Positive magnetoresistance induced by hydrodynamic fluctuations in chiral media

We analyze the combined effects of hydrodynamic fluctuations and chiral magnetic effect (CME) for a chiral medium in the presence of a background magnetic field. Based on the recently developed non-equilibrium effective field theory, we show fluctuations give rise to a CME-related positive contribution to magnetoresistance, while the early studies without accounting for the fluctuations find a CME-related negative magnetoresistance. At zero axial relaxation rate, the fluctuations contribute to the transverse conductivity in addition to the longitudinal one.


Introduction
The transport properties of a chiral medium (many-body system involving chiral fermions) and their deep connection to quantum anomalies have attracted significant interests recently. Of particular importance is the behavior of electric conductivity (or its inverse, electric resistance) under the external magnetic field B. In table-top experiments, the negative magnetoresistance is proposed as a signature of the chiral magnetic effect (CME), the anomaly-induced vector current in the presence of magnetic field and chiral charge imbalance [1][2][3][4]. Indeed, as shown in refs. [5][6][7], the balance between the axial charge density n A production due to the chiral anomaly and axial charge relaxation requires that in a steady state in the presence of electric field E, the axial charge density n A ∝ CE · B/r, where r denotes the axial charge relaxation rate and C is the anomaly coefficient. Then, with the CME, one finds an additional contribution to the longitudinal conductivity ∆σ L ∝ C 2 B 2 r . (1.1) The measurements of magnetoresistance in Weyl and Dirac semimetals have been reported in refs. [8][9][10][11]. Nevertheless, the fluctuation effects have not yet been taken into account in eq. (1.1). It is well-known that in an ordinary fluid, the fluctuations and interactions among sound and diffusive modes lead to significant effects on the behavior of transport coefficients [12][13][14][15][16][17][18]. Therefore, one may naturally ask how fluctuations would modify the magnetoresistance in a chiral medium. Addressing this question is the primary goal of the present work. For definiteness, we shall consider the fluctuations of both vector and axial charge densities. As in previous studies [5,19], we assume that r is parametrically small compared with the microscopic relaxation rate, and hence we include the axial charge density as a slow mode.
We here use the recently developed non-equilibrium effective field theory (EFT) for hydrodynamics fluctuations [20,21] (see ref. [22] for a review and refs. [23,24] for related developments), including the effects of quantum anomaly [25], to perform our analysis. Compared with the traditional methods, the EFTs are derived based on the symmetries and action principle and provide a basis for the systematic analysis. In some situations, such as the one considered in ref. [26], EFT calculations lead to different results as compared with traditional analysis. Previous work including the fluctuations of a single chiral charge and CME can be found in ref. [27]. See refs. [7,28] for the diagrammatic calculation of magnetoresistance for quark-gluon plasma (QGP) based on perturbative QCD.
In two situations, r = 0 and r = 0, we determine specific corrections to the conductivity due to the combined effects of the CME and fluctuations in the small B regime (see eq. (3.43) and eq. (4.22), respectively). Physically, those CME-related contributions have two origins. First, the CME is proportional to the axial chemical potential which generically depends on charge density non-linearly and gives rise to non-linear coupling among density fluctuations. Second, the CME modifies the dispersion relation of fluctuations modes [6,29]. Given the difference in physical origin, we should not be surprised to see that the fluctuation corrections are in marked difference from eq. (1.1). One important qualitative feature we observe is that the sign of fluctuation contributions is opposite to that of eq. (1.1), meaning they give rise to positive magnetoresistance. Moreover, we find a non-zero contribution to transverse conductivity when r = 0. As already noticed in some references [7,28,30], other mechanisms unrelated to the anomaly could cause magnetoresistance. The present work aims to demonstrate that even if one only focuses on the effects of the chiral anomaly, the contribution from fluctuations to magnetoresistance can be qualitatively different from that at "tree-level." Our results might apply to physical systems, such as the QGP created by heavy-ion collisions, Weyl semimetals, and the electroweak plasma in the primordial Universe.
This paper is organized as follows. After reviewing the construction of the EFT action in section 2, we determine the relevant Feynman rules and vertices. In sections 3 and 4, we respectively calculate the conductivity at one-loop at finite and vanishing axial charge relaxation rate. We conclude in section 5.

The action
We are interested in the fluctuation dynamics of vector charge density n V and axial charge density n A in a chiral medium. As already mentioned in the Introduction, we shall assume the relaxation rate of n A is small compared with the microscopic relaxation rate. Furthermore, we shall limit ourselves to situations that temperature is much smaller than vector chemical potential µ V and/or axial chemical potential µ A . In this regime, we could ignore the mixing of n V and n A with the energy density. We also note in electron systems including Weyl semimetals, the mean free path of momentum-relaxing scattering (e.g., impurity scattering) can typically be shorter than the mean free path of momentum-conserving scattering (electron-electron scattering). In such a situation, the momentum is not a hydrodynamic variable, and ignoring the coupling of (charge) density modes to sound/shear modes can be well justified. Therefore, in long-time and large-distance limits, we can integrate out other modes and obtain the effective action I eff describing the remaining slow modes n V and n A . In general, it is difficult to obtain I eff directly from microscopic theories. Instead, one should construct I eff based on the symmetries together with other physical requirements, as we shall do below following the formalism developed by refs. [20,21,25] (see ref. [22] for a pedagogical introduction).
We begin with the path integral representation of the generating functional on the Schwinger-Keldysh contour, Here, we have introduced the external gauge fields A r α and A a α and the dynamical fields ψ r α and ψ a α associated with charge density n α , in the "r-" and "a-" basis. One can interpret ψ r α and ψ a α as the U(1) phase rotations of each fluid element (see refs. [20,22] for more details). The r-variables are related to the physical observables, while the a-variables are the associated noise variables.
Next, we list various symmetries and consistency requirements which I eff should satisfy: and the chiral anomaly are absent. Here, U(1) α gauge transformation can be written explicitly as where φ s α is an arbitrary U(1) α phase. For a term invariant under the U(1) α symmetry, its dependence on A µ,s α and ψ s α should come through the gauge-invariant combination: In particular, the vector and axial chemical potentials are expressed as Shift symmetries: For each fluid element, it should have the freedom of making independent U(1) α phase rotations as far as those phases ζ α (x) are time-independent: Note that shift symmetries will be absent when the global U(1) α symmetry is spontaneously broken (see refs. [20,31] for further details).
3. Dynamical Kubo-Martin-Schwinger (KMS) symmetry: Suppose the microscopic theory is invariant under a Z 2 anti-unitary transformation Θ, then I eff is invariant under the KMS transformation [21], which, in the "classical" limit that quantum fluctuations are small compared with the thermodynamic fluctuations, is defined as where T is the background temperature. The dynamical KMS symmetry is motivated by the KMS condition satisfied by a thermal system and can be viewed as a definition of local thermal equilibrium. Generically, one can take Θ that includes T , i.e., Θ can be T itself, or any combination of C, P with T [21], depending on the systems of interest. The presence of background magnetic field and vector charge will break the symmetries under T and C, respectively, so we shall take Θ = CPT in this work.

Unitarity:
The unitarity of the underlying system requires that (suppressing α and µ indices) To consider the low-energy regime of the system, we also perform a derivative expansion based on the basic philosophy of EFT. For definiteness, we take the following counting scheme in this paper: where is a small expansion parameter. Now, we are ready to write down the non-equilibrium action I eff explicitly. Because of eq. (2.7), L has to contain at least one power of a-field. We shall study L up to quadratic order in a-field. More explicitly, we consider the Lagrangian density L, which is related to I eff through the standard relation and divide L into three parts: Here, L inv corresponds to L in the limit that both the effects of the axial charge damping and chiral anomaly are absent. In this case, n A is also conserved. Hence, L inv up to O( 2 ) should be of the same form as the hydrodynamic effective action with two conserved charges as derived in ref. [20] (see also ref. [26]): where σ ij αβ is the conductivity matrix which is symmetric with respect to (i, j) and (α, β). Because of the shift symmetries, L inv is independent of A r i,α . Turning to L anom , which describes the effects of the chiral anomaly, we explicitly have where C = 1/(2π 2 ) denotes the anomaly coefficient and the electric and magnetic fields are defined by E = ∇A r 0,V − ∂ t A r V and B = ∇ × A r V . To simplify the expression, we shall consider the cases in the absence of the axial gauge fields here and from now on. The first term in eq. (2.12) leads to the anomaly contribution to the non-conservation of the axial current (see eq. (2.17) below). The second and the third terms correspond to the chiral separation effect (CSE) [32,33] and CME, respectively. In appendix A, we present the derivation of eq. (2.12) by generalizing the formulation for a single chiral charge in ref. [25].
Finally, we postulate to use to describe the axial charge relaxation. Here, γ denotes the axial damping coefficient, which is assumed to be O( 2 ) so that L damp contributes to the same order as the other terms in eq. (2.10). Equation (2.13) satisfies all requirements as listed above. We here point out that the equations of motion from I eff is equivalent to the (non-)conservation equations for the currents. For the vector charge density, we have since I eff only depends on the combination A a µ,V +∂ µ ψ a V but not on A a µ,V and ψ a V individually. For later purpose, we obtain the expressions for J r and J a by differentiating (2.10) with respect to A a and A r , respectively: We can also define the axial current J µ A through the variation of I eff . In that case, we find that the equation of motion for ψ a A is nothing but the non-conservation equation for the axial current due to the chiral anomaly and axial charge damping, In summary, we shall use the following effective action up to O( 2 ) for the subsequent analysis for systems with a background magnetic field B based on eqs. (2.11), (2.12), and (2.13): (2.18) For notational brevity, here and hereafter, we suppress the a-index for ψ a α . Note that B = O( ) in our counting scheme above, and we may ignore the B-dependence of σ α and γ at the level of this effective Lagrangian.
In this work, our goal is to showcase the non-trivial interplay among the axial charge density relaxation, CME, and fluctuations in the simplest possible settings. In eq. (2.18), we have assumed that at the tree level, σ ij αβ = σ α δ ij δ αβ , which is sufficient for the present illustrative purpose. In the same spirit, we shall use the susceptibility matrix χ αβ ≡ ∂n α /∂µ β which is also diagonalized, χ αβ = χ α δ αβ .
Before closing this section, we point out that the first equation in eq. (2.14) and eq. (2.17) can be matched to the standard stochastic equations for n V and n A in the presence of the CME/CSE [34][35][36]. Conversely, one might construct an action of a similar form to eq. (2.18) from the stochastic equation following the bottom-up approach of Martin-Siggia-Rose-Janssen-de Dominicis [37][38][39], as was done in ref. [36]. However, the formalism of refs. [20,21,25] as we employ here provides a basis for the systematic analysis.

Expansion around thermal equilibrium
Let us consider the fluctuations around the equilibrium state characterized by a static and homogeneous background vector and axial charge densities (n V ) 0 and (n A ) 0 , where the subscript "0" denotes those equilibrium values. In section 3.4, we shall study the situation that axial charge damping coefficient γ is finite so that (n A ) 0 = 0. In section 4, we take the limit γ = 0 and consider the systems with a finite (n A ) 0 . In both cases, we shall use δn α = n α − (n α ) 0 as the dynamical fluctuating fields for r-variable and ψ α as the dynamical a-fields; the latter vanishes in equilibrium. 1 In addition, we rescale δn α and ψ α for convenience by Note that g −2 α = T χ α is the equilibrium fluctuation of δn α per unit volume. This means that the fluctuations of the rescaled variables λ α are of the order unity, which is real motivation for the definition (2.19). Here and throughout this paper, we do not take the summation over dummy vector/axial indices (α = V, A) unless the summation symbol Σ is present.
We shall expand the Lagrangian density as where the subscript of (· · · ) denotes the number of fluctuating fields. By construction, (L) 1 is a total derivative. We shall use (L) 2 to obtain propagators and read cubic vertices from (L) 3 . The quartic vertices from (L) 4 can contribute to one-loop corrections, but such contributions are simply proportional to the UV cut-off and will not be of physical importance. In short, L 2,3 are sufficient for the computing fluctuations corrections at oneloop order. Note that if we demand σ α , γ, and |B| to be counted as g −2 α , then (L) n ∼ g n−2 α and g α can be viewed as the effective coupling constant organizing the fluctuation corrections to the tree-level results.
To determine L 2,3 , we need to expand µ V,A , σ, and γ in terms of δn V,A : we have explicitly Here, the normalizations are chosen to make the counting in terms of g α manifest in the following expressions. From now on, we omit the label (...) 0 for equilibrium quantities when doing so would not lead to confusion.
Diagrammatic representations of (a) G ra αβ , (b) G rr αβ , and the vertices (c) U and (d) V . The solid and dotted lines correspond to λ (r-field) and ψ (a-field), respectively. Substituting eq. (2.21) into eq. (2.18) and further using eq. (2.23), we arrive at the expressions: Here, r and v correspond to the bare axial relation rate and the velocity of the chiral magnetic wave (CMW) [29,40], respectively. The first and the third lines in eq. (2.25) arise from the non-linearity due to the charge diffusion and axial charge relaxation, respectively. Since µ V,A is generically a non-linear function of n V and n A , the CME/CSE induce nonlinear couplings among fluctuating fields, as is shown in the second line of eq. (2.25). In the cubic action (2.25), the terms involving two a-fields correspond to the multiplicative noise contribution, which is necessary to ensure the KMS invariance.

Propagators and vertices
We now define the two-point correlation functions of the fields: while ψ α (x)ψ β (0) = 0 by causality.
To perform the diagrammatic analysis, we shall consider the free propagators G rr αβ , G ra αβ , and G ar αβ , which are G rr αβ , G ra αβ , and G ar αβ at the tree level, respectively. Suppressing the indices α and β, we can read their expressions from (L) 2 given by eq. (2.24) as where in the Fourier space with K µ = (ω, k), Note that M T = M . From eq. (2.28), we obtain The diagrammatic representations of G ra αβ and G rr αβ are given by figures 1 (a) and (b), respectively.
The retarded propagator G ra is the basic building block in the subsequent diagrammatic computations, whereas G ar and G rr can be expressed in terms of G ra : as one can verify explicitly from eqs. (2.30) and (2.29). A particular useful form for G ra is that in a Laurent expansion: where m = +, − labels two independent collective modes with In the limit r = 0 and for k ·B > 0, these two modes (m = +, −) correspond to the CMW propagating in the same/opposite directions to the magnetic field, respectively. Next, we define the interaction vertices from L 3 as where ψ k α ≡ ψ(t, k) and λ k α ≡ λ(t, k). There are two types of vertices: U couples one a-field with two r-fields; V couples two a-fields with one r-field. They can be read from the cubic action (2.25) Note, the first term in the second line of eq. (2.37b) and the last term in eq. (2.37d) arise from the fact that the axial damping coefficient γ depends on n V and n A . The graphic presentations of U k 1 ,k 2 ,k 3 αβγ and V k 1 ,k 2 ,k 3 αβγ are shown in figures 1 (c) and (d), respectively. With the propagators and vertices at hand, we are ready to compute one-loop corrections to the conductivity.
3 Conductivity at finite axial relaxation rate

Conductivity
From the symmetrized correlator of the vector current J µ , C µν ∼ J µ J ν , we can determine the (AC) conductivity tensor through the standard Kubo formula Alternatively, we can extract σ ij from the retarded correlator (see eq. (4.1) in section 4). The conductivity tensor in the presence of the external magnetic field can be decomposed as where σ and σ ⊥ are the longitudinal and transverse conductivity, respectively. In this work, we will not consider the Hall conductivity. From the Ward-Takahashi identity, we also have lim k→0 1 2T which allows us to extract σ and σ ⊥ from the small k behavior of C 00 . In what follows, we determine C 00 and hence σ ij (ω) from the relation, (3.5) It then follows from eqs. (3.3) and (3.2) that which reproduces the well-known CME-induced negative magnetoresistance [5]. However, the tree level result (3.6) does not take into account the fluctuation effects. We shall study the one-loop corrections to σ ij by computing G rr VV dressed by the self-energies.

Self-energies
We start from the full propagators which are dressed by self-energies Σ ar , Σ ra , and Σ aa through the Dyson equation: or equivalently, To determine the small k behavior of G rr VV from eq. (3.9), we shall first consider the behavior of the self-energies in this limit. We note Σ aa αβ can be connected to two external legs associated with ψ α and ψ β . Since ψ V is always combined with ∇ in the cubic Lagrangian density (L) 3 in eq. (2.25), we have Σ aa αβ ∼ k δ αV +δ βV and Σ ar αβ ∼ k δ αV , where δ αβ is the Kronecker delta. Then, the relevant components of the self-energies can be parametrized as where . . . are the terms suppressed by small k. Note that ∆γ, ∆r, and ∆v A can be viewed as the finite frequency corrections to γ, r, and v, respectively. To see this, one should keep in mind that Σ aa and Σ ar enter as the correction to −N and iM , respectively, in eqs.
meaning the loop corrections to the conductivity can be expressed in terms of the following ω-dependent functions: which we shall compute in section 3.4. Intuitively, we may understand eq. (3.11) by replacing (σ V ) 0 , γ, r, and v of eq. (3.6) into those including fluctuation corrections (3.12).

One-loop
In this subsection, we provide general expressions for computing the self-energies Σ ar αβ and Σ aa αβ at the one-loop level. We will give the derivation for the former and present only the results for the latter, which can be derived similarly (see appendix B for details).
The self-energy Σ ar αβ consists of three pieces: whose diagrammatic representations are given by figures 3 (a)-(c), respectively. First of all, the contribution from (Σ ar αβ ) III in figure 3 (c) vanishes. This is because where the integrand has poles only in the upper complex q 0 -plane so that the contour integral vanishes. Here and hereafter, we use the notation q ± = q ± k/2. For the first term in eq. (3.13), we have To obtain the second line from the first line, we have replaced G rr (q 0 , q + ) with iG ra (q 0 , q + ) using eq. (2.32) together with eq. (3.14). Turning to (Σ ar αβ ) II , we have Adding (Σ ar αβ ) I and (Σ ar αβ ) II , the self-energy becomes where we have defined two-by-two matrices: We now carry out the integration over q 0 in eq. (3.17) using the Cauchy residue theorem, yielding where we have used eq. (2.33) and introduced the notation, . (3.20) Substituting eq. (3.19) into eq. (3.17) leads to the final result where we have defined One can also analyze Σ aa αβ following the similar treatment (see appendix B for details). For our purpose, only the diagonal components of Σ aa αβ are needed; they read where we have defined Expressions (3.21) for Σ ar αβ and (3.23) for Σ aa αα are the main results of this subsection. These are to be used in the following subsection.

Results
We now compute one-loop contributions to the conductivity tensor. We first note that in the absence of the background axial charge density, n A;AA = n V;VA = n V;AV = n A;VV = 0, σ V;A = σ A;A = 0, γ ;A = 0 . (3.26) Let us further assume, for the present illustrative purpose, Then, the vertices are parameterized by three independent parameters: In this subsection, we will drop the subscript in D, v, and g because of eq. (3.27).
To evaluate eqs. (3.21) and (3.23), we first consider relevant Γ a α , Γ a α , and Γ r α , which, in the small k limit, are simplified as  (3.31) We first evaluate Σ aa VV which in turn determines ∆σ (ω) and ∆σ ⊥ (ω). From eqs. (3.29) and (3.24), we have so that eq. (3.23) becomes (3.33) Comparing eq. (3.33) with eq. (3.10a), we find Here and below, we use the dimensional regularization to perform the integration over q, Evaluating loop integrals using dimensional regularization is essentially picking up the contribution from the IR scale, which only depends on the physical inputs but does not depend on the unphysical UV cut-off Λ of the EFT (see, e.g., ref. [41]). For instance, in eqs. (3.35), the IR momentum scale is given by which arises from the competition between axial relaxation and diffusion at finite r. Such IR momentum scale can be interpreted as the characteristic momentum of fluctuation modes and has to be parametrically smaller than the cut-off scale of the EFT, which, in the present case, is Λ ∼ l −1 mfp . Here, l mfp denotes the mean free path. Indeed, from D ∼ l mfp , it is easy to verify that q * r Λ. Note that at r = 0, q * r ∼ ω/D, which leads to hydrodynamic long-time tail behavior [12][13][14].
Turning to Σ aa AA , a similar treatment results in (3.37) Therefore, ∆γ(ω), defined by eq. (3.10b), reads We shall interpret our results for ∆γ(ω) at the end of this section. Finally, we turn to Σ ar VA and Σ ar AA , which can be obtained from the familiar steps. They are which, upon using eqs. (3.10c) and (3.10d), gives the expressions, We now have all the ingredients needed to compute the conductivity from eq. (3.11). By the substitution of eqs. (3.34), (3.38), (3.41), and (3.42) into eq. (3.11), we find σ ⊥ does not receive any loop corrections while σ at the one-loop order is given by where the first and second terms in [. . .] correspond to the CME related tree contribution (see eq. (3.6)) and the one-loop correction, respectively. Note that the fluctuation correction to σ is of the opposite sign to the tree one and lead to a positive-magnetoresistance contribution. Equation (3.43) is the main result of this section (see section 5 for further discussions). Before closing this section, we remark that although the loop corrections to σ are resulting from the combination of the set of functions listed in eq. (3.12), our expression for each of them might be of interest on its own. Let us give two examples below.
For the first example, we recall that for a chiral medium microscopically described by non-Abelian gauge theories, γ is referred to as Chern-Simons (CS) diffusion rate. In the weak-coupling regime, the CS diffusion rate is computed by accounting for contributions at the microscopic length scale g 2 T and time scale g 4 T [42]. Our result (3.38) can be interpreted as the additional contribution to the CS diffusive rate from the macroscopic length scale ∼ r/D.
As for the second example, we consider the pole of G ra AA in the limit k → 0, which, at the tree level, is located at ω = −ir. The one-loop corrections to this relaxation pole can be determined using the Dyson equation (3.8), Upon substituting eq. (3.41), we find This should be contrasted with the naive expectation that the correction is of the order g 2 .

Conductivity at zero relaxation rate
To complement the analysis in the previous section, we shall work on the limit that the axial relaxation rate γ vanishes in this section. By sending r → 0 in the tree-level result (3.6), we notice that the CME-related contribution to the conductivity vanishes. However, as we shall show below, there is a CME-related contribution to the conductivity due to the effects of the fluctuations. Instead of extracting conductivity tensor σ ij from relevant self-energies as was done in the previous section and in appendix D, in this section, we will employ a complementary approach which determines σ ij from the retarded correlator of the vector current: Here, we have used the fluctuation-dissipation relation Let us begin with the retarded correlator expressed in terms of the vector currents J r and J a : Here and hereafter, . . . denotes the average weighted by the path-integral (2.1) with γ = 0. At one-loop order, C ij R is given by the correlation of J r and J a expanded to the quadratic order in the fluctuating fields. In details, we use the expressions given by (2.15) and (2.16) with E = 0 and have Here, O(∇) denotes terms which would vanish in the limit k → 0.
To proceed further, we first show that the first two terms in eq. (4.4) do not contribute to eq. (4.3) for k → 0: To see this, we consider Here, I int = x (L) 3 and ... 0 denotes the average weighted by the Gaussian part of the effective action, as this suffices to the present one-loop calculations. In eq. (4.7), we have also used which can be shown by causality (see ref. [43] for a general discussion) and the fact that both δI int /δψ β and (J a ) 2 contain at least one power of the a-field. On the other hand, in the absence of γ, one can confirm from eq. (2.18) that δI int /δψ β = ∇ · (J r β ) 2 , meaning the Fourier transform of eq. (4.7) vanishes in the small k limit. Therefore, the first term in eq. (4.4), which can be expressed as a linear combination of λ α (x) (J a ) 2 (x ) , should also vanish in this limit. A similar analysis applies to the second term in eq. (4.4). As a consequence, at one-loop order, eq. (4.4) is reduced to To compute eq. (4.9), we rewrite (J r ) 2 and (J a ) 2 in terms of the rescaled fields (2.19) using eq. (2.22): where we have further assumed the absence of the background vector chemical potential, Noting the correlation between the last term in eq. (4.10) and eq. (4.11) vanishes by causality, we now have two remaining contributions to eq. (4.9): the correlations between (J a ) 2 and the first term (CME part) and the second term (diffusive part) of eq. (4.10). We shall first show that the former contribution vanishes for k → 0. Indeed, 14) and similarly, Substituting eq. (4.16) into eqs. (4.14) and (4.15) leads to The one-loop corrections to the conductivity tensor now becomes Thus, we finally have In our counting scheme B = O( ), it follows that |v| 1, and one can again verify that q * B Λ. Finally, by comparing eq. (4.20) with eq. (3.2), we find the main results of this section, See the subsequent section for the discussion of eq. (4.22). In ref. [27], the authors consider the fluctuation effects of a single chiral charge in the presence of the CME. For spatial dimension d = 3, they find finite corrections to the AC conductivity (ω = 0) but vanishing DC conductivity (ω = 0). The differences between theirs and the present results mainly arise from the fact that we have considered the couplings between the axial and vector charge densities (see also refs. [16,26,44] on the studies of fluctuation dynamics with multiple conserved charges).

Positive magnetoresistance
We presented in this paper the diagrammatic calculation of the modifications of conductivity (the inverse of resistance) for a chiral medium with the chiral magnetic effect (CME) based on the non-equilibrium effective field theory (EFT) approach. We consider a generic vector charge density and axial charge density as slow variables and study the intertwined effects from their fluctuations and CME. For the first time, we obtain the CME-related modifications to the conductivity tensors due to fluctuations for systems with finite and vanishing axial relaxation rate r, as summarized in eq. (3.43) and eq. (4.22), respectively.
Contrary to the common statement that the CME leads to a negative magnetoresistance, we find that CME-related effects due to fluctuations give rise to a positive magnetoresistance. Whether the net magnetoresistance is positive or negative is determined by the competition between the two terms inside [...] in eq. (3.43). Note, to apply our one-loop results, we should require the fluctuation contribution to be much smaller than the treelevel expression, which includes both the first term and second term on the right-hand side of eq. (3.43) (c.f. eq. (3.6)). Since we are working in the weak B limit, the tree-level contribution is dominated by (σ) 0 , which is indeed much larger than the one-loop correction, i.e., the third term on the right-hand side of eq. (3.43). However, this does not necessarily mean that the third term has to be smaller than the second term. Therefore, the net CME-related contribution can in principle be dominated by the fluctuations effects. We have made a number of simplifications in our analysis, but we hope this qualitative feature might have some implication to real physical systems. Interestingly, a positive magnetoresistance might have already been seen in Weyl semimetals when the magnetic field is small (e.g., ref. [8]).
It is truly striking that the CME contributes to the conductivity even in the limit r = 0 (see eq. (4.22)). This is in a marked difference from the result, which does not account for fluctuations that the CME contribution vanishes in this limit. Moreover, the fluctuation modifies both longitudinal and transverse conductivities, while at finite r, only longitudinal conductivity receives the correction from the CME.
The parametric behavior of the ratio of fluctuation effects to the tree-level contribution is very instructive. From eqs. where q * is characteristic momentum of fluctuating modes. For the case with r = 0, q * = r/D, resulting from the competition between the diffusion of vector charge and the damping of the axial charge, whereas for r = 0, q * = |v|/D originating from the competition between diffusion and propagation of the chiral magnetic wave (CMW). Equation (5.1) indicates that the relative importance of fluctuation corrections is determined by two factors, a) the strength of non-linearity (T u) 2 and b) the ratio between the phase space volume of the long-wavelength fluctuating modes, q 3 * , and that of the whole system, g −2 = χT .
Let us end this section by comparing the role the CME played in two cases under study. For r = 0, the CME gives rise to non-linear couplings among charge fluctuations but plays no role in determining the IR scale. In contrast, for r = 0 and at vanishing background vector charge, the non-linearity relevant to the finite corrections to the conductivity solely comes from the density-dependence of diffusive constant and conductivity but does not rely on the CME. However, the competition between CMW propagation and charge diffusion leads to the emergent IR scale. Due to the differences explained above, C|B| dependence of ∆σ is different. The correction to the conductivity scales as (CB) 2 in the former case and scales as (C|B|) 3 in the latter case.

Remarks on hydrodynamic fluctuations
Another motivation of the present study is to advance our understanding of general aspects of hydrodynamic fluctuations. Here, we employ the recently developed non-equilibrium EFT for the present studies. Our exercise here demonstrates this EFT approach allows us to use powerful (and familiar) field theory techniques to analyze hydrodynamic fluctuations. By construction, the EFT automatically takes into account the constraints from symmetries. For example, in appendix D, we show explicitly how fluctuations-dissipation theorem and the Ward-Takahashi identity are satisfied at one-loop order. In the traditional method, special care is needed to ensure those relations (see the recent work [45] for the former).
Remarkably, the fluctuation contribution to the conductivity is finite due to the CME. This should be contrasted with the case of an ordinary fluid that fluctuation corrections to transport coefficients (at zero frequency limit) are typically zero. Such difference is related to the emergent IR momentum scale behavior in the loop integration, q * . Generically, the corrections to transport coefficients should scale with q * to an appropriate (and positive) power for d = 3. For a normal fluid, q * ∼ √ ω, giving rise to the renowned long-time tail phenomena [12][13][14], and consequently vanishes in the limit ω → 0 (see also refs [46][47][48] for related discussion). Therefore, to obtain finite corrections to transport coefficients in this limit, there must be additional soft scales. Such scales are generated by the magnetic field and/or axial relaxation rate in the present study. Given the generality of the discussion above, we anticipate that the CME and hydrodynamic fluctuations together might contribute to other transports coefficients. A natural follow-up would be to include fluctuations from energy and momentum densities and study the effects on shear and bulk viscosities. We leave these and other extensions of this work to future studies.
where I inv is identical to the hydrodynamic action of a conserved charge (see eq. (2.11)). We shall focus on the anomaly-related action I anom from now on.
Because of the anomaly, the consistent r-current, obeys the anomaly equation where κ = ±1 correspond to the right-handed and left-handed charge, respectively. In this appendix, we shall keep the index "a" but suppress the index "r." To obtain eq. (A.3), we use the consistent anomaly equation on the two segments of Schwinger-Keldysh contour labeled by "1,2": . We have not included a contribution quadratic in a-field on the right-hand side of eq. (A. 3) since such contribution should arise from the action involving three powers of a-field. Similar to the discussion presented in the main text, the equation of motion for ψ a , δI δψ a = 0 , (A.5) should be equivalent to the consistent anomaly equation (A.3). Therefore, the anomaly part of the action takes the form Here, A a µ = A a µ + ∂ µ ψ a as given in eq. (2.3) and J cons (J cov ) denotes the contribution to the consistent (covariant) current only from I anom , distinguished with the current J from the total action I. The difference between J cons and J cov defines the Chern-Simons (CS) current as We shall consider the following form for J µ cov : where ξ(µ) is a function of µ and we have defined E µ = F µν ν and B µ =F µν ν with µ = (1, 0) denoting the frame of the medium. We shall also use below that µ = · A (see eq. (2.4)). Since I anom is invariant under the KMS transformations (2.6a) and (2.6b), The variance of the first two terms in eq. (A.9) under the KMS transformation should precisely cancel that from the last term. This requirement uniquely fixes ξ(µ), as we shall explain below. Indeed, for Θ = CPT , we have where we have used the identitỹ (A.14) Putting all pieces together, we find On the other hand, where we used the fact that ξ(−µ)B µ ∂ µ µ is a total derivative. For eq. (A.15) to cancel eq. (A.16), we have the appropriate form for the CME: Now, we generalize the discussion above to the system with both axial and vector charges. From the consistent anomaly relation, Thus, we have, in analogous to eq. (A.6), the expression for I anom : By imposing the KMS symmetry to eq. (A.20), we find Substituting eq. (A.21) into eq. (A.20) leads to the desired expression for I anom , which reduces to eq. (2.12) when the external axial gauge field is absent.
One can show that To calculate (Σ aa αβ ) I , we use eq. (2.32) and write G rr as a sum of G ra and G ar . Then, we drop the latter contribution using eq. (3.14) and carry out the q 0 integral from the former contribution using eq. (3.19). We obtain where we have used the same matrix form as eq. (3.18): By using eq. (B.4) we have (B.10) Here, (α ↔ β + c.c.) denotes the complex conjugate of the first term with the interchanged (α, β) labels. The third contribution, figure 3 (f), can be written as (B.11) Using eqs. (2.31) and (2.32), the q 0 integral relevant to (B.11) can be written as C Conductivity tensor from density-density correlator at zero axial relaxation rate We here show another derivation of the one-loop conductivity, using the symmetrized density-density correlator C 00 S , at the vanishing axial relaxation rate with zero background vector charge density but finite axial charge density. The derivation here is more parallel to the analysis in section 3.4, whereas it will give consistent results with those in section 4, namely eq. (4.22).

D Explicit verification of the fluctuation-dissipation relation and Ward-Takahashi identity at one-loop
In this appendix, we show explicitly that at one-loop order, the symmetrized currentcurrent correlator C ij is related to the retarded correlator C ij R through the fluctuationdissipation relation (4.2). Since we have already demonstrated that conductivity tensor obtained from C 00 coincides with that from C ij R in appendix C, we therefore verify the constraint imposed by Ward-Takahashi identity at one-loop order between C 00 and C ij .
For illustrative purpose, we shall focus on the diffusive part of the current at quadratic order in fluctuations, The second term in eq. (D.1), which is proportional to the a-field ψ, arises from the multiplicative noise. We shall see this multiplicative noise contribution is crucial to ensure the fluctuation-dissipation relation and the Ward-Takahashi identity at one-loop order. We begin by computing the one-loop corrections to the symmetrized correlator On the other hand, the one-loop corrections to the retarded correlator is given by = ωT (D Vũσ ) 2 F ij,ra (K) . Note that if we had ignored the multiplicative noise contributions, i.e., the last two terms in eq. (D.3), we would obtain the wrong relation (D.10) Furthermore, one would also get a wrong relation between C 00 obtained in appendix C and C ij : which contradicts with eq. (3.3) based on the Ward-Takahashi identity.