Semi-vortices and cluster-vorticity: new concepts in the Berezinskii-Kosterlitz-Thouless phase transition
DOI:
https://doi.org/10.31349/SuplRevMexFis.3.020724Keywords:
2d XY model, essential phase transition, vortices, cluster algorithmAbstract
The Berezinski˘ı-Kosterlitz-Thouless (BKT) essential phase transition in the 2d XY model is revisited. Its mechanism is usually described by the (un)binding of vortex–anti-vortex (V–AV) pairs, which does, however, not provide a clear-cut quantitative criterion for criticality. Known sharp criteria are the divergence of the correlation length and a discontinuity of the helicity modulus. Here we propose and probe a new criterion: it is based on the concepts of semi-vortices and cluster vorticity, which are formulated in the framework of the multi-cluster algorithm that we use to simulate the 2d XY model
References
V. L. Berezinski˘ı, Destruction of long range order in onedimensional and two-dimensional systems having a continuous symmetry group. I. Classical systems, Zh. Eksp. Teor. Fiz. 59 (1970) 907–920 [Sov. Phys. JETP 32 (1971) 493]; Destruction of Long-range Order in One-dimensional and Two-dimensional Systems Possessing a Continuous Symmetry Group. II. Quantum Systems, Zh. Eksp. Teor. Fiz. 61 (1971) 1144–1156 [Sov. Phys. JETP 34 (1972) 610].
J. M. Kosterlitz and D. J. Thouless, Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory), J. Phys. C 5 (1972) L124, https://doi.org/10.1088/0022-3719/5/11/002; Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6 (1973) 1181, https://doi.org/10.1088/0022-3719/6/7/010; The critical properties of the two-dimensional xy model, J. Phys. C 7 (1974) 1046, https://doi.org/10.1088/0022-3719/7/6/005.
J. V. Jose (editor), 40 Years of Berezinskii-Kosterlitz-Thouless Theory, World Scientific, 2013, https://doi.org/10.1142/8572.
D. J. Bishop and J. D. Reppy, Study of the Superfluid Transition in Two-Dimensional 4He Films, Phys. Rev. Lett. 40 (1978) 1727, https://doi.org/10.1103/PhysRevLett.40.1727.
A. F. Hebard and A. T. Fiory, Evidence for the KosterlitzThouless Transition in Thin Superconducting Aluminum Films, Phys. Rev. Lett. 44 (1980) 291, https://doi.org/10.1103/PhysRevLett.44.291. [Erratum: Phys. Rev. Lett. 44 (1980) 620 https://doi.org/10.1103/PhysRevLett.44.620.2]. K. Epstein, A. M. Goldman and A. M. Kadin, Vortex-Antivortex Pair Dissociation in Two-Dimensional Superconductors, Phys. Rev. Lett. 47 (1981) 534, https://doi.org/10.1103/PhysRevLett.47.534. D. J. Resnick, J. C. Garland, J. T. Boyd, S. Shoemaker and R. S. Newrock, KosterlitzThouless Transition in Proximity-Coupled Superconducting Arrays, Phys. Rev. Lett. 47 (1981) 1542, https://doi.org/10.1103/PhysRevLett.47.1542.
Z. Hu et al., Evidence of the Berezinskii-KosterlitzThouless phase in a frustrated magnet, Nat. Commun. 11 (2020) 5631, https://doi.org/10.1038/s41467-020-19380-x.
J. M. Kosterlitz, The critical properties of the two-dimensional xy model, J. Phys. C 7 (1974) 1046, https://doi.org/10.1088/0022-3719/7/6/005.
M. Hasenbusch, The two dimensional XY model at the transition temperature: A high precision Monte Carlo study, J. Phys. A 38 (2005) 5869, https://doi.org/10.1088/0305-4470/38/26/003. Y. Komura and Y. Okabe, Largescale Monte Carlo simulation of two-dimensional classical XY model using multiple GPUs, J. Phys. Soc. Jpn. 81 (2012) 113001, https://journals.jps.jp/doi/10.1143/JPSJ.81.11300110.1143/JPSJ.81.113001. R. G. Jha, Critical analysis of two-dimensional classical XY model, J. Stat. Mech. (2020) 083203, https://doi.org/10.1088/1742-5468/aba686.
A. M. Polyakov, A View From the Island, in The Rise of the Standard Model, Cambridge University Press (1997), L. Hoddeson, L. Brown, M. Riordan and M. Dresden (editors), p. 243- 249.
W. Bietenholz and U. Gerber, Berezinski˘ı-Kosterlitz-Thouless Transition and the Haldane Conjecture: Highlights of the Physics Nobel Prize 2016, Rev. Cub. Fís. 33 (2016) 156.
W. Bietenholz, U. Gerber, M. Pepe and U.-J. Wiese, Topological Lattice Actions, JHEP 12 (2010) 020, https://doi.org/10.1007/JHEP12(2010)020.
W. Bietenholz, M. Bögli, F. Niedermayer, M. Pepe, F. G. Rejóon-Barrera and U.-J. Wiese, Topological Lattice Actions for the 2d XY Model, JHEP 03 (2013) 141, https://doi.org/10.1007/JHEP03(2013)141.
J. Balog, F. Knechtli, T. Korzec and U. Wolff, Numerical confirmation of analytic predictions for the finite volume mass gap of the XY model, Nucl. Phys. B 675 (2003) 555, https://doi.org/10.1016/j.nuclphysb.2003.10.010.
W. Bietenholz, U. Gerber and F. G. Rejón-Barrera, Berezinskii-Kosterlitz-Thouless transition with a constraint lattice action, J. Stat. Mech. 1312 (2013) P12009, https://doi.org/10. 1088/1742-5468/2013/12/P12009.
B. S. Gómez Bravo, El papel de la vorticidad en la transición de fase de Berezinski˘ı-Kosterlitz-Thouless, B.Sc. thesis, Universidad Nacional Autónoma de México, 2021. B. D. Juárez Hernández, Escalamiento del tamaño de los clusters en el modelo XY clásico, B.Sc. thesis, Universidad Nacional Autónoma de México, in preparation.
D. R. Nelson and J. M. Kosterlitz, Universal Jump in the Superfluid Density of Two-Dimensional Superfluids, Phys. Rev. Lett. 39 (1977) 1201, https://doi.org/10.1103/PhysRevLett.39.1201. P. Minnhagen and G. G. Warren, Superfluid density of a two-dimensional fluid, Phys. Rev. B 24 (1977) 2526, https://doi.org/10.1103/PhysRevB.24.2526. J. V. José, L. P. Kadanoff, S. Kirkpatrick and D. R. Nelson, Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model, Phys. Rev. B 16 1217, https://doi.org/10.1103/PhysRevB.16.1217. [Erratum: Phys. Rev. B 17 (1978) 1477, https://doi.org/10.1103/PhysRevB.17.1477].
J. Noh, J. Lee and J. Mun, Observation of the universal jump across the Berezinskii-Kosterlitz-Thouless transition in twodimensional Bose gases, arXiv:1305.1423 [cond-mat.quantgas].
N. V. Prokof’ev and B. V. Svistunov, Two definitions of superfluid density, Phys. Rev. B 61 (2000) 11282, https://doi.org/10.1103/PhysRevB.61.11282.
U. Wolff, Collective Monte Carlo Updating for Spin Systems, Phys. Rev. Lett. 62 (1989) 361, https://doi.org/10.1103/PhysRevLett.62.36110.1103/PhysRevLett.62.361; Collective Monte Carlo updating in a high precision study of the xy model, Nucl. Phys. B 322 (1989) 759, https://doi.org/10.1016/0550-3213(89)90236-8.
W. Bietenholz, J. C. Pinto Barros, S. Caspar, M. Hornung and U.-J. Wiese, Meron- and Semi-Vortex-Clusters as Physical Carriers of Topological Charge and Vorticity, PoS LATTICE2019 (2019) 288, https://doi.org/10.22323/1.363.0288.
W. Bietenholz, A. Pochinsky and U.-J. Wiese, Meron-Cluster Simulation of the θ-Vacuum in the 2-d O(3)-Model, Phys. Rev. Lett. 75 (1995) 4524, https://doi.org/10.1103/PhysRevLett.75.4524.
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