Nonperturbative dynamics of quantum fields out of equilibrium is often described by the time evolution of a hierarchy of correlation functions, using approximation methods such as Hartree, large ${N}_{f},$ and $n\mathrm{PI}$ -effective action techniques. These truncation schemes can be implemented equally well in a classical statistical system, where results can be tested by comparison with the complete nonlinear evolution obtained by numerical methods. For a $(1+1)$ -dimensional scalar field we find that the early-time behavior is reproduced qualitatively by the Hartree dynamics. The inclusion of direct scattering improves this to the quantitative level. We show that the emergence of nonthermal temperature profiles at intermediate times can be understood in terms of the fixed points of the evolution equations in the Hartree approximation. The form of the profile depends explicitly on the initial ensemble. While the truncated evolution equations do not seem to be able to get away from the fixed point, the full nonlinear evolution shows thermalization with a (surprisingly) slow relaxation.Nonperturbative dynamics of quantum fields out of equilibrium is often described by the time evolution of a hierarchy of correlation functions, using approximation methods such as Hartree, large ${N}_{f},$ and $n\mathrm{PI}$-effective action techniques. These truncation schemes can be implemented equally well in a classical statistical system, where results can be tested by comparison with the complete nonlinear evolution obtained by numerical methods. For a $(1+1)$-dimensional scalar field we find that the early-time behavior is reproduced qualitatively by the Hartree dynamics. The inclusion of direct scattering improves this to the quantitative level. We show that the emergence of nonthermal temperature profiles at intermediate times can be understood in terms of the fixed points of the evolution equations in the Hartree approximation. The form of the profile depends explicitly on the initial ensemble. While the truncated evolution equations do not seem to be able to get away from the fixed point, the full nonlinear evolution shows thermalization with a (surprisingly) slow relaxation.

arXiv:hep-ph/0007357Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsDec 21, 200077 citations

Nonlinear Adiabatic Evolution of Quantum Systems

Jie Liu, Sheng-Chang Li, Libin Fu +1 more

Springer eBooksJan 1, 201811 citations

Nonlinear Adiabatic Evolution of Quantum Systems

Jie Liu, Sheng-Chang Li, Libin Fu +1 more

Springer eBooksJan 1, 20181 citation

Quantum Pontus-Mpemba Effect Enabled by the Liouvillian Skin Effect

Stefano Longhi

We unveil a quantum Pontus-Mpemba effect enabled by the Liouvillian skin effect in a dissipative tight-binding chain with asymmetric incoherent hopping and coherent boundary coupling. The skin effect, induced by non-reciprocal dissipation, localizes relaxation modes near the system boundaries and gives rise to non-orthogonal spectral geometry. While such non-normality is often linked to slow relaxation, we show that it can instead accelerate relaxation through a two-step protocol - realizing a quantum Pontus-Mpemba effect. Specifically, we consider a one-dimensional open chain with coherent hopping $J$ , asymmetric incoherent hoppings $J_{\rm R} \neq J_{\rm L}$ , and a controllable end-to-end coupling $ε$ . For $ε=0$ , the system exhibits the Liouvillian skin effect, with left and right eigenmodes localized at opposite edges. We compare two relaxation protocols toward the same stationary state: (i) a direct relaxation with $ε=0$ , and (ii) a two-step (Pontus) protocol where a brief coherent evolution transfers the excitation across the lattice before relaxation. Although both share the same asymptotic decay rate, the two-step protocol relaxes significantly faster due to its reduced overlap with the slow boundary-localized Liouvillian mode. The effect disappears when $J_{\rm R}=J_{\rm L}$ , i.e., when the skin effect vanishes. Our results reveal a clear connection between boundary-induced non-normality and protocol-dependent relaxation acceleration, suggesting new routes for controlling dissipation and transient dynamics in open quantum systems.We unveil a quantum Pontus-Mpemba effect enabled by the Liouvillian skin effect in a dissipative tight-binding chain with asymmetric incoherent hopping and coherent boundary coupling. The skin effect, induced by non-reciprocal dissipation, localizes relaxation modes near the system boundaries and gives rise to non-orthogonal spectral geometry. While such non-normality is often linked to slow relaxation, we show that it can instead accelerate relaxation through a two-step protocol - realizing a quantum Pontus-Mpemba effect. Specifically, we consider a one-dimensional open chain with coherent hopping $J$, asymmetric incoherent hoppings $J_{\rm R} \neq J_{\rm L}$, and a controllable end-to-end coupling $ε$. For $ε=0$, the system exhibits the Liouvillian skin effect, with left and right eigenmodes localized at opposite edges. We compare two relaxation protocols toward the same stationary state: (i) a direct relaxation with $ε=0$, and (ii) a two-step (Pontus) protocol where a brief coherent evolution transfers the excitation across the lattice before relaxation. Although both share the same asymptotic decay rate, the two-step protocol relaxes significantly faster due to its reduced overlap with the slow boundary-localized Liouvillian mode. The effect disappears when $J_{\rm R}=J_{\rm L}$, i.e., when the skin effect vanishes. Our results reveal a clear connection between boundary-induced non-normality and protocol-dependent relaxation acceleration, suggesting new routes for controlling dissipation and transient dynamics in open quantum systems.

quant-pharXiv:2601.14083Jan 21, 2026

Liouvillian Skin Effect: Slowing Down of Relaxation Processes without Gap Closing

Taiki Haga, Masaya Nakagawa, Ryusuke Hamazaki +1 more

It is highly nontrivial to what extent we can deduce the relaxation behavior of a quantum dissipative system from the spectral gap of the Liouvillian that governs the time evolution of the density matrix. We investigate the relaxation processes of a quantum dissipative system that exhibits the Liouvillian skin effect, which means that the eigenmodes of the Liouvillian are localized exponentially close to the boundary of the system, and find that the timescale for the system to reach a steady state depends not only on the Liouvillian gap $\mathrm{\ensuremath{\Delta}}$, but also on the localization length $\ensuremath{\xi}$ of the eigenmodes. In particular, we show that the longest relaxation time $\ensuremath{\tau}$ that is maximized over initial states and local observables is given by $\ensuremath{\tau}\ensuremath{\sim}{\mathrm{\ensuremath{\Delta}}}^{\ensuremath{-}1}(1+L/\ensuremath{\xi})$ with $L$ being the system size. This implies that the longest relaxation time can diverge for $L\ensuremath{\rightarrow}\ensuremath{\infty}$ without gap closing.It is highly nontrivial to what extent we can deduce the relaxation behavior of a quantum dissipative system from the spectral gap of the Liouvillian that governs the time evolution of the density matrix. We investigate the relaxation processes of a quantum dissipative system that exhibits the Liouvillian skin effect, which means that the eigenmodes of the Liouvillian are localized exponentially close to the boundary of the system, and find that the timescale for the system to reach a steady state depends not only on the Liouvillian gap $\mathrm{\ensuremath{\Delta}}$, but also on the localization length $\ensuremath{\xi}$ of the eigenmodes. In particular, we show that the longest relaxation time $\ensuremath{\tau}$ that is maximized over initial states and local observables is given by $\ensuremath{\tau}\ensuremath{\sim}{\mathrm{\ensuremath{\Delta}}}^{\ensuremath{-}1}(1+L/\ensuremath{\xi})$ with $L$ being the system size. This implies that the longest relaxation time can diverge for $L\ensuremath{\rightarrow}\ensuremath{\infty}$ without gap closing.

arXiv:2005.00824Physical Review LettersAug 13, 2021172 citations

Nonperturbative dynamics of quantum fields out of equilibrium is often described by the time evolution of a hierarchy of correlation functions, using approximation methods such as Hartree, large

{N}_{f},

and

n\mathrm{PI}

-effective action techniques. These truncation schemes can be implemented equally well in a classical statistical system, where results can be tested by comparison with the complete nonlinear evolution obtained by numerical methods. For a

(1+1)

-dimensional scalar field we find that the early-time behavior is reproduced qualitatively by the Hartree dynamics. The inclusion of direct scattering improves this to the quantitative level. We show that the emergence of nonthermal temperature profiles at intermediate times can be understood in terms of the fixed points of the evolution equations in the Hartree approximation. The form of the profile depends explicitly on the initial ensemble. While the truncated evolution equations do not seem to be able to get away from the fixed point, the full nonlinear evolution shows thermalization with a (surprisingly) slow relaxation.

J

, asymmetric incoherent hoppings

J_{\rm R} \neq J_{\rm L}

, and a controllable end-to-end coupling

ε

. For

ε=0

, the system exhibits the Liouvillian skin effect, with left and right eigenmodes localized at opposite edges. We compare two relaxation protocols toward the same stationary state: (i) a direct relaxation with

ε=0

, and (ii) a two-step (Pontus) protocol where a brief coherent evolution transfers the excitation across the lattice before relaxation. Although both share the same asymptotic decay rate, the two-step protocol relaxes significantly faster due to its reduced overlap with the slow boundary-localized Liouvillian mode. The effect disappears when

J_{\rm R}=J_{\rm L}

, i.e., when the skin effect vanishes. Our results reveal a clear connection between boundary-induced non-normality and protocol-dependent relaxation acceleration, suggesting new routes for controlling dissipation and transient dynamics in open quantum systems.

L

being the system size. This implies that the longest relaxation time can diverge for $L\ensuremath{\rightarrow}\ensuremath{\infty}$ without gap closing.