Coexistence of topological and nontopological Fermi-superfluid phases

The two-dimensional spin-imbalanced Fermi gas subject to $s$-wave pairing and spin-orbit coupling is considered a promising platform for realizing a topological chiral-$p$-wave superfluid. In the BCS limit of $s$-wave pairing, i.e., when Cooper pairs are only weakly bound, the system enters the topological phase via a second-order transition driven by increasing the Zeeman spin-splitting energy. Stronger attractive two-particle interactions cause the system to undergo the BCS-BEC crossover, in the course of which the topological transition becomes first order. As a result, topological and nontopological superfluids coexist in spatially separated domains in an extended region of phase space spanned by the strength of $s$-wave interactions and the Zeeman energy. Here we investigate this phase-coexistence region theoretically using a zero-temperature mean-field approach. Exact numerical results are presented to illustrate basic physical characteristics of the coexisting phases and to validate an approximate analytical description derived for weak spin-orbit coupling. Besides extending our current understanding of spin-imbalanced superfluid Fermi systems, the present approach also provides a platform for future studies of unconventional Majorana excitations that, according to topology, should be present at the internal interface between coexisting topological and nontopological superfluid parts of the system.

Figure 1

Mean-field phase diagram for a spin-orbit-coupled 2D Fermi gas at zero temperature and fixed density $n=m{E}_{\mathrm{F}}/\left(\pi {\hslash }^2\right)\equiv k_{\mathrm{F}}^2/\left(2\pi \right)$ in the parameter space of attractive-interaction strength (quantified in terms of the two-particle binding energy $E_{\mathrm{b}}$ in vacuum without spin-orbit coupling and Zeeman spin splitting) and Zeeman energy $h$. The blue and green curves correspond to the critical Zeeman energies $h_{<}$ and $h_{>}$ that delimit the first-order phase-transition region $h_{<}<h<h_{>}$, indicated by light-blue shading. In this region, homogeneous ground-state phases are not possible, and coexistence between domains of different superfluid phases is expected. The second-order transition between nontopological-superfluid (NSF) and topological-superfluid (TSF) phases occurs at the red curve where $h=h_{\mathrm{c}}$. Orange dashed lines illustrate the parameter range for which further data are shown in Fig. 3. Results presented here were calculated for spin-orbit-coupling strength $\lambda =0.35E_{\mathrm{F}}/k_{\mathrm{F}}$.

Figure 2

Evolution of minima in the mean-field ground-state energy density ${\mathcal{E}}_{\mathrm{gs}}^{\left(\mathrm{MF}\right)}\left(\right|\mathrm{\Delta }|,\mu )$, taken as a function of the pair-potential magnitude $\left|\mathrm{\Delta }\right|$ for fixed chemical potential $\mu$, when the Zeeman energy $h$ is increased beyond $h_{<}$. (a) For $h\lesssim h_{<}$, the global minimum (marked by the solid black circle) is at the value of $\left|\mathrm{\Delta }\right|$ that satisfies the fixed-density condition (7b) with the given value of $\mu$, thus representing a valid single-phase equilibrium ground state of the system. As $h$ is increased, the energy difference between the global minimum and another (local) minimum (marked with a cross) decreases, and (b) for $h=h_{<}$, both minima are degenerate. For $h_{<}<h<h_{>}$, the minimum associated with the self-consistent value of $\left|\mathrm{\Delta }\right|$ [indicated by the open black circle in (c)] then ceases to be the global minimum. It therefore can no longer represent a viable equilibrium ground state but remains a possible metastable state. (d) Adjusting $\mu$ to maintain degeneracy between the two minima of ${\mathcal{E}}_{\mathrm{gs}}^{\left(\mathrm{MF}\right)}$ [see Eq. (8)] determines the pair potentials $|{\mathrm{\Delta }}_{\mathrm{s}}|$ and $|{\mathrm{\Delta }}_{\mathrm{w}}|$ (marked by the solid orange circle and square, respectively) of coexisting superfluid phases. Plotted data are for $E_{\mathrm{b}}=1.0E_{\mathrm{F}}$ and $\lambda =0.75E_{\mathrm{F}}/k_{\mathrm{F}}$ (all panels) and (a) $h=1.060E_{\mathrm{F}}$, (b) $h=1.064E_{\mathrm{F}}$, and (c) and (d) $h=1.068E_{\mathrm{F}}$.

Figure 3

First-order phase transition driven by the Zeeman spin-splitting energy $h$ in a 2D Fermi gas, with fixed total particle density $n$, subject to $s$-wave pairing and linear-in-momentum spin-orbit coupling. The system has a single-phase superfluid ground state for $h<h_{<}$ and $h>h_{>}$, with corresponding values for (a)–(c) the chemical potential $\mu$ and (d)–(f) the pair-potential magnitude $\left|\mathrm{\Delta }\right|$ indicated by solid black circles (squares) if that state is a nontopological (topological) superfluid; i.e., $h<(>)\sqrt{{\mu }^2+{\left|\mathrm{\Delta }\right|}^2}$. The transition region $h_{<}<h<h_{>}$ features the coexistence of two superfluid phases, one of which (the one with larger $\left|\mathrm{\Delta }\right|$) is always nontopological, whereas the other is either nontopological or topological. The values of $\mu$ ($\left|\mathrm{\Delta }\right|$) for the coexisting phases are indicated by solid orange diamonds (solid orange circles or squares indicating their nontopological or topological character, respectively). (g)–(i) Plots of the densities of coexisting superfluid phases, together with the density fraction $x$ of the phase with smaller $\left|\mathrm{\Delta }\right|$, as a function of $h$. Results shown in the left, middle, and right columns were calculated assuming a spin-orbit-coupling strength $\lambda =0.35E_{\mathrm{F}}/k_{\mathrm{F}}$, $0.35E_{\mathrm{F}}/k_{\mathrm{F}}$, and $0.75E_{\mathrm{F}}/k_{\mathrm{F}}$ and a two-particle binding energy $E_{\mathrm{b}}=0.30E_{\mathrm{F}}$, $1.0E_{\mathrm{F}}$, and $1.0E_{\mathrm{F}}$, respectively, with $E_{\mathrm{F}}\equiv {\hslash }^2{k}_{\mathrm{F}}^2/\left(2m\right)=\pi {\hslash }^{2}n/m$. Open circles, squares, and triangles indicate values for $\mu$ and $\left|\mathrm{\Delta }\right|$ that represent self-consistent homogeneous nontopological, topological, and critical superfluid states, respectively, at the given $h$ but are not associated with a global minimum of the ground-state energy, as in the situation depicted in Fig. 2. These states are metastable and hence will not be realized in equilibrium.

Figure 4

Comparison of the approximate analytical expressions [Eqs. (11), shown as solid blue curves] for the chemical potential $\mu$ and pair-potential magnitudes $|{\mathrm{\Delta }}_{\mathrm{s}}|$ and $|{\mathrm{\Delta }}_{\mathrm{w}}|$ in the phase-coexistence region with exact numerical results (shown by symbols with the same conventions as in Fig. 3). (a) and (c) are for the same situation as depicted in Figs. 3 and 3 ($\lambda =0.35E_{\mathrm{F}}/k_{\mathrm{F}}$, $E_{\mathrm{b}}=1.0E_{\mathrm{F}}$). The results shown in (b) and (d) were calculated assuming $\lambda =0.20E_{\mathrm{F}}/k_{\mathrm{F}}$ and $E_{\mathrm{b}}=1.5E_{\mathrm{F}}$. For illustration, dashed blue curves show results for the $\lambda =0$ case according to exact analytical formulas from Ref. [16].

Figure 5

Comparison of the approximate analytical expressions (13) and (13) for the critical Zeeman energies $h_{<}$ and $h_{>}$ (shown as long-dashed magenta curves) with exact numerical results (shown as the solid blue and green curves) for the case $\lambda =0.25E_{\mathrm{F}}/k_{\mathrm{F}}$. The red curve indicates $h_{\mathrm{c}}$ defined in Eq. (1). Phase coexistence is between a TSF and an NSF when $h_{>}>h_{\mathrm{c}}$. This is the regime for which the analytical formulas were derived and where the agreement is excellent. The inset shows results for a reduced parameter range together with the $\lambda =0$ critical fields plotted as the short-dashed blue and green curves.