Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1912.03482

Exact modular $S$ matrix for the ${\mathbb Z}_k$ parafermion quantum Hall islands and measurement of non-Abelian anyons

Lachezar S. Georgiev
View original source ↗Read source document ↗Export metadata ↓

Abstract

Using the decomposition of rational conformal filed theory characters for the ${\mathbb Z}_k$ parafermion quantum Hall droplets for general $k=2,3, \ldots$, we derive analytically the full modular $S$ matrix for these states, including the $\widehat{u(1)}$ parts corresponding to the charged sector of the full conformal field theory and the neutral parafermion contributions corresponding to the diagonal affine coset models. This precise neutral-part parafermion $S$ matrix is derived from the explicit relations between the coset matrix and those for the numerator and denominator of the coset and the latter is expressed in compact form due to the level--rank duality between the affine Lie algebras $\widehat{su(k)_2}$ and $\widehat{su(2)_k}$. The exact results obtained for the $S$ matrix elements are expected to play an important role for identifying interference patterns of fractional quantum Hall states in Fabry-Pérot interferometers which can be used to distinguish between Abelian and non-Abelian statistics of quasiparticles localized in the bulk of fractional quantum Hall droplets as well as for nondestructive interference measurement of Fibonacci anyons which can be used for universal topological quantum computation.

Explore related subjects

math-phmath.MPquant-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lachezar S. Georgiev. 2019-12-07. Exact modular $S$ matrix for the ${\mathbb Z}_k$ parafermion quantum Hall islands and measurement of non-Abelian anyons. https://arxiv.org/abs/1912.03482

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Topological Orders from Reflection Positive Frustration-free Hamiltonians

We establish a framework based on reflection positivity for analyzing topologically ordered quantum spin systems and reconstructing their boundary algebras. For any reflection positive frustration-free Hamiltonian, we prove that the local topological quantum order (LTQO) condition of ground states on a disk holds, if and only if the ground state on the sphere obtained by gluing the disk with its reflection is nondegenerate. Furthermore, we show that Osterwalder-Schrader reconstruction produces the local net of boundary operator algebras from the local ground states, offering a constructive approach to topological holography through spatial reflection positivity.

math-ph↗
arXiv2026

Generalised Langevin Dynamics: Significance and Limitations of the Projection Operator Formalism

We discuss some mathematical aspects of the Mori-Zwanzig projection operator formalism. The core of the Mori-Zwanzig formalism is the generalised Langevin equation, which is typically derived from the Dyson-Duhamel identity. We recall the derivation of the projection operator formalism for Mori's projection by means of semigroup theory, and we discuss where rigorous methods fail for the case of Zwanzig's projection. For bounded perturbations of the time-evolution operator (e.g. for Mori's projection), the Dyson-Duhamel identity coincides with the variation of constants formula. For unbounded perturbations (e.g. for Zwanzigs's projection), the Dyson-Duhamel identity should be considered an equation for the orthogonal dynamics, for which the existence of unique solutions has yet to be established. Then we recall that all properties of Mori's generalised Langevin equation follow directly from the well-posedness of Volterra equations, irrespective of the projection operator formalism. Further, we discuss the use of Mori's generalised Langevin equation as a coarse-grained model. Finally, we illustrate that the memory term is a coupling term that is not necessarily related to memory. To this end, we introduce projections onto subspaces of 'fast' and 'slow' variables that are associated with the spectral decomposition of skew-adjoint operators. For these projections, the memory term vanishes.

math-ph↗
arXiv2026

Universal fusion category symmetries on tensor products of infinite-dimensional Hilbert spaces

We show that anyon chains, after stabilizing with infinite-dimensional ancilla spaces, factorize locally as tensor products of infinite-dimensional Hilbert spaces. This implies that any unitary fusion category can be realized as symmetries on a tensor product of infinite-dimensional Hilbert spaces. We then show that any two anyon chains with the same symmetry category are related by a symmetry-compatible locality-preserving unitary after stabilizing with infinite-dimensional ancilla, showing that for a fixed fusion category, there is a single stable equivalence class of symmetry realizations on the lattice via anyon chains. As a corollary of our proof, we show that the physical boundary algebras of Levin-Wen type models are bounded spread isomorphic after stabilization if and only if they have the same bulk topological order.

math-ph↗