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.
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.
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.
We enhance the pseudo-fermion functional-renormalisation-group (PFFRG) method for the investigation of highly frustrated quantum spin systems in two significant ways. On one hand, we generalise the method towards off-diagonal Dzyaloshinsky-Moriya (DM) interactions and single-ion anisotropies. On the other hand, we develop a self-consistent Fock-like mean-field technique which is based on vertex functions from a PFFRG analysis and calculates effective free spinon models for the considered systems at low energies. In Chap. 1, we introduce the fundamental mathematical concepts on which our studies rely. The so-called flow equations for the PFFRG are then derived in Chap. 3. The first part of our original work is presented in Chap. 4 where we, for the first time, extend the PFFRG scheme towards off-diagonal spin interactions and therefore confirm the quite general applicability of this method. There are two important results from our investigations of the spin-1/2 Heisenberg-DM model on the kagome lattice in this chapter. One is that a finite DM interaction enlarges magnetically ordered phases and thereby shrinks the paramagnetic phases which possibly house spin liquid states. The other important result is that the mineral herbertsmithite which inspired this project lies in close proximity to a quantum critical point. We obtain a phase diagram for the considered Hamiltonian and qualitatively good results in comparison to experimental data for herbertsmithite [68,69]. Our collaboration with a research group from Cologne is summarised in Chap. 5. Here, we investigate two different XXZ models with tetragonally split-up second-neighbour Heisenberg interactions and single-ion anisotropies on the diamond lattice in order to study the spin-1 A-site spinel NiRh2O4. We are able to confirm the existence of a paramagnetic phase for one of the considered models. However, such a paramagnetic behaviour cannot be verified for the Hamiltonians suggested in Refs. [29] and [30]. The second main part of this thesis is contained in Chap. 6. The developed PFFRG plus mean-field method enables us to characterise the effective low-energy spinon band structure of the spin-1/2 J1-J2 (nearest-neighbour) Heisenberg model on the square (kagome) lattice. This is achieved by comparing self-consistently determined overall amplitudes for different mean-field ansätze which are taken from corresponding PSG classifications in Refs. [182] and [111]. Our studies imply that the square lattice model realises a SU(2) spin liquid with Pi-fluxes through elementary plaquettes and that the spin-1/2 kagome antiferromagnet is most likely described by a Z2 spin liquid called Z2[0,Pi]Alpha. We argue that our analysed models have current experimental relevance and that the developed techniques show plenty of possibilities for future applications. A detailed summary and discussion of all methods and results is provided in Chap. 7.
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.
Abstract The observation of strongly correlated states in moiré systems has renewed the conceptual interest in magnetic systems with higher SU(4) spin symmetry, e.g., to describe Mott insulators where the local moments are coupled spin–valley degrees of freedom. Here, we discuss a numerical renormalization group scheme to explore the formation of spin–valley ordered and unconventional spin–valley liquid states at zero temperature based on a pseudo-fermion representation. Our generalization of the conventional pseudo-fermion functional renormalization group approach for su s u (2) spins is capable of treating diagonal and off-diagonal couplings of generic spin–valley exchange Hamiltonians in the self-conjugate representation of the su s u (4) algebra. To achieve proper numerical efficiency, we derive a number of symmetry constraints on the flow equations that significantly limit the number of ordinary differential equations to be solved. As an example system, we investigate a diagonal SU(2) spin spin ⊗ ⊗ U(1) valley valley model on the triangular lattice which exhibits a rich phase diagram of spin and valley ordered phases. Graphic Abstract
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.
The Kitaev model, which hosts a quantum spin liquid (QSL) in the ground state, was originally defined on a two-dimensional honeycomb lattice, but can be straightforwardly extended to any tricoordinate lattices in any spatial dimensions. In particular, the three-dimensional (3D) extensions are of interest as a realization of 3D QSLs, and some materials like β-Li2IrO3, γ-Li2IrO3, and β-ZnIrO3 were proposed for the candidates. However, the phase diagrams of the models for those candidates have not been fully elucidated, mainly due to the limitation of numerical methods for 3D frustrated quantum spin systems. Here we study the Kitaev-Heisenberg model defined on a 3D hyperhoneycomb lattice, by using the pseudofermion functional renormalization group method. We show that the ground-state phase diagram contains the QSL phases in the vicinities of the pristine ferromagnetic and antiferromagnetic Kitaev models, in addition to four magnetically ordered phases, similar to the two-dimensional honeycomb case. Our results respect the four-sublattice symmetry inherent in the model, which was violated in the previous study. Moreover, we also show how the phase diagram changes with the anisotropy in the interactions. The results provide a reference for the search of the hyperhoneycomb Kitaev materials.