Topological Spin Liquid Research Articles

Competing antiferromagnetic-ferromagnetic states in a d7 Kitaev honeycomb magnet.

The Kitaev model is a rare example of an analytically solvable and physically instantiable Hamiltonian yielding a topological quantum spin liquid ground state. Here we report signatures of Kitaev spin liquid physics in the honeycomb magnet Li3Co2SbO6, built of high-spin (Co2+) ions, in contrast to the more typical low-spin electron configurations in the presence of large spin-orbit coupling. Neutron powder diffraction measurements, heat capacity, and magnetization studies support the development of a long-range antiferromagnetic order space group of , below at . The magnetic entropy recovered between and 50 K is estimated to be , in good agreement with the value expected for systems close to a Kitaev quantum spin liquid state. The temperature-dependent magnetic order parameter demonstrates a value of 0.19(3), consistent with XY anisotropy and in-plane ordering, with Ising-like interactions between layers. Further, we observe a spin-flop-driven crossover to ferromagnetic order with space group of under an applied magnetic field of at . Magnetic structure analysis demonstrates these magnetic states are competing at finite applied magnetic fields even below the spin-flop transition. Both the compass model, a quantitative comparison of the specific heat of Li3Co2SbO6, and related honeycomb cobaltates to the anisotropic Kitaev model further support proximity to a Kitaev spin liquid state. This material demonstrates the rich playground of high-spin systems for spin liquid candidates and complements known Ir- and Ru-based materials.

Quantum trimer models and topological SU(3) spin liquids on the kagome lattice

We construct and study quantum trimer models and resonating SU(3)-singlet models on the kagome lattice, which generalize quantum dimer models and the Resonating Valence Bond wavefunctions to a trimer and SU(3) setting. We demonstrate that these models carry a Z_3 symmetry which originates in the structure of trimers and the SU(3) representation theory, and which becomes the only symmetry under renormalization. Based on this, we construct simple and exact parent Hamiltonians for the model which exhibit a topological 9-fold degenerate ground space. A combination of analytical reasoning and numerical analysis reveals that the quantum order ultimately displayed by the model depends on the relative weight assigned to different types of trimers -- it can display either Z_3 topological order or form a symmetry-broken trimer crystal, and in addition possesses a point with an enhanced U(1) symmetry and critical behavior. Our results accordingly hold for the SU(3) model, where the two natural choices for trimer weights give rise to either a topological spin liquid or a system with symmetry-broken order, respectively. Our work thus demonstrates the suitability of resonating trimer and SU(3)-singlet ansatzes to model SU(3) topological spin liquids on the kagome lattice.

Quantum spin liquid phases in the bilinear-biquadratic two-SU(4)-fermion Hamiltonian on the square lattice

We consider the phase diagram of the most general SU(4)-symmetric two-site Hamiltonian for a system of two fermions per site (i.e., self-conjugate $\mathbf{6}$ representation) on the square lattice. It is known that this model hosts magnetic phases breaking SU(4) symmetry and quantum disordered dimerlike phases breaking lattice translation symmetry. Motivated by a previous work [O. Gauth\'e, S. Capponi, and D. Poilblanc, Phys. Rev. B 99, 241112(R) (2019)], we investigate the possibility of the existence of SU(4) quantum spin liquid phases in this model, using SU(4)-symmetric projected entangled pair states (PEPS) of small bond dimensions, which can be classified according to point group and charge (C) symmetries. Among several (disconnected) families of SU(4)-symmetric PEPS, breaking or not C-symmetry, we identify critical or topological spin liquids which may be stable in some regions of the phase diagram. These results are confronted to exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations.

Gapped Z2 spin liquid in the breathing kagome Heisenberg antiferromagnet

We investigate the spin-1/2 Heisenberg antiferromagnet on the kagome lattice with breathing anisotropy (i.e. with weak and strong triangular units), constructing an improved simplex Resonating Valence Bond (RVB) ansatz by successive applications (up to three times) of local quantum gates which implement a filtering operation on the bare nearest-neighbor RVB state. The resulting Projected Entangled Pair State involves a small number of variational parameters (only one at each level of application) and preserves full lattice and spin-rotation symmetries. Despite its simple analytic form, the simplex RVB provides very good variational energies at strong and even intermediate breathing anisotropy. We show that it carries $Z_2$ topological order which does not fade away under the first few applications of the quantum gates, suggesting that the RVB topological spin liquid becomes a competing ground state candidate for the kagome antiferromagnet at large breathing anisotropy.

Magnon bound states versus anyonic Majorana excitations in the Kitaev honeycomb magnet \u03b1-RuCl3

The pure Kitaev honeycomb model harbors a quantum spin liquid in zero magnetic fields, while applying finite magnetic fields induces a topological spin liquid with non-Abelian anyonic excitations. This latter phase has been much sought after in Kitaev candidate materials, such as α-RuCl3. Currently, two competing scenarios exist for the intermediate field phase of this compound (B = 7 − 10 T), based on experimental as well as theoretical results: (i) conventional multiparticle magnetic excitations of integer quantum number vs. (ii) Majorana fermionic excitations of possibly non-Abelian nature with a fractional quantum number. To discriminate between these scenarios a detailed investigation of excitations over a wide field-temperature phase diagram is essential. Here, we present Raman spectroscopic data revealing low-energy quasiparticles emerging out of a continuum of fractionalized excitations at intermediate fields, which are contrasted by conventional spin-wave excitations. The temperature evolution of these quasiparticles suggests the formation of bound states out of fractionalized excitations.

Topological Z2 resonating-valence-bond quantum spin liquid on the ruby lattice

We construct a short-range resonating valence-bond state (RVB) on the ruby lattice, using projected entangled-pair states (PEPS) with bond dimension $D=3$. By introducing non-local moves to the dimer patterns on the torus, we distinguish four distinct sectors in the space of dimer coverings, which is a signature of the topological nature of the RVB wave function. Furthermore, by calculating the reduced density matrix of a bipartition of the RVB state on an infinite cylinder and exploring its entanglement entropy, we confirm the topological nature of the RVB wave function by obtaining non-zero topological contribution, $\gamma=-\rm{ln}\ 2$, consistent with that of a $\mathbb{Z}_2$ topological quantum spin liquid. We also calculate the ground-state energy of the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the ruby lattice and compare it with the RVB energy. Finally, we construct a quantum-dimer model for the ruby lattice and discuss it as a possible parent Hamiltonian for the RVB wave function.

Coherent propagation of quasiparticles in topological spin liquids at finite temperature

The appearance of quasiparticle excitations with fractional statistics is a remarkable defining trait of topologically ordered systems. In this work, we investigate the experimentally relevant finite temperature regime in which one species of quasiparticle acts as a stochastic background for another, more energetically costly, species that hops coherently across the lattice. The nontrivial statistical angle between the two species leads to interference effects that we study using a combination of numerical and analytical tools. In the limit of self-retracing paths, we are able to use a Bethe lattice approximation to construct exact analytical expressions for the time evolution of the site-resolved density profile of a spinon initially confined to a single site. Our results help us to understand the temperature-dependent crossover from ballistic to quantum (sub-)diffusive behaviour as a consequence of destructive interference between lattice walks. The subdiffusive behaviour is most pronounced in the case of semionic mutual statistics, and it may be ascribed to the localised nature of the effective tight-binding description, an effect that is not captured by the Bethe lattice mapping. In addition to quantum spin liquids, our results are directly applicable to the dynamics of isolated holes in the large-$U$ limit of the Hubbard model, relevant to ultracold atomic experiments. A recent proposal to implement $\mathbb{Z}_2$ topologically ordered Hamiltonians using quantum annealers provides a further exciting avenue to test our results.

Topological nematic spin liquid on the square kagome lattice

The ground state of the spin$-1/2$ kagome antiferromagnet remains uncertain despite decades of active research. Here we step aside from this debated question to address the ground-state nature of a related, and potentially just as rich, system made of corner-sharing triangles: the square-kagome lattice (SKL). Our work is motivated by the recent synthesis of a distorted SKL compound mentioned in [Morita & Tohyama, J. Phys. Soc. Japan 87, 043704 (2018)]. We have studied its spin$-1/2$ $J_{1}$-$J_{2}$ phase diagram with an unrestricted Schwinger boson mean-field theory (SBMFT). We show that, in addition of agreeing with previous studies, three original phases appear: two incommensurate orders and a topological quantum spin liquid with weak nematicity. The topological order is characterized by fluxes on specific gauge-invariant quantities and the phase is stable under anisotropic perturbations relevant for experiments. Finally, we provide dynamical structure factors of the reported phases that could be observed in inelastic neutron scattering.

Quantum dimer models emerging from large-spin ultracold atoms

We propose an experimental protocol for using cold atoms to create and probe quantum dimer models, thereby exploring the Pauling-Anderson vision of a macroscopic collection of resonating bonds. This process can allow the study of exotic crystalline phases, fractionalization, topological spin liquids, and the relationship between resonating dimers and superconductivity subjects which have been challenging to address in solid-state experiments. Our key technical development is considering the action of an off-resonant photoassociation laser on large spin atoms localized at the sites of a deep optical lattice. The resulting superexchange interaction favors nearest-neighbor singlets. We derive an effective Hamiltonian in terms of these dimer degrees of freedom, finding that it is similar to well-known quantum dimer models, which boast a rich variety of valence bond crystal and spin liquid phases. We numerically study the ground state, explain how to tune the parameters, and develop a protocol to directly measure the dimers and their resonating patterns.

Concept and realization of Kitaev quantum spin liquids

A decade ago, Alexei Kitaev proposed an exactly solvable $S$ = 1/2 model on a two-dimensional honeycomb lattice, where the spins fractionalize into Majorana fermions and form a topological quantum spin liquid (QSL) in the ground state. It was soon recognized that a family of complex iridium oxides, as well as ruthenium chloride, with honeycomb structure are magnetic insulators and accommodate essential ingredients of the Kitaev model, due to the interplay of electron correlation and spin-orbit coupling. This initiated a race to materialize the Kitaev QSL and to capture the signature of Majorana fermions. In this review, we provide a wide perspective of this rapidly growing field, including theory, materials and experiment. We first summarize the theoretical background of the Kitaev QSL ground state and its materialization using spin-orbital-entangled $J_{\rm eff}$ = 1/2 moments. This is followed by an overview of candidate materials and their magnetic properties, including Na$_2$IrO$_3$, $\alpha$, $\beta$, $\gamma$-Li$_2$IrO$_3$,$\alpha$-RuCl$_3$ and H$_3$LiIr$_2$O$_6$. Finally, we review the latest exciting progress in the search for the Kitaev QSL. In particular, H$_3$LiIr$_2$O$_6$ and $\alpha$-RuCl$_3$ in applied magnetic field show signatures of the QSL state, and $\alpha$-RuCl$_3$ has unusual magnetic excitations and thermal transport properties that are consistent with spin fractionalization.

Large thermal Hall effect in α−RuCl3 : Evidence for heat transport by Kitaev-Heisenberg paramagnons

The honeycomb Kitaev model in a magnetic field is a source of a topological quantum spin liquid with Majorana fermions and gauge flux excitations as fractional quasiparticles. We present experimental results for the thermal Hall effect of the material $\alpha$-RuCl$_{3}$ which recently emerged as a prime candidate for realizing such physics. At temperatures above long-range magnetic ordering $T\gtrsim T_N\approx8$ K, we observe with an applied magnetic field $B$ perpendicular to the honeycomb layers a sizeable positive transversal heat conductivity $\kappa_{xy}$ which increases linearly with $B$. Upon raising the temperature, $\kappa_{xy}(T)$ increases strongly, exhibits a broad maximum at around 30 K, and eventually becomes negligible at $T\gtrsim 125$ K. Remarkably, the longitudinal heat conductivity $\kappa_{xx}(T)$ exhibits a sizeable positive thermal magnetoresistance effect. Thus, our findings provide clear-cut evidence for longitudinal and transverse magnetic heat transport and underpin the unconventional nature of the quasiparticles in the paramagnetic phase of $\alpha$-RuCl$_{3}$.

Gapped SU(3) spin liquid with Z3 topological order

We construct a topological spin liquid (TSL) model on the kagome lattice, with SU(3) symmetry with the fundamental representation at each lattice site, based on Projected Entangled Pair States (PEPS). Using the PEPS framework, we can adiabatically connect the model to a fixed point model (analogous to the dimer model for Resonating Valence Bond states) which we prove to be locally equivalent to a $Z_3$ quantum double model. Numerical study of the interpolation reveals no sign of a phase transition or long-range order, characterizing the model conclusively as a gapped TSL. We further study the entanglement spectrum of the model and find that while it is gapped, it exhibits branches with vastly different velocities, with the slow branch matching the counting of a chiral $SU(3)_1$ CFT, suggesting that it can be deformed to a model with chiral $SU(3)_1$ entanglement spectrum.

Bilayer Kitaev models: Phase diagrams and novel phases

Kitaev's honeycomb-lattice spin-$1/2$ model has become a paradigmatic example for $\mathbb{Z}_2$ quantum spin liquids, both gapped and gapless. Here we study the fate of these spin-liquid phases in differently stacked bilayer versions of the Kitaev model. Increasing the ratio between the inter-layer Heisenberg coupling $J_\perp$ and the intra-layer Kitaev couplings $K^{x,y,z}$ destroys the topological spin liquid in favor of a paramagnetic dimer phase. We study phase diagrams as a function of $J_\perp/K$ and Kitaev coupling anisotropies using Majorana-fermion mean-field theory, and we employ different expansion techniques in the limits of small and large $J_\perp/K$. For strongly anisotropic Kitaev couplings, we derive effective models for the different layer stackings which we use to discuss the quantum phase transition out of the Kitaev phase. We find that the phase diagrams depend sensitively on the nature of the stacking and anisotropy strength. While in some stackings and at strong anisotropies we find a single transition between the Kitaev and dimer phases, other stackings are more involved: Most importantly, we prove the existence of two novel macro-spin phases which can be understood in terms of Ising chains which can be either coupled ferromagnetically or remain degenerate, thus realizing a classical spin liquid. In addition, our results suggest the existence of a flux phase with spontaneous inter-layer coherence.

Strain- and pressure-tuned magnetic interactions in honeycomb Kitaev materials

A range of honeycomb-lattice compounds has been proposed and investigated in the search for a topological Kitaev spin liquid. However, sizable Heisenberg interactions and additional symmetry-allowed exchange anisotropies in the magnetic Hamiltonian of these potential Kitaev materials push them away from the pure Kitaev spin-liquid state. Particularly the Kitaev-to-Heisenberg coupling ratio is essential in this respect. With the help of advanced quantum-chemistry methods, we explore how the magnetic coupling ratios depend on pressure in several honeycomb compounds (Na$_2$IrO$_3$, $\beta$-Li$_2$IrO$_3$, and $\alpha$-RuCl$_3$). We find that the Heisenberg and Kitaev terms are affected differently by uniform pressure or strain: the Kitaev component increases more rapidly than the Heisenberg counterpart. This provides a scenario where applying pressure or strain can stabilize a spin liquid in such materials.

Robust non-Abelian spin liquid and a possible intermediate phase in the antiferromagnetic Kitaev model with magnetic field

We investigate the non-Abelian topological chiral spin liquid phase in the two-dimensional (2D) Kitaev honeycomb model subject to a magnetic field. By combining density matrix renormalization group (DMRG) and exact diagonalization (ED) we study the energy spectra, entanglement, topological degeneracy, and expectation values of Wilson loop operators, allowing for robust characterization. While the ferromagnetic (FM) Kitaev spin liquid is already destroyed by a weak magnetic field with Zeeman energy $H_*^\text{FM} \approx 0.02$, the antiferromagnetic (AFM) spin liquid remains robust up to a magnetic field that is an order of magnitude larger, $H_*^\text{AFM} \approx 0.2$. Interestingly, for larger fields $H_*^\text{AFM} < H < H_{**}^\text{AFM}$, an intermediate gapless phase is observed, before a second transition to the high-field partially-polarized paramagnet. We attribute this rich phase diagram, and the remarkable stability of the chiral topological phase in the AFM Kitaev model, to the interplay of strong spin-orbit coupling and frustration enhanced by the magnetic field. Our findings suggest relevance to recent experiments on RuCl$_3$ under magnetic fields.

Topological Z2 resonating-valence-bond spin liquid on the square lattice

A one-parameter family of long-range resonating valence bond (RVB) state on the square lattice was previously proposed to describe a critical spin liquid (SL) phase of the spin-$1/2$ frustrated Heisenberg model. We provide evidence that this RVB state in fact also realises a topological (long-range entangled) $\mathbb{Z}_2$ SL, limited by two transitions to critical SL phases. The topological phase is naturally connected to the $\mathbb{Z}_2$ gauge symmetry of the local tensor. This work shows that, on one hand, spin-$1/2$ topological SL with $C_{4v}$ point group symmetry and $SU(2)$ spin rotation symmetry exists on the square lattice and, on the other hand, criticality and nonbipartiteness are compatible. We also point out that, strong similarities between our phase diagram and the ones of classical interacting dimer models suggest both can be described by similar Kosterlitz-Thouless transitions. This scenario is further supported by the analysis of the one-dimensional boundary state.

Three-dimensional anomalous twisted gauge theories with global symmetry: Implications for quantum spin liquids

Topological spin liquids can be described by topological gauge theories with global symmetry. Due to the presence of both nontrivial bulk deconfined gauge fluxes and global symmetry, topological spin liquids are examples of the so-called "Symmetry Enriched Topological phases" (SET). In this paper, we find that, in some twisted versions of topological gauge theories (with discrete Abelian gauge group $G_g$), implementing a global symmetry (denoted by $G_s$) is anomalous although symmetry charge carried by topological point-like excitations is normally fractionalized and classified by the second cohomology group. To demonstrate the anomaly, we fully gauge the global symmetry, rendering a new gauge theory that is not gauge invariant. Therefore, the SET order of the ground state is anomalous, which cannot exist in 3D system alone. Such anomalous state construction generalizes the "2D surface topological order" to 3D. A concrete example with $G_g=\mathbb{Z}_2\times\mathbb{Z}_4$ and $G_s=\mathbb{Z}_2$ is calculated.

A spin-orbital-entangled quantum liquid on a honeycomb lattice.

The honeycomb lattice is one of the simplest lattice structures. Electrons and spins on this simple lattice, however, often form exotic phases with non-trivial excitations. Massless Dirac fermions can emerge out of itinerant electrons, as demonstrated experimentally in graphene, and a topological quantum spin liquid with exotic quasiparticles can be realized in spin-1/2 magnets, as proposed theoretically in the Kitaev model. The quantum spin liquid is a long-sought exotic state of matter, in which interacting spins remain quantum-disordered without spontaneous symmetry breaking. The Kitaev model describes one example of a quantum spin liquid, and can be solved exactly by introducing two types of Majorana fermion. Realizing a Kitaev model in the laboratory, however, remains a challenge in materials science. Mott insulators with a honeycomb lattice of spin-orbital-entangled pseudospin-1/2 moments have been proposed, including the 5d-electron systems α-Na2IrO3 (ref. 5) and α-Li2IrO3 (ref. 6) and the 4d-electron system α-RuCl3 (ref. 7). However, these candidates were found to magnetically order rather than form a liquid at sufficiently low temperatures, owing to non-Kitaev interactions. Here we report a quantum-liquid state of pseudospin-1/2 moments in the 5d-electron honeycomb compound H3LiIr2O6. This iridate does not display magnetic ordering down to 0.05 kelvin, despite an interaction energy of about 100 kelvin. We observe signatures of low-energy fermionic excitations that originate from a small number of spin defects in the nuclear-magnetic-resonance relaxation and the specific heat. We therefore conclude that H3LiIr2O6 is a quantum spin liquid. This result opens the door to finding exotic quasiparticles in a strongly spin-orbit-coupled 5d-electron transition-metal oxide.

Dynamical time-reversal symmetry breaking and photo-induced chiral spin liquids in frustrated Mott insulators

The search for quantum spin liquids in frustrated quantum magnets recently has enjoyed a surge of interest, with various candidate materials under intense scrutiny. However, an experimental confirmation of a gapped topological spin liquid remains an open question. Here, we show that circularly polarized light can provide a knob to drive frustrated Mott insulators into a chiral spin liquid, realizing an elusive quantum spin liquid with topological order. We find that the dynamics of a driven Kagome Mott insulator is well-captured by an effective Floquet spin model, with heating strongly suppressed, inducing a scalar spin chirality Si · (Sj × Sk) term which dynamically breaks time-reversal while preserving SU(2) spin symmetry. We fingerprint the transient phase diagram and find a stable photo-induced chiral spin liquid near the equilibrium state. The results presented suggest employing dynamical symmetry breaking to engineer quantum spin liquids and access elusive phase transitions that are not readily accessible in equilibrium.

Direct experimental observation of the molecular Jeff\u2009=\u20093/2 ground state in the lacunar spinel GaTa4Se8

Strong spin–orbit coupling lifts the degeneracy of t2g orbitals in 5d transition-metal systems, leaving a Kramers doublet and quartet with effective angular momentum of Jeff = 1/2 and 3/2, respectively. These spin–orbit entangled states can host exotic quantum phases such as topological Mott state, unconventional superconductivity, and quantum spin liquid. The lacunar spinel GaTa4Se8 was theoretically predicted to form the molecular Jeff = 3/2 ground state. Experimental verification of its existence is an important first step to exploring the consequences of the Jeff = 3/2 state. Here, we report direct experimental evidence of the Jeff = 3/2 state in GaTa4Se8 by means of excitation spectra of resonant inelastic X-ray scattering at the Ta L3 and L2 edges. We find that the excitations involving the Jeff = 1/2 molecular orbital are absent only at the Ta L2 edge, manifesting the realization of the molecular Jeff = 3/2 ground state in GaTa4Se8.