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Tensor Networks: Phase Transition Phenomena on Hyperbolic and Fractal Geometries Phd Thesis

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Tensor Networks: Phase Transition Phenomena on Hyperbolic and Fractal Geometries Phd Thesis Comenius University in Bratislava Faculty of Mathematics, Physics and Informatics Tensor Networks: Phase Transition Phenomena on Hyperbolic and Fractal Geometries PhD Thesis Mgr. Jozef Genzor 2016 Comenius University in Bratislava Faculty of Mathematics, Physics and Informatics Tensor Networks: Phase Transition Phenomena on Hyperbolic and Fractal Geometries PhD Thesis Study programme: Theoretical Physics and Mathematical Physics Study field: 4.1.2 General Physics and Mathematical Physics Educational institution: Research Center for Quantum Information Institute of Physics, Slovak Academy of Sciences Supervisor: Mgr. Andrej Gendiar, PhD. Bratislava 2016 Mgr. Jozef Genzor 51832218 G ?$ 2$ - ! " ! # " $% & ' ( %) & *+ , - ! ./01 23 #3 $ " 4 /567! .7018 1 $4 95769:/557! .:0&' ;<% +9=65//559! .>0? 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F1F:<F1< + '-7+4" 1;FE<F1< . 3(W %4 U7 K, . % S Abstrakt Jednym´ z doleˆ zitˇ ych´ problemov´ fyziky kondenzovanych´ latok´ je pochopit’, coˇ sa odohrava´ na pozad´ı mechanizmov kvantovych´ mnohocasticovˇ ych´ systemoch.´ Ked’zeˇ existuje iba niekol’ko upln´ ych´ analytickych´ riesenˇ ´ı pre tieto systemy,´ v po- slednych´ rokoch bolo navrhnutych´ niekol’ko numerickych´ simulacnˇ ych´ metod.´ Spomedzi nich zacˇ´ınaju´ byt’popularne´ prave´ tie algoritmy, ktore´ su´ zalozenˇ e´ na princ´ıpoch tenzorovych´ siet´ı, a to najma¨ vd’aka ich aplikovatel’nosti na simulacie´ silno korelovanych´ systemov.´ Predkladana´ praca´ sa sustred’uje´ na zovseobecnenieˇ takychto´ algoritmov, ktore´ vyuzˇ´ıvaju´ algoritmus tenzorovych´ siet´ı a zarove´ nˇ su´ dostatocneˇ robustne´ na to, aby pop´ısali kriticke´ javy a fazove´ prechody multi- spinovych´ Hamiltonianov´ v termodynamickej limite. Na to je vsakˇ nevyhnutne´ zaoberat’sa so systemami´ s nekonecneˇ vel’kym´ mnozstvomˇ interagujucich´ castˇ ´ıc. Pre tento u´celˇ sme si zvolili dva algoritmy, ktore´ su´ vhodne´ pre spinove´ systemy:´ Corner Transfer Matrix Renormalization Group a Higher-Order Tensor Renorma- lization Group. V oboch algoritmoch je zakladn´ y´ stav multistavoveho´ spinoveho´ systemu´ konstruovanˇ y´ v tvare tenzoroveho´ su´cinovˇ eho´ stavu. Ciel’om tejto prace´ je zovseobecnit’ˇ tieto dva algoritmy tak, aby bolo nimi moznˇ e´ pocˇ´ıtat’ termody- namicke´ vlastnosti neeuklidovskych´ geometri´ı. Osobitne budu´ analyzovane´ ten- zorove´ su´cinovˇ e´ stavy na hyperbolickych´ geometriach´ so zapornou´ Gaussovou krivost’ou, ale aj na fraktalnych´ systemoch.´ Nasledne´ budu´ vykonane´ rozsiahle numericke´ simulacie´ multistavovych´ spinovych´ modelov. Tieto spinove´ systemy´ boli zvolene´ pre ich vhodnost’spravne´ modelovat’zakladn´ e´ vlastnosti zlozitejˇ sˇ´ıch systemov,´ akymi´ su´ socialne´ spravanie,´ neuronov´ e´ siete, holograficky´ princ´ıp, vratane´ teorie´ korespondecieˇ medzi anti-de Sitterovym´ priestorom a konformnou teoriou´ pol’a v kvantovej gravitacii.´ Tato´ praca´ obsahuje nove´ postupy aplikacie´ tenzorovych´ siet´ı a umozˇnujeˇ pochopit’fazov´ e´ prechody a kvantovu´ previazanost’ interagujucich´ systemov´ na neeuklidovskych´ geometriach.´ Budeme sa preto bliz-ˇ sieˇ venovat’nasledujucim´ trom tematickym´ oblastiam. (1) Navrhneme novy´ ter- modynamicky´ model socialneho´ vplyvu, v ktorom budeme vysetrovat’fˇ azov´ e´ pre- chody. (2) Na nekonecnejˇ mnozineˇ geometri´ı so zapornou´ krivost’ou klasifiku- jeme a analyzujeme fazov´ e´ prechody pomocou vol’nej energie. Zarove´ nˇ bude stanoveny´ vzt’ah, ktory´ dava´ do suvisu´ vol’nu´ energiu a Gaussov polomer krivosti. (3) Navrhneme novy´ algoritmus zalozenˇ y´ na tenzorovych´ siet’ach, ktory´ umoznˇ ´ı studovat’fˇ azov´ e´ prechody na nekonecneˇ vel’kych´ fraktalnych´ struktˇ urach.´ Kl’u´covˇ e´ slova:´ Klasifikacia´ fazov´ ych´ prechodov, Kvantove´ a klasicke´ spinove´ mriezkovˇ e´ modely, Tenzorove´ su´cinovˇ e´ stavy, Tenzorove´ siete, Renormalizacia´ matice hustoty Abstract One of the challenging problems in the condensed matter physics is to understand the quantum many-body systems, especially, physical mechanisms behind. Since there are only a few complete analytical solutions of these systems, several nu- merical 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 al- gorithms, 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 Renormal- ization Group and the Higher-Order Tensor Renormalization Group. The ground state of those multistate spin systems in the thermodynamic equilibrium is con- structed 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 ge- ometries. In particular, the tensor product state algorithms of hyperbolic geome- tries with negative Gaussian curvatures as well as fractal geometries will be the- oretically analyzed followed by extensive numerical simulations of the multistate spin models. These spin systems were chosen for their applicability to mimic the intrinsic properties of much more complex systems of social behavior, neural net- work, the holographic principle, including the correspondence between the anti- de Sitter and conformal field theory in quantum gravity. This work contains novel approaches in tensor networks and opens the door for the understanding of phase transition and entanglement of the interacting systems on the non-Euclidean ge- ometries. The following three topics are investigated by means of the tensor-based algorithms. (1) A new thermodynamic model of social influence is proposed, and its phase transition phenomena are studied. (2) The phase transitions are classified and analyzed by the free energy on an infinite set of the negatively curved geome- tries. A relation between the free energy and the Gaussian radius of the curvature is conjectured. (3) A unique tensor-based algorithm is proposed, which enables to treat the phase transition on infinitely large fractal structures. Keywords: Phase Transition Phenomena, Quantum and Classical Spin Lattice Models, Tensor Product States, Tensor Networks, Density Matrix Renormalization Contents Preface xi Introduction 1 1 General introduction and concepts 3 1.1 Theory of phase transitions . .3 1.1.1 Ising model . .4 1.1.2 Equilibrium statistical physics of critical phenomena . .6 1.1.3 Correlation function . .7 1.1.4 Critical exponents . .7 1.1.5 Mean-field theory of phase transitions . .9 1.2 Transfer matrices for classical systems . 10 1.2.1 The transfer matrix . 10 1.2.2 The corner transfer matrix . 14 1.3 Suzuki-Trotter mapping . 15 2 Tensor network states 18 2.1 TEBD: MPS for ground states . 18 2.1.1 Ising model . 21 2.1.2 Heisenberg model . 23 2.2 CTMRG . 25 2.3 HOTRG . 32 2.3.1 Tensor-network representation . 32 2.3.2 Coarse-graining procedure . 36 2.3.3 Free energy calculation . 38 2.3.4 Impurity tensors . 39 2.3.5 Numerical results . 40 3 Free energy on hyperbolic geometries 42 3.1 Introduction . 42 3.2 Hyperbolic CTMRG . 46 ix 3.2.1 The Lattice Model . 47 3.2.2 Recurrence Relations . 48 3.3 Phase Transition Analysis . 52 3.3.1 Asymptotic Lattice Geometries . 53 3.4 Free energy calculation . 57 3.4.1 Free energy on (5,4) lattice . 58 3.4.2 Free energy on (p;q) lattices . 60 3.5 Results . 61 3.5.1 Absence of phase
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