Simulation of anyons with tensor network algorithms

Interacting systems of anyons pose a unique challenge to condensed-matter simulations due to their nontrivial exchange statistics. These systems are of great interest as they have the potential for robust universal quantum computation but numerical tools for studying them are as yet limited. We show how existing tensor network algorithms may be adapted for use with systems of anyons and demonstrate this process for the one-dimensional multiscale entanglement renormalization ansatz (MERA). We apply the MERA to infinite chains of interacting Fibonacci anyons, computing their scaling dimensions and local scaling operators. The scaling dimensions obtained are seen to be in agreement with conformal field theory. The techniques developed are applicable to any tensor network algorithm, and the ability to adapt these ansätze for use on anyonic systems opens the door for numerical simulation of large systems of free and interacting anyons in one and two dimensions.

Figure 1

(i) Example representation of a state $|\psi ⟩$ in a fusion tree basis for a system of six anyons. Labels $a_i$ indicate charges associated with edges of the fusion tree graph, and labels $u_i$ are degeneracies associated with vertices. The structure of the tree corresponds to a choice of basis and does not affect the physical content of the theory. (ii) Braiding may be used to change the ordering of the leaves of a fusion tree basis or to represent anyon exchange. (iii) $F$ moves convert between the bases associated with different fusion trees.

Figure 2

The leaves of this fusion tree carry the charge labels $a_1$ to $a_6$. An edge which is not a leaf, labeled with charge $a_{10}$, is indicated by the large gray arrow. The portion of the fusion tree extending from edge $a_{10}$ out to the leaves is indicated by the gray ellipse. If a degeneracy index ${\mu }_{10}$ is associated with charge $a_{10}$, then for a given value of $a_{10}$, index ${\mu }_{10}$ will enumerate all compatible labelings of the highlighted portion of the fusion tree.

Figure 3

Example enumeration of states according to $a_m$ and ${\mu }_m$ for a fusion tree describing four Fibonacci anyons. The Fibonacci anyon model has one nonvacuum charge label $\left(\tau \right)$ and one nontrivial fusion rule, $\tau \times \tau \to 1+\tau$. Because the charges 1 and $\tau$ are both self-dual, no arrows are required on diagrammatic representations of Fibonacci anyon fusion trees.

Figure 4

(i) Loops are eliminated by replacing them with an equivalent numerical factor determined by the normalization convention. The factor given here corresponds to the diagrammatic isotopy convention employed in Ref. 29. (ii) Definition of a simple two-site anyonic operator. (iii) Application of an operator to a state is performed by connecting the diagrams’ free legs. By performing $F$ moves and eliminating loops (and in more complex examples, also braiding) it is possible to obtain an expression for the resulting state in the original basis.

Figure 5

An operator acting on four Fibonacci anyons. The values of the coefficients $M_{abcde}$ may be specified as a block-diagonal matrix $M_{\alpha }^{\beta }$, for example, as in Table .

Figure 6

Diagrammatic representation of operators ${\widehat{P}}^{\left(1\right)}$ (6) and ${\widehat{P}}^{\left(2\right)}$ (7). The charges on all leaves are nondegenerate.

Figure 7

(i) Diagrammatic representation of a state $|\psi ⟩$ and two-site operator $\widehat{M}$ expressed in terms of degeneracy indices. (ii) Application of $\widehat{M}$ to state $|\psi ⟩$. Gray shapes represent tensors with charge and degeneracy multi-indices, with each leg of the shape corresponding to one charge and degeneracy index pair. These diagrams represent the same state, operator, and process as Fig. 1, (i) and Fig. 4, (ii) and (iii).

Figure 8

“Raising” of an operator $\widehat{M}$ from sites $x$ to sites $x+x^{\prime }$: (i) operator $\widehat{M}$ defined only on sites denoted $x$. (ii) Resolutions of the identity are inserted above and below $\widehat{M}$, being constructed from tensors $\tilde{N}$ and ${\tilde{N}}^{†}$. The central portion of this diagram is identified as corresponding to the new matrix $M_{\alpha }^{\prime \beta }$ which describes $\widehat{M}$ on $x+x^{\prime }$. (iii) Loop and vertex factors in the central region are evaluated separately and eliminated. (iv) The tensor network corresponding to the new central portion is contracted. The $\tilde{N}$ and ${\tilde{N}}^{†}$ tensors outside the central region become vertices of the fusion and splitting trees associated with $M_{\alpha }^{\prime \beta }$. Together the trees and the matrix $M_{\alpha }^{\prime \beta }$ constitute the raised version of $\widehat{M}$.

Figure 9

(i) $F$ move and (ii) braiding, performed on a section of fusion tree in the diagrammatic notation of Sec. 4A.

Figure 10

(i) An operator $\widehat{M}$ acting on sites on a 2D lattice is defined with respect to some arbitrary linear ordering of these sites. (ii) When manipulating the tensor network, it may on occasion be computationally convenient for the lattice to be linearized according to some alternative linearization scheme. In this example, the imposed linearization scheme is indicated by the dotted line. To apply $\widehat{M}$ to a different linearization of the lattice may require braiding. The orientation of the braids can be determined by putting the fusion tree of (i) onto the 2D lattice, then smoothly deforming the lattice into a chain in accordance with the linearization prescription. (iii) The unitary matrices corresponding to the required $F$ moves and braiding operations may be absorbed into $\widehat{M}$, defining a new operator ${\widehat{M}}^{\prime }$ on the linearized lattice.

Figure 11

Vertical bending of legs (i) in the standard diagrammatic notation and (ii) in the diagrammatic notation of Sec. 4A. White triangles represent Frobenius-Schur indicator flags. (iii) Legs on the matrix representations of states and operators may also absorb bends.

Figure 12

Opposing pairs of Frobenius-Schur indicators (i) on a pair of bends equivalent to the identity and (ii) on a pair of bends such as might be used when computing a quantum trace. (iii) As an anyon model can always be specified such that the Frobenius-Schur indicators are $\pm 1$, reversing a pair of contiguous opposed Frobenius-Schur indicator flags is always free.

Figure 13

Moving a matrix across a bend in a tensor network diagram. (i) Initial diagram. (ii) Bends are absorbed into the matrix. (iii) New bends are introduced, in accordance with Fig. 12, (i). (iv) A pair of contiguous, opposed Frobenius-Schur indicators are reversed, as per Fig. 12, (iii). The initial and final matrices $M$ and $M^{\prime }$ are related as specified in Eq. (16).

Figure 14

The use of bends may permit the more efficient contraction of pairs of anyonic operators. In the sequence of events marked (i), operator $\widehat{A}$ is first raised to the space of three sites then contracted with $\widehat{B}$. In sequence (ii) the operators are instead contracted using bends. For many anyon models the latter approach offers a significant computational advantage.

Figure 15

Now that we may bend legs up and down it is customary to introduce a further type of $F$ move, derived by applying bends to the one presented in Fig. 1, (iii).

Figure 16

Construction of an anyonic tensor corresponding to a normal tensor with more than three legs. (i) The original tensor. (ii) If required, any concavities are eliminated by introducing bends. (iii) Frobenius-Schur indicators are assigned to the bends. (iv) Directions are assigned to all legs, consistent with the rest of the network. (v) Legs are collected together into fusion and splitting trees. The central object, representing degrees of freedom of the tensor, now has less than four legs. (vi) If desired, bends can be reabsorbed into the fusion and splitting trees.

Figure 17

Construction of a 1D ternary MERA on a periodic lattice from anyonic operators. (i) The top tensor, $\widehat{T}$. (ii) Isometries, $\widehat{w}$. (iii) Disentanglers, $\widehat{u}$. The fusion tree representing an anyonic state (or ket) is usually drawn with the lattice sites at the top, so this MERA has been constructed “upside down” when compared with the diagrams in Refs. 23, 27. This is unimportant, and we could equally well have decided to follow the convention usually adopted in tensor network algorithms, labeled the tensors in (i)–(iii) by $T^{†}$, $w^{†}$, and $u^{†}$, and identified diagram (i)–(iii) as a bra. (iv) Structure of a two-site term in the Hamiltonian, $\widehat{h}$, or a two-site reduced density matrix, $\widehat{\rho }$. (v) Disentanglers and (vi) isometries satisfy the relationships $\widehat{u}{\widehat{u}}^{†}=\mathbb{I}$ and $\widehat{w}{\widehat{w}}^{†}=\mathbb{I}$.

Figure 18

(i) Anyonic operator $\widehat{E}$ constitutes the environment of operator $\widehat{M}$. Factors arising from the fusion and splitting trees should be evaluated and absorbed into matrices $E$ and $M$, following which (ii) matrix $E$ constitutes the environment of matrix $M$. After (iii) updating the matrix $M$ to $M^{\prime }$, (iv) the fusion and splitting trees of $\widehat{M}$ should be reinstated, the numerical factors associated with this process being the inverse of the fusion tree factors previously absorbed into matrix $M$. Frobenius-Schur flags in (i) and (ii) are represented by white triangles and are not to be confused with the black arrows which indicate the orientation of lines in the fusion/splitting trees.

Figure 19

Definition of ${\widehat{H}}_{21}$ in terms of ${\widehat{H}}_{12}$, on the most coarse-grained lattice $\left({\mathcal{L}}_{\tau }\right)$ of the translation-invariant periodic MERA. Lattice ${\mathcal{L}}_{\tau }$ is a two-site periodic lattice.

Figure 20

Determination of eigenoperators $\left(\phi \right)$ and associated scaling dimensions $\left(\Delta \right)$ for the one-site scaling superoperator of the anyonic 1D MERA. Eigenoperators may be classified according to the charges on edges $y_1$ and $y_2$. One interpretation of these labels is that, in addition to the sites of the 1D lattice, there may exist free charges lying in front of and behind the anyon chain. The labels $y_1$ and $y_2$ then represent the transfer of charge between these regions and the 1D lattice.

Figure 21

(Color online) Scaling dimensions of leading primary operators and their descendants, computed for (i) antiferromagnetic and (ii) ferromagnetic local Hamiltonians (17) on the golden chain. Results are grouped into conformal towers, with a slight horizontal spread introduced to show the degeneracies of the descendant fields. A circled cross indicates a primary field, and a plain cross indicates a descendant. Dashed lines indicate values predicted from CFT.