Elaborating the phase diagram of spin-1 anyonic chains

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Abstract

We revisit the phase diagram of spin-1 su(2)_k anyonic chains, originally studied by Gils et. al. . These chains possess several integrable points, which were overlooked (or only briefly considered) so far.

Exploiting integrability through a combination of algebraic techniques and exact Bethe ansatz results, we establish in particular the presence of new first order phase transitions, a new critical point described by a Z_k parafermionic CFT, and of even more phases than originally conjectured. Our results leave room for yet more progress in the understanding of spin-1 anyonic chains.

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Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices

E. Bakkers, L. Kouwenhoven, S R Plissard … (2012)

Majorana fermions are particles identical to their own antiparticles. They have been theoretically predicted to exist in topological superconductors. We report electrical measurements on InSb nanowires contacted with one normal (Au) and one superconducting electrode (NbTiN). Gate voltages vary electron density and define a tunnel barrier between normal and superconducting contacts. In the presence of magnetic fields of order 100 mT we observe bound, mid-gap states at zero bias voltage. These bound states remain fixed to zero bias even when magnetic fields and gate voltages are changed over considerable ranges. Our observations support the hypothesis of Majorana fermions in nanowires coupled to superconductors.

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Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect

N Read, Dmitry Green (1999)

We analyze pairing of fermions in two dimensions for fully-gapped cases with broken parity (P) and time-reversal (T), especially cases in which the gap function is an orbital angular momentum (\(l\)) eigenstate, in particular \(l=-1\) (p-wave, spinless or spin-triplet) and \(l=-2\) (d-wave, spin-singlet). For \(l\neq0\), these fall into two phases, weak and strong pairing, which may be distinguished topologically. In the cases with conserved spin, we derive explicitly the Hall conductivity for spin as the corresponding topological invariant. For the spinless p-wave case, the weak-pairing phase has a pair wavefunction that is asympototically the same as that in the Moore-Read (Pfaffian) quantum Hall state, and we argue that its other properties (edge states, quasihole and toroidal ground states) are also the same, indicating that nonabelian statistics is a {\em generic} property of such a paired phase. The strong-pairing phase is an abelian state, and the transition between the two phases involves a bulk Majorana fermion, the mass of which changes sign at the transition. For the d-wave case, we argue that the Haldane-Rezayi state is not the generic behavior of a phase but describes the asymptotics at the critical point between weak and strong pairing, and has gapless fermion excitations in the bulk. In this case the weak-pairing phase is an abelian phase which has been considered previously. In the p-wave case with an unbroken U(1) symmetry, which can be applied to the double layer quantum Hall problem, the weak-pairing phase has the properties of the 331 state, and with nonzero tunneling there is a transition to the Moore-Read phase. The effects of disorder on noninteracting quasiparticles are considered.

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Anyons in an exactly solved model and beyond

Alexei Kitaev (2005)

A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static \(\mathbb{Z}_{2}\) gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number \(\nu\). The Abelian and non-Abelian phases of the original model correspond to \(\nu=0\) and \(\nu=\pm 1\), respectively. The anyonic properties of excitation depend on \(\nu\bmod 16\), whereas \(\nu\) itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.

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Author and article information

Journal

Title: SciPost Physics

Abbreviated Title: SciPost Phys.

Publisher: Stichting SciPost

ISSN (Electronic): 2542-4653

Publication date Created: 2017

Publication date (Electronic): February 21 2017

Volume: 2

Issue: 1

Affiliations

[1 ]Istituto Nazionale di Fisica Nucleare Sezione di Bologna

[2 ]National Institute of Nuclear Physics (at SISSA)

[3 ]CEA Saclay

[4 ]French National Centre for Scientific Research

[5 ]Sorbonne University

[6 ]École Normale Supérieure

[7 ]University of Southern California

Article

DOI: 10.21468/SciPostPhys.2.1.004

SO-VID: 4e584da7-46ec-458f-8753-f2e50c278c6c

License:

This work is licensed under a Creative Commons Attribution 4.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/