Quantum System Partitioning by Block-Diagonalization

Simon J. D. Phoenix, Faisal Shah Khan, Berihu Teklu

Adrian H. Mühlbach, Markus Reiher

We discuss the partitioning of a quantum system through subsystem separation by unitary block-diagonalization (SSUB) applied to a Fock operator. For a one-particle Hilbert space, this separation can be formulated in a very general way. Therefore, it can be applied to very different partitionings ranging from those driven by features in the molecular structure (such as a solute surrounded by solvent molecules or an active site in an enzyme) to those that aim at an orbital separation (such as core-valence separation). Our framework embraces recent developments of Manby and Miller as well as the older ones of Huzinaga and Cantu. Projector-based embedding is simplified and accelerated by SSUB. Moreover, it directly relates to decoupling approaches for relativistic four-component many-electron theory. For a Fock operator based on the Dirac one-electron Hamiltonian, one would like to separate the so-called positronic (negative-energy) states from the electronic bound and continuum states. The exact two-component (X2C) approach developed for this purpose becomes a special case of the general SSUB framework and may therefore be viewed as a system-environment decoupling approach. Moreover, for SSUB, there exists no restriction with respect to the number of subsystems that are generated-in the limit, decoupling of all single-particle states is recovered, which represents exact diagonalization of the problem. The fact that a Fock operator depends on its eigenvectors poses challenges to all system-environment decoupling approaches and is discussed in terms of the SSUB framework. Apart from improved conceptual understanding, these relations bring about technical advances as developments in different fields can immediately cross-fertilize one another. As an important example, we discuss the atomic decomposition of the unitary block-diagonalization matrix in X2C-type approaches that can inspire approaches for the efficient partitioning of large total systems based on SSUB.

Vadym Sakhno, Mykola Sakhno

The paper is a continuation of arXiv:2012.10364, where the approach was developed to constructing the exact matrix model for any generalized Ising system, and such model was constructed for certain 2d system. In this paper, the properties of the model are specified for light block diagonalization. A corresponding example is considered. For the example, general exact partition function is obtained and analysed. The analysis shows that the free energy does not depend on the amount of rows with a large amount of cells. For the example with light boundary conditions, the partition function is obtained and the specific free energy per spin is plotted.

Gu, Xia, Xianjin Xie

We propose a systematic procedure to work out systems of topological defect lines (TDLs) in minimal models. The only input of this method is the modular invariant partition function. For diagonal and permutation diagonal models, we prove there is a bijection between simple TDLs and primary fields preserving fusion rules. For block-diagonal models, we work out simple TDLs in the $3$-state Potts model as an example. The results agree with those in $3D$ topological field theory methods.

Omid Daei, Keivan Navi, Mariam Zomorodi Moghadam