## Quantum and Semi-classical Percolation and Breakdown in Disordered SolidsAsok K. Sen, Kamal K. Bardhan, Bikas K. Chakrabarti This lecture notes in physics volume mainly focuses on the semi classical and qu- tum aspects of percolation and breakdown in disordered, composite or granular s- tems. The main reason for this undertaking has been the fact that, of late, there have been a lot of (theoretical) work on quantum percolation, but there is not even a (single) published review on the topic (and, of course, no book). Also, there are many theoretical and experimental studies on the nonlinear current-voltage characteristics both away from, as well as one approaches, an electrical breakdown in composite materials. Some of the results are quite intriguing and may broadly be explained utilising a semi classical (if not, fully quantum mechanical) tunnelling between - cron or nano-sized metallic islands dispersed separated by thin insulating layers, or in other words, between the dangling ends of small percolation clusters. There have also been several (theoretical) studies of Zener breakdown in Mott or Anderson in- lators. Again, there is no review available, connecting them in any coherent fashion. A compendium volume connecting these experimental and theoretical studies should be unique and very timely, and hence this volume. The book is organised as follows. For completeness, we have started with a short and concise introduction on classical percolation. In the ?rst chapter, D. Stauffer reviews the scaling theory of classical percolation emphasizing (biased) diffusion, without any quantum effects. The next chapter by A. K. |

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### Contents

Classical Percolation | 1 |

2 Methods | 3 |

3 Quantities and Exponents | 6 |

Incipient Infinite Cluster | 9 |

5 Simple Renormalisation Group | 12 |

6 Diffusion and Percolation | 14 |

7 Summary | 18 |

Nonlinear Response Semiclassical Percolation and Breakdown in the RRTN Model | 21 |

3 Localization Effects in Quantum Percolation | 144 |

4 Percolative Effects in Advanced Materials | 154 |

5 Conclusions | 159 |

References | 160 |

Quantum Percolation in the Quantum Hall Regime | 163 |

2 The Quantized Hall Effect and Classical Percolation | 165 |

3 Network Models at the IQHE | 168 |

4 HartreeFock Approach to the IQHE | 184 |

2 The Origin of the RRTN Model and Its Percolative Aspects | 27 |

3 Nonlinear SteadyState IV Characteristics | 35 |

4 Periodic Driving and ACResponse in the RRTNCRC Model | 46 |

5 VRH and Low Temperature Conduction in the RRTN | 55 |

6 Slow Powerlaw Dynamics FarfromEquilibrium | 59 |

7 Aspects of Reversible Breakdown in the RRTN Model | 65 |

8 Dynamical Characteristics of Breakdown | 73 |

9 Summary and Further Works | 77 |

References | 78 |

Quantum Transmittance Through Random Media | 83 |

2 One Parameter Scaling Theory of Localization | 84 |

3 Transport Mechanisms in Disordered Media | 85 |

4 Some Models of Disordered Systems | 87 |

5 Some Earlier Studies on the Quantum Percolation Model | 89 |

6 The Vector Recursion Method and Its Applications | 92 |

7 Conclusions | 105 |

References | 106 |

Quantum Percolation in Two Dimensions | 108 |

2 Resonances and Phase Variations in Ordered Limit | 116 |

3 TimeIndependent Schrodinger Equation for Finite Disorder | 120 |

Sending a Wave Packet Through a 2D Cluster | 125 |

5 Summary | 132 |

References | 133 |

Quantum Percolation in Disordered Structures | 135 |

2 Local Distribution Approach | 137 |

5 Conclusions | 189 |

References | 190 |

Percolative Quantum Transport in Manganites | 195 |

2 Standard Quantum Percolation | 196 |

Phenomenology and Model | 198 |

4 Percolative Effects in a One Band Model of Phase Competition | 205 |

5 Percolation in Two Band Models with ElectronPhonon Coupling | 217 |

6 Connection with Quantum Percolation and Resistor Networks | 221 |

7 Conclusions | 225 |

Classical and Quantum Breakdown in Disordered Materials | 227 |

2 Analysis of the Fuse Problem | 229 |

3 Dielectric Breakdown Problem | 238 |

4 Zener Breakdown in Anderson Insulators | 247 |

5 Conclusions | 249 |

Nonequilibrium Quantum Breakdown in a Strongly Correlated Electron System | 251 |

2 Nonadiabatic Evolution and Pair Creation of Carriers | 256 |

References | 283 |

Percolation in Quantum Computation and Communication | 286 |

2 Percolation and Quantum Computing | 290 |

3 Quantum Repeater Networks for Quantum Communication | 307 |

4 Summary and Open Problems | 315 |

References | 317 |

321 | |

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### Common terms and phrases

amplitude Anderson average band behaviour block bond percolation breakdown field calculate Chakrabarti classical percolation coefficient conductance configurations connected correlated corresponding critical critical exponent defect density dependence dielectric breakdown dilution dimension disorder distribution dynamics edge effect electric field electron energy entangled equation evolution exponential finite finite-size fitting fractal Hamiltonian hopping Hubbard model increasing infinite Landau-Zener Landau-Zener transition LDOS leads length Lett limit linear localized manganites many-body matrix metallic microscopic Mott insulator neighbour nodes nonlinearity exponent numerical ohmic pair panel parameter particle path percolation theory percolation threshold phase Phys Physics plot potential power-law probability problem quantum computation quantum percolation model qubits random regime regions renormalization resistor RRTN model sample Sect simulation site percolation spanning cluster spin square lattice t-bonds temperature tion transition transmission transport tunnelling typical vector voltage wave function wave packet zero