Numerical treatment of spin systems with unrestricted spin length $S$: A functional renormalization group study

TL;DR

This paper introduces a generalized pseudo-fermion functional renormalization group method capable of handling arbitrary spin magnitudes, bridging quantum and classical limits, and applies it to the honeycomb Heisenberg model to analyze phase transitions.

Contribution

The authors develop a unified PFFRG framework for all spin sizes and demonstrate its effectiveness by mapping the phase diagram from quantum to classical regimes.

Findings

Quantum fluctuations decrease rapidly with increasing S.

No disordered phase observed at S=1.

Classical phase diagram recovered as S approaches infinity.

Abstract

We develop a generalized pseudo-fermion functional renormalization group (PFFRG) approach that can be applied to arbitrary Heisenberg models with spins ranging from the quantum case $S=1/2$ to the classical limit $S\rightarrow\infty$. Within this framework, spins of magnitude $S$ are realized by implementing $M=2S$ copies of spin-1/2 degrees of freedom on each lattice site. We confirm that even without explicitly projecting onto the highest spin sector of the Hilbert space, ground states tend to select the largest possible local spin magnitude. This justifies the average treatment of the pseudo fermion constraint in previous spin-1/2 PFFRG studies. We apply this method to the antiferromagnetic $J_1$-$J_2$ honeycomb Heisenberg model with nearest neighbor $J_1>0$ and second neighbor $J_2>0$ interactions. Mapping out the phase diagram in the $J_2/J_1$-$S$ plane we find that upon increasing $S$ quantum fluctuations are rapidly decreasing. In particular, already at $S=1$ we find no indication for a magnetically disordered phase. In the limit $S\rightarrow\infty$, the known phase diagram of the classical system is exactly reproduced. More generally, we prove that for $S\rightarrow\infty$ the PFFRG approach is identical to the Luttinger-Tisza method.