Homogeneous binary trees as ground states of quantum critical Hamiltonians

Many-body states whose wave functions admit a representation in terms of a uniform binary-tree tensor decomposition are shown to obey power-law two-body correlation functions. Any such state can be associated with the ground state of a translationally invariant Hamiltonian which, depending on the dimension of the systems sites, involves at most couplings between third-neighboring sites. Under general conditions it is shown that they describe unfrustrated systems which admit an exponentially large degeneracy of the ground state.

Figure 1

BT network for $16=2^n$ sites. Inset (A) shows the isometric property of $\lambda$, (B) shows the maps ${\mathcal{D}}_L$ (left) and ${\mathcal{D}}_R$, and (C) shows the map $\mathcal{S}$ of Eq. (3).

Figure 2

Unnormalized density of states of the parent Hamiltonian generated from a sample HBTS for $N=8$ sites (energy levels have been rescaled to the maximum energy eigenvalue). For this example, it occurs that the ground-state degeneracy is twice the lower bound we discuss in the article: 32-fold over 256 states. More precisely, we have shown that the ground space of the Hamiltonian in this case coincides with the direct sum $\mathfrak{S}\oplus T(\mathfrak{S})$ of the space $\mathfrak{S}$ formed by the vectors defined in Eq. (32), and by its translated version $T(\mathfrak{S})$; $T$ being the one-site translation (we verified that, in this case, the two sectors form a linearly independent subspace).