Abstract
We develop a geometric approach to operator growth and Krylov complexity in many-body quantum systems governed by symmetries. We start by showing a direct link between a unitary evolution with the Liouvillian and the displacement operator of appropriate generalized coherent states. This connection maps operator growth to a purely classical motion in phase space. The phase spaces are endowed with a natural information metric. We show that, in this geometry, operator growth is represented by geodesics, and Krylov complexity is proportional to a volume. This geometric perspective also provides two novel avenues toward computation of Lanczos coefficients, and it sheds new light on the origin of their maximal growth. We describe the general idea and analyze it in explicit examples, among which we reproduce known results from the Sachdev-Ye-Kitaev model, derive operator growth based on SU(2) and Heisenberg-Weyl symmetries, and generalize the discussion to conformal field theories. Finally, we use techniques from quantum optics to study operator evolution with quantum information tools such as entanglement and Renyi entropies, negativity, fidelity, relative entropy, and capacity of entanglement.
Highlights
The study of classical and quantum chaos is both an exciting and an inherently complicated subject
We argue that there exists a natural algebra associated with operator dynamics and Krylov complexity, and that the closure of this algebra on different levels provides another way toward finding potential sets of Lanczos coefficients
The first, and in our opinion probably the most important, is to frame the context in a way that transparently connects to the “physicist perspective.”
Summary
The study of classical and quantum chaos is both an exciting and an inherently complicated subject. The idea of Krylov complexity, which will be our main topic, was put forward in [23] In all these approaches, various notions of size were shown to evolve exponentially fast, and Lyapunov-type exponents were extracted. Chaotic systems, Krylov complexity grows at most exponentially fast with a characteristic Lyapunov exponent They pointed out that the operator growth hypothesis might lead to a new physical proof and understanding of the chaos bound; see [10]. Various aspects of this hypothesis were already investigated in [22,25,26,27,28,29,30,31,32,33,34], and Krylov complexity became a good candidate for a universal notion of complexity in interacting quantum field theories. Four Appendixes provide more technical details complementing the discussion in the main part
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