Explicit properties of the simplest inhomogeneous matrix-product-state including the Riemann metric of the MPS manifold

Cécile Monthus (1)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681

We consider the simplest inhomogeneous matrix-product-state for an open chain of N quantum spins that involves only two angles per site and two angles per bond with the following direct physical meanings. The two angles associated to the site k are the two Bloch angles that parametrize the two orthonormal eigenvectors of the reduced density matrix of the spin k alone. The two angles associated to the bond parametrize the entanglement properties of the Schmidt decomposition across the bond . Explicit results are given for the reduced density matrix of two consecutive sites that is needed to evaluate the energy of two-body Hamiltonians, and for the reduced density matrix of two sites at distance r that is needed to evaluate the spin–spin correlations at distance r. The global structure of the MPS manifold as parametrized by these angles is then characterized by its explicit Riemann metric. Finally, the generalizations to any tree-like structure without loops and to the chain with periodic boundary conditions are discussed.

Mots clés: correlation; entanglement; density matrix; reduced; site; Riemann; spin; structure; Hamiltonian; boundary condition

Domaines: Physique Générale [physics.gen-ph]; Analyse de données, Statistiques et Probabilités [physics.data-an]

Cécile Monthus. Explicit properties of the simplest inhomogeneous matrix-product-state including the Riemann metric of the MPS manifold. J.Stat.Mech., 2020, 2002 (2), pp.023105. ⟨10.1088/1742-5468/ab6b1b⟩. ⟨hal-02504620⟩