When sizable quantum fluctuations and strong frustration mechanisms act in concert to repel the formation of conventional long-range order in quantum magnets, they can make way for massively entangled spin liquid phases which may imbue the material with extraordinary properties. \nThe search for such curious phases of matter has proceeded for several decades, gaining extra momentum some fifteen years ago when Kitaev proposed an analytically solvable model for a quantum spin liquid with anyonic excitations on the basis of realistic microscopic spin exchange terms [Kitaev, Annals of Physics 321, 2 (2006)]. Attempts to identify different models or materials which harbor quantum spin liquid ground states have kept researchers -- experimentalists and theorists alike -- in suspense ever since. However, the simulation of quantum many-body systems poses a serious challenge even to modern numerical techniques, particularly in the case of frustrated quantum magnetism in three spatial dimensions. Such models evade tractability by many established approaches, leaving a methodological void. \n \nIn this thesis, we report on recent progress in cutting-edge implementations of the pseudo-fermion functional renormalization group (pf-FRG), which has originally been proposed by Reuther and Wölfle in the context of two-dimensional frustrated quantum magnetism with highly symmetric spin interactions [Reuther and Wölfle, Phys. Rev. B 81, 144410 (2010)]. Reflecting the growing interest in models with SU(2) symmetry-breaking spin exchange terms that has followed the unearthing of Kitaev's honeycomb model, we present a generalized implementation of the pf-FRG which is suited to numerically simulate arbitrary microscopic models with diagonal or off-diagonal two-spin interactions, even in three-dimensional frustrated quantum magnets. We provide insight into the inner workings of the method which has emerged over the course of the last couple of years, arguing that the pf-FRG formalism simultaneously combines aspects of a large-S expansion as well as a large-N expansion on equal footing, thus being able to resolve the subtle interplay between magnetic ordering tendencies and disruptive quantum fluctuations. \n \nMoreover, on a case by case basis we explore the stability of quantum spin liquids in paradigmatic models of frustrated quantum magnetism and elucidate the joint action of geometric frustration, exchange frustration, and quantum fluctuations to inhibit the formation of magnetic long-range order. Examples include: (i) a Heisenberg spin model on the three-dimensional diamond lattice where geometric frustration arises from antiferromagnetic next-nearest neighbor interactions. The additional competition with nearest neighbor interactions leads to an unusually large ground state degeneracy already on classical level. The quantum-to-classical transition is studied by systematically varying the spin length and the different roles of quantum fluctuations and thermal fluctuations are discussed. The theory is applied to interpret experiments on the spin-liquid candidate NiRh2O4; (ii) a Heisenberg antiferromagnet on the face centered cubic lattice which is augmented by competing bond-directional interactions, evolving around the concurrent manifestation of geometric frustration and exchange frustration. A situation is identified where both mechanisms collude to give rise to an unusually large degree of frustration over a wide parameter regime. We discuss related experimental findings on the highly frustrated iridate compound Ba2CeIrO6; (iii) a Heisenberg antiferromagnet on the kagome lattice with additional Dzyaloshinskii-Moriya interactions. The robustness of the unperturbed kagome antiferromagnet's spin liquid ground state against low-symmetry Dzyaloshinskii-Moriya interactions is investigated. Implications for the spin-liquid candidate herbertsmithite are discussed.
When sizable quantum fluctuations and strong frustration mechanisms act in concert to repel the formation of conventional long-range order in quantum magnets, they can make way for massively entangled spin liquid phases which may imbue the material with extraordinary properties. \nThe search for such curious phases of matter has proceeded for several decades, gaining extra momentum some fifteen years ago when Kitaev proposed an analytically solvable model for a quantum spin liquid with anyonic excitations on the basis of realistic microscopic spin exchange terms [Kitaev, Annals of Physics 321, 2 (2006)]. Attempts to identify different models or materials which harbor quantum spin liquid ground states have kept researchers -- experimentalists and theorists alike -- in suspense ever since. However, the simulation of quantum many-body systems poses a serious challenge even to modern numerical techniques, particularly in the case of frustrated quantum magnetism in three spatial dimensions. Such models evade tractability by many established approaches, leaving a methodological void. \n \nIn this thesis, we report on recent progress in cutting-edge implementations of the pseudo-fermion functional renormalization group (pf-FRG), which has originally been proposed by Reuther and Wölfle in the context of two-dimensional frustrated quantum magnetism with highly symmetric spin interactions [Reuther and Wölfle, Phys. Rev. B 81, 144410 (2010)]. Reflecting the growing interest in models with SU(2) symmetry-breaking spin exchange terms that has followed the unearthing of Kitaev's honeycomb model, we present a generalized implementation of the pf-FRG which is suited to numerically simulate arbitrary microscopic models with diagonal or off-diagonal two-spin interactions, even in three-dimensional frustrated quantum magnets. We provide insight into the inner workings of the method which has emerged over the course of the last couple of years, arguing that the pf-FRG formalism simultaneously combines aspects of a large-S expansion as well as a large-N expansion on equal footing, thus being able to resolve the subtle interplay between magnetic ordering tendencies and disruptive quantum fluctuations. \n \nMoreover, on a case by case basis we explore the stability of quantum spin liquids in paradigmatic models of frustrated quantum magnetism and elucidate the joint action of geometric frustration, exchange frustration, and quantum fluctuations to inhibit the formation of magnetic long-range order. Examples include: (i) a Heisenberg spin model on the three-dimensional diamond lattice where geometric frustration arises from antiferromagnetic next-nearest neighbor interactions. The additional competition with nearest neighbor interactions leads to an unusually large ground state degeneracy already on classical level. The quantum-to-classical transition is studied by systematically varying the spin length and the different roles of quantum fluctuations and thermal fluctuations are discussed. The theory is applied to interpret experiments on the spin-liquid candidate NiRh2O4; (ii) a Heisenberg antiferromagnet on the face centered cubic lattice which is augmented by competing bond-directional interactions, evolving around the concurrent manifestation of geometric frustration and exchange frustration. A situation is identified where both mechanisms collude to give rise to an unusually large degree of frustration over a wide parameter regime. We discuss related experimental findings on the highly frustrated iridate compound Ba2CeIrO6; (iii) a Heisenberg antiferromagnet on the kagome lattice with additional Dzyaloshinskii-Moriya interactions. The robustness of the unperturbed kagome antiferromagnet's spin liquid ground state against low-symmetry Dzyaloshinskii-Moriya interactions is investigated. Implications for the spin-liquid candidate herbertsmithite are discussed.
In the field of quantum magnetism, the advent of numerous spin-orbit assisted Mott insulating compounds, such as the family of Kitaev materials, has led to a growing interest in studying general spin models with nondiagonal interactions that do not retain the SU(2) invariance of the underlying spin degrees of freedom. However, the exchange frustration arising from these nondiagonal and often bond-directional interactions for two- and three-dimensional lattice geometries poses a serious challenge for numerical many-body simulation techniques. In this paper, we present an extended formulation of the pseudofermion functional renormalization group that is capable of capturing the physics of frustrated quantum magnets with generic (diagonal and off-diagonal) two-spin interaction terms. Based on a careful symmetry analysis of the underlying flow equations, we reveal that the computational complexity grows only moderately, as compared to models with only diagonal interaction terms. We apply the formalism to a kagome antiferromagnet which is augmented by general in-plane and out-of-plane Dzyaloshinskii-Moriya (DM) interactions, as argued to be present in the spin liquid candidate material herbertsmithite. We calculate the complete ground state phase diagram in the strength of in-plane and out-of-plane DM couplings, and discuss the extended stability of the spin liquid of the unperturbed kagome antiferromagnet in the presence of these couplings.
The pseudofermion functional renormalization group (pf-FRG) is one of the few numerical approaches that has been demonstrated to quantitatively determine the ordering tendencies of frustrated quantum magnets in two and three spatial dimensions. The approach, however, relies on a number of presumptions and approximations, in particular the choice of pseudofermion decomposition and the truncation of an infinite number of flow equations to a finite set. Here we generalize the pf-FRG approach to SU(N)-spin systems with arbitrary N and demonstrate that the scheme becomes exact in the large-N limit. Numerically solving the generalized real-space renormalization group equations for arbitrary N, we can make a stringent connection between the physically most significant case of SU(2) spins and more accessible SU(N) models. In a case study of the square-lattice SU(N) Heisenberg antiferromagnet, we explicitly demonstrate that the generalized pf-FRG approach is capable of identifying the instability indicating the transition into a staggered flux spin liquid ground state in these models for large, but finite, values of N. In a companion paper [Roscher et al., Phys. Rev. B 97, 064416 (2018)] we formulate a momentum-space pf-FRG approach for SU(N) spin models that allows us to explicitly study the large-N limit and access the low-temperature spin liquid phase.
Magnetic materials with predominant antiferromagnetic couplings exhibit a wide range of phenomena arising from the interplay of quantum fluctuations, competing interactions and the underlying lattice geometry. While static properties of quantum magnets have been extensively studied and are often well understood, many of the most distinctive signatures of unconventional magnetic phases – such as fractionalization, confinement, or emergent gauge structures – are encoded in their dynamical response. Predicting these responses in strongly frustrated systems, particularly in two and three dimensions, remains a major theoretical challenge, as analytic solutions are rare and numerical methods are often limited by either dimensionality or the degree of frustration. At the same time, experimental data – e.g. from inelastic neutron scattering – is increasingly available, creating a gap between experiment and theory. To bridge this gap, we develop a real-frequency extension of the PFFRG using the Keldysh formalism, enabling dynamical observables to be computed directly without relying on a numerically unstable analytical continuation required in the Matsubara PFFRG. We derive the full set of Keldysh-indexed flow equations and perform a comprehensive symmetry analysis that is used to substantially reduce the computational effort. Benchmark calculations of the Heisenberg model on the zero-dimensional dimer, the one-dimensional spin chain, and the two-dimensional square lattice as well as the honeycomb Kitaev model show that the method is best suited for systems near the paramagnetic regime, such as quantum spin liquids. We further identify methodological limitations and outline possible extensions, including the incorporation of pseudo-Majorana particles in place of complex pseudo-Fermions, for which we also derive the corresponding flow equations and symmetry relations. With these shortcomings in mind, we apply the Keldysh PFFRG to analyze quantum spin liquid systems on the trellis lattice and the Shastry-Sutherland model and compare to a projected symmetry group approach. For the trellis lattice, the Keldysh PFFRG provides insights into which mean-field Ansätze capture the different phases together with their low-energy excitation spectra. Using optimized density-functional-theory models, we reproduce the spinon-fractionalization and deconfinement signatures observed in inelastic neutron scattering for the trellis compound \( \ce{CaCu2O3} \), and we predict the low-energy dynamical spin structure factor for \(\ce{SrCu2O3}\) and \(\ce{CaV2O5}\). In the Shastry-Sutherland model, the Keldysh PFFRG reveals the same Dirac-cone splitting predicted by the projected symmetry group approach, thereby confirming the connection between the U(1) square-lattice spin-liquid Ansatz proposed by Wen et al. [PRB 65, 165113 (2002)] and the parametrically small quantum spin liquid phase of the Shastry-Sutherland model.
The pseudofermion functional renormalization group (pf-FRG) has been put forward as a semianalytical scheme that, for a given microscopic spin model, allows to discriminate whether the low-temperature states exhibit magnetic ordering or a tendency toward the formation of quantum spin liquids. However, the precise nature of the putative spin-liquid ground state has remained hard to infer from the original (single-site) pf-FRG scheme. Here, we introduce a cluster pf-FRG approach, which allows for a more stringent connection between a microscopic spin model and its low-temperature spin-liquid ground states. In particular, it allows to calculate spatially structured fermion bilinear expectation values on spatial clusters, which are formed by splitting the original lattice into several sublattices, thereby allowing for the positive identification of a family of bilinear spin-liquid states. As an application of this cluster pf-FRG approach, we consider the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N)$ Heisenberg model on a square lattice, which is a paradigmatic example for a frustrated quantum magnet exhibiting quantum spin-liquid behavior for intermediate coupling strengths. In the well-established large-N limit of this model, we show that our approach correctly captures the emergence of the $\ensuremath{\pi}$-flux spin-liquid state at low temperatures. For small N, where the precise nature of the ground state remains controversial, we focus on the widely studied case of N=2, for which we determine the low-temperature phase diagram near the strongly frustrated regime after implementing the fermion-number constraint by the flowing Popov-Fedotov method. Our results suggest that the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model does not support the formation of a fermion bilinear spin-liquid state.
In frustrated magnetism, making a stringent connection between microscopic spin models and macroscopic properties of spin liquids remains an important challenge. A recent step towards this goal has been the development of the pseudofermion functional renormalization group approach (pf-FRG) which, building on a fermionic parton construction, enables the numerical detection of the onset of spin liquid states as temperature is lowered. In this work, focusing on the SU(N) Heisenberg model at large N, we extend this approach in a way that allows us to directly enter the low-temperature spin liquid phase, and to probe its character. Our approach proceeds in momentum space, making it possible to keep the truncation minimalistic, while also avoiding the bias introduced by an explicit decoupling of the fermionic parton interactions into a given channel. We benchmark our findings against exact mean-field results in the large-N limit, and show that even without prior knowledge the pf-FRG approach identifies the correct mean-field decoupling channel. On a technical level, we introduce an alternative finite temperature regularization scheme that is necessitated to access the spin liquid ordered phase. In a companion paper [Buessen et al., Phys. Rev. B 97, 064415 (2018)] we present a different set of modifications of the pf-FRG scheme that allow us to study SU(N) Heisenberg models (using a real-space RG approach) for arbitrary values of N, albeit only up to the phase transition towards spin liquid physics.
The theoretical inception of the Kitaev honeycomb model has had defining influence on the experimental hunt for quantum spin liquids, bringing unprecedented focus onto the synthesis of materials with bond-directional interactions. Numerous Kitaev materials, which are believed to harbor ground states parametrically close to the Kitaev spin liquid, have been investigated since. Yet, in these materials the Kitaev interaction often comes hand in hand with off-diagonal $\mathrm{\ensuremath{\Gamma}}$ interactions---with the competition of the two potentially giving rise to a magnetically ordered ground state. In an attempt to aid future material investigations, we study the phase diagram of the spin-1/2 Kitaev-$\mathrm{\ensuremath{\Gamma}}$ model on the honeycomb lattice. Employing a pseudofermion functional renormalization group (pf-FRG) approach, which directly operates in the thermodynamic limit and captures the joint effect of thermal and quantum fluctuations, we unveil the existence of extended parameter regimes with emergent incommensurate magnetic correlations at finite temperature. We supplement our results with additional calculations on a finite cylinder geometry to investigate the impact of periodic boundary conditions on the incommensurate order, thereby providing a perspective on previous numerical studies on finite systems.