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.
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 functional renormalization group (FRG) approach for spin models relying on a pseudofermionic description has proven to be a powerful technique in simulating ground-state properties of strongly frustrated magnetic lattices. A drawback of the FRG framework is that it is formulated in the imaginary-time Matsubara formalism and thus only able to access static correlations, a limitation shared with most other many-body approaches. A description of the dynamical properties of magnetic systems is the key to bridging the gap between theory and neutron-scattering spectra. We take the decisive step of expanding the scope of pseudofermion FRG to the Keldysh formalism, which, while originally developed to address nonequilibrium phenomena, enables a direct calculation of the equilibrium dynamical spin structure factors on generic lattices in arbitrary dimensions. We identify the principal features characterizing the low-energy spectra of exemplary zero-, one- and two-dimensional spin-1/2 Heisenberg models as well as the Kitaev honeycomb model while identifying current limitations of the method that have to be improved upon.
Geometrical frustration in quantum magnets provides a fertile setting for unconventional phases of matter, including quantum spin liquids (QSLs). The trellis lattice, with its complex site arrangements and edge-sharing triangular motifs, presents a promising platform for such physics. In this work, we undertake a comprehensive classification of all fully symmetric QSLs on the trellis lattice using the projective symmetry group approach within the Abrikosov-fermion representation. We find 7 U(1) and 25 Z 2 short-ranged and analyze the phase diagram in the mean-field parameter space, uncovering both gapped and Dirac QSLs as well as a semi-Dirac spin liquid that emerges at the level of projective symmetry group classification and mean-field band structure, in which the spinon dispersion is linear along one momentum direction but quadratic along the orthogonal one. We demonstrate that such dispersions can occur only at high-symmetry points in the Brillouin zone with C 2 v little groups and analyze their characteristic correlation signatures. Moreover, by optimizing over all symmetry-allowed mean-field states, we map out a phase diagram—featuring six distinct phases—of the nearest-neighbor Heisenberg Hamiltonian on the trellis lattice. Among these, we find four quasi-one-dimensional QSL phases, one dimer phase, and one Dirac QSL phase. Going beyond mean field, we also assess equal-time and dynamical spin structure factors of these phases using density-matrix renormalization group and Keldysh pseudofermion functional renormalization group calculations and compare qualitative momentum-space features of these spectra with those obtained at the mean-field level. Finally, we identify four cuprate and vanadate compounds as promising experimental realizations and provide spectroscopic predictions, based on first-principles Hamiltonians, as a guide for future neutron-scattering studies on these materials.
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.
In this thesis, we will concentrate on the numerical study of classical and quantum frustrated magnets. If we focus on the full set of problems which are contained in the class of quantum spin systems, we see that the techniques which can deal with them are plenty, but that the presence of frustration introduces several methodological hindrances. Whether it is the presence of a sign problem, or the extensive resources needed to perform the computation, many of these methods are efficient only on a reduced subset of frustrated systems. We will employ the pseudofermion-Functional Renormalization Group (pfFRG) method as our method of choice to tackle two dimensional frustrated quantum magnets, all of them modeled via a Heisenberg Hamiltonian. We will extend the pfFRG formalism to include spin systems where the spin length is unrestricted, S≥1/2. With this extension we are able to study in detail what the effect of quantum fluctuations are on the system. From the extreme quantum limit (S=1/2) to the classical limit (S→∞), the large-S extension of pfFRG allows us to manipulate the strength of the quantum fluctuations and study in detail how the quantum to classical transition happens. We apply this method to the study of incommensurate phases in the Heisenberg honeycomb model. We map the phase diagram for different values of the spin length and analyze how this length, and subsequently quantum fluctuations, affect the phases found in the classical limit. Furthermore, we prove that in the classical limit, pfFRG reduces to the Luttinger-Tisza formalism. When quantum fluctuations can be neglected, we model our frustrated magnet as a classical spin system. While this limit can easily be obtained from the large-S extension of pfFRG, this methodology is constrained to the study of two point correlators at zero temperature. To collect information regarding the behavior of our classical magnet beyond the two point correlators we study classical spin systems via Monte Carlo simulations. In this case we show how the method can be applied to continuous spin systems with strong anisotropic interaction. Furthermore, we employ this technique to numerically study α-Li2IrO3, a material which exhibits an incommensurate ground state but for which a minimal model has not been determined. We study and compare the possible minimal models that have been proposed. We reduce the number of minimal models by showing that many of those that have been proposed do not reproduce the full set of experimental results. Furthermore, we predict the magnetic behavior (for those models which reproduce the experimental results) in the presence of an external magnetic field. With this study we obtain magnetization processes and propose experiments which can confirm if one of the studied models is the correct minimal model for α-Li2IrO3.
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.
Frustrated magnetism provides a rich platform for realizing exotic quantum phases in condensed matter systems. Among these, quantum spin liquids (QSLs) are a particularly intriguing class of states that exhibit no magnetic ordering and preserve all symmetries down to zero temperature. Just as conventional magnets can be thought of as spin crystals,QSLs represent their liquid analogs. This thesis explores two broad types of QSLs in two-dimensional magnets, with a focus on their dynamic properties and the nature of transitions from conventional magnetic states into QSL phases. QSLs are topological phases that host fractionalized excitations known as spinons. In these states, mobile spinons are coupled to an emergent gauge field, forming a highly entangled phase of matter. The first part of this thesis focuses on Dirac spin liquids (DSLs), a class of QSLs where spinons exhibit a gapless Dirac dispersion. To study the dynamicsof these spin liquids beyond leading-order mean-field theory, we develop a novel large-N fermionic limit of gapless spinons. In this limit, all gauge excitations are suppressed, and the leading-order corrections instead arise from order parameter fluctuations, which may condense and drive transitions into symmetry-broken phases from the parent DSL. Incorporating random phase approximation (RPA) corrections to the spin susceptibility, we find that sharp, spin-wave–like modes are recovered in the excitation spectrum. This methodology successfully captures characteristic features of the dynamics near phase transitions and in weakly symmetry-broken phases proximate to a DSL. We apply this approach to microscopic models, including the J1-J2 XXZ model on the triangular lattice and the J1-J3 XXZ model on the honeycomb lattice. These models closely represent a wide range of frustrated materials, such as triangular and honeycomb cobaltates and triangular ytterbium compounds. Our theoretical predictions show excellent agreement with experimentally measured inelastic neutron scattering spectra from these materials. Gapless spinons can also sometimes become gapped through chiral symmetry breaking, leading to the formation of a chiral spin liquid (CSL). The CSL is an archetypal bosonic realization of a fractional quantum Hall (FQH) state at half-filling, and has been proposed as a time-reversal–breaking ground state in a variety of frustrated spin models. The second part of this thesis addresses the intriguing possibility of realizing a CSL through the "melting" of a chiral spin crystal—a conventional magnet with non-coplanar ordering. Such a transition offers a physically appealing analogue to the melting of a Wigner crystal into a fermionic FQH liquid. Focusing on the kagome lattice, where chiral interactions are knownto stabilize a CSL ground state, we study the effects of additional long-range Heisenberg interactions and demonstrate a phase transition into a non-coplanar octahedral magnetic order. To describe the universal class of transition between a CSL and a non-coplanar anti-ferromagnetic order, we then develop a novel Chern-Simons–matter field theory framework, constrained by symmetry principles and symmetry fractionalization patterns. This thesis thus provides new insights into the dynamical signatures and phase transitions of quantum spin liquids, offering a bridge between theoretical models and experimental observations in frustrated magnetic systems.