Theory of zero-field diode effect in twisted trilayer graphene Harley D. Scammell, 1 J.I.A. Li, 2 and Mathias S. Scheurer 3 1 School of Physics, the University of New South Wales, Sydney, NSW, 2052, Australia 2 Department of Physics, Brown University, Providence, RI 02912, USA 3 Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria In a recent experiment [Lin et al., arXiv:2112.07841], the superconducting phase hosted by a heterostructure of mirror-symmetric twisted trilayer graphene and WSe2 was shown to exhibit sig- nificantly different critical currents in opposite directions in the absence of external magnetic fields. We here develop a microscopic theory and analyze necessary conditions for this zero-field super- conducting diode effect. Taking into account the spin-orbit coupling induced in trilayer graphene via the proximity effect, we classify the pairing instabilities and normal-state orders and derive which combinations are consistent with the observed diode effect, in particular, its field trainability. We perform explicit calculations of the diode effect in several different models, including the full continuum model for the system, and illuminate the relation between the diode effect and finite- momentum pairing. Our theory also provides a natural explanation of the observed sign change of the current asymmetry with doping, which can be related to an approximate chiral symmetry of the system, and of the enhanced transverse resistance above the superconducting transition. Our findings not only elucidate the rich physics of trilayer graphene on WSe2, but also establish a means to distinguish between various candidate interaction-induced orders in spin-orbit-coupled graphene moiré systems, and could therefore serve as a guide for future experiments as well. CONTENTS I. Introduction 1 II. Model and symmetries 2 A. Notation and continuum model 2 B. Symmetries 3 C. Effective low-energy descriptions 4 III. Superconducting order parameters 4 IV. Symmetry analysis of diode effect 6 A. Candidate normal-state orders 6 B. Zero-field diode effect 7 C. Field-induced diode effect 9 D. Diode effect without normal-state order 9 V. Model calculations 10 A. General formalism 10 B. Patch theory 12 C. Full MBZ toy models 13 D. Continuum model results 14 VI. Doping dependence of the diode effect 15 VII. Conclusion and Outlook 16 Acknowledgments 17 References 17 A. Pairing states in the opposite limit 19 B. Diode effect of the E state 20 1. Free energy 20 2. Critical current 21 C. Patch theory expansion 23 D. LG theory in the presence of strong spin-orbit coupling or intravalley pairing 23 1. Intravalley pairing 23 2. Intervalley pairing 24 I. INTRODUCTION Semiconductor diodes play an essential role in mod- ern electronics—computation, communication and sens- ing [1]. The diode generates a nonreciprocity, host- ing low resistance in one direction, and high resistance in the opposite. Recently the superconducting diode effect—induced by magnetic field [2–8], magnetic proxim- ity [8, 9], or magnetic Josephson or tunnel junctions [10– 19]—has attracted considerable attention. In a supercon- ducting diode, the critical supercurrent in one direction is larger than in the opposite. Having nonreciprocity in common with the semiconductor, yet boasting zero resis- tance, superconducting diodes have potential as building blocks for future quantum electronics. A recent study [20] (companion to this work) considers a heterostructure consisting of twisted trilayer graphene (tTLG) and WSe 2 , as depicted in Fig. 1, and demon- strates a superconducting diode effect in the absence of external magnetic fields, magnetic proximity or a mag- netic junction; for brevity, we here refer to this effect as the zero-field superconducting diode effect (ZFDE). In addition, several revealing features of the ZFDE were reported: (i) the diode effect, i.e., the asymmetry δJ c of arXiv:2112.09115v1 [cond-mat.mes-hall] 16 Dec 2021