Abstract
In this paper we examine analytically the large-N gap equation and its solution for the 2D ℂℙN −1 sigma model defined on a Euclidean spacetime torus of arbitrary shape and size (L, β), β being the inverse temperature. We find that the system has a unique homogeneous phase, with the ℂℙN −1 fields ni acquiring a dynamically generated mass (λ) ≥ Λ2 (analogous to the mass gap of SU(N ) Yang-Mills theory in 4D), for any β and L. Several related topics in the recent literature are discussed. One concerns the possibility, which turns out to be excluded according to our analysis, of a “Higgs-like” — or deconfinement — phase at small L and at zero temperature. Another topics involves “soliton-like” (inhomogeneous) solutions of the generalized gap equation, which we do not find. A related question concerns a possible instability of the standard ℂℙN −1 vacuum on R2, which is shown not to occur. In all cases, the difference in the conclusions can be traced to the existence of certain zeromodes and their proper treatment. The ℂℙN −1 model with twisted boundary conditions is also analyzed. The θ dependence and different limits involving N , β and L are briefly discussed.
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References
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].
E. Witten, Instantons, the Quark Model and the 1/n Expansion, Nucl. Phys.B 149 (1979) 285 [INSPIRE].
H. Eichenherr, SU(N ) Invariant Nonlinear σ-models, Nucl. Phys.B 146 (1978) 215 [Erratum ibid.B 155 (1979) 544] [INSPIRE].
V.L. Golo and A.M. Perelomov, Solution of the Duality Equations for the Two-Dimensional SU(N ) Invariant Chiral Model, Phys. Lett.B 79 (1978) 112 [INSPIRE].
V.A. Fateev, I.V. Frolov and A.S. Schwarz, Quantum Fluctuations of Instantons in Two-dimensional Nonlinear Theories, Sov. J. Nucl. Phys.30 (1979) 590 [INSPIRE].
B. Berg and M. Lüscher, Computation of Quantum Fluctuations Around Multi-Instanton Fields from Exact Green’s Functions: The C P n−1Case, Commun. Math. Phys.69 (1979) 57 [INSPIRE].
G. Munster, A Study of CP (N −1)Models on the Sphere Within the 1/n Expansion, Nucl. Phys.B 218 (1983) 1 [INSPIRE].
J.-L. Richard and A. Rouet, The CP 1Model on the Torus: Contribution of Instantons, Nucl. Phys.B 211 (1983) 447 [INSPIRE].
V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Two-Dimensional σ-models: Modeling Nonperturbative Effects of Quantum Chromodynamics, Phys. Rept.116 (1984) 103 [INSPIRE].
F. David, Instantons and Condensates in Two-dimensional CP (N −1)Models, Phys. Lett.B 138 (1984) 139 [INSPIRE].
M. Campostrini and P. Rossi, 1/N expansion of the topological susceptibility in the C P N −1models, Phys. Lett.B 272 (1991) 305 [INSPIRE].
M. Campostrini and P. Rossi, CP n−1models in the 1/N expansion, Phys. Rev.D 45 (1992) 618 [Erratum ibid.D 46 (1992) 2741] [INSPIRE].
E. Vicari, Monte Carlo simulation of lattice C P N −1models at large N , Phys. Lett.B 309 (1993) 139 [hep-lat/9209025] [INSPIRE].
S.I. Hong and J.K. Kim, Finite temperature Neel transition in the C P N −1model with one periodic spatial dimension, J. Phys.A 27 (1994) 1557 [INSPIRE].
P. Rossi, Effective Lagrangian of C P N −1models in the large N limit, Phys. Rev.D 94 (2016) 045013 [arXiv:1606.07252] [INSPIRE].
M. Hasenbusch, Fighting topological freezing in the two-dimensional C P N −1model, Phys. Rev.D 96 (2017) 054504 [arXiv:1706.04443] [INSPIRE].
A. Laio, G. Martinelli and F. Sanfilippo, Metadynamics surfing on topology barriers: the C P N −1case, JHEP07 (2016) 089 [arXiv:1508.07270] [INSPIRE].
T. Rindlisbacher and P. de Forcrand, Worm algorithm for the CP N −1model, Nucl. Phys.B 918 (2017) 178 [arXiv:1610.01435] [INSPIRE].
A. Flachi, M. Nitta, S. Takada and R. Yoshii, Casimir force for the ℂP N −1model, Phys. Lett.B 798 (2019) 134999 [arXiv:1708.08807] [INSPIRE].
Y. Abe, K. Fukushima, Y. Hidaka, H. Matsueda, K. Murase and S. Sasaki, Image-processing the topological charge density in the ℂP N −1model, arXiv:1805.11058 [INSPIRE].
C. Bonanno, C. Bonati and M. D’Elia, Topological properties of C P N −1models in the large-N limit, JHEP01 (2019) 003 [arXiv:1807.11357] [INSPIRE].
I. Affleck, The Quantum Hall Effect, σ Models at θ = π and Quantum Spin Chains, Nucl. Phys.B 257 (1985) 397 [INSPIRE].
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].
Z.F. Ezawa, Spin-Pseudospin Coherence and C P 3Skyrmions in Bilayer Quantum Hall Ferromagnets, Phys. Rev. Lett.82 (1999) 3512 [cond-mat/9812188] [INSPIRE].
D.P. Arovas, A. Karlhede and D. Lilliehook, SU(N ) quantum Hall skyrmions, Phys. Rev.B 59 (1999) 13147.
R. Rajaraman, CP Nsolitons in quantum Hall systems, Eur. Phys. J.B 29 (2002) 157 [cond-mat/0112491] [INSPIRE].
A. Hanany and D. Tong, Vortices, instantons and branes, JHEP07 (2003) 037 [hep-th/0306150] [INSPIRE].
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].
M. Shifman and A. Yung, NonAbelian string junctions as confined monopoles, Phys. Rev.D 70 (2004) 045004 [hep-th/0403149] [INSPIRE].
A. Gorsky, M. Shifman and A. Yung, Non-Abelian Meissner effect in Yang-Mills theories at weak coupling, Phys. Rev.D 71 (2005) 045010 [hep-th/0412082] [INSPIRE].
S.B. Gudnason, Y. Jiang and K. Konishi, Non-Abelian vortex dynamics: Effective world-sheet action, JHEP08 (2010) 012 [arXiv:1007.2116] [INSPIRE].
S. Monin, M. Shifman and A. Yung, Non-Abelian String of a Finite Length, Phys. Rev.D 92 (2015) 025011 [arXiv:1505.07797] [INSPIRE].
A. Milekhin, C P N −1model on finite interval in the large N limit, Phys. Rev.D 86 (2012) 105002 [arXiv:1207.0417] [INSPIRE].
S. Bolognesi, K. Konishi and K. Ohashi, Large-N ℂN 1σ-model on a finite interval, JHEP10 (2016) 073 [arXiv:1604.05630] [INSPIRE].
A. Milekhin, C P Nσ-model on a finite interval revisited, Phys. Rev.D 95 (2017) 085021 [arXiv:1612.02075] [INSPIRE].
D. Pavshinkin, Grassmannian σ-model on a finite interval, Phys. Rev.D 97 (2018) 025001 [arXiv:1708.06399] [INSPIRE].
I. Ichinose and H. Yamamoto, Finite Temperature C P N −1Model and Long Range Néel Order, Mod. Phys. Lett.A 5 (1990) 1373 [INSPIRE].
A. Gorsky, A. Pikalov and A. Vainshtein, On instability of ground states in 2D ℂℙN −1and ONmodels at large N , arXiv:1811.05449 [INSPIRE].
M. Nitta and R. Yoshii, Self-consistent large-N analytical solutions of inhomogeneous condensates in quantum CP N −1model, JHEP12 (2017) 145 [arXiv:1707.03207] [INSPIRE].
M. Schechter, Operator Methods in Quantum Mechanics, Dover Books on Physics, Dover Publications, Mineola U.S.A. (2003).
G. Basar and G.V. Dunne, Self-consistent crystalline condensate in chiral Gross-Neveu and Bogoliubov-de Gennes systems, Phys. Rev. Lett.100 (2008) 200404 [arXiv:0803.1501] [INSPIRE].
G. Basar and G.V. Dunne, A Twisted Kink Crystal in the Chiral Gross-Neveu model, Phys. Rev.D 78 (2008) 065022 [arXiv:0806.2659] [INSPIRE].
M. Nitta and R. Yoshii, Confining solitons in the Higgs phase of ℂP N −1model: Self-consistent exact solutions in large-N limit, JHEP08 (2018) 007 [arXiv:1803.03009] [INSPIRE].
A. Betti, S. Bolognesi, S.B. Gudnason, K. Konishi and K. Ohashi, Large-N ℂℙN −1σ-model on a finite interval and the renormalized string energy, JHEP01 (2018) 106 [arXiv:1708.08805] [INSPIRE].
S. Bolognesi, S.B. Gudnason, K. Konishi and K. Ohashi, Large-N ℂℙN −1σ-model on a finite interval: general Dirichlet boundary conditions, JHEP06 (2018) 064 [arXiv:1802.08543] [INSPIRE].
M. Nitta and R. Yoshii, Self-consistent analytic solutions in twisted CP N −1model in the large-N limit, JHEP09 (2018) 092 [arXiv:1801.09861] [INSPIRE].
G.V. Dunne and M. Ünsal, Resurgence and Trans-series in Quantum Field Theory: The C P N −1Model, JHEP11 (2012) 170 [arXiv:1210.2423] [INSPIRE].
T. Sulejmanpasic, Global Symmetries, Volume Independence and Continuity in Quantum Field Theories, Phys. Rev. Lett.118 (2017) 011601 [arXiv:1610.04009] [INSPIRE].
T. Eguchi and H. Kawai, Reduction of Dynamical Degrees of Freedom in the Large N Gauge Theory, Phys. Rev. Lett.48 (1982) 1063 [INSPIRE].
I. Affleck, The Role of Instantons in Scale Invariant Gauge Theories, Nucl. Phys.B 162 (1980) 461 [INSPIRE].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal and Temperature, JHEP05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
M. Aguado and M. Asorey, Theta-vacuum and large N limit in C P N −1σ-models, Nucl. Phys.B 844 (2011) 243 [arXiv:1009.2629] [INSPIRE].
N. Dorey, The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms, JHEP11 (1998) 005 [hep-th/9806056] [INSPIRE].
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Bolognesi, S., Gudnason, S.B., Konishi, K. et al. Large-N ℂℙN −1 sigma model on a Euclidean torus: uniqueness and stability of the vacuum. J. High Energ. Phys. 2019, 44 (2019). https://doi.org/10.1007/JHEP12(2019)044
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DOI: https://doi.org/10.1007/JHEP12(2019)044