Tensor-network states

M. Troyer, F. Verstraete. Complete-Graph Tensor Network States A New Fermionic Wave Function Ansatz for Molecules. New. Phys., 12 (2010) 103008. 

N. Nakatani, G. K.-L. Chan. Efficient tree tensor network states (TTNS) for quantum chemistry Generalizations of the density matrix renormalization group algorithm. 

In the deformed state, the variables in the Hamiltonian change from ( R , r ) to ( R , Ar ). However, the distribution p( r ) of finding the topology r depends solely on how the material is made instantaneously at thermal equilibrium (i.e., at constant temperature T, pressure p, etc.) i.e., p( r ) does not depend on the external deformation tensor A. Then, the final answer for the free energy of the deformed network is... 

Abstract A new upscaling method has been developed using 3D numerical tools (RESOBLOK 3DEC). This method has been successfully compared with standard analytical approaches in the case of a simple fracture network. This method has been applied to determine the equivalent permeability, stiffness and Biot tensor of a real fracture rock-mass at different scales. The effects of the fracture network properties and of the state of stress on the result have been investigated. 

Silver ( Ag) (f=l/2, 1/2). Silver-containing layered networks [Ag(L)] (L = 4-pyridinesulfonate or p-toluenesulfonate) were treated with primary amines in different ratios. Solid-state ° Ag, and CP/ MAS NMR experiments were used to study the interactions between the primary amines and the parent materials. Ag chemical shift tensor parameters are extremely sensitive to changes in silver environments. Reaction of Ag20 with the A-heterocyclic carbene precursors 3,5-bis[3-(2,4,6-tri-methylphenyl)imidazolium-l-ylmethyl]-lH-pyrazole bishexafluorophosphate and 3,5-bis[3-(2,6-diisopropylphenyl)imidazolium-l-ylmethyl]-lH-pyrazole bishexafluorophosphate yielded silver(I) complexes 2 and 3, respectively The identity of those species could be elucidated by detailed NMR spectroscopic studies ( H, C, N, Ag and DOSY experiments). 

The ordinate values, A n and Aj, denote the square root of the components of the molecular deformation tensor parallel and perpendicular to the direction of stretch, respectively. Results are presented in terms of o( for networks cross-linked and stretched in the dry state, i.e., v v = 1. 

In addition to the deformation of the chains associated with the alteration of their chain vectors under strain, it is necessary to consider the action of the junctions on the constraint domains surrounding them. The junction and its domain are coupled elastic elements. Non-affine transformation of the fluctuation domains of junctions in the presence of chain connectivity creates an additional strain field in the medium around each junction. A omain deformation tensor can be introduced in analogy with A. (h) relates the mean deformation (according to ensemble distribution) of the junction constraint domains in the deformed, connected network to that in the absence of connectivity. Consequently, the state of microscopic deformation in a real network is the sum of the molecular deformation tensor. A, and the junction domain deformation tensor, Qj). ... 

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