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arXiv:2401.14525 · Physical review. B./Physical review. B

Thermodynamics of vison crystals in an anisotropic quantum spin liquid

Ritwika Majumder, Onur Erten, Anamitra Mukherjee

Published Sep 30, 2024• 5 citations
arXivHTMLPDF

Abstract

Using unbiased Monte Carlo simulations and variational analysis, we present the ground state and finite-temperature phase diagrams of an exactly solvable spin-orbital model with Kitaev-type interactions on a square lattice. We show that an array of gapped and gapless vison crystals, characterized by the periodic arrangement of ${\mathbb{Z}}_{2}$ flux excitations, can be stabilized as a function of external magnetic field and exchange anisotropy. In particular, we discover a variety of ``quarter phases'' wherein different 16-site periodic patterns emerge, with only a quarter of the fluxes adopting zero-flux configurations. In contrast, the rest remain in $\ensuremath{\pi}$-flux configurations. Vison crystals break translational symmetry and undergo finite-temperature phase transitions. We investigate the finite-temperature properties of these phases and report the corresponding critical and crossover temperatures. Our results reveal an array of novel phases in exactly solvable extensions of the Kitaev model, wherein local and topological orders can coexist.

DOI
10.1103/physrevb.110.125155

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Using unbiased Monte Carlo simulations and variational analysis, we present the ground state and finite-temperature phase diagrams of an exactly solvable spin-orbital model with Kitaev-type interactions on a square lattice. We show that an array of gapped and gapless vison crystals, characterized by the periodic arrangement of Z2{\mathbb{Z}}_{2}Z2​ flux excitations, can be stabilized as a function of external magnetic field and exchange anisotropy. In particular, we discover a variety of ``quarter phases'' wherein different 16-site periodic patterns emerge, with only a quarter of the fluxes adopting zero-flux configurations. In contrast, the rest remain in \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux configurations. Vison crystals break translational symmetry and undergo finite-temperature phase transitions. We investigate the finite-temperature properties of these phases and report the corresponding critical and crossover temperatures. Our results reveal an array of novel phases in exactly solvable extensions of the Kitaev model, wherein local and topological orders can coexist.
We study the phase diagram of the Yao-Lee model with Kitaev-type spin-orbital interactions in the presence of Dzyaloshinskii-Moriya interactions and external magnetic fields. Unlike the Kitaev model, the Yao-Lee model can still be solved exactly under these perturbations due to the enlarged local Hilbert space. Through a variational analysis, we obtain a rich ground-state phase diagram that consists of a variety of vison crystals with periodic arrangements of background Z2{\mathbb{Z}}_{2}Z2​ flux (i.e., visons). With an out-of-plane magnetic field, these phases have gapped bulk and chiral edge states, characterized by a Chern number $\ensuremath{\nu}$ and an associated chiral central charge ${c}_{\ensuremath{-}}=\ensuremath{\nu}/2$ of edge states. We also find helical Majorana edge states that are protected by magnetic mirror symmetry. For the bilayer systems, we find that interlayer coupling can also stabilize new topological phases. Our results spotlight the tunability and the accompanying rich physics in exactly solvable spin-orbital generalizations of the Kitaev model.
We develop an exactly solvable model with Kitaev-type interactions and study its phase diagram on the dual lattice of the quasicrystalline Ammann-Beenker lattice. Our construction is based on the $\mathrm{\ensuremath{\Gamma}}$-matrix generalization of the Kitaev model and utilizes the cut-and-project correspondence between the four-dimensional simple cubic lattice and the Ammann-Beenker lattice to designate four types of bonds. We obtain a rich phase diagram with gapped (chiral and abelian) and gapless spin liquid phases via Monte Carlo simulations and variational analysis. We show that the ground state can be further tuned by the inclusion of an onsite term that selects 21 different vison configurations while maintaining the integrability of the model. Our results highlight the rich physics at the intersection of quasicrystals and quantum magnetism.
Building on the recent advancements on moir\'e superlattices, we propose an exactly solvable model with Kitaev-type interactions on a bilayer honeycomb lattice for both AA stacking and moir\'e superlattices. Using Monte Carlo simulations and variational analysis, we uncover a rich variety of phases where the intra and interlayer Z2{\mathbb{Z}}_{2}Z2​ fluxes (visons) are arranged in a periodic fashion in the ground state, tuned by interlayer coupling and out-of-plane external magnetic field. We further extend our model to moir\'e superlattices at various commensurate twist angles around two distinct twist centers represented by C3z{C}_{3z}C3z​ and C6z{C}_{6z}C6z​ of the honeycomb lattice. Our simulations reveal generalized arrangements of plaquette values that correlate with the AA or AB stacking regions across the moir\'e unit cell. Moreover, we find that, depending on the twist angle, twist center and interlayer coupling, moir\'e superlattices exhibit to a variety of gapped and gapless spin liquid phases and can also host corner and edge modes. Our results highlight the rich physics in bilayer and twisted bilayer models of exactly solvable quantum spin liquids.
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.
In the field of frustrated magnetism, Kitaev models provide a unique framework to study the phenomena of spin fractionalization and emergent lattice gauge theories in two and three spatial dimensions. Their ground states are quantum spin liquids, which can typically be described in terms of a Majorana band structure and an ordering of the underlying Z2{\mathbb{Z}}_{2}Z2​ gauge structure. Here we provide a comprehensive classification of the ``gauge physics'' of a family of elementary three-dimensional Kitaev models, discussing how their thermodynamics and ground state order depends on the underlying lattice geometry. Using large-scale, sign-free quantum Monte Carlo simulations we show that the ground-state gauge order can generally be understood in terms of the length of elementary plaquettes---a result which extends the applicability of Lieb's theorem to lattice geometries beyond its original scope. At finite temperatures, the proliferation of (gapped) vison excitations destroys the gauge order at a critical temperature scale, which we show to correlate with the size of vison gap for the family of three-dimensional Kitaev models. We also discuss two notable exceptions where the lattice structure gives rise to ``gauge frustration'' or intertwines the gauge ordering with time-reversal symmetry breaking. In a more general context, the thermodynamic gauge transitions in such 3D Kitaev models are one of the most natural settings for phase transitions beyond the standard Landau-Ginzburg-Wilson paradigm.
We introduce an extension of the Kitaev honeycomb model by including four-spin interactions that preserve the local gauge structure and, hence, the integrability of the original model. The extended model has a rich phase diagram containing five distinct vison crystals, as well as a symmetric π-flux spin liquid with a Fermi surface of Majorana fermions and a sequence of Lifshitz transitions. We discuss possible experimental signatures and, in particular, present finite-temperature Monte Carlo calculations of the specific heat and the static vison structure factor. We argue that our extended model emerges naturally from generic perturbations to the Kitaev honeycomb model.