Two-dimensional flat band on the (011) surface of UTe$_2$: Implication for STM measurements with a superconducting tip

TL;DR

This work addresses the puzzle of zero-energy surface states on the (011) surface of the topological superconductor candidate UTe2 by analyzing all odd-parity triplet representations ($A_u$, $B_{1u}$, $B_{2u}$, $B_{3u}$) within a minimal tight-binding BdG framework. It reveals that a two-dimensional nearly flat band, which yields a pronounced zero-bias peak in the surface density of states, arises only in the $B_{3u}$ state due to the synergy of nontrivial Berry phases at multiple high-symmetry momenta and weak spin-conservation–enabled phase winding. The authors connect this to STM measurements, deriving a nonequilibrium dc tunneling current for a junction with an $s$-wave superconducting tip; in the low-bias, weak-tunneling limit, the Andreev tunneling current is proportional to the convolution of the surface DOS and shows a sharp ZBP uniquely for $B_{3u}$. These results provide strong evidence for $B_{3u}$ pairing in UTe2 and offer a framework to interpret STM experiments with superconducting tips, while also outlining directions to resolve discrepancies with normal-tip STM and to extend the analysis to more complex multi-band and correlated scenarios.

Abstract

Scanning tunneling microscopy (STM) measurements have been extensively performed on the easily cleavable (011) surface of UTe$_2$, using both normal-metal and superconducting tips. Motivated by these experiments, we theoretically investigate the topological surface states on the (011) surface of UTe$_2$. We find that a two-dimensional nearly flat band emerges in the $B_{3u}$ state, giving rise to a pronounced zero-energy peak in the surface density of states. This flat band is supported by two key mechanisms: (i)~nontrivial Berry phases defined at multiple momenta give rise to low-energy in-gap states, and (ii)~weak spin conservation allows the gap function to acquire phase winding. Furthermore, to investigate the relation between the zero-bias peak observed in recent STM experiments with a superconducting tip and the topological surface states, we calculate the nonequilibrium dc tunneling current in a junction between an $s$-wave superconductor and the (011) surface of UTe$_2$. Our results provide crucial insights into the superconducting pairing symmetry realized in UTe$_2$.

Figures (8)

• Figure 1: (a) A naturally cleavable $(011)$ plane. (b) Cylindrical Fermi surface elongated along the $k_z$-axis, plotted in the rotated coordinate system $(k_x, k_+, k_-)$, where the $k_yk_z$-plane is rotated by $\pi/4$ around the $k_x$-axis. (c) Plot of the $k_x = 0$ plane in the first BZ (shaded area), with the Fermi surface shown as orange curves. $k_{\sharp i}$ ($i = 1 \sim 4$) denotes the time-reversal-invariant momenta (TRIMs). Nontrivial Berry phases can be defined along 1D paths in $k_+ \in [-2\pi, 2\pi]$, centered at $\Gamma = (k_x = 0, k_m = 0)$ and $M = (k_x = 0, k_m = \pi)$. Each path connects a pair of TRIMs lying along the $k_+$ direction. (d) Fermi surface plots at $k_m = 0$ (solid curves) and $k_m = \pm\pi$ (dashed curves).
• Figure 2: (a–d) Quasiparticle energy bands for the (a) $A_u$, (b) $B_{1u}$, (c) $B_{2u}$, and (d) $B_{3u}$ states. The superconducting gap $\Delta_{\mathrm{UTe}2}$ is set to $0.05$. The inset in panel (a) shows the momentum path in the surface BZ depicted in the main panels. For the $A_u$ state, zero-energy states protected by nontrivial Berry phases appear at the $M$ and $\Gamma$ points, and low-energy in-gap states smoothly connecting them emerge along the $M$–$\Gamma$ line. For the $B{1u}$ and $B_{2u}$ states, Fermi arcs protected by the one-dimensional winding number associated with mirror reflection symmetry appear. For the $B_{3u}$ state, in addition to the in-gap states along the $M$–$\Gamma$ line, Fermi arcs protected by the one-dimensional winding number associated with spin conservation emerge along the $\Gamma$–$X$ and $R$–$M$ lines. (e–h) DOS for the $(011)$ surface (red) and the bulk (black) for the (e) $A_u$, (f) $B_{1u}$, (g) $B_{2u}$, and (h) $B_{3u}$ states. For the $B_{3u}$ state, as a result of the spread of nearly zero-energy states across the 2D surface BZ, the surface DOS exhibits a pronounced zero-energy peak.
• Figure 3: Quasiparticle energy bands on the $(011)$ plane for the superconducting $B_{3u}$ state. A substantial number of ABS form a 2D nearly flat-band structure. The red cones indicate the point nodes of the bulk superconducting gap.
• Figure 4: (a) Quasiparticle energy bands for the $B_{3u}$ state with parameters $C_x = 0.05$, $C_y = 0.05$, and $C_z = 0.05$. $\Delta_{{\rm UTe}_2}$ is set to $0.05$. The $d_x$ component breaks spin conservation, resulting in a gap opening along the $\Gamma$--$X$ and $R$--$M$ lines (b) Surface density of states as a function of $C_x$.
• Figure 5: Normalized differential conductance $(dI/dV)$ of a superconducting junction between UTe$_2$ and an $s$-wave superconductor. The superconducting state in UTe$_2$ corresponds to the (a) $A_u$, (b) $B_{1u}$, (c) $B_{2u}$, and (d) $B_{3u}$ representations, respectively. The $s$-wave superconducting gap is set to $\Delta_s = 2\Delta_{\text{UTe}_2}$. (e) Color bar shows the effective transparency, $\alpha \in [0,1]$.