×

Random matrix theory of unquenched two-colour QCD with nonzero chemical potential. (English) Zbl 1301.81282

Summary: We solve a random two-matrix model with two real asymmetric matrices whose primary purpose is to describe certain aspects of quantum chromo dynamics with two colours and dynamical fermions at nonzero quark chemical potential \(\mu\). In this symmetry class the determinant of the Dirac operator is real but not necessarily positive. Despite this sign problem the unquenched matrix model remains completely solvable and provides detailed predictions for the Dirac operator spectrum in two different physical scenarios/limits: (i) the \(\epsilon\)-regime of chiral perturbation theory at small \(\mu\), where \(\mu^{2}\) multiplied by the volume remains fixed in the infinite-volume limit and (ii) the high-density regime where a BCS gap is formed and \(\mu\) is unscaled. We give explicit examples for the complex, real, and imaginary eigenvalue densities including \(N_{f} = 2\) non-degenerate flavours. Whilst the limit of two degenerate masses has no sign problem and can be tested with standard lattice techniques, we analyse the severity of the sign problem for non-degenerate masses as a function of the mass split and of \(\mu\).{ }On the mathematical side our new results include an analytical formula for the spectral density of real Wishart eigenvalues in the limit (i) of weak non-Hermiticity, thus completing the previous solution of the corresponding quenched model of two real asymmetric Wishart matrices.

MSC:

81V05 Strong interaction, including quantum chromodynamics
81T25 Quantum field theory on lattices
81R40 Symmetry breaking in quantum theory
15B52 Random matrices (algebraic aspects)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
82B30 Statistical thermodynamics
81Q80 Special quantum systems, such as solvable systems
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Y. Aoki et al., The QCD transition temperature: results with physical masses in the continuum limit II, JHEP06 (2009) 088 [arXiv:0903.4155] [SPIRES]. · doi:10.1088/1126-6708/2009/06/088
[2] P. de Forcrand, Simulating QCD at finite density, PoS(LAT2009) 010 [arXiv:1005.0539] [SPIRES].
[3] M.G. Alford, A. Schmitt, K. Rajagopal and T. Schafer, Color superconductivity in dense quark matter, Rev. Mod. Phys.80 (2008) 1455 [arXiv:0709.4635] [SPIRES]. · doi:10.1103/RevModPhys.80.1455
[4] R. Rapp, T. Schafer, E.V. Shuryak and M. Velkovsky, High-density QCD and instantons, Annals Phys.280 (2000) 35 [hep-ph/9904353] [SPIRES]. · doi:10.1006/aphy.1999.5991
[5] K. Fukushima, Phase diagrams in the three-flavor Nambu-Jona-Lasinio model with the Polyakov loop, Phys. Rev.D 77 (2008) 114028 [arXiv:0803.3318] [SPIRES].
[6] A.M. Halasz, A.D. Jackson, R.E. Shrock, M.A. Stephanov and J.J.M. Verbaarschot, On the phase diagram of QCD, Phys. Rev.D 58 (1998) 096007 [hep-ph/9804290] [SPIRES].
[7] S. Hands et al., Numerical study of dense adjoint matter in two color QCD, Eur. Phys. J.C 17 (2000) 285 [hep-lat/0006018] [SPIRES]. · Zbl 1049.81661 · doi:10.1007/s100520000477
[8] S. Hands, I. Montvay, L. Scorzato and J. Skullerud, Diquark condensation in dense adjoint matter, Eur. Phys. J.C 22 (2001) 451 [hep-lat/0109029] [SPIRES]. · Zbl 1072.81553 · doi:10.1007/s100520100836
[9] J.B. Kogut, M.A. Stephanov, D. Toublan, J.J.M. Verbaarschot and A. Zhitnitsky, QCD-like theories at finite baryon density, Nucl. Phys.B 582 (2000) 477 [hep-ph/0001171] [SPIRES]. · doi:10.1016/S0550-3213(00)00242-X
[10] J.B. Kogut, D.K. Sinclair, S.J. Hands and S.E. Morrison, Two-colour QCD at non-zero quark-number density, Phys. Rev.D 64 (2001) 094505 [hep-lat/0105026] [SPIRES].
[11] D.T. Son, Superconductivity by long-range color magnetic interaction in high-density quark matter, Phys. Rev.D 59 (1999) 094019 [hep-ph/9812287] [SPIRES].
[12] T. Schafer, QCD and the eta’ mass: Instantons or confinement?, Phys. Rev.D 67 (2003) 074502 [hep-lat/0211035] [SPIRES].
[13] T. Kanazawa, T. Wettig and N. Yamamoto, Chiral Lagrangian and spectral sum rules for dense two-color QCD, JHEP08 (2009) 003 [arXiv:0906.3579] [SPIRES]. · doi:10.1088/1126-6708/2009/08/003
[14] C.W.J. Beenakker, Applications of random matrix theory to condensed matter and optical physics, arXiv:0904.1432 to appear in The Oxford Handbook of Random Matrix Theory, G. Akemann, J. Baik and P. Di Francesco eds., Oxford University Press (2011).
[15] A. Zabrodin, Random matrices and Laplacian growth, arXiv:0907.4929 to appear in The Oxford Handbook of Random Matrix Theory, G. Akemann, J. Baik and P. Di Francesco eds., Oxford University Press (2011).
[16] J.P. Bouchaud and M. Potters, Financial A pplications of Random Matrix Theory: a short review, arXiv:0910.1205 to appear in The Oxford Handbook of Random Matrix Theory, G. Akemann, J. Baik and P. Di Francesco eds., Oxford University Press (2011).
[17] J.J.M. Verbaarschot, Handbook Article on Applications of Random Matrix Theory to QCD, arXiv:0910.4134 [SPIRES] to appear in The Oxford Handbook of Random Matrix Theory, G. Akemann, J. Baik and P. Di Francesco eds., Oxford University Press (2011).
[18] Y.V. Fyodorov and D.V. Savin, Resonance Scattering of Waves in Chaotic Systems, arXiv:1003.0702 to appear in The Oxford Handbook of Random Matrix Theory, G. Akemann, J. Baik and P. Di Francesco eds., Oxford University Press (2011).
[19] P.L. Ferrari and H. Spohn, Random Growth Models, arXiv:1003.0881 to appear in The Oxford Handbook of Random Matrix Theory, G. Akemann, J. Baik and P. Di Francesco eds., Oxford University Press (2011).
[20] S.N. Majumdar, Extreme Eigenvalues of Wishart Matrices: Application to Entangled Bipartite System, arXiv:1005.4515 to appear in The Oxford Handbook of Random Matrix Theory, G. Akemann, J. Baik and P. Di Francesco eds., Oxford University Press (2011).
[21] P. Zinn-Justin and J.B. Zuber, Knot theory and matrix integrals, arXiv:1006.1812 to appear in The Oxford Handbook of Random Matrix Theory, G. Akemann, J. Baik and P. Di Francesco eds., Oxford University Press (2011).
[22] E.V. Shuryak and J.J.M. Verbaarschot, Random matrix theory and spectral sum rules for the Dirac operator in QCD, Nucl. Phys.A 560 (1993) 306 [hep-th/9212088] [SPIRES].
[23] J.J.M. Verbaarschot and T. Wettig, Random matrix theory and chiral symmetry in QCD, Ann. Rev. Nucl. Part. Sci.50 (2000) 343 [hep-ph/0003017] [SPIRES]. · doi:10.1146/annurev.nucl.50.1.343
[24] G. Akemann, Matrix models and QCD with chemical potential, Int. J. Mod. Phys.A 22 (2007) 1077 [hep-th/0701175] [SPIRES]. · Zbl 1110.81147
[25] M.A. Stephanov, Random matrix model of QCD at finite density and the nature of the quenched limit, Phys. Rev. Lett.76 (1996) 4472 [hep-lat/9604003] [SPIRES]. · doi:10.1103/PhysRevLett.76.4472
[26] J.C. Osborn, Universal results from an alternate random matrix model for QCD with a baryon chemical potential, Phys. Rev. Lett.93 (2004) 222001 [hep-th/0403131] [SPIRES]. · doi:10.1103/PhysRevLett.93.222001
[27] G. Akemann, J.C. Osborn, K. Splittorff and J.J.M. Verbaarschot, Unquenched QCD Dirac operator spectra at nonzero baryon chemical potential, Nucl. Phys.B 712 (2005) 287 [hep-th/0411030] [SPIRES]. · Zbl 1109.81368 · doi:10.1016/j.nuclphysb.2005.01.018
[28] G. Akemann, The complex Laguerre symplectic ensemble of non-Hermitian matrices, Nucl. Phys.B 730 (2005) 253 [hep-th/0507156] [SPIRES]. · Zbl 1276.81103 · doi:10.1016/j.nuclphysb.2005.09.039
[29] G. Akemann and F. Basile, Massive partition functions and complex eigenvalue correlations in Matrix Models with symplectic symmetry, Nucl. Phys.B 766 (2007) 150 [math-ph/0606060] [SPIRES]. · Zbl 1117.81109 · doi:10.1016/j.nuclphysb.2006.12.008
[30] G. Akemann, M.J. Phillips and H. J. Sommers, Characteristic polynomials in real Ginibre ensembles, J. Phys.A 42 (2009) 012001 [arXiv:0810.1458]. · Zbl 1154.81334
[31] A.M. Halasz, J.C. Osborn and J.J.M. Verbaarschot, Random matrix triality at nonzero chemical potential, Phys. Rev.D 56 (1997) 7059 [hep-lat/9704007] [SPIRES].
[32] J. Gasser and H. Leutwyler, Thermodynamics of Chiral Symmetry, Phys. Lett.B 188 (1987) 477 [SPIRES].
[33] T. Kanazawa, T. Wettig and N. Yamamoto, Chiral random matrix theory for two-color QCD at high density, Phys. Rev.D 81 (2010) 081701 [arXiv:0912.4999] [SPIRES].
[34] G. Akemann, M.J. Phillips and H.J. Sommers, The chiral Gaussian two-matrix ensemble of real asymmetric matrices, J. Phys.A 43 (2010) 085211 [arXiv:0911.1276] [SPIRES]. · Zbl 1188.15031
[35] Y.V. Fyodorov, B.A. Khoruzhenko and H.-J. Sommers, Almost-Hermitian Random Matrices: Eigenvalue Density in the Complex Plane, Phys. Lett.A 226 (1997) 46 [cond-mat/9606173] [SPIRES]. · Zbl 0962.82501
[36] Y.V. Fyodorov, B.A. Khoruzhenko and H.-J. Sommers, Almost Hermitian Random Matrices: Crossover from Wigner-Dyson to Ginibre Eigenvalue Statistics, Phys. Rev. Lett.79 (1997) 557 [cond-mat/9703152] [SPIRES]. · Zbl 1024.82502 · doi:10.1103/PhysRevLett.79.557
[37] G. Akemann, M. Kieburg and M.J. Phillips, Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices, J. Phys.A 43 (2010) 375207 [arXiv:1005.2983] [SPIRES]. · Zbl 1213.15025
[38] B. Klein, D. Toublan and J.J.M. Verbaarschot, Diquark and pion condensation in random matrix models for two-color QCD, Phys. Rev.D 72 (2005) 015007 [hep-ph/0405180] [SPIRES].
[39] J.C. Osborn, K. Splittorff and J.J.M. Verbaarschot, Chiral symmetry breaking and the Dirac spectrum at nonzero chemical potential, Phys. Rev. Lett.94 (2005) 202001 [hep-th/0501210] [SPIRES]. · doi:10.1103/PhysRevLett.94.202001
[40] J.C. Osborn, K. Splittorff and J.J.M. Verbaarschot, Chiral Condensate at Nonzero Chemical Potential in the Microscopic Limit of QCD, Phys. Rev.D 78 (2008) 065029 [arXiv:0805.1303] [SPIRES].
[41] J.J.M. Verbaarschot, The Spectrum of the Dirac operator near zero virtuality for Nc =2 and chiral random matrix theory, Nucl. Phys.B 426 (1994) 559 [hep-th/9401092] [SPIRES]. · Zbl 1049.81667 · doi:10.1016/0550-3213(94)90021-3
[42] A.M. Halasz and J.J.M. Verbaarschot, Effective Lagrangians and chiral random matrix theory, Phys. Rev.D 52 (1995) 2563 [hep-th/9502096] [SPIRES].
[43] T. Kanazawa, T. Wettig and N. Yamamoto, Chiral Lagrangian and spectral sum rules for two-color QCD at high density, PoS(LAT2009)195 [arXiv:0910.2300] [SPIRES].
[44] K. Splittorff and J.J.M. Verbaarschot, Lessons from Random Matrix Theory for QCD at Finite Density, arXiv:0809.4503 [SPIRES].
[45] J.C. Osborn, K. Splittorff and J.J.M. Verbaarschot, Phase Diagram of the Dirac Spectrum at Nonzero Chemical Potential, Phys. Rev.D 78 (2008) 105006 [arXiv:0807.4584] [SPIRES].
[46] K. Splittorff and J.J.M. Verbaarschot, Phase of the Fermion Determinant at Nonzero Chemical Potential, Phys. Rev. Lett.98 (2007) 031601 [hep-lat/0609076] [SPIRES]. · doi:10.1103/PhysRevLett.98.031601
[47] K. Splittorff and J.J.M. Verbaarschot, The QCD sign problem for small chemical potential, Phys. Rev.D 75 (2007) 116003 [hep-lat/0702011] [SPIRES].
[48] K. Splittorff, The sign problem in the ε-regime of QCD, PoS(LAT2006)023 [hep-lat/0610072] [SPIRES].
[49] G. Akemann and G. Vernizzi, Characteristic polynomials of complex random matrix models, Nucl. Phys.B 660 (2003) 532 [hep-th/0212051] [SPIRES]. · Zbl 1030.82003
[50] J.C.R. Bloch and T. Wettig, Random matrix analysis of the QCD sign problem for general topology, JHEP03 (2009) 100 [arXiv:0812.0324] [SPIRES]. · doi:10.1088/1126-6708/2009/03/100
[51] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, 7th Edition, Academic Press, London U.K. (2007). · Zbl 1208.65001
[52] H.J. Sommers and W. Wieczorek, General Eigenvalue Correlations for the Real Ginibre Ensemble, J. Phys.A 41 (2008) 405003 [arXiv:0806.2756]. · Zbl 1149.82005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.