D-Theory Formulation of Quantum Field Theories and Application to CP (N 1) Models − Inauguraldissertation der Philosophisch-naturwissenschaftlichen Fakult¨at der Universit¨at Bern vorgelegt von St´ephane Jean Riederer von St. Gallen/SG Leiter der Arbeit: Prof. Dr. Uwe-Jens Wiese Institut f¨ur theoretische Physik Universit¨at Bern D-Theory Formulation of Quantum Field Theories and Application to CP (N 1) Models − Inauguraldissertation der Philosophisch-naturwissenschaftlichen Fakult¨at der Universit¨at Bern vorgelegt von St´ephane Jean Riederer von St. Gallen/SG Leiter der Arbeit: Prof. Dr. Uwe-Jens Wiese Institut f¨ur theoretische Physik Universit¨at Bern Von der Philosophisch-naturwissenschaftlichen Fakult¨at angenommen. Der Dekan: Bern, den 1. Juni 2006 Prof. Dr. P. Messerli Abstract D-theory is an alternative non-perturbative approach to quantum field theory formulated in terms of discrete quantized variables instead of classical fields. Classical scalar fields are replaced by generalized quantum spins and classical gauge fields are replaced by quantum links. The classical fields of a d-dimensional quantum field theory reappear as low-energy effective degrees of freedom of the discrete variables, provided the (d + 1)-dimensional D- theory is massless. When the extent of the extra Euclidean dimension becomes small in units of the correlation length, an ordinary d-dimensional quantum field theory emerges by dimensional reduction. The D-theory formulation of scalar field theories with various global symmetries and of gauge theories with various gauge groups is constructed explic- itly and the mechanism of dimensional reduction is investigated. In particular, D-theory provides an alternative lattice regularization of the 2-d CP (N 1) quantum field theory. In this formulation the continuous classical CP (N 1) fields emerge− from the dimensional − reduction of discrete SU(N) quantum spins. In analogy to Haldane’s conjecture, ladders consisting of an even number of transversely coupled spin chains lead to a CP (N 1) model with vacuum angle θ = 0, while an odd number of chains yields θ = π. In contrast− to Wilson’s formulation of lattice field theory, in D-theory no sign problem arises at θ = π, and an efficient cluster algorithm is used to investigate the θ-vacuum effects. At θ = π there is a first order phase transition with spontaneous breaking of charge conjugation sym- metry for CP (N 1) models with N > 2. Despite several attempts, no efficient cluster algorithm has been− constructed for CP (N 1) models in the standard Wilson formulation − of lattice field theory. In fact, there is a no-go theorem that prevents the construction of an efficient Wolff-type embedding algorithm. In this thesis, we construct an efficient cluster algorithm for ferromagnetic SU(N)-symmetric quantum spin systems, which provides a regularization for CP (N 1) models at θ = 0 in the framework of D-theory. We present detailed studies of the autocorrelations− and find a dynamical critical exponent that is con- sistent with z = 0. Cluster rules for antiferromagnetic spin chains, which also provide a regularization for CP (N 1) models at θ = 0 and π, are investigated in detail as well. − Contents 1 Introduction 1 1.1 Wilson’s Approach to Quantum Field Theory . .... 2 1.2 TheD-theoryApproach ............................ 6 2 D-Theory Formulation of Quantum Field Theory 9 2.1 The O(3)ModelfromD-theory . .. .. 10 2.1.1 The 2-dimensional O(3)model..................... 10 2.1.2 The 2-dimensional quantum Heisenberg model . .... 11 2.1.3 D-theory regularization and dimensional reduction . ......... 12 2.1.4 Generalization to other dimensions d =2............... 17 6 2.2 D-Theory Representation of Basic Field Variables . ......... 19 2.2.1 Realvectors............................... 19 2.2.2 Realmatrices .............................. 21 2.2.3 Complexvectors ............................ 22 2.2.4 Complexmatrices............................ 22 2.2.5 Symplectic, symmetric, and anti-symmetric complex tensors . 23 2.3 D-Theory Formulation of Various Models with a Global Symmetry. 24 2.3.1 O(N)quantumspinmodels ...................... 24 2.3.2 SO(N) SO(N) chiral quantum spin models . 25 L ⊗ R 2.3.3 U(N) U(N) and SU(N) SU(N) chiral quantum spin models 26 L ⊗ R L ⊗ R 2.4 Classical Scalar Fields from Dimensional Reduction of Quantum Spins . 28 2.4.1 O(N)models .............................. 29 2.4.2 SU(N), Sp(N) and SO(N)chiralmodels. 31 2.5 D-Theory Formulation of Various Models with a Gauge Symmetry..... 32 2.5.1 U(N) and SU(N)quantumlinkmodels . 32 2.5.2 SO(N)quantumlinkmodels ..................... 37 2.5.3 Quantum link models with other gauge groups . ... 37 2.6 Classical Gauge Fields from Dimensional Reduction of Quantum Links . 39 2.6.1 SU(N)gaugetheories ......................... 39 2.6.2 SO(N), Sp(N) and U(1)gaugetheories . 41 2.7 QCD as a Quantum Link Model: The Inclusion of Fermions . ..... 42 2.8 ConclusionsandComments . 47 i 3 The CP (N 1) Model and Topology 49 − 3.1 The 2-Dimensional CP (N 1)Model..................... 50 − 3.1.1 Introduction............................... 50 3.1.2 Classicalformulation . 51 3.2 Non-trivial Topological Structure of the CP (N 1)Model ......... 53 − 3.2.1 Topologicalaspects ........................... 53 3.2.2 Instantons and θ-vacuumstructure . 56 3.3 Topological Consequences in QCD . ... 60 4 Study of CP (N 1) θ-Vacua Using D-Theory 63 − 4.1 Introduction................................... 63 4.2 D-Theory Formulation of CP (N 1)Models................. 65 − 4.3 CP (N 1) Model at θ =0 as an SU(N)QuantumFerromagnet . 68 − 4.3.1 SU(N)quantumferromagnet . 68 4.3.2 Low-energyeffectivetheory . 69 4.3.3 Dimensionalreduction . 72 4.3.4 Equivalence with Wilson’s regularization . ...... 74 4.4 CP (N 1) Models at θ =0, π as an SU(N) Quantum Spin Ladder . 75 − 4.4.1 SU(N)quantumspinladders . 76 4.4.2 Low-energy effective theory and dimensional reduction ....... 77 4.4.3 A first order phase transition at θ = π for N > 2........... 79 4.5 CP (N 1) Models at θ = 0, π as an Antiferromagnetic SU(N) Quantum SpinChain− ................................... 83 4.5.1 Antiferromagnetic SU(N) quantum spin chains . 84 4.5.2 A first order phase transition at θ = π for N > 2........... 86 4.6 ConclusionsandComments . 88 5 Efficient Cluster Algorithms for CP (N 1) Models 91 − 5.1 Introduction................................... 91 5.1.1 The Monte Carlo method and critical slowing down . .... 91 5.1.2 Generalities on cluster algorithms . .... 93 5.2 Path Integral Representation of SU(N)QuantumFerromagnets . 94 5.3 Cluster Algorithm and Cluster Rules for SU(N) Ferromagnetic Systems . 99 5.4 Path Integral Representation for SU(N) Antiferromagnetic Spin Chains . 102 5.5 Cluster Algorithm for SU(N) Antiferromagnetic Systems . 105 5.6 Generalization to Higher SU(N) Representations for Quantum Spin Chains 107 5.6.1 Path integral representation and layer decomposition ........ 108 5.6.2 Clusterrules............................... 110 5.7 Path Integral Representation and Cluster Rules for SU(N) Spin Ladders . 112 5.8 Efficiency of the Cluster Algorithm in the Continuum Limit ........ 113 6 Conclusions 119 ii A Determination of the Low-Energy Parameters of an SU(N)-Symmetric Ferromagnet 123 Bibliography 129 Curriculum Vitæ 140 iii iv Acknowledgments First and foremost, I would like to thank Professor Uwe-Jens Wiese for giving me the opportunity of undertaking this research project under his supervision. He has always been a source of inspiration and I have largely benefited from his inexhaustible knowledge and support. His enthusiasm and kindness have been extremely precious throughout the last three years. I have often entered his office with a doubtful mind and always got out with a clearer vision. Most of what is included in this thesis would not have been possible without the help of Michele Pepe, who proved very patient and interested in what I was doing. I would like to thank him warmly. The over-sea collaboration with Bernard B. Beard has also been fruitful. I especially thank him for letting me using his “continuous time” computer codes. I also benefited from discussions with Ferenc Niedermayer, Peter Hasenfratz, Peter Minkowski, Richard Brower and Markus Moser. I would like to thank also the sysadmins for the great job they are doing and more largely all members of the institute. Special thanks to Ottilia H¨anni for helping me to find my way out of all kind of administrative questions. Thanks to Pascal, Laurent, and Adrian for entertaining me with endless sport or social debates over lunch time. Of course, nothing would have been possible without my parents France and J¨urg, who have always supported me through all these years. Sincere thanks as well to the rest of my family and friends for simply being around. Last, but certainly not least, I have no idea what I would become without the sweetest one. Thanks for all Odile. v vi Chapter 1 Introduction At a fundamental level, nature is governed by four different interactions. Gravity and elec- tromagnetism which have long range effects and two nuclear forces which act only at short distances. The weak force is implied in all disintegration processes while the strong inter- action is responsible for the cohesion of matter. The most successful theories to describe natural phenomena are nowadays general relativity and the standard model of particle physics. They have both been tested intensively and none of them has been ruled out at the moment. However, they are not compatible since a theory of quantum gravity has not yet been formulated. Hence, gravity cannot be included naturally in the standard model of particle physics, which describes all what we know about the three other fundamental forces.