Matrix product states: Symmetries and two-body Hamiltonians

We characterize the conditions under which a translationally invariant matrix product state (MPS) is invariant under local transformations. This allows us to relate the symmetry group of a given state to the symmetry group of a simple tensor. We exploit this result in order to prove and extend a version of the Lieb-Schultz-Mattis theorem, one of the basic results in many-body physics, in the context of MPS. We illustrate the results with an exhaustive search of SU(2)-invariant two-body Hamiltonians which have such MPS as exact ground states or excitations.

Figure 1

This figure represents the MPS construction. A pair of virtual spins which are connected to their neighbors via a maximally entangled state $|\Omega ⟩={\sum }_{\alpha =1}^d|\alpha \alpha ⟩$ are mapped into the physical spins (below). All properties of the state originate from the mapping between physical and virtual system.

Figure 2

The unitary $u_g$ applied on the physical level is reflected in the virtual level as a pair of unitaries $U_g$.

Figure 3

(Color online) Space of parameters of the local Hamiltonian $h$ for the AKLT state and $n=3$. The orange (small) volume represents the points where the state is the local (and hence the global) ground state. The green (large) volume represents points corresponding to excited states detected with the witness $\frac{3}{2}\oplus \frac{1}{2}$.

Figure 4

(Color online) Space of physical parameters of the global Hamiltonian $H$ corresponding to $n=3$ and the AKLT state. The points (orange) represent where the state is the local (and hence the global) GS. The surface (green) represents points corresponding to excited states detected by means of the witness $\frac{3}{2}\oplus \frac{1}{2}$.

Figure 5

(Color online) Space of physical parameters of the total Hamiltonian for $n=5$ associated to the Majumdar-Ghosh state. The points (orange) represent where the state is the local (and hence the global) ground state. The surface (green) represents points corresponding to excited states detected by means of the witness $\frac{1}{2}\oplus 1\oplus 0$.

Figure 6

(Color online) Space of parameters of the spin $\frac{3}{2}$ model. The points (orange) are obtained numerically and they represent values of the parameters where the MPS state is the GS. The volume (green) represents points corresponding to excited states detected with the witness $\frac{3}{2}\oplus 1\oplus 0$.