We explore non-unitary dynamics of
2+1 dimensional free fermions and show that the obtained steady state is critical regardless the strength of the non-unitary evolution. Numerical results indicate that the entanglement entropy has a logarithmic violation of the area-law and the mutual information between two distant regions decays as a power-law function. In particular, we provide an interpretation of these scaling behaviors in terms of a simple quasi-particle pair picture. In addition, we study the dynamics of the correlation function and demonstrate that this system has dynamical exponent
z=1. We argue the dynamics of the correlation function in terms of a classical non-linear master equation. Our method opens a door to a vast number of non-unitary random dynamics in free fermions and can be generalized to any dimensions.
We investigate the entanglement structure and wave function characteristics of continuously monitored free fermions with
U(1) symmetry in two spatial dimensions. By deriving the exact fermion replica-quantum master equation, we line out two approaches: (i) a nonlinear sigma model analogous to disordered free fermions, resulting in an
SU(R)-symmetric field theory of symmetry class AIII in (
2+1) space-time dimensions, or (ii) for bipartite lattices, third quantization leading to a non-Hermitian
SU(2R)-symmetric Hubbard model. Using exact numerical simulations, we explore the phenomenology of the entanglement transition in two-dimensional (2D) monitored fermions, examining entanglement entropy and wave function inverse participation ratio. At weak monitoring, we observe characteristic
LlogL entanglement growth and multifractal dimension
Dq=2, resembling a metallic Fermi liquid. Under strong monitoring, wave functions localize and the entanglement saturates towards an area law. Between these regimes, we identify a high-symmetry point exhibiting both entanglement growth indicative of emergent conformal invariance and maximal multifractal behavior. While this multifractal behavior aligns with the nonlinear sigma model of the Anderson transition, the emergent conformal invariance is an unexpected feature not typically associated with Anderson localization. These discoveries add a new dimension to the study of 2D monitored fermions and underscore the need to further explore the connection between nonunitary quantum dynamics in
D dimensions and quantum statistical mechanics in
D+1 dimensions.
We explore the multifractality of the steady state wave function in nonunitary random quantum dynamics in one dimension. We focus on two classes of random systems: the hybrid Clifford circuit model and the nonunitary free fermion dynamics. In the hybrid Clifford model, we map the measurement driven transition to an Anderson localization transition in an effective graph space by using properties of the stabilizer state. We show that the volume law phase with nonzero measurement rate is nonergodic in the graph space and exhibits weak multifractal behavior. We apply the same method to the hybrid Clifford quantum automaton circuit and obtain similar multifractality in the volume law phase. For the nonunitary random free fermion system with a critical steady state, we compute the moments of the probability distribution of the single-particle wave function and demonstrate that it is also weakly multifractal and has strong variations in real space.
We construct a series of one-dimensional non-unitary dynamics consisting of both unitary and imaginary evolutions based on the Sachdev-Ye-Kitaev model. Starting from a short-range entangled state, we analyze the entanglement dynamics using the path integral formalism in the large N limit. Among all the results that we obtain, two of them are particularly interesting: (1) By varying the strength of the imaginary evolution, the interacting model exhibits a first order phase transition from the highly entangled volume law phase to an area law phase; (2) The one-dimensional free fermion model displays an extensive critical regime with emergent two-dimensional conformal symmetry.
Continuous monitoring of one-dimensional free fermionic systems can generate phenomena reminiscent of quantum criticality, such as logarithmic entanglement growth, algebraic correlations, and emergent conformal invariance, but in a nonequilibrium setting. However, whether these signatures reflect a genuine phase of nonequilibrium quantum matter or persist only over finite length scales is an active area of research. We address this question in a free fermionic chain subject to continuous monitoring of lattice-site occupations. An unraveling phase φ interpolates between measurement schemes, corresponding to different stochastic unravelings of the same Lindblad master equation; for φ = 0 , measurements disentangle lattice sites, while for φ = π / 2 they act as unitary random noise, yielding volume-law steady-state entanglement. Using replica Keldysh field theory, we obtain a nonlinear sigma model describing the long-wavelength physics. This analysis shows that for 0 ≤ φ < π / 2 , entanglement ultimately obeys an area law, but only beyond the exponentially large scale ln ( l φ , * ) ∼ J / [ γ cos ( φ ) ] , where J is the hopping amplitude and γ the measurement rate. Resolving l φ , * in numerical simulations is difficult for γ / J → 0 or φ → π / 2 . However, the theory also predicts that critical-like behavior appears below a crossover scale that grows only algebraically in J / γ , making it numerically accessible. Our simulations confirm these predictions, establishing the absence of measurement- or unraveling-induced entanglement transitions in this model.
Random fractal lattices provide a geometrically disordered setting in which quantum correlations can be shaped by noninteger dimensionality rather than onsite randomness. We investigate the entanglement properties of noninteracting fermions on random fractal lattices generated by a stochastic growth algorithm. By varying the growth parameter and adding missing links with probability
p, we tune the Hausdorff and spectral dimensions while keeping the system free of onsite disorder. For ground states at different fillings, we compute the bipartite entanglement entropy of subregions defined by graph distance and analyze its scaling with subsystem size. Over a broad parameter range, we find robust power-law behavior governed primarily by the Hausdorff dimension, consistent with a generalized area law and without the logarithmic enhancement familiar from Euclidean free fermions. We also study entanglement growth following a global quench from an uncorrelated checkerboard state and uncover an asymptotic scaling collapse in which the subsystem-size dependence is governed by the Hausdorff dimension, while the temporal evolution is governed by the spectral dimension. The resulting dynamics are logarithmically slow over an extended intermediate-time window. These results show that geometric randomness alone can generate both nontrivial ground-state entanglement structure and slow quantum-information spreading in free-fermion systems.