Particle statistics impose fundamental constraints on nonequilibrium quantum dynamics, yet it remains an open question whether fractional statistics can lead to emergent universal dynamical scaling beyond the conventional Bose-Fermi paradigm. Here we investigate the far-from-equilibrium many-body relaxation of anyons in a one-dimensional lattice and uncover an unconventional yet universal scaling behavior governed by fractional statistics. Based on large-scale numerical simulations and scaling analysis, we identify a distinct separation between particle transport and information spreading: density correlations spread superdiffusively, whereas entanglement entropy grows ballistically. The anomalous particle dynamics can be interpreted intuitively from the statistical-phase-induced quantum interference, which suppresses coherent holon-doublon propagation. In contrast, the entanglement growth turns out to be dominated by its configurational component, which propagates ballistically. Our results establish anyonic statistics as a distinct source of universal nonequilibrium dynamics beyond bosons and fermions, with direct relevance to current quantum simulation experiments.
Particle statistics impose fundamental constraints on nonequilibrium quantum dynamics, yet it remains an open question whether fractional statistics can lead to emergent universal dynamical scaling beyond the conventional Bose-Fermi paradigm. Here we investigate the far-from-equilibrium many-body relaxation of anyons in a one-dimensional lattice and uncover an unconventional yet universal scaling behavior governed by fractional statistics. Based on large-scale numerical simulations and scaling analysis, we identify a distinct separation between particle transport and information spreading: density correlations spread superdiffusively, whereas entanglement entropy grows ballistically. The anomalous particle dynamics can be interpreted intuitively from the statistical-phase-induced quantum interference, which suppresses coherent holon-doublon propagation. In contrast, the entanglement growth turns out to be dominated by its configurational component, which propagates ballistically. Our results establish anyonic statistics as a distinct source of universal nonequilibrium dynamics beyond bosons and fermions, with direct relevance to current quantum simulation experiments.
"Anyonic quantum statistics is an exotic phenomenon of identical particles in quantum mechanics. When particles are confined in two spatial dimensions, exchanges of identical particles can induce phase factors in the wave function of Abelian anyons or matrix valued transformations of the wave function of non-Abelian anyons. As a result, systems of anyons may have richer properties than those of bosons and fermions in three spatial dimensions. There is strong theoretical support for the existence of anyons in some engineered two-dimensional systems such as 2D electron gases, strongly correlated spin lattices and as edge modes of nanowires. In the future, anyons could be used in topological quantum computation to perform highly efficient information processing with very small error rates. The phases of matter described by dynamically interacting anyons have recently been studied in chains where anyons interact via Heisenberg-type exchange interactions. In this thesis, new kind of anyonic interactions are studied, induced purely by braiding during free evolution. Such interactions are of topological origin, and the information about the interactions is stored non-locally. A quantum walk model is used to study the effects of these braiding interactions on the dynamical behaviour of anyons. The anyonic quantum walk is a quasi-one-dimensional generalization of the discrete-time quantum walk which allows the simulation of anyonic dynamics analytically. The moving anyon and a chain of stationary anyons interact via braiding statistics, and the behaviour of the anyon is studied in three cases. Striking differences are found between particles with conventional boson or fermion statistics and non-Abelian anyonic statistics. The random walk is a dynamical model that describes the motion of a particle on a lattice. In physics, it is used to describe Brownian motion of fluids and gases. In such systems, energy transport is diffusive, and the order of the system approaches a highly mixed state without any information content. The quantum version of the random walk, the quantum walk, has nonintuitive properties. Generally, the information content in a unitarily evolving system is fixed, which leads to unexpected transport phenomena. An initially localized particle does not propagate diffusively, but escapes the starting point with ballistic speed. Most of our results for non-Abelian anyons use the Ising model anyon which is most likely to be measured in experiments. First, when each anyonic site is occupied by one Ising anyon, the propagation of the anyon becomes diffusive. More precisely, the variance of the spatial probability distribution of the particle depends linearly on the number of time steps. This is in stark contrast to bosons, fermions and Abelian anyons which propagate ballistically, and the variance depends linearly on the square of the number of time steps. The essential reason for this slowdown is that the non-Abelian anyons possess an extra degree of freedom called the fusion Hilbert space. This space can be viewed as an environment for the normal degrees of freedom of the particle, inducing decoherence in the quantum walk. This thesis opens the line for studies of this novel kind of decoherence mechanism in quantum walks. In the second case the study is extended to more general anyon models by losing some of the information about the history of the evolution. In this case the system is subject to decoherence, and the total system evolution is not unitary. The behaviour of the particle in this model is found to be diffusive for all the non-Abelian anyon models studied, while Abelian anyons behave ballistically also in this model. One peculiarity of quantum systems is their behaviour under disorder. Quantum mechanical particles moving in random local potentials are known to freeze and not move at all. Such a phenomenon is known as Anderson localization. Studies have shown that Anderson localization happens also in quantum walks with random spatial fluctuations in the coin parameters. In the third case, the transport properties of anyons are studied under topological randomness, allowing the occupations of anyons change between experiments. Bosons and fermions would propagate ballistically under such randomness, and Abelian anyons are shown to localize. The results show that non-Abelian anyons behave diffusively at short time scales, and it is argued that they do so in the long time limit as well. In all cases, non-Abelian anyons are shown to have very different dynamical properties than bosons, fermions and Abelian anyons. A possible simulation of the anyonic quantum walk in Fractional Quantum Hall systems is also discussed.
We study the nonequilibrium dynamics of Abelian anyons in a one-dimensional system. We find that the interplay of anyonic statistics and interactions gives rise to spatially asymmetric particle transport together with a novel dynamical symmetry that depends on the anyonic statistical angle and the sign of interactions. Moreover, we show that anyonic statistics induces asymmetric spreading of quantum information, characterized by asymmetric light cones of out-of-time-ordered correlators. Such asymmetric dynamics is in sharp contrast to the dynamics of conventional fermions or bosons, where both the transport and information dynamics are spatially symmetric. We further discuss experiments with cold atoms where the predicted phenomena can be observed using state-of-the-art technologies. Our results pave the way toward experimentally probing anyonic statistics through nonequilibrium dynamics.
We study dynamical correlations of interacting hard-core anyons on a one-dimensional lattice at infinite temperature. This is a setting in which the many-body spectrum is independent of the statistical phase θ , while dynamical correlators remain sensitive to θ through nonlocal Jordan-Wigner strings. We compute single-particle Green's functions, spectral functions, and density-density correlators, thereby separating the effects of fractional statistics on one-body coherence from those on density transport in a maximally mixed ensemble. In the noninteracting case V = 0 , high-temperature averaging leads to inversion-symmetric Green's functions for all θ despite the presence of anyonic strings. Finite nearest-neighbor interactions V generate, however, a pronounced left-right asymmetry in the Green's functions for 0 < θ < π , with the strongest chirality appearing at intermediate couplings V ∼ J where interactions and hopping compete most effectively. In this regime, the Green's function decays exponentially in time with a statistical-angle-dependent decay rate. At strong coupling, the dynamics crosses over to an atomic-limit regime in which the dependence on θ is reduced. Here, the Green's function decays universally as t − 1 and the corresponding spectral function displays a three-band structure. In contrast, density-density correlations are insensitive to statistics and recover the known infinite-temperature transport regimes of the XXZ chain, including ballistic, superdiffusive, and diffusive behaviors. These results identify dynamical correlation functions as direct probes of fractional statistics in high-entropy quantum systems.
Low-dimensional quantum systems can host anyons, particles with exchange statistics that are neither bosonic nor fermionic. However, the physics of anyons in one dimension remains largely unexplored. In this work, we realize Abelian anyons in one dimension with arbitrary exchange statistics using ultracold atoms in an optical lattice, where we engineer the statistical phase through a density-dependent Peierls phase. We explore the dynamical behavior of two anyons undergoing quantum walks and observe the anyonic Hanbury Brown-Twiss effect as well as the formation of bound states without on-site interactions. Once interactions are introduced, we observe spatially asymmetric transport in contrast to the symmetric dynamics of bosons and fermions. Our work forms the foundation for exploring the many-body behavior of one-dimensional anyons.
Low-dimensional quantum systems can host anyons, particles with exchange statistics that are neither bosonic nor fermionic. Despite indications of a wealth of exotic phenomena, the physics of anyons in one dimension (1D) remains largely unexplored. Here, we realize Abelian anyons in 1D with arbitrary exchange statistics using ultracold atoms in an optical lattice, where we engineer the statistical phase via a density-dependent Peierls phase. We explore the dynamical behavior of two anyons undergoing quantum walks, and observe the anyonic Hanbury Brown-Twiss effect, as well as the formation of bound states without on-site interactions. Once interactions are introduced, we observe spatially asymmetric transport in contrast to the symmetric dynamics of bosons and fermions. Our work forms the foundation for exploring the many-body behavior of 1D anyons.
This thesis investigates universality and complexity in classical and quantum statistical physics, with a particular emphasis on nonequilibrium phenomena. While the emergence of universal behavior in equilibrium systems stands as one of the cornerstones of statistical physics, the extension of these ideas to nonequilibrium settings presents fundamental and still open challenges. The inherently diverse and system-specific nature of nonequilibrium dynamics calls for a broad spectrum of conceptual and analytical tools. Furthermore, nonequilibrium dynamics is nowadays accessible to experiments in both classical and quantum regimes. These modern experiments produce a large amount of experimental data, which calls for analysis of physics through data-driven methods, complexity theory and high-dimensional statistics. The possibility of probing and accessing correlated states of matter out of equilibrium in modern experiments raises several questions. Can we understand the output of experiments in terms of complexity theory and dimensional reduction? Is it possible to extend concepts of universality to data structures arising in simulations? Can we describe phase transitions (either classical or quantum) concepts of modern data science? How to control entanglement in quantum dynamics? Which nonequilibrium phenomena can we hope to observe in quantum simulators? The research work presented in this thesis is an attempt to answer to these questions. In this thesis, we argue that order parameters and diagnostics of criticality in nonequilibrium classical systems can be found by unsupervised learning and dimensional reduction, which allow to reduce the complexity of datasets by projections on lower-dimensional spaces, while still preserving essential universal features. We then move to the study of models of quantum dynamics relevant for quantum simulators. In particular, we show that the same concepts of dimensional reduction apply also to quantum systems, allowing for the extraction of information from outputs of quantum simulators. We then study a protocol to prepare complex, entangled states via unitary evolution and enhance entanglement via measurements in a fermionic model. Finally, we study the emergence of anomalous transport in one-dimensional chiral systems with non-Abelian symmetries.