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arXiv:1605.09178 · Physical review. B./Physical review. B

Signatures and conditions for phase band crossings in periodically driven integrable systems

Bhaskar Mukherjee, Arnab Sen, Diptiman Sen, K. Sengupta

Published Oct 13, 2016• 31 citations
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Abstract

We present generic conditions for phase band crossings for a class of periodically driven integrable systems represented by free fermionic models subjected to arbitrary periodic drive protocols characterized by a frequency ${\ensuremath{\omega}}_{D}$. These models provide a representation for the Ising and $XY$ models in $d=1$, the Kitaev model in $d=2$, several kinds of superconductors, and Dirac fermions in graphene and atop topological insulator surfaces. Our results demonstrate that the presence of a critical point/region in the system Hamiltonian (which is traversed at a finite rate during the dynamics) may change the conditions for phase band crossings that occur at the critical modes. We also show that for $d>1$, phase band crossings leave their imprint on the equal-time off-diagonal fermionic correlation functions of these models; the Fourier transforms of such correlation functions, ${F}_{{\stackrel{P\vec}{k}}_{0}}({\ensuremath{\omega}}_{0})$, have maxima and minima at specific frequencies which can be directly related to ${\ensuremath{\omega}}_{D}$ and the time at which the phase bands cross at $\stackrel{P\vec}{k}={\stackrel{P\vec}{k}}_{0}$. We discuss the significance of our results in the contexts of generic Hamiltonians with $N>2$ phase bands and the underlying symmetry of the driven Hamiltonian.

DOI
10.1103/physrevb.94.155122

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We present generic conditions for phase band crossings for a class of periodically driven integrable systems represented by free fermionic models subjected to arbitrary periodic drive protocols characterized by a frequency \ensuremathωD{\ensuremath{\omega}}_{D}\ensuremathωD​. These models provide a representation for the Ising and XYXYXY models in d=1d=1d=1, the Kitaev model in d=2d=2d=2, several kinds of superconductors, and Dirac fermions in graphene and atop topological insulator surfaces. Our results demonstrate that the presence of a critical point/region in the system Hamiltonian (which is traversed at a finite rate during the dynamics) may change the conditions for phase band crossings that occur at the critical modes. We also show that for d>1d>1d>1, phase band crossings leave their imprint on the equal-time off-diagonal fermionic correlation functions of these models; the Fourier transforms of such correlation functions, {F}_{{\stackrel{P\vec}{k}}_{0}}({\ensuremath{\omega}}_{0}), have maxima and minima at specific frequencies which can be directly related to \ensuremathωD{\ensuremath{\omega}}_{D}\ensuremathωD​ and the time at which the phase bands cross at \stackrel{P\vec}{k}={\stackrel{P\vec}{k}}_{0}. We discuss the significance of our results in the contexts of generic Hamiltonians with N>2N>2N>2 phase bands and the underlying symmetry of the driven Hamiltonian.
We show that the correlation functions of a class of periodically driven integrable closed quantum systems approach their steady-state value as ${n}^{\ensuremath{-}(\ensuremath{\alpha}+1)/\ensuremath{\beta}}$, where nnn is the number of drive cycles and $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ denote positive integers. We find that, generically, $\ensuremath{\beta}=2$ within a dynamical phase characterized by a fixed $\ensuremath{\alpha}$; however, its value can change to $\ensuremath{\beta}=3$ or $\ensuremath{\beta}=4$ either at critical drive frequencies separating two dynamical phases or at special points within a phase. We show that such decays are realized in both driven Su-Schrieffer-Heeger (SSH) and one-dimensional (1D) transverse field Ising models, discuss the role of symmetries of the Floquet spectrum in determining $\ensuremath{\beta}$, and chart out the values of $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ realized in these models. We analyze the SSH model for a continuous drive protocol using a Floquet perturbation theory which provides analytical insight into the behavior of the correlation functions in terms of its Floquet Hamiltonian. This is supplemented by an exact numerical study of a similar behavior for the 1D Ising model driven by a square pulse protocol. For both models, we find a crossover timescale nc{n}_{c}nc​ which diverges at the transition. We also unravel a long-time oscillatory behavior of the correlators when the critical drive frequency, ${\ensuremath{\omega}}_{c}$, is approached from below $(\ensuremath{\omega}<{\ensuremath{\omega}}_{c})$. We tie such behavior to the presence of multiple stationary points in the Floquet spectrum of these models and provide an analytic expression for the time period of these oscillations.
We study a class of periodically driven ddd-dimensional integrable models and show that after nnn drive cycles with frequency $\ensuremath{\omega}$, pure states with non-area-law entanglement entropy ${S}_{n}(l)\ensuremath{\sim}{l}^{\ensuremath{\alpha}(n,\ensuremath{\omega})}$ are generated, where lll is the linear dimension of the subsystem, and $d\ensuremath{-}1\ensuremath{\le}\ensuremath{\alpha}(n,\ensuremath{\omega})\ensuremath{\le}d$. The exponent $\ensuremath{\alpha}(n,\ensuremath{\omega})$ eventually approaches ddd (volume law) for large enough lll when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We identify and analyze the crossover phenomenon from an area $(S\ensuremath{\sim}{l}^{d\ensuremath{-}1}$ for $d\ensuremath{\ge}1$) to a volume $(S\ensuremath{\sim}{l}^{d})$ law and provide a criterion for their occurrence which constitutes a generalization of Hastings's theorem to driven integrable systems in one dimension. We also find that Sn{S}_{n}Sn​ generically decays to ${S}_{\ensuremath{\infty}}$ as ${(\ensuremath{\omega}/n)}^{(d+2)/2}$ for fast and ${(\ensuremath{\omega}/n)}^{d/2}$ for slow periodic drives; these two dynamical phases are separated by a topological transition in the eigenspectrum of the Floquet Hamiltonian. This dynamical transition manifests itself in the temporal behavior of all local correlation functions and does not require a critical point crossing during the drive. We find that these dynamical phases show a rich re-entrant behavior as a function of $\ensuremath{\omega}$ for d=1d=1d=1 models and also discuss the dynamical transition for d>1d>1d>1 models. Finally, we study entanglement properties of the steady state and show that singular features (cusps and kinks in d=1d=1d=1) appear in ${S}_{\ensuremath{\infty}}$ as a function of $\ensuremath{\omega}$ whenever there is a crossing of the Floquet bands. We discuss experiments which can test our theory.
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.
Floquet Majorana edge modes capture the topological features of periodically driven superconductors. We present a Kitaev chain with multiple time periodic driving and demonstrate how the avoidance of bands crossing is altered, which gives rise to new regions supporting Majorana edge modes. A one dimensional generalized method was proposed to predict Majorana edge modes via the Zak phase of the Floquet bands. We also study the time independent effective Hamiltonian at high frequency limit and introduce diverse index to characterize topological phases with different relative phase between the multiple driving. Our work enriches the physics of driven system and paves the way for locating Majorana edge modes in larger parameter space.
The periodically driven quantum Ising chain has recently attracted a large attention in the context of Floquet engineering. In addition to the common paramagnet and ferromagnet, this driven model can give rise to new topological phases. In this work, we systematically explore its quantum phase diagram by examining the properties of its Floquet ground state. We specifically focus on driving protocols with time-reversal invariant points, and demonstrate the existence of an infinite number of distinct phases. These phases are separated by second-order quantum phase transitions, accompanied by continuous changes of local and string order parameters, as well as sudden changes of a topological winding number and of the number of protected edge states. When one of these phase transitions is adiabatically crossed, the correlator associated to the order parameter is nonvanishing over a length scale which shows a Kibble-Zurek scaling. In some phases, the Floquet ground state spontaneously breaks the discrete time-translation symmetry of the Hamiltonian. Our findings provide a better understanding of topological phases in periodically driven clean integrable models.
Abstract The topological properties of periodically driven many-body systems often have no static analogs and defy a simple description based on the effective Hamiltonian. To explore the emergent edge modes in driven p-wave superconductors in two dimensions, we analysed a toy model of Kitaev chains (one-dimensional spinless p-wave superconductors with Majorana edge states) coupled by time-periodic hopping. We showed that with proper driving, the coupled Kitaev chains can turn into a fully gapped superconductor, which is analogous to the p x +ip y state but has two, rather than one, chiral edge modes. A different driving protocol turns it into a gapless superconductor with isolated point nodes and completely flat edge states at quasienergy ω=0 or π/T, with T as the driving period. The time evolution operator U(k x , k y , t) of the toy model is computed exactly to yield the phase bands. And the “topological singularities” of the phase bands are exhausted and compared to those of a periodically driven Hofstadter model, which features counter-propagating chiral edge modes. These examples demonstrate the unique edge states in driven superconducting systems and suggest driving as a potentially fruitful route to engineer new topological superconductors.