acta physica slovaca

Acta Physica Slovaca 67, No.2&3, 85 – 206 (2017) (122 pages)


Jozef Genzor a , Tomotoshi Nishino a , Andrej Gendiar b
   aDepartment of Physics, Graduate School of Science, Kobe University, Kobe 657-8501, Japan
   bInstitute of Physics, Slovak Academy of Sciences Dúbravská cesta 9, SK-845 11 Bratislava, Slovakia

Full text: ::pdf :: Received 26 April 2018, accepted 7 May 2018

Abstract: One of the challenging problems in the condensed matter physics is to understand the quantum many-body systems, especially, their physical mechanisms behind. Since there are only a few complete analytical solutions of these systems, several numerical simulation methods have been proposed in recent years. Amongst all of them, the Tensor Network algorithms have become increasingly popular in recent years, especially for their adaptability to simulate strongly correlated systems. The current work focuses on the generalization of such Tensor-Network-based algorithms, which are sufficiently robust to describe critical phenomena and phase transitions of multistate spin Hamiltonians in the thermodynamic limit. Therefore, one has to deal with systems of infinitely many interacting spin particles. For this purpose, we have chosen two algorithms: the Corner Transfer Matrix Renormalization Group and the Higher-Order Tensor Renormalization Group. The ground state of those multistate spin systems in the thermodynamic equilibrium is constructed in terms of a tensor product state Ansatz in both of the algorithms. The main aim of this work is to generalize the idea behind these two algorithms in order to be able to calculate the thermodynamic properties of non-Euclidean geometries. In particular, the tensor product state algorithms of hyperbolic geometries with negative Gaussian curvatures as well as fractal geometries will be theoretically analyzed followed by extensive numerical simulations of the multistate spin models. These spin systems were chosen for their applicability to mimic intrinsic properties of more complex systems, such as social behavior, neural network, the holographic principle, including the correspondence between the anti-de Sitter and conformal field theory of quantum gravity. This work is based on tensor-network analysis and opens doors for the understanding of phase transition and entanglement of the interacting systems on the non-Euclidean geometries. We focus on three main topics: A new thermodynamic model of social influence, free energy is analyzed to classify the phase transitions on an infinite set of the negatively curved geometries where a relation between the free energy and the Gaussian radius of the curvature is conjectured, a unique tensor-based algorithm is proposed to study the phase transition on fractal structures.

Keywords: Classical Statistical Mechanics, Phase Transitions and Criticality, Spin Systems, Tensor Networks, Density Matrix Renormalization Group, Hyperbolic Geometry, Fractal Lattices
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