Cumulants of time-integrated observables of closed quantum systems and $\mathcal{PT}$ symmetry with an application to the quantum Ising chain

We study the connection between the cumulants of a time-integrated observable of a quantum system and the $\mathcal{P}\mathcal{T}$-symmetry properties of the non-Hermitian deformation of the Hamiltonian from which the generating function of these cumulants is obtained. This non-Hermitian Hamiltonian can display regimes of broken and of unbroken $\mathcal{P}\mathcal{T}$ symmetry, depending on the parameters of the problem and on the counting field that sets the strength of the non-Hermitian perturbation. This in turn determines the analytic structure of the long-time cumulant generating function (CGF) for the time-integrated observable. We consider in particular the case of the time-integrated (longitudinal) magnetization in the one-dimensional Ising model in a transverse field. We show that its long-time CGF is singular on a curve in the magnetic field/counting field plane that delimits a regime where $\mathcal{P}\mathcal{T}$ symmetry is spontaneously broken (which includes the static ferromagnetic phase), from one where it is preserved (which includes the static paramagnetic phase). In the paramagnetic phase, conservation of $\mathcal{P}\mathcal{T}$ symmetry implies that all cumulants are sublinear in time, a behavior usually associated with the absence of decorrelation.

Figure 1

(a) Single spin which precesses around the $x$ direction. The time-integrated magnetization of interest is at an angle $\phi$ to the precession axis. (b) Time-integrated $z$ magnetization: For the case $\phi =\pi /2$, the CGF $\theta \left(s\right)$ is zero for $\left|s\right|\le 2\epsilon$ and nonzero elsewhere. Here we have chosen $\epsilon =1$ and the functional form of $\theta \left(s\right)$ shown assumes that there is finite overlap of the initial state with the eigenstates of $H_s$, i.e., $c_{\pm }\left(s\right)$ in Eq. (12) are nonzero. The singularities of $\theta \left(s\right)$ at $\pm 2\epsilon$ correspond to the breaking of the $\mathcal{P}\mathcal{T}$ symmetry of $H_s$. These are also points at which ${\kappa }_s$ is discontinuous. (c) The emergence of the singularity of $\theta \left(s\right)$ at $s=0$ in the regime where $\mathcal{P}\mathcal{T}$ is not present, $\phi \ne \pi /2$ (here $\phi =0.1$). In this parameter regime the emergence of a long-time singularity at $s=0$ is generically independent of the initial state (here we initialize the system in the state $\frac{4}{5}|\uparrow \rangle +\frac{3}{5}|\downarrow \rangle$).

Figure 2

(a) Scaled CGF $\tilde{\theta }\left(s\right)$ in the $\lambda \text{−}s$ plane (from exact diagonalization for chains of $N=9$ spins). For $\lambda >1$ there is a large region where $\tilde{\theta }\left(s\right)\approx 0$. (b), (c) Plots of the scaled CGF, dynamical order parameter, and susceptibility as a function of $s$ for $\lambda =0.8$ and $1.2$ ($N=11$ here). We see in the ferromagnetic regime a large peak in the susceptibility at $s=0$ that indicates the cumulants grow faster than $t$. In the paramagnetic regime the order parameter is vanishing near $s=0$, but beyond some critical $s$ value the $\mathcal{P}\mathcal{T}$ symmetry breaks and it acquires a finite value. This symmetry breaking is marked by peaks in the susceptibility ${\chi }_s$. (d) Scaled second cumulant of the time-integrated magnetization in both regimes.

Figure 3

Dynamical phase diagram of the TFIM, with time-integrated magnetization $M_t^x$ as the order parameter, in terms of the transverse field $\lambda$ and counting field $s$. There are two dynamical phases: a “dynamically ordered” phase, which contains the static ordered phase of $\left|\lambda \right|<{\lambda }_{\mathrm{c}}=1$, where the $\mathcal{P}\mathcal{T}$ symmetry of $H_s$ is broken, and as a consequence the cumulants of $M_t^x$ are linear or superlinear in time; and a “dynamically disordered” phase, which contains the static ordered phase of $\left|\lambda \right|>{\lambda }_{\mathrm{c}}=1$, where $\mathcal{P}\mathcal{T}$ symmetry is unbroken, and as a consequence cumulants of $M_t^x$ are sublinear in time. The two phases are delimited by a curve of first-order transitions of $\theta \left(s\right)$ corresponding to the spontaneous breaking of $\mathcal{P}\mathcal{T}$ symmetry.