Recent experiments suggest a quantum spin liquid ground state in the material PbCuTe$_2$O$_6$, where $S=1/2$ moments are coupled by antiferromagnetic Heisenberg interactions into a three dimensional structure of corner sharing triangles dubbed the hy
per-hyperkagome lattice. It exhibits a richer connectivity, and thus likely a stronger geometric frustration, than the relatively well studied hyperkagome lattice. Here, we investigate the possible quantum spin liquids in the $S=1/2$ hyper-hyperkagome magnet using the complex fermion mean field theory. Extending the results of a previous projective symmetry group analysis, we identify only two $mathbb{Z}_2$ spin liquids and a $U(1)$ spin liquid that are compatible with the hyper-hyperkagome structure. The $U(1)$ spin liquid has a spinon Fermi surface. For the $mathbb{Z}_2$ spin liquids, one has a small excitation gap, while the other is gapless and proximate to the $U(1)$ spin liquid. We show that the gapped and gapless spin liquids can in principle be distinguished by heat capacity measurements. Moreover, we calculate the dynamical spin structure factors of all three spin liquids and find that they highly resemble the inelastic neutron scattering spectra of PbCuTe$_2$O$_6$. Implications of our work to the experiments, as well as its relations to the existing theoretical studies, are discussed.
Thermal transport in topologically-ordered phases of matter provides valuable insights as it can detect the charge-neutral quasiparticles that would not directly couple to electromagnetic probes. An important example is edge heat transport of Majoran
a fermions in a chiral spin liquid, which leads to a half-quantized thermal Hall conductivity. This signature is precisely what has recently been measured in $alpha$-RuCl$_3$ under external magnetic fields. The plateau-like behavior of the half-quantized thermal Hall conductivity as a function of external magnetic field, and the peculiar sign change depending on the magnetic field orientation, has been proposed as strong evidence for the non-Abelian Kitaev spin liquid. Alternatively, for in-plane magnetic fields, it was theoretically shown that such a sign structure can also arise from topological magnons in the field-polarized state. In this work, we investigate the full implications of topological magnons as heat carriers on thermal transport measurements. We first prove analytically that for any commensurate order with a finite magnetic unit cell, reversing the field direction leads to a sign change in the magnon thermal Hall conductivity in two-dimensional systems. We verify this proof numerically with nontrivial magnetic orders as well as the field-polarized state in Kitaev magnets subjected to an in-plane field. In the case of a tilted magnetic field, whereby there exist both finite in-plane and out-of-plane field components, we find that the plateau-like behavior of the thermal Hall conductivity and the sign change upon reversing the in-plane component of the magnetic field arise in the partially-polarized state, as long as the in-plane field contribution to the Zeeman energy is significant. While these results are consistent with the experimental observations, we comment on other aspects requiring investigation in future studies.
Recently, the observation of large thermal Hall conductivities in correlated insulators with no apparent broken symmetry have generated immense interest and debates on the underlying ground states. Here, considering frustrated magnets with bond-depen
dent interactions, which are realized in the so-called Kitaev materials, we theoretically demonstrate that a large thermal Hall conductivity can originate from a classical ground state without any magnetic order. We discover a novel liquid state of magnetic vortices, which are inhomogeneous spin textures embedded in the background of polarized spins, under out-of-plane magnetic fields. In the classical regime, different configurations of vortices form a degenerate manifold. We study the static and dynamical properties of the magnetic vortex liquid state at zero and finite temperatures. In particular, we show that the spin excitation spectrum resembles a continuum of nearly flat Chern bands, which ultimately leads to a large thermal Hall conductivity. Possible connections to experiments are discussed.
The appearance of half-quantized thermal Hall conductivity in $alpha$-RuCl$_3$ in the presence of in-plane magnetic fields has been taken as a strong evidence for Kitaev spin liquid. Apart from the quantization, the observed sign structure of the the
rmal Hall conductivity is also consistent with predictions from the exact solution of the Kitaev model. Namely, the thermal Hall conductivity changes sign when the field direction is reversed with respect to the heat current, which is perpendicular to one of the three nearest neighbor bonds on the honeycomb lattice. On the other hand, it is almost zero when the field is applied along the bond direction. Here, we show that such a peculiar sign structure of the thermal Hall conductivity is a generic property of the polarized state in the presence of in-plane magnetic-fields. In this case, thermal Hall effect arises from topological magnons with finite Chern numbers and the sign structure follows from the symmetries of the momentum space Berry curvature. Using a realistic spin model with bond-dependent interactions, we show that the thermal Hall conductivity can have a magnitude comparable to that observed in the experiments. Hence the sign structure alone cannot make a strong case for Kitaev spin liquid. The quantization at very low temperatures, however, will be a decisive test as the magnon contribution vanishes in the zero temperature limit.
The appearance of nontrivial phases in Kitaev materials exposed to an external magnetic field has recently been a subject of intensive studies. Here, we elucidate the relation between the field-induced ground states of the classical and quantum spin
models proposed for such materials, by using the infinite density matrix renormalization group (iDMRG) and the linear spin wave theory (LSWT). We consider the $K Gamma Gamma$ model, where $Gamma$ and $Gamma$ are off-diagonal spin exchanges on top of the dominant Kitaev interaction $K$. Focusing on the magnetic field along the $[111]$ direction, we explain the origin of the nematic paramagnet, which breaks the lattice-rotational symmetry and exists in an extended window of magnetic field, in the quantum model. This phenomenon can be understood as the effect of quantum order-by-disorder in the frustrated ferromagnet with a continuous manifold of degenerate ground states discovered in the corresponding classical model. We compute the dynamical spin structure factors using a matrix operator based time evolution and compare them with the predictions from LSWT. We, thus, provide predictions for future inelastic neutron scattering experiments on Kitaev materials in an external magnetic field along the $[111]$ direction. In particular, the nematic paramagnet exhibits a characteristic pseudo-Goldstone mode which results from the lifting of a continuous degeneracy via quantum fluctuations.
$alpha$-RuCl$_3$ is drawing much attention as a promising candidate Kitaev quantum spin liquid. However, despite intensive research efforts, controversy remains about the form of the basic interactions governing the physics of this material. Even the
sign of the Kitaev interaction (the bond-dependent anisotropic interaction responsible for Kitaev physics) is still under debate, with conflicting results from theoretical and experimental studies. The significance of the symmetric off-diagonal exchange interaction (referred to as the $Gamma$ term) is another contentious question. Here, we present resonant elastic x-ray scattering data that provides unambiguous experimental constraints to the two leading terms in the magnetic interaction Hamiltonian. We show that the Kitaev interaction ($K$) is ferromagnetic, and that the $Gamma$ term is antiferromagnetic and comparable in size to the Kitaev interaction. Our findings also provide a natural explanation for the large anisotropy of the magnetic susceptibility in $alpha$-RuCl$_3$ as arising from the large $Gamma$ term. We therefore provide a crucial foundation for understanding the interactions underpinning the exotic magnetic behaviours observed in $alpha$-RuCl$_3$.
Recent discovery of the half quantized thermal Hall conductivity in $alpha$-RuCl$_3$, a candidate material for the Kitaev spin liquid, suggests the presence of a highly entangled quantum state in external magnetic fields. This field induced phase app
ears between the low field zig-zag magnetic order and the high field polarized state. Motivated by this experiment, we study possible field induced quantum phases in theoretical models of the Kitaev magnets, using the two dimensional tensor network approach or infinite tensor product states. We find various quantum ground states in addition to the chiral Kitaev spin liquid occupying a small area in the phase diagram. They form a band of emergent quantum phases in an intermediate window of external magnetic fields, somewhat reminiscent of the experiment. We discuss the implications of these results in view of the experiment and previous theoretical studies.
There has been a great interest in magnetic field induced quantum spin liquids in Kitaev magnets after the discovery of neutron scattering continuum and half quantized thermal Hall conductivity in the material $alpha$-RuCl$_3$. In this work, we provi
de a semiclassical analysis of the relevant theoretical models on large system sizes, and compare the results to previous studies on quantum models with small system sizes. We find a series of competing magnetic orders with fairly large unit cells at intermediate magnetic fields, which are most likely missed by previous approaches. We show that quantum fluctuations are typically strong in these large unit cell orders, while their magnetic excitations may resemble a scattering continuum and give rise to a large thermal Hall conductivity. Our work provides an important basis for a thorough investigation of emergent spin liquids and competing phases in Kitaev magnets.
A recent inelastic neutron scattering experiment on $mathrm{Yb}_2 mathrm{Ti}_2 mathrm{O}_7$ uncovers an unusual scattering continuum in the spin excitation spectrum despite the splayed ferromagnetic order in the ground state. While there exist well d
efined spin wave excitations at high magnetic fields, the one magnon modes and the two magnon continuum start to strongly overlap upon decreasing the field, and eventually they become the scattering continuum at zero field. Motivated by these observations, we investigate the possible emergence of a magnetically ordered ground state with fractionalized excitations in the spin model with the exchange parameters determined from two previous experiments. Using the fermionic parton mean field theory, we show that the magnetically ordered state with fractionalized excitations can arise as a stable mean field ground state in the presence of sufficiently strong quantum fluctuations. The spin excitation spectrum in such a ground state is computed and shown to have the scattering continuum. Upon increasing the magnetic field, the fractionalized magnetically ordered state is suppressed, and is eventually replaced by the conventional magnetically ordered phase at high fields, which is consistent with the experimental data. We discuss further implications of these results to the experiments and possible improvements on the theoretical analysis.
We propose a theoretical model for a gapless spin liquid phase that may have been observed in a recent experiment on $mathrm{H_3Li Ir_2 O_6}$. Despite the insulating and non-magnetic nature of the material, the specific heat coefficient $C/T sim 1/sq
rt{T}$ in zero magnetic field and $C/T sim T/ B^{3/2}$ with finite magnetic field $B$ have been observed. In addition, the NMR relaxation rate shows $1/(T_1T) sim (C/T)^2$. Motivated by the fact that the interlayer/in-plane lattice parameters are reduced/elongated by the hydrogen-intercalation of the parent compound $mathrm{Li_2 Ir O_3}$, we consider four layers of the Kitaev honeycomb lattice model with additional interlayer exchange interactions. It is shown that the resulting spin liquid excitations reside mostly in the top and bottom layers of such a layered structure and possess a quartic dispersion. In an applied magnetic field, each quartic mode is split into four Majorana cones with the velocity $v sim B^{3/4}$. We suggest that the spin liquid phase in these defect layers, placed between different stacking patterns of the honeycomb layers, can explain the major phenomenology of the experiment, which can be taken as evidence that the Kitaev interaction plays the primary role in the formation of a quantum spin liquid in this material.
