Scaling of von Neumann entropy at the Anderson transition

TL;DR

This paper demonstrates that von Neumann entropy exhibits non-analytic behavior at the Anderson transition, providing a new way to characterize quantum phase transitions through ground state correlations.

Contribution

It introduces von Neumann entropy as a novel indicator of quantum criticality, capturing non-analyticity at the Anderson transition where traditional ground state energy does not.

Findings

Von Neumann entropy is non-analytic at the Anderson transition.

Entropy captures spatial fluctuations of the wave function near criticality.

This approach may generalize to other quantum phase transitions.

Abstract

Extensive body of work has shown that for the model of a non-interacting electron in a random potential there is a quantum critical point for dimensions greater than two---a metal-insulator transition. This model also plays an important role in the plateau-to-plateu transition in the integer quantum Hall effect, which is also correctly captured by a scaling theory. Yet, in neither of these cases the ground state energy shows any non-analyticity as a function of a suitable tuning parameter, typically considered to be a hallmark of a quantum phase transition, similar to the non-analyticity of the free energy in a classical phase transition. Here we show that von Neumann entropy (entanglement entropy) is non-analytic at these phase transitions and can track the fundamental changes in the internal correlations of the ground state wave function. In particular, it summarizes the spatially wildly fluctuating intensities of the wave function close to the criticality of the Anderson transition. It is likely that all quantum phase transitions can be similarly described.