# Emergence and stability of spin-valley entangled quantum liquids in moir\'e heterostructures

## TL;DR

This paper investigates the conditions under which spin-valley quantum liquids emerge and remain stable in moiré heterostructures, using a theoretical model and renormalization group analysis to identify regimes of spin liquid behavior.

## Contribution

It introduces a comprehensive analysis of spin liquid stability in SU(2)$^{ ext{spin}} imes$SU(2)$^{ ext{valley}}$ Heisenberg models with extended couplings in moiré heterostructures, highlighting the role of Hund's coupling and lattice geometry.

## Key findings

- Existence of an extended parameter regime with no long-range order indicating stable spin liquids.
- Identification of magnetic orders for large Hund's coupling, including antiferromagnetic and ferromagnetic states.
- Longer-ranged exchange couplings can both stabilize and destabilize the spin liquid regime depending on their sign.

## Abstract

Twisting moir\'e heterostructures to the flatband regime allows for the formation of strongly correlated quantum states, since the dramatic reduction of the bandwidth can cause the residual electronic interactions to set the principal energy scale. An effective description for such correlated moir\'e heterostructures, derived in the strong-coupling limit at integer filling, generically leads to spin-valley Heisenberg models. Here we explore the emergence and stability of spin liquid behavior in an SU(2)$^{\mathrm{spin}}\otimes$SU(2)$^{\mathrm{valley}}$ Heisenberg model upon inclusion of Hund's-induced and longer-ranged exchange couplings, employing a pseudofermion functional renormalization group approach. We consider two lattice geometries, triangular and honeycomb (relevant to different moir\'e heterostructures), and find, for both cases, an extended parameter regime surrounding the SU(4) symmetric point where no long-range order occurs, indicating a stable realm of quantum spin liquid behavior. For large Hund's coupling, we identify the adjacent magnetic orders, with both antiferromagnetic and ferromagnetic ground states emerging in the separate spin and valley degrees of freedom. For both lattice geometries the inclusion of longer-ranged exchange couplings is found to have both stabilizing and destabilizing effects on the spin liquid regime depending on the sign of the additional couplings.