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arXiv:2103.13450 · Physical review. B./Physical review. B

Anomalous quantum information scrambling for Z3 parafermion chains

Shun-Yao Zhang, Dong-Ling Deng

Published May 27, 2021• 3 citations
arXivHTMLPDF

Abstract

Parafermions are exotic quasiparticles with non-Abelian fractional statistics that could be exploited to realize universal topological quantum computing. Here, we study the scrambling of quantum information in one-dimensional parafermionic chains, with a focus on ${\mathbb{Z}}_{3}$ parafermions in particular. We use the generalized out-of-time-ordered correlators (OTOCs) as a measure of the information scrambling and introduce an efficient method based on matrix product operators to compute them. With this method, we compute the OTOCs for ${\mathbb{Z}}_{3}$ parafermions chains up to 200 sites for the entire early growth region. We find that, in stark contrast to the dynamics of conventional fermions or bosons, the information scrambling light cones for parafermions can be both symmetric and asymmetric, even for inversion-invariant Hamiltonians involving only hopping terms. In addition, we find a deformed light cone structure with a sharp peak at the boundary of the parafermion chains in the topological regime, which gives unambiguous evidence of the strong zero modes at infinite temperature.

DOI
10.1103/physrevb.103.195156

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Parafermions are exotic quasiparticles with non-Abelian fractional statistics that could be exploited to realize universal topological quantum computing. Here, we study the scrambling of quantum information in one-dimensional parafermionic chains, with a focus on Z3{\mathbb{Z}}_{3}Z3​ parafermions in particular. We use the generalized out-of-time-ordered correlators (OTOCs) as a measure of the information scrambling and introduce an efficient method based on matrix product operators to compute them. With this method, we compute the OTOCs for Z3{\mathbb{Z}}_{3}Z3​ parafermions chains up to 200 sites for the entire early growth region. We find that, in stark contrast to the dynamics of conventional fermions or bosons, the information scrambling light cones for parafermions can be both symmetric and asymmetric, even for inversion-invariant Hamiltonians involving only hopping terms. In addition, we find a deformed light cone structure with a sharp peak at the boundary of the parafermion chains in the topological regime, which gives unambiguous evidence of the strong zero modes at infinite temperature.
A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems? We first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone. Next, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed. Finally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.
Parafermion modes are non-Abelian anyons which were introduced as ZN{\mathbb{Z}}_{N}ZN​ generalizations of Z2{\mathbb{Z}}_{2}Z2​ Majorana states. In particular, Z3{\mathbb{Z}}_{3}Z3​ parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of the theoretical work on Z3{\mathbb{Z}}_{3}Z3​ parafermions has relied on bosonization methods or parafermionic quasiparticles. In this paper, we introduce a representation of Z3{\mathbb{Z}}_{3}Z3​ parafermions in terms of purely fermionic models. We establish the equivalency of a family of lattice fermionic models written in the basis of the $t\ensuremath{-}J$ model with a Kitaev-like chain supporting free Z3{\mathbb{Z}}_{3}Z3​ parafermionic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing Z3{\mathbb{Z}}_{3}Z3​ parafermions in strongly interacting electronic systems.
Quantum many-body scarred systems host special nonthermal eigenstates that support periodic revival dynamics and weakly break the ergodicity. Here, we study the quantum information scrambling dynamics in quantum many-body scarred systems, with a focus on the ``PXP'' model. We use the out-of-time-ordered correlator (OTOC) and Holevo information as measures of the information scrambling and apply an efficient numerical method based on matrix product operators to compute them up to 41 spins. We find that both the OTOC and Holevo information exhibit a linear light cone and periodic oscillations inside the light cone for initial states within the scarred subspace, which is in sharp contrast to thermal or many-body localized systems. The periodic revivals of OTOCs and Holevo information signify unusual breakdown of quantum chaos and are not equivalent to the revival dynamics of state fidelity or local observables studied in the previous literature. To explain the formation of the linear light-cone structure, we provide a perturbation-type calculation based on a phenomenological model. In addition, we demonstrate that the OTOC and Holevo information dynamics of the PXP model can be measured using the Rydberg-atom quantum simulators with current experimental technologies and numerically identify the measurable signatures using experimental parameters.
Parafermions are exotic quasiparticles with non-Abelian fractional statistics that can be realized and stabilized in one-dimensional models that are generalizations of the Kitaev ppp-wave wire. We study the simplest generalization, i.e., the Z3{Z}_{3}Z3​ parafermionic chain. Using a Jordan-Wigner transform we focus on the equivalent three-state chiral clock model, and study its rich phase diagram using the density matrix renormalization group technique. We perform our analyses using quantum entanglement diagnostics which allow us to determine phase boundaries, and the nature of the phase transitions. In particular, we study the transition between the topological and trivial phases, as well as to an intervening incommensurate phase which appears in a wide region of the phase diagram. The phase diagram is predicted to contain a Lifshitz type transition which we confirm using entanglement measures. We also attempt to locate and characterize a putative tricritical point in the phase diagram where the three above mentioned phases meet at a single point.
We report a numerical observation where the infinite-temperature out-of-time-order correlators (OTOCs) directly probe quantum phase transitions at zero temperature, in contrast to common intuition where low-energy quantum effects are washed away by strong thermal fluctuations at high temperature. By comparing numerical simulations with exact analytic results, we determine that this phenomenon has a topological origin and is highly generic, as long as the underlying system can be mapped to a 1D Majorana chain. Using the Majorana basis, we show that the infinite-temperature OTOCs probe zero-temperature quantum phases via detecting the presence of Majorana zero modes at the ends of the chain that is associated with 1D Z2{Z}_{2}Z2​ topological order. Hence, we show that strong zero modes also affect OTOCs and scrambling dynamics. Our results demonstrate an intriguing interplay between information scrambling and topological order, which leads to a new phenomenon in the scrambling of generic nonintegrable models: topological order induced prescrambling, paralleling the notion of prethermalization of two-time correlators that defines a timescale for the restricted scrambling of topologically protected quantum information.
Abstract The action of any local operator on a quantum system propagates through the system carrying the information of the operator. This is usually studied via the out-of-time-order correlator (OTOC). We numerically study the information propagation from one end of a periodically driven spin-1/2 XY chain with open boundary conditions using the Floquet infinite-temperature OTOC. We calculate the OTOC for two different spin operators, σ x and σ z . For sinusoidal driving, the model can be shown to host different types of edge states, namely, topological (Majorana) edge states and non-topological edge states. We observe a localization of information at the edge for both σ z and σ x OTOCs whenever edge states are present. In addition, in the case of non-topological edge states, we see oscillations of the OTOC in time near the edge, the oscillation period being inversely proportional to the gap between the Floquet eigenvalues of the edge states. We provide an analytical understanding of these effects due to the edge states. It was known earlier that the OTOC for the spin operator which is local in terms of Jordan–Wigner fermions (σ z ) shows no signature of information scrambling inside the light cone of propagation, while the OTOC for the spin operator which is non-local in terms of Jordan–Wigner fermions (σ x ) shows signatures of scrambling. We report a remarkable ‘unscrambling effect’ in the σ x OTOC after reflections from the ends of the system. Finally, we demonstrate that the information propagates into the system mainly via the bulk states with the maximum value of the group velocity, and we show how this velocity is controlled by the driving frequency and amplitude.