Transport in deformed centrosymmetric networks

TL;DR

This paper introduces a new random matrix ensemble to study how varying degrees of centrosymmetry affect quantum transport, revealing phase transitions and linking symmetry measures to transfer fidelity.

Contribution

It defines the Deformed Centrosymmetric Ensemble (DCE) and connects symmetry measures to transport properties, demonstrating phase transitions and a method to control symmetry in complex networks.

Findings

Second order phase transitions at specific symmetry parameters

Ergodic transition at a critical parameter value

Maximum transfer fidelity correlates with centrosymmetry degree

Abstract

Centrosymmetry often mediates Perfect State Transfer (PST) in various complex systems ranging from quantum wires to photosynthetic networks. We introduce the Deformed Centrosymmetric Ensemble (DCE) of random matrices, $H(\lambda) \equiv H_+ + \lambda H_-$, where $H_+$ is centrosymmetric while $H_-$ is skew-centrosymmetric. The relative strength of the $H_\pm$ prompts the system size scaling of the control parameter as $\lambda = N^{-\frac{\gamma}{2}}$. We propose two quantities, $\mathcal{P}$ and $\mathcal{C}$, quantifying centro- and skewcentro-symmetry, respectively, exhibiting second order phase transitions at $\gamma_\text{P}\equiv 1$ and $\gamma_\text{C}\equiv -1$. In addition, DCE posses an ergodic transition at $\gamma_\text{E} \equiv 0$. Thus equipped with a precise control of the extent of centrosymmetry in DCE, we study the manifestation of $\gamma$ on the transport properties of complex networks. We propose that such random networks can be constructed using the eigenvectors of $H(\lambda)$ and establish that the maximum transfer fidelity, $F_T$, is equivalent to the degree of centrosymmetry, $\mathcal{P}$.