Application of the Infinite Product Representation of the R-Matrix

N. G. van Kampen

N. Abdülbaki Baykara, Ercan Gürvіt, Metin Demіralp

In this work a study on finite dimensional matrix approximations to products of quantum mechanical operators is conducted. It is emphasized that the matrix representation of the product of two operators is equal to the product of the matrix representation of each of the operators when all the fluctuation terms are ignored. The calculation of the elements of the matrices corresponding to the matrix representation of various operators, based on three terms recursive relation is defined. Finally it is shown that the approximation quality depends on the choice of higher values of n, namely the dimension of Hilbert space.

U. Fano, A. R. P. Rau

Tomaž Prosen

A canonical form for a matrix product state representation of a general finitely correlated quantum state on a one-dimensional (finite or infinite) lattice is proposed by exploring the gauge symmetry of the matrix product. This representation is unique in the sense that it is the only one which generates canonical forms for all reduced density matrices of the state.

Sergey E. Derkachov

Max F. Frenzel, Martin Bodo Plenio

We present an alternative to the conventional matrix product state representation, which allows us to avoid the explicit local Hilbert space truncation many numerical methods employ. Utilizing chain mappings corresponding to linear and logarithmic discretizations of the spin-boson model onto a semi-infinite chain, we apply the new method to the sub-ohmic spin-boson model. We are able to reproduce many well-established features of the quantum phase transition, such as the critical exponent predicted by mean-field theory. Via extrapolation of finite-chain results, we are able to determine the infinite-chain critical couplings α c at which the transition occurs and, in general, study the behaviour of the system well into the localized phase.