Taming pseudo-fermion functional renormalization for quantum spins: Finite-temperatures and the Popov-Fedotov trick

TL;DR

This paper enhances the pseudo-fermion functional renormalization group method by incorporating the Popov-Fedotov projection, enabling accurate finite-temperature studies of quantum spins and revealing unphysical state contributions at zero temperature.

Contribution

It introduces the Popov-Fedotov projection into the pseudo-fermion FRG framework, allowing for finite-temperature analysis and addressing unphysical state contributions previously neglected.

Findings

Popov-Fedotov projection improves finite-temperature calculations.

Unphysical states contribute at zero temperature in small systems.

Finite-temperature magnetic transitions can be studied via finite-size scaling.

Abstract

The pseudo-fermion representation for $S=1/2$ quantum spins introduces unphysical states in the Hilbert space which can be projected out using the Popov-Fedotov trick. However, state-of-the-art implementation of the functional renormalization group method for pseudo-fermions have so far omitted the Popov-Fedotov projection. Instead, restrictions to zero temperature were made and absence of unphysical contributions to the ground-state was assumed. We question this belief by exact diagonalization of several small-system counterexamples where unphysical states do contribute to the ground state. We then introduce Popov-Fedotov projection to pseudo-fermion functional renormalization, enabling finite temperature computations with only minor technical modifications to the method. At large and intermediate temperatures, our results are perturbatively controlled and we confirm their accuracy in benchmark calculations. At lower temperatures, the accuracy degrades due to truncation errors in the hierarchy of flow equations. Interestingly, these problems cannot be alleviated by switching to the parquet approximation. We introduce the spin projection as a method-intrinsic quality check. We also show that finite temperature magnetic ordering transitions can be studied via finite-size scaling.