Transport properties in magnetized compact stars

Transport properties of dense quark matter are discussed in the strong magnetic field, B. B dependence as well as density dependence of the Hall conductivity is discussed in the inhomogeneous chiral phase. Anomalous Hall effect is intrinsic to the inhomogeneous chiral phase and resembles the one in Weyl semimetals in condensed matter physics. Some theoretical aspects inherent in anomalous Hall effect are revealed.


Introduction
Observations of compact stars have provided us with information about highly dense matter in their cores since their first discovery in 1967. Nowadays there have appeared many papers about the possible presence of quarks and its implications on observations of neutron-star mergers, highmass stars about 2 ⊙ or magnetars [1,2]. Magnetars bear a huge magnetic field of (10 15 G) in their surface and exhibit unique thermal evolution [3], while the origin of the magnetic field and the surface temperature have not been fully understood yet. Their persistent surface temperature is very high, compared with ordinary pulsars at the same age, and resides well above the standard cooling curve [4]. In order to resolve the issue we must take into account the thermal conduction as well as the heating mechanism in the presence of the strong magnetic field. One of the authors (T.T.) have suggested a possibility of spontaneous magnetization of quark matter inside cores as a microscopic origin of huge magnetic field, based on the energetic scale of QCD [5]. Here we examine the transport properties of quark matter under the large magnetic field, which provides, we expect, a first step to understand thermal evolution of magnetars with quark core.
Since the transport properties are important for thermal evolution of ordinary pulsars, there have been many works [6]. However, the microscopic treatment of thermal conductivity may include many subtle points, such as relativistic effects, quantum mechanical effects (including the Shubnikovde Haas effect due to discretized Landau levels) in the magnetic field. Here we reexamine these points for relativistic fermions such as quarks or electrons. For example, electrons become relativistic in the inner crust of neutron stars, and we must use the Dirac equation to describe them. It should be interesting to see such Dirac electrons become important in modern condensed matter physics [7,8], where some topological materials have been also discussed. If such topological materials develop in the inner crust, we must carefully discuss the transport properties of electrons in the inner crust.
For the electric current and the energy current , phenomenological transport equations should read for charge carrying electrons: with the electric field, = −∇ . is the fictitious "gravitational" potential introduced by Luttinger [9], with which one can apply the linear response theory for thermal conductivity. The conductivity tensor and the thermal conductivity are constructed by combining the matrix elements (second rank tensors) as, The matrices are related to each other and we confirm that the Wiedemann-Franz law holds at low temperature,

= ,
with the Lorentz number, = 1 3 ( / ) 2 . Thus we need the information about the conductivity at vanishing temperature ( = 0) to obtain thermal conductivity at low temperature. In general, or has off-diagonal components, corresponding to the Hall effect. The evaluation of the transport coefficients can be done by the Boltzmann equation or the Kubo formula in a microscopic way. Here we use the Kubo formula within the linear response theory [10].
For quarks we shall see an interesting possibility of the anomalous Hall effect (AHE) [11,12] at moderately high densities. Recently possible appearance of the inhomogeneous chiral phase (iCP) has been suggested to show up in some region of the QCD phase diagram in the temperature (T)baryon-number chemical potential (µ) plane (see Figure 1). So, if quark matter is realized in the core region of compact stars, there may be developed iCP. The physical properties of iCP phase in various situations, including those in the presence of magnetic field, have been extensively studied [13][14][15][16][17][18]. The dual chiral density wave (DCDW) is one type of iCP , which is a kind of density wave and specified by the scalar and pseudoscalar condensates with spatial modulation [13], The order parameters are the amplitude Δ and the wave-vector .
We shall see the DCDW phase shares similar physical properties with Weyl semimetals (WSM) in condensed matter physics [19].

Dual chiral density wave
AHE and its implication to axion electrodynamics has been discussed with a focus on its relation to the DCDW phase [20,21]. For flavor symmetric , quark matter, the single-particle energy of (a) (b) quarks can be easily obtained in the DCDW phase, when the Nambu-Jona-Lasinio (NJL) model is used as an effective model of QCD at low energy scales [13]. The effective Lagrangian reads (2) In the chiral limit, under the mean-field approximation. One may rewrite it in a simple form, with = −2 Δ and the space-like vector = (0, ) by the use of the Weinberg transformation, We can see that the amplitude of DCDW provides the dynamical mass for the newly defined quarks (quasi-particles) described by , while the wavevector induces the axial-vector mean-field applied to them.
The single-particle energy can be easily extracted, with = � 2 + 2 for each flavor, where and denote the particle-antiparticle and spin degrees of freedom, respectively. Accordingly, this form suggests anisotropy of the Fermi sea in the momentum space: it deforms about the direction of q in the axial-symmetric manner.
In Figure 2 we plot the energy surface in the momentum space. One can see that there is a gap between negative and positive energies in the case with /2 < (Fig.1a), while in the opposite case with /2 > , there are two Weyl points = (0,0, ± 0 ), with 0 = �( /2) 2 − 2 where the gap vanishes, as is seen in Fig. 2(b). The thermodynamic potential can be derived by the single-particle energies, and the parameters ( , ) are determined by the minimization of the thermodynamic potential: it has been shown that the relation /2 > holds in the DCDW phase [13]. Here it is worth mentioning about some resemblance with WSM [19]. The effective Lagrangian (3) is of the same form as in WSM, by replacing /2 by the strength of spin-splitting and the dynamical mass by the spin-orbit coupling strength . The relation, > holds in WSM. The important difference is that the values of ( , ) are dynamically determined as a consequence of chiral symmetry breaking in the DCDW phase, while the parameters ( , ) can be controlled with the experimental setup for WSM. Positive energy states correspond to the conduction band, while the negative energy states the valence band. Since some positive energy states are filled in the DCDW phase, one might call it "Weyl metallic state" [22]. Therefore we shall discuss, hereafter, the transport properties of dense quark matter, referring WSM as a guiding principle.

Anomalous Hall effect
One of the interesting transport properties in WSM may be the anomalous Hall effect (AHE) [19]; the Hall current flows even in the absence of magnetic field in response to the external electric Figure 3 . Density dependence of � Fermi � DCDW . field. Accordingly we shall see AHE in the DCDW phase. We can derive the anomalous Hall conductivity by way of the Kubo formula, considering a linear response to a tiny electric field [20]: it is given by the integral of the Berry curvature in the momentum space ( ), with ( ) being the Fermi-Dirac distribution function. The Berry curvature is defined in terms of the eigenfunctions as and it reads with 0 = � 2 + 2 . It is interesting to see that the Berry curvature looks like the magnetic field from the Dirac monopole located at each Weyl point in the momentum space. The contribution from the negative energy sea apparently diverges and need a relevant regularization [23]. We see that it is appropriate to use the proper-time method or the heat-kernel method. Applying the proper-time regularization we find The second term is evaluated to give − 2 /(2 ) 2 , while the first term gives different values, depending on the values of the wave vector: for /2 < it vanishes, while it gives for /2 > . It is interesting to note that the quantity 2 /2 is a topological number and nothing else but the Hall conductivity for 2D quantum Hall systems with the Chern number = 1. Therefore we may regard the DCDW phase as a stack of 2D quantum Hall systems to the 3 rd direction [24]. The anomalous Hall conductivity reads where 0 2 ≡ � 2 � 2 − 2 . We can immediately see that Dirac → 0 as → 0, because in this limit 0 → /2. It is reasonable for to be vanishing for = 0, implying no AHE for the normal phase. We shall see that the origin of the first term comes from axial anomaly. On the other hand, the anomalous Hall conductivity in WSM is given only by the second term without the anomaly term: WSM = 2 0 /(2 2 ) for > , while it is vanishing for < . Actually, Goswami and Tewari have shown the boundary-bulk correspondence by explicitly constructing the surface states: the surface states exist only for > and the system is an insulator for < [25].

Fermi sea contribution
Let us now have a quick look at the Fermi sea contributions. For cases (a) < where . It is interesting to note that � Fermi � DCDW → 0 as → 0 for an arbitrary density; this means that there is no AHE in the chiral restored phase, irrespective of the value of wave number .

Anomalous Hall conductivity
Let us now discuss how the situation changes when the system is immersed in an external magnetic field [26]. The effective Hamiltonian for quasi-particles in the DCDW phase is given as with = diag( , ) ( = 2 /3, = − /3). Since it has been shown that ∥ is the most favorable configuration [27], we set magnetic field along z-axis without loss of generality.
The effective Hamiltonian can be easily diagonalized, and resulting eigenvalues are for each flavor , where = ±1, denotes the particle-antiparticle states, and = ±1 specifies the spin degree of freedom. Note that there is no spin degree of freedom for the lowest Landau level (LLL): there occurs dimensional reduction for LLL and the eigenspinor is represented by two components. The energy spectra are depicted in Fig. 4 for two cases. Note that, the external magnetic field makes the DCDW phase extended to lower densities and both cases are realized in the presence of the magnetic field [27,28], in contrast to the previous situation discussed in Chapter 3.
The conductivity tensor consists of two kinds of the matrix elements, the diagonal ones = , and the off-diagonal one . The latter one implies the Hall effect and becomes important for small impurities in the presence of magnetic field, compared with the longitudinal conductivity = .
In the following we consider the Hall conductivity.
where the first term I vanishes when the chemical potential is located between the energy gap where the density of states is vanishing. This situation occurs for topological insulators, and also in the nearly clean 2D quantum hall system in strong magnetic field. On the other hand, in the case of finite density of states at the Fermi energy, the dissipative effects are no longer to be ignored. In this case I depends on the details of matter. Actually we can see the classical Drude-Zener relation I = −ω τ , by using arbitrary energy-dependent self-energy [30], where ω is the cyclotron frequency and τ the life-time. Note that I remains finite even in the dissipation-less limit, τ → ∞.
On the other hand, the second non-dissipative term has no classical analogy, which can be expressed by the use of the number of states under the energy E, ( ), for each flavor to represent the quantum effect, It may be worth noting that for quantized Hall effect in 2D Hall systems, I = 0, II = − 2 /(2 ) with being the integer called the Landau level filling factor, that is, the number of Landau levels below the Fermi energy [30]. Noting , the formula II = − is actually satisfied. On the other hand, in 3D Dirac materials with = 0, it has been shown that = − / [7], provided that the effect of impurities can be neglected. We do not discuss I here, and mainly focus on the quantum contribution II , where we shall see some topological effect.
Writing the baryon number operator in the normal ordering form, � = 1 2 ∫ 3 � † ( ), ( )� ( = or ) we find ( ) = norm ( ) + anom by counting the number of the eigenstates below energy E. The quasi-particle density of states can be written as , (15) in the DCDW phase, and thereby norm ( ) reads Therefore the contribution to the conductivity from the Fermi sea can be written as The second one is the anomalous quark number and can be written as We can see that it originates from the spectral asymmetry, and is closely related to a topological quantity, the invariant introduced by Atiyah-Patodi-Singer [31], (18) is not well-defined as it is, and must be properly regularized to extract the physical result. Using the gauge-invariant regularization, we can evaluate the invariant [32], We can see from the first term that the invariant is related to axial anomaly. The second term correctly cancels the anomalous contribution from the first term in the limit → 0.

Axial anomaly
Introducing chemical potential as a fictitious gauge field = ( , 0) coupling with quark number, the mean-field Lagrangian now reads where the covariant derivative is given by = − ( + ). Then we find an anomaly term in the action after the Weinberg transformation [33], with = − . The first term represents the standard axial anomaly and vanishes in the absence of electric field, while the second term produces We can see anomalous quark number appears. Such anomalous quark number becomes the same as the one given by the first term of the invariant (19). Note that this result has been explicitly verified by evaluating the invariant by using the adiabatic expansion for the quark propagator a la Goldstone and Wilczek [34].
Accordingly, the anomalous Hall conductivity can be given as for = 3 . Thus we confirm that anom coincides with Eq. (9). Note that AHE has been also discussed for WSM in the magnetic field [35]: they obtained a different result, in accordance with the boundary-bulk correspondence. The difference between Eqs. (22) and (23) comes from the regularization. Technically they did not used a gauge-invariant regularization, while Eq. (22) is obtained by the gauge-invariant one.

Fermi sea contribution
Let us finally have a look at the Fermi sea contribution Fermi given in Eq. (17). Generally we must take into account many Landau levels, depending on chemical potential and the strength of the magnetic field B [22]. We can derive an analytic formula in the limit of strong magnetic field. This is performed by restricting the level summation in the density of state to the contribution from the LLL (quantum limit regime). In this approximation we have We note that when the system approaches the homogeneous limits = 0, II,Fermi,LLL → 2 2 2 . The Fermi momentum = � 2 − 2 is now proportional to the fermion density since the system is dimensionally reduced to 1D. In fact, the fermion density in this limit can be easily evaluated as, where only spin down u-quarks and spin up d-quarks in the LLL contribute to the density. Then we have the standard relation II,Fermi,LLL = 3 �= ∑ �. We can say that in this limit, the 1/B scaling of the Hall conductivity totally comes from quantum (non-dissipative) contribution II,Fermi . It should be interesting to see that the dissipative part I,Fermi has nothing to do with the Hall conductivity in this limit. This result can never be inferred from the classical relation, I = −ω τ ., which has been used for the electron conductivity for thermal evolution of neutron stars with strong magnetic fields [6]. Incidentally the dominance of II,Fermi over I,Fermi holds in the limit, → 0, as in Dirac material: a direct evaluation of I,Fermi can be easily done to give a null result in the high-field limit.
On the other hand, in the case of weak magnetic field, we may derive the approximated expression by replacing the summation over Landau levels by the continuous integration. In general, we can expand Fermi as II,Fermi = −1 ( , , ) + 0 ( , , ) + 1 ( , , ) + � 2 �.
Setting a continuous function � ( ) = 8 2 , we find the integral expression for −1 ( , , ), which turns out to vanish after the integration by parts: The first nontrivial term 0 ( , , ) should coincide with 0 ( , , ) = � Fermi � DCDW in Eq. (10). We see that Fermi sea contribution to the non-dissipative conductivity vanishes ( II,Fermi → 0) as → 0. It was shown in [7] for Dirac material ( → 0), the relation = − / holds irrespective of the value of . Then we may conclude that, in the weak field limit, the 1/ scaling of the Hall conductivity should come totally from the dissipative part I,Fermi . It is worth mentioning that, such 1/ scaling itself is natural when the system has freely moving charge carriers, as is also inferred from the Drude-Zener model [30]. This is a consequence of the fact that active fermions in conduction bands move so that, in the equilibrium state, they do no longer feel the applied electric field which disappears in the comoving frame by the Lorentz transformation: to be specific, let us consider the system with magnetic field pointing to z-direction , and electric field pointing to direction . Then switching to the frame moving to direction with velocity , we have ′ = ( + × ) and ′ = ( − × ) with is the usual gamma factor, = √1 − 2 . As a consequence, when + × = , electric charges do not feel the electric field and thus are no longer accelerated. This situation is achieved when is parallel to the direction of × and the magnitude is with | | = = / .
The electric current in this equilibrium situation (in the original frame) is = ∑ � � with and the electric charge and number density of carrier particle , respectively. In the weak field limit, ∑ = 6 ∝ 3 while in the high field limit ∑ = 3 ∝ ( ) . We anticipate that the 1/ scaling of the Hall conductivity in the weak magnetic field comes from dissipative conductivity as: . Although we are not still able to find a general expression for 1 ( , , ), in the limit of → 0, we could find a formula 1 (0, , ) =

Concluding remarks
We have discussed the transport properties of dense QCD matter in the magnetic field. In particular we have paid attention to the quantum Hall effect. If there developed a new phase in the core of compact stars, an interesting phenomenon, the anomalous Hall effect is to be activated, besides the usual Hall effect. The phase of dual chiral density wave may appear in the moderate density region, where the spectrum of quarks resembles the one of Weyl semimetal in condensed matter physics. So one may expect that some transport properties of quark matter in compact stars can be explored in the terrestrial experiments.
In the strong magnetic field B the Hall conductivity or the thermal Hall conductivity becomes comparable with the longitudinal conductivity . In such situations, the careful treatment of contributions from the Landau levels is required. We have analyzed the B dependence of as well as density dependence.
Theoretically we have found some geometric or topological effects. In particular, we have seen that spectral asymmetry plays an important role through the invariant in the presence of the magnetic field. The invariant leads to the anomalous particle number and the anomalous Hall conductivity is proportional to it.
We have put a special emphasis on a similarity between the DCDW phase and WSM, but there is a subtle difference between them; the expression of the anomalous Hall conductivity is different between them. We have seen this difference may be originated from axial anomaly, but further discussions are needed to clarify it by way of e.g. the boundary-bulk correspondence.. We also paid some attention to the matter contribution to the quantum Hall conductivity. We derived an analytic formula for non-dissipative part of the Hall conductivity. Based on this formula we have examined its -dependence in the limit, → 0. We have seen that the expected standard behavior of the Hall conductivity for conducting media, Fermi ∝ / , comes from non-dissipative part in the high field limit, while it should come totally from dissipative part in the opposite limit. When the magnetic field is strong, the Hall conductivity mostly comes from the LLL contribution and other higher levels decouple. In this situation, the fermion density is proportional to magnetic field ∝ ( ) . As a consequence, the Hall conductivity becomes independent of magnetic field and remains finite in the high field limit → ∞ as it is just proportional to the Fermi momentum. It should be also interesting to see that the classical contribution becomes vanished in this limit.
The magnetic-field dependence of the thermal Hall conductivity is phenomenologically important to understand thermal evolution of compact stars: an anisotropy of the thermal transport parallel and perpendicular to the magnetic field becomes important there. We have seen that the anomalous Hall effect should be dominant over the usual Hall effect in the strong magnetic field. We have also shown that the quantum contribution II becomes essential, compared with the classical analog I in the high-field limit. These results raise careful discussions of thermal evolutions of compact stars as a future work.
Author Contributions: All authors have read and agreed to the published version of the manuscript.
Funding: The work of H. A. was supported by JSPS KAKENHI Grant Number JP19K03868.