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

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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 \(\omega_D\). 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, \(F_{\vec k_0}( \omega_0)\), have maxima and minima at specific frequencies which can be directly related to \(\omega_D\) and the time at which the phase bands cross at \(\vec k = \vec k_0\). 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.

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Floquet generation of Majorana end modes and topological invariants

Manisha Thakurathi, Aavishkar Patel, Diptiman Sen … (2013)

We show how Majorana end modes can be generated in a one-dimensional system by varying some of the parameters in the Hamiltonian periodically in time. The specific model we consider is a chain containing spinless electrons with a nearest-neighbor hopping amplitude, a p-wave superconducting term and a chemical potential; this is equivalent to a spin-1/2 chain with anisotropic XY couplings between nearest neighbors and a magnetic field applied in the z-direction. We show that varying the chemical potential (or magnetic field) periodically in time can produce Majorana modes at the ends of a long chain. We discuss two kinds of periodic driving, periodic delta-function kicks and a simple harmonic variation with time. We discuss some distinctive features of the end modes such as the inverse participation ratio of their wave functions and their Floquet eigenvalues which are always equal to +/- 1 for time-reversal symmetric systems. For the case of periodic delta-function kicks, we use the effective Hamiltonian of a system with periodic boundary conditions to define two topological invariants. The first invariant is a well-known winding number while the second invariant has not appeared in the literature before. The second invariant is more powerful in that it always correctly predicts the numbers of end modes with Floquet eigenvalues equal to +1 and -1, while the first invariant does not. We find that the number of end modes can become very large as the driving frequency decreases. We show that periodic delta-function kicks in the hopping and superconducting terms can also produce end modes. Finally, we study the effect of electron-phonon interactions (which are relevant at finite temperatures) and a random noise in the chemical potential on the Majorana modes.

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Slow quenches in a quantum Ising chain; dynamical phase transitions and topology

Shraddha Sharma, Uma Divakaran, Anatoli Polkovnikov … (2016)

We study the slow quenching dynamics (characterized by an inverse rate, \(\tau^{-1}\)) of a one-dimensional transverse Ising chain with nearest neighbor ferromagentic interactions across the quantum critical point (QCP) and analyze the Loschmidt overlap {measured using the subsequent temporal evolution of the final wave function (reached at the end of the quenching) with the final time-independent Hamiltonian}. Studying the Fisher zeros of the corresponding generalized "partition function", we probe non-analyticities manifested in the rate function of the return probability known as dynamical phase transitions (DPTs). In contrast to the sudden quenching case, we show that DPTs survive {in the subsequent temporal evolution following the quenching across two critical points of the model for a sufficiently slow rate; furthermore, an interesting "lobe" structure of Fisher zeros emerge.} We have also made a connection to topological aspects studying the dynamical topological order parameter (\(\nu_D(t)\)), as a function of time (\(t\)) {measured from the instant when the quenching is complete. Remarkably, the time evolution of \(\nu_D(t)\) exhibits drastically different behavior following quenches across a single QCP and two QCPs. } {In the former case, \(\nu_D (t)\) increases step-wise by unity at every DPT (i.e., \(\Delta \nu_D =1\)). In the latter case, on the other hand, \(\nu_D(t)\) essentially oscillates between 0 and 1 (i.e., successive DPTs occur with \(\Delta \nu_D =1\) and \(\Delta \nu_D =-1\), respectively), except for instants where it shows a sudden jump by a factor of unity when two successive DPTs carry a topological charge of same sign.