2D Helical Twist Controls Tricritical Point in an Interacting Majorana Chain

TL;DR

This paper investigates how 2D helical twists influence the tricritical point in an interacting Majorana chain, revealing a connection between 1D phase transitions and 2D universality classes through geometrical mapping.

Contribution

It introduces a novel geometrical twist mapping of 1D Majorana chains to 2D models, uncovering the role of helical boundary conditions in controlling phase transitions.

Findings

Identification of a tri-critical point separating phases

Exact solvability at a specific coupling with zero entanglement entropy

Phase transition behavior governed by 2D tri-critical universality class

Abstract

We analyze a series of interacting Majorana Fermion chains with finite range pair interactions with coupling strength $g$ that all exhibit a tri-critical point that separates an Ising critical phase from a supersymmetric gapped phase. We first notice that the interacting models exhibit an even-odd asymmetry depending on the number of sites, $\delta$, over which the interaction ranges. The even case exhibits competing order, thereby making it numerically untractable while the odd case exhibits an exactly solvable point at $g=-0.5$ where the entanglement entropy vanishes. By introducing a swirling geometrical twist, we map our 1D $\delta$-range chains to a series of 2D $\delta/2$-width models. Our new 2D models possess a unique helical boundary condition, constructed from 1D chains with the end of one connected to the start of another. We propose that the phase transition in the 1D system can be understood as a finite-system size transition in 2D. That is, the $g_c-\delta$ behavior is controlled by a 2D tri-critical universality class at $\delta\to\infty$ limit and is predicted by finite-size scaling theory.