Quantum Network Models and Classical Localization Problems

Cardy, John

doi:10.1142/s0217979210064678

Cited by 10 publications

(8 citation statements)

References 22 publications

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“…Another example is the Chalker-Coddington model [195] for the quantum Hall (and spin quantum Hall) effect, where the transmission matrix J between two contacts a and b is given by [196,197]…”

Section: Complex Disorder and Localizationmentioning

confidence: 99%

Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles

Wiese

2021

Preprint

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Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modelled as an elastic system subject to quenched disorder. Its field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group flow involves a function, the disorder correlator ∆(w), therefore termed the functional renormalization group (FRG). ∆(w) is a physical observable, the auto-correlation function of the centre of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of these mappings requires specific techniques, which we develop, including modelling of discrete stochastic systems via coarsegrained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and nonlinear surface growth with additional KPZ terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random field magnets. Emphasis is given to numerical and experimental tests of the theory.

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Section: Complex Disorder and Localizationmentioning

confidence: 99%

Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles

Wiese

2021

Preprint

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“…In general, oriented scattering network models consist of a directed graph, composed of a set of vertices (or nodes) representing scattering matrices, which are connected to each other by directed edges (or links) over which flows a directed current [20]. At each vertex v, the number b v of incoming links is equal to the number of outgoing links to guarantee the unitarity of scattering events, which are described by a scattering matrix S v ∈ U (b v ), which relates the incoming amplitudes c in e on each incoming edge e to the outgoing amplitudes c out f on each outgoing edge f by…”

Section: B Oriented Scattering Network Modelsmentioning

confidence: 99%

“…that each vertex is connected to the same number of links. The most simple nontrivial situations is b = 2, where matrices are U (2) rotations, and it is usually possible to reduce any network model to this situation [20,21]. While network models can be used in any space dimension, we shall focus on two-dimensional systems.…”

Section: B Oriented Scattering Network Modelsmentioning

confidence: 99%

“…The quantum Hall transition essentially arises when the equipotentials of the disorder percolate; however, near the transition, the relevant equipotentials approach the saddle points of the disorder potential and become closer and closer. Hence, wave packets can tunnel from an equipotential to another giving rise to "quantum percolation" [11] (see also [20] for a pedagogical introduction). This process is described by scattering matrices, one per saddle point, within the Chalker-Coddington model [11] that distorts the equipotentials into a periodic square lattice of such scattering matrices connected by incoming and outgoing directed links, the so-called L-lattice.…”

Section: A Introductionmentioning

confidence: 99%

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Phase rotation symmetry and the topology of oriented scattering networks

2017

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We investigate the topological properties of dynamical states evolving on periodic oriented graphs. This evolution, which encodes the scattering processes occurring at the nodes of the graph, is described by a single-step global operator, in the spirit of the Ho-Chalker model. When the successive scattering events follow a cyclic sequence, the corresponding scattering network can be equivalently described by a discrete time-periodic unitary evolution, in line with Floquet systems. Such systems may present anomalous topological phases where all the first Chern numbers are vanishing, but where protected edge states appear in a finite geometry. To investigate the origin of such anomalous phases, we introduce the phase rotation symmetry, a generalization of usual symmetries which only occurs in unitary systems (as opposed to Hamiltonian systems). Equipped with this new tool, we explore a possible explanation of the pervasiveness of anomalous phases in scattering network models, and we define bulk topological invariants suited to both equivalent descriptions of the network model, which fully capture the topology of the system. We finally show that the two invariants coincide, again through a phase rotation symmetry arising from the particular structure of the network model.

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“…We consider the Manhattan pinball problem which is a network model of a quantum localization problem (see [3] and also [12, Page 237], [5]). There are equivalent ways to state the problem formally.…”

Section: Introductionmentioning

confidence: 99%

On the Manhattan pinball problem

2021

Electron. Commun. Probab.

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We consider the periodic Manhattan lattice with alternating orientations going northsouth and east-west. Place obstructions on vertices independently with probability 0 < p < 1. A particle is moving on the edges with unit speed following the orientation of the lattice and it will turn only when encountering an obstruction. The problem is that for which value of p is the trajectory of the particle closed almost surely. We prove this is true for p > 1 2 − ε with some ε > 0.

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Quantum Network Models and Classical Localization Problems

Cited by 10 publications

References 22 publications

Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles

Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles

Phase rotation symmetry and the topology of oriented scattering networks

On the Manhattan pinball problem

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