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 (2021 arXiv: 2105.10161 ) and Witten (2021 arXiv: 2111.06514 ) 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 the 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 exhibit 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.
Using the Wei-Norman theory, we obtain a time-dependent complex Riccati equation (TDCRE) as the solution of the time evolution operator (TEO) of quantum systems described by time-dependent (TD) Hamiltonians that are linear combinations of the generators of the
su(1,1),
su(2), and
so(2,1) Lie algebras. Using a recently developed solution for the time evolution of these quantum systems, we solve the TDCRE recursively as generalized continued fractions, which are optimal for numerical implementations, and establish the necessary and sufficient conditions for the unitarity of the TEO in the factorized representation. The inherited symmetries of quantum systems can be recognized by a simple inspection of the TDCRE, allowing effective quantum Hamiltonians to be associated with it, as we show for the Bloch-Riccati equation whose Hamiltonian corresponds to that of a generic TD system of the Lie algebra
su(2). As an application, but also as a consistency test, we compare our solution with the analytic one for the Bloch-Riccati equation considering the Rabi frequency driven by a complex hyperbolic secant pulse generating spin inversion, showing an excellent agreement.
Quantum Monte Carlo is one of the most powerful numerical tools for studying nonpeturbative properties of quantum many-body systems. However, its application to real-time problems is limited since the complex and highly-oscillating path-integral weight of the real-time evolution harms the important sampling. % , which is the notorious sign problem. In this Letter, we show that some real-time problems in open fermion systems can be simulated using the quantum Monte Carlo. To this end, we prescribe a mapping between a real-time problem in open quantum systems and a statistical problem in non-Hermitian quantum systems; for some cases, the latter can be solved without suffering from the complex measure problem. To explain our idea and demonstrate how it works, we compute the real-time evolution of fidelities in open fermion systems under dissipation.