Exactly solvable non-unitary time evolution in quantum critical systems I: Effect of complex spacetime metrics

Abstract

In this series of works, we study exactly solvable non-unitary time evolutions in one-dimensional quantum critical systems ranging from quantum quenches to time-dependent drivings. In this part I, we are motivated by the recent works of Kontsevich and Segal [1] and Witten [2] on allowable complex spacetime metrics in quantum field theories. In general, such complex spacetime metrics will lead to non-unitary time evolutions. In this work, we study the universal features of such non-unitary time evolutions based on exactly solvable setups. Various physical quantities including entanglement Hamiltonian and entanglement spectrum, entanglement entropy, and energy density at an arbitrary time can be exactly solved. Due to the damping effect introduced by the complex time, the excitations in the initial state are gradually damped out in time. The non-equilibrium dynamics exhibits universal features that are qualitatively different from the case of real-time evolutions. For instance, for an infinite system after a global quench, the entanglement entropy of the semi-infinite subsystem will grow logarithmically in time, in contrast to the linear growth in a real-time evolution. Moreover, we study numerically the time-dependent driven quantum critical systems with allowable complex spacetime metrics. It is found that the competition between driving and damping leads to a steady state with an interesting entanglement structure.

Figures (16)

• Figure 1: Path integral of one-point function with different spacetime metrics (see the main text for details). (a) Euclidean metric, (b) Lorentz metric, and (c) complex metric.
• Figure 2: Path integral of the reduced density matrix $\rho_A$ after a global quench in a CFT with (a) Euclidean (b) Partial Euclidean and partial Lorentz, and (c) Complex spacetime metrics. The fields living on the upper and lower edges of the brunch cut (in blue) correspond to the rows and columns of $\rho_A$.
• Figure 3: The path integral of $\rho_A=\text{Tr}_{\overline A} \rho(\tau_1,\tau_2)$ for $A=[0,+\infty)$ in the Euclidean space. The width of the strip is $\beta/2+2\epsilon \tau_1$, with the two boundaries located along $\operatorname{Im} z=-\beta/4-\epsilon \tau_1$ and $\beta/4+\epsilon \tau_1$ respectively. The branch cut corresponding to subsystem $A$ is along $C=\{i\tau_2+x, \, x\ge 0\}$. A small disk of radius $\epsilon_0$ is removed at the entanglement point $z_0=0+i\tau_2$, with a conformal boundary condition $|a\rangle\rangle$ imposed along the boundary of this removed disk. After a conformal mapping $w=f(z)$, the strip (left) is mapped to a cylinder (right) of length $W$ in the $\operatorname{Re} w$ direction. The circumference of the cylinder along the $\operatorname{Im} w$ direction is $2\pi$.
• Figure 4: Comparison of complex time evolution of entanglement entropy $S_A(t)$ after a global quench in lattice system and CFT calculations. The lattice system is defined on $[0, L]=[0, 800]$, and the subsystem is $A=[0,400]$. The mass term in the lattice model is set as $m=1/2$ in \ref{['gap_Hamiltonian']}. From top to bottom, we choose $\epsilon$ (which characterizes the imaginary part of the complex time) as $\epsilon=0$ (Lorentz metric), $0.01$, $0.05$, and $0.1$ The CFT result is plotted according to \ref{['SA_general']} and \ref{['W:Global1']}, and the fitting parameters are $\beta=3.75$ and $\epsilon_0=0.1$.
• Figure 5: Comparison of energy density evolution $E(x,t)-E_G(x)$ after a global quench in complex spacetime metric in a free fermion lattice system (green circles) and in a CFT calculation (red dashed lines). Here $E_G(x)$ is the energy density in the ground state of a CFT. The lattice system is defined on $[0, L]=[0, 800]$. The mass term used in the initial state of the lattice model is set as $m=1/2$ in \ref{['gap_Hamiltonian']}. The energy density is calculated by considering the average $\frac{1}{l}\int_{L/2-l/2}^{L/2+l/2} E(x,t)dx$ with $l=100$ in the lattice calculation. From top to bottom, we have $\epsilon=0$, $0.01$, $0.05$, and $0.1$. The fitting parameter in CFT is $\beta=2.96$.