Spin-polarized superconductivity: order parameter topology, current dissipation, and multiple-period Josephson effect Eyal Cornfeld,1 Mark S. Rudner,2 and Erez Berg1 1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel 2Niels Bohr International Academy and Center for Quantum Devices, University of Copenhagen, 2100 Copenhagen, Denmark We discuss transport properties of fully spin-polarized triplet superconductors, where only elec- trons of one spin component (along a certain axis) are paired. Due to the structure of the order parameter space, wherein phase and spin rotations are intertwined, a configuration where the super- conducting phase winds by 4π in space is topologically equivalent to a configuration with no phase winding. This opens the possibility of supercurrent relaxation by a smooth deformation of the order parameter, where the order parameter remains non-zero at any point in space throughout the entire process. During the process, a spin texture is formed. We discuss the conditions for such processes to occur and their physical consequences. In particular, we show that when a voltage is applied, they lead to an unusual alternating-current Josephson effect whose period is an integer multiple of the usual Josephson period. These conclusions are substantiated in a simple time-dependent Ginzburg-Landau model for the dynamics of the order parameter. One of the potential applications of our analysis is for moir´esystems, such as twisted bilayer and double bilayer graphene, where superconductivity is found in the vicinity of ferromagnetism. I. INTRODUCTION Spin-triplet superconductors (SCs) and superfluids are predicted to exhibit rich phenomena owing to the inter- play between the spin and phase degrees of freedom of their order parameters. A celebrated example is super- fluidity in 3He [1,2]. Triplet superconductivity remains scarce in electronic systems, however; possible examples include uranium heavy-fermion compounds where super- conductivity is found to coexist with ferromagnetism [3{ 5], and Sr2RuO4 [6], although the latter has recently been contested [7]. Two-dimensional moir´ematerials, such as twisted bi- layer graphene (TBG) and heterostructures based on other van der Waals materials have recently emerged as a fertile ground for novel correlated-electron phenom- ena [8{29]. In particular, the phase diagrams of these sys- tems include spin and valley polarized states [19, 22, 30] residing in proximity to superconducting states, raising the possibility of spin-triplet superconductivity. More- over, since these systems have multiple valleys in their band structures and can be made relatively clean, they may avoid the pair-breaking effect of disorder that in- hibits triplet superconductivity in many materials. In carbon-based materials, spin-orbit coupling is expected to be negligible, opening the possibility of a non-trivial FIG. 1. (a) A depiction of the proposed experimental setup; a fully spin-polarized triplet superconductor is connected to a intertwining of the gapless magnetic and superconduct- arXiv:2006.10073v3 [cond-mat.supr-con] 15 Jan 2021 DC voltage source and drain via the left and right leads. We ing phase degrees of freedom. Intriguingly, recent experi- overlay a current-carrying configuration of the order param- ments in twisted double-bilayer graphene (TDBG) with a eter. (b) A visualization of the order parameter triad where perpendicular electric field show possible signs of triplet d(1) and d(2) are depicted by the red and blue arrows and superconductivity [31, 32]. the pairing polarization m is depicted by the gray arrow. (c) These remarkable findings motivate us to reexamine Schematics of the current response of the system portray AC the physics of triplet SCs in systems with negligible spin- Josephson effect period doubling. orbit coupling. We focus on a particular state which is natural for TBG and related graphene-based moir´e materials: a fully spin-polarized triplet superconductor, in which only electrons of one spin component (along a 2 spontaneously chosen quantization axis) are paired. This exchange both energy and spin angular momentum. For phase is analogous to the A1 and β phases discussed in currents larger than the critical current density men- 3 the context of superfluid He [1,2, 33]. The order param- tioned above, Jc, the model provides a prediction for the 2 eter of such a superconductor is specified by the spin di- current-voltage relation: V / (J − Jc) , up to logarith- rection of the condensate and its phase; topologically, the mic corrections, where J is the DC component of the order parameter space is equivalent to the space of three- current. When an external Zeeman field is applied (e.g., dimensional rotation matrices, SO(3) [34]; see Fig.1(b). an in-plane magnetic field in a two-dimensional system), This property crucially determines the topology of the it pins the direction of the spin magnetization, and the order parameter space, and hence the ability of the su- properties of the system rapidly cross over to those of perconductor to carry stable supercurrents. an ordinary SC. We hope that these predictions will pro- In a ring geometry, the superconducting phase of an vide guidance to experiments in novel exotic SCs, such as ordinary (e.g., singlet) SC can wind any integer number TDBG, where they can be used to confirm or invalidate of times around the hole of the ring. Mathematically, the existence of a fully spin-polarized triplet SC. this is expressed by the homotopy group π1[U(1)] = Z. This paper is organized as follows. In Sec.II we review Changing the winding number requires creating a topo- the properties of a fully-spin polarized SC, its distinc- logical defect at which the magnitude of the order pa- tion from other triplet SC phases, and the topology of rameter is suppressed to zero, such as a phase slip or a its order parameter space. We then describe the physi- vortex [35, 36]. Since such defects are energetically sup- cal picture and summarize the main results of the paper. pressed, a configuration with a non-zero winding number Sec. III describes the Ginzburg-Landau (GL) model. In (corresponding to a supercurrent) is metastable and may Sec.IV we study the energy landscape of the model, ei- persist over a very long time. ther with or without an applied Zeeman field, deriving In contrast, the homotopy group the order param- the maximum supercurrent the system can carry in a eter of a fully spin-polarized triplet SC on a ring is metastable state. Sec.V introduces a time-dependent extension of the GL model. This model allows us to π1[SO(3)] = Z2. Physically, this means that in such a SC, two vortices can always annihilate each other, re- study dynamic phenomena, such as the properties of the gardless of their vorticity. A configuration in which the system in the presence of a finite applied voltage. The SC phase winds twice, either around a point or in a ring results are discussed in Sec.VI. The Appendices contain geometry, can be \untwisted" back to a uniform configu- technical details of the solution of the time-dependent ration continuously, without diminishing the magnitude Ginzburg-Landau (TDGL) equations. of the order parameter at any location. The untwisting process involves a temporary change in the system's mag- netization; thus, the timescale for this process to occur II. PHYSICAL PICTURE & MAIN RESULTS depends on the coupling of the SC condensate to either an intrinsic or extrinsic bath with which it can exchange In order to gain physical intuition of spin-polarized spin angular momentum. triplet superconductivity, we begin with a short review The purpose of this work is to explore the unusual of the relevant order parameters. Those who are already transport properties of fully spin-polarized triplet SCs familiar with this material may opt to skip to the results that result from the topology of their order parameter which are presented in Sec. IIB. space. We find that in such SCs, a supercurrent-carrying state is much more fragile than in an ordinary SC. The critical current density, Jc, that the system can sustain in A. Review of triplet superconductivity and a metastable state depends on the applied Zeeman field, application to multi-valley systems B. For B = 0, we find that Jc scales inversely with the system size along the current direction, and depends on 1. Order parameter both the spin and phase stiffnesses. A small voltage ap- plied across the SC results in a time-dependent current with a direct-current (DC) component of magnitude close The order parameter of a triplet SC can be repre- to this critical current, and an alternating-current (AC) sented in terms of the so-called d-vector, which is a complex vector defined as d = h y iσ σ y i, where component with a fundamental frequency! ~J = eV=~, k k 2 −k y y y i.e., half of the usual Josephson frequency across a SC k = ( k"; k;#), and the Pauli matrices σ = (σ1; σ2; σ3) weak link; see Fig.1(c). This doubled periodicity is di- act on the spin degrees of freedom ("; #). The Pauli prin- rectly related to the Z2 topological structure of the order ciple forces dk = −d−k.[37] parameter space. For sufficiently large appliedp Zeeman The existence of multiple valleys in systems such as field, the critical current scales as Jc / B indepen- TBG and TDBG enables the possibility of a valley-singlet dently of system size. state which does not require the order parameter to be We demonstrate these phenomena within a time- momentum dependent within each valley. (Exotic spin- dependent Ginzburg-Landau model, where we assume singlet order parameters have been proposed in mono- that the system is coupled to a bath with which it can layer graphene [38]; here, we focus on spin-triplet pair- 3 ing.) Considering a system at a finite density away from (i) Non-spin-polarized, unitary \spin nematic" phase.{ charge neutrality with a finite Fermi surface, we project In this phase, the pairing polarization vanishes, the pairing potential to Bloch states near the Fermi sur- m = 0, corresponding to d(1) jj d(2).