Time-reversal invariant topological superfluids in Bose-Fermi mixtures

TL;DR

This paper proposes a realistic cold atomic system setup using Bose-Fermi mixtures to realize time-reversal invariant topological superfluids, analyzing phase transitions driven by inter- and intra-layer pairing interactions.

Contribution

It introduces a mixed dimensional Bose-Fermi system as a platform for topological superfluids with TRS and explores the phase transitions involved.

Findings

Intra-layer p-wave pairing dominates at large layer separation.

Decreasing layer separation induces inter-layer pairing, leading to TRS breaking or trivial phases.

The system provides a feasible experimental route to realize topological superfluids with TRS.

Abstract

A mixed dimensional system of fermions in two layers immersed in a Bose-Einstein condensate (BEC) is shown to be a promising setup to realise topological superfluids with time-reversal symmetry (TRS). The induced interaction between the fermions mediated by the BEC gives rise to a competition between p-wave pairing within each layer and s-wave pairing between the layers. When the layers are far apart, intra-layer pairing dominates and the system forms a topological superfluid either with or without TRS. With decreasing layer separation or increasing BEC coherence length, inter-layer pairing sets in. We show that this leads either to a second order transition breaking TRS where the edge modes gradually become gapped, or to a first order transition to a topologically trivial s-wave superfluid. Our results provide a realistic roadmap for experimentally realising a topological superfluid with TRS in a cold atomic system.

Equations

$$\mathcal{Z}=\int \mathcal{D}({\bar{\psi }}_F,{\psi }_F,{\psi }_B^{\ast },{\psi }_B)e^{-\left(S_{\text{F}}+S_{\text{B}}+S_{\text{int}}\right)},$$

$$S_B=\beta \sum _{\mathbf{p}\ne 0,l}{\gamma }_p^{\ast }(-i{\omega }_l+E_{\mathbf{p}}){\gamma }_p$$

$$S_F=\beta \sum _{\sigma }\sum _{{\mathbf{k}}_{\perp },j}{\bar{a}}_{k_{\perp }\sigma }(-i{\omega }_j+{\xi }_{{\mathbf{k}}_{\perp }})a_{k_{\perp }\sigma }$$

$$S_{\text{int}}=g\int d^{3}r{\int }_0^{\beta }d\tau {\bar{\psi }}_F{\psi }_F{\psi }_B^{\ast }{\psi }_B,$$

$$S_{\text{int}}=\frac{g}{T}\sqrt{\frac{n_B}{\mathcal{V}}}\sum _{\begin{array}{c}\mathbf{p}\ne 0\\ l,\sigma \end{array}}\sqrt{\frac{{ϵ}_{\mathbf{p}}}{E_{\mathbf{p}}}}({\gamma }_p+{\gamma }_p^{\ast }){\rho }_{p_{\perp }\sigma }{e}^{-i{p}_{z}d(\sigma -1)}$$

$$S_{\text{eff}}=S_F+\frac{\beta }{2\mathcal{A}}\sum _{\begin{array}{c}{p}_{\perp }\\ \sigma ,{\sigma }^{\prime }\end{array}}{\rho }_{-p_{\perp }\sigma }{V}_{\text{ind}}^{\sigma {\sigma }^{\prime }}(p_{\perp }){\rho }_{p_{\perp }{\sigma }^{\prime }},$$

$$V_{\text{ind}}^{\sigma {\sigma }^{\prime }}(p_{\perp })=g^2\int \frac{d{p}_z}{2\pi }{e}^{i{p}_{z}d(\sigma -{\sigma }^{\prime })}{\chi }_{\text{BEC}}(p).$$

$$V_{\text{ind}}^{\sigma {\sigma }^{\prime }}({\mathbf{p}}_{\perp },0)=-\frac{2g^2{n}_B{m}_B}{\sqrt{{\mathbf{p}}_{\perp }^2+2/{\xi }_B^2}}{e}^{-d|\sigma -{\sigma }^{\prime }|\sqrt{{\mathbf{p}}_{\perp }^2+2/{\xi }_B^2}},$$

$$H_{\text{BCS}}=\frac{1}{2}\sum _{\mathbf{p}}{\Psi }^{†}(\mathbf{p})\mathcal{H}(\mathbf{p})\Psi (\mathbf{p}),$$

$$\mathcal{H}(\mathbf{p})=\left[\begin{array}{cccc}{\xi }_{\mathbf{p}}& {\Delta }_{11}(\mathbf{p})& 0& {\Delta }_{12}(\mathbf{p})\\ {\Delta }_{11}^{\ast }(\mathbf{p})& -{\xi }_{\mathbf{p}}& -{\Delta }_{12}^{\ast }(\mathbf{p})& 0\\ 0& -{\Delta }_{12}(\mathbf{p})& {\xi }_{\mathbf{p}}& {\Delta }_{22}(\mathbf{p})\\ {\Delta }_{12}^{\ast }(\mathbf{p})& 0& {\Delta }_{22}^{\ast }(\mathbf{p})& -{\xi }_{\mathbf{p}}\end{array}\right].$$

$${\Delta }_{\sigma {\sigma }^{\prime }}(\mathbf{p})=-\sum _{\mathbf{k}}{V}_{\text{ind}}^{\sigma {\sigma }^{\prime }}(\mathbf{p}-\mathbf{k},0)⟨a_{\mathbf{k}\sigma }{a}_{-\mathbf{k}{\sigma }^{\prime }}⟩.$$

$${\Delta }_{\nu }(\mathbf{p})$$

$$V_{\nu {\nu }^{\prime }}(\mathbf{p}-\mathbf{k})\equiv \frac{1}{2}[V_{\text{ind}}^{11}(\mathbf{p}-\mathbf{k})+\mathrm{sgn}(\nu ,{\nu }^{\prime })$$

$$\times e^{-i\left[{\varphi }_2(\mathbf{p})-{\varphi }_1(\mathbf{p})\right]/2}{V}_{\text{ind}}^{12}(\mathbf{p}-\mathbf{k})e^{i\left[{\varphi }_2(\mathbf{k})-{\varphi }_1(\mathbf{k})\right]/2}],$$

$$E_{\mathrm{BCS}}-\mu N=\frac{1}{2}\sum _{\nu ,\mathbf{p}}[{\xi }_{\mathbf{p}}-E_{\mathbf{p},\nu }+|{\Delta }_{\mathbf{p},\nu }{|}^2/2E_{\mathbf{p},\nu }].$$

$$\mathcal{T}(a_{\mathbf{p}1},a_{\mathbf{p}2}){\mathcal{T}}^{-1}=(a_{-\mathbf{p}2},-a_{-\mathbf{p}1}),$$

$$H=\int d^2\mathbf{r}{\Psi }^{†}(\mathbf{r})\mathcal{H}(\mathbf{r})\Psi (\mathbf{r})$$

$$\mathcal{H}(\mathbf{r})=\left(\begin{array}{cccc}-\mu (\mathbf{r})& {\Delta }_{11}{e}^{-i{\varphi }_0}(-{\partial }_x+i{\partial }_y)& 0& {\Delta }_{12}\\ {\Delta }_{11}{e}^{i{\varphi }_0}({\partial }_x+i{\partial }_y)& \mu (\mathbf{r})& -{\Delta }_{12}^{\ast }& 0\\ 0& -{\Delta }_{12}& -\mu (\mathbf{r})& {\Delta }_{11}{e}^{i{\varphi }_0}(-{\partial }_x-i{\partial }_y)\\ {\Delta }_{12}^{\ast }& 0& {\Delta }_{11}{e}^{-i{\varphi }_0}({\partial }_x-i{\partial }_y)& \mu (\mathbf{r})\end{array}\right)$$

$$\chi (\mathbf{r})=\kappa e^{in\theta }\left(\begin{array}{c}{e}^{-i\varphi /2}[A(r)+iB(r)]\\ e^{i\varphi /2}[A(r)-iB(r)]\\ e^{i\varphi /2}[C(r)+iD(r)]\\ e^{-i\varphi /2}[-C(r)+iD(r)]\end{array}\right)$$

$$B(r)$$

$$C(r)$$

$$\alpha =\frac{{\Delta }_{11}n}{|{\Delta }_{12}|R}-\sqrt{{\left(\frac{{\Delta }_{11}n}{|{\Delta }_{12}|R}\right)}^2+1}.$$

Full text

I Introduction

The search for superfluids/superconductors with non-trivial topological properties has experienced an explosion of activities in recent years. One reason is that these systems can host gapless edge (Majorana) modes with possible applications in quantum computation Alicea (2012); Nayak et al. (2008). Excitingly, evidence for topological superconductivity and gapless edge states have been reported in nano-wires Mourik et al. (2012); Deng et al. (2012a); Das et al. (2012); Rokhinson et al. (2012); Finck et al. (2013); Nadj-Perge et al. (2014); Albrecht et al. (2016). So far the focus has predominantly been placed on superfluids foo without TRS, which belong to the symmetry class D in the 10-fold classification scheme of topological insulators/superfluids Schnyder et al. (2008); Kitaev (2009); Altland and Zirnbauer (1997). However, superfluids with TRS, belonging to the class DIII, can also host gapless Majorana mode pairs, which are protected by Kramers theorem. There are also several proposals to realise such systems in laboratory, both in condensed matter systems Wong and Law (2012); Deng et al. (2012b); Zhang et al. (2013); Nakosai et al. (2013); Keselman et al. (2013); Gaidamauskas et al. (2014); Klinovaja et al. (2014); Schrade et al. (2015) and in cold atomic systems Yan et al. (2013); Yang et al. (2013); Huang et al. (2015). One example of such intriguing systems is the superfluid 3He B phase, whose topological properties have been studied recently Volovik (2009); Makhlin et al. (2014). However, one has yet to observe a topological superfluid with TRS in a cold atomic system.

Recently, we showed that a mixed dimensional atomic gas system consisting of a two-dimensional (2D) layer of fermions immersed in a 3D BEC constitutes a promising system for realising a $\mathbb{Z}$ topological superfluid in class D with a high critical temperature Wu and Bruun (2016); Midtgaard et al. (2016). Here, we show that an analogous system with two layers of fermions, first studied in Ref. Nishida (2010), is naturally suited to realise a $\mathbb{Z}_{2}$ topological superfluid with TRS. Fermions in the layers interact attractively via an induced interaction mediated by the BEC. The relative strengths of the intra- and inter-layer induced interaction results in a competition between $p_{x}\pm ip_{y}$-wave pairing involving fermions in the same layer, and $s$-wave pairing involving fermions in different layers. For large distance between the layers, intra-layer pairing dominates and one has either a $(p_{x}+ip_{y})\times(p_{x}-ip_{y})$ system with TRS or a $(p_{x}+ip_{y})\times(p_{x}+ip_{y})$ without TRS. With decreasing layer distance or increasing BEC coherence length, we show that inter-layer $s$-wave pairing occurs in a second order transition for the $(p_{x}+ip_{y})\times(p_{x}-ip_{y})$ system, which breaks TRS thereby gradually gapping the edge modes without closing the bulk gap. For short layer distance, the system ends up in a topologically trivial $s$-wave superfluid, resembling the case of a single layer with two spin components Anzai and Nishida (2017). On the other hand, the transition from the topological $(p_{x}+ip_{y})\times(p_{x}+ip_{y})$ to the trivial $s$-wave superfluid is of the first order.

II Model

We consider identical (spin polarised) fermions of mass $m$ in two layers located at $z=0$ and $z=d$, see Fig. 1. The fermions are immersed in a 3D gas of bosons with mass $m_{B}$ and density $n_{B}$. The partition function of the system at temperature $T$ is

[TABLE]

where $\psi_{B}({\mathbf{r}},\tau)$ and $\psi_{F}({\mathbf{r}},\tau)$ are the bosonic and fermionic fields at point ${\mathbf{r}}$ and imaginary time $\tau$. The bosons form a weakly interacting BEC that be described by Bogoliubov theory, which yields

[TABLE]

for the bosonic part of the action, where $\beta=1/T$, $\omega_{l}=2l\pi T$ with $l=0,\pm 1,\pm 2,\ldots$ are the Bose Matsubara frequencies and $\gamma_{p}$ describes the quasi-particle with momentum ${\bf p}=(p_{x},p_{y},p_{z})$ and energy $E_{\bf p}$. Here we have defined $p\equiv(\mathbf{p},i\omega_{l})$. The Bogoliubov spectrum is $E_{\mathbf{p}}=\sqrt{\epsilon_{\mathbf{p}}(\epsilon_{\mathbf{q}}+2g_{B}n_{B})}$, where $\epsilon_{\mathbf{p}}=\mathbf{p}^{2}/2m_{B}$ and $g_{B}=4\pi a_{B}/m_{B}$, where $a_{B}$ is the boson scattering length. The fermion part of the action is

[TABLE]

where $a_{k_{\perp}\sigma}$ are the Grassmann fields for the fermions in layer $\sigma=1,2$. The effective 2D action for the fermions results from the fact that the vertical trapping potentials are sufficiently tight that the fermions reside only in the lowest trap levels $\phi_{0}(z)$ and $\phi_{0}(z-d)$ along the $z$-direction. We have defined $k_{\perp}\equiv(\mathbf{k}_{\perp},i\omega_{j})$ with $\mathbf{k}_{\perp}=(k_{x},k_{y})$ as the in-plane momentum, $\omega_{j}=(2j+1)\pi T$ with $j=0,\pm 1,\pm 2,\ldots$ are the Fermi Matsubara frequencies, and $\xi_{\mathbf{k}_{\perp}}=\mathbf{k}_{\perp}^{2}/2m-\mu$ where $\mu$ is the chemical potential of the fermions. We take $\mu$ to be the same in each layer which contains an equal number of fermions. Finally, the Bose-Fermi interaction is

[TABLE]

where $g$ is the boson-fermion interaction strength. Using the Bogoliubov theory to write $\psi_{B}({\mathbf{r}},\tau)=\mathcal{V}^{-1/2}\sum_{p}(u_{\mathbf{p}}\gamma_{p}-v_{\mathbf{p}}\gamma^{*}_{-p})\exp[i(\mathbf{p}\cdot\mathbf{r}-\omega_{l}\tau)]$ with $u_{\mathbf{p}}^{2},v_{\mathbf{p}}^{2}=[({\epsilon_{\mathbf{p}}+g_{B}n_{B}})/{E_{\mathbf{p}}}\pm 1]/2$, and expanding the fermonic fields as $\psi_{F}({\mathbf{r}},\tau)=\sum_{p_{\perp},\sigma}a_{p_{\perp}\sigma}\exp[i(\mathbf{p}\cdot\mathbf{r}_{\perp}-\omega_{j}\tau)\phi_{0}(z-(\sigma-1)d)/\sqrt{\mathcal{A}}$, we find

[TABLE]

where $\mathcal{V}$ is the BEC volume, $\mathcal{A}$ is the area of the Fermi layer, $\rho_{p_{\perp}\sigma}=\sum_{k_{\perp}}\bar{a}_{k_{\perp}-p_{\perp}\sigma}a_{k_{\perp}\sigma}$ and $p_{\perp}\equiv(\mathbf{p_{\perp}},i\omega_{l})$.

Integrating out the quadratic Bose fields in the action in (1) yields the effective action

[TABLE]

where the induced interaction between the fermions, mediated by the bosons, is

[TABLE]

Here, $\chi_{\text{BEC}}(p)=n_{B}\mathbf{p}^{2}m_{B}^{-1}/[(i\omega_{l})^{2}-E_{\mathbf{p}}^{2}]$ is the density-density correlation function for the BEC and the $p_{z}$-integration in (7) is due to the fact that the momentum along the $z$-direction is not conserved in the boson-fermion scattering due to the mixed dimensional setup. We note that the induced interaction in Eq. (7) is obtained with the assumption that the 3D BEC is not affected by the 2D Fermi gases. This is justified in our mixed dimensional setup because the properties of the 3D BEC will only be affected locally in the vicinity of the 2D layers. Since the induced interaction between the fermions is determined by the overall bulk properties of the BEC, we expect that these local effects on the 3D BEC will only lead to small corrections to the induced interaction given by Eq. (7). For zero frequency, $i\omega_{l}=0$, performing the $p_{z}$ integrals yields

[TABLE]

where $\xi_{B}=(8\pi n_{B}a_{B})^{-1/2}$ is the BEC coherence length. The inter-layer ($\sigma\neq\sigma^{\prime}$) interaction is suppressed compared to the intra-layer ($\sigma=\sigma^{\prime}$) interaction by an exponential factor related to the layer distance $d$. Fourier transforming (8) yields a Yukawa interaction $V(\mathbf{r})=-g^{2}n_{B}m_{B}\pi^{-1}\exp(-\sqrt{2}r/\xi_{B})/r$ in real space with a range determined by $\xi_{B}$ Bijlsma et al. (2000); Viverit et al. (2000); Midtgaard et al. (2016); Suchet et al. (2017). Here $r=|\mathbf{r}|$ is the distance between the particles, which can reside in the same or in different planes.

III Gap equations

Since the induced interaction given by (8) is attractive, fermions with opposite momenta can form Cooper pairs within each layer (intra-layer pairing) as well as between different layers (inter-layer pairing). The BCS Hamiltonian describing such parings is

[TABLE]

where $\Psi(\mathbf{p})=(a_{\mathbf{p}1},a^{\dagger}_{\mathbf{-p}1},a_{\mathbf{p}2},a^{\dagger}_{\mathbf{-p}2})^{T}$ and

[TABLE]

Here the $\perp$-subscript is dropped since we are dealing only with 2D momenta of the fermions from now on, and $a_{\mathbf{p}\sigma}$ are the fermi annihilation operators for layer $\sigma=1,2$. We neglect retardation effects and use only the zero frequency component of the induced interaction. Retardation effects are small when the Fermi velocity $v_{F}$ in the layers is much smaller than the speed of sound in the BEC, while for larger $v_{F}$ they suppress the magnitude of the pairing without changing the qualitative behavior Wu and Bruun (2016). The pairing fields are determined self-consistently as

[TABLE]

We take the inter-layer pairing to be $s$-wave so that $\Delta_{12}(\mathbf{p})=\Delta_{12}(-\mathbf{p})=-\Delta_{21}(\mathbf{p})$ and the Fermi anti-symmetry dictates that $\Delta_{\sigma\sigma}(\mathbf{p})=-\Delta_{\sigma\sigma}(-\mathbf{p})$ for the intra-layer pairing. Since the system has rotational symmetry with respect to the $z$-axis, we take the intra-layer pairing to be of the $p_{x}\pm ip_{y}$ form, as this fully gaps the Fermi surface Anderson and Morel (1961), i.e. $\Delta_{\sigma\sigma}(\mathbf{p})=\Delta_{\sigma}(|\mathbf{p}|)e^{i\phi_{\sigma}({\bf p})}$ where $\phi_{\sigma}({\bf p})=\phi_{0\sigma}\pm\varphi_{\bf p}$ with $\varphi_{\bf p}$ being the azimuthal angle of $\mathbf{p}$. Furthermore, for identical layers we assume that $\Delta_{1}(|{\bf p}|)=\Delta_{2}(|{\bf p}|)$ and we thus have $\Delta_{22}({\bf p})=\Delta_{11}({\bf p})e^{i\left[\phi_{2}({\bf p})-\phi_{1}({\bf p})\right]}$. We diagonalise (9) by introducing new pairing fields $\Delta_{\pm}({\bf p})=\Delta_{11}({\bf p})\pm\Delta_{12}({\bf p})e^{-i\left[\phi_{2}({\bf p})-\phi_{1}({\bf p})-\pi\right]/2}$. Equation (11) then yields a set of gap equations in a symmetrical form as

[TABLE]

Here $\nu=\pm$, $E_{{\bf p},\pm}=\sqrt{\xi_{\bf p}^{2}+|\Delta_{\pm}({\bf p})|^{2}}$, and

[TABLE]

where ${\rm sgn}(\nu,\nu)=1$ and ${\rm sgn}(\nu,-\nu)=-1$. Finally the number equation is $N=\sum_{\nu,{\bf p}}[1-\xi_{\bf p}\tanh(E_{{\bf p},\nu}/2T)/E_{{\bf p},\nu}]/2$ and the BCS ground state energy is

[TABLE]

We note that when the $s$- and $p$-wave order parameters co-exist, their relative phase is important. It cannot be gauged away contrary to the case of a single order parameter. The relative phase therefore has physical consequences, and we shall see that it determines whether the system has a time-reversal symmetry or not.

IV Symmetries and topological properties

The topological properties of the bi-layer system are determined by its symmetries and 2D dimensionality Schnyder et al. (2008); Kitaev (2009); Altland and Zirnbauer (1997). Consider first the limit where the two layers are uncoupled, which corresponds to the layer distance being much larger than the range of the induced interaction given by the BEC coherence length, i.e. $d\gg\xi_{\text{B}}$. There is then only particle-hole symmetry for each layer, and they each form a topological $p_{x}\pm ip_{y}$ superfluid in symmetry class D, which supports chiral edge states. Consider now the case when the two layers are brought closer to each other so that they interact. The topological properties and the fate of the edge states then depend on whether the Cooper pairs in the two layers have opposite or the same angular momentum, corresponding to $(p_{x}+ip_{y})\times(p_{x}-ip_{y})$ or $(p_{x}+ip_{y})\times(p_{x}+ip_{y})$ pairing respectively.

For $(p_{x}+ip_{y})\times(p_{x}-ip_{y})$ pairing illustrated in Fig. 1 (a), which we refer to as the $(+,-)$ case, the system possesses in addition to particle-hole symmetry the time-reversal symmetry:

[TABLE]

which swaps particles in the two layers. Note that this anti-unitary symmetry is different from the usual time-reversal symmetry, which flips the spin of the particles. Here, the layer index plays the role of a pseudo-spin. Since $\mathcal{T}^{2}=-1$, the bi-layer system is then in symmetry class DIII, and its ground state is a $\mathbb{Z}_{2}$ topological superfluid, which supports helical edge modes in analogy with the quantum spin Hall state Kane and Mele (2005a, b); Qi et al. (2009). The counter propagating edge modes in the two layers are related by TRS and protected by Kramers theorem. However, when the layers are sufficiently close together, the $s$-wave inter-layer pairing ($\Delta_{12}(\mathbf{p})\neq 0$) will dominate, and the system forms a topologically trivial $s$-wave superfluid. Thus, the edge states must become gapped at some critical inter-layer distance. Without solving the gap equation, one can envision two ways this can happen: either the inter-layer pairing explicitly breaks TRS thereby gapping the edge modes as soon as $\Delta_{12}(\mathbf{p})\neq 0$, or the inter-layer pairing respects TRS and the edge states become gapped only when the bulk energy gap is closed. By analysing the properties of the inter-layer gap under time-reversal, we find that these two scenarios correspond to $\Delta_{12}(\mathbf{p})$ being imaginary and real respectively. Our numerical results (see later) show that $\Delta_{12}(\mathbf{p})$ is in fact imaginary and the first scenario describes the physical transition.

For $(p_{x}+ip_{y})\times(p_{x}+ip_{y})$ pairing illustrated in Fig. 1 (b), which we refer to as the $(+,+)$ case, the system only has the particle-hole symmetry and is a $\mathbb{Z}$ topological superfluid in class D, which supports chiral edge modes propagating in the same direction in the two layers. When the layer distance is decreased, the possible onset of inter-layer pairing co-existing with the intra-layer pairing will not gap these edge modes as long as the bulk gap remains non-zero, since this pairing does not break any symmetry. However, we shall see later that such a co-existing scenario does not occur for the $(+,+)$ case. Similar to the $(+,-)$ case, the system ends up in the topologically trivial inter-layer $s$-wave superfluid for small inter-layer distances. We shall demonstrate below that this happens via a first order phase transition.

The topological $\mathbb{Z}$ and $\mathbb{Z}_{2}$ invariants of class D and DIII respectively can be calculated from the two energy bands $E_{\mathbf{p},+}$ and $E_{\mathbf{p},-}$ of the bilayer system de Lisle et al. (2014). If the two layers are uncoupled, these bands are degenerate and the invariant for class D is simply given by the sum $C=C_{1}+C_{2}$ of the Chern numbers $C_{\sigma}$ of each layer, whereas it is given by the difference $\nu=C_{1}-C_{2}\,(\text{mod}\,2)$ for class DIII. This is consistent with the fact that the $(+,-)$ state has $C_{1}=-1,C_{2}=1$ and is therefore topological in class DIII, whereas it is trivial in class D. Therefore, if the TR symmetry is broken for the $(+,-)$ state by an imaginary $\Delta_{12}(\mathbf{p})$ that mixes the two bands, the system is in class D and it is no longer topological.

V Edge states

In this section, we show explicitly how the edge states of the $(+,-)$ system become gapped with the onset of interlayer $s$-wave pairing $\Delta_{12}(\mathbf{p})$ which is imaginary. We consider the following low-energy hamiltonian in real space,

[TABLE]

where $\Psi(\mathbf{r})=(\psi_{1},\psi^{\dagger}_{1},\psi_{2},\psi^{\dagger}_{2})^{T}$ and

[TABLE]

Assuming that we can apply a local density approximation, we take $\mu(\mathbf{r})=\mu(r)$ to be positive within the radius $R$, and negative outside. Solutions to the eigenvalue equation $\mathcal{H}(\mathbf{r})\chi(\mathbf{r})=E\chi(\mathbf{r})$ with definite angular momentum can then be found, and we use the following ansatz in the usual polar coordinates,

[TABLE]

where $\kappa$ is a normalization constant. The real functions $A,B,C,D$ satisfy a set of 4 coupled equations, and for a large system with tightly confined edge modes, we can find solutions with the energy $\pm E=\pm\sqrt{\left(\Delta_{11}n/R\right)^{2}+|\Delta_{12}|^{2}}$. Here, $n$ is a half-integer related to the angular momentum of the edge state. These solutions requires $\Delta_{12}$ to be real. If it is imaginary, it is possible to show that the edge modes do not acquire a gap. When finding the specific solutions for the edge states, care should be taken to choose the solution that is normalizable and confined to the edge. As an example, consider the physical, positive branch of energies, $+E$. A possible solution is given by $A(r)=D(r)=0$ and

[TABLE]

with

[TABLE]

We see that the edge states lowest in energy are localized on both layers when the two gap parameters coexist. The states higher in energy approach the solutions for uncoupled layers, which are only localized on a single layer.

VI Numerical solution of the gap equation

We now numerically solve the gap equations (12) along with the number equation at $T=0$. The $(+,-)$ case corresponds to $\phi_{2}(\mathbf{p})-\phi_{1}(\mathbf{p})=\pi-2\varphi_{\mathbf{p}}$, while the $(+,+)$ case corresponds to $\phi_{2}(\mathbf{p})-\phi_{1}(\mathbf{p})=\pi$.

VI.1 $(+,-)$ system

In Fig. 2 (a), we plot the magnitude of the pairing fields at the Fermi surface as a function of the layer distance $d$ for the $(+,-)$ system. Here $k_{F}=\sqrt{4\pi n_{F}}$ is the Fermi momentum with $n_{F}$ the density of fermions in each layer. We have chosen a relatively weak Bose-Fermi coupling strength $g=2\pi a/\sqrt{m_{r}m_{B}}$ with $k_{F}a=0.1$, where $a$ is the 2D-3D mixed dimensional scattering length Nishida and Tan (2008). The gas parameter of the BEC is $(n_{B}a_{B}^{3})^{1/3}=0.01$ and the ratio of the Fermi and Bose interparticle distances is $n_{F}^{1/2}/n_{B}^{1/3}=0.2$. The energy of the system is plotted in Fig. 2 (b).

For layer distances $d\gtrsim 0.754\xi_{B}$, there is no inter-layer pairing and the two layers are uncoupled each realising a $p_{x}\pm ip_{y}$ topological superfluid. The corresponding edge states, illustrated in Fig. 2 (c), propagate in opposite directions in the two layers and are related by the TRS operator $\mathcal{T}$. We have chosen a circular boundary with radius $R$ to illustrate the typical geometry formed by the harmonic trap in an atomic gas experiment. As the layer distance decreases, inter-layer pairing sets in for $d\lesssim 0.754\xi_{B}$ via a second order transition and it co-exists with the intra-layer pairing. We find numerically that the inter-layer pairing $\Delta_{12}(\mathbf{k})$ is purely imaginary and it therefore breaks TRS. The edge modes in the two layers mix and become gapped as illustrated in Fig. 2 (c). More precisely, the dispersion of the edge modes is $E=\sqrt{(\Delta_{11}n/R)^{2}+|\Delta_{12}|^{2}}$, where $|\Delta_{12}|\simeq|\Delta_{12}(0)|$ and $\Delta_{11}(\mathbf{p})\simeq\Delta_{11}(p_{x}+ip_{y})$ give the magnitude of the inter- and intra-layer pairing at low momenta, and $n$ is a half-integer proportional to the angular momentum of the edge state, as seen above. The low-energy edge states with small $n$ are hybridised between the two layers; for larger $n$, the edge states become increasingly localized in a single layer, approaching those for the uncoupled layers. Finally, for layer distances $d\lesssim 0.747\xi_{B}$ the intra-layer pairing is completely suppressed by the inter-layer pairing and the system is a topologically trivial $s$-wave superfluid with no edge modes. We have not been able to find a numerical solution with a real inter-layer pairing co-exisiting with intra-layer pairing, which would preserve TRS and support the gapless edge modes. While the co-existence region shown here is quite narrow, the width can be tuned by altering the parameters (see below).

VI.2 $(+,+)$ system

For the $(+,+)$ system, our numerical results show that the transition between the topological and trivial phase is first order. The transition occurs at the critical layer distance $d\simeq 0.751\xi_{B}$ when the phases with only one type of pairing have the same energy, as indicated by the vertical line in Fig. 2(b). We do not find numerical solutions with both types of pairing coexisting. Instead, the intra-layer pairing and the associated gapless edge modes disappear and the inter-layer pairing appears abruptly.

VII Varying the coherence length

Experimentally, it might be easier to change the BEC coherence length, which determines the range of the induced interaction, by varying $a_{B}$ using a Feshbach resonance, instead of changing the layer distance. To examine this case, we plot in Fig. 3 the magnitudes of the intra- and inter-layer pairings as a function of $\xi_{B}$ with $k_{F}a=0.12$, $k_{F}d=1.0$, and $n_{F}^{1/2}/n_{B}^{1/3}=0.2$. The coherence length is varied by changing $a_{B}$ keeping $n_{B}$ fixed. For a small $\xi_{B}$, the two layers are uncoupled forming the $(+,-)$ or the $(+,+)$ topological superfluid. The $(+,-)$ system undergoes a second order phase transition to a state where intra- and inter-layer pairing co-exist for $\xi_{B}\gtrsim 1d$. Note that contrary to decreasing the distance $d$, the system does not end up in a pure $s$-wave state for large $\xi_{B}$. The reason is that for a large interaction range, the suppression of the $p$-wave channel compared to the $s$-wave channel is small, and intra- and interlayer pairing therefore co-exist. The $(+,+)$ system on the other hand again undergoes a first order transition between the topological and the trivial phases at $\xi_{B}\sim 1.05d$.

VIII Discussion

All the ingredients in the proposed setup have been realised experimentally. Bose-Fermi mixtures as well as species selective optical potentials to produce mixed dimensional systems have been reported Lamporesi et al. (2010); McKay et al. (2013); Jotzu et al. (2015). It was moreover shown in Ref. Wu and Bruun (2016) that the Berezinskii-Kosterlitz-Thouless critical temperature for the $p_{x}\pm ip_{y}$ superfluid in the present Bose-Fermi setup can be as high as $T_{\rm BKT}=\mathcal{E}_{F}/16$, which is within experimental reach Onofrio (2017). We expect the critical temperature of the phase with $s$-wave pairing to be even higher. The edge modes can be observed for instance by direct imaging or by the response to an external drive in analogy with topological insulators Goldman et al. (2013); Tran et al. (2017); Goldman et al. (2016).

An intriguing question concerns the robustness of the edge modes beyond mean-field BCS theory. To investigate this, one could analyse the coupling between the edge modes forming a Luttinger liquid Xu and Moore (2006); Tanaka and Nagaosa (2009), which is an interesting future project.

IX Conclusion

We demonstrated that a mixed dimensional system consisting of two layers of fermions in a BEC is a powerful setup to realise topological superfluids with TRS. The induced interaction between the fermions mediated by the BEC leads to a competition between $p$-wave pairing within each layer and $s$-wave pairing between the layers. For large layer separation or short BEC coherence length, intra-layer pairing dominates and the system forms a topological superfluid either with or without TRS. In the case of TRS, the system goes from a $\mathbb{Z}_{2}$ topological superfluid to a topologically trivial superfluid via a second order transition where $s$-wave pairing gradually gaps the edge modes. When there is no TRS, the transition from a $\mathbb{Z}$ topological superfluid to a topologically trivial superfluid is first order. These results show how cold atomic gases offer a realistic path to realising topological superfluids with TRS.

Acknowledgements.

We wish to acknowledge the support of the Villum Foundation via grant VKR023163 and the Danish Council of Independent Research — Natural Sciences via Grant No. DFF - 4002-00336. We acknowledge useful discussions with Alexandre Dauphin and Alessio Recati.