Exact dynamics in dual-unitary quantum circuits

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 reaches infinite temperature after a time t ∝ , 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 β, which, as β 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 β, and highlight the differences from the case of solvable MPSs.

Entanglement Membrane in Chaotic Many-Body Systems

“…We expect that the pairings, and the membrane description, continue to make sense in systems more general and more realistic than random circuits, for the heuristic reason described in the next paragraph. In support of this expectation, there is numerical evidence that a consistent membrane tension can be defined in a generic spin chain [18] and in a dual-unitary circuit [35,36]. There are analytic computations in random Floquet circuits in the large Hilbert space dimension (large q) limit [11], showing that the domain wall structure makes sense in that limit.…”

“…In our discussion of the numerical results, it is sometimes useful to use the following parameterization of the two-site gate u, which is an arbitrary SUð4Þ matrix [36,47,70,71]:…”

“…In special limits, such as h z ¼ 0 or h x ¼ 0, the model becomes integrable: In those limits, we do not expect our algorithm to succeed. Interestingly, this model also has a "self-dual" line in parameter space [35,36,[46][47][48]72]:…”

“…This fact is related to the property that, for this choice of parameters, the tensor u remains unitary if it is rotated so as to exchange the roles of space and time [36,46,47], which is referred to as "dual unitarity." Though this property is highly fine-tuned, the model is believed to remain chaotic for generic values of h z on this line [36,72]. The fact that v E ¼ 1 for the dual-unitary models implies v B ¼ 1 and that the line tension is flat as a function of velocity [18]:…”

“…We also investigate a special case of the kicked-Ising model that has maximal entanglement growth [35,36] and the property of "dual unitarity" which, remarkably, allows a range of exact computations [36,[46][47][48][49].…”