Thermalization in open many-body systems and KMS detailed balance

TL;DR

This paper derives a microscopic quantum master equation for open, many-body systems that does not rely on the rotating wave approximation and satisfies KMS detailed balance with respect to a renormalized Hamiltonian. By combining Born–Markov–Redfield-type derivations with Gaussian coarse-graining and time-averaging, it constructs a completely positive, quasi-local Lindbladian that reproduces thermal Gibbs convergence up to small Hamiltonian renormalizations and achieves linear-in-time error bounds. It further shows how to obtain an exactly detailed-balanced Lindbladian close to the coarse-grained dynamics, and it demonstrates that these generators can be efficiently simulated on quantum computers, enabling Gibbs sampling and thermodynamic analyses in large quantum many-body systems. The work clarifies the distinction between KMS and GNS detailed balance, explains the limitations of Davies maps in dense spectra, and provides a rigorous, practically implementable model of many-body thermalization with potential implications for quantum algorithms and thermodynamics of open systems.

Abstract

Starting from a microscopic description of weak system-bath interactions, we derive from first principles a quantum master equation that does not rely on the well-known rotating wave approximation. This includes generic many-body systems, with Hamiltonians with vanishingly small energy spacings that forbid that approximation. The equation satisfies a general form of detailed balance, called KMS, which ensures exact convergence to the many-body Gibbs state. Unlike the more common notion of GNS detailed balance, this notion is compatible with the absence of the rotating wave approximation. We show that the resulting Lindbladian dynamics not only reproduces the thermal equilibrium point up to a small renormalization of the system Hamiltonian, but it also approximates the true system evolution with an error that grows at most linearly in time, giving an exponential improvement upon previous estimates. This master equation has quasi-local jump operators, can be efficiently simulated on a quantum computer, and reduces to the usual Davies dynamics in the limit of a coarse-graining time much larger than the inverse of the smallest frequency difference. With it, we provide a rigorous model of many-body thermalization relevant to both open quantum systems and quantum algorithms.

Figures (2)

• Figure 1: (a) The principle of detailed balance says that at equilibrium, if the $j$-th state is more probable than the $i$-th one, then the probability of the transition $p(j\leftarrow i)$ is equally more frequent than $p(i\leftarrow j)$; (b) Simplified depiction of the spectrum of a generic many-body Hamiltonian. Since the maximum energy is expected to scale with the size of the system (say $N$), in order to accommodate an exponential amount of different energy levels (as the dimension of the Hilbert space scales as $d^N$, where $d$ is the local dimension), the minimum spacing must become exponentially small.
• Figure 2: Intuition behind Theorem \ref{['thm:coarseGrain']}. In order to illustrate the effect of the smoothing, consider the differential equation $f'(t) =F(t,\alpha):=- e^{-t/2} \left(1+\cos \left(\frac{t}{\alpha ^2}\right)\right)$. We define $f^{s}(t)$ to be the solution to the differential equation obtained by taking the Gaussian average of $F(t+\alpha^2q,\alpha)$. In analogy with the discussion in Thm. \ref{['thm:coarseGrain']}, we choose $T(\alpha) = \alpha^{-1}$. The shaded region corresponds to an error of order $\mathcal{O}\left (\alpha\right)$ around the exact solution $f(t)$. As it can be seen, the averaged solution always falls within this region. The inset corresponds to zooming the beginning of the evolution (the region contained in the dotted box): this shows that even if $f^{s}(t)$ always stays within $\mathcal{O}\left (\alpha\right)$ of $f(t)$, this distance can be larger than $\mathcal{O}\left (\alpha^2\right)$.

Theorems & Definitions (23)

• Theorem 1: GNS $\iff$ RW
• Theorem 2: Structural characterization of KMS channels
• Theorem 3: Structural characterization of KMS Lindbladians
• Theorem 4: Fastest rate in the system
• Theorem 5: Propagation of errors
• Theorem 6: Born approximation
• Theorem 7: Markov approximation
• Theorem 8: Redfield approximation
• Theorem 9: Smoothing of evolution
• Theorem 10: Complete positivity