Trajectory phase transitions and dynamical Lee-Yang zeros of the Glauber-Ising chain

We examine the generating function of the time-integrated energy for the one-dimensional Glauber-Ising model. At long times, the generating function takes on a large-deviation form and the associated cumulant generating function has singularities corresponding to continuous trajectory (or “space-time”) phase transitions between paramagnetic trajectories and ferromagnetically or antiferromagnetically ordered trajectories. In the thermodynamic limit, the singularities make up a whole curve of critical points in the complex plane of the counting field. We evaluate analytically the generating function by mapping the generator of the biased dynamics to a non-Hermitian Hamiltonian of an associated quantum spin chain. We relate the trajectory phase transitions to the high-order cumulants of the time-integrated energy which we use to extract the dynamical Lee-Yang zeros of the generating function. This approach offers the possibility to detect continuous trajectory phase transitions from the finite-time behavior of measurable quantities.

Figure 1

Trajectory phase diagram and closed curves of critical points. (a) Trajectory phase diagram in the plane of the inverse temperature $\beta$ and the real part of the counting field $\mathrm{Re}\left\{s\right\}$. (Changing the sign of the interaction $J\to -J$ is equivalent to the sign changes $s\to -s$ and $\beta \to -\beta$.) The $k=\pi$ and $k=0$ modes separate paramagnetically (PM) ordered trajectories from antiferromagnetic (AFM) and ferromagnetic (FM) trajectories, respectively. (b)–(d) Closed curves of trajectory phase transition points in the complex plane of the counting field for different inverse temperatures, $\beta =1$, 0.5, and 0.

Figure 2

Mode-resolved cumulants and dynamical Lee-Yang zeros. (a) Natural Logarithm of mode-resolved cumulants of order $n=6$–9 as functions of time with $k=3\pi /4$ and $\beta =0.5$. Exact results (full lines) together with the approximation (dashed lines) based on the extracted pair of Lee-Yang zeros. (b) Motion of the Lee-Yang zeros (open circles) extracted from the high-order cumulants. The Lee-Yang zeros move toward the phase transitions points (filled circles) on the closed curve of transition points.

Figure 3

Dynamical Lee-Yang zeros at different temperatures for a chain with $N=20$. (a) At low temperatures ($\beta =1$), the fluctuations of the time-integrated energy are dominated by the long-wavelength modes, whose singularities are closest to $s=0$. The extracted pair of Lee-Yang zeros (open circles) moves with time toward these points. (b) In the high-temperature limit ($\beta =0$), all modes contribute to the fluctuations and the corresponding singularities are equally distant from $s=0$. In this case, the extracted pair of Lee-Yang zeros (open circles) does not move toward the phase transition points. The dynamical Lee-Yang zeros were extracted from the cumulants of order $n=6,7,8,9$.