# Real Space Theory for Electron and Phonon Transport in Aperiodic Lattices via Renormalization

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^{2}

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## Abstract

**:**

^{23}coupled Schrödinger equations requires unavailable yottabytes (YB) of memory just for recording the atomic positions. In contrast, the real-space renormalization method (RSRM) uses an iterative procedure with a small number of effective sites in each step, and exponentially lessens the degrees of freedom, but keeps their participation in the final results. In this article, we review aperiodic atomic arrangements with hierarchical symmetry investigated by means of RSRM, as well as their consequences in measurable physical properties, such as electrical and thermal conductivities.

## 1. Introduction

## 2. Fibonacci Chains

## 3. Aperiodic Chains besides Fibonacci

**M**):

**M**has the following eigenvalues (${\lambda}_{\pm}$):

**M**,

**M**are satisfied [64]. On the contrary, the GF sequences with $v\ne 1$ do not satisfy the unit-determinant requirement and thus they are not quasiperiodic. When $u=v=1$, the sequence is called golden mean or the standard Fibonacci one, while the cases $u=2$ and $u=3$ are named silver and bronze means, respectively, when $v=1$, which are also known as the precious means. In addition, the metallic means stand for the sequences with $u=1$ and $v>1$ [65].

## 4. Multidimensional Aperiodic Lattices

## 5. Vibrational Excitations

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

YB | Yottabytes |

RSRM | Real-space renormalization method |

CPA | Coherent potential approximation |

CPU | Central processing unit |

1D | One-dimensional |

2D | Two-dimensional |

3D | Three-dimensional |

DC | Direct current |

AC | Alternating current |

DOS | Density of states |

IDOS | Integrated density of states |

GF | Generalized Fibonacci |

TM | Thue–Morse |

PD | Period doubling |

NW | Nanowires |

IR | Infrared |

DNA | Deoxyribonucleic acid |

A | Adenine |

C | Cytosine |

G | Guanine |

T | Thymine |

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**Figure 1.**(color online) (

**a**,

**b**) Density of states (DOS) and (

**c**,

**d**) zero-temperature direct current (DC) conductivity (σ) versus the chemical potential (μ) for two bond-disordered Fibonacci chains (

**b**,

**d**) of $n=57$ and (

**a**,

**c**) with a unit cell of $n=15$. Insets (

**a’**–

**d’**) are the respective magnifications of (

**a**–

**d**) spectra.

**Figure 2.**(color online) Schematic representations of one-dimensional (

**a**) Fibonacci, (

**b**) Thue-Morse, (

**c**) branched, and (

**d**) molecular chains, as well as two-dimensional (

**e**) Penrose, (

**f**) Fibonacci, (

**g**) labyrinth, and (

**h**) Poly(G)-Poly(C) lattices.

**Figure 3.**(color online) A log–log plot of single-energy Kubo conductivity computing time versus the total number of atoms in a quasiperiodic nanowire with a cross section of 5×5 atoms schematically illustrated in the inset, where the calculations were performed using a Fortran inversion subroutine (blue squares) and the renormalization plus convolution method of [17] (red circles).

**Figure 4.**Lattice thermal conductance (K) as a function of temperature (T) for periodic (circles), Fibonacci (squares), period doubling (up triangles), and Thue–Morse (down triangles) chains. Insets: the corresponding phononic transmittance spectra of (

**a**) Fibonacci, (

**b**) period doubling, and (

**c**) Thue–Morse chains, as well as (

**d**) an amplification of K(T)/K

_{0}−T at the low-temperature zone.

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Sánchez, V.; Wang, C.
Real Space Theory for Electron and Phonon Transport in Aperiodic Lattices via Renormalization. *Symmetry* **2020**, *12*, 430.
https://doi.org/10.3390/sym12030430

**AMA Style**

Sánchez V, Wang C.
Real Space Theory for Electron and Phonon Transport in Aperiodic Lattices via Renormalization. *Symmetry*. 2020; 12(3):430.
https://doi.org/10.3390/sym12030430

**Chicago/Turabian Style**

Sánchez, Vicenta, and Chumin Wang.
2020. "Real Space Theory for Electron and Phonon Transport in Aperiodic Lattices via Renormalization" *Symmetry* 12, no. 3: 430.
https://doi.org/10.3390/sym12030430