Skip to main content

Critical 1- and 2-point spin correlations for the O(2) model in 3d bounded domains

A preprint version of the article is available at arXiv.


We study the critical properties of the 3d O(2) universality class in bounded domains through Monte Carlo simulations of the clock model. We use an improved version of the latter, chosen to minimize finite-size corrections at criticality, with 8 orientations of the spins and in the presence of vacancies. The domain chosen for the simulations is the slab configuration with fixed spins at the boundaries. We obtain the universal critical magnetization profile and two-point correlations, which favorably compare with the predictions of the critical geometry approach based on the Yamabe equation. The main result is that the correlations, once the dimensionful contributions are factored out with the critical magnetization profile, are shown to depend only on the distance between the points computed using a metric found solving the corresponding fractional Yamabe equation. The quantitative comparison with the corresponding results for the Ising model at criticality is shown and discussed. Moreover, from the magnetization profiles the critical exponent η is extracted and found to be in reasonable agreement with up-to-date results.


  1. M. Creutz, Quarks, gluons and lattices, Cambridge Univ. Press, Cambridge, U.K. (1985).

  2. H.J. Rothe and K.D. Rothe, Classical and quantum dynamics of constrained hamiltonian systems, World Scientific, Singapore (2010).

    Book  Google Scholar 

  3. E. Fradkin, Field theories of condensed matter physics, Cambridge University Press, Cambridge, U.K. (2013).

    Book  Google Scholar 

  4. J.B. Kogut and D.K. Sinclair, Evidence for O(2) universality at the finite temperature transition for lattice QCD with 2 flavors of massless staggered quarks, Phys. Rev. D 73 (2006) 074512 [hep-lat/0603021] [INSPIRE].

  5. P. Springer and B. Klein, O(2)-scaling in finite and infinite volume, Eur. Phys. J. C 75 (2015) 468 [arXiv:1506.00909] [INSPIRE].

    ADS  Article  Google Scholar 

  6. T. Matsubara and H. Matsuda, A lattice model of liquid helium, I, Prog. Theor. Phys. 16 (1956) 569.

    ADS  Article  Google Scholar 

  7. D.D. Betts and J.R. Lothian, Comparison of the critical properties of the s = 1/2 XY model and liquid helium near the lambda transition, Can. J. Phys. 51 (1973) 2249.

    ADS  Article  Google Scholar 

  8. J.A. Lipa et al., Specific heat of helium confined to a 57 μm planar geometry near the lambda point, Phys. Rev. Lett. 84 (2000) 4894 [INSPIRE].

    ADS  Article  Google Scholar 

  9. J.A. Lipa, D.R. Swanson, J.A. Nissen, T.C.P. Chui and U.E. Israelsson, Heat capacity and thermal relaxation of bulk helium very near the lambda point, Phys. Rev. Lett. 76 (1996) 944 [INSPIRE].

    ADS  Article  Google Scholar 

  10. M. Hasenbusch, Monte Carlo study of an improved clock model in three dimensions, Phys. Rev. B 100 (2019) 224517 [arXiv:1910.05916] [INSPIRE].

    ADS  Article  Google Scholar 

  11. S.M. Chester et al., Carving out OPE space and precise O(2) model critical exponents, JHEP 06 (2020) 142 [arXiv:1912.03324] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  12. J.M. Kosterlitz and D.J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6 (1973) 1181 [INSPIRE].

    ADS  Article  Google Scholar 

  13. J.M. Kosterlitz, Nobel lecture: topological defects and phase transitions, Rev. Mod. Phys. 89 (2017) 040501.

  14. S.R. Shenoy, Vortex-loop scaling in the three-dimensional XY ferromagnet, Phys. Rev. B 40 (1989) 5056 [INSPIRE].

    ADS  Article  Google Scholar 

  15. A. Forrester and G.A. Williams, Vortex-loop calculation of the specific heat of superfluid 4He under pressure, Phys. Rev. E 100 (2019) 060104.

  16. V. Cvetkovic and J. Zaanen, Vortex duality: observing the dual nature using order propagators, Phys. Rev. B 74 (2006) 134504 [cond-mat/0511586] [INSPIRE].

  17. H.W. Diehl, The theory of boundary critical phenomena, Int. J. Mod. Phys. B 11 (1997) 3503 [cond-mat/9610143] [INSPIRE].

  18. K. Binder, Critical behaviour at surfaces, in Phase transitions and critical phenomena, volume 8, C. Domb ed., Elsevier, The Netherlands (2000).

  19. O. Vasilyev, A. Gambassi, A. Maciołek and S. Dietrich, Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations, Phys. Rev. E 79 (2009) 041142.

  20. A.J. Bray and M.A. Moore, Critical behaviour of semi-infinite systems, J. Phys. A 10 (1977) 1927.

    ADS  MathSciNet  Article  Google Scholar 

  21. H. Diehl and S. Dietrich, Scaling laws and surface exponents from renormalization group equations, Phys. Lett. A 80 (1980) 408.

    ADS  MathSciNet  Article  Google Scholar 

  22. H.W. Diehl and S. Dietrich, Field-theoretical approach to static critical phenomena in semi-infinite systems, Z. Phys. B 42 (1981) 65 [INSPIRE].

    ADS  Article  Google Scholar 

  23. T.C. Lubensky and M.H. Rubin, Critical phenomena in semi-infinite systems. I. ε expansion for positive extrapolation length, Phys. Rev. B 11 (1975) 4533.

  24. D. Gruneberg and H.W. Diehl, Thermodynamic Casimir effects involving interacting field theories with zero modes, Phys. Rev. B 77 (2008) 115409 [arXiv:0710.4436] [INSPIRE].

    ADS  Article  Google Scholar 

  25. H.W. Diehl and S.B. Rutkevich, The three-dimensional O(n) ϕ4 model on a strip with free boundary conditions: exact results for a nontrivial dimensional crossover in the limit n → ∞, Theor. Math. Phys. 190 (2017) 279 [arXiv:1512.05892] [INSPIRE].

    Article  Google Scholar 

  26. A. Gambassi and S. Dietrich, Critical dynamics in thin films, J. Statist. Phys. 123 (2006) 929.

    ADS  MathSciNet  Article  Google Scholar 

  27. A. Gambassi and S. Dietrich, Comment on “the Casimir effect for the Bose-gas in slabs” by P.A. Martin and V.A. Zagrebnov. Relation between the thermodynamic Casimir effect in Bose-gas slabs and critical Casimir forces, Europhys. Lett. 74 (2006) 754.

  28. O. Vasilyev, A. Gambassi, A. Maciołek and S. Dietrich, Monte Carlo simulation results for critical Casimir forces, Europhys. Lett. 80 (2007) 60009.

    ADS  Article  Google Scholar 

  29. J.L. Cardy, Scaling and renormalization in statistical physics, in Cambridge lecture notes in physics, volume 5, Cambridge University Press, Cambridge, U.K. (1996).

  30. A. Gambassi et al., Critical Casimir effect in classical binary liquid mixtures, Phys. Rev. E 80 (2009) 061143.

  31. V.M. Vassilev, D.M. Dantchev and P.A. Djondjorov, Order parameter profiles in a system with Neumann-Neumann boundary conditions, MATEC Web Conf. 145 (2018) 01009.

  32. N.F. Bafi, A. Maciołek and S. Dietrich, Tricritical Casimir forces and order parameter profiles in wetting films of 3He-4He mixtures, Phys. Rev. E 95 (2017) 032802.

  33. A. Maciołek, M. Krech and S. Dietrich, Phase diagram of a model for 3He-4He mixtures in three dimensions, Phys. Rev. E 69 (2004) 036117.

  34. W. Deng and W. Zimmermann Jr., Parallel-plate capacitor measurements of the superfluid wall-film thickness in a 3He/4He mixture of 3He mole fraction x = 0.75, J. Phys. Conf. Ser. 150 (2009) 032018.

  35. C. Hertlein, L. Helden, A. Gambassi, S. Dietrich and C. Bechinger, Direct measurement of critical Casimir forces, Nature 451 (2008) 172.

    ADS  Article  Google Scholar 

  36. M.A. Metlitski, Boundary criticality of the O(N) model in d = 3 critically revisited, arXiv:2009.05119 [INSPIRE].

  37. G. Gori and A. Trombettoni, Geometry of bounded critical phenomena, J. Stat. Mech. 2020 (2020) 063210.

  38. J. Hove and A. Sudbo, Criticality versus q in the 2 + 1-dimensional Zq clock model, Phys. Rev. E 68 (2003) 046107 [cond-mat/0301499] [INSPIRE].

  39. M. Creutz, L. Jacobs and C. Rebbi, Monte Carlo study of Abelian lattice gauge theories, Phys. Rev. D 20 (1979) 1915 [INSPIRE].

    ADS  Article  Google Scholar 

  40. U. Wolff, Collective Monte Carlo updating for spin systems, Phys. Rev. Lett. 62 (1989) 361 [INSPIRE].

    ADS  Article  Google Scholar 

  41. J. Kent-Dobias and J.P. Sethna, Cluster representations and the Wolff algorithm in arbitrary external fields, Phys. Rev. E 98 (2018) 063306 [arXiv:1805.04019] [INSPIRE].

  42. H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka J. Math. 12 (1960) 21.

    MathSciNet  MATH  Google Scholar 

  43. S.-Y.A. Chang and M. del Mar González, Fractional laplacian in conformal geometry, Adv. Math. 226 (2011) 1410.

    MathSciNet  Article  Google Scholar 

  44. A. Galvani, G. Gori and A. Trombettoni, Magnetization profiles at the upper critical dimension as solutions of the integer Yamabe problem, Phys. Rev. E 104 (2021) 024138 [arXiv:2103.12449] [INSPIRE].

  45. M. del Mar González Nogueras and J. Qing, Fractional conformal laplacians and fractional Yamabe problems, Anal. PDE 6 (2013) 1535.

  46. C. Cosme, J.M.V.P. Lopes and J. Penedones, Conformal symmetry of the critical 3D Ising model inside a sphere, JHEP 08 (2015) 022 [arXiv:1503.02011] [INSPIRE].

    ADS  Article  Google Scholar 

  47. G. Gori and A. Trombettoni, Conformal invariance in three dimensional percolation, J. Stat. Mech. 2015 (2015) P07014.

  48. K.A. Brakke, The surface evolver, Exper. Math. 1 (1992) 141.

    MathSciNet  Article  Google Scholar 

  49. P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer-Verlag, New York, NY, U.S.A. (1997) [INSPIRE].

    Book  Google Scholar 

  50. K. Binder, Finite size scaling analysis of Ising model block distribution functions, Z. Phys. B 43 (1981) 119 [INSPIRE].

    ADS  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Alessandro Galvani.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2108.03488

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Galvani, A., Gori, G. & Trombettoni, A. Critical 1- and 2-point spin correlations for the O(2) model in 3d bounded domains. J. High Energ. Phys. 2021, 106 (2021).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI:


  • Boundary Quantum Field Theory
  • Lattice Quantum Field Theory
  • Conformal Field Theory