Geometry of reduced density matrices

Constraints placed on a reduced density matrix or density by the requirement that it correspond to an eigenstate or ensemble of eigenstates of some operator are investigated. Linear conditions, including expectation value, commutator, hypervirial, and hierarchy equation conditions, may lead to linear constraints on lower-order reduced density matrices and densities or leave them unconstrained. A zero-dispersion condition for a $p$ -electron operator defines a parabolic trough on which reduced density matrices of order $2p$ must lie, but usually does not restrict lower-order reduced density matrices or the density. Some properties of symmetrized and antisymmetrized matrix products, related to their reduction behavior, are also investigated.Constraints placed on a reduced density matrix or density by the requirement that it correspond to an eigenstate or ensemble of eigenstates of some operator are investigated. Linear conditions, including expectation value, commutator, hypervirial, and hierarchy equation conditions, may lead to linear constraints on lower-order reduced density matrices and densities or leave them unconstrained. A zero-dispersion condition for a $p$-electron operator defines a parabolic trough on which reduced density matrices of order $2p$ must lie, but usually does not restrict lower-order reduced density matrices or the density. Some properties of symmetrized and antisymmetrized matrix products, related to their reduction behavior, are also investigated.

Physical Review AJul 1, 198418 citations

Properties of Reduced Density Matrices

Theoretical chemistryJan 1, 19760 citations

Theory of reduced density matrices

Eugene S. Kryachko

Theoretical and Mathematical PhysicsSep 1, 19751 citation

Geometry of reduced density matrices for symmetry-protected topological phases

Ji-Yao Chen, Zhengfeng Ji, Zheng-Xin Liu +2 more

In this paper, we study the geometry of reduced density matrices for states with symmetry-protected topological (SPT) order. We observe ruled surface structures on the boundary of the convex set of low-dimensional projections of the reduced density matrices. In order to signal the SPT order using ruled surfaces, it is important that we add a symmetry-breaking term to the boundary of the system---no ruled surface emerges in systems without a boundary or when we add a symmetry-breaking term representing a thermodynamic quantity. Although the ruled surfaces only appear in the thermodynamic limit where the ground-state degeneracy is exact, we analyze the precision of our numerical algorithm and show that a finite-system calculation suffices to reveal the ruled surface structures.In this paper, we study the geometry of reduced density matrices for states with symmetry-protected topological (SPT) order. We observe ruled surface structures on the boundary of the convex set of low-dimensional projections of the reduced density matrices. In order to signal the SPT order using ruled surfaces, it is important that we add a symmetry-breaking term to the boundary of the system---no ruled surface emerges in systems without a boundary or when we add a symmetry-breaking term representing a thermodynamic quantity. Although the ruled surfaces only appear in the thermodynamic limit where the ground-state degeneracy is exact, we analyze the precision of our numerical algorithm and show that a finite-system calculation suffices to reveal the ruled surface structures.

arXiv:1508.03886Physical Review AJan 7, 201613 citations

Reduced Density Matrices

Elsevier eBooksJan 1, 19782 citations

p

-electron operator defines a parabolic trough on which reduced density matrices of order

2p

must lie, but usually does not restrict lower-order reduced density matrices or the density. Some properties of symmetrized and antisymmetrized matrix products, related to their reduction behavior, are also investigated.