[Paper Review] Exact solution for the Lindbladian dynamics for the open XX spin chain with boundary dissipation

Kohei Yamanaka, Tomohiro Sasamoto|arXiv (Cornell University)|Apr 23, 2021

Quantum many-body systems4 references4 citations

TL;DR

This paper presents an exact analytical solution for the time evolution of magnetization and spin current in an open XX spin chain with boundary dissipation using the third quantization method. By mapping the Lindbladian dynamics to a non-Hermitian $N\times N$ matrix diagonalization, the authors derive exact formulas that reveal a short-time plateau regime due to light-cone propagation of boundary effects and a long-time $O(N^{-3})$ decay to steady state governed by the Liouvillian gap, with the coefficient fully determined analytically.

ABSTRACT

We obtain exact formulas for the time-dependence of a few physical observables for the open XX spin chain with Lindbladian dynamics. Our analysis is based on the fact that the Lindblad equation for an arbitrary open quadratic system of $N$ fermions is explicitly solved in terms of diagonalization of a $4N imes4N$ matrix called structure matrix by following the scheme of the third quantization. We mainly focus on the time-dependence of magnetization and spin current. As a short-time behavior at a given site, we observe the plateau regime except near the center of the chain. Basic features of this are explained by the light-cone structure created by propagations of boundary effects from the initial time, but we can explain their more detailed properties analytically using our exact formulas. On the other hand, after the plateau regime, the magnetization and spin current exhibit a slow decay to the steady state values described by the Liouvillian gap. We analytically establish its $O(N^{-3})$ scaling and also determine its coefficient.

Motivation & Objective

• To derive exact time-dependent formulas for physical observables in an open quantum spin chain under Lindbladian dynamics.
• To analyze the short-time dynamics, particularly the emergence of a plateau regime due to boundary-induced light-cone effects.
• To analytically determine the long-time decay behavior toward the steady state, focusing on the Liouvillian gap scaling.
• To demonstrate the applicability of third quantization to quadratic open fermionic systems with non-Hermitian structure matrices.
• To provide explicit analytical expressions for magnetization and spin current using Bessel function expansions and saddle-point approximations.

Proposed method

• The Lindblad equation for the open XX spin chain is mapped to a quadratic fermionic system using the Jordan-Wigner transformation.
• The dynamics is solved via the third quantization method, reducing the problem to diagonalizing a $4N\times 4N$ structure matrix, which further simplifies to an $N\times N$ non-Hermitian matrix.
• Exact time-dependent formulas for magnetization and spin current are derived using contour integrals and generating functions in the Liouvillian-Fock space.
• Asymptotic analysis via saddle-point approximation is applied to understand long-time behavior, particularly the $O(N^{-3})$ decay scaling.
• Bessel function representations are used to express the time evolution functions, enabling both analytical and numerical evaluation.
• The method accounts for boundary-localized modes and confirms their negligible contribution to the studied observables.

Experimental results

Research questions

• RQ1 What is the exact time evolution of magnetization and spin current in an open XX spin chain with boundary dissipation?
• RQ2 How do boundary effects propagate through the chain, and what causes the observed plateau regime in the short-time dynamics?
• RQ3 What is the analytical form of the Liouvillian gap, and how does it scale with system size $N$?
• RQ4 Can the third quantization method be effectively applied to derive exact time-dependent observables in open quadratic fermionic systems?
• RQ5 What role do localized boundary modes play in the dynamics of bulk observables like magnetization and current?

Key findings

• The short-time dynamics exhibit a plateau regime in magnetization and spin current, except near the chain center, due to light-cone propagation of boundary effects.
• The plateau regime is analytically explained using exact formulas derived from contour integrals and Bessel function expansions.
• After the plateau, both magnetization and spin current decay slowly to steady state with a decay rate scaling as $O(N^{-3})$.
• The coefficient of the $O(N^{-3})$ scaling is analytically determined and confirmed through asymptotic analysis of the Liouvillian spectrum.
• The long-time behavior is governed by the Liouvillian gap, which is found to scale as $O(N^{-3})$ with a precise prefactor derived from the non-Hermitian matrix structure.
• The method successfully reduces the $4N\times 4N$ Lindbladian problem to an $N\times N$ non-Hermitian matrix, enabling exact analytical solutions for physical observables.