Operator growth in open quantum systems: lessons from the dissipative SYK

Abstract

We study the operator growth in open quantum systems with dephasing dissipation terms, extending the Krylov complexity formalism of [1]. Our results are based on the study of the dissipative q-body Sachdev-Ye-Kitaev (SYKq) model, governed by the Markovian dynamics. We introduce a notion of “operator size concentration” which allows a diagrammatic and combinatorial proof of the asymptotic linear behavior of the two sets of Lanczos coefficients (an and bn) in the large q limit. Our results corroborate with the semi-analytics in finite q in the large N limit, and the numerical Arnoldi iteration in finite q and finite N limit. As a result, Krylov complexity exhibits exponential growth following a saturation at a time that grows logarithmically with the inverse dissipation strength. The growth of complexity is suppressed compared to the closed system results, yet it upper bounds the growth of the normalized out-of-time-ordered correlator (OTOC). We provide a plausible explanation of the results from the dual gravitational side.

On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

“…This direction of the study was initiated in our previous exploration [1] and continued in other following works [2,18] (see [19][20][21][22][23][24][25][26] for some recent works on op-erator growth and chaos on non-Hermitian and open quantum systems). Unlike the closed systems, the open system operator evolution (under the realm of Markovian dynamics [27]) is characterized by a non-unitary evolution through the exponentiated non-Hermitian Lindbladian L o ⋅ = [H, ⋅ ] + iT , where the second term represents the non-Hermitian part [28,29].…”

“…Unlike the closed systems, the open system operator evolution (under the realm of Markovian dynamics [27]) is characterized by a non-unitary evolution through the exponentiated non-Hermitian Lindbladian L o ⋅ = [H, ⋅ ] + iT , where the second term represents the non-Hermitian part [28,29]. 2 One usually assumes the knowledge of the system Hamiltonian and some interactions with an environment in terms of certain couplings and system Lindblad operators. In such cases, the previous study of spin chains [1] and dissipative Sachdev-Ye-Kitaev (SYK) [2] generalize the result of [3], by proposing two sets of Lanczos coefficients, characterizing the operator growth in generic systems.…”

“…2 One usually assumes the knowledge of the system Hamiltonian and some interactions with an environment in terms of certain couplings and system Lindblad operators. In such cases, the previous study of spin chains [1] and dissipative Sachdev-Ye-Kitaev (SYK) [2] generalize the result of [3], by proposing two sets of Lanczos coefficients, characterizing the operator growth in generic systems. 3 As shown in [1], for open systems, the straightforward application of the Lanczos algorithm does not result in physically meaningful conclusions [1].…”

“…Due to such a finitely large number of coefficients (however small they are), the computation of the complexity becomes increasingly difficult. However, some analytic results have been obtained in dissipative SYK [2], where the complexity appears to be suppressed and inversely proportional to the dissipation parameter for a certain class of Lindbladian [33]. Although the results feature the dynamics of an inherently chaotic system, a substantial class of dissipative integrable systems remains largely unexplored.…”

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Continuing the previous initiatives [1,2], we pursue the exploration of operator growth and Krylov complexity in dissipative open quantum systems. In this paper, we resort to the bi-Lanczos algorithm generating two bi-orthogonal Krylov spaces, which individually generate non-orthogonal subspaces.

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