Quench dynamics of the Ising field theory in a magnetic field

Abstract

We numerically simulate the time evolution of the Ising field theory after quenches starting from the $E_8$ integrable model using the Truncated Conformal Space Approach. The results are compared with two different analytic predictions based on form factor expansions in the pre-quench and post-quench basis, respectively. Our results clarify the domain of validity of these expansions and suggest directions for further improvement. We show for quenches in the $E_8$ model that the initial state is not of the integrable pair state form. We also construct quench overlap functions and show that their high-energy asymptotics are markedly different from those constructed before in the sinh/sine-Gordon theory, and argue that this is related to properties of the ultraviolet fixed point.

Figures (26)

• Figure 2.1: Illustration of quenches considered in this work in the $M-h$ parameter space. The lower (green) dot on the axis represents the post-quench Hamiltonian for Type I quenches and the pre-quench Hamiltonian for Type II quenches which were performed in both (negative and positive $M$) directions.
• Figure 3.1: Loschmidt echo $\mathcal{L}(t)$ after a quench of size $\xi=(h_i-h_f)/h_f=0.5$ at five different volumes in the range $r=30\dots50.$ Volume dependence is almost completely absent for $\mathcal{L}(t)^{1/r}.$ Time and volume is measured in units of the inverse mass $m_1^{-1}$ of the lightest particle, $r=m_1R.$
• Figure 3.2: Time evolution of the $\sigma$ operator after a small quench of size $\xi=(h_i-h_f)/h_f=0.005.$ Comparison of the TCSA data (black dots) with the first order perturbative quench expansion result \ref{['eq:Delfsig']},\ref{['eq:C2']} (red line) and the prediction of the post-quench expansion \ref{['eq:Schsig']} (magenta dashed line). In panel (b) the curves are shifted on top of each other. TCSA data are for volume $r=40$ and are extrapolated in the truncation level. Time is measured in units of the inverse mass $m_1^{-1}$ of the lightest particle. Expectation values are measured in units of $m_1^{1/8}.$
• Figure 3.3: Time evolution of the $\sigma$ operator after a small quench of size $\xi=0.05.$ In panel (b) both theoretical results are shifted to the diagonal ensemble value obtained from TCSA. TCSA data are for volume $r=40$ and are extrapolated in the truncation level. Notations and units are as in Fig. \ref{['fig:sigcomp']}.
• Figure 3.4: Time evolution of the $\sigma$ operator after a quenche of size $\xi=0.5.$ In panel (b) both theoretical results are shifted to the diagonal ensemble value obtained from TCSA. TCSA data are for volume $r=40$ and are extrapolated in the truncation level. Notations and units are as in Fig. \ref{['fig:sigcomp']}.