Eur. Phys. J. B manuscript No. (will be inserted by the editor) Moments and multiplets in moir´ e materials A pseudo-fermion functional renormalization group for spin-valley models Lasse Gresista a , Dominik Kiese, Simon Trebst 1 Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany Received: date / Accepted: date Abstract The observation of strongly-correlated states in moir´ e systems has renewed the conceptual interest in magnetic systems with higher SU(4) spin symmetry, e.g. to describe Mott insulators where the local moments are coupled spin-valley degrees of freedom. Here, we discuss a numerical renormalization group scheme to explore the formation of spin-valley ordered and unconventional spin-valley liquid states at zero temperature based on a pseudo-fermion representation. Our generalization of the conventional pseudo-fermion functional renormalization group approach for su(2) spins is capable of treating diagonal and off-diagonal couplings of generic spin-valley exchange Hamiltonians in the self-conjugate representation of the su(4) algebra. To achieve proper numerical efficiency, we derive a number of symmetry constraints on the flow equations that significantly limit the number of ordinary differential equations to be solved. As an example system, we investigate a diagonal SU(2) spin ⊗ U(1) valley model on the triangular lattice which exhibits a rich phase diagram of spin and valley ordered phases. 1 Introduction Moir´ e materials that exhibit flat bands such as twisted bilayer graphene (tBG) or certain van der Waals het- erostructures such as hexagonal boron nitride (TLG/h- BN) have recently been established as novel, highly tunable platforms for the study of strongly correlated electrons. Relative to an almost vanishing bandwidth, residual interactions in these materials can induce a plethora of different many-body phenomena ranging from the formation of correlated insulators [1–4] and super- conductors [5–7] to anomalous quantum Hall effects [8]. However, a microsopic description of these phenomena is a formidable challenge as the number of of low-energy degrees of freedom is often increased [9–11] in compar- ison to conventional Mott insulators. More specifically, it has been argued [12, 13], that multi-orbital Hubbard models can describe the flat band physics in e.g. TLG/h-BN within the topologically triv- ial regime, where fully symmetric Wannier states may be constructed [14]. The proposed interaction terms for the corresponding Hamiltonians usually include a a e-mail: gresista@thp.uni-koeln.de generalized Hubbard U [12, 13, 15] as well as Hund’s type couplings. Performing a strong coupling expan- sion where one treats the interactions as the dominant energy scale, these extended Hubbard models can then be mapped to su(4) 1 spin-valley Hamiltonians that may be used as a starting point to investigate the nature of the correlated insulating states. The so-derived su(4) models bear a close resemblance to Kugel-Khomskii models [16] that have a long history in the study of transition metal oxides, where they are used to capture the Jahn-Teller physics of intertwined spin and orbital degrees of freedom. Increasing the number of relevant microscopic degrees of freedom (in comparison to con- ventional quantum magnets) has been particlularly ap- preciated to boost quantum fluctuations independent of, e.g., lattice geometries [17], which has made Kugel- Khomskii models a recurring target in the search for un- usual many-body states such as quantum spin-orbital liquids [18–21]. As such, one might expect the su(4) spin-valley physics relevant to the correlated insulating states of moir´ e materials to hold similar promise for the observation of spin-valley liquid states with macro- 1 With su(4) we refer to the Lie algebra of the Lie group SU(4). arXiv:2202.05029v1 [cond-mat.str-el] 10 Feb 2022