Condensation-driven phase transitions in perturbed string nets

We develop methods to probe the excitation spectrum of topological phases of matter in two spatial dimensions. Applying these to the Fibonacci string nets perturbed away from exact solvability, we analyze a topological phase transition driven by the condensation of non-Abelian anyons. Our numerical results illustrate how such phase transitions involve the spontaneous breaking of a topological symmetry, generalizing the traditional Landau paradigm. The main technical tool is the characterization of the ground states using tensor networks and the topological properties using matrix-product-operator symmetries. The topological phase transition manifests itself by symmetry breaking in the entanglement degrees of freedom of the quantum transfer matrix.

Figure 1

PEPS tensor $A$ is invariant under the MPO projector (blue).

Figure 2

MPO injectivity requires the existence of a tensor (black) that acts as the pseudoinverse on the space determined by the MPO projector.

Figure 3

Pulling through property.

Figure 4

Excitation, implemented in the PEPS language by the modified green tensor, is of one definite anyon type if it lives exactly in the support of one of the elementary idempotents of the theory. These idempotents are constructed by closing an MPO loop with a fixed, specific tensor (blue oval), one for each anyon type.

Figure 5

Conditions to have a well defined, particlelike, anyon excitation. The left condition expresses that the excitation is not condensed; the right one that it is not confined.

Figure 6

Norm of a PEPS is given by the full contraction of the double layer network (left). We can take one column of the double layer and treat it as an operator from left to right, the transfer matrix (right).

Figure 7

Fundamental symmetries of the transfer matrix (red) are given by MPOs (blue) that commute both on the ket and bra layer.

Figure 8

Fixed point of the transfer matrix, represented as MPS (left). Applying an MPO to a fixed point (center) gives us another fixed point, as a possibly noninjective MPS. It can be expanded as a combination of the injective fixed points with integer coefficients $n_i$ (right).

Figure 9

Contraction scheme to calculate the norm of an excitation; see the main text for details.

Figure 10

Contraction scheme to calculate the overlap of the ground state and an excited state; see the main text for details.

Figure 11

Conditions necessary to have a well-defined excitation with blue MPO string. The figures on the left and right are to be interpreted as matrices from the upper to the lower indices.

Figure 12

Norm of an excitation, created by acting with a local operator thereby turning a red ground state tensor in a green excited tensor. We make a momentum superposition of this state with momentum $k=(k_x,k_y)$.

Figure 13

Ansatz for the topologically trivial excitations of the transfer matrix is obtained by changing one single tensor (green) and making a momentum superposition (left). Topologically nontrivial excitations, corresponding to anyons, are obtained by attaching an MPO string to the perturbed tensor (center). This is equivalent to a kink excitation, where we have a perturbed tensor (green) but different fixed point tensors (red and cyan) on either side of it (right).

Figure 14

Contraction scheme to calculate the norm of an excitation; see the main text for details. The figure does not show the momentum superpositions for the sake of clarity.

Figure 15

Eigenvector equation for the regular transfer matrix (left). An equivalent equation, using the pulling through property (right).

Figure 16

Eigenvalue equation for the mixed transfer matrix (left). This is the regular transfer matrix with a extra MPO through its ket (or bra) layer. The excitation ansatz consists now of local perturbations on the trivial fixed points, but the perturbed tensors (green) have an extra MPO index. We can naturally use the idempotents to define operators in the mixed formalism (right).

Figure 17

Mixed transfer matrix has a block decomposition induced by all the idempotents. Note the difference in the notation between the mixed transfer matrix (blue disk) and an idempotent (blue oval).

Figure 18

Network that gives the matrix element of a central idempotent with respect to some excitations at a given energy and momentum. As this is a 1D network it can be contracted efficiently.

Figure 19

Correlation length of the $exp(-{\beta }_{z}Z)$ interpolation (a), an interpolation to the Fendley model (c), and $exp(-{\beta }_{x}X)$ filtering (e). The points where the dispersion relations are given are shown in red. The dispersion relation of the string net model with $exp(-{\beta }_{z}Z)$ filtering at ${\beta }_z=0.14$ (b), the Fendley model (d), and $exp(-{\beta }_{x}X)$ filtering at ${\beta }_x=0.7$ (f).