arXiv:1408.4573v2 [hep-th] 21 Jan 2015 eraie nWy eieas[ semimetals Weyl in realized be [ charge electric large to with due ions with created of field is motion relative scale magnetic the hadronic huge to the comparable strength and produced plasma the in transitions quark-gluon topological cre- to be due might locally imbalance ated chirality heavy- the non-central where in collisions, it observe ion to possibility the by lated ∗ different of points Weyl energies. the different if have observed chiralities be [ can energy CME in points or The Weyl momentum in right-handed either and as separated being left- described with are fermions, excitations Weyl low-energy which in matter [ its instability where helical helium, the liquid is in manifestation realized experimental be also can CME The different [ fermions with chiral system right-handed a and in cur- left- field electric of numbers magnetic of Chiral generation the the the along rent is - (CME) phenomena Effect such Magnetic of examples well-known n r hsasn ncasclsses[ systems classical anomalies in quantum absent thus from chiral are stem of and which features trans- phenomena, anomalous fascinating in so-called the port research the of existence intense of the is of One fermions subject years. a recent become have fermions pnaeu hrlsmer raigadteCia Magnet Chiral the and breaking symmetry chiral Spontaneous lcrncaddress: Electronic ae ni a enraie htteCEcudalso could CME the that realized been has it on Later stimu- extend large a to been has CME to interest The ti omnsaeetta h rnpr coeffi- transport the that statement common a is It rnpr rpriso togyitrcigchiral interacting strongly of properties Transport raig enfil approximation semim Weyl mean-field fermions, breaking, Dirac transport, Anomalous chiral Keywords: of 72.20.-i,05.30.Rt,12.39.-x presence numbers: the PACS the in on fermions comment Dirac we of Finally, energy relevan vacuum the matter. the discuss QCD We interacting patt considered. strongly saturat also General is is polarizations. imbalance effect chiral transverse magnetic mixed chiral with the pha mesons) coupled regime strongly coupling the a strong in phases and different regime in weak-coupling conductivity transit bative magnetic the chemi chiral and the chiral increased, transitio study is renormalized the potential chemical for the chiral strength of the coupling critical increase The the lowebreaking. to imbalance imbalan chiral leads that chiral out which of point We presence the potential. chemical in interactions contact with erpr nama-edsuyo pnaeu raigo chi of breaking spontaneous of study mean-field a on report We .INTRODUCTION 1. [email protected] 1 nttt fTertclPyis nvriyo Regensbur of University Physics, Theoretical of Institute 5 -35 emn,Rgnbr,Universit¨atsstraße 31 Regensburg, Germany, D-93053 ia emoswt hrlimbalance chiral with fermions Dirac – 10 oe hs of phase novel a – ] 1 .Oeo the of One ]. Dtd uut2,2014) 20, August (Dated: .V Buividovich V. 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Since the derivation fields which probe the CME. It turns out that this more of [24] relies on the QCD chiral Lagrangian, it is also not systematic treatment predicts the enhancement of CME directly applicable to Weyl semimetals. due to interactions, in contrast to the dielectric screening Apart from spontaneous chiral symmetry breaking, found previously in [34]. Another difference of our study anomalous transport coefficients can receive purely radia- from the studies of QCD effective models is that the pa- tive corrections if the corresponding currents/charges are rameter which controls the breaking of chiral symmetry coupled to dynamical gauge fields [27–29]. Since in Weyl is the interaction strength rather than the temperature. semimetals the interactions are naturally associated with Because of the non-conservation of the axial charge, electric charges, one can expect that even perturbatively the chiral chemical potential µA is not a chemical po- the CME current might receive some corrections. Finally, tential in the usual sense. For instance, its value might since the chiral chemical potential itself is coupled to a be renormalized due to interactions. Moreover, it has non-conserved axial charge, it might also be subject to been shown that in the presence of nonzero µA chiral some non-trivial renormalization in interacting theories fermions coupled to electromagnetism become unstable (there are some subtle points in this statement which we towards the formation of magnetic background with non- address a bit later in this Section). trivial Chern-Simons number (also known as magnetic In this work we study spontaneous chiral symmetry helicity in plasma physics), which effectively reduces the breaking in a system of interacting Dirac fermions with chiral chemical potential [41–43]. These facts suggest chiral imbalance, addressing in particular the renormal- that the fully self-consistent description of chirally imbal- ization of chiral chemical potential and the fate of the anced matter should be dynamical and should allow for Chiral Magnetic Effect in a phase with broken chiral sym- spontaneous breaking of translational invariance. Since metry. Since we mostly keep in mind the application of such a dynamical description might be quite complicated our results to Dirac quasiparticles in Weyl semimetals, beyond the kinetic theory/hydrodynamical approxima- we consider a single flavour of Dirac fermions with in- tion, here we partly neglect the coupling of fermions to stantaneous on-site interactions between electric charges. dynamical electromagnetism and assume that there is a However, since the consequences of the chiral symmetry spatially homogeneous stationary ground state even in breaking are to a large extent independent of the un- the presence of chiral imbalance. derlying interactions, we believe that our results should In the case of Weyl semimetals, such an approximation be also at least partly applicable to strongly interacting might be justified by the fact that in condensed matter matter and quark-gluon plasma. systems interactions with magnetic field are suppressed 2 Our main tool in this work will be the mean-field ap- by a factor vF as compared to electrostatic interactions, proximation which incorporates possible condensation of where vF is the Fermi velocity (in units of the speed of all fermionic bilinear operators. While this approxima- light). Since the decay of chiral imbalance necessarily tion might eventually break down at sufficiently strong involves the generation of magnetic fields, one can ex- coupling due to large fluctuations of fermionic conden- pect that the typical decay time will be enhanced by a 2 sates, we justify our approach by the fact that in most factor of 1/vF as compared to the time scales at which cases mean-field approximation correctly predicts possi- electrostatic interactions are important, and the station- ble patterns of spontaneous symmetry breaking and the ary ground state might be a good approximation at such types of the phase transitions, while the exact position short time scales. Another possible situation which can of the transition points might be incorrect. On the other be described in terms of the (quasi-)stationary ground hand, at weak coupling mean-field approximation simply state is when chirality is pumped into the system at a con- reproduces an infinite chain of one-loop diagrams. stant rate which compensates for its decay rate. Experi- The chiral imbalance in our study is implemented in mental realization of such “chirality pumping” in parallel terms of a nonzero chiral chemical potential µA cou- electric and magnetic fields have been recently discussed pled to the non-conserved axial charge QA = ψγ¯ 0γ5ψ. in [44–46]. The term µAQA in the single-particle Dirac Hamiltonian The main results of the present work are, first, the shifts the energies of left- and right-handed Weyl nodes quick growth of the renormalized chiral chemical poten- to µA. We note that while chiral symmetry breaking tial in a phase with spontaneously broken chiral symme- for± Weyl semimetals with spatial momentum separation try. The corresponding phase transition itself turns into between the Weyl nodes (which corresponds to the term a crossover and is shifted to slightly smaller values of the biψγ¯ iγ5ψ) have been studied in details [30–33], to our interaction potential in the presence of chiral imbalance. knowledge the case of energy separation between Weyl Second, we find that the CME is significantly enhanced nodes has been previously considered only in the context due to interactions, both in the weak- and in the strong- of effective QCD models [34–40]. In contrast to these coupling regimes. However, we do not find any disconti- works, here we will explore possible fermionic conden- nuity of the chiral magnetic conductivity across the phase sates more systematically, in particular taking into ac- transition. Third, we find that in the strongly-coupled count the renormalization of the chiral chemical poten- regime the CME current is saturated by the parity-even tial. We will also explicitly take into account the varia- particle-antiparticle bound state of spin one (or vector 3 meson in QCD terminology). In the presence of chiral Hamiltonian hx,α;y,β. V is the on-site interaction po- imbalance, such states with orthogonal transverse polar- ˆ† ˆ tential andq ˆx = ψx,αψx,α 2 is the charge operator izations are mixed with each other, thus giving rise to α − the parity-odd CME response. Moreover, such vector- at point x. The subtractionP of 2 from the charge oper- like bound states are mixed with pseudo-vector ones. Fi- ator mimics the background charge of ions in any real nally, we comment on very different responses of Dirac crystalline lattice which can support Dirac quasiparticle fermions to chiral imbalance in the context of condensed excitations. This addition, however, does not play any matter systems, where the number of states in the Dirac role in our calculation. While at the next stages of our sea is always finite, and in the context of quantum field calculations we will use the continuum Dirac Hamilto- theories which typically require some regularization of nian, for the derivation of the mean-field approximation the vacuum energy. While in the former case the vac- which we present in this Section it is more convenient to uum energy is always lowered by chiral imbalance, in the assume that x and y in (1) take discrete values. In order to represent the partition function = latter case the contribution of regulator fermions leads to Z the opposite effect. tr exp βHˆ (where β T −1 and T is the tempera- − ≡ The structure of the paper is the following: in Section ture) in the form suitable for mean-field calculations, we 2 we introduce the model Hamiltonian which we consider start with the Suzuki-Trotter decomposition and by applying the Hubbard-Stratonovich transforma- tion bring the corresponding partition function into the tr exp βHˆ = form suitable for the mean-field calculation. In Section − ˆ ˆ ˆ  ˆ  3 we study spontaneous breaking of chiral symmetry in = lim tr e−∆τH0 e−∆τHI e−∆τH0 e−∆τHI ... , (2) the presence of chiral imbalance by numerically minimiz- ∆τ→0   ing the mean-field free energy. In Section 4 we study the where the Euclidean time interval τ [0,β] is split into Chiral Magnetic Effect within the linear response approx- infinitely small intervals of size ∆τ.∈ The representation imation and comment on the mixing between different (2) is exact up to corrections of order of O ∆τ 2 . The particle-hole bound states at nonzero chiral chemical po- next step is to apply the Hubbard-Stratonovich transfor- tential. In Section 5 we speculate on the role of chiral mation to the terms involving the interaction Hamilto-
chemical potential in different regularizations or lattice nian HˆI . Since charge operatorsq ˆx in HˆI commute at implementations of Dirac fermions. Finally, in Section different points x, we can write 6 we conclude with a general discussion of the obtained results and an outlook for future work. exp ∆τHˆ = exp ∆τ V qˆ2 . (3) − I − x x   Y
2. THE MODEL: DIRAC HAMILTONIAN WITH There are now several possibilities to perform the ON-SITE INTERACTIONS AND CHIRAL Hubbard-Stratonovich transformation on each of the fac- CHEMICAL POTENTIAL tors on the right-hand side of (3), corresponding to dif- ferent grouping of four fermionic operators in (3) into two fermionic bilinears. If we were able to perform the The starting point of our analysis is the many-body integration over the Hubbard field exactly, all these rep- Hamiltonian of the following general form: resentations would be of course equivalent. However, the (0) mean-field approach which we use in this work is the Hˆ = Hˆ + Hˆ , Hˆ = ψˆ† h ψˆ , 0 I 0 x,α x,α;y,β y,β saddle-point approximation for the integral over the Hub- x,y X bard field, and the validity of this approximation might ˆ 2 HI = V qˆx. (1) strongly depend on the choice of the integration variables. x Here we perform the Hubbard-Stratonovich transforma- X tion in a way which allows to treat the chiral condensate ˆ ˆ Here H0 and HI denote the free and the interaction parts and also all other fermionic bilinear condensates as ex- ˆ† ˆ of the Hamiltonian and ψx,α and ψy,α are the fermionic trema of the corresponding effective action. To this end creation and annihilation operators at points x and y, we first rewrite which can be either points in continuous space or the sites 2 ˆ† ˆ ˆ† ˆ of some lattice. Correspondingly, the sum denotes ei- qˆx = ψx,αψx,α 2 ψx,βψx,β 2 = x,y − −  ˆ† ˆ ˆ† ˆ ˆ† ˆ ther integration over continuous coordinatesP or summa- = ψx,αψx,βψx,βψx,α + ψx,αψx,α, (4) tion over lattice sites. Small Greek letters α, β, ... label − where from now on we shorten the notation by assuming Dirac spinor components. h(0) is the (bare) single- x,α;y,β summation over repeated spinor indices. Inserting this particle Dirac Hamiltonian, which again can be either representation ofq ˆ2 into (3), we can separate the two the continuum Dirac Hamiltonian or some lattice Hamil- x summands in the second line of (4) into two exponents, tonian which at low energies describes Dirac fermions. 2 (0) making an irrelevant error of order O ∆τ . Then we The bare chiral chemical potential µA is introduced as transform the exponent containing the four fermionic op- a term of the form µ(0) (γ ) δ in the one-particle erators into the exponent of a fermionic bilinear
operator − A 5 αβ xy 4 by representing it in terms of a Gaussian integral over the matrix in spinor space: Hubbard-Stratonovich field Φx,αβ, which is a Hermitian

∆τ exp V ∆τψˆ† ψˆ ψˆ† ψˆ = dΦ exp Φ Φ ∆τΦ ψˆ† ψˆ . (5) x,α x,β x,β x,α x,αβ − 4V x,αβ x,βα − x,αβ x,α x,β   Z   Finally, we can again combine all the factors in the Suzuki-Trotter decomposition (2) into a single time-ordered exponent, neglecting the error of order of O ∆τ 2 , which yields

β 1 = Φ (τ)exp dτ Φ (τ)Φ (τ) Z D x,αβ −4V x,αβ x,βα  × x Z Z0 X β   tr exp dτ Hˆ + (Φ (τ)+ Vδ ) ψˆ† ψˆ . (6) × T − 0 x,αβ αβ x,α x,β  x ! Z0 X  

We thus have formulated the problem in terms of the summands in the second line of (8) with ǫi > 0 are zero, fermionic partition function which corresponds to the ef- and all the terms with ǫi < 0 are equal to ǫi/T . We thus fective time-dependent single-particle Hamiltonian of the see that the fermionic contribution to the free energy following form: = T log is simply equal to the (negative) energy ofF the− DiracZ sea, which is just the sum of all the energy (0) hx,α;y,β = hx,α;y,β + Vδxyδαβ +Φx,αβ (τ) δxy, (7) levels below zero. Combining this contribution with the quadratic action of the Hubbard-Stratonovich field, we with background field Φx,αβ (τ) which depends on Eu- conclude that in the static mean-field approximation we clidean time τ. have to minimize the following functional with respect to In the mean-field approximation, we replace the inte- Φx,αβ: gral over Φx,αβ (τ) in (6) by the value of the integrand at its minimum. If the minimum corresponds to some Φx,αβΦx,βα = ǫi + . (9) nonzero value of Φx,αβ (τ), one says that the fermionic F 4V ǫi<0 x condensate ψˆ† ψˆ Φ is formed. Additionally, X X h x,α x,β i∼ x,αβ one can take into account the corrections by integrat- Before proceeding with the actual minimization of the ing over Gaussian fluctuations around the saddle point, above functional, let us consider the effective chemical which are interpreted as propagating bound states of two potential term Vδαβδxy in the effective single-particle fermions. Hamiltonian (7). Its appearance might seem strange In order to find the minimum, we will assume that at the first sight, since it might induce nonzero electric the invariance under shifts of Euclidean time is not bro- charge in an initially electrically neutral system. How- ken, and thus the value of the Hubbard-Stratonovich field ever, the Hubbard-Stratonovich field Φx,αβ also contains Φx,αβ (τ) Φx,αβ at the minimum does not depend on the constant component proportional to δ which can ≡ αβ τ. Then the trace of the time-ordered exponent in the mimic the chemical potential. Let us explicitly sepa- integrand in (6) can be rewritten as simply the partition rate this term by writing Φx,αβ = µδαβ + Φ˜ x,αβ, with function of a free fermion gas with single-particle Hamil- Φ˜ x,αα = 0. The functional (9) can be then written as tonian (7) with τ-independent Φx,αβ: x P = (˜ǫi + µ + V )+ † F tr exp β ψˆ h ψˆ = ǫ˜ <−µ−V − x,α x,α;y,β y,β i X x,y,α,β 2 3 X Φ˜ x,αβΦ˜ x,βα µ L   + + , (10) 4V V −ǫi/T x = exp log 1+ e , (8) X i ! X   where L is the spatial size of the system andǫ ˜i are the where the sum goes over all energy levels ǫi of this single- eigenvalues of the single-particle Hamiltonian (7) with particle Hamiltonian. In this work we will be interested Φx,αβ replaced by Φ˜ x,αβ and without the Vδαβδxy term. in the limit of zero temperature. In this case all the In order to find the mean-field value of µ, we have to 5

∂ solve the equation ∂µ = 0, which can be written as we expect that the continuum approximation which we F make here nevertheless captures some features of the low- θ ( ǫ˜ µ V )+2L3µ/V =0, (11) energy behavior of lattice models. Preliminary studies − i − − i with Wilson-Dirac Hamiltonian, which will be published X elsewhere, qualitatively confirm the predictions obtained where θ is the Heaviside step function, which simply in the continuum approximation. counts the number of energy levels ǫi =ǫ ˜i + µ + V be- In order to calculate the mean-field diagram for the ef- low zero. One can show that for Dirac Hamiltonians fective single-particle Hamiltonian (12), we assume that with chiral chemical potential either in the continuum or external fields are absent and translational and rota- on the lattice the energy levels ˜ǫi are symmetric around tional symmetries are not broken, so that Φ Φ x,αβ ≡ αβ zero, so that every positive energy levelǫ ˜i > 0 is ac- is constant in space. Moreover, Lorentz symmetry al- companied by the negative energy level ǫ˜ . The total − i lows only the following structure of fermionic condensates number of energy levels is N = 4L3 and is equal to the † ψˆ ψˆx,β Φαβ: total number of degrees of freedom living on the lattice h x,α i∼ (4-component Dirac spinors on every lattice site). There- (0) 3 Φ= m cos θγ0 + im sin θγ0γ5 + γ5 µA µA , (13) fore there are 2L energy levelsǫ ˜i below zero. Taking this − observation into account, it is easy to see that µ = V   − where m is the effective mass induced due spontaneous is the stable solution of the equation (11). We conclude chiral symmetry breaking, θ is the complex phase of the therefore, that the effective mean-field chemical potential mass and µA is the renormalized chiral chemical poten- exactly compensates the term Vδαβδxy in (7). Therefore tial. Further we will see that the mean-field free energy we can simply discard this term and assume that the does not depend on θ, which is thus the progenitor of a Hubbard-Stratonovich field Φx,αβ is traceless on average: Goldstone mode. It is easy to calculate the energy levels Φαα = 0. x ǫs,σ ~k of the effective single-particle Hamiltonian (7) P with such Hubbard-Stratonovich field Φ: 3. MEAN-FIELD STUDY OF SPONTANEOUS ǫs,σ ~k = εs,σ ~k F ~k, Λ , CHIRAL SYMMETRY BREAKING IN THE PRESENCE OF CHIRAL IMBALANCE     2  2 εs,σ ~k = s vF ~k σµA + m (14) r | |− In this Section we study the phase diagram of our     model by explicitly minimizing the free energy (9). In where s, σ = 1, F ~k, Λ = d3xei~k·~xF (~x, Λ) is the what follows, we will use the continuum regularized Dirac ±   ~ Hamiltonian for the effective single-particle Hamiltonian factor which imposes UV cutoff andR εs,σ k denotes the hx,α;y,β in (7): energy levels of unregularized continuum  Dirac Hamil- tonian. In what follows, we assume that the func- x (0) hx;y = ivF αi + µ γ5 +Φx F˜ (x y, Λ) , (12) tion F ~k, Λ is equal to one for ~k < Λ and for − ∇i A − | |   ~k > Λ it approaches zero very quickly, so that for suffi- where vF is the Fermi velocity, x and y are now the con- | | x ciently smooth integrands A ~k the integrals of the form tinuum coordinates, i = ∂x,i + iAx,i is the covariant derivative over x which∇ includes external gauge field A x,i d3kF ~k, Λ A ~k can be written as d3kA ~k . ˜ and F (x y, Λ) is some ultraviolet regularization of the |~k|<Λ delta-function− δ (x y), which very quickly decays if x R     R   − Such UV regularization makes the energy of the Dirac and y are separated by the distance larger than the in- sea (the first summand in the mean-field free energy (9)) −1 verse UV cutoff scale Λ . αi = γ0γi = diag (σi, σi) are finite and mimics the finiteness of the Dirac sea in more the Dirac α-matrices, γ = diag (I, I) is the generator− 5 − realistic lattice models, for which Λ can be associated of chiral rotations, γµ are the Dirac gamma-matrices, σi with inverse lattice spacing. are the Pauli matrices and I is the 2 2 identity matrix. × In continuum space the action of the Hubbard field, We also suppress the spinor indices of the Dirac matrices which is the second part of the mean-field functional (9), and Φx,αβ for the sake of brevity. can be in general written as The main reason to use the continuum Dirac opera- tor in this exploratory study is to preserve exact chiral 1 cΛ3 Φ Φ = d3x Φ Φ , (15) symmetry of the action. First of all, this makes our re- 4V x αβ x βα 4V x αβ x βα x sults potentially applicable to a wider range of physi- X Z cal systems, including also QCD. Second, the mean-field where c is some constant which depends on the details of calculations in the continuum approximation are much UV regularization. This formula can be obtained, e.g., simpler and also more instructive. While for more realis- by taking the naive continuum limit in some lattice reg- tic lattice Dirac Hamiltonians the chiral symmetry is al- ularization. Clearly, the only effect of the constant c is ways broken at high energies due to lattice artifacts [67], some renormalization of the interaction potential V . 6

Inserting now the expressions (14), (13) and (15) into The very slow monotonic growth of the effective mass (9), we obtain the following expression for the mean-field m (V ) at small V is of course expectable, since finite free energy which should be minimized with respect to particle-like and hole-like Fermi surfaces at k = µA the effective mass m and the renormalized chiral chemical trigger Cooper-type instability towards the formation| | | of| potential µA: particle-hole bound states even at arbitrarily small inter- electron interaction potential V . The effective mass in d3k 2 F = v ~k σµ + m2 + this case should have an essential singularity in V of the L3 − 3 F | |− A (2π) σ=±1 r form |~k|Z<Λ X   B cΛ3 2 m = AV α exp , (18) + m2 + µ µ(0) (16) −µ2 V V A − A  A      where the factor µ2 is proportional to the density of In order to study spontaneous chiral symmetry break- A ing, we perform numerical minimization of this func- states at the Fermi level [68] (see e.g. Chapter 6 of [47]) and A, α and B are some constants. In order to check tional. To this end we set c = 1 and Λ = 1 in what follows, thus expressing all other quantities in units of UV this essential singularity scaling of the effective mass, on Fig. 2 we plot the inverse logarithm of the effective mass cutoff. Instead of fixing the Fermi velocity v to some F 1/ log (m) rescaled by µ2 as a function of V . Due to particular value, it is convenient to rescale the mean-field − A free energy as ¯ = /v and express it in terms of the finite precision of our numerical minimization procedure F F F the effective mass m contains large systematic numer- rescaled fermionic condensates and interaction potential (0) (0) (0) ical errors at sufficiently small V , and for smaller µ m¯ = m/v ,µ ¯ = µ /v ,µ ¯ = µ /v and V¯ = V/v . A F A A F A A F F the numerical errors typically set in at larger values of It is easy to check that in terms of these new variables, the mean-field free energy (16) does not depend on v : V . For this reason we only plot m in the range of V F where numerical errors are negligible. The results of nu- ¯ d3k 2 merical minimization are shown on Fig. 2 with points. F = ~k σµ¯ +m ¯ 2 + L3 − 3 | |− A One can see that indeed 1/ log (m) tends to zero in the (2π) σ=±1 r − |~k|Z<Λ X   limit of zero V . In order to quantify this tendency, we 3 2 fit the V dependence of 1/ log (m) with the function cΛ (0) 2 − 2 µAV + m¯ + µ¯A µ¯A . (17) 2 2 , as suggested by the scaling law V¯ − c−µAV log A−αµAV log V     (18). These fits are shown on Fig. 2 with solid lines. We thus see that after minimizing the above expression We find that the results of numerical minimization are the dependence of the final results on the Fermi velocity indeed well fitted by such functions, with the values of can be obtained by a simple rescaling. the parameters c, A and α coinciding within the fitting Saddle point values of the effective mass m and the uncertainty for all values of µ(0). renormalized chiral chemical potential µ are plotted on A A We therefore conclude that due to Cooper instability Fig. 1 as functions of the interaction potential V for dif- (0) (0) the second-order phase transition at µA = 0 turns into ferent values of the bare chiral chemical potential µA . If (0) the chiral chemical potential is absent in the bare Hamil- a crossover at µA > 0, and the true chiral phase transi- tonian, we observe the standard picture of second-order tion at which the effective mass m becomes different from quantum phase transition associated with spontaneous zero is shifted to V = 0. Since in realistic lattice models chiral symmetry breaking. Namely, the effective mass is the chiral symmetry is anyway broken by the nonlinear- identically zero for V/vF < Vc/vF = 39.5 and rapidly ity of the lattice dispersion relation away from the Dirac grows at V > Vc. The chiral chemical potential is iden- points and the pattern of spontaneous symmetry break- tically zero for all values of V in this case. ing might become quite different from the one in the con- However, at nonzero bare chiral chemical potential tinuum theory, we do not study here this crossover transi- (0) tion in more detail. Let us only mention that recently we µA the situation becomes more interesting. First, the second-order transition changes to some sort of crossover have performed a similar mean-field study of the phase or an infinite-order phase transition already at very diagram for Wilson-Dirac Hamiltonian with chiral imbal- ance, where the chiral phase transition is replaced by the small values of µ(0), so that the effective mass m (V ) A transition to the Aoki phase with condensed pions [48]. It still quickly rises for V larger than some pseudo-critical turned out that in this case the order of the phase transi- value but does not show any discontinuity of derivatives. tion does not change in the presence of chiral imbalance, Rather, m (V ) very slowly approaches zero as V goes to and there are no indications of Cooper-type instabilities. zero. The renormalized chiral chemical potential µA is (0) Otherwise, all the qualitative results of [48] are similar very close to the bare value for V . Vc µA but starts to those obtained in the present paper. (0) For crossover transitions the usual way to define the quickly growing at V > V µ . The derivatives of c A observable-dependent critical interaction potential is to   (0) µA (V ) also do not exhibit any discontinuities. As µA associate it with the inflection point of the approximate increases, the transition becomes more and more soft. order parameter. From Fig. 1 one can see that the critical 7

0.9 0.6 µ(0) /v =0.00 µ(0) /v =0.00 µ(0)A F µ(0)A F 0.8 A/vF=0.05 A/vF=0.05 µ(0) /v =0.10 0.5 µ(0) /v =0.10 0.7 µ(0)A F µ(0)A F A/vF=0.20 A/vF=0.20 0.6 0.4 F F 0.5 /v 0.3 A m/v 0.4 µ 0.3 0.2 0.2 0.1 0.1 0 0 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70

V/vF V/vF

FIG. 1: Mean-field values of the effective mass (on the left) and the renormalized chiral chemical potential (on the right) as (0) functions of interaction potential V at different values of the bare chiral chemical potential µA . All numbers are given in units of the UV cutoff scale Λ.

6 potential, which is not renormalized by virtue of conser- µ(0) A/vF=0.15 vation of electric charge (see the discussion at the end µ(0) /v =0.20 5 µ(0)A F of Section 2), chiral chemical potential is effectively en- A/vF=0.30 ) µ(0) hanced by interactions. A simple intuitive explanation F A/vF=0.40 4 of this observation is that the chiral imbalance, much like the Dirac mass term, lowers the vacuum energy and

Log(m/v 3 is thus energetically favourable. We discuss this prop- erty in more details for both continuum and lattice Dirac (0) 2

A fermions in Section 5. µ 2 -1/( 1 4. CHIRAL MAGNETIC EFFECT IN THE 0 LINEAR RESPONSE APPROXIMATION 0 5 10 15 20 25 30 V/v F In this Section we study how the chiral symmetry breaking discussed in the previous Section 3 affects the FIG. 2: Inverse logarithm of the effective mass m at small CME. Here we consider CME as a stationary process, values of the inter-electron interaction potential V . Solid lines that is, a response of static, permanently flowing elec- are fits to the essential singularity scaling (18). tric current to a static magnetic field. On the one hand, this approximation might be too idealistic to be real- ized in real physical systems, for which chirality pumping (0) and the application of external magnetic field are always coupling Vc µA defined as the inflection point of the dynamical processes [6, 44–46]. On the other hand, it effective mass becomes smaller as the bare chiral chemical   is probably the possibility to have charge transport in potential grows. This finding is in agreement with the the ground state of the system which makes the idea of results obtained from QCD-inspired effective models, for CME so attractive. Thus we assume that such a station- which the deconfinement temperature becomes lower at ary regime can be at least approached under the same nonzero µ [34–38]. Let us also note that the inflection A assumptions under which the system of Dirac fermions point of the derivative dµ /dV can be also used to define A with chiral imbalance can be considered as a stationary the “critical” interaction potential, which turns out to be state (see the discussion in the introductory Section 1). slightly different from the critical value obtained from the inflection point of the effective mass, but also decreases In the linear response approximation, one can charac- terize such a static CME response by the chiral magnetic for larger µ(0). This difference in the critical values of A ~ ~ V obtained from different observables also indicates that conductivity σCME k , where k is the wave vector which in the presence of chiral chemical potential spontaneous describes the spatial modulation of sufficiently weak mag- breaking of chiral symmetry is a crossover, at least in the netic field which causes the CME. In order to match the mean-field approximation. hydrodynamical description of the chirally imbalanced We see that in contrast to the conventional chemical plasma one has to take the limit ~k 0. We assume → 8 that ~k is parallel to the third coordinate axis: ~k = k3~e3. i = 1, 2, 3 are the Pauli matrices with chiral indices L, Then σ (k ) can be found from the following Kubo R. We will also use the notation Γ τ σ , where CME 3 A ≡ A ⊗ A formula [49–51]: depending on the value of A τA and σA can be the cor- responding Pauli matrices or the identity matrices. The i 1 σ (k )= eik3(x3−y3) 0 ˆj ˆj 0 = matrices ΓA are normalized as tr (ΓAΓB)=4δAB, so CME 3 −k L3 h | x,1 y,2 | i 3 x,y that the action of the Hubbard-Stratonovich field (sec- X ond summand in the mean-field free energy (9)) reads i 1 δ2 1 Φ Φ = 1 Φ2 . = eik3(x3−y3) F (19) 4V x,αβ x,βα V x,A −k L3 δA δA x x 3 x,y x,1 y,2 Ax,i=0 AfterP some manipulationsP with derivatives of implicit X functions, which are summarized in Appendix A, we ar- ˆ ˆ† ˆ where jx,i = ψxαiψx is the current density operator, rive at the following general expression for the second- A is the external Abelian gauge field and [A ] = order variation of the mean-field free energy with respect x,i F x,i T log [A ] is the free energy of the system which de- to external gauge field Ax,i: − Z x,i pends on the external gauge field Ax,i. δ2 ∂2 In the mean-field approximation the free energy F = F [A ] is approximated by its value at some saddle-point δAx,i δAy,j ∂Ax,i ∂Ay,j − F x,i value of the Hubbard field Φx,αβ, which we denote as ∂2 ∂2 ⋆ G F F , (23) Φ [Ax,i]: z,A;t,B x,αβ − ∂Ax,i ∂Φz,A ∂Ay,j ∂Φt,B z,A,t,BX [A ]= Φ⋆ [A ] , A , (20) F x,i F x,αβ x,i x,i where is the mean-field free energy (9) calculated in the presenceF of external gauge field A and G is where the saddle-point value Φ⋆ [A ] is the solution x,i z,A;t,B x,αβ x,i the propagator of the Hubbard-Stratonovich field defined of the equation by the identity

∂ 2 [Φx,αβ, Ax,i]=0 (21) ∂ ∂Φx,αβ F Gx,A;y,B F = δxzδAC. (24) ∂Φy,B∂Φz,C Xy,B and [Φx,αβ, Ax,i] is now the mean-field free energy (9) F To proceed, we now restrict our analysis to the con- in the presence of external magnetic field B~ = ~ A~ tinuum theory with the effective single-particle Hamil- described by the gauge vector field A . Since we∇× con- x,i tonian (12) and the action of the Hubbard-Stratonovich sider the static CME response, all quantities in these field given by (15). An important property of the Hamil- equations are completely time-independent. Note also tonian (12) which will significantly simplify the calcula- that since the mean-field free energy (9) in general de- tions is that the external gauge field A enters it and pends on the external electromagnetic field, the saddle- x,i ⋆ thus the fermionic part point value of the Hubbard field Φx,αβ [Ax,i] also becomes some functional of the gauge field Ax,i. This is sim- ǫ ǫ = ǫ = i − | i| (25) ply the reflection of the change of fermionic condensates S i 2 ǫ <0 i in external gauge field Ax,i. The variations of the ex- Xi X tremum value of the mean-field free energy [A ] with F x,i of the mean-field free energy (9) in exactly the same Ax,i should then take into account both the variation of way as the component of the Hubbard-Stratonovich field the functional [Φx,αβ, Ax,i] itself and the variation of which corresponds to ΓA = τ3 σi = γ0γi = αi (up to the saddle-pointF value of the Hubbard-Stratonovich field 0 ⊗ ⋆ a trivial factor of vF ), and thus we can write Φx,αβ [Ax,i], which is in general spatially inhomogeneous. Since the spinor structure of the fermionic condensates ∂2 ∂2 in the presence of external gauge field might be quite F = vF S . (26) ∂Ax,i ∂Φz,A ∂Φx,A ∂Φz,A complicated, for further notational convenience we de- 0 compose the spinor part of Φx,αβ in the full basis ΓA, To simplify the notation, we continue to use the symbol A =1 ... 16 of hermitian spinor operators: of the partial derivative ∂ also in the continuum theory, although now it should be understood as a functional 16 derivative. Φx,αβ = Φx,AΓA,αβ, It is important to note that the identity (26) holds AX=1 only for the specific cutoff regularization (12) of the con- ΓA = I I, I σi, τ1 I, τ1 σi, tinuum Dirac Hamiltonian without any covariant point- { ⊗ ⊗ ⊗ ⊗ τ I, τ σ , τ I, τ σ , i =1, 2, 3, (22) splitting regularization for the covariant derivative with 2 ⊗ 2 ⊗ i 3 ⊗ 3 ⊗ i} vector gauge field Ax,i. For lattice regularizations, the where the first factors in the direct products are the ma- gauge field will be associated with lattice links and the trices with chiral indices L (eft) and R (ight), the second Hubbard-Stratonovich field - with lattice sites. There- factors are the matrices with spin indices , and τ , fore in general they will enter the action in different ↑ ↓ i 9 ways. On the one hand, our cutoff regularization im- the coordinate representation of the fermionic one-loop plies the loss of invariance under gauge transformations correction to the self-energy of the Hubbard-Stratonovich of Ax,i. On the other hand, with such a regularization field Φx,A. The equation (27) can be then written simply the bare Hamiltonian (12) remains invariant under chi- as ral rotations h e−iγ5θheiγ5θ. This property is also → δ2 inherited by the vector current operator ˆjx,i which can 2 F = vF Πx,A0;y,B0 . (29) be obtained as a variation of (12) with respect to Ax,i. δAx,i δAy,j This violation of vector current conservation at the ex- pense of maintaining invariance under chiral rotations After some simple algebraic manipulations, the equation is the usual redistribution of the axial anomaly between (27) can be brought into the following form: vector and axial currents [19, 51–53] and is (sometimes 2 −1 2 4 1 implicitly) used in many derivations of the CME which Π=Σ Σ Σ+ Σ= . (30) − V V − V 2 Σ+ 2 rely on cutoff regularization, see e.g. [2, 34, 35, 54, 55].   V Often the role of the cutoff is played by the Fermi sur- Using now the identity face at ǫ = µA, which is also the property of a partic- ular regularization [19]. It is well known that for free 2 δxy δAB fermions cutoff regularization yields the finite answer for Σx,A;y,B + = V ~ µA the chiral magnetic conductivity σCME k 0 = 2 2 → 2π ∂ −1 = F = (Gx,A;y,B) , (31) [2, 19, 54–56]. The conservation of vector current can be ∂Φx,A∂Φy,B always restored by adding a suitable Bardeen countert- 4 erm δS µA d xǫijk Ax,i∂j Ax,k to the effective action where the inverse is again the operator inverse, we can of the theory∼ (which is in our case the mean-field free finally rewrite the equation (27) as energy (9)).R In this case, the chiral magnetic conduc- 2 2 2 tivity should vanish in the limit of zero momentum [53]. δ 2 vF δ (x y) δij 4 vF F = − 2 Gx,A0;y,B0 . (32) In this work, we will perform all the calculations in the δAx,i δAy,j V − V cutoff regularization, which is much more convenient for our purposes, and then restore the vector current conser- Let us now discuss the physical meaning of this equa- vation with the help of the Bardeen counterterm. The tion. We see that the Hubbard-Stratonovich field Φx,αβ possibility to use other regularizations for our problem can mimic any local term in the Dirac Hamiltonian. Thus will be discussed in more details in Section 5. if the interaction potential V is very large and we can ne- We now use (26) to express the propagator of the glect the effect of the quadratic action term of this field Hubbard-Stratonovich field in terms of and rewrite the (second summand in (9)), any local perturbation δhx,y = second variation of the mean-field freeS energy (23) as φx,AΓAδxy of the Dirac Hamiltonian by some exter- A nal field φ can be screened by forming the fermionic δ2 ∂2 P x,A 2 ˆ† ˆ ⋆ F = vF S condensate ψx,αψx,β Φx,AΓA,αβ = φx,AΓA,αβ. As δAx,i δAy,j ∂Φx,A0 ∂Φy,B0 − h i ∼ a result, the free energy does not depend on φx,A and 2 −1 2 ∂ 2 δ (z t) δAB the second variation (32) vanishes. From (32) one can vF S + − − ∂Φz,A ∂Φt,B V × see that for large but finite V the response to exter- z,A,t,BX   nal perturbations scales as 1/V . The linear operator ∂2 ∂2 which describes such a response consists of a contact term S S ,(27) ×∂Φ ∂Φ ∂Φ ∂Φ 2δxy δij x,A0 z,A y,B0 t,B V and a nontrivial term with the propagator of the Hubbard-Stratonovich field. Remembering that in the where Γ = τ σ = γ γ = α , the inverse in the B0 3 j 0 j j strong-coupling regime the fluctuations of the Hubbard- second line is the⊗ operator inverse defined as in (24) and ⋆ Stratonovich field around the saddle point correspond to all the derivatives should be taken at Φx,A = Φx,A and the propagation of fermionic bound states, we conclude Ax,i = 0. We have also set c = 1 and Λ = 1, as in the that this nontrivial term describes particle-hole bound previous Section. The above expression can be simplified states excited by the external field. even further by rewriting it in terms of the operators In the weak-coupling regime we can expand (30) in δ2 powers of V to obtain Πx,A;y,B = F , δφx,Aδφy,B V V V Π=Σ Σ Σ+Σ Σ Σ+ .... (33) ∂2 − 2 2 2 Σx,A;y,B = S , (28) ∂Φx,A∂Φy,B ⋆ Φ As we will see from what follows, the operator Σx,A;y,B

corresponds to the fermionic loop with the insertion of where Πx,A;y,B describes the mean-field response to some spinor operators ΓA and ΓB at points x and y. Then external field φx,A which enters the Dirac Hamiltonian the expansion (33) can be readily interpreted as a sum of (12) as δhx,y = φx,AΓAF˜ (x y, Λ) and Σx,A;y,B is A − an infinite number of chain-like diagrams, as illustrated P 10

energy levels which cross zero in the presence of exter- nal perturbations. The second summand is the usual one-loop fermionic contribution to the self-energy of the Hubbard-Stratonovich field. The summation over i is re- stricted to occupied energy levels with ǫi < 0, while sum- mation over j goes over all energy levels which do not coincide with ǫi. For our continuum regularized Dirac Hamiltonian (12), the eigenstates Ψi are labelled by FIG. 3: Diagrams which contribute to the mean-field linear two discrete indices s, σ = 1 as well| asi the momentum response operator (32) in the weak-coupling regime. ± ~k, and the energies ǫi are given by (14). The derivative ∂h is given by ∂Φx,A on Fig. 3. At sufficiently large interaction potential V geometric series which describe such a sum diverge, and ∂hx;y = δ (x z) F˜ (x y, Λ)ΓA. (36) one has to re-sum them and re-interpret the result in ∂Φz,A − − terms of particle-hole bound states. The last ingredient which we need for the calculation of We relegate the details of the calculation of Σx,A;y,B to the mean-field CME response is the operator Σx,A;y,B = Appendix B. Since the exact analytic calculation turned 2 ∂ S in explicit form. Since we are interested in out to be very complicated at nonzero effective mass m ∂Φx,A ∂Φy,B the static response functions at zero temperature, the and chiral chemical potential µA, we have used numeri- cal integration to sum over all states in (35). We then most direct way to arrive at the desired result is to use the standard quantum-mechanical perturbation theory perform the Fourier transform of Σx,A;y,B with respect ~ for the effective single-particle Hamiltonian (12) and to to x and y as in (19) and assume that the momentum k ~ expand each of its energy levels to the second order in the is parallel to the 3rd coordinate axis, k = k3~e3. ⋆ Hubbard-Stratonovich field Φx,A around Φx,A = Φx,A. In order to discuss the structure of ΣAB ~k and In the abstract bra-ket notation, the first and the second its physical implications, let us recall that according  variations of the energy level ǫi with Φ read to (31) Σx,A;y,B and hence also ΣAB ~k differs from ∂ǫi ∂h = Ψi Ψi the inverse propagator of the Hubbard-Stratonovich  field ∂Φ h | ∂Φ | i −1 x,A x,A (Gx,A;y,B) only by a term diagonal in A and B. ∂2ǫ Since in the strong-coupling regime this propagator de- i = ∂Φx,A ∂Φx,B scribes particle-hole bound states (mesons in QCD ter- minology), we can interpret the appearance of the off- Ψ ∂h Ψ Ψ ∂h Ψ i ∂Φx,A j j ∂Φy,B i = h | | ih | | i + diagonal terms in ΣAB ~k as the mixing between dif- ǫi ǫj Xj6=i − ferent bound states. Several  states are mixed already ∂h ∂h at zero chiral chemical potential due to spontaneous Ψi ∂Φ Ψj Ψj ∂Φ Ψi + h | y,B | ih | x,A | i , (34) breaking of chiral symmetry. First, by virtue of the ǫi ǫj A j6=i on-shell equation ∂ j = 2m ψγ¯ ψ there is the mixing X − µ µ 5 between the longitudinal component of the axial cur- where Ψi is the eigenstate which corresponds to the A † ¯ | i rent kiji = kiψ I σi ψ = kiψγiγ5ψ and the Nambu- energy level ǫi of the single-particle effective Hamiltonian. Goldstone mode (“pion”)⊗ which corresponds to the op- Correspondingly, the operator Σx,A;y,B can be written as † erator ψ τ2 I ψ = ψγ¯ 5ψ. Here we have used the “rel- ⊗ ¯ † ∂ǫ ∂ǫ ativistic” notation ψ = ψ γ0 in order to facilitate the Σ = δ (ǫ ) i i + x,A;y,B i ∂Φ ∂Φ identification of the bound states discussed here with i x,A y,B X meson states in QCD. Another similar on-shell equa- ∂2ǫ tion, ∂ ψ¯ [γ ,γ ] ψ = 4m ψγ¯ ψ, implies the mixing + θ ( ǫ ) i = µ µ ν ν − i ∂Φ ∂Φ between the longitudinal component of the tensor ex- i x,A y,B ¯
¯ X citations kiψ [γi,γj ] ψ and kiψ [γi,γ0] ψ and the vector ∂h ∂h = δ (ǫ ) Ψ Ψ Ψ Ψ + current and the charge density, correspondingly. i h i| ∂Φ | iih i| ∂Φ | ii i x,A y,B Nonzero chiral chemical potential µA explicitly breaks X ∂h ∂h parity and hence induces mixing between parity-odd and Ψi ∂Φ Ψj Ψj ∂Φ Ψi + h | x,A | ih | y,B | i + parity-even states. First, the states created by the op- ǫ ǫ † ¯ i:ǫ <0 i j erators ψ τ3 σi ψ = ψγiψ (vector current fluctuations, Xi Xj6=i − ⊗ ∂h ∂h or vector mesons in QCD terminology) are mixed with Ψi Ψj Ψj Ψi h | ∂Φy,B | ih | ∂Φx,A | i the fluctuations of magnetization which are described + . (35) † ǫi ǫj by the operators ψ τ1 σi ψ = iǫijkψ¯ [γj ,γk] ψ. Sec- i:ǫi<0 j6=i − X X ond, the scalar states which⊗ correspond to the operator The first summand on the r.h.s. of (35) originates from ψ† τ I ψ = ψψ¯ (fluctuations of the chiral condensate, 1 ⊗ 11 or σ-meson in QCD terminology) are mixed with the fluc- ductivity at zero momentum, however, now σCME (k3) tuations of the axial charge density, described by the op- approaches constant in the limit of infinite momentum † erator ψ τ3 I ψ = ψγ¯ 5γ0ψ. [19]. In what follows, the correlator of conserved vector ⊗ currents is denoted as ˜j ˜j (k ). Moreover, chiral imbalance induces the mixing be- h 1 2 i 3 tween the two transverse polarizations of the vector and Second, in order to include the Fermi velocity vF < pseudo-vector excitations. It is precisely this mixing be- 1 into our calculations, it is again convenient to ex- tween the transverse fluctuations of the vector current, press all results in terms of the rescaled variables † ¯ j1,2 = ψ τ3 σ1,2 ψ = ψγ¯ 1,2ψ, which leads to the ap- m¯ = m/vF ,µ ¯A = µA/vF and V = V/vF . As ⊗ ˆ ˆ pearance of the nonzero off-diagonal element 0 j1j2 0 discussed in Appendix B, ΣAB ~k depends on the of the vector current correlator and hence to nonzeroh | chi-| i Fermi velocity as Σ ~k Σ  ~k; µ ,m,v = ral magnetic conductivity σCME according to the Kubo AB ≡ AB A F formula (19). We therefore conclude that in the strong- v−1Σ ~k;¯µ , m,¯ v =1 . Inserting this relation into coupling phase the CME current is saturated by vector- F AB A F like bound states (ρ-mesons in QCD terminology) with (31) and (32), we see that the ratios σCME /vF and j j /v depend only on the rescaled variables. Thus mixed transverse polarizations and a small admixture of h 1 2 i F pseudo-vector states of both transverse polarizations. A in order to calculate the anomalous current-current cor- similar picture of meson mixing has been recently ad- relators and the chiral magnetic conductivity at some ar- dressed also in QCD effective models [40]. In this work it bitrary value of vF , one should simply plug the rescaled ¯ was also pointed out that the mixing between the trans- variablesm ¯ = m/vF ,µ ¯A = µA/vF and V = V/vF into verse components of vector mesons can be encoded in the the result obtained with vF = 1, and finally multiply it by Chern-Simons term in the effective action for the vector vF . Therefore we give all our numerical results in terms mesons. of the rescaled variables introduced above. It is impor- ~ tant to note that the momentum variables should not be After having calculated ΣAB k numerically, we plug rescaled. An important consequence of this simple scal- it into the equations (31) and (32 ) and calculate the ing with Fermi velocity is that for free fermions the chi- (0) (0) Fourier transforms of the anomalous current-current cor- µ¯A µA ral magnetic conductivity σ = v 2 = v 2 = CME F 2π F 2π vF relators j1j2 (k3). We then use the Kubo formula (0) h i µA (19) in order to find the chiral magnetic conductivity 2π2 does not depend on vF . σ (k ). Before presenting our results for j j (k ) CME 3 h 1 2 i 3 The results for the anomalous current-current corre- and σCME , let us make several remarks on their inter- lators at different values of the interaction potential V pretation. (0) and the bare chiral chemical potential µA are presented First, the vector current defined by the functional on Fig. 4. Left and right plots represent the correlators derivative of the free energy over the gauge field as in j1j2 (k3) and ˜j1˜j2 (k3) of the non-conserved and con- (19) is not conserved for our regularization (12) of the hservedi vector currents,h i respectively. The plots at the top Dirac Hamiltonian, see the discussion after equation (26). and on the bottom correspond to bare chiral chemical On the other hand, our regularization preserves the in- (0) (0) potential µA /vF = 0.05 and µA /vF = 0.20. One can variance of the Hamiltonian under chiral rotations. For immediately see that the anomalous current-current cor- free fermions such a regularization yields a finite an- ˜ ˜ (0) relators j1j2 (k3) and j1j2 (k3) both grow with V for µA h i h i swer σCME = 2π2 [2, 54]. Conservation of vector cur- all values of the momentum k3, both in the perturbative rent can be restored by adding the Bardeen countert- regime at V < Vc and in the strongly coupled regime at 4 erm SB d xǫijk Ax,i∂j Ax,k to the bare action, which V > Vc. The relative growth is even more pronounced for should lead∼ to vanishing chiral magnetic conductivity in the smaller value of the chiral chemical potential. This the limit ofR small momentum [53]. The second varia- enhancement of the anomalous current response for in- tion of the Bardeen counterterm over the gauge field teracting fermions is one of the main conclusions of this 2 δ SB ǫijlkl has the same form as the anomalous work. δAiδAj ∼ current-current correlator which enters the Kubo for- The slope of the anomalous correlator of non-conserved mula (19) for the chiral magnetic conductivity. Since the vector currents at small k3 is the small-momentum limit chiral magnetic conductivity should vanish in a gauge- of the chiral magnetic conductivity σCME (k3 0). It → invariant regularization [53], we conclude that the coef- is this limit which is relevant for the hydrodynami- ficient before the Bardeen counterterm should be equal cal description of chirally imbalanced medium [49–51]. σCME (k3 0) is plotted on the left plot on Fig. 5 as to the chiral magnetic conductivity σCME ~k 0 cal- → a function→ of interaction potential V for different val- culated in terms of the non-conserved current. Taking   ues of the bare chiral chemical potential µ(0). In or- into account this form of the Bardeen counterterm, we A der to illustrate the dependence of σCME (k3 0) on conclude that in order to calculate the anomalous corre- (0) → lator of conserved currents we simply have to subtract the µA , on the right plot on Fig. 5 we also plot the ra- (0) term iσCME (k3 0) k3 from j1j2 (k3). This subtrac- tios σCME (k3 0) /µA . Again we see that interac- tion clearly leads→ to the vanishingh chirali magnetic con- tions enhance the→ chiral magnetic conductivity both in 12

0.0045 0.014 V/vF = 5 V/v = 25 0.004 0.012 F V/vF = 40 0.0035 V/v = 55 0.01 (0) F 2 F µ π 0.003 F A k3/(2 vF)

)/v V/v = 5 F )/v 3 0.0025 V/vF = 25 3 0.008 >(k

V/v = 40 >(k

2 F 2 j ~ 0.002 V/v = 55 j

1 F 0.006 (0) 2 1

µ π ~ A k3/(2 vF)

0.0015 i

k3 k3

0.03 0.08 V/vF = 5 V/vF = 5 V/vF = 25 0.07 V/vF = 25 0.025 V/vF = 40 V/vF = 40 V/v = 55 V/v = 55 (0) F 2 0.06 (0) F 2

F µ π µ π 0.02 A k3/(2 vF) F A k3/(2 vF) )/v

)/v 0.05 3 3 >(k

0.015 >(k 0.04 2 2 j ~ j 1 1

~ 0.03

0.01 i

k3 k3

FIG. 4: The anomalous current-current correlator h j1j2 i (k3) which enters the Kubo relations (19) as a function of spatial (0) (0) momentum k3. At the top: at the bare chiral chemical potential µA /vF = 0.05, on the bottom: at µA /vF = 0.20. On the left: current-current correlator calculated in terms of the non-conserved vector currents j1,2, on the right: current-current correlator in terms of conserved vector currents ˜j1,2 (after the subtraction of the Bardeen counterterm). Thin lines illustrate the asymptotic behavior of h j1j2 i (k3) either in the limit of zero momentum (for non-conserved vector current) or in the limit of large momentum (for conserved current). All numbers are given in units of the UV cutoff scale Λ.

iµ(0)k the weak- and in the strong-coupling regimes. At small A 3 we can simply use the free-fermion result 2 . Insert- V σ (k 0) grows linearly. In the strong-coupling 2π CME 3 → ing this expression into (33), (29) and (19) and keeping regime this enhancement becomes much stronger. From only the first two terms of the expansion in powers of V the right plot on Fig. 5 one can again see that the rela- as well as only the leading power of µA, we obtain tive increase of the chiral magnetic conductivity is larger (0) (0) for smaller values of µ , whereas for larger µ one can ΠA B ~k =ΣA B ~k A A 0 0 0 0 − note some indications of the saturation of σ at large CME   V   V . Σ ~k Σ ~k − A0 A0 2 A0 B0 − Linear growth of ΣCME in the perturbative regime can     ~ V ~ be readily understood from the first terms in the expan- ΣA0 B0 k ΣB0 B0 k . (37) − 2 sion (33). The operator Σ ~k depends on the inter-     AB Direct calculation shows that the diagonal elements action potential V only through  the renormalized mass ~ ~ ΣA0 A0 k = ΣB0 B0 k are negative, and thus the m and the chiral chemical potential µA. Since at small (0)     ~ V their values are very close to the bare values m =0 first perturbative contribution increases ΠA0 B0 k and (0) and µA (see Fig. 2 and Fig. 1), in the weak-coupling hence by virtue of the relations (29) and (19) also the regime we can neglect the V dependence of ΣAB ~k . In chiral magnetic conductivity. In the strong-coupling regime, the growth of the chiral ~   particular, for the off-diagonal part of ΣA0 B0 k which magnetic conductivity is a nontrivial interplay between corresponds to the anomalous current-current correlator the “screening” factor 1/V 2 in the second summand of 13

(32) and the growth of renormalized chiral chemical po- then restore by adding the Bardeen counterterm to the tential with V , which makes the off-diagonal elements action. It is therefore compelling to check whether our

GA0B0 larger. results are still valid in more consistent regularizations which automatically preserve gauge invariance (and also preferably the chiral symmetry) of the theory. Here we 5. VACUUM ENERGY IN THE PRESENCE OF will check whether the vacuum energy is still lowered by CHIRAL CHEMICAL POTENTIAL the chiral chemical potential µA for several different reg- ularizations of the continuum Dirac Hamiltonian. We From the mean-field analysis of Section 3 we have seen show the plots of the vacuum energy (defined as in (25)) that interactions tend to increase the chiral chemical po- as a function of µA on Fig. 7. The vacuum energy at tential µA. This behavior can be readily explained using µA = 0 is subtracted from the result. For comparison the following simple argument. For simplicity let us as- with different regularizations, we also plot the vacuum sume that the Dirac mass is zero, so that the energy levels energy for our continuum Hamiltonian (12). of the Dirac Hamiltonian h0 = iαi i + µ5γ5 read Let us first consider local lattice regularizations, − ∇ which are natural in the context of condensed matter physics. For instance, a simple model description of Weyl εs,σ ~k = s ~k σµA , (38) || |− | semimetals is provided by the Wilson-Dirac Hamiltonian   [9, 32, 59] of the following form: where we have assumed that the Fermi velocity vF is equal to unity for the sake of brevity. We now calculate k/ µ ∆ ~k the fermionic contribution to the free energy (25): A hW D ~k,µA = − , (41)  ∆ ~k k/ + µA  3~   − d k ~     = 3 ε−1,σ k . (39) 3 F − (2π) −1 ~ Z σ=±1   where k/ = a σi sin (aki), ∆ k = X i=1 3   ~ −1 2 P Let us group the summands with the same momentum k 2a sin (aki/2) is the Wilson term (for sim- and different values of σ: i=1 plicity,P we set the Wilson parameter ρ and the Fermi ε−1,+1 ~k + ε−1,1 ~k = velocity vF to unity), the momenta ki belong to the cubic Brillouin zone ki [ π/a,π/a] and a is the   ~   ∈ − ~ ~ 2µA, k <µA lattice spacing. Here we have introduced the chiral = k µA k + µA = − ~ |~| , (40) chemical potential µ by simply adding the term µ γ −|| |− | − || | | ( 2 k , k µA A A 5 − | | | |≥ to the conventional Wilson-Dirac Hamiltonian. Explicit 3 After such rearrangement of summands the sum in (39) calculation of the vacuum energy on the 20 lattice looks similarly to the free energy of massless Dirac shows that it also decreases as µA is increased, see fermions without any chemical potentials, however, now Fig. 7. the tip of the Dirac cone is effectively chopped off at the Next it is interesting to consider lattice regulariza- tions with exact chiral symmetry, for instance, overlap level ǫ = µA (see Fig. 6 for an illustration). This means that effectively all the energy levels of Dirac fermions be- fermions. Overlap Dirac Hamiltonian without chemical came lower, and hence the vacuum energy also decreased. potentials was introduced in ([60]): Therefore it is energetically advantageous for a system of h(0) = a−1 (γ + γ γ sign (γ γ (ah ρ))) , (42) Dirac fermions to develop nonzero chiral chemical poten- ov 0 0 5 5 0 W D − tial, or just to increase its value if the bare value µ(0) where ρ [0, 2]. The corresponding axial charge operator A ∈ is different from zero. However, in the absence of bare which commutes with the Hamiltonian (44) reads chiral chemical potential a more efficient way of lowering γ5 sign (γ5γ0 (ahW D ρ)) the vacuum energy is simply the generation of effective QA = − − . (43) mass, and the system prefers to spontaneously break chi- 2 ral symmetry. It is interesting to note that although at We then define the overlap Hamiltonian at finite chiral (0) chemical potential as µA = 0 nonzero value of µA cannot be generated, there can be still large fluctuations of µA. It was conjectured h (µ )= h(0) + µ Q (44) recently in [57, 58] that these fluctuations might be the ov A ov A A origin of the so-called inverse magnetic catalysis in QCD. and after going to momentum space explicitly calculate However, this simple argument is based on the Hamil- its energy levels on the 203 lattice. We again observe tonian with unbounded dispersion relation, for which the that the vacuum energy is lowered by the chiral chemical vacuum energy is divergent. We therefore have to regu- potential. larize the problem in some consistent way. In this work It might seem tempting to use the lattice Hamiltoni- our choice was the cutoff regularization (12). However, ans (44) or (41) for our study in order to avoid the am- cutoff regularization breaks gauge invariance, which we biguities of UV regularization. Let us note, however, 14

0.12 1.2 µ(0) µ(0) A/vF=0.05 A/vF=0.05 µ(0) /v =0.10 µ(0) /v =0.10 0.1 µ(0)A F 1 µ(0)A F A/vF=0.20 A/vF=0.20 A F (0)

0.08 µ 0.8 -> 0)/v

3 0.06 0.6 -> 0)/ 3 (k (k

CME 0.04 0.4 CME σ σ 0.02 0.2

0 0 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70

V/vF V/vF

FIG. 5: Chiral magnetic conductivity σCME (k3 → 0) (calculated in terms of the non-conserved vector current) in the limit of (0) zero momentum as a function of interaction potential V at different values of bare chiral chemical potential µA . On the right (0) plot σCME is rescaled by the bare value of the chiral chemical potential µA . All numbers are given in units of the UV cutoff scale Λ.

4 0.003 ε (k) ε ε -1,+/-1 3.5 ( -1,+1(k) + -1,-1(k))/2 |k| 0.002 (k2 + m2)1/2, m = µ

3 A 3 s 0.001 2.5 0

(k) 2 ε ) - F(0))/L A µ 1.5 -0.001 Ov.op. (F( WD.op. 1 -0.002 WD.Ham. Ov.Ham. 0.5 Dir.Ham.x104 -0.003 0 -0.1 -0.05 0 0.05 0.1 -3 -2 -1 0 1 2 3 µ A k

FIG. 6: Schematic illustration of the dispersion relation in FIG. 7: Free energy as a function of the chiral chemical po- the presence of chiral chemical potential. tential µA for different regularizations of the Dirac opera- tor/Hamiltonian. “Ov.op” is for the overlap Dirac opera- tor, “WD.op.” is for the Wilson-Dirac operator with zero bare mass, “WD.Ham.” is for the Wilson-Dirac Hamiltonian, that since lattice chiral rotations for the chirally invari- “Ov.Ham” is for the overlap Hamiltonian and “Dir.Ham.” is ant overlap Hamiltonian (44) are necessarily non-local for the continuum Dirac Hamiltonian (12). [60], the on-site four-fermion interaction term in (1) will anyway break lattice chiral symmetry. Therefore if one would like to study lattice chiral fermions in the Hamil- beyond the scope of this paper, and we postpone it for tonian formalism, one should necessarily consider more further work. complicated interactions than in (1), for example, gauge- In the context of relativistic quantum field theory it is mediated interactions. The problem with the Wilson- more usual to work with the Dirac operator rather than Dirac Hamiltonian (41) is that the chiral symmetry is the Dirac Hamiltonian. For some Dirac operator on 3 D broken from the very beginning by the Wilson term. Be- the Lt L lattice we define the vacuum energy as × s cause of that, one cannot really observe the standard picture of chiral symmetry breaking for Wilson-Dirac = log det ( )/(aLt). (45) S − D fermions. Instead, one finds the strongly-coupled phase with broken parity (Aoki phase, [61]). While a detailed As the temporal lattice extent Lt , this definition → ∞ study of the influence of chiral imbalance on this phase becomes equivalent to (25) up to finite-spacing artifacts. is certainly interesting in relation with transport proper- We continue our comparative study with the Wilson- ties of Weyl semimetals and topological insulators, it is Dirac operator at finite chiral chemical potential, which 15 was first introduced in [62]: Our observations here suggest that there are two inter- pretations of the chiral chemical potential for a system of

W D k0,~k,µA = Dirac fermions. One is valid for condensed-matter-style D models, for which the Dirac sea consists of a finite num-   ~ 2 2 ak0 i ber of levels with real fermions occupying them. This ∆ k + a sin 2 a sin (ak0 iµA)+ k/ = − .(46) is the case for our regularization 12, the Wilson-Dirac i 2 2 ak0  sin  (ak0 + iµA) k/
∆ ~k + sin  a − a 2 Hamiltonian (41), the corresponding Wilson-Dirac oper-   
 ator (46) and the overlap Dirac Hamiltonian (44). In this A practical advantage of this operator is that the chiral case the vacuum energy is always lowered in the pres- chemical potential does not break time-reversal invari- ence of chiral imbalance. Another interpretation might ance and thus allows for efficient Monte-Carlo simula- be more relevant for relativistic quantum field theories tions which are free of the sign problem [62]. Explicit with chiral fermions. In this case one has to subtract the calculation of the vacuum energy (45) for this operator infinite contribution of the Dirac sea in the energy den- on the 20 203 lattice shows that again the chiral chem- sity by using either finite or infinite number of regulator × ical potential lowers the vacuum energy, see Fig. 7. This fermions. After this subtraction the contribution of regu- result is not surprising, as the Wilson-Dirac operator is lator fermions leads to the increase of the vacuum energy naturally obtained as one goes from Hamiltonian to the in the presence of chiral imbalance. Of course, one can path integral formalism for Wilson-Dirac fermions [63]. expect that if we switch to this another interpretation Now let us consider lattice Dirac operator which pre- of the chiral chemical potential, the results presented in serves exact chiral symmetry. An example is the overlap this work might change. Dirac operator at finite chiral chemical potential, which was introduced in [56]. In contrast to previous findings, it turns out that for this operator the chiral chemical 6. CONCLUSIONS AND DISCUSSION potential increases the vacuum energy (45), see Fig. 7. In order to understand somehow this quite unexpected result, let us remember that overlap fermions can be In this work we have reported on a mean-field study considered as a limiting case of the Pauli-Villars regu- of spontaneous breaking of chiral symmetry for Dirac larization with infinitely many regulator fields [64, 65]. fermions with chiral imbalance and contact interactions For simplicity, let us consider only one Pauli-Villars reg- between charges. Our main conclusion is the enhance- ulator for the continuum Dirac operator (µ ,m) = ment of the renormalized chiral chemical potential and D A the chiral magnetic conductivity due to interactions. γµ∂µ + γ0γ5µA + m. We then have to replace the loga- rithm of the determinant of the Dirac operator in (45) by This enhancement is found both in the weak-coupling perturbative regime and in the strongly-coupled phase the difference log det ( (µA,m)) log det ( (µA,M)), where M is the very largeD mass− of the regulatorD field. with broken chiral symmetry. We believe that this effect Note that there is no consistent many-particle Hamil- is a combination of both perturbative radiative correc- tonian associated with the regularized action (formally, tions and non-perturbative effects associated with spon- regulator fields can be thought of as bosons which vio- taneous breaking of chiral symmetry. lated the spin-statistics relation). Calculating now the Perturbative corrections to anomalous transport coef- ∂2F ficients are in general allowed if the corresponding cur- derivative 2 at µA = 0 for this Pauli-Villars regular- ∂µA rents are coupled to dynamical gauge fields [27–29], which ization, we arrive at the following result: are in our case mimicked by the contact interactions be- tween electric charges. As discussed in Section 4, our ∂2 F = mean-field analysis in fact corresponds to summation of − ∂µ2 A µA=0 an infinite number of chain-like diagrams. At the level of the single-particle Dirac Hamiltonian another possible dk d3~k m2 + k2 ~k2 M 2 + k2 ~k2 = 0 0 − 0 − = source of corrections to the CME current is the renor- 4π4 2 2 ~ 2 − 2 2 ~ 2 Z m + k0 + k M + k0 + k ! malization of the Fermi velocity, which is not taken into dkk2 m2 M 2 account in this work. = .(47) Non-perturbative corrections due to spontaneous chi- π2 2 2 3/2 − 2 2 3/2 Z (m + k ) (M + k ) ! ral symmetry breaking can be expected since the appear- ance of the massless Goldstone modes and the effective The integral in the last line is divergent and requires fur- mass term in general invalidate the assumptions under- ther regularization. However, it is easy to see that the lying the proofs of the non-renormalization of anomalous integrand is predominantly negative if m M, there- transport coefficients [13, 14, 16–18]. Another argument ∂2F ≪ fore the sign of 2 should be positive. This calculation ∂µA is that the chiral magnetic conductivity can be related to shows that also for the Pauli-Villars regularization the a certain correlator of two vector and one axial currents, chiral chemical potential increases the vacuum energy. which is not renormalized in massless QCD but can get The reason is simply the contribution of the regulator nontrivial non-perturbative corrections in a phase with fermions, which also feel the chiral chemical potential. broken chiral symmetry [19, 22, 23]. It should be stressed 16 that the enhancement of the chiral magnetic conductiv- Acknowledgments ity cannot be explained by the enhancement of the chiral imbalance alone. We also note that a somewhat similar The author would like to thank G. Dunne, T. Kalay- study has been previously reported in [34], where dielec- dzhyan, D. Kharzeev and A. Sadofyev for interest- tric screening, rather than the enhancement, of the chi- ing discussions which stimulated this work, and to ral magnetic conductivity was found. We note, however, B. Roy and the anonymous referee for their remarks on that the mean-field analysis of [34] did not systemati- Cooper-type instability. This work is supported by the cally take into account all possible channels of fermion S. Kowalewskaja award from the Alexander von Hum- condensation in the presence of external magnetic field. boldt foundation. In particular, the renormalization of the chiral chemical potential was not taken into account. Appendix A: Variation of the mean-field free energy We also find that even at the smallest values of the (0) with respect to external fields bare chiral chemical potential µA the transition to the phase with broken chiral symmetry turns into a soft In the mean-field approximation one replaces the free crossover due to Cooper-type instability at small interac- energy /T = log Φ e−S(Φ,φ) by the value of the tion strength. This crossover becomes softer and softer A action SF(Φ⋆, φ)− at the saddleD point Φ⋆ of the correspond- as µ(0) is increased. R A A ing path integral over some bosonic field ΦA. We have Let us note that while all our conclusions rely on the also assumed that the action depends on some external assumption of spatially homogeneous fermionic conden- fields φa. For the sake of brevity, in this Appendix we sates (in the absence of external magnetic fields), holo- use condensed index notation, so that the indices A and graphic studies in the Sakai-Sugimoto model suggest that a of the field variables include both spatial coordinates and also any internal indices of the fields Φ and φ. We for sufficiently large bare chiral chemical potential µ(0) A also assume summation over repeated indices. the true ground state is not homogeneous [41]. It would ∂S The linear response of the observables ja = to be very interesting to find whether the corresponding ∂φa the perturbations of the external fields φa is given by the inhomogeneous ground state also exists in our lattice 2 second derivative ∂ (F/T ) . In this Appendix we calculate model, which we leave for the further work. Let us note, ∂φa ∂φb however, that direct numerical analysis of the Hessian this derivative in the mean-field approximation, replacing 2 ⋆ matrix ∂ F of the mean-field free energy (9) in /T with S (Φ , φ). Since the saddle-point equation ∂Φx,A ∂Φy,B F the vicinity of the homogeneous condensate configura- ∂S (Φ, φ) tion shows that the homogeneous condensate is at least = 0 (A1) ∂Φ ⋆ a local minimum - that is, there are no flat or unstable di- A Φ rections in the space of inhomogeneous condensates Φx,A in general depends on the external fields φa, the saddle- near Φx,A ΦA which would allow for immediate decay ⋆ ⋆ ≡ point value of ΦA is also a function of φa: ΦA =ΦA (φ). into a non-homogeneous condensate. Therefore when the free energy /T is replaced by the ⋆ F Our analysis indicates also that in the strongly cou- saddle-point action S (Φ (φ) , φ), one also has to replace the partial derivatives ∂ with the variation pled regime the CME current is saturated by vector-like ∂φa bound particle-hole bound states (vector mesons) with δ ∂ ∂Φ⋆ (φ) ∂ mixed transverse polarizations. On the other hand, in = + A . (A2) [24–26] the CME current in the phase with broken chi- δφa ∂φa ∂φa ∂ΦA ral symmetry was derived from the contact terms in the Then one can write for the second derivative of the free low-energy chiral Lagrangian. It is an interesting ques- energy over the external fields: tion whether the contribution of vector mesons found in our work corresponds somehow to these contact terms in ∂2 ( /T ) δ δ the low-energy effective action or it should be addition- F = S = ∂φa ∂φb δφa δφb ⋆ ally taken into account. Φ 2 ⋆ 2 ∂ S ∂ΦA ∂ S = + + Our study should be most relevant for condensed mat- ∂φ ∂φ ∂φ ∂Φ ∂φ  a b a A b ter systems with Dirac or Weyl fermions as low-energy ∂Φ⋆ ∂2S ∂Φ⋆ ∂Φ⋆ ∂2S excitations. As discussed in Section 5, for relativistic + A + A B , (A3) ∂φ ∂Φ ∂φ ∂φ ∂φ ∂Φ ∂Φ ⋆ quantum field theories with chiral imbalance there is a b A a a b A B  Φ certain regularization ambiguity, which might potentially where we have omitted the arguments of the action change our conclusions. However, we believe that for S (Φ, φ) and of the saddle-point field Φ⋆ (φ) for the sake instance the mixing of transverse vector mesons in the ∂S(Φ,φ) of brevity. Due to the identity ⋆ = 0, there is ∂ΦA Φ presence of chiral imbalance should also be present in 2 | ⋆ QCD, irrespective of whether chiral symmetry breaking no term with the second derivative ∂ ΦA in the above ∂φa∂φb is enhanced or suppressed by chiral chemical potential. equation. 17

⋆ s,σ ~ The derivative of the saddle-point field ΦA over φ can and (36). The wave functions Ψx k which correspond be calculated by differentiating the saddle-point equation ~   (A1) over φ: to the energy levels ǫs,σ k are   2 ⋆ 2 ∂ S ∂ΦB ∂ S i~k·~x + =0. (A4) s,σ ~ ~ e ∂Φ ∂φ ⋆ ∂φ ∂Φ ∂Φ ⋆ Ψx k = ϕs,σ k , A a Φ a A B Φ √L3     ~ It is now convenient to define the propagator GAB of the 1 σvF |k|−µA ~ 2 + ~ ησ k field ΦA through the identity 2εs,σ (k) ϕs,σ ~k =  r  , (B1) σv |~k|−µ   2 1 F A ~ ∂ S s 2 ~ ησ k    − 2εs,σ (k)  GAB = δAC , (A5)  r  ∂Φ ∂Φ ⋆   B C Φ  

where ησ ~k is the Weyl spinor which is the normal- so that the equation (A4) can be written as ized eigenstate  of the operator kiσi with eigenvalue σ ~k : ⋆ 2 | | ∂ΦA ∂ S ~ ~ ~ = GAB . (A6) kiσiησ k = σ k ησ k . We note also that the energy ∂φa ⋆ − ∂ΦB∂φa ⋆ | | Φ Φ     levels εs,σ ~k which enter the spinor part ϕs,σ ~k of Inserting this equation into (A3) and using (A6) once the wave functions  (B1) are the energy levels (14) of the again, we finally arrive at the desired result: unregularized Dirac Hamiltonian. ∂2 ( /T ) δ δ We now insert these expressions into (35) and perform F = S = the Fourier transform with respect to x and y, as in (19): ∂φ ∂φ δφ δφ ⋆ a b a b Φ 2 2 2 ∂ S ∂ S ∂ S ~ 1 i~k·(~x−~y) = GAB . (A7) ΣAB k = 3 e Σx,A;y,B. (B2) ∂φ ∂φ ⋆ − ∂Φ ∂φ ∂Φ ∂φ ⋆ L a b Φ A a B b Φ x,y   X We begin with the calculation of the contribution of ~ Appendix B: One-loop fermionic contribution to the the second summand on the r.h.s. of (35) to ΣAB k , self-energy of the Hubbard-Stratonovich field (2) ~   which we denote as ΣAB k , and later consider the first  (1) ~ In this Appendix we give the details of the calculation contribution (denoted as ΣAB k ) associated with en- ∂2S of the operator Σx,A;y,B = , where given by ergy levels crossing zero.   ∂Φx,A ∂Φy,B S (25) is the fermionic contribution to the mean-field free Explicitly performing the Fourier transform (B2), we (2) ~ energy (9). Our starting point are the expressions (35) obtain the following expression for ΣAB k :  

d3l ϕ¯ (~q)Γ ϕ (~p)ϕ ¯ (~p)Γ ϕ (~q) F (~p, Λ) F (~q, Λ) Σ(2) ~k = s1,σ1 A s2,σ2 s2,σ2 B s1,σ1 + AB 3 ε (~q) F (~q, Λ) ε (~p) F (~p, Λ) s ,σ ,σ (2π) s1,σ1 s2,σ2   2X1 2 Z − d3l ϕ¯ (~p)Γ ϕ (~q)ϕ ¯ (~q)Γ ϕ (~p) F (~p, Λ) F (~q, Λ) + s1,σ1 B s2,σ2 s2,σ2 A s1,σ1 , (B3) 3 ε (~p) F (~p, Λ) ε (~q) F (~q, Λ) s ,σ ,σ (2π) s1,σ1 s2,σ2 2X1 2 Z −

where ~p = ~l + ~k/2, ~q = ~l ~k/2 and the “loop momen- it is easy to see that due to a specific combination of − tum” variable ~l was introduced in order to satisfy the F (~p, Λ) and F (~q, Λ) the integrand of (B3) vanishes if constraints of momentum conservation ~k + ~q ~p = 0. ~p = ~l + ~k/2 > Λ or ~q = ~l ~k/2 > Λ. Correspond- | | | | | | | − | In order to impose the constraint ǫ < 0 in (35−), we set ingly, in a region with ~p < Λ and ~q < Λ one can simply i | | | | s1 = 1, thus excluding this index from summations in replace these factors by unity, which significantly simpli- the above− expression. fies the calculations. In particular, it is convenient to Remembering that the regulating factor F (~p, Λ) is perform an explicit summation over s2 and σ2 using the equal to one for ~p < Λ and is very small for ~p > Λ, identity | | | | 18

ϕ (~p)ϕ ¯ (~p) s2,σ2 s2,σ2 = (ε (~q) h (~p))−1 = ε (~q) ε (~p) s1,σ1 − s ,σ s1,σ1 s2,σ2 X2 2 − 1 εs1,σ1 (~q)+ σ2vF ~p µA m = 2 2 | |− σ2 (~p) ,(B4) m εs1,σ1 (~q) σ2vF ~p + µA ⊗P σ =±1 (vF ~q σ1µA) (vF ~p σ2µA) 2X | |− − | |−  − | |  where h (~p) = vF αipi + mγ0 + µAγ5 is the Fourier-transformed effective single-particle Hamiltonian (12) without the regulating factor, the first matrix factor in the last line has chiral indices L, R and σ2 (~p) = ησ2 (~p)¯ησ2 (~p) = 1+σ2σipi/|~p| P 2 is the projection operator in the spin space which projects the spin on the direction of momentum ~p with sign σ2. Similar identity can be also obtained for the second line of (B3) upon the replacement ~p ~q. Now it is convenient to completely factor out the chiral and the spin indices into direct products and to rewrite↔ the equation (B3) as

d3l tr ( σ σ ) Σ(2) ~k = Pq APp B AB 3 r2 r2 × σ ,σ (2π) p q   X1 2|~p|,|Z~q|<Λ − rq ǫp m rp ǫq m ϕ¯p τB − τA ϕp ϕ¯q τA − τB ϕq , (B5) × m rq ǫp − m rp ǫq   − −   − −   where we have introduced the following short-hand no- integrand of (B5). Explicit calculation of the “chiral” tations in order to make the expressions more compact: part (the trace which involves τA and τB ) shows that the integrand is only nonzero if τA = τB (diagonal terms) rp = σ1vF ~p µA, rq = σ2vF ~q µA, | |− | |− or if τA = I, τB = τ2 and vice versa or τA = τ1 and 1+ σ1σipi/ ~p 1+ σ2σiqi/ ~q τB = τ3 and vice versa (off-diagonal terms). Assuming p = | |, q = | |, P 2 P 2 that the momentum ~k is parallel to the 3rd coordinate 2 2 axis, ~k = k3~e3, one can also see that the “spin” part is εp = εs1,σ1 (~p) = rp + m , | | only nonzero if σ = σ (diagonal terms) or if σ = I, q A B A 2 2 σ = σ and vice versa or if σ = σ , σ = σ and εq = εs1,σ2 (~q) = rq + m , B 3 A 1 B 2 | | vice versa. Some of the off-diagonal terms become zero q only after integrating out the loop momentum. The ap- 1 + σ1vF |~p|−µA 1 rp pearance of these off-diagonal terms both in the “chiral” 2 2ε (~p) 2 2εp s1,σ1 − ϕp = = , and the ”spin” parts of the self-energy of the Hubbard-  sq 1 σ1vF |~p|−µA   q 1 + rp  1 2 2ε (~p) 2 2εp − s1,σ1 − Stratonovich field implies an interesting picture of mixing  q1 σ2vF |~q|−µA   q1 rq  between different particle-hole bound states (“mesons” in + 2 2ε 2 2εs1,σ2 (~q) q the language of QCD) of different spin/parity and polar- ϕq = = − .  sq 1 σ2vF |~q|−µA   q 1 + rq  izations in the presence of parity-breaking chiral chemical 1 2 2ε (~q) 2 2εq − s1,σ2 − potential µ . We discuss this picture in more details in  q   q  A When defining the energies εp, εq and the chiral wave the Section 4 of the main text. functions ϕp and ϕq we have explicitly taken into account After explicitly calculating the integrand of (B5), we that s = 1. By expressing the integrand of (B5) in rewrite the integral over the loop momentum l in the 1 − terms of the rescaled variablesµ ¯A = µA/vF ,m ¯ = m/vF cylindrical coordinates ~l = l cos(ϑ) ,l sin (ϑ) ,l . It { ⊥ ⊥ 3} (see Eq. (17) in Section 3) and correspondinglyr ¯p,q = is important that the integration region specified by rp,q/vF ,ε ¯p,q = εp,q/vF one can completely eliminate the ~p < Λ, ~q < Λ is also cylindrically symmetric around Fermi velocity in the integrand of (B5). It only appears the| | 3rd coordinate| | axis. Integration over ϑ can be then −1 as an overall factor of vF in front of (B5). Thus in performed analytically and additionally removes many order to calculate ΣAB ~k at some vF < 1, one should terms in the integrand. We are then left with the inte- gral over l and l , which can be performed analytically simply substitute the rescaled variablesµ ¯ andm ¯ into ⊥ 3   A at zero mass [19]. At nonzero mass, this is no longer pos- the expression obtained with v = 1, and multiply the F sible and we use the Cubature C package [66] to calculate result by v−1. F the integral. The above representation of Σ(2) ~k is especially con- AB Let us now turn to the second contribution to Σx,A;y,B, venient for both analytic transformations  and numerical which is associated with the energy levels which cross calculations. First, we see that the traces over the “chi- zero in external field. This contribution comes from the ral” and the “spin” indices completely factorize in the first summand on the r.h.s. of (35). We denote it as 19

(1) (1) Σx,A;y,B. First we note that obviously this contribution Σ contribution is proportional to the delta-function at is only nonzero if the effective mass m is zero and there is zero momentum: Σ(1) ~k δ ~k . Moreover, it is only no gap in the spectrum. If the spectrum has finite gap, in- ∼ nonzero for σ = σ and τ = I, τ and τ = I, τ . This finitely small perturbations cannot move any energy level A B   A  1 B 1 delta-function singularity will result in the usual “ballis- down to zero. Thus we should consider only the phase tic” contribution to the transport coefficients and will not with unbroken chiral symmetry. Inserting the explicit affect the anomalous transport. Therefore we disregard form of the eigenstates (B1) into (35) and performing it in this work. the Fourier transform (B2), one can immediately see that

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