# Transport Properties in Dense QCD Matter

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## Abstract

**:**

## 1. Introduction

## 2. Inhomogeneous Chiral Phase in Dense QCD and Weyl Semimetal

#### 2.1. Dual Chiral Density Wave

**q**. We, hereafter, focus on this type, because the phase degree of freedom is indispensable for manifestation of topological effects in iCP. Moreover, it may be the most favorable configuration in the presence of the magnetic field [6,7,8].

**q**.

#### 2.2. Similarity with Weyl Semimetal

## 3. Anomalous Hall Effect in iCP

#### 3.1. Hall Conductivity

**,**$b\left(k\right)={\nabla}_{k}\times a\left(k\right)$. Note that the factor ${e}^{2}$ comes from the cancellation due to the different directions of the wave vector for u and d quarks. Once the eigenfunctions $|{u}_{k}\rangle $ are known for any Hamiltonian $H\left(k\right)$, the Berry connection renders

#### 3.2. AHE in the DCDW Phase

## 4. Transport Properties in the Presence of the Magnetic Field

#### 4.1. Hall Conductivity

#### 4.2. Spectral Asymmetry and AHE

## 5. Summary and Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Friman, B.; Hoehne, C.; Knoll, J.; Leupold, S.; Randrup, J.; Rapp, R. The CBM Physics book. Lect. Notes Phys.
**2011**, 814, 1. [Google Scholar] - De Forcrand, P. Simulating QCD at finite density. PoS LAT
**2009**, 2009, 010. [Google Scholar] - Nakano, E.; Tatsumi, T. Chiral symmetry and density waves in quark matter. Phys. Rev.
**2005**, D71, 114006. [Google Scholar] [CrossRef] [Green Version] - Nickel, D. How manty phases meet at the chiral critical point? Phys. Rev. Lett.
**2009**, 103, 072301. [Google Scholar] [CrossRef] - Nickel, D. Inhomogeneous phases in the Nambu-Jona-Lasinio and quark-meson model. Phys. Rev.
**2009**, D80, 074025. [Google Scholar] [CrossRef] [Green Version] - Buballa, M.; Carignano, S. Inhomogeneous chiral condensates. Prog. Part. Nucl. Phys.
**2015**, 81, 39. [Google Scholar] [CrossRef] [Green Version] - Tatsumi, T. Inhomogeneous Chiral Phase in Quark Matter. JPS Conf. Proc.
**2018**, 20, 011008. [Google Scholar] - Abuki, H. Chiral crystallization in an external magnetic background: Chiral spiral versus real kink crystal. Phys. Rev.
**2018**, D98, 054006. [Google Scholar] [CrossRef] [Green Version] - Armitage, N.P.; Mele, E.J.; Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys.
**2018**, 90, 015001. [Google Scholar] [CrossRef] [Green Version] - Nagaosa, N.; Sinova, J.; Onoda, S.; MacDonald, A.H.; Ong, N.P. Anomalous Hall effect. Rev. Mod. Phys.
**2010**, 82, 1539. [Google Scholar] [CrossRef] [Green Version] - Xiao, D.; Chang, M.-C.; Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys.
**2010**, 82, 1959. [Google Scholar] [CrossRef] [Green Version] - Potekhin, A.Y.; Pons, J.A.; Page, D. Neutron Stars-Cooling and Transport. Space Sci. Rev.
**2015**, 191, 239. [Google Scholar] [CrossRef] [Green Version] - Konye, V.; Ogata, M. Magnetoresistance of a three-dimensional Dirac gas. Phys. Rev.
**2018**, B98, 195420. [Google Scholar] [CrossRef] [Green Version] - Karasawa, S.; Tatsumi, T. Variational approach to the inhomogeneous chiral phase in quark matter. Phys. Rev.
**2015**, D92, 116004. [Google Scholar] [CrossRef] [Green Version] - Baser, G.; Dunne, G.V. Twisted kink crystal in the chiral Gross-Neveu model. Phys. Rev.
**2008**, D78, 065002. [Google Scholar] - Basar, G.; Dunne, G.V.; Thies, M. Inhomogeneous condensates in the thermodynamics of the chiral NJL2 model. Phys. Rev.
**2009**, D79, 105012. [Google Scholar] - Fulde, P.; Ferrel, R.A. Superconductivity in a strong spin-exchange field. Phys. Rev.
**1964**, 135, A550. [Google Scholar] [CrossRef] - Larkin, A.L.; Ovchinnikov, Y.N. Inhomogeneous state of supercomductors. Sov. Phys. JETP
**1965**, 20, 762. [Google Scholar] - Wilczek, F. Two applications of axion electrodynamics. Phys. Rev. Lett.
**1987**, 58, 1799. [Google Scholar] [CrossRef] - Ferrer, E.J.; de la Incera, V. Dissipationless Hall current in dense quark matter in a magnetic field. Phys. Lett.
**2017**, B769, 208. [Google Scholar] [CrossRef] - Tatsumi, T.; Yoshiike, R.; Kashiwa, K. Anomalous Hall effect in dense QCD matter. Phys. Lett.
**2018**, B785, 46. [Google Scholar] [CrossRef] - Thouless, D.J.; Kohmoto, M.; Nightingale, M.P.; den Nijs, M. Quantised Hall Conductance in a two-dimensional periodic potential. Phys. Rev. Lett.
**1982**, 49, 405. [Google Scholar] [CrossRef] [Green Version] - Grushin, A.G. Consequences of a condensed matter realization of Lorentz-violating QED in Weyl semi-metals. Phys. Rev.
**2012**, D86, 045001. [Google Scholar] [CrossRef] [Green Version] - Goswani, P.; Tewari, S. Axionic field theory of (3+1)-dimensional Weyl semimetals. Phys. Rev.
**2013**, B88, 245107. [Google Scholar] [CrossRef] [Green Version] - Frolov, I.E.; Zhukivsky, V.C.; Klimenko, G.K. Chiral density waves in quark matter within the Nambu-Jona-Lasinio model in an external magnetic field. Phys. Rev.
**2010**, D82, 076002. [Google Scholar] [CrossRef] [Green Version] - Bastin, A.; Lewiner, C.; Betbeder-Matibet, O.; Nozieres, P. Quantum oscillations of the Hall effect of a Fermion gas with random impurity scattering. J. Phys. Chem, Solids
**1971**, 32, 1811. [Google Scholar] [CrossRef] - Tatsumi, T.; Abuki, H. Hall effect in the inhomogeneous chiral phase.
**2020**. (in preparation). [Google Scholar] - Streda, P. Theory of quantized Hall conductivity in two dimensions. J. Phys.
**1982**, C15, L717. [Google Scholar] - Tatsumi, T.; Nishiyama, K.; Karasawa, S. Novel Lifshitz point for chiral transition in the magnetic field. Phys. Lett.
**2015**, B743, 66. [Google Scholar] [CrossRef] [Green Version] - Yoshiike, R.; Nishiyama, K.; Tatsumi, T. Spontaneous magnetization of quark matter in the inhomogeneous chiral phase. Phys. Lett.
**2015**, B751, 123. [Google Scholar] [CrossRef] [Green Version] - Niemi, A.J.; Semenoff, G.W. Fermion number fractionization in quantum field theory. Phys. Rept.
**1986**, 135, 99. [Google Scholar] [CrossRef] - Zyuzin, A.A.; Burkov, A.A. Topological response in Weyl semimetals and the chiral anomaly. Phys. Rev.
**2012**, B86, 115133. [Google Scholar] [CrossRef] [Green Version] - Fukushima, K.; Khazrzeev, D.E.; Warringa, H.J. Chiral magnetic effect. Phys. Rev.
**2008**, D78, 074033. [Google Scholar] [CrossRef] - Klinkhamer, F.R.; Volovik, G.E. Emergent CPT violation from the splitting of Fermi points. Int. J. Mod. Phys.
**2005**, A20, 2795. [Google Scholar] [CrossRef] [Green Version] - Harutyunyan, A.; Rischke, D.H.; Sedrakian, A. Transport coefficients of two-flavor quark matter from the Kubo formalism. Phys. Rev.
**2017**, D95, 114021. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(

**a**) QCD phase diagram including the dual chiral density wave (DCDW) phase in the density–temperature plane from Karasawa and Tatsumi [14]. The triple point LP is called the Lifshitz point, where the spontaneously symmetry-broken (SSB) phase, the QGP phase, and iCP coexist. (

**b**) Density dependence of the order parameters in the unit of the cut-off parameter $\mathsf{\Lambda}$, from Nakano and Tatsumi [3]. The dynamical mass in the usual chiral transition is also depicted by $M/\mathsf{\Lambda}\left(q=0\right)$. Here we take $\mathit{q}=\left(0,0,\mathrm{q}\right),\mathrm{q}0$.

**Figure 2.**Schematic view of the energy surface of the quasi-particles with $s=-1$ in the momentum space. The Weyl points are located at $\pm {K}_{0}$ on the ${p}_{z}$ axis, around which the excitations of the quasi-particles are described by the Weyl Hamiltonian.

**Figure 3.**Integral along the ${p}_{z}$ axis. The function $s{E}_{0}+q/2$ changes its sign at the Weyl points for $s=-1$, while it is constant for $s=1$.

**Figure 4.**Energy spectra as functions of ${p}_{z}$. Thick lines denote the lowest Landau level (LLL).

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**MDPI and ACS Style**

Tatsumi, T.; Abuki, H.
Transport Properties in Dense QCD Matter. *Symmetry* **2020**, *12*, 366.
https://doi.org/10.3390/sym12030366

**AMA Style**

Tatsumi T, Abuki H.
Transport Properties in Dense QCD Matter. *Symmetry*. 2020; 12(3):366.
https://doi.org/10.3390/sym12030366

**Chicago/Turabian Style**

Tatsumi, Toshitaka, and Hiroaki Abuki.
2020. "Transport Properties in Dense QCD Matter" *Symmetry* 12, no. 3: 366.
https://doi.org/10.3390/sym12030366