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Journal of Mathematical Chemistry

Interaction of a Single State with a Known Infinite System Containing One-Parameter Eigenvalue Band

Tomislav P. Živković

Published Aug 1, 2002• 6 citations
Publisher
DOI
10.1023/a:1021219426380

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The interaction of an isolated state |Theta> with a known infinite-dimensional quantum system S-infinity(b) containing multiple eigenvalue bands is considered. An example is the interaction of an isolated molecular state with the electromagnetic field. This problem is the main subject of spectroscopy. Another example is the interaction of an isolated molecular state of a molecule situated on the surface of a solid with this solid. This is the subject of the surface state physics. Such problems are usually treated within the formalism of the perturbation expansion. Recently a new mathematical method for the treatment of such problems was suggested. This method produces correct solution of the combined system, however strong the interaction between the state |Theta> and the system S-infinity(b). Key quantities in this method are characteristic functions f(v)(epsilon) and functions omega(v)(epsilon). Functions f(v)(epsilon) are constructed from the quantities describing unperturbed system S-infinity(b) and from the known interaction of this system with the state |Theta>. Functions omega(v)(epsilon) are then expressed in terms of the characteristic functions f(v)(epsilon). An important mathematical problem is the calculation of the functions omega(v)(epsilon), once functions f(v)(epsilon) are known. A general mathematical solution to this problem is obtained for the case when characteristic functions f(v)(epsilon) are polynomials. Since each function which is continuous in a finite interval can be approximated (to any desired degree of accuracy) with some polynomial, this solution presents an important mathematical tool for the practical application of the suggested method. Obtained results are illustrated with few model examples of the interaction of an isolated state with an infinite quantum system.

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The interaction of an isolated state |Theta> with a known infinite-dimensional quantum system S-infinity(b) containing multiple eigenvalue bands is considered. An example is the interaction of an isolated molecular state with the electromagnetic field. This problem is the main subject of spectroscopy. Another example is the interaction of an isolated molecular state of a molecule situated on the surface of a solid with this solid. This is the subject of the surface state physics. Such problems are usually treated within the formalism of the perturbation expansion. Recently a new mathematical method for the treatment of such problems was suggested. This method produces correct solution of the combined system, however strong the interaction between the state |Theta> and the system S-infinity(b). Key quantities in this method are characteristic functions f(v)(epsilon) and functions omega(v)(epsilon). Functions f(v)(epsilon) are constructed from the quantities describing unperturbed system S-infinity(b) and from the known interaction of this system with the state |Theta>. Functions omega(v)(epsilon) are then expressed in terms of the characteristic functions f(v)(epsilon). An important mathematical problem is the calculation of the functions omega(v)(epsilon), once functions f(v)(epsilon) are known. A general mathematical solution to this problem is obtained for the case when characteristic functions f(v)(epsilon) are polynomials. Since each function which is continuous in a finite interval can be approximated (to any desired degree of accuracy) with some polynomial, this solution presents an important mathematical tool for the practical application of the suggested method. Obtained results are illustrated with few model examples of the interaction of an isolated state with an infinite quantum system.
The entanglement spectrum of a pure state of a bipartite system is the full set of eigenvalues of the reduced density matrix obtained from tracing out one part. Such spectra are known in several cases to contain important information beyond that in the entanglement entropy. This paper studies the entanglement spectrum for a variety of critical and near-critical quantum lattice models in one dimension, chiefly by the infinite time evolving block decimation (iTEBD) numerical method, which enables both integrable and non-integrable models to be studied. We find that the distribution of eigenvalues in the entanglement spectra agrees with an approximate result derived by Calabrese and Lefevre to an accuracy of a few per cent for all models studied. This result applies whether the correlation length is intrinsic or generated by the finite matrix size accessible in iTEBD. For the transverse Ising model, the known exact results from Peschel and Eisler for the entanglement spectrum are used to confirm the validity of the iTEBD approach. For more general models, no exact result is available but the iTEBD results directly test the hypothesis that all moments of the reduced density matrix are determined by a single parameter.