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:: 2016-06-23

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.

Copy to clipboard


                    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

Copy to clipboard


                    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.

Copy to clipboard


                    
@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         = {2016},

  month        = {6}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1007/JHEP06(2016)145

Other availability

Search WorldCat to find libraries that may hold this journal

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

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

Kuusalo, T.
The Quarterly Journal of Mathematics, Vol. 54, Issue 3
DOI: 10.1093/qmath/hag016

How good is the villain approximation?
journal, January 1986

Janke, W.; Kleinert, H.
Nuclear Physics B, Vol. 270
DOI: 10.1016/0550-3213(86)90549-3

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

Friess, Joshua J.; Gubser, Steven S.
Nuclear Physics B, Vol. 750, Issue 3
DOI: 10.1016/j.nuclphysb.2006.05.008

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

Michalogiorgakis, Georgios; Gubser, Steven S.
Nuclear Physics B, Vol. 757, Issue 1-2
DOI: 10.1016/j.nuclphysb.2006.08.026

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
DOI: 10.1016/j.nuclphysb.2015.01.024

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
DOI: 10.1142/S0217979211101831

Similar Records in DOE PAGES and OSTI.GOV collections:

On the theory of point vortices in two-dimensional Bose liquids

Miscellaneous Speliotopoulos, A D

The physics and structure of the Kosterlitz-Thouless phase transition, as it is applied to superfluidity in two dimensions, will be studied by looking at the origins and properties of point vortices in a Bose Liquid. A lagrangian for the two-dimensional vortex gas is derived from a general microscopic lagrangian for {sup 4}He atoms on an arbitrary compact Riemann Surface without boundary. In the contrast density limit the vortex hamiltonian obtained from this lagrangian is found to be the same as the Kosterlitz and Thouless coulombic interaction hamiltonian. The dynamics and symmetries of the vortex gas on compact Riemann Surfaces aremore »« less
Vortices in the SU(N) x SU(N) spin systems in two dimensions

Thesis/Dissertation Kares, R J.D.

The SU(N) x SU(N) or chiral spin systems in two dimensions with spin variables in both the fundamental and the adjoint representations of SU(N) are considered. In the adjoint representation the chiral models are found to possess topologically stable, classical vortex solutions which carry a Z(N) topological charge. A relationship is established between the chiral models and massive Yang-Mills theory in two dimensions. This relationship is exploited to prove the asymptotic freedom of the chiral models and to find their weak coupling mass gap. The connection between the vortices of the chiral models and those of the massive Yang-Mills theorymore »« less
Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons

Journal Article Chaoming, Jian ; Hui, Zhai - Physical Review. B, Condensed Matter and Materials Physics

In this Rapid Communication we study finite-temperature properties of spin-1/2 interacting bosons with spin-orbit coupling in two dimensions. When the ground state has stripe order, we show that thermal fluctuations will first melt the stripe order and lead to a superfluid of boson pairs if the spin-orbit coupling is isotropic or nearly isotropic. Such a phase supports fractionalized quantum vortices. The Kosterlitz-Thouless transition from superfluid to normal state is driven by proliferation of half vortices. When the ground state is a plane-wave state, the transition to normal state is driven by conventional Kosterlitz-Thouless transition. However, the critical temperature will dropmore »« less
https://doi.org/10.1103/PHYSREVB.84.060508
An alternative to wave mechanics on curved spaces

Journal Article Tomaschitz, R - International Journal of Theoretical Physics

Geodesic motion in infinite spaces of constant negative curvature provides for the first time an example where a basically quantum mechanical quantity, a ground-state energy, is derived from Newtonian mechanics in a rigorous, non-semiclassical way. The ground state energy emerges as the Hausdorff dimension of a quasi-self-similar curve at infinity of three-dimensional hyperbolic space H{sup 3} in which our manifolds are embedded and where their universal covers are realized. This curve is just the locus of the limit set {Lambda}({Gamma}) of the Kleinian group {Gamma} of covering transformations, which determines the bounded trajectories in the manifold; all of them liemore »« less
https://doi.org/10.1007/BF00673252
Vortices on the string and superstring world sheets

Journal Article Abrikosov, Jr, A A ; Kogan, Y I - International Journal of Modern Physics A; (United States)

This paper considers the world-sheet dynamics of the first quantized string propagating in nonsimply connected space. Presence of the vortices on the world sheet leads to Berezinsky-Kosterlitz-Thouless (BKT) phase transition. Bosonic and superstring cases are discussed. It is shown that Hagedorn transition can be described as BKT transition both in bosonic and superstring cases.
https://doi.org/10.1142/S0217751X91000794

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