Matrix Product States, Entanglement and Lattice Momentum

Abstract:

A major obstacle with describing a many-body system is the exponentially growing number of parameters with the system size. The matrix product states (MPS) are a class of states that can describe 1D many-body systems by replacing the complex coefficients of the states with products of finite-dimensional matrices associated with each site. MPS has caught attention because they are numerically manageable (with only linear growth of parameters) while they can efficiently approximate ground states of local, gapped Hamiltonians. The properties of MPS with site-independent matrices, and thus translational symmetry, are well-studied; however, the situation becomes less clear when the matrices become site-dependent, even when the translational symmetry of the state is preserved. 

In this project, we proved that an MPS with normal tensors in canonical form that can be diagonalized by a unitary matrix must be short-range entangled, hence if it has translational symmetry, the translation eigenvalue cannot have a non-trivial phase. We also showed that any MPS with translation symmetry (up to a phase) can be transformed into a representation with all matrices being identical except for one. The current challenge is to reduce the redundancy in the MPS representation mentioned above.