Recent experiments show that the ground state of some layered materials with localized moments is in close proximity to the Kitaev spin liquid, calling for a proper model to describe the measurements. The Kitaev-Hubbard (KHu) model is the minimal model that captures the essential ingredients of these systems; it yields the Kitaev-Heisenberg spin model at the strong coupling limit and contains the charge fluctuations present in these materials as well. Despite its relevance, the phase diagram of the KHu model has not been rigorously revealed yet. In this work, we study the full phase diagram of the KHu model using the ${\mathbb{Z}}_{2}$ slave-spin mean-field theory as well as the auxiliary field quantum Monte Carlo on rather large systems and at low temperatures. The Mott transition is signaled by a vanishing quasiparticle weight evaluated using the slave-spin construction. Moreover, we demonstrate that there are multiple magnetic phase transitions within the Mott phase including magnetically ordered phases and most notably a quantum spin liquid phase for $1.0\ensuremath{\lesssim}{t}^{\ensuremath{'}}/t\ensuremath{\lesssim}1.11$ at $U/t=5$.
Recent experiments show that the ground state of some layered materials with localized moments is in close proximity to the Kitaev spin liquid, calling for a proper model to describe the measurements. The Kitaev-Hubbard (KHu) model is the minimal model that captures the essential ingredients of these systems; it yields the Kitaev-Heisenberg spin model at the strong coupling limit and contains the charge fluctuations present in these materials as well. Despite its relevance, the phase diagram of the KHu model has not been rigorously revealed yet. In this work, we study the full phase diagram of the KHu model using the Z2 slave-spin mean-field theory as well as the auxiliary field quantum Monte Carlo on rather large systems and at low temperatures. The Mott transition is signaled by a vanishing quasiparticle weight evaluated using the slave-spin construction. Moreover, we demonstrate that there are multiple magnetic phase transitions within the Mott phase including magnetically ordered phases and most notably a quantum spin liquid phase for 1.0\ensuremath≲t\ensuremath′/t\ensuremath≲1.11 at U/t=5.
We explore the nature of the quantum phase transition between a magnetically ordered state with collinear spin pattern and a gapless Z2 spin liquid in the Heisenberg-Kitaev model. We construct a slave particle mean-field theory for the Heisenberg-Kitaev model in terms of complex fermionic spinons. It is shown that this theory, formulated in the appropriate basis, is capable of describing the Kitaev spin liquid as well as the transition between the gapless Z2 spin liquid and the so-called stripy antiferromagnet. Within our mean-field theory, we find a discontinuous transition from the Z2 spin liquid to the stripy antiferromagnet. We argue that subtle spinon confinement effects, associated with the instability of gapped U(1) spin liquid in two spatial dimensions, play an important role at this transition. The possibility of an exotic continuous transition is briefly addressed.
The Kitaev model, whose ground state is a quantum spin liquid (QSL), was originally conceived for spin S=1/2 moments on a honeycomb lattice. In recent years, the model has been extended to higher S from both theoretical and experimental interests, but the stability of the QSL ground state has not been systematically clarified for general S, especially in the presence of other additional interactions, which inevitably exist in candidate materials. Here we study the spin-S Kitaev-Heisenberg models by using an extension of the pseudofermion functional renormalization group method to general S. We show that, similar to the S=1/2 case, the phase diagram for higher S contains the QSL phases in the vicinities of the pristine ferromagnetic and antiferromagnetic Kitaev models, in addition to four magnetically ordered phases. We find, however, that the QSL phases shrink rapidly with increasing S, becoming vanishingly narrow for $S\ensuremath{\ge}2$, whereas the phase boundaries between the ordered phases remain almost intact. Our results provide a reference for the search for higher-S Kitaev materials.
The Kitaev spin liquid, stabilized as the ground state of the Kitaev honeycomb model, is a paradigmatic example of a topological Z2 quantum spin liquid. The fate of the Kitaev spin liquid in presence of an external magnetic field is at present, a subject of intense scrutiny due to recent experimental signatures pointing to a Z2 topological phase in certain materials, as well as theoretical studies predicting the emergence of a quantum spin liquid phase of debated nature. In this work, we employ hierarchical mean-field theory, a method based on the use of clusters preserving relevant symmetries and short-range quantum correlations, to investigate the quantum phase diagram of the antiferromagnetic Kitaev's model in a [111] magnetic field. By using clusters of 24 sites, we predict that the Kitaev spin liquid transits through two intermediate phases, characterized by emerging stripe and chiral orders, respectively, before entering the trivial partially polarized phase, differing from previous studies. We assess our results by performing exact diagonalization and computing the scaling of different observables, including the many-body Chern number and other topological quantities, thus establishing hierarchical mean-field theory as a method to study models of frustrated quantum magnetism potentially hosting topological quantum spin liquids.
The long-sought quantum spin liquid is a quantum-entangled magnetic state leading to the fractionalization of spin degrees of freedom. Quasiparticles emergent from the fractionalization affect not only the ground-state properties but also thermodynamic behavior in a peculiar manner. Here we investigate how the spin dynamics evolves from the high-temperature paramagnet to the quantum spin-liquid ground state, for the Kitaev spin model describing the fractionalization into itinerant matter fermions and localized Z2 fluxes. Beyond the previous studies [J. Yoshitake, J. Nasu, and Y. Motome, Phys. Rev. Lett. 117, 157203 (2016); J. Yoshitake, J. Nasu, Y. Kato, and Y. Motome, Phys. Rev. B 96, 024438 (2017)], in which the mean-field nature of the cluster dynamical mean-field theory prevented us from studying low-temperature properties, we develop a numerical technique by applying the continuous-time quantum Monte Carlo (CTQMC) method to statistical samples generated by the quantum Monte Carlo (QMC) method in a Majorana fermion representation. This QMC+CTQMC method is fully unbiased and enables us to investigate the low-temperature spin dynamics dominated by thermally excited Z2 fluxes, including the unconventional phase transition caused by Z2 flux loops in three dimensions, which was unreachable by the previous methods. We apply this technique to the Kitaev model in both two and three dimensions. Our results clearly distinguish two cases: while the dynamics changes smoothly through the crossover in the two-dimensional honeycomb case, it exhibits singular behaviors at the phase transition in the three-dimensional hyperhoneycomb case. We show that the low-temperature spin dynamics is a sensitive probe for thermally fluctuating Z2 fluxes that behave very differently between two and three dimensions.
In the field of frustrated magnetism, Kitaev models provide prototypical examples of exactly solvable quantum spin liquids, in which the spin degrees of freedom fractionalize into Majorana fermions coupled to an emergent Z2 gauge field. While the ground states of these models are already well understood as exhibiting different kinds of Majorana (semi)metals, it has still been an open task to achieve a detailed understanding of the thermodynamics, and, namely, the low-temperature ordering of the gauge field into different Z2 flux configurations. In this thesis, we investigate the thermodynamics of Kitaev systems in two and three spatial dimensions with sign-problem-free quantum Monte Carlo simulations. In a first study of elementary 3D Kitaev models, we verify that the ground state Z2 flux sectors of these systems are entirely determined by their elementary plaquette length – a result which shows the validity of Lieb’s theorem for lattice geometries which lack the geometric requirements for its proof. We closely investigate the low-temperature phase transition associated with gauge-ordering, which is a particular realization of an inverted Ising phase transition, as it occurs in general lattice Z2 gauge theories. Our results corroborate the understanding of this transition as separating different regimes in terms of vison-loop excitations, by showing a clear correlation between the critical temperature and the vison gap. We also introduce the concept of “gauge frustration”, which, for a particular 3D Kitaev model, leads to the suppression of the phase transition and a more complex physical behavior at low temperatures. A second quantum Monte Carlo study focuses on a generalized 2D Kitaev model on a five-coordinated lattice system. Here, the gauge ordering is accompanied by the spontaneous breaking of time-reversal symmetry, a scenario which not only allows for a variety of topological ground states, but also for the occurrence of a phase transition in two spatial dimensions. We show that this model exhibits such a transition at a particularly high temperature scale. Moreover, the system possesses a number of phases where the Z2 fluxes are only partially ordered.
We present thermodynamic phase diagrams showing magnetic analog of ``three states of matter,'' namely, spin liquid, paramagnetic, and magnetically ordered phases, obtained by unbiased quantum Monte Carlo simulations. Our simulations are carried out for Kitaev's toric codes in two and three dimensions, i.e., the simplest realizations of gapped topological Z2 spin liquids, extended by a nearest-neighbor ferromagnetic Ising coupling. We find that the ordered phase borders on the spin liquid by a discontinuous transition line in three dimensions, while it grows continuously from the quantum critical point in two dimensions. In both cases, our results elucidate peculiar proximity effects to the nearby spin liquids in the high-temperature paramagnetic phase, even when the ground state is magnetically ordered. The thorough study of magnetic three states of matter is achieved by introducing the ``fictitious vertex'' method into the directed loop algorithm. This provides a generic prescription to simulate models with off-diagonal multispin interactions, in which the conventional scheme may suffer from intrinsic ergodicity breakdown as in the present case.