Determining topological order from infinite projected entangled pair states

Abstract

We present a method of extracting information about the topological order from the ground state of a strongly correlated two-dimensional system computed with the infinite projected entangled pair state (iPEPS). For topologically ordered systems, the iPEPS wrapped on a torus becomes a superposition of degenerate, locally indistinguishable ground states. Projectors in the form of infinite matrix product operators (iMPO) onto states with well-defined anyon flux are used to compute topological S and T matrices (encoding mutual-and self-statistics of emergent anyons). The algorithm is shown to be robust against a perturbation driving string-net toric code across a phase transition to a ferromagnetic phase. Our approach provides accurate results near quantum phase transition, where the correlation length is prohibitively large for other numerical methods. Moreover, we used numerically optimized iPEPS describing the ground state of the Kitaev honeycomb model in the toric code phase and obtained topological data in excellent agreement with theoretical prediction.Topologically ordered phases [1] have in recent years attracted significant attention, mostly due to the fact that they support anyonic excitations -exotic quasiparticles that obey fractional statistics. They are of interest not only from a fundamental perspective but also because of the possibility of realizing fault-tolerant quantum computation [2] based on the braiding of non-Abelian anyons. An important challenge is to identify microscopic lattice Hamiltonians that can realize such exotic phases of matter. Apart from a number of exactly solvable models [2-4], verifying whether a given microscopic Hamiltonian realizes a topologically ordered phase and accessing its properties has traditionally been regarded as an extremely hard task.A leading computational approach is to use Density Matrix Renormalization Group (DMRG) [5, 6] on a long cylinder [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. In the limit of infinitely long cylinders, DMRG naturally produces ground states with welldefined anyonic flux, from which one can obtain full characterization of a topological order, via so-called topological S and T matrices [21]. Since the proposal of Ref.[21], the study of topological order by computing the ground states of an infinite cylinder with DMRG has become a common practice [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].The cost of a DMRG simulation grows exponentially with the width of cylinder, effectively restricting this approach to thin cylinders. Instead, (infinite) Projected Entangled Pair States (iPEPS) allow for much larger systems [40][41][42]. However, (variationally optimized) iPEPS naturally describe ground states with a superposition of anyonic fluxes. Here we show, starting with one such PEPS, how 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 ground states. * correspondi...

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

“…This has been demonstrated in Ref. [48] by examples of toric code and double semions per-turbed away from a fixed point towards a ferromagnetic phase as well as for the numerical iPEPS representing the ground state of the Kitaev model in the gapped phase. The last example shows that the method does not require restoring the symmetries by suitable gauge transformations of a numerical iPEPS, a feat that was accomplished in Ref.…”

“…Here we generalize the approach of Ref. [48] to non-Abelian topological order and show how to produce * corresponding author: anna.francuz@uj.edu.pl FIG. 1.…”

“…The approach of Ref. [48] does not assume clean realization of certain symmetries on the bond indices, in contrast to [49][50][51][52]. This has been demonstrated in Ref.…”

“…Instead, infinite projected entangled pair states (iPEPS) in principle allow for much longer correlation lengths [43][44][45]. A unique ground state on an infinite lattice can be represented by an iPEPS that is either a variational ansatz [46] or a result of numerical optimization [47,48]. When wrapped on a cylinder the iPEPS becomes a superposition of degenerate ground states with definite anyonic fluxes.…”

“…Then a series of subsections follows illustrating the general concept with a series of examples: the Abelian toric code (to make contact with Ref. [48]), Fibonacci string net, and Ising string net. In the end the algorithm is summarized in section IX.…”