Spontaneous chiral symmetry breaking and the Chiral Magnetic Effect for interacting Dirac fermions with chiral imbalance P. V. Buividovich1, ∗ 1Institute of Theoretical Physics, University of Regensburg, D-93053 Germany, Regensburg, Universit¨atsstraße 31 (Dated: August 20, 2014) We report on a mean-field study of spontaneous breaking of chiral symmetry for Dirac fermions with contact interactions in the presence of chiral imbalance, which is modelled by nonzero chiral chemical potential. We point out that chiral imbalance lowers the vacuum energy of Dirac fermions, which leads to the increase of the renormalized chiral chemical potential upon chiral symmetry breaking. The critical coupling strength for the transition to the broken phase is slightly lowered as the chiral chemical potential is increased, and the transition itself becomes milder. Furthermore, we study the chiral magnetic conductivity in different phases and find that it grows both in the pertur- bative weak-coupling regime and in the strongly coupled phase with broken chiral symmetry. In the strong coupling regime the chiral magnetic effect is saturated by vector-like bound states (vector mesons) with mixed transverse polarizations. General pattern of meson mixing in the presence of chiral imbalance is also considered. We discuss the relevance of our study for Weyl semimetals and strongly interacting QCD matter. Finally, we comment on the ambiguity of the regularization of the vacuum energy of Dirac fermions in the presence of chirality imbalance. PACS numbers: 72.20.-i,05.30.Rt,12.39.-x Keywords: Anomalous transport, Dirac fermions, Weyl semimetals, Chiral Magnetic Effect, Chiral symmetry breaking, mean-field approximation 1. INTRODUCTION cients which correspond to the CME as well as to other anomalous transport phenomena are universal and do Transport properties of strongly interacting chiral not change when the interactions between fermions are fermions have become a subject of intense research in switched on. This statement, however, relies on a number recent years. One of the fascinating features of chiral of nontrivial assumptions on the properties of the under- fermions is the existence of the so-called anomalous trans- lying quantum field theory, such as the existence of a port phenomena, which stem from quantum anomalies Fermi surface with well-defined quasiparticle excitations and are thus absent in classical systems [1]. One of the around it [13–15] or the finiteness of the static screen- well-known examples of such phenomena is the Chiral ing length [16–18]. It is easy to see that both of these Magnetic Effect (CME) - the generation of electric cur- assumptions are violated when spontaneous chiral sym- rent along the magnetic field in a system with different metry breaking occurs. First, the emergence of massless numbers of left- and right-handed chiral fermions [2]. Goldstone bosons leads to the infinite static screening length. Second, the effective mass term generated due The interest to CME has been to a large extend stimu- to spontaneous symmetry breaking invalidates the Fermi lated by the possibility to observe it in non-central heavy- liquid picture at finite chiral chemical potential [19]. The ion collisions, where the chirality imbalance might be cre- generalization of the chiral kinetic equations of [14, 15] to ated locally due to topological transitions in the produced massive fermions was discussed recently in [20, 21], how- quark-gluon plasma and the huge magnetic field with arXiv:1408.4573v2 [hep-th] 21 Jan 2015 ever, this results are still valid only in the realm of the strength comparable to hadronic scale is created due to applicability of kinetic theory, that is, for weakly coupled the relative motion of ions with large electric charge [3]. dilute plasmas, for which one do not expect any sponta- The CME can also be realized in liquid helium, where its neous symmetry breaking. Let us also mention that the experimental manifestation is the helical instability [4]. asymptotic behavior of the chiral magnetic conductivity Later on it has been realized that the CME could also at high momenta can be related to a certain correlator of be realized in Weyl semimetals [5–10] – a novel phase of two vector and one axial current, which is not renormal- matter in which low-energy excitations are described as ized in massless QCD only if the chiral symmetry is not Weyl fermions, with left- and right-handed Weyl points spontaneously broken [19, 22, 23]. being separated either in momentum or in energy [11, 12]. The CME can be observed if the Weyl points of different While the universal value of the chiral magnetic con- chiralities have different energies. ductivity can be formally derived from the low-energy It is a common statement that the transport coeffi- chiral Lagrangian [24–26], the role of the chiral chemical potential in this derivation is played by the time deriva- tive of the axion field, which makes its interpretation in the Euclidean finite-temperature path integral formalism ∗Electronic address: [email protected] quite unclear [9]. In particular, it is not clear how to de- 2 scribe the stationary CME current as a response to static tion of fermionic condensates in external electromagnetic magnetic field in such a framework. 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.