TY - JOUR

T1 - Nonlinear sigma models with compact hyperbolic target spaces

AU - Gubser, Steven

AU - Saleem, Zain H.

AU - Schoenholz, Samuel S.

AU - Stoica, Bogdan

AU - Stokes, James

N1 - Publisher Copyright:
© 2016, The Author(s).

PY - 2016/6/1

Y1 - 2016/6/1

N2 - 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.

AB - 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.

KW - Effective field theories

KW - Integrable Field Theories

KW - Lattice Quantum Field Theory

KW - Matrix Models

UR - http://www.scopus.com/inward/record.url?scp=84978891123&partnerID=8YFLogxK

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U2 - 10.1007/JHEP06(2016)145

DO - 10.1007/JHEP06(2016)145

M3 - Article

AN - SCOPUS:84978891123

SN - 1126-6708

VL - 2016

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

IS - 6

M1 - 145

ER -