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, $h_0$, the initial transverse field, $h_{\mathrm{i}}$, and the equilibrium quantum critical point $h_c$, which we find to generate a sequence of non-equilibrium critical points $h_{\ast l}=h_c+l{\omega }_0/2$, with $l$ integer. When $h_{\mathrm{i}}\ne h_c$, we find a "universal" edge singularity in $P(W)$ at a threshold value of $W_{\mathrm{th}}=2|h_{\mathrm{i}}-h_c|$ which is entirely determined by $h_{\mathrm{i}}$. The form of that singularity --- Dirac delta derivative or square root --- depends on $h_0$ being or not at a non-equilibrium critical point $h_{\ast l}$. On the contrary, when $h_{\mathrm{i}}=h_c$, $G(u)$ decays as a power-law for large $u$, leading to different types of edge singularity at $W_{\mathrm{th}}=0$. 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.

24 pages, 5 figures, published in Jour. Stat. Mech. Corrected error in the demonstration of appendix C

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