Search

The Anderson model for a magnetic impurity in a one-dimensional quasicrystal is studied using the numerical renormalization group (NRG). The main focus is elucidating the physics at the critical point of the Aubry-Andre (AA) Hamiltonian, which exhibits a fractal spectrum with multifractal wave functions, leading to an AA Anderson (AAA) impurity model with an energy-dependent hybridization function defined through the multifractal local density of states at the impurity site. We first study a class of Anderson impurity models with uniform fractal hybridization functions that the NRG can solve to arbitrarily low temperatures. Below a Kondo scale $T_K$, these models approach a fractal strong-coupling fixed point where impurity thermodynamic properties oscillate with $\log_b T$ about negative average values determined by the fractal dimension of the spectrum. The fractal dimension also enters into a power-law dependence of $T_K$ on the Kondo exchange coupling $J_K$. To treat the AAA model, we combine the NRG with the kernel polynomial method (KPM) to form an efficient approach that can treat hosts without translational symmetry down to a temperature scale set by the KPM expansion order. The aforementioned fractal strong-coupling fixed point is reached by the critical AAA model in a simplified treatment that neglects the wave-function contribution to the hybridization. The temperature-averaged properties are those expected for the numerically determined fractal dimension of $0.5$. At the AA critical point, impurity thermodynamic properties become negative and oscillatory. Under sample-averaging, the mean and median Kondo temperatures exhibit power-law dependences on $J_K$ with exponents characteristic of different fractal dimensions. We attribute these signatures to the impurity probing a distribution of fractal strong-coupling fixed points with decreasing temperature., 29 pages, 19 figures

We show that the classical (mean-field) description of far from equilibrium superconductivity is exact in the thermodynamic limit for local observables but breaks down for global quantities, such as the entanglement entropy or Loschmidt echo. We do this by solving for and comparing exact quantum and exact classical long-time dynamics of a BCS superconductor with interaction strength inversely proportional to time and evaluating local observables explicitly. Mean field is exact for both normal and anomalous averages (superconducting order) in the thermodynamic limit. However, for anomalous expectation values, this limit does not commute with adiabatic and strong coupling limits and, as a consequence, their quantum fluctuations can be unusually strong. The long-time steady state of the system is a gapless superconductor whose superfluid properties are only accessible through energy resolved measurements. This state is nonthermal but conforms to an emergent generalized Gibbs ensemble. Our study clarifies the nature of symmetry-broken many-body states in and out of equilibrium and fills a crucial gap in the theory of time-dependent quantum integrability., Comment: 30 pages, 10 figures, new title

Structural vulnerability of a network can be studied via two key notions in graph theory: articulation points (APs) and bridges, representing nodes and edges whose removal will disconnect the network, respectively. Fundamental properties of APs and bridges in classical random networks have been studied recently. Yet, it is unknown if those properties still hold in quantum networks. Quantum networks allow for the transmission of quantum information between physically separated quantum systems. They play a very important role in quantum computing, quantum communication, and quantum sensing. Here we offer an analytical framework to study the structural vulnerability of quantum networks in terms of APs and bridges. In particular, we analytically calculate the fraction of APs and bridges for quantum networks with arbitrary degree distribution and entangled qubits in pure states. We find that quantum networks with swap operations have lower fractions of APs and bridges than their classical counterparts. Moreover, we find that quantum networks under low-degree swap operations are substantially more robust against AP attacks than their classical counterparts. These results help us better understand the structural vulnerability of quantum networks and shed light on the design of more robust quantum networks.

Covering problems are classical computational problems concerning whether a certain combinatorial structure 'covers' another. For example, the minimum vertex covering problem aims to find the smallest set of vertices in a graph so that each edge is incident to at least one vertex in that set. Interestingly, the computational complexity of the minimum vertex covering problem in graphs is closely related to the core percolation problem, where the core is a special subgraph obtained by the greedy leaf removal procedure. Here, by generalizing the greedy leaf removal procedure in graphs to hypergraphs, we introduce two generalizations of core percolation in graphs to hypergraphs, related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs, respectively. We offer analytical solutions of these two core percolations for random hypergraphs with arbitrary vertex degree and hyperedge cardinality distributions. We also compute these two cores in several real-world hypergraphs, finding that they tend to be much smaller than their randomized counterparts. This result suggests that both the minimum hyperedge cover problem and the minimum vertex cover problem in those real-world hypergraphs can actually be solved in polynomial time. Finally, we map the minimum dominating set problem in graphs to the minimum hyperedge cover problem in hypergraphs. We show that our generalized greedy leaf removel procedure significantly outperforms the state-of-the-art method in solving the minimum dominating set problem.

We study the Anderson orthogonality catastrophe, and the corresponding x-ray edge problem, in systems that are at a localization transition driven by a deterministic quasiperiodic potential. Specifically, we address how the ground state of the Aubry-Andre model, at its critical point, responds to an instantaneous local quench. At this critical point, both the single-particle wavefunctions and the density of states are fractal. We find, numerically, that the overlap between post-quench and pre-quench wavefunctions, as well as the "core-hole" Green function, evolve in a complex, non-monotonic way with system size and time respectively. We interpret our results in terms of the fractal density of states at this critical point. In a given sample, as the post-quench time increases, the system resolves increasingly finely spaced minibands, leading to a series of alternating temporal regimes in which the response is flat or algebraically decaying. In addition, the fractal critical wavefunctions give rise to a quench response that varies strongly from site to site across the sample, which produces broad distributions of many-body observables. Upon averaging this broad distribution over samples, we recover coarse-grained power laws and dynamical exponents characterizing the x-ray edge singularity. We discuss how these features can be probed in ultra-cold atomic gases using radio-frequency spectroscopy and Ramsey interference., Comment: 12 pages, 10 figures

A bridge in a graph is an edge whose removal disconnects the graph and increases the number of connected components. We calculate the fraction of bridges in a wide range of real-world networks and their randomized counterparts. We find that real networks typically have more bridges than their completely randomized counterparts, but very similar fraction of bridges as their degree-preserving randomizations. We define a new edge centrality measure, called bridgeness, to quantify the importance of a bridge in damaging a network. We find that certain real networks have very large average and variance of bridgeness compared to their degree-preserving randomizations and other real networks. Finally, we offer an analytical framework to calculate the bridge fraction , the average and variance of bridgeness for uncorrelated random networks with arbitrary degree distributions., 18 pages, 10 figures