Drag Suppression in Anomalous Chiral Media

Drag suppression in anomalous chiral media

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As Published http://dx.doi.org/10.1103/PhysRevD.93.125026

Publisher American Physical Society

Version Final published version

Citable link http://hdl.handle.net/1721.1/107705

Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 93, 125026 (2016) Drag suppression in anomalous chiral media

† Andrey V. Sadofyev1,2,* and Yi Yin3, 1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2ITEP, B. Cheremushkinskaya 25, Moscow 117218, Russia 3Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA (Received 9 January 2016; revised manuscript received 18 March 2016; published 20 June 2016) We study a heavy impurity moving longitudinal with the direction of an external magnetic field in an anomalous chiral medium. Such system would carry a nondissipative current of chiral magnetic effect associated with the anomaly. We show, by generalizing Landau’s criterion for superfluidity, that the “anomalous component” which gives rise to the anomalous transport will not contribute to the drag experienced by an impurity. We argue on a very general basis that those systems with a strong magnetic field would exhibit an interesting transport phenomenon—the motion of the heavy impurity is frictionless, in analogy to the case of a superfluid. We demonstrate and confirm our general results with two complementary examples: weakly coupled chiral fermion gases and strongly interacting chiral liquids.

DOI: 10.1103/PhysRevD.93.125026

I. INTRODUCTION Specifically, we put those chiral systems in the presence of a background magnetic field B. Those systems would Recently, novel transport phenomena of chiral (parity- exhibit a CME current (1) that is originated from the violating) media tied with the quantum anomaly has “anomalous component” (which we identify explicitly attracted a lot of interest. A prominent example is the below) in the media.1 We then consider a heavy impurity chiral magnetic effect (CME) [1,2], the presence of the Bˆ vector (electric) current along the direction of the magnetic moving longitudinal to the direction. Will the motion of a field B: heavy impurity experience any drag? We shall show, by generalizing Landau’s criterion for superfluidity [8], that the “anomalous component” which jV ¼ CAμAB: ð1Þ contributes to CME current does not contribute to the C ¼ 1=2π2 friction acting on the impurity. This implies that CME Here A is the coefficient in front of the chiral current will not be destroyed by the presence of impurities. ∂ jμ ¼ C E B μ anomaly: μ A A · and A is the axial chemical We further argue in such a strong magnetic field that chiral potential. The broad set of systems exhibiting CME and media are populated with the “anomalous component,” other related anomalous transport phenomena is widely these systems would exert no drag on an impurity moving discussed in the literature and includes the primordial parallel with B. While frictionless motion of a heavy- electroweak plasma in early universe (see e.g., [3]), the impurity in a chiral medium is naturally connected to the QCD matter created in heavy-ion collisions (see e.g., [4]) nondissipative nature of the anomalous transport, this novel — and newly discovered condensed matter systems Weyl phenomenon has not been reported to the extent of our and Dirac semimetals (see e.g., [5,6]). knowledge. We emphasize that the necessary condition to One salient feature of those anomaly-induced currents is realize this drag-free motion is the presence of a strong that they are dissipationless. In this respect, a chiral magnetic field. In the weak magnetic field regime as medium is similar to a superfluid. In contrast to the considered in Refs. [9,10], there will still be a nondissi- Ohm current, both the CME current and the flow of the pative anomalous current but drag force is nonzero in those superfluid component do not produce entropy. Another situations. interesting feature of an ordinary superfluid is that the Our general results will be illustrated and supported by motion of a heavy impurity through a superfluid is two examples. We show that the drag force is absent for a frictionless. What would happen if a heavy impurity is weakly-coupled chiral fermion gas when the impurity moving in an anomalous chiral medium? velocity satisfies v<1 and for a strongly coupled chiral In this paper, we wish to obtain further insights into the fluid when v

2470-0010=2016=93(12)=125026(6) 125026-1 © 2016 American Physical Society ANDREY V. SADOFYEV and YI YIN PHYSICAL REVIEW D 93, 125026 (2016) v E ðk Þ¼k : ð Þ magnetic wave (CMW) [11]. Remarkably, χ will also LLL z z 3 approach 1 in the strong magnetic field limit [11], therefore results in those two examples coincide. Throughout the It is well known that HLL will not contribute to the CME paper, we set the Fermi velocity (or the speed of light) and current jz as the contribution from a fermion with kz cancels the electric charge e to unity. that with −kz [16]. On the other hand, the contribution from LLL is nonzero and precisely gives CME current in (1).We ’ therefore identify LLL as the “anomalous component” and II. LANDAU S CRITERION FOR SUPERFLUIDITY “ ” AND ANOMALOUS TRANSPORT HLL as the normal component of the system. Let us consider the motion of a heavy impurity along Bˆ . Let us begin by reviewing the pertinent ingredients of We see immediately, by extending Landau’s criterion that Landau’s criterion for superfluidity. We consider a heavy LLL would not contribute to the friction. Indeed, if a chiral impurity moving through a medium along z-direction fermion at LLL gains momentum qz from the impurity, the (without loss of generality) with velocity v. If this motion energy of that chiral fermion increases, according to the q E ðk Þ¼k is accompanied by friction, a part of momentum z carried dispersion relation of LLL: LLL z z, by the amount by the heavy impurity will be transferred to the medium ΔE ¼ qz, i.e., ΔE=qz ¼ 1. Obviously, the kinematic con- and the heavy impurity will lose kinetic energy Ω ¼ vqzþ strain (2) will not be satisfied for a heavy-impurity moving 2 Oðqz Þ. As usual we assume that qz is much smaller than the at v<1. One can show similarly that the transition from momentum of the heavy impurity Mv. In other words, we LLL to HLL is also energetically impossible for v<1. assume that the impurity is so heavy that it takes many The fact that LLL is chiral plays a crucial role in this collisions to change its momentum substantially. This is the analysis. For a dispersion relation which is even with condition that motion of the heavy impurity can be respect to kz, the kinematic constraint can always be described as a random walk and has been widely used satisfied via backward scattering. This difference is intui- in previous studies (cf., Ref. [12]). Meanwhile, by gaining tively clear: when colliding with heavy particles, light momentum qz, the energy of the medium will increase by particles will be bounced back. However, such backward the amount ΔEðqzÞ which equals to Ω by the conservation scattering is forbidden for chiral fermions at LLL as their of energy. This transition will be possible only if chirality is slaved to the direction of their momentum along z-direction. Therefore, those chiral fermions will pass ΔEðqzÞ=qz ¼ Ω=qz ¼ v: ð2Þ through the heavy particle without transferring momentum. One could extend the analysis above to other chiral ΔEðq Þ=q v Thus, if z z has a finite minimum, lim, the media in the presence of magnetic field B. In those systems, vchiral anomaly and CME [17]. directly related to the excitation spectrum of the system. Since those zero modes are P-odd, expanding their This is a familiar picture about the ordinary superfluid. It is dispersion relation around the level crossing point, one ’ ϵ ðk Þ ∝ k worth pointing out here that the applicability of Landau s will always find zero z z which is chiral. Therefore, criterion for superfluidity, which relates the vanishing of the existence of the zero modes with chiral dispersion the drag force to the specific features of the spectrum of relation should be universal, independent of the micro- the media, is not limited to the superfluid. Furthermore, the scopic details of the system.2 Consequently, our previous drag suppression also does not necessarily imply that the analysis on LLL can be readily applied to zero modes of medium is a superfluid, see Refs. [13,14] and references other chiral media with the anomaly. therein for examples. The concept of zero modes is of topological nature. We now turn to an anomalous chiral medium, the central Hence it is well-expected that they would not contribute topic of this paper. Let us first consider the simplest to the dissipation. In fact, the dissipationless nature of example of such a system: a gas of right-handed chiral the anomalous transport has been used as a guiding fermions. In the presence of a static and homogeneous principle in Refs. [18,19] to derive the anomalous hydro- magnetic field B along, say z-direction, the energy spec- dynamics. Despite this, a simple microscopic physical trum of those chiral fermions is given by Landau levels interpretation presented in this paper, which connects the (LL) [15]. The higher Landau levels (HLL) are even with chiral dispersion of zero modes and Landau’s criterion for respect top theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi momentum along z-direction kz, i.e., superfluidity, has not been revealed till now. E ðk Þ¼ 2nB þ k2 n ¼ 1; 2; 3… n z z for . In contrast, the pffiffiffiffi lowest Landau Level (LLL) is odd in kz and it has a level 2For systems with size smaller than 1= B, zero modes would crossing point at kz ¼ 0: be absent and we do not consider this possibility.

125026-2 DRAG SUPPRESSION IN ANOMALOUS CHIRAL MEDIA PHYSICAL REVIEW D 93, 125026 (2016) III. DRAG SUPPRESSION IN CHIRAL MEDIA all orders in medium-medium couplings), can be derived using Fermi’s golden rule [20,21]: We now consider a chiral medium at finite density and 2 temperature T in a strong external magnetic field eB ≫ μ , dR ρðω; qÞ 2 3 2 T ð2πÞ ¼jUqj : ð5Þ that the medium is dominant by the zero modes. For d3q 1 − e−ω=T such systems with one chirality (i.e., with right or left- handed fermions only), it is clear that a heavy impurity U UðxÞ ρðω; qÞ ˆ Here q is the Fourier transform of and is the moving longitudinally to B will not experience any drag R spectral density. As usual, ρðω; qÞ¼−2ImG ðω; qÞ with force for v smaller than a characteristic value, determined R G ðω; qÞ ∼ hnVnVi the retarded Green’s function. The drag by the chiral dispersion of the zero modes. In analogy to the force is related to the momentum transfer rate: F ¼ ordinary superfluid that the scarcity of low lying excited R R dP=dt ¼ − RðΩ ; qÞq ≡ d3q=ð2πÞ3 states implies the “superfluidity,” the lack of “nonzero q q where q . By noting modes” for chiral media in the strong magnetic field limit is ρðω; qÞ¼−ρð−ω; −qÞ, we then have: the physical origin of the drag suppression. Z dP The situation with more than one chirality, say with both 2 F ¼ ¼ − qjUqj ρðΩ ¼ v · q; qÞ: ð6Þ right-handed R and left-handed L fermions, is however dt q more subtle. When the medium-impurity coupling is weak, the dominant process for energy transfer involves only one The general expression (6) has been widely used in chirality, e.g., the two-to-two scattering such as HL → HL studying the drag force in the condensed-matter literature and HR → HR where H denotes the heavy impurity. Thus, (cf., [21]). (6) can also be matched to the computation our previous conclusion on the absence of drag force still of the drag force in quark-gluon plasma via perturbation holds. Of course, there are multiscattering processes theory [12]. which satisfy the kinematic constraint (2). Nevertheless, As the first example, we consider weakly coupled chiral this type of process will be suppressed as far as the fermions in the presence of B. ρðω; qÞ can be computed by coupling between the impurity and the medium is weak. considering a polarization loop with fermion propagator in Indeed, there are examples that Landau’s criterion is not the presence of magnetic field. In strong B limit, only LLL applicable if the medium-impurity interaction is not weak contributions are needed [22]: (cf. Refs. [13]). In general, the chirality flipping rate Γχ is nonzero yet eB −q2 =2B2 ρLLLðω; qÞ¼ e ⊥ ρ2Dðω;qzÞ; ð7Þ 1=Γχ is typically much longer than the characteristic time 2π scales of other relaxation processes. In a Weyl semimetal, the inverse of Γχ is given by the intervalley scattering mean where ρ2Dðω;qzÞ denotes the spectral density for a free free time while in quark-gluon plasma, Γχ is linked to the chiral-fermion in 1 þ 1 dimension (2D) and q⊥ denotes ˆ sphaleron transition rate. With finite Γχ, there could be the momentum transverse to B. (7) is expected from the scattering processes HL → HR in which L gains momen- dimensional reduction. Since 2D anomaly receives no tum and becomes R that could satisfy the kinematic thermal corrections, ρ2Dðω;qzÞ is independent of temper- constrain (2). Nevertheless, the probability of those proc- ature and density [23]: esses should be suppressed when Γχ is small. We will 2 ω2 − q2 discuss the effects of finite Γχ later. ρ ðω; qÞ¼− z 2D Im 2 2 π ðω þ iϵÞ − qz

IV. EXAMPLES ¼ ω½δðω − qzÞþδðω þ qzÞ: ð8Þ We now support our general results by the explicit computation of the drag force. Below, we will assume As expected, the delta function in (8) is related to the chiral δðω q Þ the coupling between the medium and impurity is weak, but dispersion of LLL with z corresponding to right- the interaction among chiral fermions can be strong. handed and left-handed chiral fermions. It is clear by v < 1 Let us start with a generic interacting Hamiltonian substituting (8), (7) into (6) that for z the drag force F between the fermion density nVðxÞ and the impurity density z vanishes due to the delta function in (8). This manifests n ðxÞ the kinematic constraint as we discussed earlier. Note that if imp : Z Z the heavy impurity is moving along the direction transverse Bˆ H ¼ d3x d3x0n ðx;tÞUðx − x0Þn ðx0;tÞ; ð Þ to , drag force is in general nonzero [24]. I V imp 4 As another example, we now consider a strongly interacting chiral fluid in the presence of a background where Uðx − x0Þ is the interacting potential. The momen- magnetic field B in which the mean free path is small. tum transfer rate between the impurity and the medium, at We could then assume that characteristic momentum the leading order in the medium-impurity coupling (but to transfer q between the impurity and the medium is in

125026-3 ANDREY V. SADOFYEV and YI YIN PHYSICAL REVIEW D 93, 125026 (2016) the hydrodynamic regime. Consequently, the form of Such B-dependence has also been observed in holographic R −1 G ðω; qÞ as well as ρðω; qÞ will be completely fixed by models [31]. By noting DT;L ¼ χ σT;L we will take hydrodynamics [25]. −1 −3 We start with the constitute relation for the vector and vχ → 1; χ → CAB; DL ∼ B ;DT ∼ B ; axial currents jV; jA. The CME current (1) and its cousin in ð13Þ the axial current—the charge separation effect [26] (jA ∼ CAμVB) modifies the constitute relation (see e.g., for the discussion below. In this limit, we could neglect DT [18,27]): Γ Γ ≈ Γ ≡ dependence of q and make approximation q qz 2 l l lm lm DLqz. JV ¼ CAμAB þ σ Em − D ∇mnV; ð9Þ We now read the relation between δnA and δA0 from (11) l l lm and match the results to the definition of the retarded Green JA ¼ CAμVB − D ∇mnA;l;m¼ 1; 2; 3; ð10Þ function: δnV ¼ GRðω; qÞδA0. This gives (see also Ref. [32]): where Dij is the diffusion coefficient tensor which is related ij to the susceptibility χ and the conductivity tensor σ by ωðω þ iΓ þ iΓ Þ ij ij ij qz χ Einstein’s relation σ ¼ χD . D may be decomposed GRðω;qzÞ¼χ 1 − ; ð14Þ ij ij ˆ i ˆ j ˆ i ˆ j Δðω;qzÞ [28] as D ¼ DTðδ − B B ÞþDLB B . We similarly introduce the transverse and longitudinal conductivities Δðω;q Þ¼ðω þ iΓ Þðω þ iΓ þ iΓ Þ − v2q2; ð Þ z qz qz χ χ z 15 σT;L ¼ χDT;L. We also note that the constitute relation depends on the choice of fluid frame. In (9), (10), we will where due to (13), q⊥-dependence is suppressed. use the “no-drag” frame, the frame in which the drag force Δðω;qzÞ¼0 determines the dispersion relation of will be absent if the impurity is at rest, that has been collective excitations of the system. We first consider recently discussed in Refs. [9,10]. the situation that Γq ≫ Γχ. We then have ωðqzÞ¼ We now restrict our discussion to a neutral chiral fluid, z vχqz − iΓq . This is nothing but CMW [11,27] and ρ ¼ i.e., background μV;A ¼ 0. This also allows us to neglect z −2 mG possible axial current induced by electric field [29] and I R gives: B additional nonlinear term in as considered in Ref. [30], ωχΓ ρ ðω;q Þ¼ qz þðv → −v Þ : ð Þ and perturb the system by imposing a space-time dependent hyd z 2 2 χ χ 16 −iωtþiq·x ðω − v q Þ þ Γ gauge potential δA0 ∝ e . That perturbation will χ z qz lead to currents and chemical potentials δμV;A fluctuations. μ μ As a result of (13), (16) becomes: Substituting (9), (10) into ∂μjV ¼ 0 and ∂μjA ¼ CAE · B − ΓχnA where we have incorporated the finite chirality ρ ðω; qÞ ≈ ωχπ½δðω − vχqzÞþδðω þ vχqzÞ: ð17Þ flipping rate Γχ, we have: hyd

! Again, the drag force vanishes for v

δn ¼ Here, we use the linearized equation of state V;A A. Effects of chirality flipping rate 2 χδμV;A þ Oðδμ Þ and introduce We now return to the effects of a finite chirality Γ Γ ≡ q Dijq ¼ðD q2 þ D q2 Þ;v≡ ðC BÞ=χ: flipping rate χ using (14). It is instructive to first take q i j L z T ⊥ χ A Γ ≪ Γ the opposite limit that χ qz . The dispersion relation ð12Þ from Δðω;qzÞ¼0 then leads to the “anomalous” diffusive mode [33]: One immediately finds that vχ is the speed of CMW [11]. 2 2 Before proceeding further, let us discuss vχ and DT, DL ωðqzÞ¼−iDχqz ;Dχ ¼ vχ=Γχ; ð18Þ in large B limit. As shown in Ref. [11], vχ approaches 1 in large B limit. This also implies that χ → CAB in this limit. where Dχ is directly related to the anomaly-induced 2 Under the Drude approximation, σT ∼ 1=B as the motion negative magnetoresistance σχ [34] via the Einstein relation of charge carriers in transverse plane is suppressed by σχ ¼ χDχ. The behavior of ρ is then dominated by this large B. Meanwhile, σL shows no dependence on B as diffusive mode. The drag force now becomes nonzero even charge carriers moving along Bˆ feels no Lorentz force. for a very small v. However, the drag force coefficient η,

125026-4 DRAG SUPPRESSION IN ANOMALOUS CHIRAL MEDIA PHYSICAL REVIEW D 93, 125026 (2016) anomalous chiral medium in the strong magnetic field B limit exhibits a novel transport phenomena—a heavy impurity moving longitudinal to Bˆ becomes frictionless. It is worth noting that this generic result is insensitive to the specific realization of the chiral medium and particular types of impurities. For example, in a recent publication. [24], the heavy quark drag force coefficient of quark gluon plasma (QGP) with strong magnetic field has been computed at the leading order in αs, where αs is the strong coupling constant. The suppression of drag force longitudinal to the magnetic field direction has also been observed. We now discuss potential applications of our results. A very strong magnetic field is present at early times of the FðvÞ Bˆ FIG. 1. Drag force of a heavy impurity moving along - QGP created in heavy-ion collisions. The suppression of direction with velocity v in an anomalous chiral fluid with a longitudinal drag will lead to a strong anisotropy in the drag strong magnetic field. We employ a toy potential (see text for force coefficient. As discussed in detail in Ref. [24], the details). Dash-dotted black curve plots Mη0v. Here, we intro- duced η0, which is η with Γχ=DLΛ ¼ 20 to normalize FðvÞ. motion of heavy quarks in QGP and thus D-meson spectrum as measured in experiments will be influenced by such anisotropy. In addition, it would also be interesting FðvÞ¼−ηMv v which enters in for small is tied to the to explore the consequence of the drag suppression in chiral Γ v ≪ v ρðω; qÞ chirality flipping rate χ. Indeed, for χ, medium in condensed matter systems such as Weyl and obtained from (14) gives ρðω ¼ vqz;qzÞ¼ð2vχΓχÞ= Dirac semimetals. 2 ðvχqzÞ. Substituting it into (6),wehaveη ∝ Γχ. In the work, we consider the situation that the coupling To illustrate the effects due to intermediate Γχ, i.e., Γχ ∼ of the impurity to the medium is weak. It would be Γ qz and summarize the above two sections, we compute the interesting to extend the current analysis to the case when drag force FðvÞ using a toy potential UðqÞ. We consider a the impurity is strongly coupled to the medium. We defer −q2=Λ2 potential of the form: UðqÞ¼U⊥ðq⊥Þe z where Λ sets this for future study. the cutoff scale for hydrodynamic regime. FðvÞ is computed using (6), (14) with different Γχ ≪ Λ. In numerics, ACKNOWLEDGMENTS 2 we set DLΛ ¼ 0.001 to mimic large B limit (13). As Fig. 1 The authors would like to thank K. Rajagopal, M. demonstrates, the suppression of the drag force is trans- Stephanov, Naoki Yamamoto for very helpful comments parent for small Γχ. For sufficiently large Γχ, FðvÞ grows on the draft and K. Fukushima, J. Goldstone, K. Hattori, R. linearly for small v with drag force coefficient η propor- Jackiw, D. Kharzeev, L. McLerran, F. Wilczek, Ho-Ung. tional to Γχ. Yee, V. I. Zakharov for useful conversations. We also thank the referees for providing constructive comments and help V. CONCLUSIONS AND OUTLOOK in improving the presentations of this paper. The work of In this paper, we observe that Landau’s criterion for Y. Y. was supported by DOE Contract No. DE-SC0012704. superfluidity can reveal the microscopic origin of the The part of the work of A. S. (Secs. III and IV)was nondissipative nature of the anomalous transport: energy supported by Russian Science Foundation Grant No 16-12- and momentum exchange between an heavy impurity and 10059, the remaining part of the work of A. S. was zero modes is kinematically forbidden. We show that an supported by DOE Contract No. DE-SC0011090.

[1] A. Vilenkin, Phys. Rev. D 22, 3080 (1980). [4] D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, Nucl. [2] D. E. Kharzeev, K. Landsteiner, A. Schmitt, and H.-U. Yee, Phys. A803, 227 (2008); D. Kharzeev and A. Zhitnitsky, Nucl. Lect. Notes Phys. 871, 1 (2013); D. E. Kharzeev, Prog. Part. Phys. A797,67(2007); J. Liao, Pramana 84, 901 (2015). Nucl. Phys. 75, 133 (2014). [5] D. Ciudad, Nat. Mater. 14, 863 (2015). [3] M. Giovannini and M. E. Shaposhnikov, Phys. Rev. D 57, [6] M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian, C. 2186 (1998). Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, H. Lin,

125026-5 ANDREY V. SADOFYEV and YI YIN PHYSICAL REVIEW D 93, 125026 (2016) A. Bansil, F. Chou, and M. Z. Hasan, Nat. Commun. 5, 3786 [20] M. L. Bellac, Thermal Field Theory (Cambridge University (2014). Press, Cambridge, England, 2011). [7] M. Lublinsky and I. Zahed, Phys. Lett. B 684, 119 (2010); [21] P. Nozieres and D. Pines, Theory Of Quantum Liquids, A. V. Sadofyev, V. I. Shevchenko, and V. I. Zakharov, Phys. Advanced Books Classics Series (Westview Press, 1999); S. Rev. D 83, 105025 (2011). Caron-Huot and G. D. Moore, Phys. Rev. Lett. 100, 052301 [8] I. Khalatnikov, Advanced Books Classics Series, Advanced (2008). Book Program (Perseus, Cambridge, 2000). [22] E. V. Gorbar, V. A. Miransky, and I. A. Shovkovy, Phys. [9] K. Rajagopal and A. V. Sadofyev, J. High Energy Phys. 10 Rev. D 83, 085003 (2011); Phys. Rev. B 89, 085126 (2014); (2015) 018. K. Fukushima, Phys. Rev. D 83, 111501 (2011). [10] M. A. Stephanov and H.-U. Yee, Phys. Rev. Lett. 116, [23] R. Baier and E. Pilon, Z. Phys. C 52, 339 (1991). 122302 (2016). [24] K. Fukushima, K. Hattori, H.-U. Yee, and Y. Yin, arXiv: [11] D. E. Kharzeev and H.-U. Yee, Phys. Rev. D 83, 085007 1512.03689. (2011). [25] L. P. Kadanoff and P. C. Martin, Ann. Phys. (N.Y.) 24, 419 [12] G. D. Moore and D. Teaney, Phys. Rev. C 71, 064904 (1963). (2005). [26] D. T. Son and A. R. Zhitnitsky, Phys. Rev. D 70, 074018 [13] A. Y. Cherny, J.-S. Caux, and J. Brand, Front. Phys. 7,54 (2004). (2012). [27] G. M. Newman, J. High Energy Phys. 01 (2006) 158. [14] V. B. Bobrov, S. A. Trigger, and D. I. Litinski, arXiv: [28] E. Lifshitz, L. Pitaevskii, and L. Landau, Physical Kinetics 1407.6184; V. B. Bobrov and S. A. Trigger, J. Low Temp. (Pergamon Press, Oxford, 1981), Vol. 60. Phys. 170, 31 (2013); V. B. Bobrov and S. A. Trigger, Prog. [29] X.-G. Huang and J. Liao, Phys. Rev. Lett. 110, 232302 Theor. Exp. Phys. 2013, 043I01 (2013). (2013). [15] L. Landau and E. Lifshits, Quantum Mechanics: [30] E. V. Gorbar, I. A. Shovkovy, S. Vilchinskii, I. Rudenok, A. Non-relativistic Theory (Butterworth-Heinemann, London, Boyarsky, and O. Ruchayskiy, Phys. Rev. D 93, 105028 1977). (2016). [16] M. A. Metlitski and A. R. Zhitnitsky, Phys. Rev. D 72, [31] M. Ammon, T. H. Ngo, and A. O’Bannon, J. High Energy 045011 (2005). Phys. 10 (2009) 027. [17] H. Nielsen and M. Ninomiya, Phys. Lett. 130B, 389 [32] Y. Yin, Phys. Rev. C 90, 044903 (2014). (1983). [33] K. Landsteiner, Y. Liu, and Y.-W. Sun, J. High Energy Phys. [18] D. T. Son and P. Surowka, Phys. Rev. Lett. 103, 191601 03 (2015) 127; M. Stephanov, H.-U. Yee, and Y. Yin, Phys. (2009). Rev. D 91, 125014 (2015). [19] D. E. Kharzeev and H.-U. Yee, Phys. Rev. D 84, 045025 [34] D. T. Son and B. Z. Spivak, Phys. Rev. B 88, 104412 (2011). (2013).

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