Session L4: Focus Session: Quantum Spin Liquids

Finite-temperature phase transition to a quantum spin liquid in a 3D Kitaev model

The Kitaev model has recently attracted considerable attention due to the spin-liquid ground states. This model is defined on a honeycomb lattice, and is exactly solvable due to the Ising conserved quantities associated with each hexagon. In this study, we investigate the thermodynamic properties of a three-dimensional (3D) generalization of the Kitaev model defined on a hyperhoneycomb lattice, which was introduced in Ref. [1]. Although this model has spin-liquid ground states similar to the 2D model, the excited states are contrasting as they are described by Ising conserved quantities forming a loop-like structure on the pyrochlore lattice. We analyze this model in the limit where one of the inequivalent bonds is stronger than the others, where a classical Monte Carlo simulation is applicable [2]. As a result, we find a phase transition at a finite temperature between the gapped quantum spin liquid and paramagnet. This phase transition is of second order and belongs to the 3D Ising universality class. We provide a topological characterization of the phase transition in terms of a flux density. We also calculate the temperature dependence of the magnetic susceptibility. [1] S. Mandal and N. Surendran, Phys. Rev. B \textbf{79}, 024426 (2009). [2] J. Nasu et al., arXiv:1309.3068.   

Density-Matrix Renormalization Group Study of Effective Spin Model for Na$_2$IrO$_3$

The Kitaev-Heisenberg honeycomb lattice model has recently been proposed to describe magnetic properties in A$_2$IrO$_3$(A=Na,Li). The model includes an isotropic Heisenberg term and strongly anisotropic Kitaev terms. The Kitaev terms give a spin-liquid ground state. With increasing the strength of Heisenberg coupling, the ground state first turns into a stripy antiferromagnetic phase and then into a Neel antiferromagnet. The x-ray and neutron scattering experiments indicate that the ground state of Na$_2$IrO$_3$ is most likely characterized by a zig-zag spin structure. However, this type of magnetic order cannot be theoretically explained by the Kitaev-Heisenberg model. It is necessary to introduce further neighbor Heisenberg couplings and/or trigonal distortion of the oxygen octahedra. We study an extended Kitaev-Heisenberg model including these additional terms, by using two-dimensional density-matrix renormalization group method. Calculating spin correlation functions, we find that the zigzag order appears in a parameter regime relevant to Na$_2$IrO$_3$.   

Heisenberg-Kitaev model on the hyperhoneycomb lattice

Motivated by recent experiments on $\beta$-Li$_2$IrO$_3$, we study the phase diagram of the Heisenberg-Kitaev model on a three dimensional lattice of tri-coordinated Ir$^{4+}$, dubbed the hyperhoneycomb lattice. The lattice geometry of this material, along with Ir$^{4+}$ ions carrying $J_{eff}=1/2$ moments, suggests that the Heisenberg-Kitaev model may effectively capture the low energy spin physics of the system in the strong-coupling limit. Using a combination of semiclassical analysis, exact solution, and slave-fermion mean field theory, we find a spin-liquid and four different magnetically ordered phases---the Neel, the polarized ferromagnet, the skew-stripy, and the skew-zig-zag. The three dimensional $Z_2$ spin liquid, which extends over an extended parameter regime around the exactly solvable Kitaev point, has a gapless Majorana mode with a deformed Fermi-circle (co-dimensions, $d_c=2$). We discuss the effect of magnetic field and finite temperature on various phases that may be relevant for future experiments.   

Magnetic Anisotropy of a 3-Dimensional Honeycomb Iridate Lattice

We present magnetic anisotropy measurements of a 3-dimensional honeycomb iridate lattice. The large spin-orbit coupling and the edge-shared octahedra create the possibility for large spin-anisotropic Kitaev exchange. The structure preserves the connectivity of the honeycomb lattice indicating its potential as a spin-liquid candidate. A complete temperature and angular dependence of torque measurements provides evidence for highly spin-anisotropic exchange interactions. At high temperature, the geometry of the octahedral environment and the iridium g-factor anisotropy constrain the susceptibility. Upon lowering temperature, we unambiguously identify a reordering of the principle components of susceptibility. An order of magnitude increase in anisotropy with field orientation at low temperature highlights the strong orbital character of the coupling.   

Kitaev's honeycomb model on a buckyball

We study the effect of disclinations in the Kitaev's honeycomb model [1] by examining the effective tight binding hamiltonian of Majorana fermions on Buckminsterfullerene [2]. Disclinations are realized by the 12 pentagons which shape the buckyball. We found that the ground state of the system with isotropic nearest neighbor coupling $t_1$, corresponds to a uniform flux sector of the $Z_2$ gauge field, where hexagons are flux free and pentagons have the same fluxes. Inclusion of second neighbor couplings $t_2$, preserve the projective symmetries of the truncated icosahedron as long as fluxes through all plaquettes (triangles, pentagons, and hexagons) related by symmetries are the same. For $t_1/t_2$ smaller than $1/2$, the local density of states reorganizes suggesting that the zero energy Majorana modes localize at the disclinations. The robustness of this quantum state against noise is examined. \\[4pt] [1] A. Kitaev, Ann. Phys. 321, 2 (2006).\\[0pt] [2] Kroto, Harold W., et al., Nature 318.6042 (1985).   

Confinementr and deconfinement of topological excitations in Na$_{2}$IrO$_{3}$

Using phase sensitive heterodyne transient grating technique we establish that in the limit of low pumping fluences the optical response of Na-213 iridate system below the antiferromagnetic ordering temperature T$_{\mathrm{N}}$ is dominated by Hubbard excitons (HE). Unpaired single particle excitations (SE) constituting HE are strongly suppressed, appearing only above T$_{\mathrm{N}}$. We argue that this is due to the interplay between the frustrated Kitaev term in the Hamiltonian and the weak Heisenberg term responsible for the antiferromagnetic order below T$_{\mathrm{N}}$, which mediates an effective interaction between spin singlet SE excitations. This interaction grows linearly with distance resulting in a sudden increase of the exciton binding energy as the system enters the ordered state. This is a solid state realization of a phenomenon known in high energy physics as confinement.   

Non-abelian anyons on dislocations in Kitaev's honeycomb spin liquid

Kitaev's honeycomb model [1] is an exactly solvable model of a quantum spin liquid. Its gapped phase exhibits $Z_2$ topological order and has low-energy excitations in the form of $Z_2$ fluxes (visons). Previous studies [2] have demonstrated that even trivial lattice defects such as vacancies induce free magnetic moments with peculiar properties. We show [3] that certain kinds of lattice dislocations and bond defects in this system carry even more exotic excitations: unpaired Majorana fermions. Each pair of such defects (known as twists [4]) gives rise to a non-local physical (complex) fermion mode made out of two Majorana (real) fermions connected by a $Z_2$ gauge string. Their interaction decays exponentially with the distance. The non-local fermion can be created or annihilated by winding a vortex around a dislocation. The vortex also changes its topological charge in this process. The model remains exactly solvable in the presence of such defects and reveals a crucial role of the emergent gauge field in the physics of Majorana modes. \\[4pt] [1] A. Kitaev, Ann. Phys. \textbf{321}, 2 (2006). \\[0pt] [2] A. J. Willans, J. T. Chalker, and R. Moessner, Phys. Rev. B \textbf{84}, 115146 (2011). \\[0pt] [3] O. Petrova, P. Mellado, and O. Tchernyshyov, Phys. Rev. B \textbf{88}, 140405 (2013). \\[0pt] [4] H. Bombin, Phys. Rev. Lett. \textbf{105}, 030403 (2010).   

Loops, sign structures and emergent Fermi statistics in three-dimensional quantum dimer models

We introduce and study three-dimensional quantum dimer models with positive resonance terms. We demonstrate that their ground state wavefunctions exhibit a nonlocal sign structure that can be exactly formulated in terms of loops, and as a direct consequence, monomer excitations obey Fermi statistics. The sign structure and Fermi statistics in these ``signful'' quantum dimer models can be naturally described by a novel parton construction, which becomes exact at the solvable point.   

Magnetic Correlations in a Frustrated Ni$^{3+}$ - Based Spin 1/2 Honeycomb Lattice

We have studied the magnetic properties, via thermodynamic probes and inelastic neutron scattering, of the new spin-1/2 honeycomb material Na$_{0.95}$Ni$_2$SbO$_6$ $\cdot$ 1.5D$_2$O [1]. This hydrated compound hosts well separated honeycomb layers of nickel ions in the unusual Ni$^{3+}$ oxidation state, which produces S=1/2 magnetic moments. While a Curie-Weiss temperature of -13K indicates overall anti-ferromagnetic interactions, specific heat and neutron scattering reveal the presence of ferromagnetic correlations with coherent spin excitations that build up gradually upon cooling below 10K. No transition to long range order is observed down to 2 K, as evidenced by specific heat and neutron scattering, although AC susceptibility measurements indicate a dramatic change in dynamics near 4.2K. The results indicate the presence of frustration arising from competing interactions between ions in the layers. This compound, along with potential isostructural analogs, opens a new route to study the phase diagram of spin 1/2 honeycomb lattice models with competing interactions. \\[4pt] [1] J.H. Roudebush and R.J. Cava. J. Solid State Chem, 204. 178-185 (2013)   

Generic spin model for the honeycomb iridates beyond the Kitaev limit

Recently, realizations of Kitaev physics have been sought in the A$_2$IrO$_3$ family of honeycomb iridates, originating from oxygen-mediated exchange through edge-shared octahedra. However, for the $j_{\rm eff} = 1/2$ Mott insulator in these materials exchange from direct $d$-orbital overlap is relevant, and it was proposed that a Heisenberg term should be added to the Kitaev model. Here we provide the generic nearest-neighbour spin Hamiltonian when both oxygen-mediated and direct overlap are present, containing a bond dependent off-diagonal exchange in addition to Heisenberg and Kitaev terms. We analyze this complete model using a combination of classical techniques and exact diagonalization. Near the Kitaev limit we find new magnetic phases: 120${}^\circ$ and incommensurate spiral order, as well as extended regions of zigzag and stripy order. Possible applications to Na$_2$IrO$_3$ and Li$_2$IrO$_3$ are discussed.   

Emergent magnetic scale in an exotic lithium iridate compound

We study the low temperature magnetic properties of an exotic iridate compoud. Iridium-oxide materials show a range of interesting magnetic driven by their strong spin-orbit coupling and structural anisotropy, and may realize exotic magnetic behavior arising from Kitaev interactions. Newly synthesized single crystals exhibit a 3D structure and strongly anisotropic magnetic properties. We observe a kink in the low-temperature magnetization at a field $H^*$, corresponding to an induced moment of 0.2 $\mu_ B$. We will discuss the appearance and evolution of this new field scale, and its connections to magnetic order in this new family of materials.   

Effective field theory, edge states and classification of symmetric Z2 spin liquids

Growing numerical evidence for gapped Z2 spin liquids in physically realistic spin models provides strong motivation for a deeper theoretical understanding of their properties. In particular the interplay of symmetry and topological order is known to lead to distinct phases of matter, symmetry enriched topological states, which differ in the action of symmetry on the topological excitations. In this work we present a Chern-Simons field theory description of symmetric spin liquids, which allows for a complete classification of these states as well as access to their physical properties such as edge states and quasiparticle quantum numbers. As an application we show that there are 6 distinct classes of Z2 spin liquids in the presence of a global Ising (Z2) symmetry.   

Phases of the frustrated XY model on the honeycomb lattice

We study the phase diagram of the frustrated XY model on the honeycomb lattice by using accurate correlated wave functions and variational Monte Carlo simulations. Our results suggest that a spin-liquid state is energetically favorable in the region of intermediate frustration, intervening between two magnetically ordered phases. We briefly discuss our results in the light of recent DMRG simulations where instead of a spin liquid, an unsual magnetically ordered state is found.