Advertisement

Large-N ℂℙN − 1 sigma model on a finite interval: general Dirichlet boundary conditions

  • Stefano Bolognesi
  • Sven Bjarke Gudnason
  • Kenichi Konishi
  • Keisuke Ohashi
Open Access
Regular Article - Theoretical Physics
  • 32 Downloads

Abstract

This is the third of the series of articles on the large-N two-dimensional ℂℙN − 1 sigma model, defined on a finite space interval L with Dirichlet boundary conditions. Here the cases of the general Dirichlet boundary conditions are studied, where the relative ℂℙN − 1 orientations at the two boundaries are generic, and numerical solutions are presented. Distinctive features of the ℂℙN − 1 sigma model, as compared e.g., to an O(N) model, which were not entirely evident in the basic properties studied in the first two articles in the large N limit, manifest themselves here. It is found that the total energy is minimized when the fields are aligned in the same direction at the two boundaries.

Keywords

Effective Field Theories 1/N Expansion Nonperturbative Effects 

Notes

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.

References

  1. [1]
    A. D’Adda, M. Lüscher and P. Di Vecchia, A 1/n Expandable Series of Nonlinear σ-models with Instantons, Nucl. Phys. B 146 (1978) 63 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    E. Witten, Instantons, the Quark Model and the 1/n Expansion, Nucl. Phys. B 149 (1979) 285 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    I. Affleck, The Quantum Hall Effect, σ Models at θ = π and Quantum Spin Chains, Nucl. Phys. B 257 (1985) 397 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    S.L. Sondhi, A. Karlhede, S.A. Kivelson and E.H. Rezayi, Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies, Phys. Rev. B 47 (1993) 16419 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    Z.F. Ezawa, Spin-Pseudospin Coherence and CP 3 Skyrmions in Bilayer Quantum Hall Ferromagnets, Phys. Rev. Lett. 82 (1999) 3512 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    D.P. Arovas, A. Karlhede and D. Lilliehook, SU(N) quantum Hall skyrmions, Phys. Rev. B 59 (1999) 13147.ADSCrossRefGoogle Scholar
  7. [7]
    R. Rajaraman, CP N solitons in quantum Hall systems, Eur. Phys. J. B 29 (2002) 157 [cond-mat/0112491] [INSPIRE].
  8. [8]
    S. Bolognesi, K. Konishi and K. Ohashi, Large-NP N − 1 sigma model on a finite interval, JHEP 10 (2016) 073 [arXiv:1604.05630] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    A. Betti, S. Bolognesi, S.B. Gudnason, K. Konishi and K. Ohashi, Large-NP N − 1 sigma model on a finite interval and the renormalized string energy, JHEP 01 (2018) 106 [arXiv:1708.08805] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    R. Auzzi, S. Bolognesi, J. Evslin and K. Konishi, NonAbelian monopoles and the vortices that confine them, Nucl. Phys. B 686 (2004) 119 [hep-th/0312233] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  11. [11]
    K. Konishi, A. Michelini and K. Ohashi, Monopole-vortex complex in a theta vacuum, Phys. Rev. D 82 (2010) 125028 [arXiv:1009.2042] [INSPIRE].ADSGoogle Scholar
  12. [12]
    M. Cipriani, D. Dorigoni, S.B. Gudnason, K. Konishi and A. Michelini, Non-Abelian monopole-vortex complex, Phys. Rev. D 84 (2011) 045024 [arXiv:1106.4214] [INSPIRE].ADSGoogle Scholar
  13. [13]
    C. Chatterjee and K. Konishi, Monopole-vortex complex at large distances and nonAbelian duality, JHEP 09 (2014) 039 [arXiv:1406.5639] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    A. Hanany and D. Tong, Vortices, instantons and branes, JHEP 07 (2003) 037 [hep-th/0306150] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, NonAbelian superconductors: Vortices and confinement in N = 2 SQCD, Nucl. Phys. B 673 (2003) 187 [hep-th/0307287] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    M. Shifman and A. Yung, NonAbelian string junctions as confined monopoles, Phys. Rev. D 70 (2004) 045004 [hep-th/0403149] [INSPIRE].ADSGoogle Scholar
  17. [17]
    A. Milekhin, CP(N-1) model on finite interval in the large N limit, Phys. Rev. D 86 (2012) 105002 [arXiv:1207.0417] [INSPIRE].ADSGoogle Scholar
  18. [18]
    S. Monin, M. Shifman and A. Yung, Non-Abelian String of a Finite Length, Phys. Rev. D 92 (2015) 025011 [arXiv:1505.07797] [INSPIRE].ADSMathSciNetGoogle Scholar
  19. [19]
    A. Milekhin, CP N sigma model on a finite interval revisited, Phys. Rev. D 95 (2017) 085021 [arXiv:1612.02075] [INSPIRE].ADSGoogle Scholar
  20. [20]
    A. Flachi, M. Nitta, S. Takada and R. Yoshii, Sign Flip in the Casimir Force for Interacting Fermion Systems, Phys. Rev. Lett. 119 (2017) 031601 [arXiv:1704.04918] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    M. Nitta and R. Yoshii, Self-consistent large-N analytical solutions of inhomogeneous condensates in quantumP N − 1 model, JHEP 12 (2017) 145 [arXiv:1707.03207] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    A. Flachi, M. Nitta, S. Takada and R. Yoshii, Casimir Force for theP N − 1 Model, arXiv:1708.08807 [INSPIRE].
  23. [23]
    D. Pavshinkin, Grassmannian σ-model on a finite interval, Phys. Rev. D 97 (2018) 025001 [arXiv:1708.06399] [INSPIRE].ADSGoogle Scholar
  24. [24]
    M. Nitta and R. Yoshii, Self-consistent Analytic Solutions in TwistedP N − 1 Model in the Large-N Limit, arXiv:1801.09861 [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Stefano Bolognesi
    • 1
    • 2
  • Sven Bjarke Gudnason
    • 3
  • Kenichi Konishi
    • 1
    • 2
  • Keisuke Ohashi
    • 4
  1. 1.Department of Physics “E. Fermi”University of PisaPisaItaly
  2. 2.INFN, Sezione di PisaPisaItaly
  3. 3.Institute of Modern PhysicsChinese Academy of SciencesLanzhouChina
  4. 4.Research and Education Center for Natural SciencesKeio UniversityYokohamaJapan

Personalised recommendations