Higher-order topological phases for time-reversal-symmetry breaking superconductivity in UTe$_2$

Abstract

The recent discovery of heavy-fermion superconductor UTe$_2$ has broadened the possibility of realizing exotic time-reversal-symmetry-breaking superconductivity. However, a comprehensive understanding of the topological phases in the superconducting states of UTe$_2$ is still lacking. Here, we present an exhaustive classification of topological phases for all time-reversal symmetry breaking pairing symmetries of UTe$_2$. Using the K theoretical classification approach, we uncover that 25 out of 36 possible pairing states are classified as higher-order topological phases, with some demonstrating hybrid-order topology through an intricate interplay of hinge and corner states. Furthermore, under the weak-coupling condition of the pair potentials, the possible pairing symmetries are constrained to $B_{ju} + i B_{ku}$, $A_{u} + i B_{j u}$, and $B_{j g} + iA_u$ ($j,k = 1,2,3$; $j \neq k$), where these symbols denote the irreducible representations of the point group $D_{2h}$. For these pairing states, the topological invariants are related to the Fermi surface topology via the Fermi-surface formula, enabling us to systematically diagnose higher-order topological phases. Using a tight-binding model, we demonstrate the higher-order topological phases of the mixed-parity $A_u + iB_{1g}$ superconductors, where the second-order and hybrid-order topological phases emerge as the number of Fermi surfaces enclosing the time-reversal invariant momentum evolves from two to four. The findings suggest that UTe$_2$ serves as a compelling platform for exploring higher-order topological superconductors with diverse topological surface states.

Figures (7)

• Figure 1: Figure panels illustrate possible topological surface states in 3D TRSB superconductors with pairing symmetry (a) $B_{1g}+i B_{1g}$, (b) $A_{u}+i A_{u}$, (c) $B_{1u}+iB_{1u}$, (d) $B_{2g}+i B_{3g}$, (e) $A_u+i B_{1u}$, (f) $B_{2u}+i B_{3u}$, and (g) $B_{1g}+i A_{u}$. Each panel shows the topological invariants and configurations of topological surface states. The configurations are depicted by gray spheres, representing system boundaries, and colored lines and points, which indicate hinge and corner states, respectively. For instance, in case (a), two mirror Chern numbers, $\text{Ch}_1[m_x]$ and $\text{Ch}_1[m_y]$, define the HOTPs. The two surface state configurations shown correspond to topological invariants $\{(2,2),(2,-2)\}$, respectively. Here, the set notation $\{\}$ denotes a minimal set of topological invariants (see Appendix \ref{['app:generator']}). Elements of the classifying groups are generated by the combinations of these minimal sets. In case (c), two topological invariants, $\text{Ch}_1[m_z]$ and $\kappa[I]_+^{m_z}$, correspond to second and third order topological phases, respectively. The minimal set {$(2,2)$} indicates coexistence of Majorana hinge and corner states. Doubling this set yields $(4,4)=(4,0)$, because $\kappa[I]_+^{m_z} = 4 =0 \mod 4$. This phase therefore supports only four Majorana hinge states. In cases (b), (e), and (g), diverse surface state patterns result from the coexistence of multiple topological invariants. In cases (d) and (f), an alternative configuration shown in the dotted box is possible through the addition of extrinsic topological surface states (see Appendix \ref{['app:boundary']}).
• Figure 2: The 3D Fermi surface (a) [(d)], topological invariants (b) [(e)], and density of states for in-gap states (c) [(f)] are shown, where we use the parameters in Table \ref{['tab: parameter']} when #FS=2 [#FS=4]. We plot the density of states of eigenstates satisfying $|\epsilon_n| \lesssim 0.25|E_{\text{gap}}|$, which is described as $|\Psi(\bm{x})|^2=(1/450)\sum^{450}_{n=1}|u_n(\bm{x})|^2$ for #FS=2 and $|\Psi(\bm{x})|^2=(1/150)\sum^{150}_{n=1}|u_n(\bm{x})|^2$ for #FS=4, where $u_n(\bm{x})$ and $\epsilon_n$ are eigenstates and eigenvalues of Eq. (\ref{['eq:bdg_open']}), and $E_{\text{gap}}$ is the minimum of the bulk energy gap.
• Figure 3: (a) The definition of unit vectors. (b) The lattice configuration of an octahedron with $L=5$, where $\alpha=x', y'$.
• Figure 4: Schematic illustration of bulk cellular decomposition (a) and surface decorations [(b),(c)] for the case of the magnetic point group $M=2$ and $B$ pairing state. The gray sphere represents a 3D system that is compatible with the magnetic point group symmetry. In (b) and (c), we start from trivial states that have no boundary mode, and then construct an extrinsic topological surface state by pasting 2D (1D) TSCs on each cell. The pasted topological states are illustrated by the blue places (lines). The chiral edge modes of 2D TSCs are depicted by the red and blue arrows, whose direction indicates the propagating direction of chiral edge modes. The edge states of the 1D TSCs are denoted by the red points. If there is a symmetry-preserving mass term, the configuration of boundary states change.
• Figure 5: Schematic illustration of the surface decorations for the case with the magnetic point group $M=m'm'm$ and $B_u$ pairing state. The symbols are the same as those in Figure \ref{['fig: Decoration-rotation']}.