Dynamical quantum phase transitions of quantum spin chains with a Loschmidt-rate critical exponent equal to $\frac{1}{2}$

We describe a new universality class of dynamical quantum phase transitions of the Loschmidt amplitude of quantum spin chains off equilibrium criticality. We demonstrate that in many cases it is possible to change the conventional linear singularity of the Loschmidt rate function into a smooth peak by tuning one parameter of the quench protocol. Exactly at the point when this change-over occurs, the singularity of the Loschmidt rate function persists, with a critical exponent equal to $\frac{1}{2}$. The nonequilibrium renormalization group fixed point controlling this universality class is described. An asymptotically exact renormalization group recursion relation is derived around this fixed point to obtain the critical exponent.

Figure 1

The rate function of the three-state Potts chain. Left: The rate function for $J=0.9J_c,1.1J_c$, and $J_c$, obtained through the iTEBD algorithm. Right: The log-log plot of the rate function at $J_c$ for $t\in [t_c-4.832\times {10}^{-6},t_c-1\times {10}^{-6}]$ and $t\in [t_c+1\times {10}^{-6},t_c+1.88\times {10}^{-5}]$, where $t_c=0.9449044833$. A linear fit is performed on this plot to obtain the critical exponents.

Figure 2

The fixed-point transfer matrix of the Potts chain for $J=0.9J_c,1.1J_c$, and $J_c$. Left: The real part of the matrix element of ${\mathbf{T}}^{*}$ in row 2 and column 2. Right: The $|1+{\mathbf{v}}^T\mathbf{u}|$ of ${\mathbf{T}}^{*}$. The inset is a blowup at $J_c$ for $t$ near $t_c$. ${\mathbf{T}}^{*}$ is taken as the renormalized transfer matrix after 500 RG iterations.

Figure 3

The modulus of the leading and the subleading eigenvalues, ${\lambda }_1$ and ${\lambda }_2$, of the transfer matrix of the XY Ising chain with $h_0=3,h_1=3,{\gamma }_0=3$. ${\gamma }_1={\gamma }_c-0.1$ on the left, ${\gamma }_c$ in the middle, and ${\gamma }_c+0.1$ on the right. The transfer matrix is obtained with iTEBD.