Correlations between superconducting and resistive anisotropies

TL;DR

This work develops a unified framework to relate rotational-symmetry breaking in the normal state to transport anisotropies in the corresponding superconducting phase, focusing on intervalley and intravalley pairing in moiré graphene and related systems. By combining a general formalism for inter-valley pairing with explicit model calculations on a triangular lattice, the authors compare normal-state resistivity anisotropy and superconducting critical-current anisotropy across scenarios including $C_{3z}$-breaking nematic normal states, vestigial nematic order from nematic superconductivity, and finite-momentum intravalley pairing with vestigial stripe orders. Central results show that the relative orientation of maximal resistivity and maximal critical current, captured by $\zeta_{xy}$, depends sensitively on the source of symmetry breaking and on pairing parity, providing a diagnostic tool to constrain pairing symmetry and the role of vestigial orders in experiments on twisted graphene and TMD heterostructures. The analysis, including Aslamazov-Larkin paraconductivity and two-quanta vestigial phases, highlights how transport measurements along different crystallographic directions can reveal the underlying intertwined orders and influence the interpretation of nematic superconductivity in two-dimensional moiré materials.

Abstract

There are multiple possible origins of transport anisotropies in metals and superconductors. For instance, rotational symmetry can be spontaneously broken in the normal state as a result of electronic nematic order inducing anisotropies in an otherwise $s$-wave superconducting phase. Another possibility is that the dominant source of rotational symmetry breaking is the superconductor itself and its vestiges that may survive in the normal state. We here theoretically analyze the correlations of transport anisotropies in the normal and the corresponding superconducting phase for different scenarios of broken symmetry, either coming solely from the normal state, solely from the superconductor and its vestiges in the metallic regimes, or from both simultaneously. We further include both zero-momentum and finite-momentum pairing; we develop a theory of vestigial order for the latter, characterized by broken rotational and translational symmetry. Our findings reveal that the relative transport anisotropies in the normal and superconducting phases sensitively depend on the scenario, including the form of vestigial order and, in some cases, the parity of the superconducting order parameter. As such, measuring the directional dependence of the critical current and resistivity can provide strong constraints on the origin of rotational symmetry breaking. We demonstrate our findings in minimal models relevant to twisted multilayer graphene, rhombohedral graphene, and twisted transition metal dichalcogenides.

Figures (7)

• Figure 1: (a) Fermi surfaces (FSs) for the two-valley model. The presence of nematic order along $x$ ($\theta = 0$) breaks the $C_{3z}$ symmetry in the normal state (b) shows the resistivity (in blue) and critical current (in red) for $\Phi = 0.2$. The rest of the parameters are given in Caption1.
• Figure 2: Nematic SC arising from $C_{3z}$ breaking in the normal state. (i) Weak nematic/strain order ($\Phi_{V} = 0.03$); (ii) strong nematic/strain order ($\Phi_{V} = 0.3$). (a) Fermi surfaces (FSs) for the two-valley model. The presence of nematic order along $x$ ($\theta = 0$) breaks the $C_{3z}$ symmetry in the normal state. The critical currents (in red) and resistivity (in blue) are shown for (b) odd and (c) even form factors. Explicit parameters are same as Caption1 except $\phi = -0.7\pi$.
• Figure 3: Critical current of a nematic superconductor with (a) odd form factor, (b) even form factor. Panel (c) depicts the angular dependence of resistivity arising from a vestigial nematic order. For both the even and odd cases, the angular dependence $\rho(\Omega)$ looks qualitatively identical. Parameters are the same as Caption1.
• Figure 4: (a) Depiction of the Aslamazov-Larkin diagram. (b) Resistivity anisotropy, $\rho_{xx}-\rho_{yy}$ vs vestigial order $\tilde{\Phi}_x^0$, as computed via the Aslamazov-Larkin diagram.
• Figure 5: Normal state vestigial phases and their resistivities. In (a,c) we show the Fermi surfaces and resistivities for the cases with $\beta=0$, and in (b,d) with $\tilde{\alpha}(\boldsymbol{k})=0,\beta\neq 0$. In (a,c) the green FS and resistivity curve indicates the case without nematicity, whereas the dark blue curves are with $\Lambda_{1}= 0, \Lambda_{2}=\Lambda_{3}=\Lambda\neq 0$. The Brillouin-zone strip is indicated by the dashed lines in (b).