# Multi-critical Behavior in Topological Phase Transitions

## TL;DR

This paper investigates the critical behavior of topological phase transitions by analyzing edge mode penetration depths in generalized SSH models, revealing multi-critical points and different universality classes.

## Contribution

It introduces a numerical method to determine correlation length exponents directly from edge mode penetration depths in generalized topological models.

## Key findings

- Identification of different universality classes of topological transitions.
- Discovery of a multi-critical point where behavior depends on the approach path.
- Validation of results through a scaling approach to the Berry connection.

## Abstract

Topological phase transitions can be described by the theory of critical phenomena and identified by critical exponents that define their universality classes. This is a consequence of the existence of a diverging length at the transition that has been identified as the penetration depth of the surface modes in the non-trivial topological phase. In this paper, we characterize different universality classes of topological transitions by determining their correlation length exponents directly from numerical calculations of the penetration length of the edge modes as a function of the distance to the topological transition. We consider generalizations of the topological non-trivial Su-Schrieefer-Heeger (SSH) model, for the case of next nearest neighbors hopping and in the presence of a synthetic potential. The latter allows the system to transit between two universality classes with different correlation length and dynamic critical exponents. It presents a multi-critical point in its phase diagram since the behavior of the Berry connection depends on the path it is approached. We compare our results with those obtained from a scaling approach to the Berry connection.