Emergent superconducting phases in unconventional $p$-wave magnets: Topological superconductivity, Bogoliubov Fermi surfaces and superconducting diode effect

Abstract

The recent discovery of unconventional momentum-dependent magnetic orders has expanded the landscape of magnetism beyond conventional ferromagnetism and antiferromagnetism. Among them, $p$-wave magnets ($p$WMs) represent a novel class of odd-parity, non-collinear compensated magnetic order that generates spin-split electronic bands. In this work, our theoretical investigation establishes $p$WMs as a versatile platform for realizing intriguing superconducting phases including topological superconductivity (TSC), Bogoliubov Fermi surfaces (BFSs), and superconducting diode effect (SDE), within a unified microscopic framework. Employing a minimal model incorporating $p$-wave magnetic order, exchange coupling, and Zeeman fields, we perform a self-consistent mean-field analysis and uncover a rich phase diagram featuring unconventional finite-momentum Fulde-Ferrell (FF) and Larkin-Ovchinnikov (LO) superconducting phases. Remarkably, we also show that $p$WMs can undergo a transition to a TSC phase anchoring Majorana flat edge modes, a hallmark of two-dimensional TSCs, even without Rashba spin-orbit coupling and Zeeman field. Upon applying a Zeeman field, gapless FF and LO phases emerge with BFSs characterized by the appearance of finite zero-energy quasiparticle density of states. Furthermore, we demonstrate that SDE arises naturally in the asymmetric FF phase. Our analysis manifests that $p$WMs serve as a unique and novel platform to host TSC phase, gapless superconducting states, and non-reciprocal transport phenomena.

Figures (7)

• Figure 1: Illustration of the unconventional superconducting phases and the normal state spectral properties.(a) Schematic illustration for possible phases in $p$WM in the presence of an external Zeeman field $B$, along with the MFEMs at the boundary. In panels, (b)-(f) [(g)-(k)] bulk energy-spectrum [Fermi surfaces] along with spin-polarization $\braket{S_z}$ (in units of $\hbar/2$) have been demonstrated. Other model parameter values are chosen as (b),(g)$(\alpha,J_{sd},B)=(0,0,0)$, (c),(h)$(\alpha,J_{sd},B)=(t,0,0)$, (d),(i)$(\alpha,J_{sd},B)=(t,0.3t,0)$, (e),(j)$(\alpha,J_{sd},B)=(t,0,0.3t)$, (f),(k)$(\alpha,J_{sd},B)=(t,0.3t,0.3t)$ with $t=1$ and $\mu=-2t$.
• Figure 2: Phase diagram for the superconducting ground state: Behavior of the (a) superconducting order parameter $\Delta^{\!c}/\Delta_0$, (b)$q_x^c$ and (c)$q_y^c$ are depicted across the parameter space spanned by $J_{sd}$ and $B$. In panel (a), we highlight the unconventional superconducting phases, TSC along with gapped FF, gapless FF, and LO pairing states. Inset of (b) and (c) display the condensation energy density, $\Omega(\mathbf{q},\Delta_{\mathbf{q}}^{\rm FF})$, in the $q_x- q_y$ plane, corresponding to the gapless FF [$(J_{sd},B)=(0.5,0.7)\Delta_0$] and gapless LO [$(J_{sd},B)=(0,0.75)\Delta_0$] state, respectively.
• Figure 3: Emergence of topological superconductivity:(a) Bulk energy spectrum is shown in the TSC phase in the $k_x-k_y$ plane exhibiting four isolated nodal points. (b) Eigenvalue spectrum (left axis) and winding number (right axis) are shown as a function of $k_y$ considering finite size along $x$-direction. (c) LDOS profile at zero energy is shown in the co-ordinate space for the rectangular geometry. Inset displays the variation of energy-eigenvalues $E_n$ as a function of state index, $n$, manifesting the flat zero energy modes. (d) Variation of winding number, $\mathcal{N}_x$, is shown in the $J_{sd}$-$k_y$ plane. In panels (a)-(c), we choose $(J_{sd},B)=(1.4\Delta_0,0)$ corresponding to the TSC phase of Fig. \ref{['Fig_2']}(a) and set $B=0$ in panel (d).
• Figure 4: Emergence of BFSs driven by finite-momentum Cooper pairs: Demonstration of the appearance of BFSs in the $k_x-k_y$ plane corresponding to the gapless (a) LO and (b) FF phase. (c) Normalized single-particle density of states is shown as a function of $E/\Delta_0$ in the gapless finite-momentum superconductivity channels. (d) Phase diagram of the zero energy density of states is displayed in the $J_{sd}-B$ plane, highlighting the presence of BFSs in the gapless superconducting phases.
• Figure 5: Manifestation of SDE in the FF pairing channel: In panels (a) and (b), the variation of supercurrent $J_x(\mathbf{q})$ and $J_y(\mathbf{q})$, respectively, are shown in the $q_x-q_y$ plane corresponding to the FF pairing state with $(J_{sd}/\Delta_0,B/\Delta_0)=(0.5,0.7)$. Panels (c) and (d) display the Diode efficiency factor, $\eta$, as a function of the external Zeeman field, $B$ for various choices of $\mu$ and $J_{sd}$, respectively. We choose the other model parameters as $(J_{sd},\alpha)=(0.5t,t)$ in panel (c) and $(\mu,\alpha)=(-2.2t,t)$ in panel (d).