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

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.

36 pages

References in corpus (13)

• The density-matrix renormalization group in the age of matrix product states
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Many-body localization edge in the random-field Heisenberg chain
• Tensor networks for complex quantum systems
• Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography
• Many-body localisation implies that eigenvectors are matrix-product states
• Rényi entropy of a line in two-dimensional Ising models
• Participation spectroscopy and entanglement Hamiltonian of quantum spin models
• Spectral tensor networks for many-body localization
• Improving entanglement and thermodynamic Rényi entropy measurements in quantum Monte Carlo
• Information theoretic aspects of the two-dimensional Ising model
• Universal behavior of the Shannon and Rényi mutual information of quantum critical chains
• Generalized mutual informations of quantum critical chains