Abstract
We present a quantum circuit with measurements and post-selection that exhibits a panoply of space- and/or time-ordered phases, from ferromagnetic order to spin-density waves to time crystals. Unlike the time crystals that have been found in unitary models, those that occur here are \emph{incommensurate} with the drive frequency. The period of the incommensurate time-crystal phase may be tuned by adjusting the circuit parameters. We demonstrate that the phases of our quantum circuit, including the inherently non-equilibrium dynamical ones, correspond to complex-temperature equilibrium phases of the exactly solvable square-lattice anisotropic Ising model.
Highlights
For a many-body quantum system with Hamiltonian operator H, there is an evident formal similarity between the unitary time-evolution operator, e−iHt/h, and the density operator for a thermal equilibrium state, e−βH
In the present work we have exploited the correspondence between nonunitary quantum circuits and complex temperature statistical mechanics to construct a simple quantum circuit that has a surprisingly rich phase diagram, including a phase with incommensurate temporal order
Such incommensurate time crystals do not seem to occur in closed systems, nor do they occur in 1D open quantum systems with shortrange interactions, for entropic reasons
Summary
For a many-body quantum system with Hamiltonian operator H , there is an evident formal similarity between the unitary time-evolution operator, e−iHt/h , and the density operator for a thermal equilibrium state, e−βH. At least one of these latter phases bears a phenomenological resemblance to the time crystals recently discussed in the context of unitary dynamics of isolated many-body localized systems [31] It does not fit into the classification presented in that work, since the circuits we consider are nonunitary, and the no-go theorems [32] forbidding timecrystalline order do not apply. The present work applies tools from the complex-temperature statistical mechanics literature [47,48] to discuss the unexplored physics of spatiotemporal correlations in nonunitary quantum circuits These circuits have primarily been studied for their entanglement properties; we demonstrate here that even their conventional correlation functions can exhibit striking phenomena that would be forbidden by unitarity (in closed systems) or by dimensionality (in open systems [33,34]). We conclude with a summary and a discussion of open problems
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