Generalizing the O(N)-field theory to N-colored manifolds of arbitrary internal dimension D |
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| Institution: | 1. Fachbereich Physik, Universität GH Essen, 45117 Essen, Germany;2. Department of Physics, MIT, Cambridge, MA 02139, USA;1. Hannover Medical School (MHH), Institute for History, Ethics, and Philosophy of Medicine, Germany;2. University of Duisburg-Essen, Ethics Committee of the Medical Faculty, Germany;3. University of Gießen, Research Ethics Committee, Germany;4. Ludwig Maximilians Universität München (LMU), Ethics Committee, Germany;5. University of Kiel, Ethics Committee of the Medical Faculty, Germany;6. Humboldt-Universität zu Berlin, Faculty of Law, Germany;7. University Hospital of Wuerzburg, Interdisciplinary Bank of Biomaterials and Data Wuerzburg (ibdw), Germany;8. University of Kiel, Chair of Public and Medical Law, Germany;9. Technische Universität München (TUM), University Hospital Klinikum Rechts der Isar, Germany;10. University of Mannheim, Professorship for Civil Law, Law of Civil Procedure, Private International Law and Comparative Law, Germany;11. Hannover Medical School (MHH), Research Ethics Committee, Germany;1. Department of Physiology, JiangXi Medical College of Nanchang University, Nanchang, Jiangxi, China;2. Undergraduate Student of Clinical Medical College, JiangXi Medical College of Nanchang University, Nanchang, Jiangxi, China;3. Department of Science and Education, Chest Hospital of Jiangxi Province, Nanchang, Jiangxi, China;4. Department of Pathology, Jiangxi Medical College of Nanchang University, Nanchang, Jiangxi, China;5. Undergraduate Student of Basic Medical College, JiangXi Medical College of Nanchang University, Nanchang, Jiangxi, China;6. Department of Nursing, The Second Affliated Hospital of Nanchang University, Nanchang, Jiangxi, China;1. Centre for Biomedical Ethics and Law, Department of Public Health and Primary Care, KU Leuven, Leuven, Belgium;2. Centre for Health Equity, Melbourne School of Population and Global Health, The University of Melbourne, Parkville, Australia;3. Genetics Education and Health Research, Murdoch Childrens Research Institute, The Royal Children''s Hospital and Department of Paediatrics, The University of Melbourne, Parkville, Australia;1. Section of Genetics, Nationwide Children''s Hospital, Columbus, OH;2. Department of Pediatrics, Ohio State University, Columbus, OH;3. Division of Human Genetics, Cincinnati Children''s Hospital Medical Center, Cincinnati, OH;4. Department of Pediatrics, University of Cincinnati College of Medicine, Cincinnati, OH;5. Division of Biostatistics and Epidemiology, Cincinnati Children''s Hospital Medical Center, Cincinnati, OH;6. Suburban Pediatric Associates, Inc, Mason, OH |
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| Abstract: | We introduce a geometric generalization of the O(N)-field theory that describes N-colored membranes with arbitrary dimension D. As the O(N)-model reduces in the limit N → 0 to self-avoiding polymers, the N-colored manifold model leads to self-avoiding tethered membranes. In the other limit, for inner dimension D → 1, the manifold model reduces to the O(N)-field theory. We analyze the scaling properties of the model at criticality by a one-loop perturbative renormalization group analysis around an upper critical line. The freedom to optimize with respect to the expansion point on this line allows us to obtain the exponent ν of standard field theory to much better precision that the usual 1-loop calculations. Some other field theoretical techniques, such as the large N limit and Hartree approximation, can also be applied to this model. By comparison of low- and high-temperature expansions, we arrive at a conjecture for the nature of droplets dominating the 3d Ising model at criticality, which is satisfied by our numerical results. We can also construct an appropriate generalization that describes cubic anisotropy, by adding an interaction between manifolds of the same color. The two parameter space includes a variety of new phases and fixed points, some with Ising criticality, enabling us to extract a remarkably precise value of 0.6315 for the exponent ν in d = 3. A particular limit of the model with cubic anisotropy corresponds to the random bond Ising problem; unlike the field theory formulation, we find a fixed point describing this system at 1-loop order. |
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