Session K2: Kitaev Spin Liquid Physics in Honeycomb and Related Lattice Materials

The magnetic ground state and relationship to Kitaev physics in~$\alpha $-RuCl$_{\mathrm{3}}$

The 2D Kitaev candidate alpha-RuCl$_{\mathrm{3}}$~consists of stacked honeycomb layers weakly coupled by Van der Waals interactions.~Here we report the measurements of bulk properties and neutron diffraction in both powder and single crystal samples.~~ Our results show that the full three dimensional magnetic ground state is highly pliable with at least two dominant phases corresponding to two different out-of-plane magnetic orders. They have different Neel temperatures dependent on the stacking of the 2D layers, such as a broad magnetic transition at T$_{\mathrm{N}} \quad =$ 14 K as observed in phase-pure powder samples, or a sharp magnetic transition at a lower T$_{\mathrm{N}} \quad =$ 7 K as observed in homogeneous single crystals with no evidence for stacking faults.~ The magnetic refinements of the neutron scattering data [1] will be discussed, which in all cases shows the in-plane magnetic ground state is the zigzag phase common in Kitaev related materials including the honeycomb lattice Iridates. Inelastic neutron scattering in all cases shows that this material consistently exhibit strong two-dimensional magnetic fluctuations leading to a break-down of the classical spin-wave picture [2]. [1] H.B. Cao, A. Banerjee, J-Q. Yan, C.B. Bridges, M. Lumsden, B.C. Chakoumakos, D.G. Mandrus, D.A. Tennant, S.E. Nagler, \textit{Low-temperature crystal and magnetic structure of alpha-RuCl}$_{3}$, (manuscript in preparation). [2] A. Banerjee \textit{et al.,} \textit{arxiv:}1504.08037 (2015);   

How to identify and resolve beyond-geometrical frustration

In this talk, we will discuss recent theoretical developments triggered by the experimental discoveries of iridium oxides $\alpha,\beta,\gamma$-Li$_2$IrO$_3$. In these polytypes, spin-orbit-coupled J=1/2 moments form 2D and 3D lattices (honeycomb, hyperhoneycomb and stripyhoneycomb) which generalize the 2D honeycomb lattice. Scattering experiments on these compounds have uncovered a peculiar non-coplanar incommensurate magnetic order, involving spirals which counter-rotate across neighboring sites. We discuss the emergence of this ordering, and the striking similarities visible across the three Li$_2$IrO$_3$ structures. The model Hamiltonians that capture the materials indicate strong magnetic frustration, which arises from spin-orbit coupling. Tuning the frustration, perhaps by just a $10\%$ Hamiltonian perturbation, exposes a fractionalized phase: Kitaev's three-dimensional quantum spin liquid (QSL). What is its range of stability to the competing Hamiltonian terms which occur in the materials, such as antiferromagnetic Heisenberg exchange? The frustration prohibits direct computations. Instead, we demonstrate a viable approach by numerically solving the model in a fully quantum infinite-dimensional approximation, which captures both the magnetically ordered and the QSL phases. Finally, we discuss the phenomenology of the QSL phase, including the role of its emergent magnetic-like field lines in stabilizing its deconfined fermion excitations to finite temperatures. The resulting phase transition is a signature unique to three-dimensional fractionalization.   

3D Kitaev spin liquids

The Kitaev honeycomb model has become one of the archetypal spin models exhibiting topological phases of matter, where the magnetic moments fractionalize into Majorana fermions interacting with a Z$_2$ gauge field. In this talk, we discuss generalizations of this model to three-dimensional lattice structures. Our main focus is the metallic state that the emergent Majorana fermions form. In particular, we discuss the relation of the nature of this Majorana metal to the details of the underlying lattice structure. Besides (almost) conventional metals with a Majorana Fermi surface\footnote{M. Hermanns and S. Trebst, PRB {\bf 89}, 235102 (2014).}, one also finds various realizations of Dirac semi-metals, where the gapless modes form Fermi lines or even Weyl nodes\footnote{M. Hermanns, K. O'Brien, and S. Trebst, PRL {\bf 114}, 157202 (2015).}. We introduce a general classification of these gapless quantum spin liquids using projective symmetry analysis. Furthermore, we briefly outline why these Majorana metals in 3D Kitaev systems provide an even richer variety of Dirac and Weyl phases than possible for electronic matter and comment on possible experimental signatures. \\ Work done in collaboration with Kevin O'Brien and Simon Trebst.   

Hyperhoneycomb iridate beta-Li2IrO3 as a platform for Kitaev spin liquid

Realization of quantum spin liquid has been a long-sought dream in condensed matter physics, where exotic excitations and unconventional superconductivity upon doping are expected. Honeycomb iridates recently emerged as a possible materialization of Kitaev spin liquid with frustrated ``$bond-dependent\; ferromagnetic\; interaction$''. However, the real materials, $\alpha$-Na$_2$IrO$_3$ and $\alpha$-Li$_2$IrO$_3$, undergo antiferromagnetic ordering likely due to the presence of other dominant magnetic interactions and lattice distortion. We discovered a new form of Li$_2$IrO$_3$, $\beta$-Li$_2$IrO$_3$, which comprises a three-dimensional analogue of honeycomb lattice dubbed as "hyperhoneycomb". Each Ir$^{4+}$ ion of the hyperhoneycomb lattice has three neighboring like ions rotated by 120$^{\circ}$ and thus the local structure is identical with 2D honeycomb, indicating that the hyperhoneycomb lattice is a new platform for Kitaev physics. $\beta$-Li$_2$IrO$_3$ diplays a spiral magnetic order below 38 K, which likely originates from dominance of ferromagnetic Kitaev interaction. We argure that $\beta$-Li$_2$IrO$_3$ locates in a close proximity to Kitaev spin liquid. We also discuss the spin liquid behavior observed in a new honeycomb iridate obtained by chemical modulation.   

Magnetic ``three states of matter'' in two and three dimensions: a quantum Monte Carlo study of the extended toric codes

The possibility of quantum spin liquids, characterized by nontrivial entanglement properties or a topological nonlocal order parameter, has long been debated both theoretically and experimentally. Since candidate systems (e.g., frustrated quantum magnets or 5$d$ transition metal oxides) may host other competing phases including conventional magnetic ordered phases, it is natural to ask what types of global phase diagrams can be anticipated depending on coupling constants, temperature, dimensionality, etc. In this talk, by considering an extension of the Kitaev toric code Hamiltonians by Ising interactions on 2D (square) and 3D (cubic) lattices, I will present thermodynamic phase diagrams featuring magnetic ``three states of matter,'' namely, quantum spin liquid, paramagnetic, and magnetically ordered phases (analogous to liquid, gas, and solid, respectively, in conventional matter) obtained by unbiased quantum Monte Carlo simulations [YK, Y. Kato, J. Nasu, and Y. Motome, PRB 92, 100403(R) (2015)]. We find that the ordered phase borders on the spin liquid around the exactly solvable point by a discontinuous transition line in 3D, while it grows continuously from the quantum critical point in 2D. In both cases, peculiar \textit{proximity effects} to the nearby spin liquid phases are observed at high temperature even when the ground state is magnetically ordered. Such proximity effects include flux-shrinking and a tricritical behavior in 3D and a ``fractionalization'' of the order parameter field at the quantum critical point in 2D, both of which can be detected by measuring critical exponents. (*) Work done in collaboration with Yasuyuki Kato, Joji Nasu, and Yukitoshi Motome