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Pressure-Induced Gapless Quantum Spin Liquids on the Honeycomb Lattice

Jiucai Wang, Zheng-Xin Liu

Published Dec 12, 2019• 0 citations
OpenAlex

Abstract

Exerting high pressure is a practical way to tune the spin-spin interactions in Mott insulators and can be applied to prepare for quantum spin liquid (QSL) ground states. We illustrate this possibility by studying a realistic spin model which contains the Kitaev and $\Gamma$-interactions through extensive analytic and numerical calculations. As a consequence of the external pressure, the interaction intensities in our model are spatially anisotropic. While the anisotropic pure Kitaev model contains only two phases, the presence of non-Kitaev interactions results in a much richer phase diagram which contains several new gapless $Z_2$ QSL phases. These QSLs are characterized by a certain number of cones in the low-energy excitation spectrum, which can be reflected in the dynamic structure factors. We provide a complete classification of nodal $Z_2$ QSLs. Furthermore, a small out-of-plane magnetic field gaps out the cones in these gapless QSLs and results in Abelian or non-Abelian chiral spin liquids (CSLs), all belonging to the Kitaev's 16-fold classification. From the dynamic structure factor and the thermal Hall conductance of the field-induced CSLs, all the QSLs can be experimentally distinguished. Our results are instructive to search for QSL phases in related materials.

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Exerting high pressure is a practical way to tune the spin-spin interactions in Mott insulators and can be applied to prepare for quantum spin liquid (QSL) ground states. We illustrate this possibility by studying a realistic spin model which contains the Kitaev and Γ\GammaΓ-interactions through extensive analytic and numerical calculations. As a consequence of the external pressure, the interaction intensities in our model are spatially anisotropic. While the anisotropic pure Kitaev model contains only two phases, the presence of non-Kitaev interactions results in a much richer phase diagram which contains several new gapless Z2Z_2Z2​ QSL phases. These QSLs are characterized by a certain number of cones in the low-energy excitation spectrum, which can be reflected in the dynamic structure factors. We provide a complete classification of nodal Z2Z_2Z2​ QSLs. Furthermore, a small out-of-plane magnetic field gaps out the cones in these gapless QSLs and results in Abelian or non-Abelian chiral spin liquids (CSLs), all belonging to the Kitaev's 16-fold classification. From the dynamic structure factor and the thermal Hall conductance of the field-induced CSLs, all the QSLs can be experimentally distinguished. Our results are instructive to search for QSL phases in related materials.
By including a material-relevant off-diagonal interaction called the $\mathrm{\ensuremath{\Gamma}}$ term into the Kitaev model and introducing spatial anisotropy in the interaction strength on the honeycomb lattice, we obtain a series of nodal Z2{Z}_{2}Z2​ quantum spin liquids (QSLs) from the parton approach. These QSLs share the same projective symmetry group and are characterized by certain numbers of symmetry-protected Majorana cones in their low-energy excitation spectrum. We illustrate that the physical properties of the QSLs are dependent on the information of the cones. Using the $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ method, we analyze the chirality of every cone with respect to mass generating perturbations. Especially, for an applied external magnetic field, we provide the maximum-mass field orientation for every cone. Thus, for arbitrarily oriented weak magnetic fields, we can immediately read out the Chern number of the system and the properties of the resultant chiral spin liquids. The new gapless QSLs predicted in our phase diagrams are promising to be realized experimentally by exerting uniaxial pressure to tune the anisotropy of the interaction strength. We further show that all these QSLs can be distinguished by measurable quantities. Based on the study of these QSL phases, we conclude that a complete classification of nodal QSLs with certain symmetry should include not only the projective symmetry groups, but also the information of the cones, i.e., their total number, their chiralities, and the way in which they are symmetry related.
In addition to the Kitaev (K) interaction, candidate Kitaev materials also possess Heisenberg (J) and off-diagonal symmetric (Γ) couplings. We investigate the quantum (S=1/2) K-J-Γ model on the honeycomb lattice by a variational Monte Carlo method. In addition to the "generic" Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The PKSL is a gapless Z_{2} state with 14 Majorana cones, which in contrast to the KSL has a gapless spin response. In a magnetic field applied normal to the honeycomb plane, it realizes two of Kitaev's gapped chiral spin-liquid phases, of which one is non-Abelian with Chern number ν=5 and the other is Abelian with ν=4. These two phases could be distinguished by their thermal Hall conductance.
Kitaev's model of spins interacting on a honeycomb lattice describes a quantum spin liquid, where an emergent static Z2{\mathbb{Z}}_{2}Z2​ gauge field is coupled to Majorana fermions. In the presence of an external magnetic field and for a range of interaction strengths, the system behaves as a gapped, non-Abelian quantum spin liquid. In this phase, the vortex excitations of the emergent Z2{\mathbb{Z}}_{2}Z2​ gauge field have Majorana zero modes bound to them. Motivated by recent experimental progress in measuring and characterizing real materials that could exhibit spin-liquid behavior, we analytically calculate the dynamical spin structure factor in the non-Abelian phase of the Kitaev's honeycomb model. In particular, we treat the case of quenched disorder in the vortex configurations. Our calculations reveal a peak in the low-energy dynamical structure factor that is a signature of the spin-liquid behavior. We map the effective Hamiltonian to that of a chiral ppp-wave superconductor by using the Jordan-Wigner transformation. Subsequently, we analytically calculate the wave functions of the Majorana zero modes, the energy splitting for finite separation of the vortices and, finally, the dynamical structure factor in presence of quenched disorder.
Abstract A family of spin–orbit coupled honeycomb Mott insulators offers a playground to search for quantum spin liquids (QSLs) via bond-dependent interactions. In candidate materials, a symmetric off-diagonal Γ term, close cousin of Kitaev interaction, has emerged as another source of frustration that is essential for complete understanding of these systems. However, the ground state of honeycomb Γ model remains elusive, with a suggested zigzag magnetic order. Here we attempt to resolve the puzzle by perturbing the Γ region with a staggered Heisenberg interaction which favours the zigzag ordering. Despite such favour, we find a wide disordered region inclusive of the Γ limit in the phase diagram. Further, this phase exhibits a vanishing energy gap, a collapse of excitation spectrum, and a logarithmic entanglement entropy scaling on long cylinders, indicating a gapless QSL. Other quantities such as plaquette-plaquette correlation are also discussed.
The ruby lattice is a four-valent lattice interpolating between honeycomb and triangular lattices. In this work we investigate the topological spin-liquid phases of a spin Hamiltonian with Kitaev interactions on the ruby lattice using exact diagonalization and perturbative methods. The latter interactions combined with the structure of the lattice yield a model with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ gauge symmetry. We mapped out the phase diagram of the model and found gapped and gapless spin-liquid phases. While the low-energy sector of the gapped phase corresponds to the well-known topological color code model on a honeycomb lattice, the low-energy sector of the gapless phases is described by an effective spin model with three-body interactions on a triangular lattice. A gap is opened in the spectrum in small magnetic fields, where we showed that the ground state has a finite topological entanglement entropy. We argue that the gapped phases could be possibly described by exotic excitations, and their corresponding spectrum is richer than the Ising phase of the Kitaev model.
Variational Monte Carlo (VMC) studies of extended Kitaev honeycomb models reveal a series of multinode quantum spin liquids (QSLs). They have an emergent Z 2 gauge structure, a discrete number of symmetry-protected Majorana cones, and form Abelian or non-Abelian chiral spin liquids in applied magnetic fields. The large number of Z 2 projective symmetry groups for spin–orbit-coupled states on the honeycomb lattice suggests that multinode QSLs can be found experimentally in future work on proximate Kitaev materials.