Disentanglement, disorder lines, and Majorana edge states in a solvable quantum chain

TL;DR

This paper analyzes an exactly solvable 1D quantum chain model, revealing its phase diagram, disorder lines, and Majorana edge states through complex zeros of the spectrum, with implications for topological phases and quantum criticality.

Contribution

It introduces the first identification of disorder lines in this model and links the zeros of the spectrum to phase transitions, disentanglement, and Majorana edge states.

Findings

Disorder lines of two types identified in the model.

Ground state is separable and disentangled on certain disorder lines.

Majorana edge states are characterized by the zeros of the spectrum.

Abstract

We study the exactly solvable 1D model: the dimerized $XY$ chain with uniform and staggered transverse fields, equivalent upon fermionization to the noninteracting dimerized Kitaev-Majorana chain with modulation. The model has three known gapped phases with local and nonlocal (string) orders, along with the gapless incommensurate (IC) phase in the $U(1)$ limit. The criticality is controlled by the properties of zeros of model's partition function, analytically continued onto the complex wave numbers. In the ground state they become complex zeros of the spectrum of the Hamiltonian. The analysis of those roots yields the phase diagram which contains continuous quantum phase transitions and weaker singularities known as disorder lines (DLs) or modulation transitions. The latter, reported for the first time in this model, are shown to occur in two types: DLs of the first kind with continuous appearance of the IC oscillations, and DLs of the second kind corresponding to a jump of the wave number of oscillations. The salient property of zeros of the spectrum is that the ground state is shown to be separable (factorized) and the model is disentangled on a subset of the DLs. From analysis of those zeros we also find the Majorana edge states and their wave functions.