Spin-Polarized Superconductivity: Order Parameter Topology, Current Dissipation, and Multiple-Period Josephson Effect

Home , Andrew Peter Mackenzie, Bilayer graphene, Rafi Bistritzer

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 electrons 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 superconducting 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 moirésystems, 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 superconductivity is found to coexist with ferromagnetism [3– 5], and Sr2RuO4 [6], although the latter has recently been contested [7]. Two-dimensional moirématerials, such as twisted bilayer graphene (TBG) and heterostructures based on other van der Waals materials have recently emerged as a fertile ground for novel correlated-electron phenomena [8–29]. In particular, the phase diagrams of these systems 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 moiré 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 magnetization; 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 applied 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 = hψ† iσ σψ† i, where component with a fundamental frequencyω ˜J = eV/~, k k 2 −k † † † i.e., half of the usual Josephson frequency across a SC ψk = (ψk↑, ψk,↓), 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 applied√ 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) || d(2). The symme- faces of the two valleys. try is reduced to the group of SO(2)S spin rotations In this context, we denote creation operators of elec- around a preferred direction and the [Z2]S+φ oper- trons at the K+ and K− valleys by the two-component ation of a simultaneous π spin rotation around an † † † † † † axis perpendicular to d1,2 and a π gauge transfor- spinors ψ+ = (ψ+,↑, ψ+,↓) and ψ− = (ψ−,↑, ψ−,↓), respectively. An inter-valley, valley-singlet superconduct- mation. The order parameter degeneracy space is ing order parameter corresponds to [39] given by

† † SO(3)S × U(1)φ 2 d = hψ+iσ2σψ−i, (1) = [S /Z2]S × U(1)φ. (5) SO(2)S × [Z2]S+φ where d is independent of momentum within each val- This phase has the same pattern of spin and gauge ley. For simplicity, we will focus on this type of state symmetry breaking as the A and B phases of henceforth. 2 3He [1, 33]. We note, however, that the phases The d-vector may be conveniently described using two 3 (1,2) of He differ by the breaking of orbital symme- real vectors d , try which is absent for the valley-singlet phase described here. d = d(1) + id(2), (2) (ii) Fully spin-polarized non-unitary triplet.– In this We define the pairing polarization m and superfluid den- phase d · d = 0, corresponding to d(1) ⊥ d(2) and sity ρ as |d(1)| = |d(2)|. The condition d·d = 0 can be inter- preted as the vanishing of a charge 4e scalar order 1 m = d∗ × d = d(1) × d(2), ρ = d∗ · d, (3) parameter ∆4e = d·d. The symmetry is reduced to 2i that of U(1)S+φ rotations around the pairing polarization direction, m, matched with gauge transfor- see Fig.1(b). At temperatures far below the mean-field mations by the same phase. The order parameter superconducting phase transition, we may treat the prob- degeneracy space is given by lem within the London limit where the magnitude of the order parameter is essentially fixed, SO(3)S × U(1)φ = SO(3)S+φ. (6) ∗ 2 U(1)S+φ d · d = 2d0. (4) This situation is similar to the A1 and β phases of Note that the pairing polarization m arises from the 3He [1, 33], up to orbital symmetries (see above). anomalous expectation value d in Eq. (1). At equilibrium, the pairing polarization is coupled to the physical (iii) Partially spin-polarized non-unitary triplet.– In this magnetization (spin polarization) density, and propor- phase neither m nor ∆4e vanish. The symmetry tional to it. This relation also holds for slowly varying is completely broken and the degeneracy space is magnetization textures. Thus, below we will discuss the given by behavior of the pairing polarization interchangeably with that of the magnetization density. SO(3)S × U(1)φ. (7)

This situation is similar to the B and A2 phases of 3He [1, 33], up to orbital symmetries (see above). 2. Symmetry classification of valley-singlet states Recent theoretical works [40, 41] have pointed out that Triplet superconducting phases have been studied in the order parameter of TDBG might be in a fully spin- great detail in the context of superfluid 3He; see Refs.1 polarized state, at least in a part of the phase diagram. and2 for a review. These works naturally focused on Motivated by this possibility, together with a fundamen- p-wave phases. However, as discussed above, in multi- tal interest in the novel properties that we uncover below, valley materials, there is a possibility of a valley-singlet in the remainder of this work we will focus on the proper- phase. Nevertheless, a similar symmetry classification of ties of fully spin-polarized triplet SCs [phase (ii) above]. the possible phases can be carried out. Neglecting spin-orbit coupling, the free energy of the B. Supercurrent decay mechanisms system is invariant under transformations in the SO(3)S group of global spin rotations and in the group of U(1)φ gauge transformations. We thus distinguish three dis- The robustness of persistent supercurrents in ordinary tinct triplet phases, according to their patterns of broken SCs has a topological origin: in a superconducting ring, symmetries: the winding number of the phase of the order parameter 4

FIG. 2. Continuous trajectory for supercurrent relaxation. We present the trajectory of Eq. (31) relaxing a supercurrent of (1) (2) J = 2J0 at t = 0 to a supercurrent of J = 0 at t = τ. The order parameter components, d and d , are depicted by the red and blue arrows, the gray arrows depict the pairing polarization, m; see Fig.1(b).

around the ring is an integer that cannot be changed rent direction, B is the applied Zeeman field, and TBKT by a small, local perturbation. For fully spin-polarized is the Berezinskii-Kosterlitz-Thouless (BKT) transition triplet SCs, the SO(3) topology of the order parameter temperature. Intuitively, in for a small Zeeman field, we configuration space [42], find only a mesoscopic contribution that scales inversely with system size, corresponding to a finite number of π [SO(3)] = Z , (8) 1 2 windings. On the contrary, a large Zeeman field acts to has important consequences for the stability of the super- lock the pairing polarization, increasing the critical cur- current. Specifically, consider an initial current-carrying rent. For a large enough Zeeman field, Jc exceeds the configuration with a constant pairing polarization, and intrinsic (microscopic) critical current density; continu- a certain number n of windings of the superconducting ous unwinding events then become irrelevant, and the phase across the system. A continuous relaxation event, system behaves as an ordinary SC. such as the one illustrated in Fig.2, can reduce the phase winding to n−δ, for some even integer, δ. The unwinding event may involve a finite free energy barrier, which de- A direct physical manifestation of the order parame- pends on the initial number of windings, n, on the ratio ter topology can be observed when a voltage is applied between the phase and pairing polarization stiffnesses, across a fully spin-polarized SC [Fig.1(a)]. The volt- and on the applied Zeeman field. The barrier originates age causes the superconducting phase at the right lead from the fact that the unwinding process involves twist- to wind relative to the phase at the left lead, at a con- ing the pairing polarization in space and changing its stant rate ω = 2eV/ . Once the phase difference across average direction. J ~ the system has increased by 4π, there is no topological Throughout this paper we assume the presence of suf- obstruction to continuously “unwind” the phase twist, ficiently large fluctuations such that the system explores relaxing the supercurrent. After the unwinding event, its configuration space and that any energetically favor- the phase difference (and hence the supercurrent) starts able configuration (without an intermediate energy bar- growing linearly in time again. Assuming that the dis- rier) is quickly explored; we discuss relaxing this assump- sipation rate is large compared to the Josephson fre- tion at Sec.VIB. We analyze the energy landscape of the quency (a condition to be discussed further in Sec.VIC), continuous unwinding process in Sec.IV. The result is we find that the current undergoes periodic oscillations that the energy barrier vanishes above a certain current with frequencyω ˜ = 2eV/ , i.e., half of the ordinary density, which we define as the critical current J . The J ~ c AC Josephson frequency [see Fig.1(c)]. In fact, we find critical current has two contributions, that at large magnetic fields it is energetically favorable √ ( 2 to unwind more than two windings at once thus further e kBTBKTµB µB kBTBKT/Lx, Jc ∼ 2 (9) fractionalizing the ordinary AC Josephson frequency, i.e., ekBTBKT/Lx µB kBTBKT/Lx. ω˜J = 2eV/δ~ with δ ≥ 2 an even integer. This is one of where µ is the magnetic moment density in the ma- our main results. We substantiate it by solving a time- terial, Lx is the length of the sample along the cur- dependent Ginzburg-Landau model in Sec.V. 5

III. GINZBURG-LANDAU MODEL expressed in Eq. (8) follows directly from this property (see discussion in Sec.IV). This description holds in the 2 2 A. Complex vector description London limit where |d| = 2d0. Gauge transformations are implemented as

The starting point of our analysis of a spin-polarized iqΛ i q Λσ A 7→ A + ∇Λ, d 7→ e d, u 7→ ue 2 3 , (15) triplet SC is most generic gauge-invariant form of the of the free energy, expanded in long wavelengths up to where Λ(r) is a scalar function. On the other hand, a second order in derivatives, as a function of the complex spin rotation of angle θ around an axis nˆ acts by d-vector order parameter: θ i 2 nˆ·σ Z κ κ 2 u 7→ e u. (16) F [d] = d2r d |(∇ − iqA)d|2 + m ∇( 1 d∗ × d) 2d2 2d4 2i 0 0 The most general form of the free energy invariant un- µ 1 ∗ der gauge and spin rotation transformations [Eq. (10)] is − 2 B · ( 2i d × d) + U(d) . (10) d0 given by

Here, κd is a generalized phase stiffness, κm is the excess Z κ − κ F [u] = d2r m d Tr ∇(uσ u†) · ∇(uσ u†) spin stiffness, B is the applied Zeeman field, µ is the mag- 4 3 3 netic moment density, and the potential U(d) can be any † q † q gauge-invariant function[43]. In order for the free energy + 2κdTr ∇u + i 2 Aσ3u ∇u − i 2 Auσ3 to be bounded from below, the generalized phase stiff- µ − B · Tr uσ u†σ . (17) ness, κd, and the excess spin stiffness, κm, must satisfy 2 3 κd ≥ 0 and κm ≥ −κd. We have opted for the Weyl- gauge, i.e., zero electric scalar potential, φ = 0, such This unitary matrix description automatically enforces ∗ 2 2 that the system is coupled to the magnetic vector poten- both d ·d = 2d0 and |d·d| = 0, hence, we have dropped tial A; for generality, we keep the charge, q, generic (in the constant scalar potential, U(d) = const.. Within this electronic SCs q = −2e). description, the supercurrent density [Eq. (13)] and the The Poisson bracket between the components of d pairing polarization [Eq. (3)] are given by would be useful is the study of the dynamics (Sec.V), δF ∗ 0 0 J = − = −iqκ Tr u†∇uσ − uσ ∇u† − 2q2κ A, {di(r) , dj(r )} = 2iδijδ(r − r ). (11) δA d 3 3 d 1 These give rise to the following relations: m = d2 Tr uσ u†σ. (18) 0 2 3 0 0 {mi(r), mj(r )} = εijkmk(r)δ(r − r ), {d (r), m (r0)} = ε d (r)δ(r − r0), i j ijk k IV. ENERGY LANDSCAPE {d(r), ρ(r0)} = −2id(r)δ(r − r0), (12) consistent with m and ρ being the generators of spin We now consider the energetics of the continuous un- rotations and gauge transformations, respectively. winding process for supercurrent relaxation. In general, The supercurrent density is given by a supercurrent carrying configuration would be energetically less favorable than a configuration with a smaller ∗ ∗ ∗ δF d · ∇d − d · ∇d 2 d · d supercurrent. This less favorable configuration may be J = − = qκd 2 −q κd 2 A. (13) δA 2id0 d0 either a metastable or an unstable state. As long as the higher-energy configuration is metastable, either thermally activated or quantum tunneling events would act to B. Unitary matrix description relax the current; however, such processes are expected to be exponentially suppressed. Therefore, as the supercur- A very useful description of the SO(3) topology for the rent gradually increases (e.g., in response to an applied purpose of explicit computations is through its SU(2) DC voltage; see Fig.1), the system should stabilize on a double cover. This description is applicable only for value of the current close to its lowest unstable state. the fully spin-polarized phase (see Sec. IIA2), where it captures the three-dimensional nature of the degeneracy space. For every matrix u ∈ SU(2) we can associate a A. Linear stability complex d-vector We seek the lowest possible supercurrent for which a 1 † d = d0 Tr u(σ1 + iσ2)u σ . (14) supercurrent carrying state ceases to be metastable (i.e., 2 where an instability develops). The double cover is evident by the identical order param- We consider an applied Zeeman field in the plane of the eters corresponding to both u and −u. The homotopy SC, hence, without loss of generality, we set B = Bxˆ. In 6 the presence of this Zeeman field, the system always has spontaneous polarization texture should form. Neverthe- as a fixed point, δF/δd = 0, where the d-vector exhibits less, this does not guarantee the existence of a trajectory a uniform configuration described by in configuration space which connects the system to a new stable fixed-point without traversing an energy bar- d0 = (0, 1, i)d0. (19) rier. In the next section, we thus estimate the possible energy barriers and seek out these trajectories. This configuration may be set to carry arbitrarily high 1 currents by applying a vector potential A = − 2 J, 2q κd with a current J = Jxˆ; see Eq. (13). In search of an B. Unwinding trajectories and energy barriers instability, we study small deviations, d = d0 + δd, from the configuration d in Eq. (19), with 0 We move on to estimate the free energy barrier for re- δd(1)/d = (−η , − 1 (η2 + η2), η ), laxing two windings of the superconducting phase contin- 0 2 2 2 3 3 uously, as allowed by the topology of the order parameter (2) 1 2 2 δd /d0 = (η1, −η3, − 2 (η1 + η3)), (20) [Eq. (8)]. (1) (2) 2 1 2 2 This is easiest to describe using the unitary matrix δ(d × d )/d = (− (η + η ), η2, −η1), (21) 0 2 1 2 presentation. Moreover, as the only applied electromag- where η = (η1, η2, η3) is real, and |η| 1. Here we netic field is the in-plane Zeeman field, we opt to pick the have taken the most general deviation respecting the fully gauge of A = 0 at all positions within the plane of the ∗ 2 spin-polarized SO(3) structure, d ·d = 2d0, and d·d = 0, SC. In this gauge and presentation, a uniform persistent up to second order in η. Expanding the free energy to supercurrent carrying state is given by second order in the deviation η yields Jx iσ3 4qκ Z u(r) = u0e d , (27) 2 κm + κd 2 2 2 δF [η] = d r (∇η1) + (∇η2) + κd(∇η3) 2 where u0 is any SU(2) matrix. We consider a system with 1 µB cylindrical geometry of height Ly and circumference Lx. − J · (η ∇η − η ∇η ) + (η2 + η2) . (22) 2q 1 2 2 1 2 1 2 The order parameter, d, thus has periodic boundary conditions along x, which force either periodic or antiperi- We seek the least stable direction, defined with respect odic boundary conditions for u. These correspond to the to the curvature of δF , and find that it takes the form two Z2 classes of maps from the cylinder to the order parameter space; see Eq. (8) and discussion in Sec.IIIB. As η = (η cos kx, η sin kx, 0). (23) a consequence, the supercurrent is quantized to J = J0n, where n ∈ Z and For such a deviation, the corresponding change of the free energy is given by 4πqκd J0 = (28) Lx Z 2 h i δF [η] = d2r η (κ + κ )k2 − 1 Jk + µB . (24) 2 m d q is the fundamental current; this configuration is overlaid in Fig.1(a). The wavenumber, k, for which the free energy is minimal, From the Z order parameter topology, we know that is thus k = J . However, mesoscopic finite size 2 2q(κm+κd) for any even integer, δ, there exist trajectories u(r, t) with effects must be taken into consideration. Specifically, a n windings at t = 0 and n−δ windings at some later time system on a cylinder of circumference Lx cannot support t = τ, in which the order parameter magnitude remains fluctuations smaller than the fundamental wavelength, constant throughout the deformation. Among these, we i.e., seek the trajectory with the minimal energy barrier. Since B = Bxˆ, we look for trajectories with uniform 2π J k ' max , , (25) pairing polarization m = d0xˆ at initial and final times Lx 2q(κm + κd) t = 0 and t = τ, respectively: where strictly speaking, the wavenumber, k, must take iπσ n x u(r, t = 0) = u e 3 Lx , (29) the closest positive integer multiple of 2π . We thus find 0 Lx x iπσ3(n−δ) the critical current for the onset of instability, Jc, as the u(r, t = τ) = u0e Lx , (30) value of J for which the curvature of the quadratic de- π −i σ2 pendence of δF on η [Eq. (24)] becomes negative: with u0 = e 4 . We study the following variational

2 family of trajectories u(r, s1, s2), which connect these ini- ( p 4π (κm+κd) 2q (κm + κd)µB µB ≥ 2 , tial and ﬁnal states, and parametrically depend on time Lx Jc ' 2 (26) 2πq(κm+κd) qLxµB 4π (κm+κd) via s1,2(t = 0) = 0 and s1,2(t = τ) = 1: + µB < 2 . Lx 2π Lx

π 1 x π 1 x −i σ1s2 iπσ3 δ i σ1s1 iπσ3(n− δ) For J ≥ Jc, the uniform pairing-polarization ﬁxed-point u(r, s1, s2) = u0e 2 e 2 Lx e 2 e 2 Lx . conﬁguration in Eq. (19) becomes a saddle point, and a (31) 7

Comparing Eqs. (29)-(31) with Eq. (27), it is evident that For J < Jc, current relaxation by continuous unwinding this trajectory connects a supercurrent-carrying state processes is possible, but requires traversing a barrier. In with n windings to a state with n − δ windings. For this case, relaxation is expected to be much slower than example, the trajectory s1 = t/τ, s2 = 0 is shown in relaxation for J ≥ Jc, especially at low temperatures. Fig.2. Note, that for κm ≤ κd, we must consider trajec- By inserting the ansatz for u(r, t) in Eq. (31) into tories where s1,2 in Eq. (31) are varied simultaneously Eq. (18), one finds that the variation of s1 (for s2 held and the variational bounds we obtain are less strict, i.e., var fixed at 0) acts to dissipate the current but flips the pair- Jc Jc . Nevertheless, these bounds portray the same ing polarization relative to its initial orientation parallel functional dependence on B and Lx as Eq. (26) and are to the external field. The variation of s2 acts to re-align thus omitted for simplicity’s sake. Particularly, for B = 0 var the pairing polarization. An upper bound on the free en- we still find that Jc equals Jc. ergy barrier can be obtained by considering an example Moreover, as we show in Sec.V, using dynamical sim- trajectory where s1 rises from 0 to 1 strictly before s2 ulations, Eq. (31) provides a good approximation for the does so. Such a trajectory has the appealing property continuous unwinding trajectories that naturally emerge that the free energy density is spatially homogeneous, from the dynamics of the system, and the critical current and is given by is indeed well captured by Eq. (26).

1 4π2 h 2 F [u(r, s1, 0)] = −µB cos(πs1) + 2 κdn LxLy Lx i C. Competing mechanisms 2 1 2 2 πs1 2 πs1 − κd(2nδ − δ ) − (κm − κd) 2 δ cos ( 2 ) sin ( 2 ) , 1 4π2 2 Throughout this analysis, we have assumed that the L L F [u(r, 1, s2)] = µB cos(πs2) + L2 κd(n − δ) . x y x order parameter is uniform in the y direction. There also (32) exist trajectories where the unwinding process occurs in This enables us to obtain the maximal energy barrier a “domain wall” whose width in the y direction is smaller along this trajectory, than Ly. The domain wall then propagates along y and def unwinds the phase twist in the entire system. However, ∆F [u] = max {F [u(r, t2)] − F [u(r, t1)]} , (33) 0≤t1≤t2≤τ the optimal width of the domain wall is proportional to L , and thus its formation is only favorable when L is which provides an upper bound on the energy barrier x y def sufficiently larger than Lx. amongst all possible trajectories, ∆F = minu{∆F [u]}. Another competing mechanism for current relaxation Beyond a certain critical current, there is no energy is by vortex motion perpendicular to the current direc- barrier for supercurrent relaxation. Through the de- tion. As discussed in Sec.VI, we find the continuous pendence of the current on the number of windings, unwinding mechanism is dominant at sufficiently low cur- J = J0n, each variational trajectory yields an upper var rents and low temperatures. bound Jc ≤ Jc (δ) at which the free energy barrier for the continuous unwinding process vanishes. By optimiz- ing over the number of unwindings δ, we find the best V. DYNAMIC TRANSPORT var def var variational estimate, Jc = minδ{Jc (δ)}. Particularly, for κ > κ , our variational trajectories m d So far, we have explored current relaxation from an turn out to provide a strict estimate: energetic point of view. In this section, we support the var πq(κm+κd)δ qLxµB predictions from the analysis of Sec.IV using an unbiased Jc (δ) = L + πδ , (34) x model for the order parameter dynamics that enables the var def var Jc = minδ{Jc (δ)} = Jc. (35) system to explore its configuration space. var Remarkably, Jc equals Jc found by our linear stability analysis [Eq. (26)]. This also provides us with an expression for the optimal number of simultaneous unwindings, A. Time dependent model L r µB δ ' x , (36) To capture the dynamics of the current relaxation π κm + κd process, we use a Time-Dependent Ginzburg-Landau where strictly speaking, δ, must be the closest positive (TDGL) formulation supplemented by a stochastic noise even integer. As discussed in Sec.IIB, the number of term [44–46]. Crucially for the continuous unwinding simultaneous unwindings, δ, determines the periodicity process, we must include a mechanism for the system of the AC Josephson effect in our system,ω ˜J = qV/δ~. to exchange magnetization with its environment (relax- At currents close to the critical current, J ∼ Jc, the ing the constraint of angular momentum conservation). free energy barrier in Eq. (33) satisfies: To this end, we assume that the system is coupled to an ( external spin bath; for further discussion see Sec.VIB. 0 J ≥ Jc, Within the stochastic TDGL approach, the order pa- ∆F ≤ LxLy 2 (37) 2 (J − Jc) J < Jc. 2q (κm−κd) rameter evolves according to a Langevin-type equation 8 of motion of the form Within our stochastic model, we thus seek the mean current, hJ(t)iζ (averaged over all realizations of the noise, ∂d(α) δF δF i = εαβ − Γgα gβ + gα ζ . (38) ζ), that flows due to a uniform, constant electric field ∂t (β) ik jk (β) ij j δdi δdj A(r, t) = −Et. (43)

Here, we have utilized the Einstein summation conven- Here, E = V xˆ/Lx. Since the TDGL equations [Eq. (40)] tion, with α, β ∈ {1, 2} and i, j, k ∈ {1, 2, 3}. The are nonlinear stochastic partial differential equations, three terms on the right-hand side of Eq. (38) are the their solutions are cumbersome to write. Therefore, we kinetic (dissipationless) term, the dissipative term, and leave the details of the solution to AppendicesB andC the source (noise) term, respectively. We take ζj(r, t) to and present here the main results. be a real Gaussian-distributed stochastic field with zero We study the dependence on the model parameters, mean and correlation function i.e., the stiffnesses κd and κm, sample dimensions Lx and Ly, fluctuation coefficient Γ, temperature T , and 0 0 0 0 hζi(r, t)ζj(r , t )i = 2kBT Γδijδ(r − r )δ(t − t ). (39) applied voltage V . We focus on certain analytically tractable limits of physical interest. First, to avoid the α α We choose gik(d) = εijkdj , which corresponds to a spin domain-wall formation discussed in Sec.IV, we keep Ly bath acting as a spatiotemporally-fluctuating Zeeman sufficiently small compared to Lx. Current relaxation field ζ that induces precession of the order parameter. by transverse motion of vortices is neglected, assuming The form of Eq. (38) guarantees that the fluctuation- that the temperature is sufficiently low (see discussion dissipation theorem is satisfied at thermal equilibrium; in Sec.VIC). Under these conditions, the order param- see AppendixA. eter may be assumed to be independent of y. Second, In terms of the complex d-vector, The Langevin equa- for simplicity, we study the case of large fluctuation and −2 tion takes the form dissipation, Γ d0 , such that the kinetic term in Eq. (40) is negligible compared to the dissipation and ∂d 1 δF 1 δF = − d · d∗ noise terms. The applied Zeeman field is moreover set ∂t i δd∗ d∗ · d δd∗ to zero. Finally, in order to work in the regime where δF δF relaxation occurs through well-separated-in-time individ- − Γ (d · d∗) + (d · d) δd∗ δd ual unwinding events, we focus on low temperatures such that k T L pκ qV/Γ; see Eq. (B25) in AppendixB. δF δF B y d ∗ Below, we present the time dependent solution of the − d · ∗ d − d · d + d × ζ. δd δd TDGL equations, Eq. (40), as well as the the long-time- (40) averaged mean supercurrent, Z τ As our system is in the fully spin-polarized state, the ¯ 1 J = lim dt hJ(t)iζ. (44) potential in the free energy [Eq. (10)] must include a po- τ→∞ τ 0 2 larizing term, U(d) = λ |d · d| + ..., with large coupling Note that, as the system is self averaging, the long-time λ. Taking the limit λ → ∞ generates the correction to averaged current takes the value J¯ for each generic real- the kinetic term in the first line of Eq. (40). 1 R τ ization, J¯ = limτ→∞ dt J(t). Importantly, Eq. (40) is manifestly gauge invariant, τ 0 and, as constructed, leads to the conservation laws: 1. Energy landscape predictions ∂(d · d) = 0, (41) ∂t Our analysis of the continuous supercurrent dissipation ∗ ∂(d · d) 1 ∗ δF δF mechanism and estimates of the critical supercurrent in = d · ∗ − d · = −∇ · J. ∂t d·d=0 i δd δd Sec.IV enable us to give predictions for the DC transport (42) setup discussed above. Suppose we start from a configuration with a uniform pairing polarization and apply a 1 ∗ By design, the pairing polarization m = 2i d × d, which voltage. As long as this configuration is locally stable, is proportional to the spin magnetization at equilibrium, the current rises linearly with time. This proceeds un- is not conserved. til the current exceeds Jc. Then, the energy barrier for spontaneous dissipation vanishes, and a dissipation event would initiate. This process should repeat itself and the B. Solutions system would oscillate back and forth between a current close to the critical current J ∼ Jc and the relaxed state To compute transport in the system, we consider a with J ∼ Jc − 2J0; see Fig.1. Thus the observable mean current would be maintained close to cylindrical geometry with circumference Lx and height

Ly, where a current ﬂows due to the presence of an ap- def 2πq(κm − κd) J¯c = Jc − J0 = . (45) plied electromotive force, equivalent to a DC voltage, V . Lx 9

FIG. 3. Numerics versus analytics: Current-voltage curve for κm = κd. Each circle depicts the average supercurrent from 18 instances of numerical simulations of Eq. (40); the gray area depict 95% statistical conﬁdence. The dashed line de- κdΓ picts the analytic results of Eq. (49). Here, V1 = 2 ; exact qLx details of the simulation are found in AppendixC. Remark- ably, there are no ﬁtting parameters.

At a finite voltage, the current may overshoot the critical value, since the uniform configuration is a saddle point of the free energy (despite being unstable). The escape process is triggered by thermals fluctuations and takes a finite time to occur. The larger the voltage, the more the current overshoots Jc, and hence the average current increases. Hence we anticipate FIG. 4. Supercurrent profiles J(t) for infinite dissipation; see ¯ ¯ ¯ β Sec. VB2. The average supercurrent J is depicted by the J ' Jc + O(V ), (46) dashed line; the shaded areas depict the deviation from 0 supercurrent. Here, t = 2π is the Josephson period. with some exponent β. Note, that this estimate only 0 qV holds for J¯c ≥ 0, since otherwise, Jc is lesser than one winding. In this limit of large dissipation, the time-averaged su- In the following sections, we present both analytic and percurrent is independent of voltage and temperature, numeric results (see Figs.3 and4) showing these generic predictions are indeed realized in our specific TDGL ( ¯ 0 κm ≤ κd model. J = 2πq(κm−κd) (47) κm > κd. Lx Due to the effectively infinite dissipation coefficient, the 2. Large dissipation limit unwinding event begins immediately when the current reaches Jc, exactly matching our predictions of Eq. (26): Γκd The simplest case to analyze is the case of 2 → ∞, there is no “overshoot,” even at a finite voltage. There- LxqV where the dissipation rate is large compared with the fore, the finite voltage correction discussed in Sec.VB1 ¯ ¯ Josephson frequency. In this case, the dependence of the vanishes and we indeed get J ' Jc; cf. Eq. (45). Note current on time can be entirely found analytically (see that the cusps in current vs. time are special to the in- AppendixB). finite dissipation limit, and are smoothed out for finite In Fig.4 we show time-dependent supercurrent profiles dissipation; see Fig.5 in AppendixB. J(t) for various values of κm/κd, together with the corresponding values of the time-averaged supercurrent J¯. 3. Finite dissipation For κm ≤ κd the current profile is an asymmetric triangle wave pattern which averages out to 0, but for κm > κd we have a shifted saw-tooth wave pattern which averages In the generic case of finite dissipation rate, we obtain to a finite value. These patterns always exhibit a period a power-law relation between the time-averaged current ¯ of 2t0 = 2(h/2eV ), exactly matching our predictions for J and the applied voltage. We find that the doubling of the AC Josephson effect for small B; see q ¯ ¯ κd Sec.IIB and Eq. (36). J − Jc ∝ q Γ qV , (48) 10 cf. Eq. (46). This result holds for the supercurrent range i.e., hdi 6= 0. This is, of course, never strictly true in ¯ q 1 κm J Lx qV ¯ ¯ two spatial dimensions at any non-zero temperature. In of ( + 3) − 2 , and J > |Jc|; see 2 κd J0 4π Γκd the absence of a Zeeman field and any other breaking ¯ Eq. (B37) in AppendixB. The dependence of J(V ) on of spin rotational symmetry, the system is always in a temperature is (sub-)logarithmic; the power-law depen- disordered phase at all T > 0 [47], in accordance with dence on V in Eq. (48) may similarly be modified by the Mermin-Wagner theorem. However, the spin corre- logarithmic voltage corrections. Such logarithmic correc- lation length grows rapidly with decreasing temperature, tions are explicitly presented in the following subsection. ξ ∼ e2π(κm+κd)/(kB T ); hence, at sufficiently low temperatures, ξ max(Lx,Ly) and our treatment should apply. If κm is larger than κd, then there is a finite temperature 4. Transition point crossover for finite systems that resembles a BKT transition. In the presence of a Zeeman field that pins the In the limit of infinite dissipation rate, Eq. (47) dis- direction of the pairing polarization, the system under- ¯ plays a non-analytic dependence of J on the ratio κm/κd, goes a genuine finite-temperature BKT transition. with a critical (transition) point at κm = κd. For finite We note in passing that, in contrast to a fully polarized ¯ dissipation rates, the full equations for J are somewhat triplet SC [phase (ii) in Sec.IIA2], a partially polarized simplified at this transition point. At this point we can phase [phase (iii) in Sec. IIA2] undergoes a true BKT obtain a complete solution; we find a characteristic volt- transition at finite temperature even for B = 0, at which age and temperature dependence of the average super- ∆4e becomes quasi-long range ordered [47]. This phase is current, interesting in its own right, as it supports half quantum vortices (carrying flux h/4e). We leave a more detailed v ! u r discussion of this phase and its physical implications for ¯ uκdqV 8Ly κdqV J = qt ln . (49) future work. πΓ kBT Γ

This result is in excellent agreement with numerical simulations of Eq. (40), as seen in Fig.3; the details of the B. Mechanism for magnetization relaxation simulations are given in AppendixC. Note that our result for the average current in Eq. (49) The continuous supercurrent relaxation process dis- na¨ıvely diverges at T → 0. Although the current exceeds cussed in this work requires a mechanism for the pairing a level where there is no longer an energy barrier for the polarization to dynamically change in time (both locally continuous unwinding process, at T = 0 there are no and globally). Since the pairing polarization is pinned to fluctuations to trigger decay within the classical model the physical spin polarization, this requires a mechanism of Eq. (40). However, at exponentially low temperatures for the system to exchange spin angular momentum with where J¯ J0, spontaneous quantum decay process un- its environment. In our time-dependent model (Sec.V), captured by our analysis are expected to dominate the we assumed the existence of a spin bath on empirical relaxation. grounds. In practice, the maximum rate at which the continuous relaxation process can proceed, 1/τs, depends on the physical mechanism of spin relaxation [48]. The VI. DISCUSSION AND CONCLUSIONS system’s spin can relax either through spin-orbit coupling (which also introduces anisotropic terms in spin space We end with a discussion of various points regard- into the free energy), a spin bath due to internal or sub- ing the implications of our results in the context of po- strate impurities or nuclear spins, or through the bound- larized triplet SCs. In the following, we discuss the aries, in cases where the system is coupled to metallic thermal phase diagram of two-dimensional spin-polarized leads. SCs, possible mechanisms for relaxation of the magneti- In the context of TBG and related materials, spin- zation (necessary for our topological mechanism for con- orbit coupling is expected to be weak, and nuclear spins tinuous current relaxation), the role of conventional su- are rare (unless 13C impurities are introduced intention- percurrent relaxation by vortex-antivortex dissociation, ally). In the absence of other magnetic impurities, the and estimated parameters for triplet superconductivity most dominant source of spin relaxation is likely by spin in graphene-based moirématerials. Finally, we point out transport to the metallic leads. This mechanism is ab- some topics worthy of future studies. sent in our simple dynamical model of Sec.V and we leave its detailed study to follow up work. An appro- priate description of this mechanism might be the use of A. Thermodynamics of 2D spin-polarized either the solitons of the degeneracy space [49, 50] or the superconductors Goldstone modes of the order parameter in order to carry the magnetic texture to the leads. Throughout our discussion we have assumed that, at Here, we make a rough estimate of the spin relaxation equilibrium, our system is essentially long-range ordered, time due to this mechanism. For simplicity’s sake, we 11 assume that there is no applied Zeeman field and set energy barrier. Hence, the continuous unwinding mech- B = 0. Given that the spin stiffness in our model is anism is dominant when V Vvm(Jc). Therefore, as κm +κd [see Eq. (B4)], we expect on dimensional grounds long as Jc < Jc,0, one sees that Vvm(Jc) → 0 at low tem- 2 (κm+κd)ξm perature, since the vortex-antivortex unbinding process that 1/τs ∼ 2 , where ξm is a microscopic “mag- Lx 2 is thermally activated. netic coherence length” (essentially, µB/ξm is the magnetization density) and Lx is the distance between the leads. This estimate for τs can be understood as the time it takes for a quadratically dispersing magnon with D. Twisted double-bilayer graphene parameters wavevector |k| ∼ 1/Lx to travel across the system. For a sufficiently small applied voltage such that eV As discussed in the Introduction, recent experiments 4π~/τs, the continuous unwinding process can take place in TDBG with a perpendicular electric field revealed a within one (doubled) Josephson period, with spin carried ferromagnetic ground state at half-filling of the moirélat- in and out of the system through its connection to the tice [31, 54]. Upon changing the density away from half- leads. For larger voltages, the continuous unwinding pro- filling, the resistance drops dramatically, possibly due to cess is limited by the rate of spin relaxation. The study superconductivity [31, 32]. Most strikingly, the temper- of this case goes beyond our present analysis. A possi- ature at which the resistivity drops increases linearly as ble scenario, in this case, is that the phase accumulates a function of an in-plane magnetic field for small fields, a more than two windings between the continuous unwind- signature of triplet superconductivity [31, 40, 41, 55–57]. ing events, decreasing the effective Josephson frequency Note, however, that an interpretation of these observa- belowω ˜J = eV/~. tions in terms of a non-superconducting state has also been proposed [58]. To get a rough estimate of 1/τs due to spin dissipation at the boundaries in graphene-based moirématerials, we Specifically for this case of TDBG, we may estimate assume that the spin stiffness κm + κd is of the order of the values of the various parameters in our model using 0.1 − 1 meV and ξm is of the order of a few times the the experimental data of Ref. 31. The sample dimensions moirélattice spacing a ≈ 10 nm. For example, suppose in these experiments were of the order of a few µm, with 12 −2 that κm + κd = 1 meV and ξm = 5a. Then, for a system a carrier density of n ≈ 2 × 10 cm . As discussed in of size Lx = 1 µm, the above considerations give that the Sec.IIA, we focus on dynamics far below the critical tem- crossover voltage is V ≈ 4π /(eτs) ≈ 30 µV. perature TBKT ≈ 3.5 K. For an order of magnitude esti- ~ 1 1 mate, we set κd ∼ 2 κs(TBKT) = π kBTBKT ≈ 0.1 meV. We furthermore estimate the magnetic moment density as µ ≈ nµ . C. Phase unwinding due to vortex motion B Using these estimated parameter values, we assess the two regimes of critical current behavior in Eq. (26). We In our analysis of the supercurrent relaxation through nA q B continuous unwinding, we have neglected the ordinary find Jc ≈ 300 µm × mT for large applied Zeeman field, Ly mechanism of vortex-antivortex dissociation and motion B, and Ic ≈ 500 nA × for small B. The crossover Lx of free vortices. In two spatial dimensions, this mecha- µm2 between the two regimes occurs at B ≈ 3 mT × 2 . nism gives rise to the celebrated nonlinear I-V charac- Lx teristics [35, 36, 51–53] associated with the BKT transition: V ∝ Iα with α(T ) = πκs(T ) + 1, where the phase kB T 2 E. Conclusions and future directions stiffness κs satisfies κs(TBKT) = π kBTBKT such that 1 α(TBKT) = 3. In our model, κd = 2 κs, this provides us with the estimates in Eq. (9). In this paper, we have studied the transport proper- In order to determine which mechanism dominates the ties of fully spin-polarized triplet SCs. Interestingly, we supercurrent relaxation, we need to compare the rate of found that the topology of the order parameter degen- phase unwinding (i.e., the voltage) generated by vortex eracy space can have dramatic consequences, given that motion to the rate of the continuous unwinding process. a mechanism for spin dissipation is available. Basically, The voltage due to vortex motion is estimated as [35] two windings of the order parameter are topologically equivalent to zero windings, and hence a supercurrent α(T )−1 can decay via a smooth deformation of the order param- ξ J eter that involves a transient inhomogeneous spin texture Vvm ≈ 2ρnJLx (α(T ) − 3) max , . L Jc,0 but does not involve creating topological defects (such as (50) vortices). We expect that this mechanism of supercurrent Here, ρn is the normal state resistivity, J is the cur- relaxation may become dominant at sufficiently low tem- rent density, ξ is the coherence length, L = min(Lx,Ly), peratures, where vortex configurations are suppressed. e kB TBKT and Jc,0 ∼ is microscopic critical current den- A direct observable manifestation of the continuous su- ~ξ sity. The continuous unwinding mechanism occurs when percurrent relaxation mechanism is that a sample sub- J & Jc, whereby it does not involve passing through an jected to a DC voltage exhibits an oscillatory current 12 with a fundamental frequency which is a fraction of the function W ({X}, t) is [46]: Josephson frequency. Furthermore, an applied Zeeman field suppresses the continuous supercurrent decay mech- ∂W ∂ ({X}, t) = − [hI ({X})W ] anism, since it pins the direction of the system’s magne- ∂t ∂XI tization. ∂ ∂ A possible extension of our analysis is the study of + kBT Γ gIµ({X}) [gJµ({X})W ] . (A2) ∂XI ∂XJ the equilibrium properties of the system in the presence of a perpendicular magnetic field. At high applied field Applying this rule to our Langevin equation [Eq. (38)] one would find the standard vortex lattice groundstate. (α) with XI 7→ di (r), we obtain: However, at low magnetic fields, one would suspect a stabilization of a vortex-free spin texture; this would be ∂W ({d}, t) = reminiscent of Fig.2 in space rather than in time. In ∂t (" # ) addition, similar physics to that discussed here might ∂ δF δF − εαβ − Γgα gβ W arise in superconducting states that emerge out of more (α) (β) ik jk (β) exotic spin-valley ferromagnets, such as the skyrmion SC ∂di δdi δdj ( ) discussed in Ref. [59]. We leave these directions to future ∂ ∂ h i +k T Γ gα gβ W . (A3) work. B (α) ik (β) jk ∂di ∂dj

Substituting W = e−F [{d}]/kB T /Z and using the fact ACKNOWLEDGMENTS α (α) that for our choice gik = εijkdj [see discussion below Eq. (38)]: We are grateful for illuminating discussions with A. Auerbach, J. Ruhman, O. Golan, S. Kivelson, ∂gβ jk = 0, (A4) D. Podolsky, T. Senthil, and A. Vishwanath. EB (β) and EC were supported by the European Research ∂dj Council (ERC) under grant HQMAT (Grant Agreement we ﬁnd that the Gibbs distribution is a stationary No. 817799) and the US-Israel Binational Science Foun- solution of Eq. (A3). Hence the FDT is satisﬁed. dation (BSF). EB and MR acknowledge support from CRC 183 of the Deutsche Forschungsgemeinschaft. MR gratefully acknowledges the support of the European Re- search Council (ERC) under the European Union Hori- Appendix B: Analytic solutions to the TDGL zon 2020 Research and Innovation Programme (Grant equations Agreement No. 678862), and the Villum Foundation. In this appendix, we present the details of the analytic solution to Eq. (40). Appendix A: Fluctuation-dissipation theorem

In this Appendix, we show that the Langevin equa- 1. Preliminaries tion [Eq. (38)] satisfies the fluctuation-dissipation theo- −2 rem (FDT) at equilibrium. Recall that a sufficient con- We focus on the case of large dissipation Γ d0 . In dition for a stationary stochastic process to satisfy the this case there are three frequency scales in the system. FDT is that the Gibbs distribution, W ∝ e−F/kB T , is The inverse of the Josephson period, time-independent under the stochastic process [60]. This 1 qEL means that the Gibbs distribution is a solution of the cor- = x , (B1) responding Fokker-Planck equation. t0 2π We consider a set of dynamical variables XI which sat- the dissipation rate, isfy general Langevin equations of the form (using the 2 Stratonovich convention [46]) 4π κdΓ γ = 2 , (B2) Lx ∂X I = h ({X}) + g ({X})ζ (t), (A1) and the fluctuation rate, ∂t I Iµ µ 2k T Γ E = B . (B3) where repeated indices are summed over; hI , gIµ are dif- LxLy ferentiable functions, and ζµ(t) are independent Gaussian noise sources with zero mean, satisfying hζµ(t)ζν (t)i = Whenever possible we shall express our equations and 0 kBT Γδµν δ(t − t ). The corresponding Fokker-Planck results in terms of these quantities. equation that describes the evolution of the distribution 13

2. The equations allows one to rewrite the free energy as

We explicitly take d · d = 0, characteristic of the fully Z spin-polarized phase, as discussed in Sec.IIA2; this con- 2 κd + κm 2 κm 2 F [d] = d r 2 |(∇ − iqA) d| − 2 J , dition is conserved by the equations of motion. We fur- 2d0 4κd ∗ 2 thermore take the London limit of d · d = 2d0. These (B4) conditions imply U(d) = const and ∇·J = −ρ˙ = 0. This and to explicitly write the equations of motion [Eq. (40)],

∂d κd + κm 2 1 2 ∗ 1 2 ∗ = Γ 2 ∇ − 2iqA · ∇ d − 2 d · ∇ − 2iqA · ∇ d d − 2 d · ∇ + 2iqA · ∇ d d ∂t 2 d0 d0 κm 1 ∗ − iΓ 2∇d + 2 (d · ∇d ) d J + d × ζ. (B5) 2κd d0

π This nonlinear partial differential equation is best solved discussed above, i.e., f± = ± 2 + 2π`, where ` ∈ Z. by the method of guessing the solution. We use an ansatz However, when t → ∞ only f+ are stable fixed-points inspired by the unwinding trajectory of Sec.IV, under small perturbations while f− are unstable; when t → −∞ only f are stable fixed-points while f are un- φ − + x ∆ 1 π x φn iσ3 π∆ L + 2 iσ1( f(t)+ ) iσ3(−πn + ) u = u0e x e 2 4 e Lx 2 . stable. Therefore, a solution to this equation describes a (B6) trajectory from f− to f+. This trajectory describes relaxation of 2∆ windings, and the fundamental 2-windings Here, n and ∆ are integers, while φn and φ∆ are some relaxation trajectory is attained for ∆ = 1. arbitrary phases, and u0 is an arbitrary SU(2) matrix. At an intermediate time, tinit, when J(tinit) = Jc [see The current is given by Eq. (26)], the stable fixed point f− at t → −∞ becomes a saddle-point. Any small deviation would thus send the h t i J(t) = J0 − (n + ∆ sin f(t)) . (B7) system on a trajectory towards the stable fixed point f t0 + at t → ∞, which itself ceases being a saddle-point only This trajectory connects two families of constant con- at some other intermediate time (tfin discussed below). π figurations; f− = − 2 + 2π` with n − ∆ windings, and However, at T = 0, there is no source for small devi- π f+ = 2 + 2π` with n + ∆ windings, where ` ∈ Z. ations and the system would thus remain frozen at f−. Note that these configurations have uniform pairing- Moreover, even given some initial perturbation, it could polarization and hence the winding number is unam- only trigger the first unwinding event and the system biguously defined. Both configurations have supercur- would eventually remain asymptotically close to f+ with t rent profiles growing linearly with time, J±(t) = J0[ − t0 nothing to trigger the next unwinding event. Therefore (n ± ∆)]. it is crucial to study the effects of finite temperature. The dynamics of the system consist of a series of unwinding events with different values of parameters n, ∆, φn,∆, u0. The pairing-polarization and supercur- 4. Effective model for finite temperatures rent at the end of an unwinding event determine u0 and n − ∆ of the next event. We assume the system is ini- At finite temperatures, T > 0, the thermal fluctua- tialized with J(t = 0) = 0 such that n = ∆ for the first tions would nudge the system from its saddle-point and unwinding event. initiate an unwinding trajectory. The system may spontaneously choose any random fluctuation direction v(x) within the manifold of configurations escaping the saddle- 3. Zero temperature point. Without loss of generality, we pick such a direction v(x) compatible with our ansatz [Eq. (B6)], First, we analyze the case of T = 0, whereby our ansatz [Eq. (B6)] reduces the equations of motion to an ordinary  x  cos 2π∆ L + φ∆ differential equation (ODE),  x  v(x) = x . (B9) − sin 2π∆ + φ∆  n o  Lx  f˙(t) = γ∆ 2 t − n − ∆ 1 − κm sin f(t) cos f(t). t0 κd 0 (B8) There are two families of fixed-point solutions to this This is supported by numerical simulations of Eq. (B5) equation, corresponding to the constant configurations for finite temperatures. 14

α α This choice of direction enables us to drastically sim- gij(d)ζj(r, t) 7→ εijkdj vk(x)ζ(t), with plify the functional vector noise term in Eq. (38) and 0 0 replace it with an eﬀective uniform scalar noise term, hζ(t)ζ(t )i = 2kBT Γ∆δ(t − t ). (B10)

The corresponding equations of motion are modiﬁed ac- cordingly (see AppendixA),

Z ∂d(r, t) κd + κm 2 0 ∗ 2 2 ∗ = Γ∆ 2 d r (d × v) · ∇ − 2iqA · ∇ d + (d × v) · ∇ + 2iqA · ∇ d r0 (d(r, t) × v(x)) ∂t 2d0 Z κm 2 0 ∗ ∗ − iΓ∆ 2 d r [((d × v) · ∇d − (d × v) · ∇d ) J]r0 [d(r, t) × v(x)] + d(r, t) × v(x)ζ(t). (B11) 2d0κd

As we show in AppendixB4a, consistency requires Γ ∆ = 5. Solutions of the effective model Γ/LxLy. Remarkably, this effective noise term propagates well We are now finally in a position to derive the results into our ansatz [Eq. (B6)], of Sec.VB. We solve Eq. (B12) and use Eq. (B7) and ¯ n o Eq. (44) to obtain the average supercurrent J for vari- f˙(t) = γ∆ 2 t − n − ∆ 1 − κm sin f(t) cos f(t) ous ratios of the rates t−1, γ, E and various ratios of the t0 κd 0 stiffnesses κ , κ . + ζ(t), (B12) d m thus reducing the equations of motion to a stochas- a. Infinite dissipation tic ODE. As seen in Fig.6 in AppendixC, this effective model exquisitely captures the numerical detail of Eq. (40). The simplest solution is when t0γ → ∞. In such cases the solution is  −1 t < t ,  init a. Evaluation of the diffusion rate t−tinit κm < κd : sin f(t) = −1 + 2 tinit ≤ t ≤ tfin, tfin−tinit +1 t > t , Given our original 2+1 dimensional vector model, fin ( −1 t < tinit, 0 0 0 0 hζi(r, t)ζj(r , t )i = 2kBT Γδijδ(r − r )δ(t − t ), (B13) κm ≥ κd : sin f(t) = (B17) +1 t > tinit, and an effective 0+1 dimensional scalar model fluctuating where we have defined in the v(x) direction, ∆ κm 0 0 tinit = t0 n − 1 − , hζ(t)ζ(t )i = 2kBT Γ∆δ(t − t ), (B14) 2 κd ∆ κm we wish to find the value of Γ such that the effec- tfin = t0 n + 1 − . (B18) ∆ 2 κ tive scalar model best approximates the original vector d model. By matching the initial and final conditions discussed Hence, we project ζ(r, t) onto v(x), i.e., in AppendixB2 we find that all unwinding events have ∆ = 1 (corresponding to the fundamental 2 windings re- R Ly R Lx dy dx v(x) · ζ(r, t) laxation event), and that n = 1, 3, 5,... . For κm ≤ κd ζ(t) = 0 0 , (B15) R Lx the current profile is an asymmetric triangle wave pat- Ly dx v(x) · v(x) 0 tern which averages out to 0, but for κm > κd we have a shifted saw-tooth wave pattern for J(t) and J¯ = such that [tinit]n=1−t0 1 κm J0 = J0 − 1 . We thus conclude: RR 2 RR 2 0 0 0 0 t0 2 κd 0 d r d r v(x) · hζ(r, t)ζ(r , t )i · v(x ) hζ(t)ζ(t )i = 2 2 ( LxLy 0 κm ≤ κd, J¯ = (B19) 1 κm 1 0 J0 − 1 κm > κd. = 2kBT Γδ(t − t ), (B16) 2 κd LxLy This is precisely Eq. (47) of the main text, with the cor- and thus find Γ∆ = Γ/LxLy. responding behavior plotted in Fig.4. 15

b. Finite dissipation

For ﬁnite t0γ, we linearize Eq. (B12) around f(t) = π − 2 + h(t) with small h(t) such that t − t h˙ (t) ' 2γ∆ init h(t) + ζ(t). (B20) t0 This is solved by t−t γ∆ 2 Z init γ∆ 02 (t−tinit) 0 − t 0 h(t) = e t0 dt e t0 ζ(t ), (B21) −∞ γ∆ 2 2 2 (t−tinit) h(t) = e t0 t−t t−t Z init Z init 0 00 − γ∆ (t02+t002) 0 00 × dt dt e t0 Eδ(t − t )

−∞ −∞ t−t γ∆ 2 Z init γ∆ 02 2 (t−tinit) 0 −2 t = Ee t0 dt e t0 , (B22) −∞ 1rπ r t h2 = h(t )2 = E 0 . (B23) init init 2 2 γ∆ Here we have approximated h(t → −∞) → 0. This approximation requires contributions from previous un- FIG. 5. Supercurrent proﬁles J(t) for various voltages; see winding events, which end at time tprev, to decay before ¯ the initiation of a new event: Sec. VB3. The average supercurrent J is depicted by the dashed line; the shaded areas depict the deviation from the q 2π t0 inﬁnite dissipation limit; see Fig.4. Here, t1 = ; exact 2 qV1 h(tprev) 2πγ∆ details of the simulation are found in AppendixC. 1 2 ' . (B24) hh(tinit) i tinit − tprev

We evaluate tprev and check our solutions for this condi- We may hence easily evaluate the average supercurrent, tion at AppendixB5d. 1 Z ∞ t Moreover, the linearization requires J¯ ' dt J(t) − J − 2 2t 0 t h π, where h ∼ E1/2(t /γ)1/4. (B25) 0 t0 0 init init 0 Z ∞ J0 0 1 = dt 2γ This is the low-temperature condition discussed in t02 2 h 2t0 −∞ 1 + e t0 tan init Sec.VB. It ensures there are well separated unwinding 2 r events as the temperature is not so high that the system π 2 hinit = −J0 Li 1 − cot . (B28) escapes the stable fixed-points at any time by thermal 8γt0 2 2 fluctuations. Here, Lis(z) is the polylogarithm function. As we have 1 taken hinit π we may use Lis(−z) ' − Γ(s+1) ln z and c. Exactly solvable point get v ¯ u 1 8 The nonlinear ordinary differential equation Eq. (B12) J ' J0u ln q . (B29) t2γt0 p π t0 becomes an exact differential equation at κm = κd. E 2 γ First, we set ∆ = 1 as in the t0γ → ∞ case (see discussion in AppendixB5d). Next, we use the linearized This is Eq. (49) of the main text. equation to find the initial displacement [Eq. (B23)]. π Then, starting from f(tinit) ' − 2 + hinit, we propagate in time, neglecting the effects of the noise term, d. Generic case

t − tinit f˙(t) ' 2γ cos f(t), (B26) At large but finite dissipation we wish to find the ap- t0 proximate escape times t∗ > tinit when the linearization γ 2 π f(t > tinit) = 2 arctan tanh (t − tinit) breaks down and h(t∗) ∼ 2 is no longer small. 2t0 We focus on π − h 2 init r t − arctanh tan . (B27) t − t 0 , (B30) 2 ∗ init 2γ 16 where one finds The supercurrent profile J(t) is numerically simulated and plotted in Fig.5 for various voltages. To proceed with our analytic analysis, we evaluate the difference r r γ∆ 2 2 2 π t0 2 (t∗−tinit) h = h(t ) 'E e t0 , from the solution at infinite dissipation J (t) given by ∗ ∗ 2 γ∆ ∞ v 2  t u t h t < tinit, 0 ∗  t0 t∗ ' tinit + u ln . (B31)  t t−t t q t init 2γ∆ p π 0 κm < κd : J∞(t) = J0 t − 2 t −t tinit < t < tfin, E 2 γ∆ 0 fin init  t−2t0  tfin < t, t0 ( t t < tinit, The first unwinding event satisfies n = ∆. The follow- κ ≥ κ : J (t) = J t0 (B33) m d ∞ 0 t−2t0 t > tinit. ing cascade of unwinding events is very complicated. Al- t0 though this case looks interesting, we suspend its analysis to further research. We thus stick for now with the as- We approximate the unwinding events as immediate transitions to J at time t ; this makes our following sumption that ∆ = 1, which requires t∗|∆=1 t∗|∆≥2. ∞ ∗ This holds when results hold up to order-one corrections:

Z t∗ 2 1 t h √ 2 J¯ ∼ dt J0 − J∞(t) ∗ κm ln 4 2 1 + t0γ. (B32) 2t0 0 t0 q κd p π t0 2 E 2 γ ( (t∗−tinit) 2(t −t )t t∗ ≤ tﬁn, = J0 fin init 0 (B34) t∗−t0 t∗ ≥ tﬁn. t0 Note that this condition is only violated for exponentially small temperatures. Plugging Eq. (B31) into Eq. (B34) immediately yields

 −1 2 r 2 κm 1 h∗ 1 h∗ κm  1 − ln √ q for ln √ q < 1 − ,  κd 4t0γ E π t0 2t0γ E π t0 κd J¯ ∼ J 2 γ 2 γ (B35) 0 r 2 r 2 1 κm 1 h∗ 1 h∗ κm  − 1 + ln √ q for ln √ q ≥ 1 − .  2 κd 2t0γ π t0 2t0γ π t0 κd  E 2 γ E 2 γ

π Here, recall that h∗ ∼ 2 , and note, that for κm = κd, This is the condition on moderately low currents pre- the exact√ solution Eq. (B29) matches the form above for sented in Sec.VB4. h∗ = 2 2. To evaluate the consistency condition in Eq. (B24), we t∗ J¯ use tprev = max{t∗, tfin} − 2t0 as well as ∼ + 1 for t0 J0 Our results in Eq. (B35) may be re-expressed using t ≥ t . We thus find ∗ fin 2πq(κm−κd) the original model parameters and J¯c = . Lx ( κm This yields the characteristic I-V curves of the model 1 1 + t∗ ≤ tfin, √ κd (B36) in Eq. (48), 1 κm J¯ 2πt0γ 3 + − t∗ ≥ tfin. 2 κd J0

 q ¯ Lxκd ¯ ¯ κm Lx qV J ∝ q qV for J < |Jc| and + 1 2 ,  Γ(κd−κm) κd 4π Γκd κm < κd : ¯ q ¯ ¯ p κd ¯ ¯ 1 κm J Lx qV J − Jc ∝ q qV for J > |Jc| and ( + 3) − 2 , Γ 2 κd J0 4π Γκd n ¯ q ¯ ¯ p κd 1 κm J Lx qV κm ≥ κd : J − Jc ∝ q qV for ( + 3) − 2 . (B37) Γ 2 κd J0 4π Γκd

Appendix C: Numeric solutions to the TDGL We numerically evaluate Eq. (B5) for a time interval equations

In this Appendix, we provide the details of the numerical simulations presented in Sec.VB. 17

where Nx,Ny,Nt are integers. Here the noise Fourier components {ck,ω} are random variables satisfying

∗ hci,k,ωcj,k0,ω0 i = Eδijδk,k0 δω,ω0 . (C2)

This formulation ensures that

hζi(r, t)ζj(r, t)i = Eδij(2Nx + 1)(2Ny + 1)(2Nt + 1), (C3)

Z Lx Z Ly Z τ 0 0 dx dy dthζi(r, t)ζj(r , t )i = Eδij. (C4) 0 0 0

FIG. 6. Logarithmic corrections to the current-voltage curve for κm = κd; see Fig.3. Each circle depicts the average supercurrent from 18 instances of numerical simulations of 1. Exact parameter values for the figures Eq. (40); the gray area depict 95% statistical confidence. The dashed line depicts the analytic results of Eq. (49). Here, κdΓ V1 = qL2 . There are no fitting parameters. In Fig.3 we choose Ly/Lx = 1, κm/κd = 1, E/γ = x 1 −7 1 −2 4π 4π2 × 10 , qV1/γ = 4π2 , V/V1 ∈ [10 , 1], τ = qV , Nx = 2, Ny = 0, Nt = 100. We compare the results to t ∈ [0, τ]. We use a stochastic noise term the analytic expression of Eq. (49); note that there are no fitting parameters. Nx Ny Nt 1 X X X In Fig.5 we choose Ly/Lx = 1, κm/κd = 0, E/γ = ζ(x, t) = 1 −9 1 4π p 2 ×10 , qV1/γ = 2 , V/V1 ∈ [1, 4], τ = , Nx = 8, LxLyτ 4π 2π qV1 kx=−Nx ky =−Ny ω=−Nt Ny = 0, Nt = 100. ky y 1 ∗ 2πi kxx 2πi 2πi ωt Lx Ly τ In Fig.6 we normalize Fig.3 by the leading power- √ ck,ω + c−k,−ω e e e , 2 law behavior and find a very good agreement with our (C1) predicted logarithmic corrections.

[1] Dieter Vollhardt and Peter Wolfle, The superfluid phases parameter in sr 2 ruo 4 from oxygen-17 nuclear magnetic of helium 3 (Courier Corporation, 2013). resonance,” Nature 574, 72–75 (2019). [2] Grigory E Volovik, The universe in a helium droplet, Vol. [8] J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. 117 (Oxford University Press on Demand, 2003). Castro Neto, “Graphene bilayer with a twist: Electronic [3] SS Saxena, P Agarwal, K Ahilan, FM Grosche, RKW structure,” Phys. Rev. Lett. 99, 256802 (2007). Haselwimmer, MJ Steiner, E Pugh, IR Walker, SR Ju- [9] Rafi Bistritzer and Allan H MacDonald, “Moirébands in lian, P Monthoux, et al., “Superconductivity on the bor- twisted double-layer graphene,” Proceedings of the Na- der of itinerant-electron ferromagnetism in uge 2,” Na- tional Academy of Sciences 108, 12233–12237 (2011). ture 406, 587–592 (2000). [10] Guohong Li, A Luican, JMB Lopes Dos Santos, AH Cas- [4] Dai Aoki, Andrew Huxley, Eric Ressouche, Daniel tro Neto, A Reina, J Kong, and EY Andrei, “Observa- Braithwaite, Jacques Flouquet, Jean-Pascal Brison, Elsa tion of van hove singularities in twisted graphene layers,” Lhotel, and Carley Paulsen, “Coexistence of supercon- Nature Physics 6, 109–113 (2010). ductivity and ferromagnetism in urhge,” Nature 413, [11] Yuan Cao, Valla Fatemi, Ahmet Demir, Shiang 613–616 (2001). Fang, Spencer L Tomarken, Jason Y Luo, Javier D [5] N. T. Huy, A. Gasparini, D. E. de Nijs, Y. Huang, J. C. P. Sanchez-Yamagishi, Kenji Watanabe, Takashi Taniguchi, Klaasse, T. Gortenmulder, A. de Visser, A. Hamann, Efthimios Kaxiras, et al., “Correlated insulator be- T. Görlach, and H. v. Löhneysen,“Superconductivity on haviour at half-filling in magic-angle graphene superlat- the border of weak itinerant ferromagnetism in ucoge,” tices,” Nature 556, 80 (2018). Phys. Rev. Lett. 99, 067006 (2007). [12] Yuan Cao, Valla Fatemi, Shiang Fang, Kenji Watanabe, [6] Andrew Peter Mackenzie and Yoshiteru Maeno, “The su- Takashi Taniguchi, Efthimios Kaxiras, and Pablo Jarillo- perconductivity of sr2ruo4 and the physics of spin-triplet Herrero, “Unconventional superconductivity in magic- pairing,” Rev. Mod. Phys. 75, 657–712 (2003). angle graphene superlattices,” Nature 556, 43–50 (2018). [7] Andrej Pustogow, Yongkang Luo, Aaron Chronister, Y-S [13] Alexander Kerelsky, Leo J McGilly, Dante M Kennes, Su, DA Sokolov, Fabian Jerzembeck, Andrew P Macken- Lede Xian, Matthew Yankowitz, Shaowen Chen, zie, Clifford William Hicks, Naoki Kikugawa, Srinivas K Watanabe, T Taniguchi, James Hone, Cory Dean, Raghu, et al., “Constraints on the superconducting order et al., “Maximized electron interactions at the magic 18

angle in twisted bilayer graphene,” Nature 572, 95–100 et al., “Mapping the twist-angle disorder and landau lev- (2019). els in magic-angle graphene,” Nature 581, 47–52 (2020). [14] Matthew Yankowitz, Shaowen Chen, Hryhoriy Polshyn, [26] Yuan Cao, Debanjan Chowdhury, Daniel Rodan- Yuxuan Zhang, K. Watanabe, T. Taniguchi, David Graf, Legrain, Oriol Rubies-Bigorda, Kenji Watanabe, Takashi Andrea F. Young, and Cory R. Dean, “Tuning super- Taniguchi, T. Senthil, and Pablo Jarillo-Herrero, conductivity in twisted bilayer graphene,” Science 363, “Strange Metal in Magic-Angle Graphene with near 1059–1064 (2019). Planckian Dissipation,” Phys. Rev. Lett. 124, 076801 [15] Xiaobo Lu, Petr Stepanov, Wei Yang, Ming Xie, Mo- (2020). hammed Ali Aamir, Ipsita Das, Carles Urgell, Kenji [27] Yuan Cao, Daniel Rodan-Legrain, Jeong Min Park, Watanabe, Takashi Taniguchi, Guangyu Zhang, et al., Fanqi Noah Yuan, Kenji Watanabe, Takashi Taniguchi, “Superconductors, orbital magnets and correlated states Rafael M Fernandes, Liang Fu, and Pablo Jarillo- in magic-angle bilayer graphene,” Nature 574, 653–657 Herrero, “Nematicity and competing orders in su- (2019). perconducting magic-angle graphene,” arXiv:2004.04148 [16] Yuhang Jiang, Xinyuan Lai, Kenji Watanabe, Takashi (2020). Taniguchi, Kristjan Haule, Jinhai Mao, and Eva Y. An- [28] Emma C Regan, Danqing Wang, Chenhao Jin, drei, “Charge order and broken rotational symmetry in M Iqbal Bakti Utama, Beini Gao, Xin Wei, Sihan Zhao, magic-angle twisted bilayer graphene,” Nature (London) Wenyu Zhao, Zuocheng Zhang, Kentaro Yumigeta, et al., 573, 91–95 (2019). “Mott and generalized wigner crystal states in wse 2/ws [17] Yonglong Xie, Biao Lian, Berthold Jäck, Xiaomeng Liu, 2 moirésuperlattices,” Nature 579, 359–363 (2020). Cheng-Li Chiu, Kenji Watanabe, Takashi Taniguchi, [29] Guorui Chen, Aaron L Sharpe, Eli J Fox, Ya-Hui Zhang, B Andrei Bernevig, and Ali Yazdani, “Spectroscopic sig- Shaoxin Wang, Lili Jiang, Bosai Lyu, Hongyuan Li, Kenji natures of many-body correlations in magic-angle twisted Watanabe, Takashi Taniguchi, et al., “Tunable correlated bilayer graphene,” Nature 572, 101–105 (2019). chern insulator and ferromagnetism in a moirésuperlat- [18] Youngjoon Choi, Jeannette Kemmer, Yang Peng, Alex tice,” Nature 579, 56–61 (2020). Thomson, Harpreet Arora, Robert Polski, Yiran Zhang, [30] Zhe Liu, Yu Li, and Yi-Feng Yang, “Possible nodeless Hechen Ren, Jason Alicea, Gil Refael, et al., “Electronic s±-wave superconductivity in twisted bilayer graphene,” correlations in twisted bilayer graphene near the magic Chinese Physics B 28, 077103 (2019). angle,” Nature Physics 15, 1174–1180 (2019). [31] Xiaomeng Liu, Zeyu Hao, Eslam Khalaf, Jong Yeon [19] U. Zondiner, A. Rozen, D. Rodan-Legrain, Y. Cao, Lee, Yuval Ronen, Hyobin Yoo, Danial Haei Na- R. Queiroz, T. Taniguchi, K. Watanabe, Y. Oreg, F. von jafabadi, Kenji Watanabe, Takashi Taniguchi, Ashvin Oppen, Ady Stern, E. Berg, P. Jarillo-Herrero, and Vishwanath, et al., “Tunable spin-polarized correlated S. Ilani, “Cascade of phase transitions and Dirac revivals states in twisted double bilayer graphene,” Nature 583, in magic-angle graphene,” Nature (London) 582, 203– 221–225 (2020). 208 (2020). [32] Cheng Shen, Yanbang Chu, QuanSheng Wu, Na Li, [20] Dillon Wong, Kevin P Nuckolls, Myungchul Oh, Biao Shuopei Wang, Yanchong Zhao, Jian Tang, Jieying Liu, Lian, Yonglong Xie, Sangjun Jeon, Kenji Watanabe, Jinpeng Tian, Kenji Watanabe, et al., “Correlated states Takashi Taniguchi, B Andrei Bernevig, and Ali Yazdani, in twisted double bilayer graphene,” Nature Physics 16, “Cascade of electronic transitions in magic-angle twisted 520–525 (2020). bilayer graphene,” Nature 582, 198–202 (2020). [33] Christoph Bruder and Dieter Vollhardt, “Symmetry and [21] Petr Stepanov, Ipsita Das, Xiaobo Lu, Ali Fahimniya, stationary points of a free energy: The case of superfluid Kenji Watanabe, Takashi Taniguchi, Frank H. L. Kop- 3He,” Phys. Rev. B 34, 131–146 (1986). pens, Johannes Lischner, Leonid Levitov, and Dmitri K. [34] N. D. Mermin, “The topological theory of defects in or- Efetov, “Untying the insulating and superconducting ordered media,” Rev. Mod. Phys. 51, 591–648 (1979). ders in magic-angle graphene,” Nature (London) 583, [35] BI Halperin and David R Nelson, “Resistive transition 375–378 (2020). in superconducting films,” Journal of low temperature [22] Aaron L. Sharpe, Eli J. Fox, Arthur W. Barnard, Joe physics 36, 599–616 (1979). Finney, Kenji Watanabe, Takashi Taniguchi, M. A. Kast- [36] Vinay Ambegaokar, B. I. Halperin, David R. Nelson, ner, and David Goldhaber-Gordon, “Emergent ferro- and Eric D. Siggia, “Dynamics of superfluid films,” Phys. magnetism near three-quarters filling in twisted bilayer Rev. B 21, 1806–1826 (1980). graphene,” Science 365, 605–608 (2019). [37] In a rotationally symmetric system, this implies that [23] Guorui Chen, Lili Jiang, Shuang Wu, Bosai Lyu, Cooper pairs carry an odd orbital angular momentum. Hongyuan Li, Bheema Lingam Chittari, Kenji Watan- [38] Bruno Uchoa and A. H. Castro Neto, “Superconducting abe, Takashi Taniguchi, Zhiwen Shi, Jeil Jung, et al., states of pure and doped graphene,” Phys. Rev. Lett. 98, “Evidence of a gate-tunable mott insulator in a trilayer 146801 (2007). graphene moirésuperlattice,” Nature Physics 15, 237– [39] Note that taking k to −k exchanges K+ with K−. 241 (2019). [40] Jong Yeon Lee, Eslam Khalaf, Shang Liu, Xiaomeng Liu, [24] Guorui Chen, Aaron L Sharpe, Patrick Gallagher, Ilan T Zeyu Hao, Philip Kim, and Ashvin Vishwanath, “Theory Rosen, Eli J Fox, Lili Jiang, Bosai Lyu, Hongyuan Li, of correlated insulating behaviour and spin-triplet super- Kenji Watanabe, Takashi Taniguchi, et al., “Signatures conductivity in twisted double bilayer graphene,” Nature of tunable superconductivity in a trilayer graphene moiré communications 10, 1–10 (2019). superlattice,” Nature 572, 215–219 (2019). [41] Mathias S. Scheurer and Rhine Samajdar, “Pairing in [25] Aviram Uri, Sameer Grover, Yuan Cao, John A Crosse, graphene-based moiré superlattices,” Phys. Rev. Re- Kousik Bagani, Daniel Rodan-Legrain, Yuri Myasoedov, search 2, 033062 (2020). Kenji Watanabe, Takashi Taniguchi, Pilkyung Moon, [42] The existence of Z2 defects has been studied in the con- 19

text of superfluid 3He; see Ref.1. [53] RS Newrock, CJ Lobb, U Geigenmüller,and M Octavio, [43] Note that the term proportional to κd itself gives an en- “The two-dimensional physics of josephson junction ar- ergy cost for twists of the pairing polarization; there- rays,” Solid state physics (New York. 1955) 54, 263–512 fore κm captures the excess spin stiffness beyond that (2000). included in the generalized phase stiffness term. [54] Yuan Cao, Daniel Rodan-Legrain, Oriol Rubies-Bigorda, [44] P. C. Hohenberg and B. I. Halperin, “Theory of dy- Jeong Park, Kenji Watanabe, Takashi Taniguchi, namic critical phenomena,” Rev. Mod. Phys. 49, 435–479 and Pablo Jarillo-Herrero, “Tunable correlated states (1977). and spin-polarized phases in twisted bilayer–bilayer [45] Paul M Chaikin, Tom C Lubensky, and Thomas A graphene,” Nature , 1–6 (2020). Witten, Principles of condensed matter physics, Vol. 10 [55] Rhine Samajdar and Mathias S. Scheurer, “Microscopic (Cambridge university press Cambridge, 1995). pairing mechanism, order parameter, and disorder sen- [46] H Risken and TK Caugheyz, The fokker-planck equa- sitivity in moirésuperlattices: Applications to twisted tion: Methods of solution and application (Springer- double-bilayer graphene,” Phys. Rev. B 102, 064501 Verlag Berlin Heidelberg, 1996). (2020). [47] Subroto Mukerjee, Cenke Xu, and J. E. Moore, “Topo- [56] Yi-Ting Hsu, Fengcheng Wu, and S. Das Sarma, “Topo- logical defects and the superfluid transition of the s = 1 logical superconductivity, ferromagnetism, and valley- spinor condensate in two dimensions,” Phys. Rev. Lett. polarized phases in moiré systems: Renormalization 97, 120406 (2006). group analysis for twisted double bilayer graphene,” [48] Within the time-dependent model discussed in Sec.V, Phys. Rev. B 102, 085103 (2020). 1/τs corresponds to the dissipation rate, Eq. (B2) in Ap- [57] Fengcheng Wu and Sankar Das Sarma, “Identification pendixB. of superconducting pairing symmetry in twisted bilayer [49] J. Tjon and Jon Wright, “Solitons in the continuous graphene using in-plane magnetic field and strain,” Phys. heisenberg spin chain,” Phys. Rev. B 15, 3470–3476 Rev. B 99, 220507 (2019). (1977). [58] Minhao He, Yuhao Li, Jiaqi Cai, Yang Liu, K Watanabe, [50] Austen Lamacraft, “Persistent currents in ferromagnetic T Taniguchi, Xiaodong Xu, and Matthew Yankowitz, condensates,” Phys. Rev. B 95, 224512 (2017). “Tunable correlation-driven symmetry breaking in [51] K. Epstein, A. M. Goldman, and A. M. Kadin, “Vortex- twisted double bilayer graphene,” arXiv:2002.08904 antivortex pair dissociation in two-dimensional supercon- (2020). ductors,” Phys. Rev. Lett. 47, 534–537 (1981). [59] Eslam Khalaf, Shubhayu Chatterjee, Nick Bultinck, [52] A. M. Kadin, K. Epstein, and A. M. Goldman, “Renor- Michael P Zaletel, and Ashvin Vishwanath, “Charged malization and the kosterlitz-thouless transition in a two- skyrmions and topological origin of superconductivity in dimensional superconductor,” Phys. Rev. B 27, 6691– magic angle graphene,” arXiv:2004.00638 (2020). 6702 (1983). [60] Wikipedia contributors, “Fluctuation-dissipation theorem,” Wikipedia (2020).