Topological Entanglement Entropy with a Twist

Abstract

Defects in topologically ordered models have interesting properties that are reminiscent of the anyonic excitations of the models themselves. For example, dislocations in the toric code model are known as twists and possess properties that are analogous to Ising anyons. We strengthen this analogy by using the topological entanglement entropy as a diagnostic tool to identify properties of both defects and excitations in the toric code. Specifically, we show, through explicit calculation, that the toric code model including twists and dyon excitations has the same quantum dimensions, the same total quantum dimension, and the same fusion rules as an Ising anyon model.A fascinating class of many-body quantum systems are those that exhibit topological order [1]. Such systems are characterized by a gapped ground state manifold, with a degeneracy that depends on the boundary conditions, and have anyonic quasi-particle excitations. The degeneracy is robust to local perturbations, and therefore such systems are promising candidates for storing and manipulating quantum information [2][3][4][5].The structure of topologically-ordered systems can be further enriched by the use of domain walls or defects, across which quasi-particles transform nontrivially. We will be most interested in studying these defects in explicit lattice models [6][7][8][9][10], and in particular the model introduced by Bombin [6]. Essentially the same defects have also been studied in Chern-Simons theories [11][12][13]. In both of these settings, it has been shown that such defects can be viewed as having anyonic-like properties that are not associated with the underlying model [6,9,[11][12][13]. More specifically, in a particular two-dimensional topologically ordered spin lattice model known as the toric code, such a lattice dislocation leads to interesting behaviour: the points where dislocations terminate, known as twists, interact with the anyons of the toric code to reproduce properties, such as fusion rules, of nonabelian (specifically, Ising) anyons [6]. This is particularly surprising, as all excitations of the toric code are abelian anyons.In this paper, we further interrogate the analogy between twists and Ising anyons by using topological entanglement entropy (TEE) [14,15] as a diagnostic. Specifically, for a toric code model containing twists, we use the TEE to determine: (i) the total quantum dimension of the lattice with twists; (ii) the quantum dimensions of the objects (quasi-particles and defects) on the lattice, and (iii) the quantum dimensions of all of their fusion products, allowing us to reconstruct the fusion rules for twists and excitations. Our results coincide precisely with those of the Ising anyon model, lending further support to the analogy between the latter and the toric code with twists. We note that TEE is a particularly useful quantity in the context of the toric code, as both the parent Hamiltonian and the modified one with twists are described within the stabilizer formalism and so allow the TEE to be calculated e...

Anyonic entanglement and topological entanglement entropy

“…(159)-(162), but the partition boundary components are now allowed to carry topological charges corresponding to quasiparticles or defects from the G-crossed MTC describing the system. This has been confirmed in the case of "twist defects" in the toric code model [21].…”

“…[14,15], TEE has received a significant amount of attention. Theoretical studies have investigated the connections between TEE and ground state degeneracy [22], derived the TEE for Chern-Simons theories on higher genus surfaces [17,24], derived TEE for certain systems with topological defects [21], and explored the TEE in the context of (3 + 1)-dimensional topological phases [18,20,23,25]. In numerical studies, TEE has become a useful quantity for identifying topological phases [26,27,28,29,30,31,33,34,35,36,37,38,39] (though it has been demonstrated that the accuracy of numerical extractions of TEE requires some caution [29,32]).…”

“…The topological entanglement entropy (TEE) [14,15] is a signature of topological order that has been the focus of numerous theoretical [16,17,18,19,20,21,22,23,24,25] and numerical studies [26,27,28,29,30,31,32,33,34,35,36,37,38,39]. Despite these efforts, an intuitive understanding of the origin and form of the TEE has remained elusive and only an inchoate connection between the TEE and the anyonic excitations of the system has been established.…”

Demonstrating non-Abelian statistics of Majorana fermions using twist defects

“…With a mean-field treatment, twists are shown to carry unpaired Majorana fermions 9 . More recently, the topological entanglement entropy was calculated to show that twist defects have the same quantum dimension and fusion rules as Ising anyons 11 . The twists have also been proposed as qubits for quantum computation in a surface code implementation to reduce the space-time cost 12 .…”

Collision dynamics at medium and relativistic energies

“…In particular, the lowdensity behavior represented in the figure is evinced from the isospin diffusion analysis discussed in Section 5.1.1 [4] and from constraints associated with the energy of the Isabaric Analog State (IAS) in several nuclei [167]. The black points correspond to constraints coming from structure properties (ground state features of doubly magic nuclei [168] and neutron skin thickness of heavy nuclei [169]). Figure 27: (Color online) Constraints deduced for the density dependence of the symmetry energy from the ASY-EOS data [10] in comparison with the FOPI-LAND result of Ref.…”