Nonlinear sigma models with compact hyperbolic target spaces

Gubser, Steven; Saleem, Zain H.; Schoenholz, Samuel S.; Stoica, Bogdan; Stokes, James

doi:10.1007/JHEP06(2016)145

Title: 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 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. 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:

Gubser, Steven ^[1]; Saleem, Zain H. ^[2]; Schoenholz, Samuel S. ^[3]; Stoica, Bogdan ^[4]; Stokes, James ^[3]

Princeton Univ., Princeton, NJ (United States)
Univ. of Pennsylvania, Philadelphia, PA (United States); Quaid-e-Azam Univ. Campus, Islambad (Pakistan)
Univ. of Pennsylvania, Philadelphia, PA (United States)
California Inst. of Technology (CalTech), Pasadena, CA (United States)

Publication Date:: Thu Jun 23 00:00:00 EDT 2016

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:: AC02-76ER03071; FG02-05ER46199; 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 1029-8479

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.

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                    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

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                    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.

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@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 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. 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         = {Thu Jun 23 00:00:00 EDT 2016},

  month        = {Thu Jun 23 00:00:00 EDT 2016}

}

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Works referenced in this record:

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journal, September 2003

Kuusalo, T.
The Quarterly Journal of Mathematics, Vol. 54, Issue 3
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How good is the villain approximation?
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Janke, W.; Kleinert, H.
Nuclear Physics B, Vol. 270
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Non-linear sigma models with anti-de Sitter target spaces
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Friess, Joshua J.; Gubser, Steven S.
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Michalogiorgakis, Georgios; Gubser, Steven S.
Nuclear Physics B, Vol. 757, Issue 1-2
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De Sitter space and eternity
journal, July 2008

Polyakov, A. M.
Nuclear Physics B, Vol. 797, Issue 1-2
DOI: 10.1016/j.nuclphysb.2008.01.002

The curious case of large-N expansions on a (pseudo)sphere
journal, April 2015

Polyakov, Alexander M.; Saleem, Zain H.; Stokes, James
Nuclear Physics B, Vol. 893
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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
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Similar Records

Title: Nonlinear sigma models with compact hyperbolic target spaces

Abstract

Citation Formats

Weierstrass Points of Genus-2 Surfaces with Regular Fundamental Domains journal, September 2003

How good is the villain approximation? journal, January 1986

Non-linear sigma models with anti-de Sitter target spaces journal, August 2006

Heterotic non-linear sigma models with anti-de Sitter target spaces journal, November 2006

De Sitter space and eternity journal, July 2008

The curious case of large-N expansions on a (pseudo)sphere journal, April 2015

Directed Random walk on the Lattices of Genus two journal, October 2011

Weierstrass Points of Genus-2 Surfaces with Regular Fundamental Domains
journal, September 2003

How good is the villain approximation?
journal, January 1986

Non-linear sigma models with anti-de Sitter target spaces
journal, August 2006

Heterotic non-linear sigma models with anti-de Sitter target spaces
journal, November 2006

De Sitter space and eternity
journal, July 2008

The curious case of large-N expansions on a (pseudo)sphere
journal, April 2015

Directed Random walk on the Lattices of Genus two
journal, October 2011