Detecting topological order with ribbon operators

Abstract

We introduce a numerical method for identifying topological order in two-dimensional models based on one-dimensional bulk operators. The idea is to identify approximate symmetries supported on thin strips through the bulk that behave as string operators associated to an anyon model. We can express these ribbon operators in matrix product form and define a cost function that allows us to efficiently optimize over this ansatz class. We test this method on spin models with abelian topological order by finding ribbon operators for Z d quantum double models with local fields and Ising-like terms. In addition, we identify ribbons in the abelian phase of Kitaev's honeycomb model which serve as the logical operators of the encoded qubit for the quantum error-correcting code. We further identify the topologically encoded qubit in the quantum compass model, and show that despite this qubit, the model does not support topological order. Finally, we discuss how the method supports generalizations for detecting nonabelian topological order.Despite the apparent simplicity of quantum spin models, they can exhibit a wide variety of interesting and potentially useful phenomena. These range from conventional magnetic order to the more novel topological [1] and symmetry-protected[2] and symmetry-enriched [3] topological orders which are of interest in both condensed matter physics [4] and quantum information theory. [5] These states are disordered in the sense of LandauGinzburg-Wilson, however they do exhibit properties distinct from the usual disordered phases. For example, topological phases possess quasiparticle excitations, known as anyons, whose braid relations can be far more exotic than those of fermions or bosons.[6] These phases also have ground state degeneracy which depends on the topology of the lattice. [7] This protected degeneracy has prompted the investigation of topologically ordered models as quantum memories. [5,8] Quantum information stored in the degenerate subspace can be protected from arbitrary local noise when error correction techniques are employed. [9][10][11] Distinguishing topological phases can be an especially challenging task precisely because of their topological nature: there is no broken symmetry and no local order parameter signalling the phase transition.[1] There has been a large amount of previous work which attempts to identify topological order (TO). The existing key techniques, such as the topological entanglement entropy, [12][13][14][15][16][17] [26,27] have been highly successful in various domains of applicability. We review these methods below. A common feature of these methods is that they utilize the ground state of the model, which is unfortunately a challenging computational task in general.Here we propose a numerical method that we call the ribbon operators method for identifying TO in the ground state of a given 2D Hamiltonian using only the Hamiltonian, without reference to the ground state. We reduce the search for TO to a 1D problem through the bulk of the material, and we p...

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