Nonlinear sigma models with compact hyperbolic target spaces
Abstract
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 genus2 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 KosterlitzThouless 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. In conclusion, 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.
 Authors:

 Princeton Univ., Princeton, NJ (United States)
 Univ. of Pennsylvania, Philadelphia, PA (United States); QuaideAzam Univ. Campus, Islambad (Pakistan)
 Univ. of Pennsylvania, Philadelphia, PA (United States)
 California Inst. of Technology (CalTech), Pasadena, CA (United States)
 Publication Date:
 Research Org.:
 Univ. of Pennsylvania, Philadelphia, PA (United States); California Institute of Technology (CalTech), Pasadena, CA (United States); Princeton Univ., NJ (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), High Energy Physics (HEP)
 OSTI Identifier:
 1326979
 Grant/Contract Number:
 AC0276ER03071; FG0205ER46199; SC0011632; SC0007968
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of High Energy Physics (Online)
 Additional Journal Information:
 Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2016; Journal Issue: 6; Journal ID: ISSN 10298479
 Publisher:
 Springer Berlin
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Effective field theories; Integrable Field Theories; Lattice Quantum Field Theory; Matrix Model
Citation Formats
Gubser, Steven, Saleem, Zain H., Schoenholz, Samuel S., Stoica, Bogdan, and Stokes, James. Nonlinear sigma models with compact hyperbolic target spaces. United States: N. p., 2016.
Web. doi:10.1007/JHEP06(2016)145.
Gubser, Steven, Saleem, Zain H., Schoenholz, Samuel S., Stoica, Bogdan, & Stokes, James. Nonlinear sigma models with compact hyperbolic target spaces. United States. https://doi.org/10.1007/JHEP06(2016)145
Gubser, Steven, Saleem, Zain H., Schoenholz, Samuel S., Stoica, Bogdan, and Stokes, James. Thu .
"Nonlinear sigma models with compact hyperbolic target spaces". United States. https://doi.org/10.1007/JHEP06(2016)145. https://www.osti.gov/servlets/purl/1326979.
@article{osti_1326979,
title = {Nonlinear sigma models with compact hyperbolic target spaces},
author = {Gubser, Steven and Saleem, Zain H. and Schoenholz, Samuel S. and Stoica, Bogdan and Stokes, James},
abstractNote = {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 genus2 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 KosterlitzThouless 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. In conclusion, 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.},
doi = {10.1007/JHEP06(2016)145},
journal = {Journal of High Energy Physics (Online)},
number = 6,
volume = 2016,
place = {United States},
year = {2016},
month = {6}
}
Works referenced in this record:
Weierstrass Points of Genus2 Surfaces with Regular Fundamental Domains
journal, September 2003
 Kuusalo, T.
 The Quarterly Journal of Mathematics, Vol. 54, Issue 3
How good is the villain approximation?
journal, January 1986
 Janke, W.; Kleinert, H.
 Nuclear Physics B, Vol. 270
Nonlinear sigma models with antide Sitter target spaces
journal, August 2006
 Friess, Joshua J.; Gubser, Steven S.
 Nuclear Physics B, Vol. 750, Issue 3
Heterotic nonlinear sigma models with antide Sitter target spaces
journal, November 2006
 Michalogiorgakis, Georgios; Gubser, Steven S.
 Nuclear Physics B, Vol. 757, Issue 12
De Sitter space and eternity
journal, July 2008
 Polyakov, A. M.
 Nuclear Physics B, Vol. 797, Issue 12
The curious case of largeN expansions on a (pseudo)sphere
journal, April 2015
 Polyakov, Alexander M.; Saleem, Zain H.; Stokes, James
 Nuclear Physics B, Vol. 893
Directed Random walk on the Lattices of Genus two
journal, October 2011
 Nazarenko, A. V.
 International Journal of Modern Physics B, Vol. 25, Issue 26