Session P8: Focus Session: Spin Liquids I

Quantum Ice : Experimental Signatures

``Quantum Spin Ice'' materials have attracted considerable attention as three-dimensional examples of quantum spin liquids. Recently, we have used zero-temperature Quantum Monte Carlo simulation to explore one possible scenario for these materials, confirming the possibility of a ``quantum ice'' state driven by quantum tunnelling between an extensive number of different spin-ice configurations [1]. Here we address the simple question : what would such a quantum ice look like in experiment ? We focus in particular on the fate of ``pinch point'' singularities seen in neutron scattering experiments on spin ice materials, showing how these are suppressed and ultimately eliminated as the system is cooled to its ground state [1,2]. \\[4pt] [1] N. Shannon et al., arXiv:1105.4196\\[0pt] [2] O. Benton et al., in preparation.   

Quantum Ice : A Quantum Monte Carlo Study

The magnetic ``ice'' state found in spin ice materials has recently generated great excitement for its magnetic monopole excitations. However the deconfined nature of these monopoles depends crucially on the macroscopic degeneracy of the classical ice ground state. And at very low temperatures we might expect this degeneracy to be lifted by quantum tunneling between different ice configurations. Here we present the results of large-scale Green's function Monte Carlo simulation of ice-type models which include quantum tunneling. We find compelling evidence of an extended quantum U(1)-liquid ground state with deconfined monopole excitations in both the quantum dimer model [1,2] and the quantum ice model on the diamond lattice [3]. This quantum U(1) liquid proves to be remarkably robust against the inclusion of long range dipolar interactions. \\[0pt] [1] O. Sikora {\it et al.}, Phys. Rev. Lett. {\bf 103}, 247001 (2009) [2] O. Sikora {\it et al.}, Phys. Rev. B {\bf 84}, 115129 (2011) [3] N. Shannon {\it et al.}, arXiv:1105.4196   

Three-dimensional generalized Kitaev models

We generalize Kitaev's honeycomb lattice spin model to a gamma matrix model on three-dimensional cubic octahedron and pyrochlore lattices. We find the ground state ${Z}_2$ flux configuration, reducing the problem to free Majorana fermion hopping. For the cubic octahedron lattice, which has reflection planes, the ground states must obey Lieb's theorem, i.e. the ${Z}_2$ fluxes are reflection symmetric. By adding flux-flux interaction terms, a variety of interesting phases can be stabilized, including metallic, semimetallic, and both trivial and topological insulating phases.   

High pressure sequence of Ba$_{3}$NiSb$_{2}$O$_{9}$ structural phases: new $S $= 1 quantum spin-liquids based on Ni$^{2+}$

A quantum spin-liquid (QSL) is a ground-state where strong quantum- mechanical ?uctuations prevents a phase-transition towards conventional magnetic order and makes the spin ensemble to remain in a liquid-like state. Most QSL candidates studied to date are two-dimensional frustrated magnets with either a triangular or a kagome lattice composed of $S$ = 1/2 spins. Here, we report the use of a high pressure, high temperature technique to transform the antiferromagnetically ordered ($T_{N}$ = 13.5 K) 6H-A phase of Ba$_{3}$NiSb$_{2}$O$_{9}$ into two new QSL candidates with larger $S$ = 1 (Ni$^{2+})$ moments: the 6H-B phase of Ba$_{3}$NiSb$_{2}$O$_{9}$ which crystallizes in a triangular lattice and the 3C-phase of Ba$_{3}$NiSb$_{2}$O$_{9}$ which forms a three-dimensional edge-shared tetrahedral lattice. Both compounds show no evidence for magnetic order down to $T$ = 0.35 K despite Curie-Weiss temperatures \textit{$\theta $}$_{CW}$ of -75.5 K (6H-B) and -182.5 K (3C), respectively. Below $\sim $25 K the magnetic susceptibility of the 6H-B phase is found to saturate at a constant value $\chi $ = 0.013 emu/mol which is followed below 7 K, by a linear in temperature dependence for the magnetic contribution to the specificheat ($C_{M})$ which displays a giant coefficient $\gamma $ = 168 mJ/mol-K$^{2}$ comparable to values observed in heavy-fermion metallic systems. Taken together, both observations indicate the development of a Fermi-liquid like ground-state characterized by a Wilson ratio of 5.6 in this otherwise insulating material It also points to the formation at finite temperatures of a well defined Fermi surface of $S$ = 1 spin-excitations which behave as charged quasiparticles. For the 3C phase one observes $C_{M} \quad \propto \quad T ^{2}$ indicating a unique $S$ = 1 three-dimensional QSL ground-state as previously reported for Na$_{3}$Ir$_{4}$O$_{8}$ although this later compound is composed of Ir$^{4+ }$ions having $S$ = 1/2. \\[4pt] Work done in collaboration with J. G. Cheng, G. Li, J. S. Zhou, J. B. Goodenough, C Xu and H. D. Zhou.   

Generalizations of the Kitaev-Heisenberg model and connections to experiment

Generalizations of Kitaev-Heisenberg spin models are both interesting on their own right and potentially relevant for the Mott insulating layered Iridates A$_2$IrO$_3$ (A=Na,Li) and other complex oxides of 5d transition metal compounds. To describe Na$_2$IrO$_3$ and Li$_2$IrO$_3$ we propose the Kitaev-Heisenberg-$J_2$-$J_3$ model, a combination of the Kitaev honeycomb model and the Heisenberg model with all three nearest neighbor couplings $J_1$, $J_2$ and $J_3$. A rich phase diagram is obtained at the classical level, including the experimentally suggested \textit{zigzag} ordered phase; as well as the \textit{stripy} phase, which extends from the Kitaev-Heisenberg limit to the $J_1$-$J_2$-$J_3$ one. Combining the experimentally observed spin order with the optimal fitting to the uniform magnetic susceptibility data gives an estimate of possible parameter values, which in turn reaffirms the necessity of including both the Kitaev and farther neighbor couplings in describing the materials. Generalizations for other spin-orbit coupled Mott insulating 5d transition metal oxides with Kitaev-Heisenberg type models exhibit related magnetically ordered phases as well as different spin liquid phases.   

Topological phases of the 2D AKLT like model via study of degenerate singular value spectra

We study the two-dimensional quantum spin-3/2 systems on the honeycomb lattice, which include the 2D AKLT model. We use the infinite time-evolving block decimation (iTEBD) method to numerically solve the ground state wave function by adopting the tensor product states (TPS) ansatz. We then evaluate the singular value spectra, entanglement entropy, and the Neel order by the method of tensor renormalization group (TRG). With all these results we determine the phase diagram of the spin-3/2 model. Our result shows the singular value spectra become doubly degenerate in a belt region including the AKLT point, and indicates the existence of possible topological phases. The double degeneracy of singular value spectra is related to the recent observation in [C.-Y. Huang~and~F.-L. Lin, Phys. Rev. B 84, 125110 (2011)] of using it to characterize the topological phase. We also cross check this statement by calculating the topological Renyi entropy. The 2D AKLT state has been shown to be a universal resource for measurement-based quantum computation. Our results indicate this state is topological in nature and should be robust against local perturbations.   

Finite temperature phase diagram of the classical Heisenberg-Kitaev model.

We study finite-temperature properties of classical version of the Heisenberg-Kitaev model on the honeycomb lattice. This model is a prominent example of anisotropic spin-orbital models, which can possibly describe the low-energy physics of Na$_2$IrO$_3$ and Li$_2$IrO$_3$. In these compounds, Ir$^{4+}$ ions are in a low spin $5d^5$ configuration and form weakly coupled hexagonal layers. Our main result is a finite-temperature phase diagram obtained by classical Monte Carlo simulations. Because of highly anisotropic Kitaev interaction, the spin symmetry of the model is reduced to the discrete symmetry, which may be regarded as $Z_3 x Z_2$. As the discrete symmetry can be broken at finite temperature even at 2D, the model undergoes phase transitions as a function of temperature. At low temperature phase, thermal fluctuations induce order-by-disorder, just as the quantum fluctuations do at zero temperature. As a result, magnetically ordered ground states of the Heisenberg-Kitaev model persist up to a certain critical temperature. Finally, we discuss the relevance of obtained results for experimental findings in real compounds.   

Dynamical Jahn-Teller effect in spin-orbital coupled system

Orbital degree of freedom is one of the most attractive themes in strongly correlated electron system. A coupling between the orbital and the lattice vibration is known as a Jahn-Teller effect (JTE). The dynamical aspect of the Jahn-Teller interaction is often neglected in solid, because it is strongly suppressed by the cooperative JTE. Recently, Ba$_{3}$CuSb$_{2}$O$_{9}$ has been reported as a candidate of the spin liquid. A Cu$^{2+}$ has the $e_{g}$ orbital degree of freedom and is surrounded by the O$^{2-}$ octahedron. The octahedra on the neighboring sites do not have the common O ions. This fact implies that the cooperative JTE is weak, and the dynamical JTE is expected to play some key roles on orbital and magnetic properties. The purpose of this research is to study the dynamical JTE in a spin-orbital coupled system. In particular, we focus on the competitive or cooperative phenomena between the superexchange interaction and the dynamical JTE. The superexchange interactions are derived from the $d-p$ model on a honeycomb lattice. We have confirmed this interaction stabilizes the antiferro-spin and ferro-orbital configurations for the realistic parameters. The dynamical JTE described as the orbital-lattice coupling is obtained by extracting the low energy states of the vibronic Hamiltonian. We analyze the model including the two kinds of interactions by using the Bethe approximation. We find that the magnetic order is unstable in wide parameter region and the spin-dimer state with the orbital order is realized. Furthermore the orbital order is strongly suppressed by the dynamical JTE.   

Quantum spin liquid in frustrated one dimensional LiCuSbO$_4$

A quantum magnet, LiCuSbO$_4$, with chains of edge-sharing $S\!=\!1/2$ CuO$_6$ octahedra is reported. Short-range ordering is observed while no phase transition or spin freezing occurs down to 100 mK in zero magnetic field. Specific heat indicates a distinct low-temperature high-field phase near the 12 T saturation field. Neutron scattering shows incommensurate spin correlations with $q\!=\!0.47\pm0.01~\pi/a$ and places an upper limit of 70~$\mu$eV on a potential spin gap. Exact diagonalization of easy plane $S\!=\!1/2$ chains with competing nearest neighbor ferro- and next-nearest neighbor antiferromagnetic interactions (J$_1\!=\!-75$~K, J$_2\!=\!34$~K) accounts for the $T\!>\!2$~K bulk and neutron data. Close to a quantum critical point, free from long-range order and with an achievable saturation field, LiCuSbO$_4$ is a promising candidate material to test long-standing predictions for chiral and nematic states in quantum spin chains   

Exotic S=1 spin liquid state with fermionic excitations on triangular lattice

Motivated by recent experiments on the material Ba$_3$NiSb$_2$O$_9$ we consider a spin-one quantum antiferromagnet on a triangular lattice with the Heisenberg bilinear and biquadratic exchange interactions and a single-ion anisotropy. Using a fermionic ``triplon'' representation for spins, we study the phase diagram within mean-field theory. In addition to a fully gapped spin-liquid ground state, we find a state where one gapless triplon mode with Fermi surface coexists with $d + id$ topological pairing of the other triplons. Despite the existence of a Fermi surface, this ground state has fully gapped bulk spin excitations. Such a state has linear in-temperature specific heat and constant in-plane spin susceptibility, with an unusually high Wilson ratio.   

Spin Liquid Phases for Spin-1 systems on the Triangular lattice

Motivated by recent experiments on material Ba$_3$NiSb$_2$O$_9$, we propose two novel spin liquid phases (A and B) for spin-1 systems on a triangular lattice. At the mean field level, both spin liquid phases have gapless fermionic spinon excitations with quadratic band touching, thus in both phases the spin susceptibility and C$_v$/T saturate to a constant at zero temperature, which are consistent with the experimental results on Ba3NiSb2O9. On the lattice scale, these spin liquid phases have Sp(4) $\sim$ SO(5) gauge fluctuation; while in the long wavelength limit this Sp(4) gauge symmetry is broken down to U(1)xZ$_2$ in type A spin liquid phase, and broken down to Z$_4$ in type B phase. We also demonstrate that the $A$ phase is the parent state of the ferro-quadrupole state, nematic state, and the noncollinear spin density wave state.   

Stability of the U(1) Dirac spin liquid state on the kagome lattice

The ground state of the spin 1/2 nearest neighbor Heisenberg antiferromagnet on the kagome lattice is still unknown. Recent DMRG calculations[1] have challenged the proposal[2] that this ground state is an algebraic spin liquid with Dirac fermions and photons as elementary excitations. We numerically study all time reversal invariant Gutzwiller projected variational wave functions for this system and find a state with mild symmetry breaking as the lowest energy state. To avoid getting stuck in a local minimum, we begin our Monte-Carlo calculation from all Z2 spin liquid states cataloged in Ref. [3]. Analyzing the resulting wave function, we find it lies very close to the U(1) Dirac state proposed in Ref. [2] in terms of its gauge fluxes, but with lower energy. We believe these results strongly emphasize the dominance and stability of the U(1) Dirac spin liquid state among Gutzwiller projected wave functions. However, our state has higher energy than the DMRG matrix product state, so correlations beyond those captured by it must play a fundamental role in the characterization of the ground state. [1]S. Yan, D.A. Huse, and S.R. White, Science 332, 1173 (2011) [2]Y. Ran, M. Hermele, P.A. Lee, and X.-G. Wen, PRL, 98, 117205 (2007) [3]Y.-m. Lu, Y. Ran, and P. Lee, PRB, 83, 12 (2011)   

Theory of spin liquids in integer spin pyrochlores

Rare earth pyrochlores, with a chemical formula A2B2O7, exhibit many interesting features in A site spin system. Depending on A site rare earth elements, spin ice and magnetically ordered phases are shown in several experiments. Moreover, they have been also focused as possible candidates of U(1) spin liquid. In order to explore such versatile phases, we study the pseudospin-1/2 model, which is quite generic to describe rare earth pyrochlores with integer spins, in the presence of spin-orbit coupling and crystalline electric field. Using a new ``gauge mean field theory,'' we show the possible ground states, corresponding to several phases listed above. We also briefly discuss the experimental suggestions based on our theory.