Nonlinear sigma models with compact hyperbolic target spaces

Journal Article · Thu Jun 23 04:00:00 UTC 2016 · Journal of High Energy Physics (Online)

DOI: https://doi.org/10.1007/JHEP06(2016)145 · OSTI ID:1326979

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

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1. Princeton Univ., Princeton, NJ (United States); None
2. Univ. of Pennsylvania, Philadelphia, PA (United States); Quaid-e-Azam Univ. Campus, Islambad (Pakistan)
3. Univ. of Pennsylvania, Philadelphia, PA (United States)
4. California Inst. of Technology (CalTech), Pasadena, CA (United States)

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.

• Research Organization: Univ. of Pennsylvania, Philadelphia, PA (United States); California Inst. of Technology, Pasadena, CA (United States); Princeton Univ., NJ (United States)
• Sponsoring Organization: USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)
• Grant/Contract Number: AC02-76ER03071; FG02-05ER46199; SC0011632; SC0007968
• OSTI ID: 1326979
• Journal Information: Journal of High Energy Physics (Online), Journal Name: Journal of High Energy Physics (Online) Journal Issue: 6 Vol. 2016; ISSN 1029-8479
• Publisher: Springer BerlinCopyright Statement
• Country of Publication: United States
• Language: English