathrm{Li}6Li with two hyperfine levels of atoms paired in the lattice. We find the long-range antiferromagnetism of such a three-component Fermi-Hubbard model can be much stronger than that of a typical two-component Fermi-Hubbard model at low temperature. Our results may be useful for experimental investigations of the quantum phase transition and quantum criticality of the 1D and 2D quantum Ising models with ultracold neutral atoms.
We explore the dynamical properties of a one-dimensional Bose-Hubbard model, where two bosonic species interact via Feshbach resonance. We focus on the region in the phase diagram which is described by an effective, low-energy ferromagnetic Ising model in both transverse and longitudinal fields. In this regime, we numerically calculate the dynamical structure factor of the Bose-Hubbard model using the time-evolving block decimation method. In the ferromagnetic phase, we observe both the continuum of excitations and the bound states in the presence of a longitudinal field. Near the Ising critical point, we observe the celebrated E8{E}_{8}E8​ mass spectrum in the excited states. We also point out possible measurements which could be used to detect these excitations in an optical lattice experiment.
We study ultracold atoms in an optical lattice with two local minima per unit cell and show that the low-energy states of a multiband Bose-Hubbard (BH) Hamiltonian with only pairwise interactions are equivalent to an effective single-band Hamiltonian with strong three-body interactions. We focus on a double-well optical lattice with a symmetric double well along the xxx axis and single-well structure along the perpendicular directions. Tunneling and two-body interaction energies are obtained from an exact band-structure calculation and numerically constructed Wannier functions in order to construct a BH Hamiltonian spanning the lowest two bands. Our effective Hamiltonian is constructed from the ground state of the NNN-atom Hamiltonian for each unit cell obtained within the subspace spanned by the Wannier functions of the two lowest bands. The model includes hopping between ground states of neighboring unit cells. We show that such an effective Hamiltonian has strong three-body interactions that can be easily tuned by changing the lattice parameters. Finally, relying on numerical mean-field simulations, we show that the effective Hamiltonian is an excellent approximation of the two-band BH Hamiltonian over a wide range of lattice parameters, in both the superfluid and Mott insulator regions.
We study the nonequilibrium dynamics of a one-dimensional Bose-Hubbard model in a gradient potential and a superlattice, beginning from a deep Mott insulator regime with an average filling of one particle per site. Studying a quench that is near resonance to tunneling of the particles over two lattice sites, we show how a spin model emerges consisting of two coupled Ising chains that are coupled by interaction terms in a staggered geometry. We compare and contrast the behavior in this case with that in a previously studied case where the resonant tunneling was over a single site. Using optimized tensor network techniques to calculate finite-temperature behavior of the model, as well as finite-size scaling for the ground state, we conclude that the universality class of the phase transition for the coupled chains is that of a tricritical Ising point. We also investigate the out-of-equilibrium dynamics after the quench in the vicinity of the resonance and compare dynamics with recent experiments realized without the superlattice geometry. This model is directly realizable in current experiments and reflects a general way to realize spin models with ultracold atoms in optical lattices.
For systems of interacting, ultracold spin-zero neutral bosonic atoms, harmonically trapped and subject to an optical lattice potential, we derive an Extended Bose Hubbard (EBH) model by developing a systematic expansion for the Hamiltonian of the system in powers of the lattice parameters and of a scale parameter, the lattice attenuation factor. We identify the dominant terms that need to be retained in realistic experimental conditions, up to nearest-neighbor interactions and nearest-neighbor hoppings conditioned by the on-site occupation numbers. In the mean field approximation, we determine the free energy of the system and study the phase diagram both at zero and at finite temperature. At variance with the standard on site Bose Hubbard model, the zero-temperature phase diagram of the EBH model possesses a dual structure in the Mott insulating regime. Namely, for specific ranges of the lattice parameters, a density wave phase characterizes the system at integer fillings, with domains of alternating mean occupation numbers that are the atomic counterparts of the domains of staggered magnetizations in an antiferromagnetic phase. We show as well that in the EBH model, a zero-temperature quantum phase transition to pair superfluidity is, in principle, possible, but completely suppressed at the lowest order in the lattice attenuation factor. Finally, we determine the possible occurrence of the different phases as a function of the experimentally controllable lattice parameters.
Using a similarity Hamiltonian renormalization procedure, we determine an effective spin-12\frac{1}{2}21​ representation of the Bose-Hubbard model at half-integer filling and at a finite onsite interaction energy UUU. By means of bosonization, we are able to recast the effective Hamiltonian as that of a spin-12\frac{1}{2}21​ XXZ magnetic chain with pertinently renormalized coupling and anisotropy parameters. We use this mapping to provide analytical estimates of the correlation functions of the Bose-Hubbard model. We then compare such results with those based on density matrix renormalization group numerical simulations of the Bose-Hubbard model for various values of UUU and for a number LLL of lattice sites as low as $L\ensuremath{\sim}30$. We find an excellent agreement up to 1010%10 between the output of analytical and numerical computations, even for relatively small values of UUU. Our analysis implies that, also at finite UUU, the one-dimensional Bose-Hubbard model with suitably chosen parameters may be seen as a quantum simulator of the XXZ chain.