# Quantum phase transition and criticality in quasi one-dimensional spinless Dirac fermions

## TL;DR

This paper investigates quantum phase transitions in quasi-one-dimensional spinless Dirac fermions on a pi-flux square lattice, revealing continuous Ising and Gaussian criticalities and their relation via bosonization.

## Contribution

It uncovers the nature of quantum critical points in this system, showing their classification and connection through a unified bosonization framework.

## Key findings

- Quantum phase transitions are continuous and belong to the (1+1)-D Ising universality class for certain circumferences.
- Other circumferences exhibit Gaussian transitions from gapless Dirac fermions to charge density wave states.
- A critical line with central charge c=1/2 emerges from a Gaussian transition, indicating an Ising transition involving Majorana fermions.

## Abstract

We study quantum criticality of spinless fermions on the quasi one dimensional $\pi$-flux square lattice in cylinder geometry, by using the infinite density matrix renormalization group and abelian bosonization. For a series of the cylinder circumferences $L_y=4n+2=2, 6, \cdots$ with the periodic boundary condition, there are quantum phase transitions from gapped Dirac fermion states to charge density wave (CDW) states. We find that the quantum phase transitions for such circumferences are continuous and belong to the (1+1)-dimensional Ising universality class. On the other hand, when $L_y=4n=4, 8, \cdots$, there are gapless Dirac fermions at the non-interacting point and the phase transition to the CDW state is Gaussian. Both of these two criticalities are described in a unified way by the bosonization. We clarify their intimate relationship and demonstrate that a central charge $c=1/2$ Ising transition line arises as a critical state of an emergent Majorana fermion from the $c=2$ Gaussian transition point.