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 $ømega_0$, 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, $h_i$, and the equilibrium quantum
critical point $h_c$, which we find to generate a sequence of non-equilibrium
critical points $h_*l=h_c+lømega_0/2$, with $l$ integer. When $h_i\neq
h_c$, we find a üniversal" edge singularity in $P(W)$ at a threshold value of
$W_th=2|h_i-h_c|$ which is entirely determined by $h_i$. The
form of that singularity --- Dirac delta derivative or square root --- depends
on $h_0$ being or not at a non-equilibrium critical point $h_*l$. On the
contrary, when $h_i=h_c$, $G(u)$ decays as a power-law for large $u$,
leading to different types of edge singularity at $W_th=0$. 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.
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