Session P37: Honeycomb Lattice and Other Low-D Models

Quantum Spin Liquid phases in extended Kitaev models on the Honeycomb Lattice

Kitaev materials have attracted great attention thanks to their potential for realizing the Kitaev model, an exactly solvable spin model that hosts topological quantum spin liquids. However, most Kitaev materials undergo phase transition into a magnetically ordered phase at low temperatures, indicating that non-Kitaev interactions are non-neglectible. Recent experiments on the Kitaev material α-RuCl₃ showed that, under very low temperature an intermediate in-plane magnetic field can suppress the static zigzag magnetic order and induce a liquid-like disordered phase. The nature of the field-induced disordered phase was not fully understood. Motivated by this issue, we studied the K-Γ model (K =6.8 meV and Γ= 9.5meV) which was the minimal model prosed to describe the low-energy physics of α-RuCl₃. From variational Monte Carlo calculations, we found that in-plane magnetic field can indeed cause a phase transition from a zigzag ordered phase into a gapless U(1) Dirac spin liquid phase [1]. Our results semi-quatitatively explain the observed temperature dependence of spin-lattice relaxation rate 1/T₁~T³ in a recent nuclear magnetic resonance experiment [2]. It was also shown that an out-of-plane magnetic field can induce a Kalmeyer-Laughlin-type abelian chiral spin liquid phase, which would show an integer-quantized thermal Hall effect. Our theory hopefully captures the low energy physics of α-RuCl₃ and provides a clue to undertand the experiments in related materials.

[1] Zheng-Xin Liu and B. Normand, Phys. Rev. Lett. 120, 187201 (2018).
[2] J. Zheng, K. Ran, T. Li, J.Wang, P.Wang, B. Liu, Z.-X. Liu, B. Normand, J. Wen, and W. Yu, Phys. Rev. Lett. 119, 227208
(2017).   

Field-orientation-dependent spin dynamics of the Kitaev honeycomb model

The main question we address is how to probe the fractionalized excitations of a quantum spin-liquid (QSL), for example, in the Kitaev honeycomb model. From analyzing the energy spectrum and entanglement entropy, for antiferromagnetic couplings and a field along either [111], or [-110], we find a new gapless QSL phase, sandwiched between the non-Abelian Kitaev QSL and polarized phases. Rotating the field towards [001] destroys this intermediate QSL phase and results in a considerable reduction in the number of frequency modes. In certain parameter regimes, we observe a beating pattern in the local dynamical correlations, possibly observable in pump-probe experiments. Finally, we explore the field strength and orientation dependence of the longitudinal and transverse thermal conductivities of this model as a function of temperature.   

Chiral spin liquid near the ferromagnetic Kitaev regime under a magnetic field

The Kitaev model is a rare example of an exactly solvable model which exhibits a spin liquid ground state, and hosts non-Abelian anyon excitations. Recent research has focused on material realization of Kitaev physics, with α-RuCl₃ emerging as a leading candidate. Compelling evidence for a chiral spin liquid (CSL) is a half-quantized thermal Hall conductivity in α-RuCl₃, a signature of Majorana edge currents. However, it remains a non-trivial task to explain the existence of a CSL in α-RuCl₃ from a microscopic model, due to dominant ferromagnetic (FM) Kitaev interactions. This is because the FM Kitaev phase almost immediately transitions to the polarized phase in an applied field, with no intervening phase. Here we present a microscopic theory describing the existence of a CSL near the FM Kitaev regime under a magnetic field. Further implications related to the quantized thermal Hall conductivity are also discussed.   

Classical Kitaev model in a magnetic field

We study the classical version of Kitaev's honeycomb model in the presence of a magnetic field. For an anti-ferromagnetic Kitaev coupling a classical spin liquid phase persists over a large window of field strength, independent of field direction. We characterize this phase using analytic arguments, a coarse-grained theory and via Monte Carlo simulations, finding the Coulomb correlations present at zero-field are immediately destroyed when the field is applied. We further probe the nature of this phase via the introduction of vacancies. Over a wide field range we find removing a site leaves the magnetization unchanged, the liquid exactly compensating the missing spin. Finally, we speculate on implications for the quantum model in the semi-classical limit.   

Plaquette order in classical spin liquid stabilized by strong off-diagonal exchange

We report a new classical spin liquid in which the collective flux degrees of freedom break the translation symmetry of the honeycomb lattice. This exotic phase exists in frustrated spin-orbit magnets where a dominant off-diagonal exchange, the so-called Γ term, results in a macroscopic ground-state degeneracy at the classical level [1]. We demonstrate that the system undergoes a phase transition driven by thermal order-by-disorder at a critical temperature T_c ≈ 0.04 |Γ|. Interestingly, while the cubic symmetry is broken at T < T_c, spins in the low temperature phase remain disordered. We show that this phase transition actually corresponds to plaquette ordering of hexagonal fluxes. By introducing a proper order parameter to describe this translational symmetry breaking, we performed extensive Monte Carlo simulations and finite-size analysis to investigate the nature of the plaquette-ordering transition. We also study the dynamical behavior of fluxes and the influence of other types of interactions on the phase transition.[1] I. Rousochatzakis and N. B. Perkins Phys. Rev. Lett. 118, 147204 (2017).   

Designing S = 1 Kitaev materials

As the spin S = 1/2 Kitaev model is gaining increasing attention, there have been theoretical efforts to understand if S = 1/2 Kitaev features appear in generalized Kitaev models with higher spins. In particular, numerical studies on the S = 1 Kitaev model show interesting results, but there is no microscopic understanding on how to generate such a bond-dependent S = 1 interaction in solid state materials. Here we present a microscopic mechanism and necessary ingredients to design the S = 1 Kitaev interaction. Candidate materials are also discussed.   

Dynamical and finite temperature properties of Kitaev magnet beta-Li2IrO3 in magnetic field

We present theoretical study of the field-induced magnetic phases in the three-dimensional, hyperhoneycomb Kitaev compound beta-Li2IrO3. We show that a relatively weak magnetic field along the crystallographic b-axis drives the system from its incommensurate counter-rotating order to a correlated paramagnet, with a significant uniform ‘zigzag’ component superimposing the magnetization along the field. We compute the magnon excitation spectra, the dynamical spin structure factors and their polarization dependence and argue that our results provide additional distinctive fingerprint that can be checked experimentally. Finally, using extensive Monte Carlo simulations, we study the evolution of the magnetic orders with temperature and provide the finite temperature phase diagram of field-induced magnetic phases.   

Using light to tune magnetism on the honeycomb lattice

The search for spin liquid phases in condensed matter systems has generated a lot of interest in recent years. In practice, however, such phases are usually restricted to small regions of phase diagrams, which makes them challenging to access in experiments. Concerning this issue, circularly polarized Floquet potentials have been shown to generate chiral fields on the kagome lattice, making it possible to tune between Z2 and chiral spin liquids [1]. Given the potential deconfined criticality and chiral spin liquid phases on honeycomb lattices, here we address the effects of different Floquet potentials in these lattices. We discuss, in particular, how periodic optical drives can be used to manipulate different spin exchange couplings. We also comment on how our results can be relevant to experiments.

[1] Martin Claassen, Hong-Chen Jiang, Brian Moritz, and Thomas P. Devereaux. Dynamical time-reversal symmetry breaking and photo-induced chiral spin liquids in frustrated Mott insulators. Nature Communications, 8(1):1192, 2017   

Classification of spin liquids on the stuffed honeycomb lattice

We introduce the stuffed honeycomb lattice (a honeycomb lattice coupled to its dual - a triangular lattice) that interpolates between the triangular and the honeycomb lattices. We consider S = 1/2 Heisenberg spins. Our classical phase diagram reveals a multi-critical point on the triangular lattice axis, with two new neighboring noncollinear phases appearing only off axis. Our quantum phase diagram found via exact diagonalization hosts a large spin liquid (SL) region that eats up most of the phase space of the new classical phases around the multi-critical point. We present a projective symmetry group analysis of all possible symmetric SLs on the stuffed honeycomb lattice and attempt to probe the possible nature (gapless, nematic, etc) to compare to the SL region found in exact diagonalization. Among these SLs, we focus on characterizing the competitive ones found in a variational Monte Carlo analysis.   

Field-induced Spin Liquid-like State in a Magnetic Honeycomb Lattice

Quantum fluctuations in magnetic lattices can yield a quantum spin liquid (QSL) state, where no long-range order appears even at zero temperature. The variety of mechanisms that can generate the spin liquid state and the more exotic QSL state remain unclear, however. Here, we report a new magnetic honeycomb system, BaCo₂(P_1-xV_x)₂O₈, in which the spin correlations can be tuned by the disorder, leading to different magnetic behaviors. At low x, the material has a spin glass ground state that appears to be due to coexisting and competing correlations. We have found that an external magnetic field can introduce spin liquid-like behavior for some members of the solid solution, testified by the magnetic and thermodynamic experiments. Our results suggest that structural geometry, chemical disorder and external field may help enhance quantum fluctuations in magnetic honeycomb materials.
  

Ripple state in the frustrated honeycomb-lattice antiferromagnet

Rich nontrivial magnetic structures in multiple-q states have attracted much attention these days. One of the most celebrated example might be a skyrmion-lattice state known as a triple-q state with forming periodic vortex-crystal structure. We have found a new type of multiple-q state, a "ripple state" in the J₁-J₂ classical honeycomb-lattice Heisenberg antiferromagnet by means of extensive Monte Carlo simulations. This honeycomb model has been known to have an infinite ring-like degeneracy in the ground state. In the ripple state, surprisingly, order-by-disorder mechnism does not work and all wavevectors on the degenerate ring equally contribute to its order. The real-space spin texture of the ripple state does not form a crystal in spite of the breaking of the translational symmetry and seems to be like a "water ripple" observed when we throw a stone on the surface of water. We will talk about the detail of the intringuing properties of the ripple state and discuss the possible realization in the honeycomb-lattice spin-liquid material, Bi₃Mn₄O₁₂(NO₃).   

AKLT-like valence bond solid state in the frustrated ferromagnetic J1-J2 chain

Frustrated one-dimensional (1D) magnets are known as ideal playgrounds for new exotic quantum phenomena. We consider an elementary frustrated 1D system: the spin-1/2 ferromagnetic (J₁) Heisenberg chain with next-nearest-neighbor antiferromagnetic (AFM) (J₂) interactions. On the basis of density-matrix renormalization group calculations we show the existence of a finite spin gap at J₂/|J₁| > 1/4 and we find the ground state in this region to be a valence bond solid (VBS) with spin-singlet dimerization between third-neighbor sites. The VBS is the consequence of spontaneous symmetry breaking through order by disorder. Quite interestingly, this VBS state has a Affleck-Kennedy-Lieb-Tasaki-type topological order. We also investigate the possibility of multipolar physics induced by AFM interchain coupling.   

Critical properties of six-state clock model on randomly frustrated 2D lattices

We study the antiferromagnetic six-state clock model
on bond-diluted triangular lattice, which is randomly frustrated.
The corresponding pure system of the model experiences two separated
phase transitions, i.e., the Kosterlitz-Thouless (KT) transition of
magnetic ordering and the chiral ordering transition which occurs
at slightly higher temperature. Due to discrete symmetry of spins
there exists an additional KT transition at lower temperature,
separating the true (magnetic) and the quasi-long range order (QLRO).
As randomness is important for phase transition, it is desirable to
probe the role played by dilution in affecting the existing KT
transitions as well as the chiral ordering transition.
With the presence of bond dilution in the triangular lattice the system
is transformed from the fully frustrated into a randomly frustrated case.
As the energy of the model is represented by the multiple of $J/2$,
where $J$ is the coupling constant, we use Wang-Landau algorithm of
Monte Carlo method. Various concentrations of depleted bonds were simulated.
We observed a systematic decrease of critical temperatures,
both for the KT transition and chiral transition,
as the concentration of dilution increases. We obtained the phase diagram of the system.