Kitaev honeycomb tensor networks: Exact unitary circuits and applications

The Kitaev honeycomb model is a paradigm of exactly solvable models, showing nontrivial physical properties such as topological quantum order, Abelian and non-Abelian anyons, and chirality. Its solution is one of the most beautiful examples of the interplay of different mathematical techniques in condensed matter physics. In this paper, we show how to derive a tensor network (TN) description of the eigenstates of this spin-1/2 model in the thermodynamic limit, and in particular for its ground state. In our setting, eigenstates are naturally encoded by an exact 3d TN structure made of fermionic unitary operators, corresponding to the unitary quantum circuit building up the many-body quantum state. In our derivation we review how the different “solution ingredients” of the Kitaev honeycomb model can be accounted for in the TN language, namely, Jordan-Wigner transformation, braidings of Majorana modes, fermionic Fourier transformation, and Bogoliubov transformation. The TN built in this way allows for a clear understanding of several properties of the model. In particular, we show how the fidelity diagram is straightforward both at zero temperature and at finite temperature in the vortex-free sector. We also show how the properties of two-point correlation functions follow easily. Finally, we also discuss the pros and cons of contracting of our 3d TN down to a 2d projected entangled pair state (PEPS) with finite bond dimension. The results in this paper can be extended to generalizations of the Kitaev model, e.g., to other lattices, spins, and dimensions.

Figure 1

(a) Original honeycomb lattice, and (b) topologically equivalent brick wall lattice. The nearest-neighbor spin-spin interaction depends on the orientation of the bonds.

Figure 2

A plaquette of the honeycomb lattice. The conserved quantity ${\widehat{B}}_p$ is constructed as a product of Pauli matrices at every site, where the specific matrix is given by the type of “outgoing” bond.

Figure 3

String associated with the ordering of fermionic modes in the Jordan-Wigner transformation of Eq. (4). Boundary terms can be neglected in the thermodynamic limit.

Figure 4

Brick wall lattice with unit cell and plane vectors.

Figure 5

Phase diagram of the Kitaev honeycomb model. A point in the triangle corresponds to relative magnitudes of the spin-spin interactions $J_x,J_y$, and $J_z$, which are represented by a normalized vector $\vec{J}=(J_x,J_y,J_z)$ with $J_x+J_y+J_z=1$. The three different A phases ${\text{A}}_x,{\text{A}}_y$, and ${\text{A}}_z$ correspond to the case in which one of the couplings predominates over the other two.

Figure 6

(a) Matrix product state (MPS). (b) Matrix product operator (MPO). (c) Projected entangled pair state (PEPS). (d) Projected entangled pair operator (PEPO). (e) Tree tensor network (TTN). (f) 1d binary multiscale entanglement renormalization ansatz (MERA), with unitaries $u$ and isometries $w$. The physical dimension is $q$ in all cases, whereas the bond dimension is $\chi$ for MPS, MPO, TTN, and MERA, and $D$ for PEPS and PEPO (though this is a convention).

Figure 7

TN fermionization rules: (a) parity invariance of tensors, where $P$ is a representation of the ${\mathbb{Z}}_2$ parity symmetry operator; (b) crossings are replaced by fermionic swaps.

Figure 8

Product state in momentum space corresponding to the Bogoliubov vacuum on a $2\times 2$ square lattice. Dotted line is for reference.

Figure 9

Modes along the black arrow are coupled to the ones on the right-hand side with opposite momenta, following the ordering indicated by the arrow. The value of $\vec{k}$ for the modes along the arrow also determines the input parameter for the Bogoliubov transformation.

Figure 10

Entangled state for the $2\times 2$ square lattice, obtained after applying the Bogoliubov transformation. The construction can be easily scaled up to the thermodynamic limit using the prescriptions described in the text. This is an entangled state of fermionic momentum modes. Dotted lines are for reference.

Figure 11

A possible planar projection of the TN in Fig. 10, accounting for a specific fermionic order in second quantization (from left to right). To satisfy the fermionic anticommutation relations, fermionic SWAP gates (black squares) are introduced each time two fermions are swapped.

Figure 12

Quantum circuit, or spectral TN, for the Fourier transformation for eight fermionic modes, implemented entirely in terms of one- and two-body gates. The upper side of the diagram corresponds to momentum space, and the lower part to real space. Crossings of wires correspond to fermionic SWAP gates.

Figure 13

(a) TN for the four-site Fourier transformation with a bit-reversal permutation at the end. This 1d structure is repeated along $x$ and $y$ directions to achieve a two-dimensional Fourier transformation, such as the one in (b) for a $4\times 4$ square lattice. In (b), one-body gates and two-site MPOs are contracted into two-body gates (for simplicity of the diagram), a bit-reversal permutation is performed at the bottom, and fermionic SWAPs are not explicitly highlighted also for simplicity.

Figure 14

Majorana braiding operations to correctly recombine the different modes on the left, and the corresponding action on the Fock space of the two Dirac fermions on the right. FT refers to the modes out of the Fourier transformation, and CQ to the modes coming from the conserved quantities. The one-body gate braids Majorana modes within a Dirac fermion, whereas the two-body gate braids Majorana modes of different Dirac fermions. Valid patterns of clockwise/anticlockwise braidings are explained in the main text.

Figure 15

Fermionic tensor network to reintroduce the conserved quantities and braid the Majorana fermions. The transformation $T$ is given in Fig. 14 for the case of an unoccupied “vortex” fermion. Fermionic SWAPs are not included for simplicity.

Figure 16

A different projection of the TN in Fig. 15, where wires do not cross any gate. The distinction between the modes coming from the Fourier transformation and those coming from the conserved quantities (“vortex” modes) is also made explicit.

Figure 17

3d unitary TN for the ground state of the Kitaev model, for an 8-site honeycomb lattice with periodic boundary conditions. Physical degrees of freedom are spins, but essentially the whole network is fermionic. Fermionic SWAP gates are not included, for simplicity of the figure. Dotted lines are for reference.

Figure 18

Fermionic part of the TN in Fig. 17, but projected differently.

Figure 19

Trajectory in the parameter space given by $J_x=J_y=0.5(1-J_z)$.

Figure 20

Simplification in the contraction of (a) unitary and (b) isometric tensors. Notice that the Majorana braiding operators $\widehat{T}$ in our 3d TN are like in (b).

Figure 21

TN diagram for the ground-state fidelity.

Figure 22

Fidelity per lattice site for (a) 32, (b) 512, (c) 8192, and (d) 131 072 spins on the honeycomb lattice, corresponding respectively to $4\times 4,16\times 16,64\times 64$, and $256\times 256$ square lattices of Bogoliubov momenta.

Figure 23

TN diagram for $e^{-\beta {\widehat{H}}_{\text{vf}}/2}$, where state $|\alpha \rangle$ is our unitary 3d TN for the ground state, but for an excited configuration $\alpha$ of the Bogoliubov momenta.

Figure 24

TN diagram for the numerator of Eq. (57). For different couplings and temperatures, the left-hand side is to be considered. If the couplings are the same and only temperatures differ, then this can be further simplified, and one obtains the diagram on the right-hand side.

Figure 25

Thermal fidelity per lattice site in the vortex-free sector for 128 spins ($8\times 8$ square lattice of Bogoliubov momenta), along the trajectory in Fig. 19, and inverse temperatures $\tilde{\beta }=\beta =$ (a) 50, (b) 100, (c) 200, and (d) 400.

Figure 26

Thermal fidelity in the vortex-free sector and different inverse temperatures for (a) 2048 spins ($32\times 32$ Bogoliubov fermions) and $J_x=J_y=0.1,J_z=0.8$ (A phase), (b) 32 768 spins ($128\times 128$ Bogoliubov fermions) and $J_x=J_y=0.1,J_z=0.8$ (A phase), (c) 2048 spins ($32\times 32$ Bogoliubov fermions) and $J_x=J_y=J_z=1/3$ (B phase), and (d) 32 768 spins ($128\times 128$ Bogoliubov fermions) and $J_x=J_y=J_z=1/3$ (B phase).

Figure 27

Several relations between the fermionic SWAP gate and operators $X,Y$, and $Z$. Notice that $Z$ is the fermionic parity operator.

Figure 28

Matrix elements of operators $X,Y$, and $Z$ when sandwiched with the isometry $T$, resulting from the two-body gate $T$ after fixing one of its indices to 0.

Figure 29

8-fermion example, in quantum circuit form (i.e., projected on the plane of the paper). For a $z$ link between fermions 2 and 5, using the relations in Figs. 27 and 28, the only nonzero correlator is the one between Pauli-$z$ operators [case (c)].

Figure 30

Fourier transformation for a $4\times 4$ square lattice, redrawn in a different way. Each crossing in the mode-ordering tensor as well as in the 4-site Fourier transformation ${\widehat{F}}_4$ needs to be accounted for by fermionic SWAP gates.

Figure 31

$4\times 4$ lattice in momentum space. The boundary conditions in both directions shape the lattice into a torus. Interacting terms at the boundary are signalled in blue, and need to be treated differently in the Hamiltonian as compared to the bulk interaction terms, as explained in the text.