Finite-temperature ferromagnetic transition in coherently coupled Bose gases

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.

Figure 1

Schematic of the phase diagram of the ground state of a homogeneous coherently coupled condensate $(\mathrm{\Omega }\ne 0)$ at zero temperature. The solid blue line separates the paramagnetic from the ferromagnetic regime. For $T\ne 0$ and $\mathrm{\Omega }\ne 0$, the transition is expected to get shifted and broadened. A detailed analysis for uncoupled ($\mathrm{\Omega }=0$) Bose condensed mixtures at $T\ne 0$ has been carried out in our previous work [66].

Figure 2

Magnetization $Z=(N_1-N_2)/(N_1+N_2)$ vs relative detuning $\delta /\mathrm{\Omega }$, calculated by solving the Gross-Pitaevskii equation (1) at $T=0$ for a cycle in which the value of $\delta$ is changed in time from ${\delta }_0$ to $-{\delta }_0$ and then back to ${\delta }_0$ again, at the same rate. Here we fix ${\delta }_0/\mathrm{\Omega }=5$ and $\mathrm{\Omega }=0.1gn$. If the initial configuration is paramagnetic, as for $g_{12}/{\overline{g}}_{12}=0.75$, the magnetization curve is the same in both directions and shows no magnetization at zero detuning. Conversely, if the initial configuration is ferromagnetic, as for $g_{12}/{\overline{g}}_{12}>1$, it exhibits an hysteresis curve with finite magnetization at zero detuning.

Figure 3

(a–c) Typical magnetization density profiles obtained by evolving Eq. (5) for 3 s, with zero detuning ($\delta =0$). Each simulation starts from a purely random $c$ field. The temperature is the same, $T/T_g=0.2$, while the interaction parameter is $g_{12}/{\overline{g}}_{12}=0.95,1.01$ and 1.02, respectively. The color scale refers to the quantity $(n_1-n_2)/(n_1+n_2)$. (d) Magnetization $Z$ during the full time evolution. The markers at the end of the trajectories correspond to the snapshots in (a)–(c).

Figure 4

Results of SGPE simulations at finite temperature and zero detuning for (a) the average magnetization $\mathcal{Z}$ [see Eq. (8)], (b) its variance ${\left(\mathrm{\Delta }Z\right)}^2$, and (c) the dominant relaxation frequency ${\omega }_{\mathrm{M}}$. In the main plots, these quantities are given as a function of the interaction parameter $g_{12}/g_{12}^c\left(0\right)$, where $g_{12}^c\left(0\right)$ is the critical value at $T=0$ defined in Eq. (10). The correspondence between temperature, in units of $T_g=gn/k_B$, and type of markers is given in the legend in (c). The black dashed line in (a) is the $T=0$ mean-field magnetization obtained from Eq. (4) for a homogeneous gas. (d) Variation of the critical interaction parameter $g_{12}^c\left(T\right)$ with temperature, as extracted from the location of the maxima of ${\left(\mathrm{\Delta }Z\right)}^2$ in (b) and the minima of ${\omega }_{\mathrm{M}}$ in (c); the difference between the two estimates is smaller than the error bar, which is identified as the grid spacing in the parameter space of SGPE simulations; the dashed line is a linear fit. The inset in (a) shows the magnetization vs the relative distance from the critical point, defined in Eq. (11), with a power-law rescaling as in (13), with $a=3/5$, producing a collapse of all points on a single curve in the critical region; in the inset of (b) and (c), a similar rescaling is applied to ${\left(\mathrm{\Delta }Z\right)}^2$, according to Eq. (12), and to ${\omega }_{\mathrm{M}}$, respectively, with the same exponent $a$. The dashed line in the inset of (c) corresponds to the ${\mathrm{\Delta }}_{{}_{\mathrm{para}}}^2$, Eq. (18), and the solid line on the right corresponds to ${\mathrm{\Delta }}_{{}_{\mathrm{ferro}}}^2$, Eq. (19).

Figure 5

Average magnetization $\mathcal{Z}$ [see Eq. (8)] as a function of $T/T_g$ for fixed values of $g_{12}/g_{12}^c\left(0\right)$. Points correspond to the results of SGPE as in Fig. 4, while lines are a guide to the eye. The same data are shown in the inset, but plotted as a function of the relative distance from the critical temperature, $\delta T=(T-T_c)/T_g$, where $T_c$ is extracted from the linear fit to the critical points in Fig. 4. The dashed line represents the power law ${\left|\delta T\right|}^{1/2}$.

Figure 6

Relation between ${\left(\mathrm{\Delta }Z\right)}^2$ and the magnetic susceptibility $\chi$; dashed lines represent the prediction of the fluctuation-dissipation theorem according to Eq. (21).