A bound on approximating non-Markovian dynamics by tensor networks in the time domain

Spin-boson (SB) model plays a central role in studies of dissipative quantum dynamics, both due its conceptual importance and relevance to a number of physical systems. Here we provide rigorous bounds of the computational complexity of the SB model for the physically relevant case of a zero temperature Ohmic bath. We start with the description of the bosonic bath via its Feynman-Vernon influence functional (IF), which is a tensor on the space of spin's trajectories. By expanding the kernel of the IF functional via a sum of decaying exponentials, we obtain an analytical approximation of the continuous bath by a finite number of damped bosonic modes. We bound the error induced by restricting bosonic Hilbert spaces to a finite-dimensional subspace with small boson numbers, which yields an analytical form of a matrix-product state (MPS) representation of the IF. We show that the MPS bond dimension $D$ scales polynomially in the error on physical observables $\epsilon$, as well as in the evolution time $T$, $D\propto T^4/\epsilon^2$. This bound indicates that the spin-boson model can be efficiently simulated using polynomial in time computational resources.Comment: 10 pages, 0 figure

Charge, spin and pseudospin in graphene

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2008.Includes bibliographical references (p. 167-180).Graphene, a one-atom-thick form of carbon, has emerged in the last few years as a fertile electron system, highly promising for both fundamental research and applications. In this thesis we consider several topics in electronic and spin properties of graphene, with a particular emphasis on the quantum Hall effect (QHE) regime, where this material exhibits most interesting behavior. We shall start with analyzing general properties of the two-terminal conductance for graphene mono- and bilayer samples. Using conformal invariance and the theory of conformal mappings, we characterize the dependence of conductance on the sample shape. We identify the features which distinguish monolayers and bilayers and illustrate the use of the two-terminal conductance as a tool for sample diagnostic. Next, we present a microscopic study of the edge states in the QHE regime. This analysis provides a simple and general explanation of the half-integer Hall quantization in graphene. We discuss the edge states dispersion for different orientations of the boundary, and propose a way to image the edge states using STM spectroscopy. Then, we extend the picture of edge states to describe QHE in spatially nonuniform systems, recently demonstrated p-n and p-n-p devices. We show that the bipolar p-n and p-n-p junctions can host counter-circulating QHE edge states, which mix at the p-n interfaces, giving rise to fractional and integer quantization of the two-terminal conductance, observed in this structures. Graphene exhibits interesting spin- and valley-polarized QH ferromagnetic (FM) states. We show that spin-polarized QH state at zero doping hosts counter-circulating edge states carrying opposite spins, and propose to use this regime as a vehicle to study spin transport. We study ordering in the valley-polarized QH state.(cont.) Coupling of valley QHFM order parameter to random strain-induced vector potential yields an easy-plane-type ordering of the valley QHFM, giving rise to Berezinskii-Kosterlitz-Thouless transition, with fractionally charged vortices (merons) in the ordered state.by Dmitry A. Abanin.Ph.D