Abstract
The nearest-neighbor Villain, or periodic Gaussian, model is a useful tool to understand the physics of the topological defects of the two-dimensional nearest-neighbor XY model, as the two models share the same symmetries and are in the same universality class. The long-range counterpart of the two-dimensional XY has been recently shown to exhibit a non-trivial critical behavior, with a complex phase diagram including a range of values of the power-law exponent of the couplings decay, σ, in which there are a magnetized, a disordered and a critical phase [1]. Here we address the issue of whether the critical behavior of the two-dimensional XY model with long-range couplings can be described by the Villain counterpart of the model. After introducing a suitable generalization of the Villain model with long-range couplings, we derive a set of renormalization-group equations for the vortex-vortex potential, which differs from the one of the long-range XY model, signaling that the decoupling of spin-waves and topological defects is no longer justified in this regime. The main results are that for σ < 2 the two models no longer share the same universality class. Remarkably, within a large region of its the phase diagram, the Villain model is found to behave similarly to the one-dimensional Ising model with 1/r2 interactions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Giachetti, N. Defenu, S. Ruffo and A. Trombettoni, Berezinskii-Kosterlitz-Thouless Phase Transitions with Long-Range Couplings, Phys. Rev. Lett. 127 (2021) 156801 [arXiv:2104.13217] [INSPIRE].
N. Goldenfeld, Lectures on phase transitions and the renormalization group, CRC Press (1992) [INSPIRE].
J.M. Kosterlitz and D.J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6 (1973) 1181 [INSPIRE].
J. Michael Kosterlitz, Nobel Lecture: Topological defects and phase transitions, Rev. Mod. Phys. 89 (2017) 040501.
R. Savit, Duality in Field Theory and Statistical Systems, Rev. Mod. Phys. 52 (1980) 453 [INSPIRE].
J. Villain, Theory of one-dimensional and two-dimensional magnets with an easy magnetization plane. 2. The Planar, classical, two-dimensional magnet, J. Phys. (France) 36 (1975) 581 [INSPIRE].
J.V. José, L.P. Kadanoff, S. Kirkpatrick and D.R. Nelson, Renormalization, vortices, and symmetry breaking perturbations on the two-dimensional planar model, Phys. Rev. B 16 (1977) 1217 [INSPIRE].
H. Kleinert, Villain Approximation and Villain Model, in Gauge fields in condensed matter, chapter 7, vol. 2, World Scientific (1989) [https://doi.org/10.1142/9789814415606_0014].
P. Dario and W. Wu, Massless Phases for the Villain model in d ≥ 3, arXiv:2002.02946 [INSPIRE].
J. Fröhlich and T. Spencer, The Kosterlitz-thouless Transition in Two-dimensional Abelian Spin Systems and the Coulomb Gas, Commun. Math. Phys. 81 (1981) 527 [INSPIRE].
O. Kapikranian, B. Berche and Y. Holovatch, Interplay of topological and structural defects in the two-dimensional XY model, Phys. Lett. A 372 (2008) 5716.
J. T Haraldsen and R. S Fishman, Spin rotation technique for non-collinear magnetic systems: application to the generalized Villain model, J. Phys. Condens. Matter 21 (2009) 216001.
M. Aizenman, M. Harel, R. Peled and J. Shapiro, Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models, arXiv:2110.09498].
M. Gabay, T. Garel, G. N. Parker and W. M. Saslow, Phase diagram for the generalized Villain model, Phys. Rev. B 40 (1989) 264.
P. Gorantla, H.T. Lam, N. Seiberg and S.-H. Shao, A modified Villain formulation of fractons and other exotic theories, J. Math. Phys. 62 (2021) 102301 [arXiv:2103.01257] [INSPIRE].
W. Janke and H. Kleinert, How Good Is the Villain Approximation?, Nucl. Phys. B 270 (1986) 135 [INSPIRE].
W. Janke and K. Nather, High precision MonteCarlo study of the 2-dimensional XY Villain model, Phys. Rev. B 48 (1993) 7419 [INSPIRE].
W. Janke, Logarithmic corrections in the two-dimensional XY model, Phys. Rev. B 55 (1997) 3580 [hep-lat/9609045] [INSPIRE].
M. Hasenbusch, The Two dimensional XY model at the transition temperature: A High precision Monte Carlo study, J. Phys. A 38 (2005) 5869 [cond-mat/0502556] [INSPIRE].
T. Surungan, S. Masuda, Y. Komura and Y. Okabe, Berezinskii-Kosterlitz-Thouless transition on regular and Villain types of q-state clock models, J. Phys. A 52 (2019) 275002 [arXiv:1901.03936] [INSPIRE].
W. Witczak-Krempa, E. Sørensen and S. Sachdev, The dynamics of quantum criticality via Quantum Monte Carlo and holography, Nature Phys. 10 (2014) 361 [arXiv:1309.2941] [INSPIRE].
C. Dasgupta and B.I. Halperin, Phase Transition in a Lattice Model of Superconductivity, Phys. Rev. Lett. 47 (1981) 1556 [INSPIRE].
E. Onofri, SU(N) Lattice Gauge Theory With Villain’s Action, Nuovo Cim. A 66 (1981) 293 [INSPIRE].
M. Romo and M. Tierz, Unitary Chern-Simons matrix model and the Villain lattice action, Phys. Rev. D 86 (2012) 045027 [arXiv:1103.2421] [INSPIRE].
Y. Choi et al., Noninvertible duality defects in 3 + 1 dimensions, Phys. Rev. D 105 (2022) 125016 [arXiv:2111.01139] [INSPIRE].
O. Borisenko, V. Chelnokov, M. Gravina and A. Papa, Deconfinement and universality in the 3D U(1) lattice gauge theory at finite temperature: study in the dual formulation, JHEP 09 (2015) 062 [arXiv:1507.00833] [INSPIRE].
B. Sathiapalan, Duality in Statistical Mechanics and String Theory, Phys. Rev. Lett. 58 (1987) 1597 [INSPIRE].
A. Campa, T. Dauxois, D. Fanelli and S. Ruffo, Physics of Long-Range Interacting Systems, Oxford University Press (2014).
N. Defenu et al., Long-range interacting quantum systems, arXiv:2109.01063 [INSPIRE].
H. Haffner, C. Roos and R. Blatt, Quantum computing with trapped ions, Phys. Rept. 469 (2008) 155.
T. Lahaye et al., The physics of dipolar bosonic quantum gases, Rept. Prog. Phys. 72 (2009) 126401.
M. Saffman, T. G. Walker and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys. 82 (2010) 2313.
H. Ritsch, P. Domokos, F. Brennecke and T. Esslinger, Cold atoms in cavity-generated dynamical optical potentials, Rev. Mod. Phys. 85 (2013) 553.
H. Bernien et al., Probing many-body dynamics on a 51-atom quantum simulator, Nature 551 (2017) 579.
C. Monroe et al., Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys. 93 (2021) 025001 [arXiv:1912.07845] [INSPIRE].
F. Mivehvar, F. Piazza, T. Donner and H. Ritsch, Cavity QED with Quantum Gases: New Paradigms in Many-Body Physics, Adv. Phys. 70 (2021) 1 [arXiv:2102.04473] [INSPIRE].
N.D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17 (1966) 1133 [INSPIRE].
F.J. Dyson, Existence of a phase transition in a one-dimensional Ising ferromagnet, Commun. Math. Phys. 12 (1969) 91 [INSPIRE].
D.J. Thouless, Long-Range Order in One-Dimensional Ising Systems, Phys. Rev. 187 (1969) 732 [INSPIRE].
J.M. Kosterlitz, Phase Transitions in Long-Range Ferromagnetic Chains, Phys. Rev. Lett. 37 (1976) 1577 [INSPIRE].
J. L Cardy, One-dimensional models with 1/r2 interactions, J. Phys. A 14 (1981) 1407.
N. Defenu, A. Trombettoni and A. Codello, Fixed-point structure and effective fractional dimensionality for O(N) models with long-range interactions, Phys. Rev. E 92 (2015) 052113 [arXiv:1409.8322] [INSPIRE].
J. Sak, Recursion Relations and Fixed Points for Ferromagnets with Long-Range Interactions, Phys. Rev. B 8 (1973) 281.
E. Luijten, Monte Carlo Simulation of Spin Models with Long-Range Interactions, Ph.D. Thesis, Technische Universiteit Delft (1997).
N. Defenu, A. Codello, S. Ruffo and A. Trombettoni, Criticality of spin systems with weak long-range interactions, J. Phys. A 53 (2020) 143001 [arXiv:1908.05158] [INSPIRE].
G. Giachetti, A. Trombettoni, S. Ruffo and N. Defenu, Berezinskii-Kosterlitz-Thouless transitions in classical and quantum long-range systems, Phys. Rev. B 106 (2022) 014106 [arXiv:2201.03650] [INSPIRE].
G. Giachetti, N. Defenu, S. Ruffo and A. Trombettoni, Self-consistent harmonic approximation in presence of non-local couplings, EPL 133 (2021) 57004 [arXiv:2012.14896] [INSPIRE].
M. Chiara Angelini, G. Parisi and F. Ricci-Tersenghi, Relations between short-range and long-range Ising models, Phys. Rev. E 89 (2014) 062120.
M. Kac, G.E. Uhlenbeck and P.C. Hemmer, On the van der Waals Theory of the Vapor-Liquid Equilibrium. I. Discussion of a One-Dimensional Model, J. Math. Phys. 4 (1963) 216.
P. Minnhagen, The two-dimensional Coulomb gas, vortex unbinding, and superfluid-superconducting films, Rev. Mod. Phys. 59 (1987) 1001 [INSPIRE].
A.O. Gogolin, A.A. Nersesian and A.M. Tsvelik, Bosonization and strongly correlated systems, Cambridge University Press (2004) [INSPIRE].
T. Giamarchi, Quantum Physics in One-Dimension, Clarendon Press (2004).
G. Bighin et al., Berezinskii-Kosterlitz-Thouless Paired Phase in Coupled XY Models, Phys. Rev. Lett. 123 (2019) 100601 [arXiv:1907.06253] [INSPIRE].
P. G. Maier and F. Schwabl, Ferromagnetic ordering in the two-dimensional dipolar XY model, Phys. Rev. B 70 (2004) 134430.
A.Y. Vasiliev et al., Universality of the Berezinskii-Kosterlitz-Thouless type of phase transition in the dipolar XY-model, New J. Phys. 16 (2014) 053011 [arXiv:1303.4915].
L. Jacobs and R. Savit, Self-duality and the logarithmic gas in three dimensions, Annals N. Y. Acad. Sci. 410 (1983) 281.
N. Defenu, A. Trombettoni and D. Zappalà, Topological phase transitions in four dimensions, Nucl. Phys. B 964 (2021) 115295 [arXiv:2003.04909] [INSPIRE].
S. R. Shenoy and B. Chattopadhyay, Anisotropic three-dimensional xy model and vortex-loop scaling, Phys. Rev. B 51 (1995) 9129.
M. Ibáñez Berganza and L. Leuzzi, Critical behavior of the xy model in complex topologies, Phys. Rev. B 88 (2013) 144104.
F. Cescatti, M. Ibáñez-Berganza, A. Vezzani and R. Burioni, Analysis of the low-temperature phase in the two-dimensional long-range diluted XY model, Phys. Rev. B 100 (2019) 054203 [arXiv:1905.06688] [INSPIRE].
J. Fröhlich and T. Spencer, Massless phases and symmetry restoration in abelian gauge theories and spin systems, Commun. Math. Phys. 83 (1982) 411 [INSPIRE].
P. W. Anderson, G. Yuval and D. R. Hamann, Exact Results in the Kondo Problem. II. Scaling Theory, Qualitatively Correct Solution, and Some New Results on One-Dimensional Classical Statistical Models, Phys. Rev. B 1 (1970) 4464.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2209.11810
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.
About this article
Cite this article
Giachetti, G., Defenu, N., Ruffo, S. et al. Villain model with long-range couplings. J. High Energ. Phys. 2023, 238 (2023). https://doi.org/10.1007/JHEP02(2023)238
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2023)238