Determining non-Abelian topological order from infinite projected entangled pair states

Abstract

We generalize the method introduced in Phys. Rev. B 101, 041108 (2020) of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS) to non-Abelian topological order. When wrapped on a torus the unique iPEPS becomes a superposition of degenerate and locally indistinguishable ground states. We find numerically symmetries of the iPEPS, represented by infinite matrix product operators (MPO), and their fusion rules. The rules tell us how to combine the symmetries into projectors onto states with well defined anyon flux. A linear structure of the MPO projectors allows for efficient determination for each state its second Renyi topological entanglement entropy on an infinitely long cylinder directly in the limit of infinite cylinder's width. The same projectors are used to compute topological S and T matrices encoding mutual-and self-statistics of emergent anyons. The algorithm is illustrated by examples of Fibonacci and Ising non-Abelian string net models.

Characterization of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters

“…As shown in Fig. 1(c), in order to obtain an MES with a well-defined horizontal anyonic flux, one needs to insert a vertical idempotent into the PEPS [30]. The idempotents come from the tube algebra [9,31], and consist of the MPO tensors together with specific tensors inserted at the crossing point, see Appendix A for their definitions.…”

Variational methods for characterizing matrix product operator symmetries

“…51 but notice that here the cost function ( 7) is more demanding because it has two terms instead of just one as in Ref. 51. Numerically there is a freedom of the global phase of the eigenvectors v U,D i , which can be partially eliminated (up to minus sign) by requiring their Hermiticity (when applicable).…”

“…When wrapped on a cylinder the iPEPS becomes a superposition of degenerate ground states with definite anyonic fluxes. In the realm of the string-net models it is possible 49,51 to produce a PEPS-like tensor network for each ground state with well-defined flux. Such tensor networks are suitable for extracting topological S and T matrices by computing overlaps between the ground states.…”

“…The same approch was generalized to non-Abelian topological order in Ref. 51. The method does not require restoring the symmetries by suitable gauge transformations of a numerical iPEPS, a feat that was accomplished in Ref.…”

“…In contrast to Ref. 49 and 51, here they are obtained by algebraic manipulation from the F -symbols, the calculation of which is a purely 1D problem, which significantly reduces the complexity of the numerical algorithms whereas the calculations of S and T matrices via overlaps 49,51 between different MES is done on an infinite 2D lattice. Thus the route via the F -symbols is an alternative that is potentially more stable numerically.…”