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arXiv:2212.12253 · Physical Review A

Finite-temperature ferromagnetic transition in coherently coupled Bose gases

Arko Roy, Miki Ota, F. Dalfovo, Alessio Recati

Published Apr 3, 2023• 5 citations
arXivHTMLPDF

Abstract

A paramagnetic-ferromagnetic quantum phase transition is known to occur at zero temperature in a two-dimensional coherently coupled Bose mixture of dilute ultracold atomic gases provided the interspecies interaction strength is large enough. Here we study the fate of such a transition at finite temperature by performing numerical simulations with the stochastic (projected) Gross-Pitaevskii formalism, which includes both thermal and beyond mean-field effects. By extracting the average magnetization, the magnetic fluctuations and characteristic relaxation frequency (or critical slowing down), we identify a finite-temperature critical line for the transition. We find that the critical point shifts linearly with temperature and, in addition, the three quantities used to probe the transition exhibit a temperature power-law scaling. The scaling of the critical slowing down is found to be consistent with thermal critical exponents and is very well approximated by the square of the spin excitation gap at zero temperature.

DOI
10.1103/physreva.107.043301

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A paramagnetic-ferromagnetic quantum phase transition is known to occur at zero temperature in a two-dimensional coherently coupled Bose mixture of dilute ultracold atomic gases provided the interspecies interaction strength is large enough. Here we study the fate of such a transition at finite temperature by performing numerical simulations with the stochastic (projected) Gross-Pitaevskii formalism, which includes both thermal and beyond mean-field effects. By extracting the average magnetization, the magnetic fluctuations and characteristic relaxation frequency (or critical slowing down), we identify a finite-temperature critical line for the transition. We find that the critical point shifts linearly with temperature and, in addition, the three quantities used to probe the transition exhibit a temperature power-law scaling. The scaling of the critical slowing down is found to be consistent with thermal critical exponents and is very well approximated by the square of the spin excitation gap at zero temperature.
We investigate the ferromagnetic-paramagnetic phase transition in coherently (Rabi) coupled Bose-Einstein condensates at zero and finite temperatures, exploring different routes to the transition by tuning the Rabi coupling or increasing the temperature at a fixed coupling. Using the Hartree-Fock-Bogoliubov theory within the Popov approximation, we map out the finite-temperature phase diagram of a three-dimensional homogeneous condensate and identify the critical line through the softening of the spin gap. Magnetization and the spin dispersion branch reveal the progressive suppression of the ferromagnetic order with increasing temperature. In quasi-one-dimensional harmonic traps, the transition, driven by Rabi coupling, is inferred through the softening of the spin breathing mode with its minimum shifting to lower coupling values with increasing temperature. Notably, the thermally driven transition causes monotonic hardening of all the spin modes. For both coupling and temperature-driven transition, the hybridized density modes in the ferromagnetic phase acquire more density character while approaching the critical point.
The miscibility condition for a binary mixture of two interacting Bose-Einstein condensates is shown to be deeply affected by interaction driven thermal fluctuations. These give rise to a first order phase transition to a demixed phase with full spatial separation of the two condensates, even if the mixture is miscible at zero temperature. Explicit predictions for the isothermal compressibility, the spin susceptibility, and the phase transition temperature T_{M} are obtained in the framework of Popov theory, which properly includes beyond mean-field quantum and thermal fluctuations in both the spin and density channels. For a mixture of two sodium condensates occupying the hyperfine states |F=1⟩,|m_{F}=1⟩ and |F=1,m_{F}=-1⟩, respectively, T_{M} is predicted to occur at about 0.7 times the usual BEC critical temperature.
This paper develops and implements the stochastic projected Gross-Pitaevskii equation for spin-1 Bose gases, addressing key considerations for numerical simulations. As an application of the theory we explore equilibrium phases in a two-dimensional spin-1 gas, where quasi-long-range order emerges via a Berezinskii-Kosterlitz-Thouless transition. Our analysis includes the definition of superfluid densities for both mass and spin degrees of freedom in a manner suitable for implementation within a stochastic projected Gross-Pitaevskii equation simulation. We present a finite-temperature phase diagram for the ferromagnetic spin-1 Bose gas and identify three distinct superfluid phases: two exhibiting conventional Berezinskii-Kosterlitz-Thouless-like behavior and an additional phase that simultaneously supports independent mass and spin superflows. As temperature increases, the stability region of this additional phase shrinks. We provide a characterization of the phase transitions through consideration of the spin-component densities and the unbinding of multiple types of vortices. This work provides a foundation for further studies of nonequilibrium and finite-temperature phenomena in spinor Bose gases.
We investigate condensation and finite-temperature magnetic phase transitions in a coherently (Rabi) coupled Bose gas confined in a three-dimensional harmonic trap. For the noninteracting system, we derive analytical expressions for the critical temperature and condensate fraction using a semiclassical description of the single-particle spectrum and density of states. At fixed particle number, coherent coupling enhances the critical temperature relative to the uncoupled system, and this enhancement decreases with increasing particle number NNN. The finite-size correction, in contrast, lowers the transition temperature, with its effect diminishing for larger NNN. We then incorporate repulsive interactions within the Hartree--Fock--Bogoliubov--Popov framework to investigate the finite-temperature phase diagram in the temperature--Rabi coupling plane. The system undergoes successive transitions from a ferromagnetic to a paramagnetic condensate and, at higher temperature, to a thermal gas. The transition boundaries obtained from the vanishing of the condensate fraction show excellent agreement with the analytical Hartree-Fock predictions. We find that the interactions substantially suppress the variation in the condensation temperature with an increase in the coherent coupling.
Abstract At temperatures well below the Fermi temperature T F , the coupling of magnetic fluctuations to particle-hole excitations in a two-component Fermi gas leads to non-analytic corrections in the renormalized free energy and makes the transition to itinerant ferromagnetism first order. On the other hand, despite that larger symmetry often introduces larger degeneracies in the low-lying states, here we show that for a Fermi gas with SU(N > 2)-symmetry in three space dimensions the ferromagnetic phase transition is first order in agreement with the predictions of Landau’s mean-field theory in its minimal formulation, which contains a cubic term in the free-energy (Cazalilla et al 2009 New J. Phys. 11 103033). By performing unrestricted Hartree–Fock calculations for an SU(N > 2)-symmetric Fermi gas with short range interactions, we find the order parameter undergoes a finite jump across the transition. In addition, for SU(N > 2) we do not observe any tri-critical point up to temperatures T ≃ 0.5 T F , for which the thermal smearing of the Fermi surface and thermal fluctuations are substantial. Going beyond mean-field theory, we find that the coupling of magnetic fluctuations to particle-hole excitations makes the transition more abrupt and further enhances the tendency of the gas to become fully polarized for smaller values of N and the gas parameter k F a s . In our study, we also clarify the role of time reversal symmetry in the microscopic Hamiltonian and obtain the temperature dependence of Tan’s contact. For the latter, the presence of the tri-critical point for N = 2 leads to a more pronounced temperature dependence around the transition than for SU(N > 2)-symmetric gases. Although the results are obtained for a microscopic model relevant to ultracold atomic gases, they may also apply to models of solid state systems with SU(N > 2) symmetry provided Coulomb interactions are sufficiently well screened.
We find nearly all the exact ground states of a mixture of two species of spin-1 atoms with both interspecies and intraspecies spin exchanges in the absence of a magnetic field. The quantum phase diagram in the three-dimensional parameter space and its two-dimensional cross sections are described. The boundaries where the ground states are either continuous or discontinuous are determined, with the latter identified as where quantum phase transitions take place. The two species are always disentangled if the interspecies spin coupling is ferromagnetic or zero. Quantum phase transitions occur when the interspecies spin coupling varies between antiferromagnetic and zero or ferromagnetic while the two intraspecies spin couplings both remain ferromagnetic. On the other hand, by tuning the interspecies spin coupling from zero to antiferromagnetic and then back to zero, one can circumvent the quantum phase transition due to sign change of the intraspecies spin coupling of a single species, which is spin decoupled with the other species with ferromagnetic intraspecies spin coupling. Generally speaking, interplay among interspecies and two intraspecies spin exchanges significantly enriches quantum phases of spinor atomic gases.