Bifurcation in entanglement renormalization group flow of a gapped spin model

Abstract

We study entanglement renormalization group transformations for the ground states of a spin model, called cubic code model H[subscript 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[subscript A] on a lattice of lattice spacing a to the tensor product of ground spaces of two independent Hamiltonians H[subscript A] and H[subscript B] on lattices of lattice spacing 2a. We further find a disentangling unitary for the ground space of H[subscript 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[subscript A] and H[subscript B] represent distinct phases of matter.