Localization of the Grover Walks on Spidernets and Free Meixner Laws

Konno, Norio; Obata, Nobuaki; Segawa, Etsuo

doi:10.1007/s00220-013-1742-x

Cited by 19 publications

(13 citation statements)

References 34 publications

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“…10),(3.11) and(3.5), one finds the same quasi-periodicities as above also hold for the partition functionsW M,N (u 1 , . .…”

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confidence: 66%

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Elliptic supersymmetric integrable model and multivariable elliptic functions

Motegi¹

2017

Progress of Theoretical and Experimental Physics

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We investigate the elliptic integrable model introduced by Deguchi and Martin, which is an elliptic extension of the Perk-Schultz model. We introduce and study a class of partition functions of the elliptic model by using the Izergin-Korepin analysis. We show that the partition functions are expressed as a product of elliptic factors and elliptic Schurtype symmetric functions. This result resembles the recent works by number theorists in which the correspondence between the partition functions of trigonometric models and the product of the deformed Vandermonde determinant and Schur functions were established. *

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“…10),(3.11) and(3.5), one finds the same quasi-periodicities as above also hold for the partition functionsW M,N (u 1 , . .…”

supporting

confidence: 66%

“…The most famous one is the Baxter's eight-vertex model [7]. Investigating the underlying algebraic structures, several versions of the elliptic quantum groups have been formulated and studied [8,9,10,11,12].…”

Section: Introductionmentioning

confidence: 99%

Elliptic supersymmetric integrable model and multivariable elliptic functions

Motegi¹

2017

Progress of Theoretical and Experimental Physics

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“…Various types of the underlying graphs have been studied, e.g. quantum walks on general graphs [17], hypercubes [18,19], trees [20], honeycombs [21,22], spidernets [23] or fractal structures [24] (see also review [4]). One of the main driving forces of this research activities is interest in asymptotic properties of quantum walks, including limiting position distributions, speed of walker's propagation, and structure of trapped states.…”

Section: Introductionmentioning

confidence: 99%

Percolated quantum walks with a general shift operator: From trapping to transport

2019

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We present a generalized definition of discrete-time quantum walks convenient for capturing a rather broad spectrum of walker's behavior on arbitrary graphs. It includes and covers both: the geometry of possible walker's positions with interconnecting links and the prescribed rule in which directions the walker will move at each vertex. While the former allows for the analysis of inhomogeneous quantum walks on graphs with vertices of varying degree, the latter offers us to choose, investigate, and compare quantum walks with different shift operators. The synthesis of both key ingredients constitutes a well-suited playground for analyzing percolated quantum walks on a quite general class of graphs. Analytical treatment of the asymptotic behavior of percolated quantum walks is presented and worked out in details for the Grover walk on graphs with maximal degree 3. We find, that for these walks with cyclic shift operators the existence of an edge-3-coloring of the graph allows for non-stationary asymptotic behavior of the walk. For different shift operators the general structure of localized attractors is investigated, which determines the overall efficiency of a source-to-sink quantum transport across a dynamically changing medium. As a simple nontrivial example of the theory we treat a single excitation transport on a percolated cube.

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“…Investigations, aiming first at the quantum walker on the line, have gradually broadened the scope of their interest to different graph geometries like e.g. cycles [6], hypercubes [9,10], trees [11], honeycombs [12,13], spidernets [14] or fractal structures [15] (for more see review [1]).…”

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confidence: 99%

Quantum walk transport on carbon nanotube structures

2020

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We study source-to-sink excitation transport on carbon nanotubes using the concept of quantum walks. In particular, we focus on transport properties of Grover coined quantum walks on ideal and percolation perturbed nanotubes with zig-zag and armchair chiralities. Using analytic and numerical methods we identify how geometric properties of nanotubes and different types of a sink altogether control the structure of trapped states and, as a result, the overall source-to-sink transport efficiency. It is shown that chirality of nanotubes splits behavior of the transport efficiency into a few typically well separated quantitative branches. Based on that we uncover interesting quantum transport phenomena, e.g. increasing the length of the tube can enhance the transport and the highest transport efficiency is achieved for the thinnest tube. We also demonstrate, that the transport efficiency of the quantum walk on ideal nanotubes may exhibit even oscillatory behavior dependent on length and chirality.Discrete coined quantum walks (CQWs) have become a standard tool in studying transport phenomena in the quantum domain [1]. Over the last two decades, quantum walker's behavior has been analyzed in the context of recurrence phenomena [2,3], state transfer [4, 5], speed of wave packet propagation [6], hitting times [7], or e.g. topological phenomena [8]. Investigations, aiming first at the quantum walker on the line, have gradually broadened the scope of their interest to different graph geometries like e.g. cycles [6], hypercubes [9, 10], trees [11], honeycombs [12, 13], spidernets [14] or fractal structures [15] (for more see review [1]).Analysis of complex graph structures has revealed that if vertices with degree at least three are present, the walker's dynamics may exhibit localisation originating from the presence of so-called trapped states [16][17][18][19][20][21][22]. They appear in various quantum systems and are known, in a different context, also as localized invariant or dark states [23][24][25][26]. These are eigenstates of the given dynamics, whose support does not spread over the whole position space. Due to that, the initial states having overlap with the trapped states can not fully propagate through the medium and the efficiency of quantum transport may be significantly reduced [27,28].On the other hand, trapped states were found to be fragile with respect to certain decoherence mechanisms arising in the presence of random external perturbations [29]. When the quantum walker moves on a changing graph whose edges are randomly and repeatedly closed and open again, we arrive at so-called dynamically percolated coined quantum walks (PCQWs) [30,31] capable to destroy some trapped states of the original nonpercolated CQW [32]. Recently, it was shown how the underlying geometry of Grover PCQW controls the structure of walker's trapped states [33]. The detailed analytical recipe for constructing the basis has essentially contributed to identify interesting counterintuitive quantum transport phenomena [34]. In partic...

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Localization of the Grover Walks on Spidernets and Free Meixner Laws

Cited by 19 publications

References 34 publications

Elliptic supersymmetric integrable model and multivariable elliptic functions

Elliptic supersymmetric integrable model and multivariable elliptic functions

Percolated quantum walks with a general shift operator: From trapping to transport

Quantum walk transport on carbon nanotube structures

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