Abstract
We solve a multitrace matrix model approximating the real quartic scalar field theory on the fuzzy sphere and obtain its phase diagram. We generalize this method to models with modified kinetic terms and demonstrate its use by investigating models related to the removal of the UV/IR mixing. We show that for the fuzzy sphere a modification of the kinetic part of the action by higher derivative term can change the phase diagram of the theory such that the triple point moves further from the origin.
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H.S. Snyder, Quantized space-time, Phys. Rev. 71 (1947) 38 [INSPIRE].
H. Grosse, C. Klimčík and P. Prešnajder, Towards finite quantum field theory in noncommutative geometry, Int. J. Theor. Phys. 35 (1996) 231 [hep-th/9505175] [INSPIRE].
D. Karabali and V.P. Nair, Quantum Hall effect in higher dimensions, matrix models and fuzzy geometry, J. Phys. A 39 (2006) 12735.
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
C. Sochichiu, Matrix models, Lect. Notes Phys. 698 (2006) 189 [hep-th/0506186] [INSPIRE].
H.C. Steinacker, On the quantum structure of space-time, gravity and higher spin in matrix models, Class. Quant. Grav. 37 (2020) 113001 [arXiv:1911.03162] [INSPIRE].
H. Steinacker, Non-commutative geometry and matrix models, PoS(QGQGS2011)004 [arXiv:1109.5521] [INSPIRE].
S. Doplicher, K. Fredenhagen and J.E. Roberts, The quantum structure of spacetime at the Planck scale and quantum fields, Comm. Math. Phys. 172 (1995) 187.
S. Kováčik and D. O’Connor, Triple point of a scalar field theory on a fuzzy sphere, JHEP 10 (2018) 010 [arXiv:1805.08111] [INSPIRE].
F. Garcia Flores, X. Martin and D. O’Connor, Simulation of a scalar field on a fuzzy sphere, Int. J. Mod. Phys. A 24 (2009) 3917 [arXiv:0903.1986] [INSPIRE].
B. Ydri, New algorithm and phase diagram of noncommutative 𝜙4 on the fuzzy sphere, JHEP 03 (2014) 065 [arXiv:1401.1529] [INSPIRE].
F. Lizzi and B. Spisso, Noncommutative field theory: numerical analysis with the fuzzy disc, Int. J. Mod. Phys. A 27 (2012) 1250137 [arXiv:1207.4998] [INSPIRE].
J. Medina, W. Bietenholz and D. O’Connor, Probing the fuzzy sphere regularisation in simulations of the 3d lambda 𝜙 4 model, JHEP 04 (2008) 041 [arXiv:0712.3366] [INSPIRE].
H. Mejía-Díaz, W. Bietenholz and M. Panero, The continuum phase diagram of the 2d non-commutative λ𝜙 4 model, JHEP 10 (2014) 056 [arXiv:1403.3318] [INSPIRE].
D. Prekrat, K.N. Todorović-Vasović and D. Ranković, Detecting scaling in phase transitions on the truncated Heisenberg algebra, arXiv:2002.05704 [INSPIRE].
M. Panero, The numerical approach to quantum field theory in a non-commutative space, PoS(CORFU2015)099 [arXiv:1601.01176] [INSPIRE].
S.S. Gubser and S.L. Sondhi, Phase structure of noncommutative scalar field theories, Nucl. Phys. B 605 (2001) 395 [hep-th/0006119] [INSPIRE].
K. Hatakeyama, A. Tsuchiya and K. Yamashiro, Renormalization on the fuzzy sphere, PTEP 2018 (2018) 063B05 [arXiv:1805.03975] [INSPIRE].
K. Hatakeyama, A. Tsuchiya and K. Yamashiro, Renormalization on the fuzzy sphere, PoS(LATTICE2018)045 [arXiv:1811.10806] [INSPIRE].
J.L. Karczmarek and P. Sabella-Garnier, Entanglement entropy on the fuzzy sphere, JHEP 03 (2014) 129 [arXiv:1310.8345] [INSPIRE].
S. Okuno, M. Suzuki and A. Tsuchiya, Entanglement entropy in scalar field theory on the fuzzy sphere, PTEP 2016 (2016) 023B03 [arXiv:1512.06484] [INSPIRE].
H.Z. Chen and J.L. Karczmarek, Entanglement entropy on a fuzzy sphere with a UV cutoff, JHEP 08 (2018) 154 [arXiv:1712.09464] [INSPIRE].
D. O’Connor and C. Sämann, Fuzzy scalar field theory as a multitrace matrix model, JHEP 08 (2007) 066 [arXiv:0706.2493] [INSPIRE].
C. Sämann, The multitrace matrix model of scalar field theory on fuzzy CPn , SIGMA 6 (2010) 050 [arXiv:1003.4683] [INSPIRE].
C. Sämann, Bootstrapping fuzzy scalar field theory, JHEP 04 (2015) 044 [arXiv:1412.6255] [INSPIRE].
M. Ihl, C. Sachse and C. Sämann, Fuzzy scalar field theory as matrix quantum mechanics, JHEP 03 (2011) 091 [arXiv:1012.3568] [INSPIRE].
S. Rea and C. Sämann, The phase diagram of scalar field theory on the fuzzy disc, JHEP 11 (2015) 115 [arXiv:1507.05978] [INSPIRE].
A.P. Polychronakos, Effective action and phase transitions of scalar field on the fuzzy sphere, Phys. Rev. D 88 (2013) 065010 [arXiv:1306.6645] [INSPIRE].
H. Steinacker, A non-perturbative approach to non-commutative scalar field theory, JHEP 03 (2005) 075 [hep-th/0501174] [INSPIRE].
V.P. Nair, A.P. Polychronakos and J. Tekel, Fuzzy spaces and new random matrix ensembles, Phys. Rev. D 85 (2012) 045021 [arXiv:1109.3349].
J. Tekel, Asymmetric hermitian matrix models and fuzzy field theory, Phys. Rev. D 97 (2018) 125018 [arXiv:1711.02008] [INSPIRE].
A.P. Balachandran, S. Kurkcuoglu and S. Vaidya, Lectures on fuzzy and fuzzy SUSY physics, hep-th/0511114 [INSPIRE].
B. Ydri, Lectures on matrix field theory, Lect. Notes Phys. 929 (2017) pp.1 [arXiv:1603.00924] [INSPIRE].
B. Eynard, T. Kimura and S. Ribault, Random matrices, arXiv:1510.04430 [INSPIRE].
J. Tekel, Phase strucutre of fuzzy field theories and multitrace matrix models, Acta Phys. Slov. 65 (2015) 369 [arXiv:1512.00689] [INSPIRE].
M. Šubjaková, J. Tekel, Fuzzy field theories and related matrix models, to appear in PoS(CORFU2019).
D. Anninos and B. Mühlmann, Notes on Matrix Models, arXiv:2004.01171 [INSPIRE].
H. Grosse, A. Hock and R. Wulkenhaar, Solution of all quartic matrix models, arXiv:1906.04600 [INSPIRE].
Y. Shimamune, On the phase structure of large N matrix models and gauge models, Phys. Lett. 108B (1982) 407 [INSPIRE].
J. Madore, The fuzzy sphere, Class. Quant. Grav. 9 (1992) 69 [INSPIRE].
S. Minwalla, M. Van Raamsdonk and N. Seiberg, Noncommutative perturbative dynamics, JHEP 02 (2000) 020 [hep-th/9912072] [INSPIRE].
C.S. Chu, J. Madore and H. Steinacker, Scaling limits of the fuzzy sphere at one loop, JHEP 03 (2005) 075.
B.P. Dolan, D. O’Connor and P. Prešnajder, Matrix 𝜙4 models on the fuzzy sphere and their continuum limits, JHEP 03 (2002) 013 [hep-th/0109084] [INSPIRE].
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Šubjaková, M., Tekel, J. Second moment fuzzy-field-theory-like matrix models. J. High Energ. Phys. 2020, 88 (2020). https://doi.org/10.1007/JHEP06(2020)088
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DOI: https://doi.org/10.1007/JHEP06(2020)088