Transfer matrix approach to quantum systems subject to certain Lindblad evolution

TL;DR

Solving the time evolution of many-body systems under Lindblad dynamics is typically intractable due to exponential state-space growth. The authors extend the transfer-matrix formalism to a class of Hermitian Lindblad evolutions that close the two-point correlator hierarchy and apply it to the XX spin chain with local dephasing, obtaining a Laplace-domain Green's function expressible as a product of small transfer matrices. This yields analytical access to the thermodynamic limit and reveals a ballistic-to-diffusive crossover with diffusion constant $D=2J^2/\gamma$, as well as a universal long-time decay of off-diagonal correlators $|C_{x+l,x}(t)| \sim t^{-(\lceil l/2\rceil+1/2)}$. The method also provides a fast numerical approach with complexity $O(L^2 r^6)$, enabling simulations of systems with up to $\sim 10^6$ spins and generalizations to other Hermitian dissipators. Overall, the work offers a powerful, scalable framework for two-point correlators in dissipative quadratic Hamiltonians and broadens applicability beyond the XX model.

Abstract

Solving for the time evolution of a many particle system whose dynamics is governed by Lindblad equation is hard. We extend the use of the transfer matrix approach to a class of Lindblad equations that admit a closed hierarchy of two point correlators. An example that we treat is the XX spin chain, i.e., free fermions, subject to the local on-site dephasing, but can be extended to other Hermitian dissipators, e.g., non-local dephasing. We find a simple expression of the Green's function in the Laplace domain. The method can be used to get analytical results in the thermodynamic limit, for instance, to get the evolution of the magnetization density and to explicitly see the crossover between ballistic and diffusive behavior, or to show that the correlations between operators at distance $l$ decay with time as $1/t^{\lceil l/2 \rceil+1/2}$. It also provides a fast numerical method to determine the evolution of the density with a complexity scaling with the system size more favorably than in previous methods, easily allowing one to study systems with $\sim 10^6$ spins.

Figures (4)

• Figure 1: Contours of integration for evaluating the Laplace inverse for the two cases namely $4\gamma>\omega(q)$ and $4\gamma<\omega(q)$. $s_-$ in (a) is shown in blue as it lies on the second Riemann sheet and therefore does not contribute in the integral.
• Figure 2: Evolution of the density, near $x=L/2$, starting from a domain wall initial state for $L=10^5$ using Eq. (\ref{['eq:Cxxfeq']}) with $\gamma=0.01,~J=1$ where the Laplace inverse has been taken numerically.
• Figure 3: (a) Variation of the transferred magnetization $M(t)$ with time at $\gamma=0.01$. We see that at short times, $\gamma t\ll1$, $M(t)\sim t$ and at long times, $\gamma t\gg1$, $M(t)\sim \sqrt{t}$. (b) Logarithmic derivative of the transferred magnetization with time. The red crosses, brown dots and the blue dashed line is $\beta(t)$ obtained by using Eq. (\ref{['eq:Cxxfeq']}). The black solid line is the data from exact diagonalization of Eq. (\ref{['eq:correq']}).
• Figure 4: Evolution of the off-diagonal elements with $\gamma=0.5$ and $L=200$. We see an agreement with the asymptotic behavior predicted by Eq. (\ref{['eq:Cxylt']}).