Criticality and entanglement in non-unitary quantum circuits and tensor networks of non-interacting fermions

TL;DR

This paper establishes a correspondence between non-unitary quantum circuits of non-interacting fermions and higher-dimensional Hamiltonian problems, revealing critical entanglement behavior linked to symmetry classes and providing new insights into quantum criticality.

Contribution

It introduces a full correspondence between non-unitary fermionic circuits and static Hamiltonian problems in higher dimensions, classifies their symmetries, and connects criticality to entanglement phases.

Findings

Criticality in symmetry classes explains entanglement properties.

Most general tensor networks correspond to class DIII Hamiltonians.

Numerical evidence of stable critical metallic phases in D=2 and D=3.

Abstract

Models for non-unitary quantum dynamics, such as quantum circuits that include projective measurements, have been shown to exhibit rich quantum critical behavior. There are many complementary perspectives on this behavior. For example, there is a known correspondence between d-dimensional local non-unitary quantum circuits and tensor networks on a D=(d+1)-dimensional lattice. Here, we show that in the case of systems of non-interacting fermions, there is furthermore a full correspondence between non-unitary circuits in d spatial dimensions and unitary non-interacting fermion problems with static Hermitian Hamiltonians in D=(d+1) spatial dimensions. This provides a powerful new perspective for understanding entanglement phases and critical behavior exhibited by non-interacting circuits. Classifying the symmetries of the corresponding non-interacting Hamiltonian, we show that a large class of random circuits, including the most generic circuits with randomness in space and time, are in correspondence with Hamiltonians with static spatial disorder in the ten Altland-Zirnbauer symmetry classes. We find the criticality that is known to occur in all of these classes to be the origin of the critical entanglement properties of the corresponding random non-unitary circuit. To exemplify this, we numerically study the quantum states at the boundary of Haar-random Gaussian fermionic tensor networks of dimension D=2 and D=3. We show that the most general such tensor network ensemble corresponds to a unitary problem of non-interacting fermions with static disorder in Altland-Zirnbauer symmetry class DIII, which for both D=2 and D=3 is known to exhibit a stable critical metallic phase. Tensor networks and corresponding random non-unitary circuits in the other nine Altland-Zirnbauer symmetry classes can be obtained from the DIII case by implementing Clifford algebra extensions for classifying spaces.