Chiral anomalous dispersion

Journal of High Energy Physics, Feb 2018

Abstract The linearized Einstein equation describing graviton propagation through a chiral medium appears to be helicity dependent. We analyze features of the corresponding spectrum in a collision-less regime above a flat background. In the long wave-length limit, circularly polarized metric perturbations travel with a helicity dependent group velocity that can turn negative giving rise to a new type of an anomalous dispersion. We further show that this chiral anomalous dispersion is a general feature of polarized modes propagating through chiral plasmas extending our result to the electromagnetic sector.

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Chiral anomalous dispersion

HJE Chiral anomalous dispersion Andrey Sadofyev 0 1 3 4 6 Srimoyee Sen 0 1 2 3 5 0 1118 E. 4th Street, Tucson, AZ 85721 , U.S.A 1 77 Massachusetts Ave Cambridge , MA 02139 , U.S.A 2 Department of Physics, University of Arizona , USA 3 Los Alamos , NM 87545 , U.S.A 4 Center for Theoretical Physics, Massachusetts Institute of Technology , USA 5 Institute for Nuclear Theory, University of Washington , USA 6 Theoretical Division, Los Alamos National Laboratory The linearized Einstein equation describing graviton propagation through a chiral medium appears to be helicity dependent. We analyze features of the corresponding spectrum in a collision-less regime above a at background. In the long wave-length limit, circularly polarized metric perturbations travel with a helicity dependent group velocity that can turn negative giving rise to a new type of an anomalous dispersion. We further show that this chiral anomalous dispersion is a general feature of polarized modes propagating through chiral plasmas extending our result to the electromagnetic sector. Anomalies in Field and String Theories; Thermal Field Theory 1 Introduction 2 3 4 5 1 Chiral matter in gravitational eld Gravitational dispersion relation Electromagnetic dispersion relation Outlook and discussion Introduction to new contributions to the stress-energy transport [18, 19]. As a simple example, one 1Note that the regular chiral e ects may also gain corrections due to the presence of a gravitational eld, see e.g. [15, 16]. { 1 { may consider an anomalous response caused by a gravitational wave (GW) propagating through the medium. Without loss of generality we choose the GW momentum to be in the z-direction denoting it as q3 and its absolute value as q, then the only non-zero components of a GW eld h = g are h11 = h22; h12 = h21. The resulting linear (P-odd) response of the stress-energy tensor can be expressed as T11 = T12 = T22 = i T (!; q)q3 h12 i T (!; q)q3 h11 : (1.2) Here T is proportional to the axial chemical potential , which is used as a measure of follows we refer to this transport as chiral gravitational e ect (CGE). While it is theoretically motivated to study the back-reaction of a chiral medium to a propagating gravitational perturbation, in general, one also could think about possible applications in the physics of early universe where chiral imbalance is often discussed in the context of axion dynamics and primordial magnetic eld generation [2]. It is also appealing since all matter elds are coupled to the gravitational sector and one may expect P-odd e ects due to the imbalance caused by a neutrino background appearing in some cosmological models, see e.g. [20{23]. Here we study the modi cation of the GW dispersion relation by a chiral medium at nite axial chemical potential and temperature T concentrating on the P-odd features of the spectrum. We nd that for time-like four momenta the dispersion relation modi ed by the P-odd part of the graviton self-energy results in a polarization dependence of GW damping. We compare this e ect with an order-of-magnitude estimate of the P-even counterpart due to the medium viscosity [24, 25] and nd that the chiral damping of GWs is suppressed compared to the helicity-independent damping. Strikingly, for space-like four momenta, we nd that the GW group velocity vg not only is helicity dependent but can turn negative for waves of particular polarization at a given sign of the chiral asymmetry. We stress that plasmons (electromagnetic modes) in chiral media exhibit the same peculiar behavior which is not discussed in the literature to the best of our knowledge.2 The corresponding modi cation in the spectrum can be seen as a new helicity-dependent realization of the anomalous dispersion which attracted considerable attention in the context of arti cial materials [28, 29]. It should be mentioned that propagation of helical waves in a chiral medium involves no gain or absorption in contrast with textbook examples of the anomalous dispersion [30]. Finally, we argue that the helicity dependent birefringence may lead to speci c phenomenological signatures which can in principle be observed. We stress that while in the case of gravity the e ects are rather small due to the weakness of the gravitational interaction, the electromagnetic dispersion relation can be studied in tabletop experiments with topological systems of condensed matter supporting relativistic spectrum. This paper is organized in the following way: in the next section we discuss an intuitive picture of the spin-gravity interaction leading to CGE. Then we turn to the details of the gravitational dispersion relation and show that the GW group velocity can turn negative. 2A detailed discussion of plasmons in chiral media can be found in [26, 27]. { 2 { Finally, this result is generalized to the well-studied case of electromagnetic excitations in chiral media. We conclude our work with an outlook and a brief discussion of possible phenomenological consequences of the helicity dependent spectrum in chiral media. 2 Chiral matter in gravitational eld In order to understand the back-reaction of a chiral medium to a metric perturbation it is instructive to review the simpler case of the magnetic response. Considering a system of massive fermions in the presence of an external magnetic eld B one expects a net nd an electric current along the magnetic eld which decays with time. Extending this argument, one would expect that CGE is sourced by a similar mechanism. Indeed, one may concentrate on the coupling of the stress-energy tensor of the medium to an external gravitational eld L = 2 h of the fermionic stress-energy tensor can be decomposed as T . Following [32], the expectation 4 hP +q=2jT (0)jP q=2i = i u(P +q=2) A(q2) f P g + 2 m B(q2)p p +C(q2) 1 (q q m g q2) u(P q=2) (2.1) where q is momentum transfered to a fermion by the external eld, f; g denotes symmetrization of Lorentz indices and (A; B; C) are gravitational form factors. The coupling of spin with a transverse-traceless (TT) metric perturbation3 can be obtained from (2.1) in a fashion analogius to the derivation of the Pauli interaction from electromagnetic form factors. Taking the limit q ! 0 and using the non-relativistic expansion for the spinors one nds the energy due to the spin-gravity interaction which can be written as E fik3 Mgkpjgq3hij : (2.2) From (2.2) one expects a perturbation in h11 to result in a preferred orthogonal combination of spin and linear momentum leading to non-zero perturbation in hT 12i according to (2.1). This e ect cancels for two helicities of fermions in a P-even set-up but one may expect a response analogous to (1.2) if there is a chiral imbalance. Note that these simple arguments based on the spin- eld interaction cannot serve as a rigorous derivation for massless fermions and should be supported by direct calculations in a chiral medium (as it is done for both magnetic [31, 34, 35] and gravitational [19] responses). However, we think that this picture can be helpful for understanding of the origin of CGE. 3The interaction between spin and a GW was recently discussed in [33]. { 3 { given by q reaction takes the form where T is the scalar response function. One can see that (3.1) explicitly satis es the Ward identity which constraints the graviton self-energy and for the P-odd contribution is Equipped with the response (3.1), the linearized Einstein equation including backLet us now analyse the spectrum of a TT metric perturbation propagating in a chiral plasma. The stress-energy tensor induced by the medium response (1.2) modi es Einstein's equation for h . Following the standard Kubo formalism it can be expressed as h T i = h , where is the retarded graviton self-energy. The P-odd TT component of the response function is derived in [19] and its tensorial structure reads T (!; q) = i T (!; q)u Q f P g + ( T $ ) ; (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) where is the gravitational coupling, T pert and Gpert are the perturbations of the stressenergy tensor and the Einstein tensor. Note that in general T pert contains also P-even contributions. For a TT metric perturbation propagating along z-axis the linearised Einstein tensor takes a particularly simple form Gipjert = (!2 q2)hij with i(j) = 1; 2. The equation (3.2) is homogeneous with respect to h and has a non-trivial solution only if its determinant is zero. As usual, this constraint de nes the spectrum of perturbations which is given by ! 2 q2 + 1 m2p S(!; q) = q m2p T (!; q) where the signs corresponds to two helicities, we express the gravitational coupling through the Planck mass = mp 2, and S is introduced to stress the presence of P-even counterpart of the graviton self-energy. The leading contribution to T in the derivative expansion j!j2; q2 2; T 2 was obtained in [19] and is given by T (!; q) = 1 i ( 2 + 2T 2) ! tivated limits. Starting with the quasi-static regime j!j contributions to the response function (3.4) as q we can write the leading m2pq ( 2+ 2T 2) 1 and characteristic wavelengths are expected to be larger than the horizon size mp=T 2 even if the instability has a realization. Note that in the strictly static limit T is non-zero beyond the leading order in the derivative expansion [18, 19], the rst non-trivial contribution appears at the second order and takes the form T (0; q) = 1 One can readily nd that for momenta satisfying q2 type of chiral instability. To be speci c, in the presence of the P-even damping, say due to viscosity, in the quasi-static limit, the spectrum could exhibit a negative imaginary part of the frequency, depending on the sign of the chiral imbalance leading to a growing mode. This instability is chiral in its nature and corresponds to an exponentially growing mode for a helicity xed by the sign of . However, its realization requires trans-Planckian momenta q > m2p in contrast with the chiral plasma instability leading beyond the applicability of the classical gravity and to a breakdown of the derivative expansion. We now turn to the regime of propagating waves j!j q. Anticipating the imaginary part of the frequency !Im to be much smaller than the real part !Re in the expansion Q2=q2, the dispersion relation can be solved for the real part of the frequency with q2q3 192 2m2p < 0 there may appear a new It is convenient to expand around the linear spectrum ! = q + ! with !Re=Im being the real/imaginary part of the correction, then one nds 2 !Re q2 = q m2P Re[ T (!Re; q)]: !Re = 1 damping of modes in the quasi-static regime. It should be mentioned that eq. (3.7) implies The two opposite chiralities result in distinct situations with !Re 7 q depending on the sign of . Indeed, neglecting the imaginary part of !, one expects that !Re < q supports !Im 6= 0 while !Re > q results in the theta function to be zero and !Im = 0. The leading contribution to the imaginary part of the frequency !Im is given by !Im = q Im[ T (q; !)] 2m2P !Re 1 q2 2 + 2T 2 !3 m2P : { 5 { (3.7) (3.8) (3.9) (3.10) (3.11) Although the dimensionful quantity multiplying the step function in the expression for T can be positive or negative depending on the sign of the chemical potential , the step function is non-zero only for one of the two polarizations (at given sign of ) xing the sign of the imaginary part. Note that for positive helicity, a positive produces a negative !Re and also sets the imaginary part of the frequency to be positive. This complex frequency corresponds to a suppression for one of polarized modes. The opposite helicity at produces a positive !Re leading to !Im = 0 due to the theta function in L(!; q). In the presence of P-even contributions to the graviton self-energy, eq. (3.11) leads to a helicity dependent damping of GWs similarly to the quasi-static limit. While the helicity dependent damping seems to be suppressed by additional powers of mP , it is instructive to roughly compare its magnitude with the e ect of the P-even response in the same limit. One expects that in a P-even setup, both helicities of GWs are damped either by a Landau damping in the collision-less limit or by the viscous damping due to the shear viscosity of the medium, for additional details see the discussion in [25]. For an estimate we concentrate on the e ect of the shear viscosity or more precisely on the upper bound for the damping rate !Im m2P [36]. For a rough estimate one can take T 3 and set T reducing the number of scales in the problem. Then the helicity dependent contribution to the imaginary part of the frequency (3.11) is comparable with the viscous damping contribution only if T 3 m2P 1 q2 mT2P3 . In the limit T ! mP the relation (3.12) results in a characteristic momentum q of the Planck scale. For modes of the horizon size l the polarization dependent damping is suppressed comparing to the upper bound on the viscous damping by a small factor of T 2=m2P decreasing for shorter wavelengths. In the long-wavelength regime j!j=q 1 the leading GW dispersion relation is expected to be plasmon-like !2 = !p2l + q2, see e.g. [37]. The plasma frequency can be estimated as T and we focus on this limit. In the presence of a chiral asymmetry the spectrum is modi ed by a helicity-dependent P-odd contribution to the response function !pl mT2P4 for resulting in (3.12) mP =T 2, (3.13) (3.14) T (q0; q) = Note that the linear term in powers of momentum q, on the r.h.s. of (3.14), is much smaller than !p2l for !; q ; T insuring stability of the system. Propagation of long wavelength GWs through a chiral medium can be illustrated with a simple analysis of a wave-packet behavior. The presence of the linear term in the dispersion relation points to the dependence of group and phase velocities on the GW polarization. { 6 { For simplicity we consider modes with positive circular polarization recovering results for the opposite helicity by the parity transformation (and changing the sign of ). Then the group velocity corresponding to the dispersion relation (3.14) is given by vg = d! dq = T 2 Remarkably, for q < 1210 m2P the GW group velocity (3.15) turns negative for cating an anomalous dispersion for the positive polarization. One can see that the phase < 0 indiis suppressed by ( q)=T 2 comparing to the leading P-even term. velocity vp !q derived from (3.14) is also helicity dependent but the helical contribution Before we delve into the implications of the chiral anomalous dispersion, let us estimate the order of magnitude of the terms involved in (3.14) in the regime where group velocity turns negative. The corresponding momentum is given by q T 2=m2p and satis es both q ( ; T ) and q !pl for ( ; T ) mP . In this limit one nds that ! !pl and the expansion parameter q=! q=!pl =mp is small as expected. It is well known that Gaussian packets of propagating waves travel with the group velocity which, if negative, can lead to a peculiar behaviour. The scalar part of the Fourier amplitude for a gaussian wave-packet centered around the momentum qc is given by e An inverse Fourier-transform produces the following amplitude in coordinate space (q qc)2 . ei(qcz !0t)e (z vg(qc)t)2 4 (3.15) after expanding the spectrum around q = qc. Thus, if the group velocity changes sign, the wave-packets for two polarizations propagate in the opposite directions. Typically a negative group velocity is associated with an absorptive medium. The simplest example [30] involves modeling the response of atoms or molecules in ordinary matter to an external perturbation of electromagnetic eld. The atoms or molecules gain dipole moments and are treated as damped simple harmonic oscillators with some characteristic absorption frequency. In this case the group velocity and the refractive index n are complex for propagating modes with a real wave-vector. Near the characteristic frequency, the real part of the index of refraction jumps from a positive to a negative value. The slope of the real part of the refractive index becomes negative with increasing frequency, causing the group velocity to become negative and at times in nite vg = (n + ! dd!n ) 1. However, the chiral anomalous dispersion for GWs obtained in this paper is not associate with any absorption. The frequency, the index of refraction and the group velocity remain real. Although a negative group velocity in chiral medium will require q < 120 mT2p2 , chiral 1 T 2 splitting of group velocities is present for arbitrary q < !. If q & 1210 m2p the group velocity is positive for both helicities but is still polarization dependent. In order to estimate the upper bound on the helical correction to vg one can set q !pl T 2=mp. Then the chiral contribution to the group velocity goes as =mP while the leading term is of order !pl=q 1. An accurate estimate of the chiral splitting of the group velocity requires numerical study of the spectrum including the full self-energy and we leave it for future work. { 7 { Here we brie y discuss propagation of electromagnetic modes in a chiral plasma in the presence of non-zero and show that the chiral anomalous dispersion is a general feature of helical excitations in such systems. The dispersion relation for plasmons can be obtained in a manner similar to the GW case discussed in the previous section. Maxwell's equations with the back-reaction taken into account produce det (q2 !2) ij kikj + ij = 0 ; and we use A0 = 0 gauge following conventions of [4]. Here the response function the retarded photon self-energy in a chiral medium derived in [38]. At nite temperature and density there is a preferred reference frame, the polarization operator has a more involved structure than in vacuum and can be decomposed into longitudinal, transverse, and antisymmetric parts where the P-odd contribution corresponds to the presence of a chiral imbalance. Using this decomposition one can reduce (4.1) to with two signs corresponding to di erent polarizations. The explicit form the three contributions to the photon self-energy are given by L = m2D !q22 L~(!; q) ; T = m2D 2 q2 1+ q 2 !2 L~(!; q) ; A = q 1 ij = LPLiij + T PT iij + APAiij PLij = qiqj q2 ; PT = ij PLij ; P Aij = i ijkqk q ; (4.1) ij is (4.2) (4.3) !2 q2 q 3 !3 L~(!; q) ; (4.4) : (4.5) (4.6) where m2D = e 2 T62 + 2 22 , L~(!; q) long-wavelength limit j!=qj 1, z-direction. In analogy with the gravity sector, we consider the photon self-energy in the L(!; q) + 1 and we take the wave-vector in the 1 L ' 3 m2D + O q 2 !2 ; 1 T ' 3 m2D + O q 2 !2 ; A ' 3 q + O Substituting (4.5) into (4.3) one nds the dispersion relation in the long-wavelength limit As previously in the GW case, the dispersion relation is helicity dependent and one of the two polarizations acquires a negative group velocity for q < 3 . Thus, we have shown that the chiral anomalous dispersion discussed in this papers exists for the electromagnetic modes in chiral media as well as for gravitational ones. Emphasizing a point mentioned earlier, a milder constraint ! > q is satis ed for q !pl. Although this regime is not { 8 { accessible analytically, a numerical analysis is expected to reproduce the chiral splitting of the group velocity and we leave it for further study. The relatively larger value of the electromagnetic coupling makes the anomalous dispersion of plasmons in chiral media to be in principle observable in tabletop experiments based on condensed matter systems with relativistic linear spectrum such as Dirac semi-metals. This proposal is additionally supported by the recent experimental observation of CME in those systems, see [3]. 5 Outlook and discussion In this paper we study how circularly polarized electromagnetic and gravitational waves propagate through a chiral medium. The response of a chiral medium to the electromagnetic and gravitational elds is given by photon and graviton retarded self-energies and we use that to construct linearized equations of motion for corresponding waves. These equations determine the spectrum of excitations propagating in a chiral plasma. We start with the gravitational sector and consider CGE | the chiral medium response to an external TT gravitational eld in the stress-energy tensor. We consider the back reaction of this e ect to gravitational perturbations propagating through the medium and the resulting spectrum in several limiting regimes. Concentrating on P-odd features of the dynamics we mostly omit the P-even part of the graviton self-energy where it is possible and qualitatively restore it if required. In the quasi-static limit the electromagnetic response results in the chiral magnetic instability which corresponds to a transfer of the microscopic chirality to the macroscopic helicity of magnetic eld, see e.g. [4]. Following this intuition we start the consideration of GWs in a chiral medium with the quasi-static limit j!j=q 1 and nd that a gravitational chiral instability would require wavelength above the horizon size. We further argue that this regime is also considerably modi ed by P-even contributions and the unstable behavior vanishes. This is expected since the leading order CGE disappears in the exact static limit in contrast with CME. The rst non-trivial CGE contribution in the static limit appears at the 3d order in the derivative expansion (3.8). Taking it j!j=q into account one can nd that the modi ed dispersion relation exhibits another instability (in the UV limit) which, however, requires trans-Planckian momenta and goes beyond applicability of the classical gravity (and this consideration). In the regime of relativistic propagating waves ! q the spectrum indicates helicity dependent attenuation which leads to an unequal damping of polarized GWs. It is however highly suppressed for shorter wavelengths. We illustrate this result comparing the helicity dependent contribution with the textbook example of the GW damping by the shear viscosity. Maximizing the Podd e ect we consider modes of the horizon size and nd that it is still suppressed by a small factor of T 2=m2P compared to the P-even part. Turning to the long-wavelength limit 1 we nd an interesting result | the GW group velocity appears to be helicity dependent and may become negative giving rise to a new type of anomalous dispersion. We believe that the mutual e ect of the polarization dependent GW propagation can turn into an overall helical asymmetry in distinct regions of space which in principle could lead to helical distributions of matter and radiation. One should expect this asymmetry to be rather small but it would be interesting to study whether its signatures could at { 9 { least in principle be observed. We also note that an additional chiral imbalance may be caused by a background electromagnetic helicity4 and axions which should be added to this consideration. Inspired by the group velocity behavior in the gravitational case we turn to reviewing electromagnetic excitations in chiral media. The plasmon spectrum in a medium with a chiral asymmetry is studied in the literature in great details, for a recent discussion see [26, 27]. Focusing on the long-wavelength limit we nd that the leading correction to the plasmon spectrum, indeed, results in a similar phenomenon: the plasmon group velocity of a particular helicity can turn negative. The electromagnetic coupling is much larger resulting in a stronger e ect than that in the gravitational sector and one may expect that this birefringence of chiral media may be observed in experiments with Dirac semi-metals. In addition, phenomenological e ects of the chiral anomalous dispersion can be studied for a radiation produced in and passing through a cosmic P-odd background such as a supernova where a chiral asymmetry may be generated. Then, one can study the di erence between polarized light signals searching for helical asymmetries. Acknowledgments We would like to thank Yi Yin for initializing the discussion on the GW propagation in chiral media. We would like to thank D.B. Kaplan, V. Kirilin, L. McLerran, S. Reddy, I. Shovkovy, O. Teryaev, and V.I. Zakharov for useful discussions and comments. AS is supported by the LANL/LDRD Program and, at the beginning of this work, by U.S. Department of Energy under grant Contract Number DE-SC0011090. AS is grateful for travel support from RFBR grant 17-02-01108 at the beginning of this work. SS is supported by U.S. Department of Energy under grant Contract Number DE-FG02-00ER41132 and, at the beginning of this work, by U.S. Department of Energy under grant Contract Number DE-FG02-04ER41338. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [arXiv:1511.04050] [INSPIRE]. [2] M. 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Andrey Sadofyev, Srimoyee Sen. Chiral anomalous dispersion, Journal of High Energy Physics, 2018, 99, DOI: 10.1007/JHEP02(2018)099