Abstract
We introduce a class of hybrid quantum circuits, with random unitaries and projective measurements, which host long-range order in the area law entanglement phase of the steady state. Our primary example is circuits with unitaries respecting a global Ising symmetry and two competing types of measurements. The phase diagram has an area law phase with spin glass order, which undergoes a direct transition to a paramagnetic phase with volume law entanglement, as well as a critical regime. Using mutual information diagnostics, we find that such entanglement transitions preserving a global symmetry are in new universality classes. We analyze generalizations of such hybrid circuits to higher dimensions, which allow for coexistence of order and volume law entanglement, as well as topological order without any symmetry restrictions.
Highlights
A major frontier of quantum many-body physics is understanding what types of order can be stabilized in nonequilibrium settings
Much progress has stemmed from many-body localization [1,2], which is characterized by the breakdown of thermalization and the restricted growth of entanglement
An alternative approach for restricting entanglement growth has been proposed in hybrid quantum circuits involving both unitary evolution and measurements [6,7,8,9,10,11]
Summary
A major frontier of quantum many-body physics is understanding what types of order can be stabilized in nonequilibrium settings. We present a class of hybrid circuits which hosts long-range quantum order within the area-law phase. We demonstrate the existence of long-range spin-glass order in the area-law phase of a class of hybrid circuits with unitaries respecting a global Z2 symmetry and two competing types of measurements. For two neighboring qubits i, i + 1, we define M1 to be the projective measurement of ZiZi+1 and M2 to be the projective measurement of Xi. As is the case in previous works, it is convenient for scalable simulation to choose the random unitary from an ensemble of Clifford gates, which have the property of mapping a string of Pauli operators to another Pauli string (in the Heisenberg picture). The scaling of this order parameter with system size (constant versus linear) can be used to identify the spin-glass phase After averaging, these quantities SA, O depend only on the parameters of the circuit ensemble p, r
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