Bifurcation in entanglement renormalization group flow of a gapped spin model

We study entanglement renormalization group transformations for the ground states of a spin model, called cubic code model $H_A$ in three dimensions, in order to understand long-range entanglement structure. The cubic code model has degenerate and locally indistinguishable ground states under periodic boundary conditions. In the entanglement renormalization, one applies local unitary transformations on a state, called disentangling transformations, after which some of the spins are completely disentangled from the rest and then discarded. We find a disentangling unitary to establish equivalence of the ground state of $H_A$ on a lattice of lattice spacing $a$ to the tensor product of ground spaces of two independent Hamiltonians $H_A$ and $H_B$ on lattices of lattice spacing $2a$. We further find a disentangling unitary for the ground space of $H_B$ with the lattice spacing $a$ to show that it decomposes into two copies of itself on the lattice of the lattice spacing $2a$. The disentangling transformations yield a tensor network description for the ground state of the cubic code model. Using exact formulas for the degeneracy as a function of system size, we show that the two Hamiltonians $H_A$ and $H_B$ represent distinct phases of matter.

Figure 1

Simple cubic lattice of lattice spacing of $2a$ and the unit cell. There are 16 qubits labeled by 0, $A$, or $B$ in the unit cell. Those that are labeled by 0 are in the trivial product state. $|{\psi }_A\rangle$ and $|{\psi }_B\rangle$ in Eq. (1) are states of the system of the qubits labeled by $A$ and $B$, respectively.

Figure 2

Equivalence between the Wen plaquette model and the toric code model can be observed, only when a unit cell is properly chosen.

Figure 3

Honeycomb lattice with qubits numbered within a unit cell.