Constructing effective free energies for dynamical quantum phase transitions in the transverse-field Ising chain

The theory of dynamical quantum phase transitions represents an attempt to extend the concept of phase transitions to the far from equilibrium regime. While there are many formal analogies to conventional transitions, it is a major question to which extent it is possible to formulate a nonequilibrium counterpart to a Landau-Ginzburg theory. In this paper we take a first step in this direction by constructing an effective free energy for continuous dynamical quantum phase transitions appearing after quantum quenches in the transverse-field Ising chain. Due to unitarity of quantum time evolution this effective free energy becomes a complex quantity transforming the conventional minimization principle of the free energy into a saddle-point equation in the complex plane of the order parameter, which as in equilibrium is the magnetization. We study this effective free energy in the vicinity of the dynamical quantum phase transition by performing an expansion in terms of the complex magnetization and discuss the connections to the equilibrium case. Furthermore, we study the influence of perturbations and signatures of these dynamical quantum phase transitions in spin correlation functions.

Figure 1

Absolute values of the eigenvalues ${\varepsilon }_1,{\varepsilon }_2$ of the transfer matrix $T$ (18) for $\mu =0$. Vertical lines are at $t=\pi /4$ and $t=3\pi /4$ where the two eigenvalues cross each other and consequently the system undergoes a DQPT.

Figure 2

Dynamical analog of the equation of state for the complex magnetization density $m(\mu ,t)$ as a function of the complex magnetic field $\mu$ for different times $t$. Left: Real part of the magnetization density as a function of $Jt$ and $\mu t$. Crossing the vertical line $Jt=\pi /4$ the magnetization density has a jump if the external field $\mu$ is different from zero. This reflects the nature of the DQPT occurring at $Jt=\pi /4$, which is continuouslike at vanishing field $\mu$ and first order otherwise. The sudden jump comes from a switching of the saddle point in Eq. (31), which affects $m$ via Eqs. (36) and (37). Right: Imaginary part of the magnetization density as a function of $Jt$ and $\mu t$. This quantity does not have any nonanalyticities in the $Jt\text{−}\mu t$ plane.

Figure 3

Real part of the effective free energy at different times in the complex magnetization density plane. Within the range of the displayed $m\in \mathbb{C}$ the effective free energy $f(m,t)$ does not exhibit sudden jumps as happens in Fig. 7. This is a consequence of the magnetization range which is chosen in such a way that $g(\mu ,t)$ in Eq. (32) and therefore $f(m,t)$ is obtained through either ${\varepsilon }_1$ or ${\varepsilon }_2$ which are continuous functions of their variables. The plot on the left is obtained with $t<t_c$, in particular $t=\frac{\pi }{4}-\frac{\pi }{32}$, while the plot on the right with $t>t_c$, in particular $t=\frac{\pi }{4}+\frac{\pi }{32}$.

Figure 4

Absolute value of the fixed point of the $y$ variable. The blue points indicate the couple of couplings ($\mu t,Jt$) which flow to the unstable fixed point.

Figure 5

Real part of the magnetization density $m\left(t\right)$ obtained from the EoS (36) as a function of $Jt$ for different values of $\mu$. The jump of $m\left(t\right)$ at the critical time $Jt=\pi /4$ decreases linearly with the field $\mu$. This feature outlines the nature of the DQPT which is first-order-like at nonzero field $\mu$, while it is continuous at vanishing field $\mu =0$.

Figure 6

Fluctuations of the magnetization $C\left(t\right)$ as a function of $Jt$. The result has been normalized in such a way that the maximum value is equal to one. We observe that the curve exhibits its maximum when a DQPT occurs: at $t=\pi /4$ and $t=3\pi /4$.

Figure 7

Absolute value of the effective free energy $f(m,t)$ for the following values of $Jt$: first row $Jt=\pi \frac{73}{320}$ and $Jt=\pi \frac{79}{320}$. Second row: $Jt=\pi \frac{81}{320}$ and $Jt=\pi \frac{87}{320}$. The jumps of the effective free energy $f(m,t)$ are manifested in the plot as sudden changes of the colors dark red and yellow.

Figure 8

Left: comparison between the exact result (blue continuous line) and the one obtained through the expansion (40) (red dotted line) of the real part of the dominant effective free energy (39). Right: same analysis but of the imaginary part.

Figure 9

Real part of the expansion of effective free energy density given by Eq. (40) at different times in the complex magnetization density plane. The times chosen and the interval of the complex magnetization density $m$ are the same as the ones of Fig. 3 to allow a direct comparison with that figure. The value of $\alpha$ coming from the best fitting procedure is: $\alpha =0.98+0.5i$.