Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum criticality in non-unitary dynamics of 2+1d free fermions

Qicheng Tang, Xiao Dong Chen, W. Zhu

Published Jan 13, 2021• 1 citation
OpenAlex

Abstract

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.

Related papers

Emergent conformal symmetry in nonunitary random dynamics of free fermions

Xiao Chen, Yaodong Li, Matthew P. A. Fisher +1 more

We present random quantum circuit models for nonunitary quantum dynamics of free fermions in one spatial dimension.Numerical simulations reveal that the dynamics tends toward steady states with logarithmic violations of the entanglement area law and power law correlation functions.Moreover, starting with a short-range entangled many-body state, the dynamical evolution of entanglement and correlations quantitatively agrees with the predictions of two-dimensional conformal field theory with a spacelike time direction.We argue that this behavior is generic in nonunitary free quantum dynamics with time-dependent randomness, and we show that the emergent conformal dynamics of two-point functions arises out of a simple "nonlinear master equation."

arXiv:2004.09577•Physical Review Research•Jul 6, 2020•164 citations

Entanglement transition in unitary system-bath dynamics

Bo Xing, Giuliano Chiriacò, Paola Cappellaro +2 more

The evolution of a system coupled to baths is commonly described by a master equation that, in the long-time limit, yields a steady-state density matrix. However, when the same evolution is unraveled into quantum trajectories, it is possible to observe a transition in the scaling of entanglement within the system as the system-bath coupling increases - a phenomenon that is invisible in the trajectory-averaged reduced density matrix of the system. Here, we go beyond the paradigm of trajectories from master equations and explore whether a qualitatively analogous entanglement-scaling transition emerges in a single unitary evolution of the combined system-bath setup, without monitoring the dynamics of the system. We investigate the scaling of entanglement in a unitary quantum setup composed of a two-dimensional lattice of free fermions, where each site is coupled to a fermionic bath. As the system-bath coupling increases, the logarithmic fermionic negativity reveals an entanglement transition from logarithmic-law to area-law scaling. This occurs while the system's steady-state properties are trivial, highlighting that the signatures of these different scalings are within the bath-bath correlations. Evidence of the transition is also found in the mutual information and the correlations of the full system-bath setup, suggesting that the entanglement transition is underpinned by a change in the spatial structure of quantum information.

quant-pharXiv:2512.06081•Physical Review Letters•Dec 9, 2025•1 citation

Measurement-induced phase transitions for free fermions in a quasiperiodic potential

Toranosuke Matsubara, Kazuki Yamamoto, Akihisa Koga

We study the dynamics under continuous measurements for free fermions in a quasiperiodic potential by using the Aubry-Andr\'e-Harper model with hopping rate $J$ and potential strength $V$. On the basis of the quantum trajectory method, we obtain the phase diagram for the steady-state entanglement entropy and demonstrate that robust logarithmic system-size scaling emerges up to a critical potential strength ${V}_{c}/J\ensuremath{\sim}2.3$. Moreover, we find that the measurement induces entanglement phase transitions from the logarithmic-law phase to the area-law phase for the potential strength $V<{V}_{c}$, while any finite measurement stabilizes the area-law phase for $V>{V}_{c}$. This result is distinct from the entanglement scaling in the unitary limit, where volume-law and area-law phases undergo a transition at $V/J=2$. To further support the phase diagram, we analyze the connected correlation function and find that it shows algebraic decay in the logarithmic-law phase, while it decays quickly in the area-law phase. Our results can be tested in ultracold atoms by introducing quasiperiodic potentials and continuously monitoring the local occupation number with an off-resonant probe light.

arXiv:2503.23807•Physical review. B./Physical review. B•Aug 1, 2025•4 citations

Magic transition in monitored free fermion dynamics

Yucheng Wang, Zhi-Cheng Yang, Tianci Zhou +1 more

We investigate magic and its connection to entanglement in 1+1 dimensional random free fermion circuits, with a focus on hybrid free fermion dynamics that can exhibit an entanglement phase transition. To quantify magic, we use the Stabilizer Rényi Entropy (SRE), which we compute numerically via a perfect sampling algorithm. We show that although the SRE remains extensive as the system transitions from a critical phase to an area-law (disentangled) phase, the structure of magic itself undergoes a delocalization phase transition. This transition is characterized using the bipartite stabilizer mutual information, which exhibits the same scaling behavior as entanglement entropy: logarithmic scaling in the critical phase and a finite constant in the area-law phase. Additionally, we explore the dynamics of SRE. While the total SRE becomes extensive in $O(1)$ time, we find that in the critical phase, the relaxation time to the steady-state value is parameterically longer than that in generic random circuits. The relaxation follows a universal form, with a relaxation time that grows linearly with the system size, providing further evidence for the critical nature of the phase.

arXiv:2507.10688•Jul 14, 2025•0 citations

Entanglement transitions and multifractality in monitored free-fermions with random long-range hopping

Bo Fan, Antonio M. García-García

We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent $α$. For $α\lesssim 1$, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size $L$ is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small $α\lesssim 1/2$. For $α> 3/2$, the EE is in the area-law phase, namely, no scaling with $L$, for any monitoring strength. For $1 < α\lesssim 3/2$, we identify an $α$-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.

quant-pharXiv:2607.23871•Jul 28, 2026•0 citations

Universal Entanglement Transitions of Free Fermions with Long-range Non-unitary Dynamics

Pengfei Zhang, Chunxiao Liu, Shao-Kai Jian +1 more

Non-unitary evolution can give rise to novel steady states classified by their entanglement properties. In this work, we aim to understand the effect of long-range hopping that decays with r−α in non-Hermitian free-fermion systems. We first study two solvable Brownian models with long-range non-unitary dynamics: a large- N SYK 2 chain and a single-flavor fermion chain and we show that they share the same phase diagram. When α>0.5 , we observe two critical phases with subvolume entanglement scaling: (i) α>1.5 , a logarithmic phase with dynamical exponent z=1 and logarithmic subsystem entanglement, and (ii) 0.5<α<1.5 , a fractal phase with z=2α−12 and subsystem entanglement SA∝LA1−z , where LA is the length of the subsystem A . These two phases cannot be distinguished by the purification dynamics, in which the entropy always decays as L/T . We then confirm that the results are also valid for the static SYK 2 chain, indicating the phase diagram is universal for general free-fermion systems. We also discuss phase diagrams in higher dimensions and the implication in measurement-induced phase transitions.

arXiv:2105.08895•Quantum•May 27, 2022•48 citations
We explore non-unitary dynamics of 2+12+12+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=1z=1z=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 present random quantum circuit models for nonunitary quantum dynamics of free fermions in one spatial dimension.Numerical simulations reveal that the dynamics tends toward steady states with logarithmic violations of the entanglement area law and power law correlation functions.Moreover, starting with a short-range entangled many-body state, the dynamical evolution of entanglement and correlations quantitatively agrees with the predictions of two-dimensional conformal field theory with a spacelike time direction.We argue that this behavior is generic in nonunitary free quantum dynamics with time-dependent randomness, and we show that the emergent conformal dynamics of two-point functions arises out of a simple "nonlinear master equation."
The evolution of a system coupled to baths is commonly described by a master equation that, in the long-time limit, yields a steady-state density matrix. However, when the same evolution is unraveled into quantum trajectories, it is possible to observe a transition in the scaling of entanglement within the system as the system-bath coupling increases - a phenomenon that is invisible in the trajectory-averaged reduced density matrix of the system. Here, we go beyond the paradigm of trajectories from master equations and explore whether a qualitatively analogous entanglement-scaling transition emerges in a single unitary evolution of the combined system-bath setup, without monitoring the dynamics of the system. We investigate the scaling of entanglement in a unitary quantum setup composed of a two-dimensional lattice of free fermions, where each site is coupled to a fermionic bath. As the system-bath coupling increases, the logarithmic fermionic negativity reveals an entanglement transition from logarithmic-law to area-law scaling. This occurs while the system's steady-state properties are trivial, highlighting that the signatures of these different scalings are within the bath-bath correlations. Evidence of the transition is also found in the mutual information and the correlations of the full system-bath setup, suggesting that the entanglement transition is underpinned by a change in the spatial structure of quantum information.
We study the dynamics under continuous measurements for free fermions in a quasiperiodic potential by using the Aubry-Andr\'e-Harper model with hopping rate JJJ and potential strength VVV. On the basis of the quantum trajectory method, we obtain the phase diagram for the steady-state entanglement entropy and demonstrate that robust logarithmic system-size scaling emerges up to a critical potential strength ${V}_{c}/J\ensuremath{\sim}2.3$. Moreover, we find that the measurement induces entanglement phase transitions from the logarithmic-law phase to the area-law phase for the potential strength V<VcV<{V}_{c}V<Vc​, while any finite measurement stabilizes the area-law phase for V>VcV>{V}_{c}V>Vc​. This result is distinct from the entanglement scaling in the unitary limit, where volume-law and area-law phases undergo a transition at V/J=2V/J=2V/J=2. To further support the phase diagram, we analyze the connected correlation function and find that it shows algebraic decay in the logarithmic-law phase, while it decays quickly in the area-law phase. Our results can be tested in ultracold atoms by introducing quasiperiodic potentials and continuously monitoring the local occupation number with an off-resonant probe light.
We investigate magic and its connection to entanglement in 1+1 dimensional random free fermion circuits, with a focus on hybrid free fermion dynamics that can exhibit an entanglement phase transition. To quantify magic, we use the Stabilizer Rényi Entropy (SRE), which we compute numerically via a perfect sampling algorithm. We show that although the SRE remains extensive as the system transitions from a critical phase to an area-law (disentangled) phase, the structure of magic itself undergoes a delocalization phase transition. This transition is characterized using the bipartite stabilizer mutual information, which exhibits the same scaling behavior as entanglement entropy: logarithmic scaling in the critical phase and a finite constant in the area-law phase. Additionally, we explore the dynamics of SRE. While the total SRE becomes extensive in O(1)O(1)O(1) time, we find that in the critical phase, the relaxation time to the steady-state value is parameterically longer than that in generic random circuits. The relaxation follows a universal form, with a relaxation time that grows linearly with the system size, providing further evidence for the critical nature of the phase.
We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent ααα. For α≲1α\lesssim 1α≲1, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size LLL is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small α≲1/2α\lesssim 1/2α≲1/2. For α>3/2α> 3/2α>3/2, the EE is in the area-law phase, namely, no scaling with LLL, for any monitoring strength. For 1<α≲3/21 < α\lesssim 3/21<α≲3/2, we identify an ααα-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.
Non-unitary evolution can give rise to novel steady states classified by their entanglement properties. In this work, we aim to understand the effect of long-range hopping that decays with r−α in non-Hermitian free-fermion systems. We first study two solvable Brownian models with long-range non-unitary dynamics: a large- N SYK 2 chain and a single-flavor fermion chain and we show that they share the same phase diagram. When α>0.5 , we observe two critical phases with subvolume entanglement scaling: (i) α>1.5 , a logarithmic phase with dynamical exponent z=1 and logarithmic subsystem entanglement, and (ii) 0.5<α<1.5 , a fractal phase with z=2α−12 and subsystem entanglement SA∝LA1−z , where LA is the length of the subsystem A . These two phases cannot be distinguished by the purification dynamics, in which the entropy always decays as L/T . We then confirm that the results are also valid for the static SYK 2 chain, indicating the phase diagram is universal for general free-fermion systems. We also discuss phase diagrams in higher dimensions and the implication in measurement-induced phase transitions.