Asymptotic work statistics of periodically driven Ising chains

Abstract

We study the work statistics of a periodically-driven integrable closed quantum system, addressing in particular the role played by the presence of a quantum critical point. Taking the example of a one-dimensional transverse Ising model in the presence of a spatially homogeneous but periodically time-varying transverse field of frequency ${{\omega}_{0}}$ , we arrive at the characteristic cumulant generating function G(u), which is then used to calculate the work distribution function P(W). By applying the Floquet theory we show that, in the infinite time limit, P(W) converges, starting from the initial ground state, towards an asymptotic steady state value whose small-W behaviour depends only on the properties of the small-wave-vector modes and on a few important ingredients: the time-averaged value of the transverse field, h0, the initial transverse field, ${{h}_{\text{i}}}$ , and the equilibrium quantum critical point ${{h}_{\text{c}}}$ , which we find to generate a sequence of non-equilibrium critical points ${{h}_{*l}}={{h}_{\text{c}}}+l{{\omega}_{0}}/2$ , with l integer. When ${{h}_{\text{i}}}\ne {{h}_{\text{c}}}$ , we find a 'universal' edge singularity in P(W) at a threshold value of ${{W}_{\text{th}}}=2|{{h}_{\text{i}}}-{{h}_{\text{c}}}|$ which is entirely determined by ${{h}_{\text{i}}}$ . The form of that singularity—Dirac delta derivative or square root—depends on h0 being or not at a non-equilibrium critical point h*l. On the contrary, when ${{h}_{\text{i}}}={{h}_{\text{c}}}$ , G(u) decays as a power-law for large u, leading to different types of edge singularity at ${{W}_{\text{th}}}=0$ . Generalizing our calculations to the case in which we initialize the system in a finite temperature density matrix, the irreversible entropy generated by the periodic driving is also shown to reach a steady state value in the infinite time limit.