Variational study of phase diagrams of spin–orbit coupled bosons

Abstract

We use a variational order method to explore the ground state phase diagrams of spin-orbit coupled Bose atoms in optical lattices. By properly parameterizing the order with an ansatz function for each possible phase and comparing their energies which are minimized with respect to the variational parameters, we can effectively identify the lowest energy states and depict the phase diagram. While the Mott-insulator-superfluid phase boundary is computed by the usual Gutzwiller method, the spinordered phase structures of the deep Mott regime as well as the superfluid regime are studied by the variational order method. For the isotropic spin-orbit coupling, the phase diagram in the deep Mott insulator regime is qualitatively similar to results derived by the classical Monte-Carlo simulations or other numerical methods. The spiral phases with different spatial periods are discussed in detail. We find three new spin-ordered phases in the deep Mott insulator regime with anisotropic spin-orbit coupling. We also identify the uniform superfluid, stripe and checkerboard supersolid phases with exotic spin orders in the superfluid regime. ( | | ) ] ( )ˆˆˆÊ J J J J J J

Magnetic phase transitions of insulating spin-orbit coupled Bose atoms in one-dimensional optical lattices

“…σ1 Tr M σ1[1] † M σ1 [1] , (25) where the dominate is the norm square of the state. When reduced to single-site and two-site cases, the calculation is greatly simplified by making use of the normalization conditions.…”

“…It may adjust the anisotropic couplings and then lead to the so-called Dzyaloshinskii-Moriya (DM) [19,20] exchange interaction, which induces exotic magnetic phases. In two-dimensional (2D) systems, besides normal ferromagnetic and antiferromagnetic phases, the coexistence of DM interaction and anisotropy give spiral phases, vortex crystal structure and novel Skymion [21][22][23][24][25]. Classical phase diagrams have been obtained by Monte Carlo simulations [22,24], steepest descent minimization * Emails: lichaoh2@mail.sysu.edu.cn, chleecn@gmail.com method [23] and variational mean-field approach [25].…”

“…In two-dimensional (2D) systems, besides normal ferromagnetic and antiferromagnetic phases, the coexistence of DM interaction and anisotropy give spiral phases, vortex crystal structure and novel Skymion [21][22][23][24][25]. Classical phase diagrams have been obtained by Monte Carlo simulations [22,24], steepest descent minimization * Emails: lichaoh2@mail.sysu.edu.cn, chleecn@gmail.com method [23] and variational mean-field approach [25]. In one-dimensional systems, the effective model reduces into an anisotropic Heisenberg chain with one-component DM interaction.…”

Magnetic supersolid phases of two-dimensional extended Bose–Hubbard model with spin–orbit coupling

“…The effective super-exchange spin model with the Dzyaloshinskii-Moriya type interaction can be obtained by the second-order perturbation theory [51,52] in the MI regime of two-dimensional (2D) spin-orbit coupled BH model. The spiral, vortex crystal and skyrmion crystal magnetic structures, are found by applying the classical Monte-Carlo (MC) simulations, bosonic dynamical mean-field theory, variational order and tensor network states methods [53][54][55][56][57][58][59][60][61][62][63][64]. The effects of the strength and symmetry of SOC on the SF phase and MI-SF phase transition are also investigated.…”

Supersolid phases in Bose gases with three-dimensional spin–orbit coupling

“…The occurrences of the supersolid (SS) phase have been carried out in atomic gases with dipole-dipole [40][41][42][43], and soft core, finite range interactions [44][45][46]. The SS phase is also predicted in the two-component BEC with SOC, the translational symmetry of the space is broken, and the spontaneous density modulation that realizes the SS phase [7,8,12,[47][48][49]. The stability of the SS phase in the two-component BEC with SOC are proven by the Bogoliubov theory and the Lee-Huang-Yang correction [49][50][51][52][53].…”