Topological and quantum critical properties of the interacting Majorana chain model

TL;DR

This paper explores the complex phase diagram of an interacting Majorana chain model, revealing six phases, topological properties, and critical behaviors, challenging previous assumptions about the nature of the Luttinger liquid phase.

Contribution

It provides the first detailed phase diagram of the model, identifies the universality classes of critical points, and analyzes topological features with Majorana zero modes.

Findings

Six distinct phases identified, including four gapped and two gapless.

The Luttinger liquid phase ends at a weaker interaction than the Ising critical point.

Topological phases exhibit Majorana zero modes and parity switches.

Abstract

We study Majorana chain with the shortest possible interaction term and in the presence of hoping alternation. When formulated in terms of spins the model corresponds to the transverse field Ising model with nearest-neighbor transverse and next-nearest-neighbor longitudinal repulsion. The phase diagram obtained with extensive DMRG simulations is very rich and contains six phases. Four gapped phases include paramagnetic, period-2 with broken translation symmetry, $\mathbb{Z}_2$ with broken parity symmetry and the $\mathbb{Z}_2$ period-2 phase with both symmetries broken. In addition there are two floating phases: gapless and critical Luttinger liquid with incommensurate correlations, and with an additional spontaneously broken $\mathbb{Z}_2$ symmetry in one of them. By analyzing an extended phase diagram we demonstrate that, in contrast with a common belief, the Luttinger liquid phase along the self-dual critical line terminates at a weaker interaction strength than the end point of the Ising critical line that we find to be in the tri-critical Ising universality class. We also show that none of these two points is a Lifshitz point terminating the incommensurability. In addition, we analyzed topological properties through Majorana zero modes emergent in the two topological phases, with and without incommensurability. In the weak interaction regime, a self-consistent mean-field treatment provides a remarkable accuracy for the description of the spectral pairing and the parity switches induced by the interaction.