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Nonlinear Sigma Models with Compact Hyperbolic Target Spaces

Gubser, Steven and Saleem, Zain H. and Schoenholz, Samuel S. and Stoica, Bogdan and Stokes, James (2016) Nonlinear Sigma Models with Compact Hyperbolic Target Spaces. Journal of High Energy Physics, 2016 (06). Art. No. 145. ISSN 1126-6708. doi:10.1007/JHEP06(2016)145.

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We explore the phase structure of nonlinear sigma models with target spaces corresponding to compact quotients of hyperbolic space, focusing on the case of a hyperbolic genus-2 Riemann surface. The continuum theory of these models can be approximated by a lattice spin system which we simulate using Monte Carlo methods. The target space possesses interesting geometric and topological properties which are reflected in novel features of the sigma model. In particular, we observe a topological phase transition at a critical temperature, above which vortices proliferate, reminiscent of the Kosterlitz-Thouless phase transition in the O(2) model [1, 2]. Unlike in the O(2) case, there are many different types of vortices, suggesting a possible analogy to the Hagedorn treatment of statistical mechanics of a proliferating number of hadron species. Below the critical temperature the spins cluster around six special points in the target space known as Weierstrass points. The diversity of compact hyperbolic manifolds suggests that our model is only the simplest example of a broad class of statistical mechanical models whose main features can be understood essentially in geometric terms.

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Additional Information:© 2016 The Authors. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Article funded by SCOAP3. Received: December 10, 2015; Revised: May 14, 2016; Accepted: June 3, 2016; Published: June 23, 2016. J.S., Z.H.S. and S.S.S. would like thank Randall Kamien, Hernan Piragua and Alexander Polyakov for discussions. B.S. would like to thank Hirosi Ooguri for useful discussions, and the Institute for Advanced Study, Princeton University, and the Simons Center for Geometry and Physics for hospitality. B.S. also gratefully acknowledges support from the Simons SummerWorkshop 2015, at which part of the research for this paper was performed. J.S. is supported in part by NASA ATP grant NNX14AH53G. Z.H.S. is supported in part by DOE Grant DOE-EY-76-02-3071. S.S.S. is supported by DOE DE-FG02-05ER46199. B.S. is supported in part by the Walter Burke Institute for Theoretical Physics at Caltech and by U.S. DOE grant DE-SC0011632.
Group:Walter Burke Institute for Theoretical Physics
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Department of Energy (DOE)DOE-EY-76-02-3071
Department of Energy (DOE)DE-FG02-05ER46199
Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
Department of Energy (DOE)DE-SC0011632
Subject Keywords:Effective field theories, Integrable Field Theories, Lattice Quantum Field Theory, Matrix Models
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Issue or Number:06
Record Number:CaltechAUTHORS:20151022-123136159
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:61426
Deposited By: Joy Painter
Deposited On:22 Oct 2015 20:05
Last Modified:10 Nov 2021 22:48

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