Fibonacci anyon excitations of one-dimensional dipolar lattice bosons

Abstract

We study a system of dipolar bosons in a one-dimensional optical lattice using exact diagonalization and density matrix renormalization group methods. In particular, we analyze low energy properties of the system at an average filling of 3/2 atoms per lattice site. We identify the region of the parameter space where the system has non-Abelian Fibonacci anyon excitations that correspond to fractional domain walls between different charge-density-waves. When such onedimensional systems are combined into a two-dimensional network, braiding of Fibonacci anyon excitations has potential application for fault tolerant, universal, topological quantum computation. Contrary to previous calculations, our results also demonstrate that super-solid phases are not present in the phase diagram for the discussed 3/2 average filling. Instead, decreasing the value of the nearest-neighbor tunneling strength leads to a direct, Berezinskii-Kosterlitz-Thouless, super-fluid to charge-density-wave quantum phase transition.

Introduction to topological quantum computation with non-Abelian anyons

“…One way to mitigate the effect of these errors is in using topological quantum computing (Collins, 2006a;Freedman, 1998;Nayak et al, 2008;Pachos, 2012;Stanescu, 2017;Wang, 2010). In contrast to locally encoding information and computation using, for example, the spin of an electron (Castelvecchi, 2018;Kane, 1998;Loss and DiVincenzo, 1998;Reilly et al, 2008), the energy levels of an ion (Cirac and Zoller, 1995;Leibfried et al, 2003), optical modes containing one photon (Knill et al, 2001), or superconducting Josephson junctions (Shnirman et al, 1997), topological quantum computers encode information using global, topological properties of a quantum system, which are resilient to local perturbations (Bombin and Martin-Delgado, 2008;Bombin and Martin-Delgado, 2011;Nayak et al, 2008;Pachos and Simon, 2014). These topological quantum computers can be implemented using non-Abelian anyons, which are quasiparticles in two-dimensional systems which exhibit exotic exchange statistics, beyond a simple phase change (Pachos, 2012).…”

“…There has also been considerable study into candidate physical systems which could contain non-Abelian anyons. Most notable candidate for finding Fibonacci anyons is the fractional quantum Hall effect at ν = 12/5 (Ardonne and Schoutens, 2007;Bonderson et al, 2006;Brennen and Pachos, 2008;Mong et al, 2017;Nayak et al, 2008;Rezayi and Read, 2009;Sarma et al, 2006;Trebst et al, 2008;Wu et al, 2014), although other candidates exist (Brennen and Pachos, 2008;Ðurić et al, 2017;Cooper et al, 2001;Fendley et al, 2013). Meanwhile, significant effort is directed toward finding Ising anyons in nanowires hosting Majorana zero modes (Alicea, 2012;Sarma et al, 2015;Zhang et al, 2018) In this work, we have explicitly carried out a quantum algorithm, specifically the AJL algorithm, by simulating the braiding of Fibonacci anyons.…”

Quantum phases of dipolar bosons in one-dimensional optical lattices

“…These correlations might be also important at fractional filling. For instance, they might affect the properties of the Fibonacci anyonic excitations expected at ρ = 3/2 for low tunnelling rates [53]. The phase diagrams are calculated by means of a DMRG numerical program using the ITensor C++ library [54] (see also Ref.…”

Critical entanglement for the half-filled extended Hubbard model

“…As demonstrated in Ref. 41 and citations to that article [62][63][64][65], the peak in the central charge provides a reliable, universal way of identifying BKT transitions from finitesize data. We demonstrate this approach for the EHM with two methods: extract central charge for each V with a simple curve fit that has been scaled to the middle, and the logarithmic derivative method to extract central charge for each V, as described in the Methods section, II C. The results presented below are to be compared against the most reliable, found in Refs.…”