# The theory of avoided criticality in quantum motion in a random potential in high dimensions

## TL;DR

This paper develops an analytic theory explaining how rare region effects in high-dimensional quantum systems with disorder smooth out singularities in the density of states, transforming what appears as a phase transition into a crossover.

## Contribution

It provides a theoretical framework showing how rare region fluctuations cause the rounding of critical singularities in high-dimensional disordered quantum systems.

## Key findings

- Rare region effects lead to smoothing of the density of states singularity.
- The transition is replaced by a crossover due to disorder fluctuations.
- Analytic theory aligns with numerical and renormalization group results.

## Abstract

The density of states of a three dimensional Dirac equation with a random potential as well as in other problems of quantum motion in a random potential placed in sufficiently high spatial dimensionality appears to be singular at a certain critical disorder strength. This was seen numerically in a variety of studies as well as supported by detailed renormalization group calculations. At the same time it was suggested by a number of arguments accompanied by detailed numerical simulations that this singularity is rounded off by the rare region fluctuations of random potential, and that tuning the disorder past its critical value is not a genuine phase transition but rather a crossover. Here we develop an analytic theory which explains how rare region effects indeed lead to rounding off of the singularity and to the crossover replacing the transition.