Dissipation meets conformal interface in open quantum systems: How the relaxation rate is suppressed

TL;DR

This paper introduces a novel relaxation descriptor, $c_{ ext{relax}}$, to characterize conformal interfaces in open quantum critical systems by analyzing the Liouvillian gap in a dissipative free-fermion chain. Through perturbative and exact-numerical methods, it shows that the Liouvillian gap scales as $g( ext{λ}) \\propto L^{-3}$ and that $c_{ ext{relax}}/c \\approx 1 - \\sqrt{1 - \\lambda^{2}}$ in the weak-dissipation regime, with universal inequalities $0 \\le c_{ ext{relax}} \\le c_{ ext{LR}} \\le c_{ ext{eff}} \\le c$ that tighten to equalities at endpoints $\\lambda=0,1$. The analysis leverages the third-quantization formalism to link the Liouvillian spectrum to a non-Hermitian effective Hamiltonian and derives both perturbative and finite-$\\gamma$ results for the gap, demonstrating the universality and independence from dissipation location in the weak-dissipation limit. These findings reveal a distinct, universal relaxation signature of conformal interfaces in open quantum systems, offering new insights beyond previously understood closed-system measures like $c_{ ext{LR}}$ and $c_{ ext{eff}}$. The work suggests broad avenues for future exploration, including extensions to other CFTs, holographic approaches, and potential experimental probes of dissipative interface physics.

Abstract

Conformal interfaces play an important role in quantum critical systems. In closed systems, the transmission properties of conformal interfaces are typically characterized by two quantities: One is the effective central charge $c_{\text{eff}}$, which measures the amount of quantum entanglement through the interface, and the other is the transmission coefficient $c_{\text{LR}}$, which measures the energy transmission through the interface. In the present work, to characterize the transmission property of conformal interfaces in open quantum systems, we propose a third quantity $c_{\text{relax}}$, which is defined through the ratio of Liouvillian gaps with and without an interface. Physically, $c_{\text{relax}}$ measures the suppression of the relaxation rate towards a steady state when the system is subject to a local dissipation. We perform both analytical perturbation calculations and exact numerical calculations based on a free fermion chain at the critical point. It is found that $c_{\text{relax}}$ decreases monotonically with the strength of the interface. In particular, $0\le c_{\text{relax}}\le c_{\text{LR}}\le c_{\text{eff}}$, where the equalities hold if and only if the interface is totally reflective or totally transmissive. Our result for $c_{\text{relax}}$ is universal in the sense that $c_{\text{relax}}$ is independent of (i) the dissipation strength in the weak dissipation regime and (ii) the location where the local dissipation is introduced. Comparing to the previously known $c_{\text{LR}}$ and $c_{\text{eff}}$ in a closed system, our $c_{\text{relax}}$ shows a distinct behavior as a function of the interface strength, suggesting its novelty to characterize conformal interfaces in open systems and offering insights into critical systems under dissipation.

Figures (13)

• Figure 1: Summary of the main results in this work. (a) A sketch for introducing a local dissipation to a one-dimensional quantum critical system in the presence of a conformal interface. The total system is of length $L=2N$, the conformal interface is located at $x=N$, and the dissipation is introduced at $x=n_d$. (b) The relaxation coefficient $c_{\text{relax}}$ and its comparison with $c_{\text{LR}}$ and $c_{\text{eff}}$, plotted as a function of the interface parameter $\lambda$ [see \ref{['eq:H2']}] in a free-fermion lattice model at the critical point. The markers represent the exact numerical results of $c_{\text{relax}}$ for various choices of locations $n_d$ where the dissipation is introduced (Note: markers appear on the top of each other in the figure). For example, $n_d=1$ corresponds to the left boundary, and $n_d=N-1$ corresponds to the site next to the conformal interface. The green dashed line is the analytical result of $c_{\text{relax}}$ in Eq. \ref{['eq:crelax_analytic']}, obtained from perturbative calculations. The blue and red lines are analytic results of $c_{\text{LR}}$ and $c_{\text{eff}}$ in Eqs. \ref{['eq:cLR_energy']} and \ref{['eq:ceff_entangle']} respectively, with the corresponding lattice model calculations given in the appendix. Here the dissipation strength is $\gamma = 0.05$, and the central charge of the bulk critical theory is $c = 1$. Bottom left: A summary of defining properties for $c_{\text{eff}}$, $c_{\text{LR}}$, and $c_{\text{relax}}$.
• Figure 2: Two free-fermion chains of size $N$ are connected via a conformal interface characterized by a parameter $\lambda$. A local dissipation of strength $\gamma$ is introduced at site $n_d \in [1, N]$ on the left chain. The total system size is $L=2N$.
• Figure 3: Rapidity spectrum $\{\beta_i\}$ for the free fermionic chain of length $L=2N = 20$, with dissipation strength $\gamma = 0.05$. (a) and (b): The local dissipation is added on the left boundary, i.e. $n_d = 1$, and the interface parameter is set as (a) $\lambda = 1$ (totally transmissive) and (b) $\lambda = 0.4$ (partially transmissive). (c) and (d): The local dissipation is added right next to the interface, i.e. at $n_d = N-1$, and the interface parameter is set as (c) $\lambda = 1$ (totally transmissive), and (d) $\lambda = 0.4$ (partially transmissive). Orange dots represent rapidities with smallest real part other than zero, which determines the Liouvillian gap. Green dots represent rapidities with a vanishing real part and correspond to oscillating modes. Their appearance depends on the location of dissipation $n_d$ and the system size $2N$. All higher modes (decay modes) are denoted by blue dots.
• Figure 4: Full spectrum of the Liouvillian superoperator for a total size $L=2N=8$, with dissipation strength $\gamma = 0.05$, and $\lambda = 0.8$. The local dissipation is applied at (a) $n_d=1$ and (b) $n_d=N-1$. All eigenvalues of the Liouvillian come in complex conjugate pairs with non-positive real parts. Purple dots show the steady-state solution. Green dots are the oscillatory modes with zero real part. Orange dots represent the slowest decaying modes, and blue dots represent all higher modes.
• Figure 5: The Liouvillian gap $g(\lambda)$ as a function of the total system size $L=2N$, with local dissipation applied at the left boundary ($n_d = 1$). For various values of the interface parameter $\lambda$, there is an excellent agreement between the numerical results (dots) and the analytical result from perturbative analysis (dashed line) in Eq. \ref{['eq:perturb_gap_boundary']}. Here the total system size is chosen as $L=2N = 200,\, 260, \,320, \dots, 800$, and the dissipation strength is set to be $\gamma = 0.05$.