Kitaev Quantum Spin Liquids

TL;DR

This review surveys Kitaev quantum spin liquids, focusing on how bond-directional Kitaev interactions yield fractionalized Majorana excitations and Ising topological order, and how $\alpha$-RuCl$_3$ provides a proximate realization. It outlines the Kitaev model, Jackeli–Khaliullin mechanism, and the role of non-Kitaev exchanges (J, $\Gamma$, $\Gamma'$) in driving zigzag order, along with evidence from Raman, INS, THz, IXS, NMR, and thermal transport. It discusses field- and pressure-tuned phases, including the field-induced quantum-disordered state and observed half-integer thermal Hall conductance, while highlighting debates on phonon vs Majorana contributions and disorder effects. It concludes with perspectives on material design, sample quality, and open theoretical questions toward achieving a bona fide Kitaev QSL.

Abstract

Quantum spin liquids (QSLs) represent exotic states of matter where quantum spins interact strongly yet evade long-range magnetic order down to absolute zero. Characterized by non-local quantum entanglement and resultant fractionalized excitations, QSLs have emerged as a frontier in condensed matter physics, bolstered by the recent identification of several candidate materials. This field holds profound implications for understanding strong correlations, topological order, and emergent phenomena in quantum materials. Among them, the Kitaev model, featuring bond-directional Ising interactions, provides a rare exactly solvable QSL example. Its ground state is a topological QSL, with spin degrees of freedom fractionalized into emergent Majorana fermions. Under an applied magnetic field, the Kitaev QSL transitions to a topologically non-trivial chiral spin liquid state with non-Abelian anyons, offering potential resources for topological quantum computation. The non-Abelian character of these anyons in the Kitaev QSL demonstrates a profound connection to certain topological superconductors and even-denominator fractional quantum Hall states. Since the theoretical prediction that the Kitaev model could manifest in spin-orbit-coupled materials such as honeycomb iridates and ruthenates, research has focused on identifying candidate compounds. In particular, experimental evidence suggests spin fractionalization and topological phenomena akin to the Kitaev model in the spin-orbit Mott insulator RuCl3. However, results and interpretations remain actively debated. This review begins with a brief review on QSLs in other systems, followed by a comprehensive survey of existing studies on Kitaev candidate materials, with a particular focus on RuCl3. Rather than offering conclusive remarks, our aim is to inspire future research by examining several key aspects of the current literature and perspectives.

Figures (33)

• Figure 1: Crystal structures of archetypal QSL candidates. (a) 1D chain, (b) 2D triangular, (c) 2D kagome, (d) 2D honeycomb, and (e) 3D pyrochlore lattices. The triangular, kagome, and pyrochlore lattices exhibit strong geometric frustration. We include the honeycomb lattice here, as Kitaev's bond-dependent interaction leads to spin frustration, even though it does not display geometrical frustration.
• Figure 2: The excitation spectrum of the 1D Heisenberg antiferromagnet CuSO$_4$·5D$_2$O revealed by INS measurements above 3D Néel temperature. The elementary excitations are spin-1/2 quasiparticles known as spinons, which are created exclusively in pairs. The observed excitation continuum comprises both two-spinon and four-spinon states. The experimental data are shown alongside exact theoretical calculations demonstrating the characteristic continuous spectra of these multi-spinon contributions mourigal2013fractional.
• Figure 3: (a) Structural schematic of a honeycomb lattice with crystallographic directions. The lattice comprises two interpenetrating triangular sublattices, where filled and open circles denote Ru atomic sites on distinct sublattices. Bond orientations are designated as $x$-, $y$-, and $z$-links, with corresponding Kitaev exchange interactions $K_x$, $K_y$, and $K_z$ acting along these bonds, respectively. (b) Spin orientations are depicted perpendicular to their respective exchange bonds, illustrating ferromagnetic Kitaev interactions. The system exhibits geometric frustration due to spin-orbit-coupled bond-directional interactions, where Ising-like exchange couplings are constrained by bond orientation. This configuration prevents simultaneous energy minimization across all exchange bonds, a fundamental frustration that persists in both quantum and classical spin systems. (c) Schematic representation of the honeycomb Kitaev model with bond-directional exchange couplings $K_x$, $K_y$, and $K_z$. The model's exact solution is achieved through fractionalization into four Majorana fermions: three localized species (depicted by red, blue, and green circles) and one itinerant species (represented by open and gray circles). (d) The model simplification via gauge field transformation, where itinerant Majorana fermions combine to form a static $Z_2$ gauge field ($u_{ij}$), leaving a system of non-interacting itinerant Majorana fermions (open and gray circles).
• Figure 4: (a) Ground state. $w_p=1$ for all hexagons. (b) A vison excitation ($w_p=-1$). The vison is a $Z_2$ magnetic flux excitation that occurs when the $Z_2$ gauge field ($u_{ij}$) is flipped from +1 to -1 on a plaquette. It's a gapped, localized excitation that can be understood as a violation of the plaquette operator's ground state condition.
• Figure 5: (a) In the absence of an external magnetic field, itinerant Majorana fermions on a honeycomb lattice manifest Dirac cone dispersions at the $K$- and $K'$- points (corners) of the Brillouin zone, preserving particle-hole symmetry. (b)(c) Upon application of a magnetic field, these Dirac points generally develop a topologically non-trivial gap, though this response shows marked directional dependence. When $H$ is applied within the 2D plane along the antibonding (zigzag) direction of the honeycomb lattice ($a$-axis in the $C2/m$ notation), the system develops an energy gap. The chirality of the resulting state depends critically on the field orientation: for $\bm{H} || a$, clockwise chiral thermal edge currents are induced within the honeycomb plane (a), while for $\bm{H} || -a$ counterclockwise circulation is generated (b). (d) The system exhibits markedly different behavior when $H$ is aligned parallel to the bond (armchair) direction ($b$-axis). In this configuration, the Dirac cone structure remains intact, and no chiral edge currents emerge. These demonstrate the profound influence of field orientation on the system's topological properties.