The observation of strongly correlated states in moiré 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 $${{\mathfrak {s}}}{{\mathfrak {u}}}$$
s
u
(2) spins is capable of treating diagonal and off-diagonal couplings of generic spin–valley exchange Hamiltonians in the self-conjugate representation of the $${{\mathfrak {s}}}{{\mathfrak {u}}}$$
s
u
(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)$$_{\text {spin}}$$
spin
$$\otimes $$
⊗
U(1)$$_{\text {valley}}$$
valley
model on the triangular lattice which exhibits a rich phase diagram of spin and valley ordered phases.
Graphic Abstract
Moiré systems provide a highly tunable platform for engineering band structures and exotic correlated phases. Here, we theoretically study a model for a single layer of graphene subject to a smooth moiré electrostatic potential, induced by an insulating substrate layer. For sufficiently large moiré unit cells, we find that ultra-flat bands coexist with a triangular network of chiral one-dimensional (1D) channels. These channels mediate an effective interaction between localized modes with spin-, orbital- and valley degrees of freedom emerging from the flat bands. The form of the interaction reflects the chiralilty and 1D nature of the network. We study this interacting model within an SU(4) mean-field theory, semi-classical Monte-Carlo simulations, and an SU(4) spin-wave theory, focusing on commensurate order stabilized by local two-site and chiral three-site interactions. By tuning a gate voltage, one can trigger a non-coplanar phase characterized by a peculiar coexistence of three different types of order: ferromagnetic spin order in one valley, non-coplanar chiral spin order in the other valley, and 120° order in the remaining spin and valley-mixed degrees of freedom. Quantum and classical fluctuations have qualitatively different effects on the observed phases and can, for example, create a finite spin-chirality
purely via fluctuation effects.
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