Nonlinear sigma models with compact hyperbolic target spaces

  • Steven Gubser
  • Zain H. Saleem
  • Samuel S. Schoenholz
  • Bogdan Stoica
  • James Stokes
Open Access
Regular Article - Theoretical Physics


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.


Effective field theories Integrable Field Theories Lattice Quantum Field Theory Matrix Models 


Open Access

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.


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Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Steven Gubser
    • 1
  • Zain H. Saleem
    • 2
    • 4
  • Samuel S. Schoenholz
    • 2
  • Bogdan Stoica
    • 3
  • James Stokes
    • 2
  1. 1.Joseph Henry LaboratoriesPrinceton UniversityPrincetonU.S.A.
  2. 2.Department of Physics and AstronomyUniversity of PennsylvaniaPhiladelphiaU.S.A.
  3. 3.Walter Burke Institute for Theoretical Physics, California Institute of TechnologyPasadenaU.S.A.
  4. 4.National Center for PhysicsQuaid-e-Azam University CampusIslamabadPakistan

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