NUPHB14399S0550-3213(18)30195-010.1016/j.nuclphysb.2018.07.009The AuthorsHigh Energy Physics – PhenomenologyOn magnetostatics of chiral mediaZ.V.KhaidukovaV.P.Kirilinab⁎vkirilin@princeton.eduA.V.SadofyevacdV.I.ZakharovaeaITEP, B. Cheremushkinskaya 25, Moscow, 117218, RussiaITEPB. Cheremushkinskaya 25Moscow117218RussiabDepartment of Physics, Princeton University, Princeton, NJ 08544, USADepartment of PhysicsPrinceton UniversityPrincetonNJ08544USAcCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USACenter for Theoretical PhysicsMassachusetts Institute of TechnologyCambridgeMA02139USAdTheoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USATheoretical DivisionLos Alamos National LaboratoryLos AlamosNM87545USAeSchool of Biomedicine, Far Eastern Federal University, Sukhanova str 8, Vladivostok, 690950, RussiaSchool of BiomedicineFar Eastern Federal UniversitySukhanova str 8Vladivostok690950Russia⁎Corresponding author.Editor: Tommy OhlssonAbstractWe consider magnetostatics of chiral media with a non-vanishing chiral chemical potential μ5≠0. In such media, the chiral anomaly has macroscopic manifestations which go beyond the standard classical electrodynamics. A new term in the effective action takes form of a topological three-dimensional photon mass. The topological mass turns to be imaginary and signals instability. Namely, magnetic field with non-trivial helicity is generated spontaneously. A stable state corresponds to a solution of the Beltrami equations. From the action relevant to instability, we show also that the interaction between two current loops reduces to the linking number of the loops. In addition, we observe the vanishing of the chiral magnetic conductivity in the strict static and uniform limit.1IntroductionElectrodynamics of chiral media attracted a lot of attention recently. One expects that in such media the triangle chiral anomaly, which is a pure quantum phenomenon, has macroscopic manifestations, for a review and further references see [1]. In particular, an external magnetic field Bi (i=1,2,3) induces an electric current proportional to μ5 [2–4]:(1)Ji=σBi where σ can be referred to as the chiral conductivity. The chiral conductivity can readily be evaluated on one-loop level, σ0=(e2μ5)/(2π2). Moreover, there exist beautiful theorems on non-renormalizability of σ rooted in the non-renormalizability of the triangle anomaly. As first argued in [5] the non-renormalizability of σ persists even in hydrodynamics.Another remarkable feature of the chiral magnetic effect (1) is that the current is non-dissipative, as a consequence of the time-reversal invariance of strong interactions [6]. In this respect, the chiral conductivity is similar to the Hall conductivity (for a review and references see, e.g., [7,8]). The non-dissipativity of the current (1) might become a basis for new technologies [9].Non-dissipativity is usually related to topological nature of the corresponding effects, see, e.g., [7]. It would be important therefore to clarify the topological origin, if any, of the chiral magnetic effect (CME), especially in view of the absence of any apparent temperature dependence of the chiral conductivity. At vanishing interactions there exist derivations of the CME in terms of the Berry phase in the momentum space [10–12] which shed some light on the topology related to the chiral magnetic effect.It is crucial to emphasize that the chiral magnetic effect is a feature of the equilibrium physics and, in field-theoretic language, represents a static effect. Namely, the chiral conductivity, to the linear order in the external field, is given by the antisymmetric part of the static current–current correlator at 3d momentum tending to zero, ki→0:(2)σ=limkn→0ϵijni2knΠij, where Πij=〈Ji,Jj〉. Eq. (2) is to be contrasted with the standard Kubo formulae which assume another limiting procedure, ki=0 with frequency ω→0. The chiral conductivity is actually static, suggesting the recently coined name “thermodynamic susceptibility” to describe it [10,13].In this note we concentrate, therefore, on the long-range forces in the static limit. The point is that the chiral magnetic effect (1) modifies in fact the Maxwell electrodynamics. Namely, one should take into account the backreaction of the medium, or the magnetic field generated by the current (1). The modification turns drastic at distances of order r∼(e2μ5)−1.We perform calculations using an effective action with an extra piece accounting for the effects of the anomaly:(3)δLeff=12σ0Ai∂jAnϵijn, where Ai is the vector potential of electromagnetic field. Effective action has been widely used, in particular, in thermodynamic studies of the chiral effects, see, e.g., [14]. Similar expansions are common in theory of the Hall effects. Note, however, that we do not have a gap and, therefore, the validity of the effective-action approximation, or expansion in momenta is a subtle issue, to our mind. A closer look into the dynamics is needed. Here we note only that there exist schemes where the value of the chiral magnetic effect is related to the details of ultraviolet renormalization, or polynomials in the correlator of the currents (2), see, e.g. [15]. Within such schemes the validity of the effective-action approximation is granted.Amusingly enough, in this approximation the magnetostatics of the four-dimensional (4d) electrodynamics of chiral media is described by the Euclidean version of the 3d electrodynamics with the so called topological photon mass considered in many papers, see in particular [16–18]. From our perspective, it is the topological aspects and non-renormalization theorems of the 3d theory which are most relevant. In particular, it was emphasized in [18,19] that the 3d topological photon mass corresponds to an instantaneous interaction, and this observation adds insight to understanding the non-dissipativity of the chiral magnetic effect. The non-renormalizability of the 3d topological mass [17] has already been exploited in the evaluation of the chiral vortical effect in the 4d case [20]. Below, we will consider another application of the technique of the Ref. [17] for the CME current [45].The most specific feature of the case considered is that the 3d topological photon mass turns to be imaginary, thus signaling instability [22] of the external magnetic field in chiral media. Magnetic fields are rearranged to a kind of a self-consistent distribution with nonzero field helicity H:(4)H=∫A→⋅B→d3x, which should be included into the helicity conservation law. With this rearrangement, the chiral charge Q5 of the constituents decreases, see also [23,24]. This instability and other aspects of the backreaction of the chiral effects on the medium were widely discussed in the literature, see e.g. [25–37].Furthermore, we will also demonstrate that the term iσ0ϵijlkl in the one-loop current–current correlator induces a similar term in the exact propagator for the abelian gauge field,(5)δDij(ω=0,k→)→−iϵijlklσ0k2,k→→0. The term (5) results in a topological type interaction of static current loops (k≪σ0), proportional to their linking number.2PropagatorLet us start by setting up notations. We use the metric gμν=diag(−,+,+,+). The theory we will consider is QED with chiral fermions at a non-zero value of μ5. The corresponding action is given by:(6)S=∫dtd3x[ψ¯(iγμDμ+μ5γ0γ5)ψ−14FμνFμν]. Here Dμ=∂μ−ieAμ. As is already mentioned above, all the quantities we will calculate are taken at ω=0. This puts us into an effective 3d situation. Also, at ω=0 all Minkowski-space two-point functions coincide with their Euclidean counterparts.Consider now the dressed photon propagator. Rotational and gauge invariances fix its form as:(7)Dij(0,k→)=DS(k)(δij−kˆikˆj)+DA(k)iϵijlkˆl+akˆikˆjk2, where kˆ is the unit vector along the 3d momentum and the last term is the gauge fixing while the indices S,A refer to symmetric and antisymmetric parts, respectively. Similarly, the inverse photon propagator obeys(8)Dij−1(0,k→)=PS(k)(δij−kˆikˆj)+PA(k)iϵijlkˆl+k2akˆikˆj. It is straightforward to show that(9)DS=PSPS2−PA2,DA=PAPA2−PS2. Moreover, it is well known that the difference between inverses of the exact and bare photon propagators is given by the sum of 1-particle-irreducible (1PI) graphs of the photon self-energy,(10)Dij−1=D(0)ij−1−Pij.We shall use this relation to find the infrared limit of the functions ΠS and ΠA. At one loop Pij=Πij, and one readily finds(11)PA=−σ0k+O(k2). As for the symmetric part,(12)PS=O(k2), to maintain the gauge and Lorentz invariances.If we keep only linear terms in the functions PA,PS, the propagator (7) takes a particularly simple form, corresponding to the summation of the bubble-type diagrams:(13)Dij(0,ki)=1k2−σ02(δij−kikjk2)+iσ0ϵijlklk2(k2−σ02)+akikjk4. This means that the photon has acquired an imaginary mass, m=iσ0 which results in an instability [22] which we will discuss later. Moreover, a novel pole at k2=0 appears in (13). As we show later, it is responsible for another type of topological interaction in the infrared limit.If we include terms beyond linear in (11) and (12) and all the diagrams of higher order in e2, the propagator (13) would be modified. We will now demonstrate that in spite of that the relation (5) is exact and unaffected by the interactions. It is sufficient to this end to prove that (11) is actually valid to all orders in e2. The proof is along the lines of the Coleman–Hill theorem [17,20]. The main idea is to construct all the diagrams contributing to Pij using the n-photon effective vertices(14)Γμ1...μn(n)(k(1)...,k(n)). These vertices determine the dressed propagator. Indeed, if we cut all internal photon lines then a generic diagram would be obtained by multiplying various Γ's by the corresponding propagators, and performing loop integrations when necessary.Next, we proceed by noting that it is sufficient to study Γ(n) in the Euclidean space (our external momenta are Euclidean by construction, and we can always Wick rotate the internal ones). Then we can claim the analyticity of Γ(n) as a function of its arguments. The role of an infrared regulator protecting us from the divergences which might stem from vanishing fermion mass is played by μ5. In complete analogy with [17] we conclude then(15)Γμ1...μn(n)(k(1),...,k(n))=O(k(1)...k(n)),n>2. We may now use this observation for a few purposes. First, if we start contracting the internal photon lines we would find out that for every factor k(i)−2 associated with a propagator, two factors of k(i) come from the Γ's to which the photon is attached. This protects us from additional infrared singularities.Second, and most important, we immediately see that all graphs which involve only Γ(n) with n>2 can only contribute O(k2) terms to the Pij. Indeed, if both external lines, carrying momenta k and −k are attached to a single fermion loop, then the graph is O(k2) by the formula in the preceding paragraph. If the lines are attached to different loops then a factor O(k) is associated with each of them. The internal photon propagators do not spoil this due to the arguments in the preceding paragraph. The last subtlety we need to address is the reason, why we need to consider Γ(n) only with n>2. By definition, Pij only involves 1PI diagrams, which cannot include an isolated Γ(2)(k,−k), where k is the external momenta, because that would mean we could cut one of the photon lines coming out of this Γ(2) to obtain a reducible diagram. Now all non-isolated Γ(2)'s, which carry the loop momentum k(i) would only emerge as dressing the bare propagator connecting other Γ(n), n>2 effective vertices. Upon summing up all the diagrams we would arrive to the conclusion that the worst singularity it can contribute is just k(i)−2 which would cancel with the corresponding factors of k(i) coming from effective vertices (as stated in the paragraph above).This concludes our proof that PA=−σk+O(k2). At one loop this formula holds, and all higher order corrections behave as O(k2). Similarly, PS=O(k2). This is a generalization of the well-known fact of the non-renormalization of the topological mass term in 2+1 dimensions [17]. Finally we get DA=−(σk)−1+O(1) and obtain the pole term (5). We also note that this term fixes the exact (non-1PI) antisymmetric part of the current–current correlator as:(16)ΠijA=−ik2σ0ϵijlkl+O(k4), which implies that the CME vanishes in the strict k→0 limit.3Long-range forceAs shown in [18], existence of the pole (5) implies a new interaction term for two external currents, which has the form of the linking number. To see this, we first study the long distance limit of the field produced by external sources Ji(ω=0,k→), which we take to be static and purely three-dimensional (J0=0) from the start. Using the propagator (5) we get(17)Ai(k→)=−iϵijkkkσ0k2Jj(k→)+... Furthermore, consider current of the form Ji(x)=I∫dτδ3(x→−x→(τ))x˙i(τ), essentially representing an infinitely thin static loop with the total current (total charge passing through the cross section per unit time) equal to I. Then it is straightforward to demonstrate that Eq. (17) implies that:(18)V=2II′σ0∫C∫C′dxidyjϵijk(x−y)k4π|x−y|3 The double integral here is proportional to the Gauss linking number of the two current loops.To reiterate, we have obtained that static current loops interact topologically, that is insensitive to the distance between the loops. The topological term in (13) is exact in the infrared limit. For currents varying faster this interaction does get considerable corrections. Similarly, exact values of terms having a pole at k2∼σ02 depend on details of interaction. It makes one to suggest that if theory is expanded around the proper vacuum the topological term would take the same form while pole terms will be changed to be regular.4On the consistency of the static approximationAs we already noticed, the results obtained for the photon propagator can be also neatly expressed in terms of an effective action. If we put all our fields to be static from the beginning, then the low-energy effective action obtained [46] after integrating the fermions out would look like [22]:(19)S=∫d3x[−14FijFij+σ02ϵijkAi∂jAk]. This is the Euclidean version of the well-known 3d Maxwell–Chern–Simons theory. In this case, however, the mass is imaginary. As a result, the propagator (13) has a pole at k2=σ02, i.e. in the physical region.At least naively, the presence of the pole at k2=σ02 indicates that in the true vacuum there are static waves with fixed k2. Although this result might look bizarre at first sight, fixed values of the momentum is indeed a feature of the solutions to the Beltrami equation:(20)curlB=σ0B, which can be derived from the effective action (19). Moreover, the simplest solution, the so called Arnold–Beltrami–Childress flow [38–40], is characterized by three vectors k→1,2,3 which form a left- or right-handed vector basis. Emergence of such a handedness in the true vacuum is indicated by presence of the pole at k2=σ02 (see Eq. (13)) in front of the structure ϵijlkl. The standing wave ground states are known to develop in the context of QCD, as shown in [41] in case of baryon chemical potential and in [24] for μ5. In QCD the instability sets in at a certain critical value of μ5. It is absent in our QED case because the corresponding condensed vector meson–photon is massless unlike the rho-meson in the case of QCD.The physical picture corresponding to the magnetic field rearrangement is as follows. Chiral particles move chaotically without magnetic field. With field turning on, particles momenta become ordered along the field and helicity changes macroscopically. The current of chiral particles results, in turn, in a magnetic field distribution with macroscopic nonzero helicity. Finally, a self-consistent distribution of the currents and magnetic fields is set in. Through the process of evolution to the equilibrium distribution chiral charge of the constituents decreases while helicity of magnetic field configuration is increasing.5ConclusionsIn this letter we have demonstrated the emergence of the CS-type term in the effective action for the magnetostatics in presence of axial chemical potential. We have proven that this term is infrared exact. As a consequence, static current loops interact topologically, in the form of Gauss linking number. Moreover it implies that CME current vanishes in the strict k→0 limit.The effective action relevant to the magnetostatics belongs to the same sequence as effective action relevant to the Hall effect and this observation supports the idea that the chiral magnetic effect is topological in origin. An important difference, however, is that the effective action indicates to the instability of any external magnetic field in chiral media. Furthermore, it is natural to assert that a distribution of magnetic fields, of order B∼μ52 is generated spontaneously in chiral media with μ5≠0. The total chiral charge is shared then between the charge of elementary constituents and helicity of the magnetic field. The stable state could either be a smooth macroscopic solution to the Beltrami equation or, even, a kind of a turbulent or chaotic state, as is typical for the solutions of the Beltrami equations, see, e.g., [38]. This stable state can be viewed as a physical realization of the CME.AcknowledgementsThe authors are grateful to M.I. Polikarpov and O.V. Teryaev for useful discussions. They also acknowledge helpful remarks of V.A. Rubakov. The work of AS was supported in part by the U.S. Department of Energy under grant Contract Number DE-SC0011090. The work of VK and VZ was supported by RFBR grant 17-02-01108.References[1]D.E.KharzeevK.LandsteinerA.SchmittH.-U.YeeStrongly interacting matter in magnetic fields: an overviewLect. 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