Multi-fractal geometry of finite networks of spins: Nonequilibrium dynamics beyond thermalization and many-body-localization

Abstract

Quantum spin networks overcome the challenges of traditional charge-based electronics by encoding the information into spin degrees of freedom. Although beneficial for transmitting information with minimal losses when compared to their charge-based counterparts, the mathematical formalization of the information propagation in a spin(tronic) network is challenging due to its complicated scaling properties. In this paper, we propose a geometric approach-specific to finite networks-for unraveling the information-theoretic phenomena of spin chains and rings by abstracting them as weighted graphs, where the vertices correspond to the spin excitation states and the edges represent the information theoretic distance between pair of nodes. The weighted graph representation of the quantum spin network dynamics exhibits a complex self-similar structure (where subgraphs repeat to some extent over various space scales). To quantify this complex behavior, we develop a new box counting inspired algorithm which assesses the mono-fractal versus multi-fractal properties of quantum spin networks. Besides specific to finite networks, multi-fractality is further compounded by "engineering" or "biaising" the network for selective transfer, as selectivity makes the network more heterogeneous. To demonstrate criticality in finite size systems, we define a thermodynamics inspired framework for describing information propagation and show evidence that some spin chains and rings exhibit an informational phase transition phenomenon, akin to the metal-to-insulator phase transition in Anderson localization in finite media.Many fundamental particles such as electrons, protons and certain atomic nuclei exhibit a fundamental quantum property called spin. Spin degrees of freedom have played an important role since the discovery of nuclear magnetic resonance [23] and electron spin resonance [5], which have become essential tools for characterizing chemical structure, material properties and biomedical imaging [13,25,27]. More recently, spin degrees of freedom have been in the spotlight again as potential carriers of quantum information, and the foundation of quantum spintronics [3,4].Conventional electronics, while powerful, also has drawbacks. Electrical resistance encountered by moving electrons generates heat, wasting energy and limiting integration densities and data processing speeds in conventional semiconductor devices [35]. Spintronics in its most basic form is about exploiting spin degrees of freedom, usually of electrons, to encode, process, store and transfer information. Encoding information in spin degrees of freedom such as excitations of a spin network opens up many possibilities including quantum information processing [19,21], where spintronics devices offer benefits such as generally long coherence lifetimes of spins at low temperatures. Although there are many technological challenges that remain to be solved, considerable efforts are currently under way to realize various types of spintronic devices [3,4]. For example, spin p...

Jonckheere‐Terpstra test for nonclassical error versus log‐sensitivity relationship of quantum spin network controllers

“…The latter property means that the state remains localized and would not diffuse. This means that, in some cases, our controller achieves Anderson localization . ()…”

“…The latter property means that the state remains localized and won't diffuse. This means that in some cases [5] our controller achieves Anderson localization [2,13,23]. With these significant departures from classicality, one wonders whether the fundamental error versus log sensitivity limitation is still in force.…”

“…The controller belongs to a class of quantum control systems where the controller sole authority is to modify, in a physically meaningful manner, the parameters of the Hamiltonian [8]. Even though the basic concepts developed here remain valid for the whole class of such controllers, here, however, we will more specifically consider controllers that induce energy level shifts [5,15,16,17,18,24,33]. More specifically in this paper, D = diag (D 1 , D 2 , ..., D M ) is a diagonal matrix of bias fields added to the Hamiltonian in some subspace.…”

“…Although the basic concepts developed here remain valid for the whole class of such controllers, here, we will more specifically consider controllers that induce energy level shifts. () More specifically, in this paper, D =diag ( D 1 , D 2 ,…, D M ) is a diagonal matrix of bias fields added to the Hamiltonian in some subspace. The concept is illustrated in Figure .…”

An innovative magnetic state generator using machine learning techniques

“…Conventionally, magnetic structures are calculated by solving the magnetic Hamiltonian with the help of tools such as Monte-Carlo simulations, greedy algorithm, and spin dynamics simulations based on the Landau-Lifshitz-Gilbert equation to investigate the characteristics of various magnetic structures 8,9 . However, obtaining well-ordered magnetic states 10,11 using these methods is not guaranteed because the solutions of those methods are only some of the multiple local stable states the magnetic systems can have.…”