Explicit properties of the simplest inhomogeneous Matrix-Product-State including the Riemann metric of the MPS manifold

TL;DR

This paper provides explicit parametrizations and geometric properties of a simple inhomogeneous Matrix-Product-State model for quantum spins, including reduced density matrices and the Riemann metric of the MPS manifold.

Contribution

It introduces a minimal parametrization of inhomogeneous MPS with explicit formulas for reduced density matrices and the Riemann metric, extending to tree-like and periodic structures.

Findings

Explicit formulas for two-site and multi-site reduced density matrices.

Derivation of the Riemann metric for the MPS manifold.

Generalization to tree-like and periodic chain structures.

Abstract

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 $\rho_k$ of the spin $k$ alone. The two angles associated to the bond $(k,k+1)$ parametrize the entanglement properties of the Schmidt decomposition across the bond $(k,k+1)$. Explicit results are given for the reduced density matrix $\rho_{k,k+1}$ of two consecutive sites that is needed to evaluate the energy of two-body Hamiltonians, and for the reduced density matrix $\rho_{k,k+r}$ 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 $(4N-2)$ 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.