Abstract
We consider the class of dual-unitary quantum circuits in $1+1$ dimensions and introduce a notion of ``solvable'' matrix product states (MPSs), defined by a specific condition which allows us to tackle their time evolution analytically. We provide a classification of the latter, showing that they include certain MPSs of arbitrary bond dimension, and study analytically different aspects of their dynamics. For these initial states, we show that while any subsystem of size $\ell$ reaches infinite temperature after a time $t\propto \ell$, irrespective of the presence of conserved quantities, the light-cone of two-point correlation functions displays qualitatively different features depending on the ergodicity of the quantum circuit, defined by the behavior of infinite-temperature dynamical correlation functions. Furthermore, we study the entanglement spreading from such solvable initial states, providing a closed formula for the time evolution of the entanglement entropy of a connected block. This generalizes recent results obtained in the context of the self-dual kicked Ising model. By comparison, we also consider a family of non-solvable initial mixed states depending on one real parameter $\beta$, which, as $\beta$ is varied from zero to infinity, interpolate between the infinite temperature density matrix and arbitrary initial pure product states. We study analytically their dynamics for small values of $\beta$, and highlight the differences from the case of solvable MPSs.
Highlights
The extensive study of isolated quantum matter out of equilibrium carried out in the last two decades reminded us, once again, of how tremendously complex the quantum manybody dynamics can be [1,2,3]
Even though the past decade has witnessed the development of powerful numerical techniques based on matrix product states [4] (MPSs) that are able to determine, quite generally, the dynamics of quantum many-body systems in one spatial dimension [5,6,7,8,9,10], these methods are usually limited to small or intermediate timescales [10]
This is due to the generic linear growth of the entanglement entropy [11], which is a major obstacle for the MPS representation of the time-evolving state
Summary
The extensive study of isolated quantum matter out of equilibrium carried out in the last two decades reminded us, once again, of how tremendously complex the quantum manybody dynamics can be [1,2,3]. Gives rise to qualitative differences when compared to clean, homogeneous systems In this respect, an interesting class of quantum circuits without disorder, called “dual unitary,” was recently introduced [50], for which several dynamical features could be investigated analytically. In this paper we show that there exists a much broader family of “solvable” initial states, for which the dynamics can be tackled analytically, irrespective of the choice of the dual-unitary gates This class includes MPSs of arbitrary bond dimension, and allows for the exact computation of several quantities beyond the growth of bipartite entanglement entropy, including the spreading of two-point correlation functions, and the thermalization time of finite subsystems. The most technical aspects of our study are consigned to two appendices
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