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_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 ɑ to the tensor product of ground spaces of two independent Hamiltonians H_A and H_B on lattices of lattice spacing 2ɑ. 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 2ɑ. 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.