arXiv:2003.07812 · SciPost Physics Core

Phase diagram of the $\mathbb{Z}_3$-Fock parafermion chain with pair hopping

Iman Mahyaeh, Jurriaan Wouters, Dirk Schuricht

Published Nov 12, 2020• 11 citations

Abstract

We study a tight binding model of \mathbb{Z}_3 ℤ 3 -Fock parafermions with single-particle and pair-hopping terms. The phase diagram has four different phases: a gapped phase, a gapless phase with central charge \boldsymbol{c=2} 𝐜 = 2 , and two gapless phases with central charge \boldsymbol{c=1} 𝐜 = 1 . We characterise each phase by analysing the energy gap, entanglement entropy and different correlation functions. The numerical simulations are complemented by analytical arguments.

DOI: 10.21468/scipostphyscore.3.2.011

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arXiv:2003.07812 · SciPost Physics Core

Phase diagram of the $\mathbb{Z}_3$ -Fock parafermion chain with pair hoppingPhase diagram of the $\mathbb{Z}_3$-Fock parafermion chain with pair hopping

Iman Mahyaeh, Jurriaan Wouters, Dirk Schuricht

Published Nov 12, 2020• 11 citations

arXiv HTML PDF

Abstract

DOI: 10.21468/scipostphyscore.3.2.011

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Phase diagram of the Z3 -Fock parafermion chain with pair hopping

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Jan 1, 20201 citation

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Parafermions are exotic quasiparticles with non-Abelian fractional statistics that can be realized and stabilized in one-dimensional models that are generalizations of the Kitaev $p$ -wave wire. We study the simplest generalization, i.e., the ${Z}_{3}$ parafermionic chain. Using a Jordan-Wigner transform we focus on the equivalent three-state chiral clock model, and study its rich phase diagram using the density matrix renormalization group technique. We perform our analyses using quantum entanglement diagnostics which allow us to determine phase boundaries, and the nature of the phase transitions. In particular, we study the transition between the topological and trivial phases, as well as to an intervening incommensurate phase which appears in a wide region of the phase diagram. The phase diagram is predicted to contain a Lifshitz type transition which we confirm using entanglement measures. We also attempt to locate and characterize a putative tricritical point in the phase diagram where the three above mentioned phases meet at a single point.Parafermions are exotic quasiparticles with non-Abelian fractional statistics that can be realized and stabilized in one-dimensional models that are generalizations of the Kitaev $p$-wave wire. We study the simplest generalization, i.e., the ${Z}_{3}$ parafermionic chain. Using a Jordan-Wigner transform we focus on the equivalent three-state chiral clock model, and study its rich phase diagram using the density matrix renormalization group technique. We perform our analyses using quantum entanglement diagnostics which allow us to determine phase boundaries, and the nature of the phase transitions. In particular, we study the transition between the topological and trivial phases, as well as to an intervening incommensurate phase which appears in a wide region of the phase diagram. The phase diagram is predicted to contain a Lifshitz type transition which we confirm using entanglement measures. We also attempt to locate and characterize a putative tricritical point in the phase diagram where the three above mentioned phases meet at a single point.

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Phase diagram of the

\mathbb{Z}_3

-Fock parafermion chain with pair hopping

\mathbb{Z}_3

axial next-nearest neighbour Potts model. We discuss a possible experimental realisation using hetero-nanostructures. The phase diagram contains several gapped phases, including a topological phase where the system possesses three (nearly) degenerate ground states, and a gapless Luttinger-liquid phase.

In this work, we numerically study critical phases in translation-invariant

{\mathbb{Z}}_{N}

parafermion chains with both nearest- and next-nearest-neighbor hopping terms. The model can be mapped to a

{\mathbb{Z}}_{N}

spin model with nearest-neighbor couplings via a generalized Jordan-Wigner transformation and translational invariance ensures that the spin model is always self-dual. We first study the low-energy spectrum of chains with only nearest-neighbor coupling, which are mapped onto standard self-dual

{\mathbb{Z}}_{N}

clock models. For $3\ensuremath{\le}N\ensuremath{\le}6$, we match the numerical results to the known conformal field theory(CFT) identification. We then analyze in detail the phase diagram of a

N=3

chain with both nearest and next-nearest-neighbor hopping and six critical phases with central charges being

4/5

, 1, or 2 are found. We find continuous phase transitions between

c=1

and 2 phases, while the phase transition between

c=4/5

and 1 is conjectured to be of Kosterlitz-Thouless type.

Parafermions are exotic quasiparticles with non-Abelian fractional statistics that can be realized and stabilized in one-dimensional models that are generalizations of the Kitaev

p

-wave wire. We study the simplest generalization, i.e., the

{Z}_{3}

parafermionic chain. Using a Jordan-Wigner transform we focus on the equivalent three-state chiral clock model, and study its rich phase diagram using the density matrix renormalization group technique. We perform our analyses using quantum entanglement diagnostics which allow us to determine phase boundaries, and the nature of the phase transitions. In particular, we study the transition between the topological and trivial phases, as well as to an intervening incommensurate phase which appears in a wide region of the phase diagram. The phase diagram is predicted to contain a Lifshitz type transition which we confirm using entanglement measures. We also attempt to locate and characterize a putative tricritical point in the phase diagram where the three above mentioned phases meet at a single point.