# Dualities and inhomogeneous phases in dense quark matter with chiral and isospin imbalances in the framework of effective model

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## Abstract

It has been shown in [15, 70] in the framework of Nambu-Jona-Lasinio model with the assumption of spatially homogeneous condensates that in the large-*N*_{c} limit (*N*_{c} is the number of quark colours) there exist three dual symmetries of the thermodynamic potential, which describes dense quark matter with chiral and isospin imbalances. The main duality is between the chiral symmetry breaking and the charged pion condensation phenomena. There have been a lot of studies and hints that the ground state could be characterized by spatially inhomogeneous condensates, so the question arises if duality is a rather deep property of the phase structure or just accidental property in the homogeneous case. In this paper we have shown that even if the phase diagram contains phases with spatially inhomogeneous condensates, it still possesses the property of this main duality. Two other dual symmetries are not realized in the theory if it is investigated within an inhomogeneous approach to a ground state. Based on various previously studied aspects of the QCD phase diagram of dense isospin asymmetric matter with possible inhomogeneous condensates, in the present paper a unified picture and full phase diagram of dense quark matter with isospin imbalance have been assembled. Acting on this diagram by a dual transformation, we obtained, in the framework of an approach with spatially inhomogeneous condensates and without any calculations, a full phase diagram of chirally asymmetric dense medium. This example shows that the duality is not just entertaining mathematical property but an instrument with very high predictivity power. The obtained phase diagram is quite rich and contains various spatially inhomogeneous phases.

## Keywords

Phase Diagram of QCD Chiral Lagrangians## Notes

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## References

- [1]Y. Nambu and G. Jona-Lasinio,
*Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. 1.*,*Phys. Rev.***122**(1961) 345 [INSPIRE]. - [2]Y. Nambu and G. Jona-Lasinio,
*Dynamical model of elementary particles based on an analogy with superconductivity. II*,*Phys. Rev.***124**(1961) 246 [INSPIRE]. - [3]S.P. Klevansky,
*The Nambu-Jona-Lasinio model of quantum chromodynamics*,*Rev. Mod. Phys.***64**(1992) 649 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [4]T. Hatsuda and T. Kunihiro,
*QCD phenomenology based on a chiral effective Lagrangian*,*Phys. Rept.***247**(1994) 221 [hep-ph/9401310] [INSPIRE]. - [5]M. Buballa,
*NJL model analysis of quark matter at large density*,*Phys. Rept.***407**(2005) 205 [hep-ph/0402234] [INSPIRE]. - [6]K. Fukushima, D.E. Kharzeev and H.J. Warringa,
*The Chiral Magnetic Effect*,*Phys. Rev.***D 78**(2008) 074033 [arXiv:0808.3382] [INSPIRE]. - [7]M.A. Metlitski and A.R. Zhitnitsky,
*Anomalous axion interactions and topological currents in dense matter*,*Phys. Rev.***D 72**(2005) 045011 [hep-ph/0505072] [INSPIRE]. - [8]
- [9]M. Ruggieri, G.X. Peng and M. Chernodub,
*Chiral medium produced by parallel electric and magnetic fields*,*EPJ Web Conf.***129**(2016) 00037 [arXiv:1609.04537] [INSPIRE]. - [10]M. Ruggieri, Z.Y. Lu and G.X. Peng,
*Influence of chiral chemical potential, parallel electric and magnetic fields on the critical temperature of QCD*,*Phys. Rev.***D 94**(2016) 116003 [arXiv:1608.08310] [INSPIRE]. - [11]M. Ruggieri, G.X. Peng and M. Chernodub,
*Chiral Relaxation Time at the Crossover of Quantum Chromodynamics*,*Phys. Rev.***D 94**(2016) 054011 [arXiv:1606.03287] [INSPIRE]. - [12]M. Ruggieri and G.X. Peng,
*Quark matter in a parallel electric and magnetic field background: Chiral phase transition and equilibration of chiral density*,*Phys. Rev.***D 93**(2016) 094021 [arXiv:1602.08994] [INSPIRE]. - [13]V. Skokov, A.Y. Illarionov and V. Toneev,
*Estimate of the magnetic field strength in heavy-ion collisions*,*Int. J. Mod. Phys.***A 24**(2009) 5925 [arXiv:0907.1396] [INSPIRE]. - [14]A. Bzdak and V. Skokov,
*Event-by-event fluctuations of magnetic and electric fields in heavy ion collisions*,*Phys. Lett.***B 710**(2012) 171 [arXiv:1111.1949] [INSPIRE]. - [15]T.G. Khunjua, K.G. Klimenko and R.N. Zhokhov,
*Dualities in dense quark matter with isospin, chiral and chiral isospin imbalance in the framework of the large-N*_{c}*limit of the NJL*_{4}*model*,*Phys. Rev.***D 98**(2018) 054030 [arXiv:1804.01014] [INSPIRE]. - [16]D.T. Son and M.A. Stephanov,
*QCD at finite isospin density*,*Phys. Rev. Lett.***86**(2001) 592 [hep-ph/0005225] [INSPIRE]. - [17]A.W. Overhauser,
*Structure of Nuclear Matter*,*Phys. Rev. Lett.***4**(1960) 415.ADSCrossRefzbMATHGoogle Scholar - [18]
- [19]A.B. Migdal,
*Pion Fields in Nuclear Matter*,*Rev. Mod. Phys.***50**(1978) 107 [INSPIRE].ADSCrossRefGoogle Scholar - [20]A.B. Migdal,
*Stability of vacuum and limiting fields*,*Zh. Eksp. Teor. Fiz.***61**(1971) 2209 [INSPIRE].Google Scholar - [21]A.B. Migdal,
*Pi condensation in nuclear matter*,*Phys. Rev. Lett.***31**(1973) 257 [INSPIRE].ADSCrossRefGoogle Scholar - [22]F. Dautry and E.M. Nyman,
*Pion condensation and the*σ*-model in liquid neutron matter*,*Nucl. Phys.***A 319**(1979) 323 [INSPIRE]. - [23]G. Gruner,
*The dynamics of spin-density waves*,*Rev. Mod. Phys.***66**(1994) 1 [INSPIRE].ADSCrossRefGoogle Scholar - [24]A.I. Larkin and Y.N. Ovchinnikov,
*Nonuniform state of superconductors*,*Zh. Eksp. Teor. Fiz.***47**(1964) 1136 [INSPIRE].Google Scholar - [25]P. Fulde and R.A. Ferrell,
*Superconductivity in a Strong Spin-Exchange Field*,*Phys. Rev.***135**(1964) A550 [INSPIRE].ADSCrossRefGoogle Scholar - [26]M.G. Alford, J.A. Bowers and K. Rajagopal,
*Crystalline color superconductivity*,*Phys. Rev.***D 63**(2001) 074016 [hep-ph/0008208] [INSPIRE]. - [27]D. Nickel and M. Buballa,
*Solitonic ground states in (color-) superconductivity*,*Phys. Rev.***D 79**(2009) 054009 [arXiv:0811.2400] [INSPIRE]. - [28]R. Anglani et al.,
*Crystalline color superconductors*,*Rev. Mod. Phys.***86**(2014) 509 [arXiv:1302.4264] [INSPIRE].ADSCrossRefGoogle Scholar - [29]R. Casalbuoni and G. Nardulli,
*Inhomogeneous superconductivity in condensed matter and QCD*,*Rev. Mod. Phys.***76**(2004) 263 [hep-ph/0305069] [INSPIRE]. - [30]J.A. Bowers and K. Rajagopal,
*The Crystallography of color superconductivity*,*Phys. Rev.***D 66**(2002) 065002 [hep-ph/0204079] [INSPIRE]. - [31]M. Mannarelli,
*Overview of Crystalline Color Superconductors*, in*Proceedings, Compact Stars in the QCD Phase Diagram IV (CSQCD IV)*, Prerow, Germany, September 26-30, 2014 (2015) [arXiv:1505.04888] [INSPIRE]. - [32]D.V. Deryagin, D.Y. Grigoriev and V.A. Rubakov,
*Standing wave ground state in high density, zero temperature QCD at large N*_{c},*Int. J. Mod. Phys.***A 7**(1992) 659 [INSPIRE]. - [33]M. Sadzikowski and W. Broniowski,
*Nonuniform chiral phase in effective chiral quark models*,*Phys. Lett.***B 488**(2000) 63 [hep-ph/0003282] [INSPIRE]. - [34]
- [35]S. Carignano and M. Buballa,
*Inhomogeneous islands and continents in the Nambu-Jona-Lasinio model*,*Acta Phys. Polon. Supp.***5**(2012) 641 [arXiv:1111.4400] [INSPIRE].CrossRefGoogle Scholar - [36]O. Schnetz, M. Thies and K. Urlichs,
*Phase diagram of the Gross-Neveu model: Exact results and condensed matter precursors*,*Annals Phys.***314**(2004) 425 [hep-th/0402014] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [37]G. Basar, G.V. Dunne and M. Thies,
*Inhomogeneous Condensates in the Thermodynamics of the Chiral N J L*_{2}*model*,*Phys. Rev.***D 79**(2009) 105012 [arXiv:0903.1868] [INSPIRE]. - [38]N.V. Gubina, V.C. Zhukovsky, K.G. Klimenko and S.G. Kurbanov,
*Emergence of a nonuniform pion condensate in the*(1 + 1)*-dimensional Nambu-Jona-Lasinio model*,*Phys. Atom. Nucl.***76**(2013) 1377 [INSPIRE]. - [39]V.C. Zhukovsky, K.G. Klimenko and T.G. Khunjua,
*The effect of chiral density waves on the superconducting phase in the two-dimensional Gross-Neveu model*,*Moscow Univ. Phys. Bull.***68**(2013) 105 [INSPIRE].ADSCrossRefGoogle Scholar - [40]D. Ebert, T.G. Khunjua, K.G. Klimenko and V.C. Zhukovsky,
*Suppression of superconductivity by inhomogeneous chiral condensation in the N J L*_{2}*model*,*Int. J. Mod. Phys.***A 29**(2014) 1450025 [arXiv:1306.4485] [INSPIRE]. - [41]J. Braun, S. Finkbeiner, F. Karbstein and D. Roscher,
*Search for inhomogeneous phases in fermionic models*,*Phys. Rev.***D 91**(2015) 116006 [arXiv:1410.8181] [INSPIRE]. - [42]M. Buballa and S. Carignano,
*Inhomogeneous chiral condensates*,*Prog. Part. Nucl. Phys.***81**(2015) 39 [arXiv:1406.1367] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [43]A. Heinz,
*QCD under extreme conditions: inhomogeneous condensation*, Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften, Frankfurt am Main, September (2014).Google Scholar - [44]C.-f. Mu, L.-y. He and Y.-x. Liu,
*Evaluating the phase diagram at finite isospin and baryon chemical potentials in the Nambu-Jona-Lasinio model*,*Phys. Rev.***D 82**(2010) 056006 [INSPIRE]. - [45]N.V. Gubina, K.G. Klimenko, S.G. Kurbanov and V.C. Zhukovsky,
*Inhomogeneous charged pion condensation phenomenon in the N J L*_{2}*model with quark number and isospin chemical potentials*,*Phys. Rev.***D 86**(2012) 085011 [arXiv:1206.2519] [INSPIRE]. - [46]D. Ebert, T.G. Khunjua, K.G. Klimenko and V.C. Zhukovsky,
*Competition and duality correspondence between inhomogeneous fermion-antifermion and fermion-fermion condensations in the N J L*_{2}*model*,*Phys. Rev.***D 90**(2014) 045021 [arXiv:1405.3789] [INSPIRE]. - [47]E. Nakano and T. Tatsumi,
*Chiral symmetry and density wave in quark matter*,*Phys. Rev.***D 71**(2005) 114006 [hep-ph/0411350] [INSPIRE]. - [48]V. Schon and M. Thies,
*Emergence of Skyrme crystal in Gross-Neveu and*’*t Hooft models at finite density*,*Phys. Rev.***D 62**(2000) 096002 [hep-th/0003195] [INSPIRE]. - [49]D. Ebert, N.V. Gubina, K.G. Klimenko, S.G. Kurbanov and V.C. Zhukovsky,
*Chiral density waves in the NJL*_{2}*model with quark number and isospin chemical potentials*,*Phys. Rev.***D 84**(2011) 025004 [arXiv:1102.4079] [INSPIRE]. - [50]P. Adhikari and J.O. Andersen,
*Chiral density wave versus pion condensation in the*1 + 1*dimensional NJL model*,*Phys. Rev.***D 95**(2017) 054020 [arXiv:1610.01647] [INSPIRE]. - [51]J.O. Andersen and P. Kneschke,
*Chiral density wave versus pion condensation at finite density and zero temperature*,*Phys. Rev.***D 97**(2018) 076005 [arXiv:1802.01832] [INSPIRE]. - [52]D. Nickel,
*Inhomogeneous phases in the Nambu-Jona-Lasino and quark-meson model*,*Phys. Rev.***D 80**(2009) 074025 [arXiv:0906.5295] [INSPIRE]. - [53]I.E. Frolov, V.C. Zhukovsky and K.G. Klimenko,
*Chiral density waves in quark matter within the Nambu-Jona-Lasinio model in an external magnetic field*,*Phys. Rev.***D 82**(2010) 076002 [arXiv:1007.2984] [INSPIRE]. - [54]V.C. Zhukovsky, K.G. Klimenko and I.E. Frolov,
*Density waves in quark matter within the Nambu-Jona-Lasinio model in an external magnetic field*,*Moscow Univ. Phys. Bull.***65**(2010) 539 [INSPIRE]. - [55]E.J. Ferrer and V. de la Incera,
*Novel Topological Effects in Dense QCD in a Magnetic Field*,*Nucl. Phys.***B 931**(2018) 192 [arXiv:1512.03972] [INSPIRE]. - [56]W. Broniowski,
*Chiral waves in quark matter*,*Acta Phys. Polon. Supp.***5**(2012) 631 [arXiv:1110.4063] [INSPIRE].CrossRefGoogle Scholar - [57]J. Moreira, B. Hiller, W. Broniowski, A.A. Osipov and A.H. Blin,
*Nonuniform phases in a three-flavor Nambu-Jona-Lasinio model*,*Phys. Rev.***D 89**(2014) 036009 [arXiv:1312.4942] [INSPIRE]. - [58]T. Tatsumi and T. Muto,
*Quark beta decay in the inhomogeneous chiral phase and cooling of compact stars*,*Phys. Rev.***D 89**(2014) 103005 [arXiv:1403.1927] [INSPIRE]. - [59]K. Nishiyama, S. Karasawa and T. Tatsumi,
*Hybrid chiral condensate in the external magnetic field*,*Phys. Rev.***D 92**(2015) 036008 [arXiv:1505.01928] [INSPIRE]. - [60]A. Heinz, F. Giacosa, M. Wagner and D.H. Rischke,
*Inhomogeneous condensation in effective models for QCD using the finite-mode approach*,*Phys. Rev.***D 93**(2016) 014007 [arXiv:1508.06057] [INSPIRE]. - [61]T. Kojo, Y. Hidaka, L. McLerran and R.D. Pisarski,
*Quarkyonic Chiral Spirals*,*Nucl. Phys.***A 843**(2010) 37 [arXiv:0912.3800] [INSPIRE]. - [62]T. Kojo, Y. Hidaka, K. Fukushima, L.D. McLerran and R.D. Pisarski,
*Interweaving Chiral Spirals*,*Nucl. Phys.***A 875**(2012) 94 [arXiv:1107.2124] [INSPIRE]. - [63]A. Heinz, F. Giacosa and D.H. Rischke,
*Chiral density wave in nuclear matter*,*Nucl. Phys.***A 933**(2015) 34 [arXiv:1312.3244] [INSPIRE]. - [64]Y. Takeda, H. Abuki and M. Harada,
*Novel dual chiral density wave in nuclear matter based on a parity doublet structure*,*Phys. Rev.***D 97**(2018) 094032 [arXiv:1803.06779] [INSPIRE]. - [65]M. Thies,
*Duality between quark quark and quark anti-quark pairing in*1 + 1*dimensional large N models*,*Phys. Rev.***D 68**(2003) 047703 [hep-th/0303026] [INSPIRE]. - [66]G. Cao, L. He and P. Zhuang,
*Collective modes and Kosterlitz-Thouless transition in a magnetic field in the planar Nambu-Jona-Lasino model*,*Phys. Rev.***D 90**(2014) 056005 [arXiv:1408.5364] [INSPIRE]. - [67]D. Ebert, T.G. Khunjua, K.G. Klimenko and V.C. Zhukovsky,
*Competition and duality correspondence between chiral and superconducting channels in*(2 + 1)*-dimensional four-fermion models with fermion number and chiral chemical potentials*,*Phys. Rev.***D 93**(2016) 105022 [arXiv:1603.00357] [INSPIRE]. - [68]D. Ebert, T.G. Khunjua and K.G. Klimenko,
*Duality between chiral symmetry breaking and charged pion condensation at large N*_{c}*: Consideration of an N J L*_{2}*model with baryon, isospin and chiral isospin chemical potentials*,*Phys. Rev.***D 94**(2016) 116016 [arXiv:1608.07688] [INSPIRE]. - [69]T.G. Khunjua, K.G. Klimenko, R.N. Zhokhov and V.C. Zhukovsky,
*Inhomogeneous charged pion condensation in chiral asymmetric dense quark matter in the framework of N J L*_{2}*model*,*Phys. Rev.***D 95**(2017) 105010 [arXiv:1704.01477] [INSPIRE]. - [70]T.G. Khunjua, K.G. Klimenko and R.N. Zhokhov,
*Dense baryon matter with isospin and chiral imbalance in the framework of N J L*_{4}*model at large N*_{c}*: duality between chiral symmetry breaking and charged pion condensation*,*Phys. Rev.***D 97**(2018) 054036 [arXiv:1710.09706] [INSPIRE]. - [71]C. Ratti and W. Weise,
*Thermodynamics of two-colour QCD and the Nambu Jona-Lasinio model*,*Phys. Rev.***D 70**(2004) 054013 [hep-ph/0406159] [INSPIRE]. - [72]A.A. Andrianov, D. Espriu and X. Planells,
*An effective QCD Lagrangian in the presence of an axial chemical potential*,*Eur. Phys. J.***C 73**(2013) 2294 [arXiv:1210.7712] [INSPIRE]. - [73]R. Gatto and M. Ruggieri,
*Hot Quark Matter with an Axial Chemical Potential*,*Phys. Rev.***D 85**(2012) 054013 [arXiv:1110.4904] [INSPIRE]. - [74]L. Yu, H. Liu and M. Huang,
*Spontaneous generation of local CP-violation and inverse magnetic catalysis*,*Phys. Rev.***D 90**(2014) 074009 [arXiv:1404.6969] [INSPIRE]. - [75]L. Yu, H. Liu and M. Huang,
*Effect of the chiral chemical potential on the chiral phase transition in the NJL model with different regularization schemes*,*Phys. Rev.***D 94**(2016) 014026 [arXiv:1511.03073] [INSPIRE]. - [76]M. Ruggieri and G.X. Peng,
*Critical Temperature of Chiral Symmetry Restoration for Quark Matter with a Chiral Chemical Potential*,*J. Phys.***G 43**(2016) 125101 [arXiv:1602.05250] [INSPIRE]. - [77]G. Cao and P. Zhuang,
*Effects of chiral imbalance and magnetic field on pion superfluidity and color superconductivity*,*Phys. Rev.***D 92**(2015) 105030 [arXiv:1505.05307] [INSPIRE]. - [78]V.V. Braguta, E.M. Ilgenfritz, A. Yu. Kotov, B. Petersson and S.A. Skinderev,
*Study of QCD Phase Diagram with Non-Zero Chiral Chemical Potential*,*Phys. Rev.***D 93**(2016) 034509 [arXiv:1512.05873] [INSPIRE]. - [79]N.Y. Astrakhantsev, V.V. Braguta, A.Y. Kotov and A.A. Nikolaev,
*Lattice study of QCD at finite chiral density: topology and confinement*, arXiv:1902.09325 [INSPIRE]. - [80]V.V. Braguta, E.M. Ilgenfritz, A.Y. Kotov, A.V. Molochkov and A.A. Nikolaev,
*Study of the phase diagram of dense two-color QCD within lattice simulation*,*Phys. Rev.***D 94**(2016) 114510 [arXiv:1605.04090] [INSPIRE]. - [81]E. Shuster and D.T. Son,
*On finite density QCD at large N*_{c},*Nucl. Phys.***B 573**(2000) 434 [hep-ph/9905448] [INSPIRE]. - [82]E.V. Shuryak,
*Scales and phases of nonperturbative QCD*, in*QCD: Perturbative or nonperturbative? Proceedings, 17th Autumn School*, Lisbon, Portugal, September 29-October 4, 1999, pp. 210-248 (1999) [hep-ph/9911244] [INSPIRE]. - [83]T. Schäfer and F. Wilczek,
*Superconductivity from perturbative one gluon exchange in high density quark matter*,*Phys. Rev.***D 60**(1999) 114033 [hep-ph/9906512] [INSPIRE]. - [84]S. Weinberg,
*The Quantum Theory of Field II*, Cambridge University Press, Cambridge, England (1996).Google Scholar - [85]K. Fujikawa,
*Path Integral for Gauge Theories with Fermions*,*Phys. Rev.***D 21**(1980) 2848 [*Erratum ibid.***D 22**(1980) 1499] [INSPIRE]. - [86]M. Hanada and N. Yamamoto,
*Universality of phase diagrams in QCD and QCD-like theories*, PoS(LATTICE2011)221 (2011) [arXiv:1111.3391] [INSPIRE]. - [87]M. Hanada and N. Yamamoto,
*Universality of Phases in QCD and QCD-like Theories*,*JHEP***02**(2012) 138 [arXiv:1103.5480] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [88]D.N. Voskresensky,
*On the possibility of the condensation of the charged rho meson field in dense isospin asymmetric baryon matter*,*Phys. Lett.***B 392**(1997) 262 [INSPIRE]. - [89]J.T. Lenaghan, F. Sannino and K. Splittorff,
*The Superfluid and conformal phase transitions of two color QCD*,*Phys. Rev.***D 65**(2002) 054002 [hep-ph/0107099] [INSPIRE]. - [90]F. Sannino,
*General structure of relativistic vector condensation*,*Phys. Rev.***D 67**(2003) 054006 [hep-ph/0211367] [INSPIRE]. - [91]T. Brauner and X.-G. Huang,
*Vector meson condensation in a pion superfluid*,*Phys. Rev.***D 94**(2016) 094003 [arXiv:1610.00426] [INSPIRE]. - [92]O. Aharony, K. Peeters, J. Sonnenschein and M. Zamaklar,
*Rho meson condensation at finite isospin chemical potential in a holographic model for QCD*,*JHEP***02**(2008) 071 [arXiv:0709.3948] [INSPIRE].ADSCrossRefGoogle Scholar - [93]T.G. Khunjua, K.G. Klimenko and R.N. Zhokhov,
*Chiral imbalanced hot and dense quark matter: NJL analysis at the physical point and comparison with lattice QCD*,*Eur. Phys. J.***C 79**(2019) 151 [arXiv:1812.00772] [INSPIRE]. - [94]D. Nowakowski, M. Buballa, S. Carignano and J. Wambach,
*Inhomogeneous chiral symmetry breaking phases in isospin-asymmetric matter*, in*Proceedings, Compact Stars in the QCD Phase Diagram IV (CSQCD IV)*, Prerow, Germany, September 26-30, 2014 (2015) [arXiv:1506.04260] [INSPIRE]. - [95]D. Nowakowski,
*Inhomogeneous chiral symmetry breaking in isospin-asymmetric strong-interaction matter*, Ph.D. Thesis, Technische Univ. Darmstadt (2017).Google Scholar