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Topological and quantum critical properties of the interacting Majorana chain model

by Natalia Chepiga, Nicolas Laflorencie

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Submission summary

Authors (as registered SciPost users): Natalia Chepiga · Nicolas Laflorencie
Submission information
Preprint Link: scipost_202212_00017v2  (pdf)
Date accepted: 2023-04-11
Date submitted: 2023-02-03 18:46
Submitted by: Laflorencie, Nicolas
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approach: Computational


We study Majorana chain with the shortest possible interaction term and in the presence of hopping 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 period-2-$\mathbb{Z}_2$ 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.

Author comments upon resubmission

Dear editor,

We are pleased to see that our manuscript has three positive reports and that we are only asked for minor revisions. Please find enclosed our response to all comments, as well as a summary of changes, following some referee suggestions.

Natalia Chepiga and Nicolas Laflorencie

List of changes

1/ We have rephrased “zero modes vanishing exponentially with a system size” onto “zero modes showing a vanishing (parity) gap, exponentially suppressed with the system size.”

2/ We have generated a new Fig. 7 with a panel (d) showing the N^{-3} finite size scaling of the first low-energy gaps at the M point.

3/ We have merged [15] and [35], and added [24] to the group [20-25].
[24] M. McGinley, J. Knolle and A. Nunnenkamp, Robustness of majorana edge modes and topological order: Exact results for the symmetric interacting Kitaev chain with disorder, Phys. Rev. B 96, 241113 (2017), doi:10.1103/PhysRevB.96.241113.

4/ In section 2.1.1 we have added: “Note that this phase has no topological interest for the (Ising) spin degrees of freedom for which the so-called "topological phase" there boils down to a more conventional magnetic order.”

5/ We have added a sentence in the caption of Fig. 12 “$E_0$ states for the ground-state energy and $E_1$ is the energy of the in-gap excitation.”

Published as SciPost Phys. 14, 152 (2023)

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