BKT TRANSITION IN PHYLLOTAXIS

Capa

Citar

Texto integral

Acesso aberto Acesso aberto
Acesso é fechado Acesso está concedido
Acesso é fechado Somente assinantes

Resumo

We explore a two-parameter renormalization group (RG) within the framework of the «energetic approach» introduced by L. Levitov, for the phyllotaxis model. Our focus lies on an equilibrium distribution of strongly repulsive particles situated on the surface of a finite cylinder. We investigate how these particles redistribute as the cylinder undergoes compression along its axis. Specifically, we construct the modular-invariant β-function for the system, which is explicitly expressed in terms of the Dedekind η-function. Utilizing this β-function, we derive equations that describe the RG flow near the bifurcation points, which mark the boundaries between different lattice configurations. By analyzing the structure of these RG equations, we assert the emergence of Berezinskii – Kosterlitz – Thouless transitions under significant cylinder compression.

Sobre autores

S. Nechaev

LPTMS, CNRS–Université Paris-Saclay

Autor responsável pela correspondência
Email: sergei.nechaev@gmail.com
Orsay Cedex, France

Bibliografia

  1. Livio, The Golden Ratio: The Story of φ,the World’s Most Astonishing Number, Broadway Books, New York (2008).
  2. Rothen and A.-J. Koch, Phyllotaxis, or the Properties of Spiral Lattices. -I. Shape Invariance Inder Compression, J. de Physique 50, 633 (1989).
  3. Rothen and A. J. Koch, Phyllotaxis or the Properties of Spiral Lattices. - II. Packing of Circles Along Logarithmic Spirals, J. de Physique 50, 1603 (1989).
  4. M. Kunz and F. Rothen, Phyllotaxis or the Properties of Spiral Lattices. III. an Algebraic Model of Morphogenesis, J. de Physique I 2, 2131 (1992).
  5. L. Levitov, Energetic Approach to Phyllotaxis, Europhysics Letters 14, 533 (1991).
  6. L. S. Levitov, Phyllotaxis of Flux Lattices in Layered Superconductors, Phys. Rev. Lett. 66, 224 (1991).
  7. S. Douady and Y. Couder, Phyllotaxis as a Physical Self-Organized Growth Process, Phys. Rev. Lett. 68, 2098 (1992).
  8. Nisoli, N. M. Gabor, P. E. Lammert, J. D. Maynard, and V. H. Crespi, Static and Dynamical Phyllotaxis in a Magnetic Cactus, Phys. Rev. Lett. 102, 186103 (2009).
  9. S. Gukov, RG Flows and Bifurcations, Nuclear Physics B 919, 583 (2017).
  10. B. Jepsen and F. K. Popov, Homoclinic Renormalization Group Flows, or When Relevant Operators Become Irrelevant, Phys. Rev. Lett. 127, 141602 (2021).
  11. M. M. Bosschaert, C. B. Jepsen, and F. K. Popov, Chaotic RG Flow in Tensor Models, Phys. Rev. D 105, 065021 (2022).
  12. M. Wilkinson, Critical Properties of Electron Eigenstates in Incommensurate Systems, Proc. Royal Soc. London A. Math. Phys. Sci. 391, 305 (1984).
  13. M. Wilkinson, An Exact Renormalisation Group for Bloch Electrons in a Magnetic Field, J. Phys. A: Math. Gen. 20, 4337 (1987).
  14. S. Aubry, Devil’s Staircase and Order Without Periodicity in Classical Condensed Matter, J. de Physique 44, 147 (1983).
  15. Bak, Commensurate Phases, Incommensurate Phases and the Devil’s Staircase, Rep. Progr. Phys. 45, 587 (1982).
  16. J. Bergholtz, T. H. Hansson, M. Hermanns, and A. Karlhede, Microscopic Theory of the Quantum Hall Hierarchy, Phys. Rev. Lett. 99, 256803 (2007).
  17. J. Bergholtz and A. Karlhede, Quantum Hall System in Tao-Thouless limit, Phys. Rev. B 77, 155308 (2008).
  18. D. Lundholm, Many-Anyon Trial States, Phys. Rev. A 96, 012116 (2017).
  19. M. Planat and C. Eckert, On the Frequency and Amplitude Spectrum and the Fluctuations at the Output of a Communication Receiver, IEEE Trans. Ultrasonics, Ferroelectrics, and Frequency Control 47, 1173 (2000).
  20. Trifonov, L. Pascualucci, R. Dalla-Favera, and R. Rabadan, Fractal-like Distributions over the Rational Numbers in Highthroughput Biological and Clinical Data, Sci. Rep. 1, 191 (2011).
  21. M. Middendorf, E. Ziv, and C. Wiggins, Inferring Network Mechanisms: The Drosophila Melanogaster Protein Interaction Network, PNAS 102, 3192 (2005).
  22. Altland, D. Bagrets, and A. Kamenev, Topology versus Anderson Localization: Nonperturbative Solutions in One Dimension, Phys. Rev. B 91, 085429 (2015).
  23. M. Pruisken, On Localization in the Theory of the Quantized Hall Effect: A Twodimensional Ralization of the θ-Vacuum, Nucl. Phys. B 235, 277 (1984).
  24. Levine, S. Libby, and A. Pruisken, Theory of the Quantum Hall Effect (i)-(iii), Nucl. Phys. B 240, 30 (1984).
  25. Montonen and D. Olive, Magnetic Monopoles as Gauge Particles?, Phys. Lett. B 72, 117 (1977).
  26. J. L. Cardy and E. Rabinovici, Phase Structure of ZP Models in the Presence of a θ-Parameter, Nucl. Phys. B 205, 1 (1982).
  27. K. Bulycheva and A. Gorsky, Limit Cycles in Renormalization Group Dynamics, Physics Uspekhi 57, 171 (2014).
  28. R. Musin and A. S. Tarasov, The Tammes problem for n = 14, Experimental Mathematics 24, 460 (2015).
  29. L. Altschuler, T. J. Williams, E. R. Ratner, R. Tipton, R. Stong, F. Dowla, and F. Wooten, Possible Global Minimum Lattice Cconfigurations for Thomson’s Problem of Charges on a Sphere, Phys. Rev. Lett. 78, 2681 (1997).
  30. Abrikosov, The Magnetic Properties of Superconducting Alloys, J. Phys. Chem. Solids 2, 199 (1957).
  31. Rammal, G. Toulouse, and M. A. Virasoro, Ultrametricity for Physicists, Rev. Mod. Phys. 58, 765 (1986).
  32. O’Sullivan, Formulas for Nonholomorphic Eisenstein Series and for the Riemann Zeta Function at Odd Integers, Research in Number Theory 4, 36 (2018).
  33. P. Ribeiro and S. Yakubovich, On the Epstein Zeta Function and the Zeros of a Class of Dirichlet Series (2022), arXiv:2112.10561 [math.NT]
  34. L. Siegel and S. Raghavan, Lectures on Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Mumbai, India (1961).
  35. Motohashi, A New Proof of the Limit Formula of Kronecker, Proc. Japan Academy 44, 614 (1968).
  36. T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory. Chapter 3, 41, SpringerVerlag (1990).
  37. B. Kaplan, J.-W. Lee, D. T. Son, and M. A. Stephanov, Conformality Lost, Phys. Rev. D 80, 125005 (2009).
  38. A. Lutken and G. G. Ross, Implications of Experimental Probes of the RG-Flow in Quantum Hall Systems (2009), arXiv:0906.5551 [condmat. other]
  39. Carpentier, Renormalization of Modular Invariant Coulomb Gas and Sine-Gordon Theories, and the Quantum Hall Flow Diagram, J. Phys. A: Math. Gen. 32, 3865 (1999).
  40. K. Fischer, Kosterlitz-Thouless Transition in Layered High-Tc Superconductors, Physica C: Superconductivity 210, 179 (1993).
  41. Flack, A. Gorsky, and S. Nechaev, Generalized Devil’s Staircase and RG Flows, Nucl. Phys. B 996, 116376 (2023).
  42. Cristiano Nisoli, Nathaniel M. Gabor, Paul E. Lammert, J. D. Maynard, and Vincent H. Crespi, Annealing a Magnetic Cactus into Phyllotaxis, Phys. Rev. E 81, 046107 (2010).
  43. B. Balagurov and V. Vaks, Random Walks of a Particle on Lattices with Traps, Sov. Phys. JETP 38, 968 (1974).
  44. M. Donsker, S. Varadhan, Large Deviations for Stationary Gaussian Processes, Comm. Math. Phys. 97, 187 (1985).

Declaração de direitos autorais © Russian Academy of Sciences, 2024

Este site utiliza cookies

Ao continuar usando nosso site, você concorda com o procedimento de cookies que mantêm o site funcionando normalmente.

Informação sobre cookies