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arXiv:2307.06589 · Physical review. B./Physical review. B

Dynamical freezing and switching in periodically driven bilayer graphene

Soumya Sasidharan, Naveen Surendran

Published May 1, 2023• 3 citations
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Abstract

A class of integrable models, such as the one-dimensional transverse-field Ising model, responds nonmonotonically to a periodic drive with respect to the driving parameters and freezes almost absolutely for certain combinations of the latter. In this paper, we go beyond the two-band structure of the Ising-like models studied previously and ask whether such an unusual nonmonotonic response and near-absolute freezing occur in integrable systems with a higher number of bands. To this end, we consider a tight-binding model for bilayer graphene subjected to an interlayer potential difference. We find that when the potential is driven periodically, the system responds nonmonotonically to variations in the driving amplitude ${V}_{0}$ and frequency $\ensuremath{\omega}$ and shows near-absolute freezing for certain values of ${V}_{0}/\ensuremath{\omega}$. However, the freezing occurs only in the presence of a constant bias in the driving, i.e., when $V={V}^{\ensuremath{'}}+{V}_{0}cos\ensuremath{\omega}t$. When ${V}^{\ensuremath{'}}=0$, the freezing is switched off for all values of ${V}_{0}/\ensuremath{\omega}$. We support our numerical results with analytical calculations based on a rotating wave approximation. We also give a proposal to realize the driven bilayer system via ultracold atoms in an optical lattice, where the driving can be implemented by shaking the lattice.

DOI
10.1103/physrevb.107.174301

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A class of integrable models, such as the one-dimensional transverse-field Ising model, responds nonmonotonically to a periodic drive with respect to the driving parameters and freezes almost absolutely for certain combinations of the latter. In this paper, we go beyond the two-band structure of the Ising-like models studied previously and ask whether such an unusual nonmonotonic response and near-absolute freezing occur in integrable systems with a higher number of bands. To this end, we consider a tight-binding model for bilayer graphene subjected to an interlayer potential difference. We find that when the potential is driven periodically, the system responds nonmonotonically to variations in the driving amplitude V0{V}_{0}V0​ and frequency \ensuremathω\ensuremath{\omega}\ensuremathω and shows near-absolute freezing for certain values of V0/\ensuremathω{V}_{0}/\ensuremath{\omega}V0​/\ensuremathω. However, the freezing occurs only in the presence of a constant bias in the driving, i.e., when V=V\ensuremath′+V0cos\ensuremathωtV={V}^{\ensuremath{'}}+{V}_{0}cos\ensuremath{\omega}tV=V\ensuremath′+V0​cos\ensuremathωt. When V\ensuremath′=0{V}^{\ensuremath{'}}=0V\ensuremath′=0, the freezing is switched off for all values of V0/\ensuremathω{V}_{0}/\ensuremath{\omega}V0​/\ensuremathω. We support our numerical results with analytical calculations based on a rotating wave approximation. We also give a proposal to realize the driven bilayer system via ultracold atoms in an optical lattice, where the driving can be implemented by shaking the lattice.
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 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.
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 study the long time dynamics in closed quantum systems periodically driven via time dependent parameters with two frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}=r{\ensuremath{\omega}}_{1}$. Tuning of the ratio rrr there can unleash plenty of dynamical phenomena to occur. Our study includes integrable models like Ising and XYXYXY models in d=1d=1d=1 and the Kitaev model in d=1d=1d=1 and 2 and can also be extended to Dirac fermions in graphene. We witness the wave-function overlap or dynamic freezing that occurs within some small/ intermediate frequency regimes in the $({\ensuremath{\omega}}_{1},r)$ plane (with $r\ensuremath{\ne}0$) when the ground state is evolved through a single cycle of driving. However, evolved states soon become steady with long driving, and the freezing scenario gets rarer. We extend the formalism of adiabatic-impulse approximation for many cycle driving within our two-rate protocol and show the near-exact comparisons at small frequencies. An extension of the rotating wave approximation is also developed to gather an analytical framework of the dynamics at high frequencies. Finally we compute the entanglement entropy in the stroboscopically evolved states within the gapped phases of the system and observe how it gets tuned with the ratio rrr in our protocol. The minimally entangled states are found to fall within the regime of dynamical freezing. In general, the results indicate that the entanglement entropy in our driven short-ranged integrable systems follow a genuine nonarea law of scaling and show a convergence (with a rrr dependent pace) towards volume scaling behavior as the driving is continued for a long time.
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 the dynamics of a tilted one-dimensional Bose-Hubbard model for two distinct protocols using numerical diagonalization for a finite sized system ($N\ensuremath{\le}18$). The first protocol involves periodic variation of the effective electric field EEE seen by the bosons which takes the system twice (per drive cycle) through the intermediate quantum critical point. We show that such a drive leads to nonmonotonic variation of the excitation density DDD and the wave function overlap FFF at the end of a drive cycle as a function of the drive frequency ${\ensuremath{\omega}}_{1}$, relate this effect to a generalized version of St\"uckelberg interference phenomenon, and identify special frequencies for which DDD and $1\ensuremath{-}F$ approach zero leading to near-perfect dynamic freezing phenomenon. The second protocol involves a simultaneous linear ramp of both the electric field EEE (with a rate ${\ensuremath{\omega}}_{1}$) and the boson hopping parameter JJJ (with a rate ${\ensuremath{\omega}}_{2}$) starting from the ground state for a low effective electric field up to the quantum critical point. We find that both DDD and the residual energy QQQ decrease with increasing ${\ensuremath{\omega}}_{2}$; our results thus demonstrate a method of achieving near-adiabatic protocol in an experimentally realizable quantum critical system. We suggest experiments to test our theory.