Pseudo-fermion functional renormalization group for spin models.

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. We review the 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 ofm-particle pseudofermion vertices. The article reviews the state of the art of PFFRG and PMFRG and discusses their application to exemplary domains of frustrated magnetism, but most importantly, it makes the algorithmic and implementation details of these methods accessible to everyone. By thus lowering the entry barrier to their application, we hope that this review will contribute towards establishing PFFRG and PMFRG as the numerical methods for addressing frustrated quantum magnetism in higher spatial dimensions.

Quantum Effects on Unconventional Pinch Point Singularities.

Fracton phases are a particularly exotic type of quantum spin liquids where the elementary quasiparticles are intrinsically immobile. These phases may be described by unconventional gauge theories known as tensor or multipolar gauge theories, characteristic for so-called type-I or type-II fracton phases, respectively. Both variants have been associated with distinctive singular patterns in the spin structure factor, such as multifold pinch points for type-I and quadratic pinch points for type-II fracton phases. Here, we assess the impact of quantum fluctuations on these patterns by numerically investigating the spin S=1/2 quantum version of a classical spin model on the octahedral lattice featuring exact realizations of multifold and quadratic pinch points, as well as an unusual pinch line singularity. Based on large scale pseudofermion and pseudo-Majorana functional renormalization group calculations, we take the intactness of these spectroscopic signatures as a measure for the stability of the corresponding fracton phases. We find that in all three cases, quantum fluctuations significantly modify the shape of pinch points or lines by smearing them out and shifting signal away from the singularities in contrast to effects of pure thermal fluctuations. This indicates possible fragility of these phases and allows us to identify characteristic fingerprints of their remnants.

Magnetic Field Induced Quantum Spin Liquid in the Two Coupled Trillium Lattices of K_{2}Ni_{2}(SO_{4})_{3}.

Quantum spin liquids are exotic states of matter that form when strongly frustrated magnetic interactions induce a highly entangled quantum paramagnet far below the energy scale of the magnetic interactions. Three-dimensional cases are especially challenging due to the significant reduction of the influence of quantum fluctuations. Here, we report the magnetic characterization of K_{2}Ni_{2}(SO_{4})_{3} forming a three-dimensional network of Ni^{2+} spins. Using density functional theory calculations, we show that this network consists of two interconnected spin-1 trillium lattices. In the absence of a magnetic field, magnetization, specific heat, neutron scattering, and muon spin relaxation experiments demonstrate a highly correlated and dynamic state, coexisting with a peculiar, very small static component exhibiting a strongly renormalized moment. A magnetic field B≳4 T diminishes the ordered component and drives the system into a pure quantum spin liquid state. This shows that a system of interconnected S=1 trillium lattices exhibits a significantly elevated level of geometrical frustration.

Evidence for a three-dimensional quantum spin liquid in PbCuTe2O6.

The quantum spin liquid is a highly entangled magnetic state characterized by the absence of static magnetism in its ground state. Instead, the spins fluctuate in a highly correlated way down to the lowest temperatures. Quantum spin liquids are very rare and are confined to a few specific cases where the interactions between the magnetic ions cannot be simultaneously satisfied (known as frustration). Lattices with magnetic ions in triangular or tetrahedral arrangements, which interact via isotropic antiferromagnetic interactions, can generate such a frustration. Three-dimensional isotropic spin liquids have mostly been sought in materials where the magnetic ions form pyrochlore or hyperkagome lattices. Here we present a three-dimensional lattice called the hyper-hyperkagome that enables spin liquid behaviour and manifests in the compound PbCuTe2O6. Using a combination of experiment and theory, we show that this system exhibits signs of being a quantum spin liquid with no detectable static magnetism together with the presence of diffuse continua in the magnetic spectrum suggestive of fractional spinon excitations.

Classical spiral spin liquids as a possible route to quantum spin liquids.

Quantum spin liquids are long-range entangled phases whose magnetic correlations are determined by strong quantum fluctuations. While an overarching principle specifying the precise microscopic coupling scenarios for which quantum spin-liquid behavior arises is unknown, it is well-established that they are preferably found in spin systems where the corresponding classical limit of spin magnitudes [Formula: see text] exhibits a macroscopic ground state degeneracy, so-called classical spin liquids. Spiral spin liquids represent a special family of classical spin liquids where degenerate manifolds of spin spirals form closed contours or surfaces in momentum space. Here, we investigate the potential of spiral spin liquids to evoke quantum spin-liquid behavior when the spin magnitude is tuned from the classical [Formula: see text] limit to the quantum S = 1/2 case. To this end, we first use the Luttinger-Tisza method to formulate a general scheme which allows one to construct new spiral spin liquids based on bipartite lattices. We apply this approach to the two-dimensional square lattice and the three-dimensional hcp lattice to design classical spiral spin-liquid phases which have not been previously studied. By employing the pseudofermion functional renormalization group (PFFRG) technique we investigate the effects of quantum fluctuations when the classical spins are replaced by quantum S = 1/2 spins. We indeed find that extended spiral spin-liquid regimes change into paramagnetic quantum phases possibly realizing quantum spin liquids. Remnants of the degenerate spiral surfaces are still discernible in the momentum-resolved susceptibility, even in the quantum S = 1/2 case. In total, this corroborates the potential of classical spiral spin liquids to induce more complex non-magnetic quantum phases.

Quantum Spin Liquids in Frustrated Spin-1 Diamond Antiferromagnets.

Motivated by the recent synthesis of the spin-1 A-site spinel NiRh_{2}O_{4}, we investigate the classical to quantum crossover of a frustrated J_{1}-J_{2} Heisenberg model on the diamond lattice upon varying the spin length S. Applying a recently developed pseudospin functional renormalization group approach for arbitrary spin-S magnets, we find that systems with S≥3/2 reside in the classical regime, where the low-temperature physics is dominated by the formation of coplanar spirals and a thermal (order-by-disorder) transition. For smaller local moments S=1 or S=1/2, we find that the system evades a thermal ordering transition and forms a quantum spiral spin liquid where the fluctuations are restricted to characteristic momentum-space surfaces. For the tetragonal phase of NiRh_{2}O_{4}, a modified J_{1}-J_{2}^{-}-J_{2}^{⊥} exchange model is found to favor a conventionally ordered Néel state (for arbitrary spin S), even in the presence of a strong local single-ion spin anisotropy, and it requires additional sources of frustration to explain the experimentally observed absence of a thermal ordering transition.

Quantum Paramagnet in a π Flux Triangular Lattice Hubbard Model.

We propose the π flux triangular lattice Hubbard model (π THM) as a prototypical setup to stabilize magnetically disordered quantum states of matter in the presence of charge fluctuations. The quantum paramagnetic domain of the π THM that we identify for intermediate Hubbard U is framed by a Dirac semimetal for weak coupling and by 120° Néel order for strong coupling. Generalizing the Klein duality from spin Hamiltonians to tight-binding models, the π THM maps to a Hubbard model which corresponds to the (J_{H},J_{K})=(-1,2) Heisenberg-Kitaev model in its strong coupling limit. The π THM provides a promising microscopic testing ground for exotic finite-U spin liquid ground states amenable to numerical investigation.