American Journal of Modern Physics

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Strongly Coupled Fermions in Odd Dimensions and the Running Cut-off Λd

Received: 30 October 2023    Accepted: 23 November 2023    Published: 11 January 2024
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Abstract

One may observe that the fermionic U(N) Gross-Neveu model at imaginary chemical potential and finite temperature for odd d dimensions, in the strong coupling regime, by using the gap (saddle point) equation for the fermion condensate of the model. This equation describes the phase transitions from weak to strong coupling regime. It is pointed out that the higher odd dimensional gap equations are linear combinations of the lower dimensional equations in a way that as the dimension of the model increases the lower dimensions are weaker coupled but still in the strong coupling regime. Interestingly, at a specific value of the chemical potential, exactly in the middle of the thermal windows that separate the fermionic from the bosonic (condensed) state of the fermions, it is found that the mass of the fermion condensate for d = 3, 5, 7, 9. An anomaly occurs at the 5 dimensional theory where it is stronger coupled against other theories in higher dimensions and lower energy. The main idea of this work is that the cut-off Λ regulator for the UV divergent parts of the fermion mass saddle point equation, plays the role of a physical parameter that makes the separation of the odd dimensional fermionic theories according to how deep they are in the strong coupling regime. This idea is based on the identity of the asymptotic freedom of the Gross-Neveu model as a toy model for QCD.

DOI 10.11648/j.ajmp.20241301.11
Published in American Journal of Modern Physics (Volume 13, Issue 1, February 2024)
Page(s) 1-11
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Gross-Neveu, Strong Coupling, Cut-off

References
[1] E. H. Fradkin and F. A. Schaposnik, The Fermion - boson mapping in three-dimensional quantum field theory, Phys. Lett. B338 (1994) 253–258, [http://arxiv.org/abs/hep-th/9407182].
[2] F. Wilczek, Magnetic Flux, Angular Momentum, and Statistics, Phys. Rev. Lett. 48 (1982) 1144–1146.
[3] A. M. Polyakov, Fermi-Bose Transmutations Induced by Gauge Fields, Mod. Phys. Lett. A3 (1988) 325.
[4] A. Karch and D. Tong, Particle-Vortex Duality from 3d Bosonization, Phys. Rev. X6 (2016), no. 3 031043, http://arxiv.org/abs/1606.01893].
[5] J. Murugan and H. Nastase, Particle-vortex duality in topological insulators and superconductors, JHEP 05 (2017) 159, [http://arxiv.org/abs/1606.01912].
[6] N. Seiberg, T. Senthil, C. Wang, and E. Witten, A Duality Web in 2+1 Dimensions and Condensed Matter Physics, Annals Phys. 374 (2016) 395–433, [http://arxiv.org/abs/1606.01989].
[7] S. Kachru, M. Mulligan, G. Torroba, and H. Wang, Bosonization and Mirror Symmetry, Phys. Rev. D94 (2016), no. 8 085009, [http://arxiv.org/abs/1608.05077].
[8] O. Türker, J. Van den Brink, T. Meng, FS. Nogueira, Bosonization in 2 + 1 dimensions via Chern-Simons bosonic particle-vortex duality, Phys. Rev. D102 (2020), 034506, [https://arxiv.org/abs/2004.10789].
[9] M. E. Peskin, Mandelstam’t Hooft Duality in Abelian Lattice Models, Annals Phys. 113 (1978) 122.
[10] S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S. R. Wadia, and X. Yin, Chern-Simons Theory with Vector Fermion Matter, Eur. Phys. J. C72 (2012) 2112, [http://arxiv.org/abs/1110.4386].
[11] O. Aharony, S. Giombi, G. Gur-Ari, J. Maldacena, and R. Yacoby, The Thermal Free Energy in Large N Chern-Simons-Matter Theories, JHEP 03 (2013) 121, [http://arxiv.org/abs/1211.4843].
[12] L. Alvarez-Gaume and D. Orlando and S. Reffert, Large charge at large N, Journal of High Energy Physics 12 (2019), [https://doi.org/10.1007/JHEP12(2019)142].
[13] S. Bellucci, E.R. Bezerra de Mello, and A.A. Saharian, Finite temperature fermionic condensate and currents in topologically nontrivial spaces, Phys. Rev. D89 (2014) 085002, [https://doi.org/10.1103/PhysRevD.89.085002].
[14] Justin R. David, Srijan Kumar, Thermal one-point functions: CFT’s with fermions, large d and large spin, [https://doi.org/10.48550/arXiv.2307.14847].
[15] Genolini, P.B., Tizzano, L. Instantons, symmetries and anomalies in five dimensions, J. High Energ. Phys. 2021, 188 (2021). https://doi.org/10.1007/JHEP04(2021)188
[16] Gaiotto, D., Kapustin, A., Komargodski, Z. et al. Theta, time reversal and temperature., J. High Energ. Phys. 2017, 91 (2017). https://doi.org/10.1007/JHEP05(2017)091.
[17] E. G. Filothodoros, A. C. Petkou, and N. D. Vlachos, 3d fermion-boson map with imaginary chemical potential, Phys. Rev. D95 (2017), no. 6 065029, [http://arxiv.org/abs/1608.07795].
[18] E. G. Filothodoros, Anastasios C. Petkou, Nicholas D. Vlachos, The fermion-boson map for large d, Nuclear Physics B 941 (2019) Pages 195-224, [http://arxiv.org/abs/1803.05950].
[19] E. G. Filothodoros, The fermion-boson map at imaginary chemical potential in odd dimensions, [http://ikee.lib.auth.gr/record/303052/files/GRI-2019- 23684.pdf].
[20] M. Barkeshli and J. McGreevy, Continuous transition between fractional quantum Hall and superfluid states, Phys. Rev. B89 (2014), no. 23 235116.
[21] D. Zagier, The dilogarithm function, In Frontiers in Number Theory, Physics and Geometry II, (2006, P. Cartier, B. Julia, P. Moussa, P. Vanhove (eds.), Springer- Verlag, Berlin-Heidelberg-New York) 3–65.
[22] E. G. Filothodoros, The fermion boson map for large d and its connection to lattice transformations, [https://doi.org/10.48550/arXiv.2302.07013].
[23] D. Zagier, The Bloch-Wigner-Ramakrishnan polylogarithm function, Math. Ann. 286 (1990) (1990), no. 1-3 613–624.
[24] C. D. Fosco, G. L. Rossini, and F. A. Schaposnik, Induced parity breaking term in arbitrary odd dimensions at finite temperature, Phys. Rev. D59 (1999) 085012, [http://arxiv.org/abs/hep- th/9810199hep-th/9810199].
[25] R. J. Etienne, On the Zeros and Extrema of Generalised Clausen Functions , Lecture Notes of TICMI vol. 22 (2021), 91-113, [http://www.viam.science.tsu.ge/others/ticmi/lnt/vol22/6- R. J. Etienne].
[26] A. C. Petkou and M. B. Silva Neto, On the free energy of three-dimensional CFTs and polylogarithms, Phys. Lett. B456 (1999) 147–154, [http://arxiv.org/abs/hep- th/9812166].
[27] H. R. Christiansen, A. C. Petkou, M. B. Silva Neto, and N. D. Vlachos, On the thermodynamics of the (2+1)-dimensional Gross-Neveu model with complex chemical potential, Phys. Rev. D62 (2000) 025018, [http://arxiv.org/abs/hep-th/9911177].
[28] S. Sachdev, Polylogarithm identities in a conformal field theoryinthree-dimensions, Phys. Lett. B309(1993)285– 288, [http://arxiv.org/abs/hep-th/9305131].
[29] A. D’Adda, M. Luscher, and P. D. Vecchia, A 1/n expandable series of nonlinear sigma models with instantons, Nucl. Phys. B146 (1978) 63-76.
[30] J. M. Borwein and A. Straub, Relations for Nielsen polylogarithms, J. Approx. Theor. 193 (2015) 74–88.
[31] S.K.Kolbig, Nielsen’sgeneralizedpolylogarithms, SIAM J. Math. Anal. 17 (1986) 1232–1258.
[32] J. Zinn-Justin, Quantum field theory and critical phenomena, Int. Ser. Monogr. Phys. 113 (2002).
[33] OEIS Foundation Inc. (2023), Entry A363503 in The On-Line Encyclopedia of Integer Sequences, https://oeis.org/A363503.
[34] OEIS Foundation Inc. (2023), Entry A298338 in The On-Line Encyclopedia of Integer Sequences, https://oeis.org/A298338.
[35] Daniel F. Litim, Optimisation of the exact renormalisation group, Physics Letters B 486 (2000), [https://doi.org/10.1016/S0370-2693(00)00748-6].
[36] Chu, Chong-Sun and Miao, Rong-Xin, Fermion condensation induced by the Weyl anomaly, PhysRev. D, 102.046011, [https://link.aps.org/doi/10.1103/ PhysRevD.102.046011].
[37] E. Epelbaum, W. Glöckle, Ulf-G. Meißner, Improving the convergence of the chiral expansion for nuclear forces II: low phases and the deuteron, The European Physical Journal A - Hadrons and Nuclei 19, 401-412 (2004), [https://doi.org/10.1140/epja/i2003-10129-8].
[38] Yafis Barlas, T. Pereg-Barnea, Marco Polini, Reza Asgari, and A.H. MacDonald, Chiralityand Correlations in Graphene, Phys. Rev. Lett. 98, 236601, [https://doi.org/10.1103/PhysRevLett.98.236601].
[39] Yong Tao, BCS quantum critical phenomena, Europhysics Letters 118, 5, [https://dx.doi.org/10.1209/ 0295-5075/118/57007].
[40] Nick Evans, Keun-Young Kim, Maria Magou, Holographic Wilsonian Renormalization and Chiral Phase Transitions, Phys. Rev., D84, 126016, [https://doi.org/10.1103/PhysRevD.84.126016].
[41] Saso Grozdanov, Wilsonian renormalisation and the exact cut-off scale from holographic duality, Journal of High Energy Physics, volume 2012, Article number: 79 (2012), [https://doi.org/10.1007/JHEP06%282012%29079].
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    Filothodoros, E. G. (2024). Strongly Coupled Fermions in Odd Dimensions and the Running Cut-off Λd. American Journal of Modern Physics, 13(1), 1-11. https://doi.org/10.11648/j.ajmp.20241301.11

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    Filothodoros, E. G. Strongly Coupled Fermions in Odd Dimensions and the Running Cut-off Λd. Am. J. Mod. Phys. 2024, 13(1), 1-11. doi: 10.11648/j.ajmp.20241301.11

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    AMA Style

    Filothodoros EG. Strongly Coupled Fermions in Odd Dimensions and the Running Cut-off Λd. Am J Mod Phys. 2024;13(1):1-11. doi: 10.11648/j.ajmp.20241301.11

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  • @article{10.11648/j.ajmp.20241301.11,
      author = {Evangelos Georgiou Filothodoros},
      title = {Strongly Coupled Fermions in Odd Dimensions and the Running Cut-off Λd},
      journal = {American Journal of Modern Physics},
      volume = {13},
      number = {1},
      pages = {1-11},
      doi = {10.11648/j.ajmp.20241301.11},
      url = {https://doi.org/10.11648/j.ajmp.20241301.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20241301.11},
      abstract = {One may observe that the fermionic U(N) Gross-Neveu model at imaginary chemical potential and finite temperature for odd d dimensions, in the strong coupling regime, by using the gap (saddle point) equation for the fermion condensate of the model. This equation describes the phase transitions from weak to strong coupling regime. It is pointed out that the higher odd dimensional gap equations are linear combinations of the lower dimensional equations in a way that as the dimension of the model increases the lower dimensions are weaker coupled but still in the strong coupling regime. Interestingly, at a specific value of the chemical potential, exactly in the middle of the thermal windows that separate the fermionic from the bosonic (condensed) state of the fermions, it is found that the mass of the fermion condensate for d = 3, 5, 7, 9. An anomaly occurs at the 5 dimensional theory where it is stronger coupled against other theories in higher dimensions and lower energy. The main idea of this work is that the cut-off Λ regulator for the UV divergent parts of the fermion mass saddle point equation, plays the role of a physical parameter that makes the separation of the odd dimensional fermionic theories according to how deep they are in the strong coupling regime. This idea is based on the identity of the asymptotic freedom of the Gross-Neveu model as a toy model for QCD.},
     year = {2024}
    }
    

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    Y1  - 2024/01/11
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    T2  - American Journal of Modern Physics
    JF  - American Journal of Modern Physics
    JO  - American Journal of Modern Physics
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    UR  - https://doi.org/10.11648/j.ajmp.20241301.11
    AB  - One may observe that the fermionic U(N) Gross-Neveu model at imaginary chemical potential and finite temperature for odd d dimensions, in the strong coupling regime, by using the gap (saddle point) equation for the fermion condensate of the model. This equation describes the phase transitions from weak to strong coupling regime. It is pointed out that the higher odd dimensional gap equations are linear combinations of the lower dimensional equations in a way that as the dimension of the model increases the lower dimensions are weaker coupled but still in the strong coupling regime. Interestingly, at a specific value of the chemical potential, exactly in the middle of the thermal windows that separate the fermionic from the bosonic (condensed) state of the fermions, it is found that the mass of the fermion condensate for d = 3, 5, 7, 9. An anomaly occurs at the 5 dimensional theory where it is stronger coupled against other theories in higher dimensions and lower energy. The main idea of this work is that the cut-off Λ regulator for the UV divergent parts of the fermion mass saddle point equation, plays the role of a physical parameter that makes the separation of the odd dimensional fermionic theories according to how deep they are in the strong coupling regime. This idea is based on the identity of the asymptotic freedom of the Gross-Neveu model as a toy model for QCD.
    VL  - 13
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Author Information
  • Institute of Theoretical Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece

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