Field-angle-resolved heat transport in UTe$_2$: determination of nodal positions in the superconducting order parameter

TL;DR

The study addresses identifying the superconducting order-parameter symmetry in UTe2 by performing field-angle-resolved bulk thermal transport measurements deep in the superconducting state. It combines a realistic two-band tight-binding description of the Fermi surface with a Doppler-shift (Volovik) framework to model the angular dependence of the electronic thermal conductivity, comparing single-component and two-component (TRSB) states. The key finding is that point nodes lie along the crystallographic $b$-axis, placing the zero-field ground state in the $B_{2u}$ irreducible representation; this conclusion is supported by the observed inversion of oscillations with temperature and is robust against reasonable variations in model parameters. The results provide a direct bulk diagnostic of gap symmetry, offering a clear pathway to map nodal structures in UTe2 and guiding future high-resolution angle-dependent studies of unconventional superconductors.

Abstract

One of the recurring hurdles in studying unconventional superconductivity is the challenge of efficiently and conclusively identifying the symmetry of the superconducting order parameter in a new material. Uranium ditelluride (UTe$_2$) exhibits an unprecedented number of superconducting phases as a function of pressure and magnetic field, each presumably characterized by a different symmetry of the superconducting gap function. None of these phases has had its symmetry conclusively identified so far. In this article, we report results of an extensive study of the thermal conductivity of UTe$_2$ in its low-field, low-temperature superconducting state as a function of the angle of an applied magnetic field rotated in the $b$-$c$ plane. We observe clear and substantial oscillations in the thermal conductivity as a function of field angle, which naturally suggests the existence of point nodes in the gap. Utilizing the experimentally determined Fermi surface, we are able to model this phenomenon for all the potential gap structures in UTe$_2$ and positively identify the location of these nodes as being along the crystallographic $b$-axis, implying that the superconducting order parameter belongs to the $B_{2u}$ irreducible representation of the crystal point group. The clarity of this result will accelerate the identification of other superconducting phases in UTe$_2$, and guide future studies through the use of high resolution field-angle-dependent measurements.

Figures (9)

• Figure 1: Magnetic field-angle dependence of thermal conductivity deep in the superconducting state of UTe$_2$. Panel a) presents the angle-sweep data measured at a fixed temperature of 70 mK ($\sim 0.03 T_c$), with heat current applied parallel to the $a$-axis and field rotations through the perpendicular $b$-$c$ plane at fixed fields of 100, 150 and 250 mT. Panel (b) shows fixed-angle field sweeps with the same geometric configuration at the same temperature. The field values in this plot have been normalized to the angle-dependent upper critical field $H_{c2}(\theta)$ran_extreme_2019.
• Figure 2: Model Fermi surface of UTe$_2$ and gap node arrangements for thermal conductivity calculations. Panel a) presents the three-dimensional Fermi surfaces of UTe$_2$ used to model thermal conductivity, where the color function depicts the the magnitude of the Fermi velocity normalized to its maximum value. Heat current ($\parallel a$-axis) and magnetic field rotation ($b$-$c$) plane are indicated by magenta and yellow arrows, respectively. Panel b) shows the dispersion of U and Te bands that cross the Fermi level along the high symmetry directions. Panels c) and d) present the $k_z$=0 Fermi surface cuts and nodal positions for the possible $B_{2u}$ and $B_{3u}$ order parameter states, respectively, including symmetry-imposed nodes (red squares) and accidental nodes (circles) for various values of tuning parameter $\alpha$ which controls the positions of the accidental nodes (see text). The solid lines represent the Fermi surface contours, and the dotted-dashed lines indicate the zeros of the gap function.
• Figure 3: Comparison of experimental $b$-$c$ plane field-angle sweep thermal conductivity to theoretical models. Panel (a) presents experimental data measured at 40 mK for $a$-axis heat current and $b$-$c$ plane rotation of 100 mT field. Panels b)-f) present theoretical models of the angle-dependent electronic thermal conductivity normalized to its average for the same current-field orientation, for the following cases: b) for a simple axial state on a spherical Fermi surface with linear point nodes positioned along the ${b}$ (black) and ${a}$ (red) axes; c) for all four irreducible representations of the D$_{\mathrm{2h}}$ states on realistic Fermi surfaces with equal magnitude gaps on two bands; d) for the $B_{2u}$ state with different gaps on two bands; e) for the $B_{2u}$ state with accidental nodes generated by parameter $\alpha$ that mixes nearest neighbor and next-nearest neighbor pairing ($B_{3u}$ state with only next-nearest neighbor pairing shown with dashed line); f) for the $B_{2u}$ state with variation of the coefficient of the $\hat{c}$ component of the $\mathbf{d}$-vector basis function. All theoretical calculations are done at 40 mK and 100 mT field. The impurity parameters are chosen to give $0.5\%$$T_c$ suppression.
• Figure 4: Temperature evolution of field-angle sweep $a$-axis thermal conductivity. Panel (a) plots experimental data measured at fixed temperatures ranging from 40-900 mK and $b$-$c$ plane rotation of 100 mT field. Panel (b) presents the calculated angle-dependent thermal conductivity for the $B_{2u}$ state for 100 mT applied field and various temperatures. The impurity parameter gives $0.5\%$ suppression of the transition temperature with equal gap magnitudes on two bands. The coefficients of the basis functions are set to unity and only nearest neighbor pairing is included (i.e.,$\alpha=1$).
• Figure S1: Magnetic field-angle dependence of UTe$_2$ thermal conductivity up to 1750 mT. Panel a) presents the fixed-angle field sweeps with heat current applied parallel to the $a$-axis and field angles rotated through the perpendicular $b$-$c$ plane. Panel b) shows the fixed-field angle sweeps. All data are measured at a fixed temperature of 70 mK.