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arXiv:2310.07043 · Physical review. B./Physical review. B

Information scrambling in free fermion systems with a sole interaction

Qu-Cheng Gao, Pengfei Zhang, Xiao Chen

Published Jul 15, 2024• 6 citations
arXivHTMLPDF

Abstract

It is well established that the presence of single impurity can have a substantial impact on the transport properties of quantum many-body systems at low temperature. In this work, we investigate a close analog of this problem from the perspective of quantum information dynamics. We construct Brownian circuits and Clifford circuits consisting of a free fermion hopping term and a sole interaction. In both circuits, our findings reveal the emergence of operator scrambling. Notably, the growth of the operator can be mapped to the symmetric exclusion process in the presence of a source term localized at a single point. We demonstrate that in the one-dimensional system, both the operator and entanglement exhibit diffusive scaling. Conversely, in scenarios characterized by all-to-all hopping, the operator's size undergoes exponential growth, while the entanglement exhibits a linear increase over time.

DOI
10.1103/physrevb.110.035137

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It is well established that the presence of single impurity can have a substantial impact on the transport properties of quantum many-body systems at low temperature. In this work, we investigate a close analog of this problem from the perspective of quantum information dynamics. We construct Brownian circuits and Clifford circuits consisting of a free fermion hopping term and a sole interaction. In both circuits, our findings reveal the emergence of operator scrambling. Notably, the growth of the operator can be mapped to the symmetric exclusion process in the presence of a source term localized at a single point. We demonstrate that in the one-dimensional system, both the operator and entanglement exhibit diffusive scaling. Conversely, in scenarios characterized by all-to-all hopping, the operator's size undergoes exponential growth, while the entanglement exhibits a linear increase over time.
In quantum many-body systems, interactions play a crucial role in the emergence of information scrambling. When particles interact throughout the system, the entanglement between them can lead to a rapid and chaotic spreading of quantum information, typically probed by the growth in operator size in the Heisenberg picture. In this study, we explore whether the operator undergoes scrambling when particles interact solely through a single impurity in generic spatial dimensions, focusing on fermion systems with spatial and temporal random hoppings. By connecting the dynamics of the operator to the symmetric exclusion process with a source term, we demonstrate the presence of an escape-to-scrambling transition when tuning the interaction strength for fermions in three dimensions. As a comparison, systems in lower dimensions are proven to scramble at arbitrarily weak interactions unless the hopping becomes sufficiently long-ranged. Our predictions are validated using both a Brownian circuit with a single Majorana fermion per site and a solvable Brownian SYK model with a large local Hilbert space dimension. This suggests the universality of the theoretical picture for free fermion systems with spatial and temporal randomness.
We study impurity-induced particle growth and scrambling in clean one-dimensional free-fermion systems. We show that a single local impurity can act as a branching source: particle or operator weight propagates coherently into the free bulk, returns to the impurity, and is locally converted into additional degrees of freedom. We develop this branching picture in three complementary settings: a monitored free-fermion model with feedback, a fully unitary interacting particle model, and Heisenberg operator dynamics with an interacting impurity. In the monitored model, we find a feedback-driven transition for both boundary and bulk impurities. In the unitary particle and operator models, a boundary impurity gives rise to an analogous transition from saturation to sustained growth and scrambling. These results reveal how a single impurity can generate complex many-body dynamics in an otherwise clean and free quantum system.
In closed generic many-body systems, unitary evolution disperses local quantum information into highly nonlocal objects, resulting in thermalization. Such a process is called information scrambling, whose swiftness is quantified by the operator size growth. However, the impact of couplings to the environment on the process of information scrambling remains unexplored for quantum systems embedded within an environment. Here we predict a dynamical transition in quantum systems with all-to-all interactions accompanied by an environment, which separates two phases. In the dissipative phase, information scrambling halts as the operator size decays with time, while in the scrambling phase, dispersion of information persists, and the operator size grows and saturates to an O(N) value in the long-time limit with N the number of degrees of freedom of the systems. The transition is driven by the competition between the system's intrinsic and environment propelled scramblings and the environment-induced dissipation. Our prediction is derived from a general argument based on epidemiological models and demonstrated analytically via solvable Brownian Sachdev-Ye-Kitaev models. We provide further evidence which suggests that the transition is generic to quantum chaotic systems when coupled to an environment. Our study sheds light on the fundamental behavior of quantum systems in the presence of an environment.
Abstract In generic closed quantum systems, the complexity of operators increases under time evolution governed by the Heisenberg equation, reflecting the scrambling of local quantum information. However, when systems interact with an external environment, the system-environment coupling allows operators to escape from the system, inducing a dynamical transition between the scrambling phase and the dissipative phase. This transition is known as the environment-induced information scrambling transition, originally proposed in Majorana fermion systems. In this work, we advance this discovery by investigating the transition in charge-conserved systems with space-time randomness. We construct solvable Brownian Sachdev-Ye-Kitaev models of complex fermions coupled to an environment, enabling the analytical computation of operator growth. We determine the critical dissipation strength, which is proportional to n(1−n)n(1-n)n(1−n) n ( 1 - n ) with n being the density of the complex fermions, arising from the suppression in the quantum Lyapunova exponent due to the Pauli blockade in the scattering process. We further analyze the density dependence of maximally scrambled operators at late time. Our results shed light on the intriguing interplay between information scrambling, dissipation, and conservation laws.
Free fermion models with an operator entanglement growing as O(t)O(\sqrt{t})O(t​)
We study the time evolution of the Operator Space Entanglement Entropy (OSEE) of one-dimensional number-conserving free fermions subject to a random space- and time-dependent evolution. It is known that, in free-fermion systems without randomness, the OSEE of \emph{local} operators grows at most \emph{logarithmically} in time. Here we show that, in the presence of randomness, the OSEE of local spin operators involving a Jordan-Wigner string grows as O(t)\mathcal O(\sqrt t)O(t​), for all values of the Rényi index. We establish this in two models: a Brownian Hamiltonian and a brick-wall matchgate circuit. We also consider the OSEE of the evolution operator, and we show that it also grows as O(t)\mathcal O(\sqrt{t})O(t​), in contrast with the linear growth generically expected without randomness. We analytically compute the average entropies of the two operators by mapping the quantum models to the symmetric simple exclusion process, relying on an assumption of self-averaging which we support numerically. We also consider two other types of randomness: random evolution in time that keeps spatial translation invariance, and quenched disorder. In the former case, we find that the OSEE also grows as t\sqrt{t}t​. In the latter case, we find that the entropy grows as O(log⁡log⁡t)\mathcal O( \log \log t)O(loglogt) for Hamiltonian evolution both for local operators with a Jordan-Wigner string and for the evolution operator, whereas it remains bounded at all times in the brick-wall matchgate circuit.
The physics of information scrambling in quantum many-body systems is intimately related to thermalisation and emergence of chaos. However, its standard characterisation through bipartite entanglement or operator growth remains inherently coarse-grained, obscuring the spatiotemporal anatomy of how quantum information flows between different regions of the system and across various length scales. In this work, we develop a theory for the dynamics of local information in local random unitary circuits which resolves the fine-grained microscopic structures of local information flow in such systems. Using the framework of information lattice which systematises the information content at each length scale within each subsystem, we identify a distribution of length scales at which information resides and obtain the dynamics of the distribution. For Haar-random circuits with infinite local Hilbert-space dimensions, we map this dynamics onto an exact classical stochastic process which reveals that this distribution is described by a Tracy-Widom form which moves ballistically in time ($ \propto t$) toward larger scales, accompanied by a ∼t1/3\sim t^{1/3}∼t1/3 broadening. We also find the same qualitative behaviour for random Clifford circuits acting on qubits. The similar scaling behaviour in two rather different settings hints strongly towards the universality of our results. In the case of Clifford circuits, we develop a phenomenological Markov process for the dynamics of the stabiliser generators in a specific gauge which confirms the Tracy-Widom distribution and the t1/3t^{1/3}t1/3 scaling of the fluctuations. Ultimately, our results establish the universal properties of the dynamics of local quantum information in a length scale-resolved fashion and provide a possible route towards bridging exact microscopic theories for information dynamics with emergent hydrodynamic descriptions of information flow.