Anomalous transport due to scale anomaly M. N. Chernodub∗ CNRS, Laboratoire de Math´ematiques et Physique Th´eorique UMR 7350, Universit´e de Tours, 37200 France Soft Matter Physics Laboratory, Far Eastern Federal University, Sukhanova 8, Vladivostok, 690950, Russia and Department of Physics and Astronomy, University of Gent, Krijgslaan 281, S9, B-9000 Gent, Belgium (Dated: March 28, 2016)
arXiv:1603.07993v1 [hep-th] 25 Mar 2016
We show that the scale anomaly in field theories leads to new anomalous transport effects that emerge in external electromagnetic field in inhomogeneous gravitational background. In inflating geometry the QED scale anomaly generates electric current which flows in opposite direction with respect to background electric field. In static spatially inhomogeneous gravitational background the dissipationless electric current flows transversely both to the magnetic field axis and to the gradient of the inhomogeneity. The anomalous currents are proportional to the beta function of the theory.
Anomalous transport phenomena emerge in systems with quantum anomalies that break certain classical symmetries and lead to nonconservation of associated (otherwise classically conserved) currents [1, 2]. For example, the axial symmetry of chiral (massless) fermions is broken by the axial anomaly that naturally leads to nonconservation of the axial current at the quantum level [3]: e2 Fµν Feµν , (1) 16π 2 where Fµν = ∂µ Aν − ∂ν Aµ is the field-strength tensor of an Abelian gauge field Aµ and Feµν = 12 εµναβ Fαβ . The simplest anomalous transport laws induced by the axial anomaly (1) are the Chiral Separation Effect [4, 5] and the Chiral Magnetic Effect (CME) [6, 7]: µA µV eB , jV = eB , (2) jA = 2π 2 2π 2 that generate, respectively, the axial current j A and the vector current j V along the axis of the external magnetic field B in dense (µV 6= 0) and in chirally imbalanced (µA 6= 0) medium. The chemical potential µV and the chiral chemical potential µA are thermodynamically conjugated to the total charge density jV0 and to the axial 0 , respectively. charge density jA The axial anomaly (1) is also responsible for the density-dependent contributions to the Chiral Vortical Effects [8, 9] which generate vector and axial currents, 2 µV µA T µ2V + µ2A jV = Ω, jA = + Ω , (3) π2 6 2π 2 µ ∂µ jA =
in chiral fluids that rotate with the angular velocity Ω. The temperature-dependent T 2 part of the rotationinduced axial current in Eq. (3) is a result of the mixed axial-gravitational anomaly [10]: µ ∂µ jA =−
1 eµναβ , Rµναβ R 384
(4)
that despite the anomaly (4) is formulated in a curved background, the associated anomalous transport is realized in a flat space too. In the presence of electromagnetic field in a curved background the total divergence of the axial current is given by the sum of the right hand sides of Eqs. (1) and (4). The anomalous transport laws (2) and (3) are nondissipative phenomena because these equations are invariant under the operation of time-inversion (T : t → −t) so that the corresponding currents do not generate any entropy. These phenomena play increasingly important role both in condensed matter and high energy physics [11, 12]. Besides the axial and mixed axial-gravitational anomalies, a class of physically interesting quantum field theories is also subjected to a scale anomaly1 which breaks classical scale invariance of the theory at the quantum level [3]. Since it seems now quite natural to think that at least some quantum anomalies may be associated with certain anomalous transport laws [1, 2], we would like to ask a natural question: does the conformal anomaly lead to a new anomalous transport law? Although the discussed anomalous transport effect are quite general, in this paper we consider a simplest case of a U(1) gauge theory with one massless Dirac fermion field ψ described by the following Lagrangian, 1 ¯ Dψ / , L = − Fµν F µν + ψi 4
/= where Dµ = ∂µ +ieAµ is the covariant derivative and D µ γ Dµ . This theory does not involve any characteristic length or energy scale since its Lagrangian (5) possesses only dimensionless coupling e. Therefore at a classical level the massless electrodynamics (5) is invariant under redefinition of the absolute length or energy scales. The corresponding scale transformations are generated by the dilatation current: µ jD = T µν xν ,
eµναβ is the Riemann curvature tensor of a curved where R space background and Rµναβ = 12 εµνγλ Rγλαβ . Notice
∗
[email protected]
(5)
1
Also known as the dilatation, trace or conformal anomaly.
(6)
2 where the (symmetric) energy-momentum tensor 1 T µν = −F µα Fαν + η µν Fαβ F αβ 4 i ¯ Dψ / , + ψ¯ (γ µ Dν + γ ν Dµ ) ψ − η µν ψi 2
(7)
can be obtained by the variation of the action S with respect to the background metric gµν : Z √ δS , S = d4 x −g L , (8) T µν (x) = 2 δgµν (x) with g = det(gµν ). We restore the flat space-time metric g µν → η µν = diag (+1, −1, −1, −1) after the variation. Classically, the dilatation current (6) has zero divergence because the classical equations of motion imply µ ∂µ jD = Tαα ,
(9)
while the trace of the energy-momentum tensor (7) vanishes at the classical level, Tαα = 0. Therefore the classical theory is invariant under the scale transformations. However, the scale invariance is broken by quantum fluctuations. Consequently, the quantum expectation value of the right hand side of Eq. (9) is nonzero and, consequently, on a quantum level the dilatation current (6) is no more conserved. Consider a Weyl scale transformation of the flat metric, ηµν → gµν (x), with gµν (x) = e2τ (x) ηµν ,
(10)
For small scale factor τ (x) with |τ (x)| 1 the metric perturbation is δgµν (x) = 2τ (x) ηµν and Eq. (8) implies: Z S → Sτ = S + d4 x τ (x) Tαα (x) + O(τ 2 ) , (11) where Sτ is the action (8) of the theory (5) in a background of the rescaled flat metrics (10). Therefore the expectation value of the trace of the energy-momentum tensor is given by the functional derivative hTαα (x)i =
1 1 δZ[Acl , τ ] i Z[Acl , τ ] δτ (x)
of the generating functional Z cl ¯ Z[Acl , τ ] = DA Dψ¯ Dψ eiSτ [A+A ;ψ,ψ] ,
(12)
The scale invariance should generally be broken at the quantum level because the (dimensionless) gauge coupling e = e(µ) is a function of the (dimensionful) renormalization scale µ. The Weyl scale transformation (10) changes the renormalization scale, µ → µ + δµ with δµ = µδτ and consequently shifts the coupling e → e+δe by δe = β(e)δτ where β(g) =
d e2 (µ) , d ln µ 4π
(16)
is the beta function of the theory. Therefore at the quantum level the trace of the energy-momentum tensor is nonzero2 , hTαα (x)i =
β(e) Fµν (x)F µν (x) , 2e
(17)
and the dilatation current (6) is no more conserved according to Eq. (9). In Eq. (17) the field-strength tensor Fµν corresponds to the external background field Acl µ (for the sake of simplicity we omit hereafter the superscript “cl” which refers to the classical background field). Below we show that the scale anomaly (17) leads to an unexpected (anomalous) contribution to electric current induced by external electromagnetic fields in spatially inhomogeneous or inflating/deflating gravitational backgrounds associated with dilatational perturbations of the form (10). In order to demonstrate the essence of the effect we consider the system at zero temperature and zero density so that both usual and chiral chemical potentials are zero, µV = µA = 0. In our derivation we follow the logic of Ref. [13] which we apply to the case of the scale anomaly in the coordinate space. The electric current hji induced by weak external electromagnetic field Aµ (x) and by small local dilatations of the metric τ (x) can be expanded in series over these perturbations: hj µ i = hj µ iKubo + hj µ idilat + hj µ iscale + . . . .
(18)
The terms in Eq. (18) are proportional, respectively, to the first powers of A and τ , and to their product Aτ . All higher-order terms are denoted by the ellipsis. The first term in Eq. (18) is given by the standard, linear-response Kubo formula Z hj µ (x)iKubo = −i d4 y Πµν (x, y)Aν (y) , (19)
(13) where
where we have also coupled our system to a background cl of the classical electromagnetic field Acl µ = Aµ (x). The latter allows us to express the electric (vector) current of the fermions, µ ¯ j µ (x) ≡ jVµ (x) = eψ(x)γ ψ(x) ,
Πµν (x, y) = hj µ (x)j ν (y)i0 ,
(20)
is a two-point correlation function of the electric currents. The subscript 0 in h. . .i0 indicates that the expectation
(14)
in terms of a functional derivative: 2
1 δZ[Acl , τ ] hj (x)i = i . cl Z[A , τ ] δAcl µ (x) µ
(15)
In contrast to the axial anomaly (1) the scale anomaly (17) is not a pure one-loop result. Consequently, multiloop corrections may appear in Eq. (17) (see Ref. [3] for detailed discussion).
3 value (20) is calculated in the absence of external perturbations (i.e. with Aµ = 0 and δgµν = 0). The second term in Eq. (18) corresponds to a linear response of the current to the pure dilatation (11), Z µ hj (x)idilat = i d4 y ΠµD (x, y)τ (y) , (21)
the anomalously generated current (26) is conserved: ∂µ hj µ (x)iscale = 0. We are interested in properties of the anomalous electric current far from the classical sources. Therefore, setting the classical current to zero in the region of the µ dilatation, jcl = 0, we get from Eq. (26): hj µ (x)iscale = −
where ΠµD (x, y)
= hj
µ
(x)Tαα (y)i0
,
(22)
2β(e) µν F (x)∂ν τ (x) . e
(28)
In the case of a general small perturbation around the flat metric, gµν = ηµν + δgµν , the anomalous current is:
is a two-point correlation function of the electric current (14) and the trace of the energy-momentum tensor (7). The correlation function (24) can be calculated by varying the anomalous expectation value (17) with respect to external electric field Aµ in a manner of Eq. (15), and setting Aµ = 0 after the variation. Since the anomaly (17) is quadratic in gauge field Aµ the correlation function (24) is zero. Therefore the electric current (21), induced by the dilatation, is vanishing in the linear response approximation, hj µ (x)idilat ≡ 0. In our paper we are mainly interested in the third term in Eq. (18). This term describes a scale-anomalous contribution to the expectation value of the electric current and corresponds to a mixed gauge-gravitational response in the double-linear approximation that includes one power of the electromagnetic potential Aµ and one power of the dilatational factor τ . According to Eq. (19) Z Z hj µ (x)iscale = d4 y d4 z Πµν D (x, y; z)Aν (y)τ (z), (23)
This equation may be derived from an analogue of Eq. (23) written for a general perturbation δgαβ . In Eq. (24) we would get T αβ instead of Tαα . Then we decompose the vacuum expectation value of the energymomentum tensor into the traceless “normal” part hTeµν i with hTeαα i = 0 and the “anomalous” part with nonzero trace, hT µν i = hTeµν i + 14 g µν hTαα i. We come back to the expectation value (24) for the anomalous part. In components, the anomalous current and the anomalous charge generated by the scale anomaly (28) in the background of the electric field E and the magnetic field B are, respectively, as follows:
where the three-point function
where
µ ν α Πµν D (x, y; z) = hj (x)j (y)Tα (z)i0 ,
(24)
can be evaluated by applying twice a functional differentiation with respect to the background gauge field Aµ to the right hand side of the scale anomaly relation (17): δ 2 hTαα (z)i Πµν (x, y; z) = − (25) D µ →0 δAµ (x)δAν (y) A g →η µν
=−
2β(e) µν αβ η η e
µν
∂ 2 δ(x − z)δ(y − z) − η µβ η να . ∂xα ∂y β
Substituting Eq. (25) into Eq. (23) one gets the anomalous electric current generated by the scale anomaly (17) in the presence of both the scale dilatation τ of the metric (10) and the background electromagnetic field Aµ : i 2β(e) h µ hj µ (x)iscale = −F µν (x)∂ν τ (x) + τ (x)jcl (x) . (26) e The first term in Eq. (26) is proportional to the electromagnetic field F µν which is induced by the classical electric current: µ jcl = −∂ν F µν .
(27)
The classical current makes the local contribution to the anomalous electric current given in the second term of Eq. (26). The presence of both terms guarantees that
hj µ (x)iscale = −
β(e) µν F (x)η αβ ∂ν δgαβ (x) . 4e
hj(x)i = σ(x)E(x) + F (x) × B(x) ,
0 scale j (x) scale = F (x) · E(x) , 2β(e) ∂τ (t, x) , e ∂t 2β(e) ∇τ (t, x) , F (t, x) = e σ(t, x) = −
(29)
(30) (31)
(32) (33)
and τ (x) is the local scale factor of the flat metric (10). The scalar quantity σ, given by Eq. (32) plays a role of the anomalous Ohm’s conductivity. Indeed, in spatially uniform (∇τ ≡ 0) background gµν = e2τ (t) ηµν with a time-dependent dilatational factor τ = τ (t) the scale anomaly generates the anomalous electric current (30) which takes exactly the functional form of Ohm’s law: hj(t, x)iscale = σ(t)E(t, x)
for ∇τ = 0 . (34)
In addition, Eq. (31) implies that a spatially uniform (∇τ ≡0) background (10) does not induce a nonzero elec tric charge density, j 0 (x) scale ≡ 0. The anomalous Ohm’s law (34) emerges in an open, expanding (or contracting) system which has an explicit arrow of time. Consequently, the anomalous Ohm’s law does not conserve entropy and does not, in general, describe a dissipationless phenomenon contrary to already known anomalous transport effects (2) and (3). Moreover, the power P dissipated by the anomalous electric current (34) per unit volume, P = hjiscale · E = σE 2 ,
(35)
4 may take both positive and negative values because the anomalous conductivity (32) may be both positive and negative quantity, respectively. As a result, in this open system the anomalous Ohm’s current (34) may not only heat the system but it may also cool it by absorbing heat. We illustrate the anomalous Ohm law (34) in Fig. 1.
� � � � �
FIG. 2. In a spatially nonuniform gravitational background the scale anomaly (17) generates the electric current J circumnavigating the external magnetic field B according to (36). The gradient of the scale factor F is given in Eq. (33).
β>� σ 0) gravitational background (10) the scale anomaly (17) generates the electric current J along the electric field E according to Eqs. (34) and (32). The direction of the current depends on the sign of the beta function.
The anomalous current (30) has also a contribution coming from the magnetic field B. This part may only appear due to local spatial inhomogeneities of the dilatation factor τ (x) which are encoded in the vector quantity F in Eq. (33). According to Eqs. (30) and (33), in a nonuniformly stretched static spacetime the scale anomaly generates the electric current which is transversal both to the direction of magnetic field B and to the gradient of the spatial inhomogeneity F : hj(t, x)iscale = F (x) × B(t, x)
for ∂t τ = 0 . (36)
The electric current (36) flows without dissipation similarly to the current (2) generated by the CME. However, there are two major differences between our effect (36) and the CME: (i) the scale anomaly generates the the electric current in the vacuum while the CME is realized in matter only (with µA 6= 0); (ii) the electric current (36) is transverse to the direction of magnetic field while in the CME the magnetic field and current are parallel to each other. Illustration of the anomalous transport (36) that occurs in the magnetic field background is shown in Fig. 2. In the presence of external electric field the scale anomaly should also lead to accumulation of electric charge (31) at spatial inhomogeneities of the metric (33). Using one-loop QED beta function for one species of a Dirac fermion (5), β 1-loop (e) = e3 /(12π 2 ) we get the anomalous transport coefficients (32) and (33): σ=−
e2 ∂τ , 6π 2 ∂t
F =
e2 ∇τ , 6π 2
(37)
where τ (x) is the local scale factor of the flat metric (10).
How significant this effect should be in expanding Universe? A generic homogeneous and anisotropic metric has the following interval, ds2 = dt2 − a2 (t)dx2 , where a(t) is a scale factor. It is convenient to introduce the new time coordinate t0 via the relation dt = a(t)dt0 , so that the interval can be rewritten as ds2 = a ˜2 (t0 )(dt02 − dx2 ), 0 0 where a ˜(t ) = a(t(t )). In the new reference frame (t0 , x) the metric has the flat form (10) with the dilatation factor τ˜(t0 ) = ln a ˜(t0 ). Applying our formalism in this reference frame we get the anomalous relation for the electric current j 0 = σ 0 (t)E 0 where the conductivity σ 0 is given by Eq. (32) with the substitution τ → τ˜. Since the electric fields and the electric currents in these coordinate systems are related, E = a(t)E 0 and j = j 0 , we obtain that in the expanding space the conductivity of each species of massless charged fermions gets an anomalous contribution due to the scale anomaly (we restore ~ and c here): σ(t) = −
e2 H(t) , 6π 2 ~c
(38)
where H(t) = a(t)/a(t) ˙ is the Hubble parameter. One may expect that the scale-anomalous transport effects (30)–(33) are quite generic phenomena because the anomalous term (17) – which is the starting point of our derivation – is present in wide varieties of physical models. We expect that these effects should be realized not only for massless fermions but they should also appear in systems describing both (massless and massive) bosons and massive fermions. Notice that the mass independence of the scale-anomalous transport should be contrasted with the axial-anomalous transport (2) as the latter is believed to be suppressed for massive fermions. Moreover, the scale-anomalous transport effects (30)– (33) are vacuum phenomena contrary to the axialanomalous transport which is realized in matter only. Our result implies that inflating vacuum should have a nonzero conductivity due to the scale anomaly. For example, taking the present-day value of the Hubble constant, H0 ∼ 10−18 s−1 , one obtains from Eq. (38) that the vacuum has extremely small but still finite negative conductivity σ0 ∼ −10−31 Ω−1 m−1 . Notice that this value is different from the recent result on the conductivity in inflating (de Sitter) spacetime induced by a fermionic Schwinger effect in very-weak-field limit, eE H 2 [19]
5 (see also [14–18]). This discrepancy may be related to the fact that we discuss distinct physical phenomena because the Schwinger pair production is associated with the axial anomaly [1] while the transport effects studied in this paper are generated by the scale anomaly. Summarizing, we have shown that the scale anomaly – similarly to the axial anomaly – leads to new transport effects (30)–(33) in theories with electrically charged particles. In inflating geometry the QED scale anomaly generates electric current (34) flowing in opposite direction to the electric field due to negative conductivity (38). In static but spatially inhomogeneous gravitational background the dissipationless electric current
flows transversely both to the magnetic field axis and to the gradient of the inhomogeneity (36). The generated electric currents are proportional to the appropriate beta function. Finally, we would like to point out that the scale-anomalous transport effects may be realized in solid state materials possessing relativistic quasiparticles, such as strained graphene [20] or elastically deformed Weyl/Dirac semimetals [21]. The author is grateful to K. Landsteiner, M. Vozmediano and M. Zubkov for interesting discussions, and acknowledges hospitality and support of IFT-UAM/CSIC, Madrid, as associate researcher under the Centro de Excelencia Severo Ochoa Programme grant SEV-2012-0249.
[1] G. Basar and G. V. Dunne, “The Chiral Magnetic Effect and Axial Anomalies,” Lect. Notes Phys. 871, 261 (2013) [arXiv:1207.4199]. [2] K. Landsteiner, E. Megias and F. Pena-Benitez, “Anomalous Transport from Kubo Formulae,” Lect. Notes Phys. 871, 433 (2013) [arXiv:1207.5808]. [3] M. A. Shifman, “Anomalies in Gauge Theories,” Phys. Rept. 209, 341 (1991) [4] D. T. Son and A. R. Zhitnitsky, “Quantum anomalies in dense matter,” Phys. Rev. D 70, 074018 (2004) [hepph/0405216]. [5] M. A. Metlitski and A. R. Zhitnitsky, “Anomalous axion interactions and topological currents in dense matter,” Phys. Rev. D 72, 045011 (2005) [hep-ph/0505072]. [6] K. Fukushima, D. E. Kharzeev and H. J. Warringa, “The Chiral Magnetic Effect,” Phys. Rev. D 78, 074033 (2008) [arXiv:0808.3382 [hep-ph]]. [7] A. Vilenkin, “Equilibrium Parity Violating Current In A Magnetic Field,” Phys. Rev. D 22, 3080 (1980). [8] A. Vilenkin, “Macroscopic Parity Violating Effects: Neutrino Fluxes From Rotating Black Holes And In Rotating Thermal Radiation,” Phys. Rev. D 20, 1807 (1979). [9] D. T. Son and P. Surowka, “Hydrodynamics with Triangle Anomalies,” Phys. Rev. Lett. 103, 191601 (2009) [arXiv:0906.5044]. [10] K. Landsteiner, E. Megias and F. Pena-Benitez, “Gravitational Anomaly and Transport,” Phys. Rev. Lett. 107, 021601 (2011) [arXiv:1103.5006]. [11] G.E. Volovik, “The Universe in a Helium Droplet”, Clarendon Press, Oxford (2003). [12] D. E. Kharzeev, “Topology, magnetic field, and strongly interacting matter,” Ann. Rev. Nucl. Part. Sci. 65, 193 (2015) [arXiv:1501.01336]; D. E. Kharzeev, J. Liao, S. A. Voloshin and G. Wang, “Chiral magnetic and vortical effects in high-energy nuclear collisions – A status report,” Prog. Part. Nucl. Phys. 88, 1 (2016) [arXiv:1511.04050]. [13] G. M. Newman and D. T. Son, “Response of stronglyinteracting matter to magnetic field: Some exact results,” Phys. Rev. D 73, 045006 (2006) [hep-ph/0510049]. [14] M. B. Fr¨ ob, J. Garriga, S. Kanno, M. Sasaki, J. Soda, T. Tanaka and A. Vilenkin, “Schwinger effect in de Sitter space,” JCAP 1404, 009 (2014) [arXiv:1401.4137].
[15] R. G. Cai and S. P. Kim, “One-Loop Effective Action and Schwinger Effect in (Anti-) de Sitter Space,” JHEP 1409, 072 (2014) [arXiv:1407.4569 [hep-th]]. [16] T. Kobayashi and N. Afshordi, “Schwinger Effect in 4D de Sitter Space and Constraints on Magnetogenesis in the Early Universe,” JHEP 1410, 166 (2014) [arXiv:1408.4141]. [17] C. Stahl, E. Strobel and S. S. Xue, “Fermionic current and Schwinger effect in de Sitter spacetime,” Phys. Rev. D 93, no. 2, 025004 (2016) [arXiv:1507.01686]. [18] E. Bavarsad, C. Stahl and S. S. Xue, “Scalar current of created pairs by Schwinger mechanism in de Sitter spacetime,” arXiv:1602.06556. [19] T. Hayashinaka, T. Fujita and J. Yokoyama, “Fermionic Schwinger effect and induced current in de Sitter space,” arXiv:1603.04165; T. Hayashinaka and J. Yokoyama, “Point splitting renormalization of Schwinger induced current in de Sitter spacetime,” arXiv:1603.06172. [20] M. A. H. Vozmediano, M. I. Katsnelson, F. Guinea, “Gauge fields in graphene”, Phys. Rep. 496, 109 (2010) [arXiv:1003.5179]; Fernando de Juan, Mauricio Sturla, Maria A. H. Vozmediano, “Space dependent Fermi velocity in strained graphene”, Phys. Rev. Lett. 108, 227205 (2012) [arXiv:1201.2656]; J. S. Bunch et al., “Impermeable Atomic Membranes from Graphene Sheets” Nano Lett., 8, 2458 (2008); G. E. Volovik and M. A. Zubkov, “Emergent Horava gravity in graphene,” Annals Phys. 340, 352 (2014) [arXiv:1305.4665]. [21] Onkar Parrikar, Taylor L. Hughes, and Robert G. Leigh, “Torsion, parity-odd response, and anomalies in topological states”, Phys. Rev. D 90, 105004 (2014) [arXiv:1407.7043]. M.A.Zubkov, “Emergent gravity and chiral anomaly in Dirac semimetals in the presence of dislocations”, Annals of Phys., 360, 655 (2015), [arXiv:1501.04998]; A. Cortijo, Y. Ferreirs, K. Landsteiner and M. A. H. Vozmediano, “Elastic Gauge Fields in Weyl Semimetals,” Phys. Rev. Lett. 115, no. 17, 177202 (2015) [arXiv:1603.02674]; A. Cortijo and M. A. Zubkov, “Emergent gravity in the cubic tightbinding model of Weyl semimetal in the presence of elastic deformations,” Annals Phys. 366, 45 (2016) [arXiv:1508.04462].
