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arXiv:2403.06123 · Physica Scripta

Periodically driven three-dimensional Kitaev model

Soumya Sasidharan, Naveen Surendran

Published Mar 4, 2024• 4 citations
arXivHTMLPDF

Abstract

Abstract We study the dynamics of a three-dimensional generalization of Kitaev’s honeycomb lattice spin model (defined on the hyperhoneycomb lattice) subjected to a harmonic driving of J z , one of the three types of spin-couplings in the Hamiltonian. Using numerical solutions supported by analytical calculations based on a rotating wave approximation, we find that the system responds nonmonotonically to variations in the frequency ω (while keeping the driving amplitude J fixed) and undergoes dynamical freezing, where at specific values of ω, it gets almost completely locked in the initial state throughout the evolution. However, this freezing occurs only when a constant bias is present in the driving, i.e., when J z = J ′ + J cos ω t , with J ′ ≠ 0 . Consequently, the bias acts as a switch that triggers the freezing phenomenon. Dynamical freezing has been previously observed in other integrable systems, such as the one-dimensional transverse-field Ising model.

DOI
10.1088/1402-4896/ad3030

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Abstract We study the dynamics of a three-dimensional generalization of Kitaev’s honeycomb lattice spin model (defined on the hyperhoneycomb lattice) subjected to a harmonic driving of J z , one of the three types of spin-couplings in the Hamiltonian. Using numerical solutions supported by analytical calculations based on a rotating wave approximation, we find that the system responds nonmonotonically to variations in the frequency ω (while keeping the driving amplitude J fixed) and undergoes dynamical freezing, where at specific values of ω, it gets almost completely locked in the initial state throughout the evolution. However, this freezing occurs only when a constant bias is present in the driving, i.e., when J z = J ′ + J cos ω t , with J ′ ≠ 0 . Consequently, the bias acts as a switch that triggers the freezing phenomenon. Dynamical freezing has been previously observed in other integrable systems, such as the one-dimensional transverse-field Ising model.
We show that when a quantum many-body system is subjected to coherent periodic driving, the response may exhibit exotic freezing behavior in high driving frequency $(\ensuremath{\omega})$ regime. In a periodically driven classical thermodynamic system, freezing at high $\ensuremath{\omega}$ occurs when $1/\ensuremath{\omega}$ is much smaller than the characteristic relaxation time of the system and hence the freezing always increases there as $\ensuremath{\omega}$ is increased. Here, in the contrary, we see surprising nonmonotonic freezing behavior of the response with $\ensuremath{\omega}$, showing curious peak-valley structure. Quite interestingly, the entire system tends to freeze almost absolutely (the freezing peaks) when driven with a certain combination of driving parameters values (amplitude and $\ensuremath{\omega}$) due to coherent suppression of dynamics of the quantum many-body modes, which has no classical analog. We demonstrate this new freezing phenomenon analytically (supported by large-scale numerics) for a general class of integrable quantum spin systems.
We experimentally demonstrate the phenomenon of dynamical many-body freezing in a periodically driven Ising chain. Theoretically [A. Das, Phys. Rev. B 82, 172402 (2010)], for certain values of the drive parameters all fundamental degrees of freedom contributing to the response dynamics freeze for all time and for arbitrary initial state. Also, since the condition of freezing involves only the drive parameters and not the quantization of the momentum (i.e., the system size), our simulation with a small (3-spin) chain captures all salient features of the freezing phenomenon predicted for the infinite chain. Using optimal control techniques, we realize high-fidelity cosine-modulated drive, and observe nonmonotonic freezing of magnetization at specific frequencies of modulation. Time evolution of the excitations in momentum space has been tracked directly through magnetization measurements.
We study periodically driven closed quantum systems where two parameters of the system Hamiltonian are varied periodically in time with frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}=r{\ensuremath{\omega}}_{1}$. We show that such drives may be used to tune towards dynamics-induced freezing where the wave function of the state of the system after a drive cycle at time $T=2\ensuremath{\pi}/{\ensuremath{\omega}}_{1}$ has almost perfect overlap with the initial state. We locate regions in the $({\ensuremath{\omega}}_{1},r)$ plane where the freezing is near exact for a class of integrable models and a specific nonintegrable model. The integrable models that we study encompass Ising and XY\mathit{XY}XY models in d=1d=1d=1, Kitaev model in d=2d=2d=2, and Dirac fermions in graphene and atop a topological insulator surface, whereas the nonintegrable model studied involves the experimentally realized one-dimensional tilted Bose-Hubbard model in an optical lattice. In addition, we compute the relevant correlation functions of such driven systems and describe their characteristics in the region of the $({\ensuremath{\omega}}_{1},r)$ plane where the freezing is near exact. We supplement our numerical analysis with semianalytic results for integrable driven systems within adiabatic-impulse approximation and discuss experiments which may test our theory.
The Kitaev model, which hosts a quantum spin liquid (QSL) in the ground state, was originally defined on a two-dimensional honeycomb lattice, but can be straightforwardly extended to any tricoordinate lattices in any spatial dimensions. In particular, the three-dimensional (3D) extensions are of interest as a realization of 3D QSLs, and some materials like βββ-Li2_{2}2​IrO3_{3}3​, γγγ-Li2_22​IrO3_33​, and βββ-ZnIrO3_{3}3​ were proposed for the candidates. However, the phase diagrams of the models for those candidates have not been fully elucidated, mainly due to the limitation of numerical methods for 3D frustrated quantum spin systems. Here we study the Kitaev-Heisenberg model defined on a 3D hyperhoneycomb lattice, by using the pseudofermion functional renormalization group method. We show that the ground-state phase diagram contains the QSL phases in the vicinities of the pristine ferromagnetic and antiferromagnetic Kitaev models, in addition to four magnetically ordered phases, similar to the two-dimensional honeycomb case. Our results respect the four-sublattice symmetry inherent in the model, which was violated in the previous study. Moreover, we also show how the phase diagram changes with the anisotropy in the interactions. The results provide a reference for the search of the hyperhoneycomb Kitaev materials.
The Kitaev model on the honeycomb lattice is a paradigmatic system known to host a wealth of nontrivial topological phases and Majorana edge modes. In the static case, the Majorana edge modes are nondispersive. When the system is periodically driven in time, such edge modes can disperse and become chiral. We obtain the full phase diagram of the driven model as a function of the coupling and the driving period. We characterize the quantum criticality of the different topological phase transitions in both the static and driven model via the notions of Majorana-Wannier state correlation functions and momentum-dependent fidelity susceptibilities. We show that the system hosts crossdimensional universality classes: although the static Kitaev model is defined on a two-dimensional (2D) honeycomb lattice, its criticality falls into the universality class of one-dimensional (1D) linear Dirac models. For the periodically driven Kitaev model, in addition to the universality class of prototype 2D linear Dirac models, an additional 1D nodal loop type of criticality exists owing to emergent time-reversal and mirror symmetries, indicating the possibility of engineering multiple universality classes by periodic driving. The manipulation of time-reversal symmetry allows the periodic driving to control the chirality of the Majorana edge states.
We study the Majorana modes, both equilibrium and Floquet, which can appear at the edges of the Kitaev model on the honeycomb lattice. We first present the analytical solutions known for the equilibrium Majorana edge modes for both zigzag and armchair edges of a semi-infinite Kitaev model and chart the parameter regimes in which they appear. We then examine how edge modes can be generated if the Kitaev coupling on the bonds perpendicular to the edge is varied periodically in time as periodic $\ensuremath{\delta}$-function kicks. We derive a general condition for the appearance and disappearance of the Floquet edge modes as a function of the drive frequency for a generic ddd-dimensional integrable system. We confirm this general condition for the Kitaev model with a finite width by mapping it to a one-dimensional model. Our numerical and analytical study of this problem shows that Floquet Majorana modes can appear on some edges in the kicked system even when the corresponding equilibrium Hamiltonian has no Majorana mode solutions on those edges. We support our analytical studies by numerics for a finite sized system which show that periodic kicks can generate modes at the edges and the corners of the lattice.