Functional Renormalization Group approaches for Quantum Spin Liquids

For decades, frustrated quantum magnets have been a seed for scientific progress and innovation in condensed matter. As much as the numerical tools for low-dimensional quantum magnetism have thrived and improved in recent years due to breakthroughs inspired by quantum information and quantum computation, higher-dimensional quantum magnetism can be considered as the final frontier, where strong quantum entanglement, multiple ordering channels, and manifold ways of paramagnetism culminate. At the same time, efforts in crystal synthesis have induced a significant increase in the number of tangible frustrated magnets which are generically three-dimensional in nature, creating an urgent need for quantitative theoretical modeling. This talk will present the state-of-the-art in pseudo-fermion (PF) and pseudo-Majorana (PM) functional renormalization group (FRG) and their specific ability to address higher-dimensional frustrated quantum magnetism. First developed more than a decade ago, the PFFRG interprets a Heisenberg model Hamiltonian in terms of Abrikosov pseudofermions, which is then treated in a diagrammatic resummation scheme formulated as a renormalization group flow of m-particle pseudofermion vertices. We will discuss the achievements of PFFRG in successfully predicting the spectroscopic signatures of candidate quantum spin liquid materials based on complex three-dimensional geometries with intricate Hamiltonians. These include the recently studied bi-trillium lattice S=1 K₂Ni₂(SO₄)₃ and the hyperhyperkagome lattice S=1/2 PbCuTe₂O₆ compounds. We also present the success of PMFRG in treating finite-temperature properties of three-dimensional systems by describing phase transitions and the associated critical exponents.