Analogous BMS Symmetry in QED and Quantum Anomaly in Dirac and Weyl Semimetals
AAnalogous BMS Symmetry in QED and Quantum Anomaly in Dirac and WeylSemimetals
Andrea Addazi ∗ and Antonino Marcian`o † Center for Field Theory and Particle Physics & Department of Physics, Fudan University, 200433 Shanghai, China
We show that the asymptotic infinite dimensional enlarged gauge symmetries constructed for QEDare anomalous in Weyl semimetals. This symmetry is particularly important in particle physics forits analogy with the Bondi-Metzner-Sachs (BMS) symmetries in gravity, as well as for its connectionwith QED soft IR theorems. This leads to observable effects, because of the induction of a newcurrent in the material, which carries a memory of the BMS symmetry precursor.
Introduction. —
Dirac and Weyl semimetals are an ex-citing test-bed for many important concepts and frame-works of quantum field theory. Indeed, it was arguedvery recently that the chiral currents generated in thesematerials are understood as a manifestation of a chiralanomaly term. An holographic explanation of the phe-nomena was found [1–3]. This was developed connectingthe quantum field theory description of the chiral anoma-lous phenomena to the anomalous terms of (an effective)string theory in higher dimensional anti-de Sitter bulk.Quantum anomalies are important in our understandingof high energy physics and in particular of the StandardModel of particles and interactions. As it is well known,the chiral anomaly induced by 1-loop triangles of quarks,explained the pion decay into two photons , reproducingthe correct amount for the transitions π → γγ observedin the laboratory.It is also well known that all gauge anomalies of theStandard Model vanish by virtue of a perfect balanceamong the contributions arising from all the matter par-ticles. This cancellation appears as an accidental miraclethat allows the Standard Model to be not plagued byunitarity and causality loss. Finally, an anomaly of thescale invariance and conformal symmetry is dynamicallygenerated by radiative corrections or by non-perturbativephenomena — this for instance the case of confinementin Quantum Chromodynamics. About the conformalanomaly, it was recently suggested that it may be testedin the very same Weyl-semimetals [4–8]. Indeed, for aboundary plane with a magnetic field parallelly oriented,the conformal Weyl anomaly induces a charge transportcurrent.In this letter, we propose a new paradigm of testinganomalies of infinite dimensional asymptotic symmetries.The Bondi-Metzner-Sachs (BMS) symmetry is certainlythe most popular infinite dimensional asymptotic sym-metry of gravity fields. As shown by Strominger and ∗ [email protected] † [email protected] We thank Jerzy Kowalsky-Glikman and Paolo Pani for remindingus to add this remark. collaborators in a series of papers, BMS is related to thesame Ward identities of the soft infrared gravitational ra-diation [9–11]. Recently Hawking, Perry and Stromingerproposed that the very same same black hole informa-tion loss paradox may be solved invoking a memory ef-fect from the new Noether charges associated to BMS[12]. BMS symmetry is an extension of the conformalWeyl symmetry with new infinite supertranslation sym-metries. Consequently BMS includes an infinite numberof supertranslation charges, which may provide a grav-itational memory effect. It has been also argued thatan analogous of the BMS symmetry is recovered also inQuantum Electrodynamics, as a large gauge symmetry[13–15].A large gauge symmetry incorporates an infinite num-ber of gauge symmetries [13–15]. Once again, this corre-sponds to an infinite number of Noether charges, and isexactly the symmetry that we claim to be anomalous inthe Weyl semimetals. We unveil the appearance of thisquantum anomaly in a simple set-up: a boundary planewith a constant magnetic field parallelly oriented on it.The magnetic field induces, through the analogous BMSanomaly, an electric current which leads to a character-istic potential and charge redistribution. Such an effectis related to the formation of Schwinger fermion pairs,which percolates into a characteristic electric potentialand a Debye screening of the electric field. This modi-fies the standard electrostatic and vector potentials in theasymptotic limit as the breaking of the Large AsymptoticQED symmetry.
BMS and large asymptotic gauge symmetries. —
Letus initially start from the standard vacuum theory inMinkowski space-time: ds = − dT + dZ + dY + dX , ( c = 1) . (1)We may complexify the variables through the change ofcoordinates X + Y + Z = r , T = u + r , (2) Z = r − z ¯ z z ¯ z , X + iY = 2 rz z ¯ z . (3) a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y This allows to obtain the metric of the Minkowski space-time in the future null infinity ( I + ), as shown in the Pen-rose’s diagram in Fig.1, namely ds = − du − dudr + 2 γ z ¯ z dzd ¯ z , (4)where u = T − R and γ z ¯ z = 2 / (1 + z ¯ z ) is the metric ofthe conformal sphere. The asymptotic topology is S × R at the r = ∞ boundary. Boundaries of I + at u = ±∞ are denoted as I + ± .The bulk equations are as follows ∇ ν F νµ = q j µ , (5)where F µν = ∂ µ A ν − ∂ ν A µ and j µ is the conserved matterwhere ∇ µ j µ = 0.The gauge symmetry associated to the equation of mo-tion (EoM) is δ (cid:15) A µ = ∂ µ (cid:15) . (6)We consider the retarded radial gauge, defined by A r = 0 , A u | I + = 0 . (7)Let us expand the fields around I + . To ensure a finitenon-vanishing radiation flux, (cid:82) I + F zu F uz (cid:54) = 0, a A z ∼ O (1) near I + is required. On the other hand, Eq.(7)implies that A u ∼ O (1 /r ) near I + . This implies thatfree Maxwell fields undergo the expansion A z ( r, u, z, ¯ z ) = A z ( u, z, ¯ z ) + ∞ (cid:88) n =1 A ( n ) z ( u, z, ¯ z ) r n , (8) A u ( r, u, z, ¯ z ) = 1 r A u ( u, z, ¯ z ) + ∞ (cid:88) n =1 A ( n +1) u ( u, z, ¯ z ) r n . (9)The leading terms of the field strength are F z ¯ z = O (1), F ur = O ( r − ), F uz = O (1) and F rz = O ( r − ). At lead-ing order the constraint equation is γ z ¯ z ∂ u A u = ∂ u ( ∂ z A ¯ z + ∂ ¯ z A z ) + q γ z ¯ z J u , (10)where J u ( u, z, ¯ z ) = lim r →∞ [ r j u ( r, u, z, ¯ z )] . (11)The gauge conditions for Eq.(7) leave unfixed the ar-bitrary function (cid:15) ≡ (cid:15) ( z, ¯ z ) on the conformal sphere at r = ∞ . These are called Large Gauge Transformations,specified by δ (cid:15) A z ( u, z, ¯ z ) = ∂ z (cid:15) ( z, ¯ z ) . (12)The charges associated to the gauge theory on I + are Q + (cid:15) + = (cid:90) I + dud z(cid:15) [ ∂ u ( ∂ z A ¯ z + ∂ ¯ z A z ) + q γ z ¯ z J u ] . (13)A similar expression for the charges can be found in thepast boundary I − : Q − (cid:15) − = (cid:90) I − dvd z(cid:15) − [ ∂ v ( ∂ z A ¯ z + ∂ ¯ z A z ) + q γ z ¯ z J v ] , (14) FIG. 1. The Penrose diagrams for the future (left panel) andthe past (right panel) charts of Minkowski space-time in vac-uum are displayed. In these diagrams, in correspondence ofevery point of the 2D real plane of r, u ( v ) there is a compact-ified conformal invariant complex sphere, with z, ¯ z complexcoordinates. I ± correspond to the future and past null in-finity. The BMS symmetry, as well as the Large AsymptoticGauge Symmetry, are defined on the I ± region lines. where now coefficient functions A depend on the ad-vanced coordinate v in stead of the retarded u .The invariance of the theory under the large gaugesymmetries generated by the charges casts (cid:104) OUT | ( Q + (cid:15) S − S Q − (cid:15) ) | IN (cid:105) = 0 , (15)where generically we have n particles in the in state | IN (cid:105) = | z IN1 , ..., z IN n (cid:105) and m in OUT as | OUT (cid:105) = | z OUT1 , ..., z
OUT m (cid:105) , while S is the S-matrix operator. Thisimplies that for an incoming state Q − (cid:15) | IN (cid:105) = F − [ (cid:15) ] | IN (cid:105) , (16)which means that the charges do not annihilate the vac-uum state, unless (cid:15) is fixed to a constant. From thisrelation, one derives the associated Goldstone bosons,which coincide with the φ ± fields entering A ± z ( z, ¯ z ) = ∂ z φ ± ( z, ¯ z ). The corresponding Ward identities are (cid:104) OUT | : F [ (cid:15) ] S : | IN (cid:105) = (17) (cid:104) n (cid:88) k =1 q IN k (cid:15) ( z IN k , ¯ z IN k ) − m (cid:88) l =1 q OUT l (cid:15) ( z IN l , ¯ z IN l ) (cid:105) (cid:104) OUT | S | IN (cid:105) . As we previously commented, in vacuum the conformalinvariant sphere casts ds = 2(1 + z ¯ z ) − dzd ¯ z . The identity disconnected component of the Lorentzgroup is isomorphic to
P SL (2 , C ), and acts as a confor-mal transformation to the Riemann sphere. More specif-ically, the transformation has the form z (cid:48) = az + bcz + d , where a, b, c, d are complex number parameters con-strained to ad − bc = 1. The transformation then acts on / L2345E [ x ] FIG. 2. The predicted electric field due to the BMS analo-gous QED anomaly near the boundary, is displayed for dif-ferent screening exponent parameters ν = 0 . , . . . , x coordinate is normalized as the semi-metal length L , while E is normalized to a typical scale E , dependent on the voltageapplied. the metric as dz (cid:48) d ¯ z (cid:48) (1 + z (cid:48) ¯ z (cid:48) ) = Ω ( z, ¯ z ) dzd ¯ z (1 + z ¯ z ) , where Ω( z, ¯ z ) = 1 + z ¯ z | az + b | + | cz + d | . The large asymptotic gauge symmetry T preserves theconformal invariance of the sphere, i.e. the algebra ofthe theory is T × P SL (2 , C ).Such a group can be broken at the quantum level byconformal anomalies, induced from either gauge interac-tions or gravity, retaining the form A nomaly = (cid:90) M √ g [ g µν (cid:104) T µν (cid:105) − (cid:104) g µν T µν (cid:105) ] . (18)For this class of QFT, A nomaly has the form A nomaly = (cid:90) M √ g [ b F µν F µν + R ] , (19)where R here denotes R , R µν R µν , R µναβ R µναβ , (cid:3) R . Boundary QFT and induced anomaly.—
Let us intro-duce a plane static boundary at a certain point x . Sucha boundary induces a large asymptotic gauge symmetryanomaly. In Boundary Quantum Field Theory [16–18],the renormalized current has a general structure close tothe boundary x → J Rµ = 1 x J (3) µ + 1 x J (2) µ + 1 x J (1) µ + log xJ (0) + ... (20)where the J ( n ) coefficients are dependent on the geom-etry. We focus on currents that are conserved once theanomaly term from Eq.(18) is disregarded.Gauge invariance imposes the following constraints: J (3) µ = J (2) µ = 0 , (21) J (1) µ = α F µν n ν + α D µ K + α D ν K νµ + α (cid:63)F µν n ν , (22)where F, n, D , K, h are the background field gaugestrenght, the normal boundary vector, the covariantderivative and the extrinsic curvature induced on theboundary. We have expressed n µ R µν h µν by means of theextrinsic curvature, using the standard Gauss-Codazzirelation n µ R µν h νγ = D µ K µγ − D γ K .The variation of the anomaly term Eq. (18) leads tothe relation with the integral on the bulk( δ A ) ∂M (cid:15) = (cid:16) (cid:90) M √ gδA µ J µR (cid:17) log (cid:15) − , (23)where (cid:15) is a x-regulator. Eq. (23) establishes a relationbetween the integrated anomaly A and a variation of thegauge field coupled to the J R current. The integral is UVlogarithmically divergent. The metric can be rewrittenin the Gauss normal coordinates ds = dx + dy a dy b ( h ab − rk ab + r q ab + ... ) , (24)where n µ = (1 , , ,
0) is the normal vector. Let us chosea gauge A r = 0 and expand the gauge field around theboundary as A i = a i + x i A (1) b + ... . The variation of theWeyl anomaly with respect to the gauge field leads to( δ A ) ∂M = 4 b (cid:82) ∂M √ hF bn δa b , which is related to (cid:16) (cid:82) M √ gJ µR δA µ (cid:17) log 1 /(cid:15) = (25) (cid:82) ∂M √ h ( α F bn + α D b k + α D j k jb + α (cid:63) F bn (cid:1) δa b . For α = 4 b and α , , = 0, the latter leads to J b = 4 b F bn x , (26)which can also be rewritten as J µ = 4 b F µν n ν x . (27)The BMS analogous charges are shifted to Q + (cid:15) + +∆ Q A ( r, ζ, ¯ ζ ) = Q + (cid:15) + + (cid:90) I + du d ζ (cid:15) γ ζ ¯ ζ b F bn x ( r, ζ, ¯ ζ ) ] , where x ( r, z, ¯ z ) is understood as a mapping between stan-dard Minkowski in Cartesian-coordinates and the Pen-rose diagram coordinates. This result is somehow ex-pected, since the new charge distribution on the x-axisdoes not allow to define conserved Noether charges onthe asymptotic null infinity. Indeed, the phenomenon isrelated to a change of the vector and scalar potential inthe asymptotic limit. Electric field from the anomaly, and suggestedexperiment.—
The new anomaly percolates on a newequation for the electric field ∂ x E x = − (cid:15) β e ec (cid:126) E x ( x ) x , x > , (28)which leads to an electrostatic potential and field, theprofile of which is φ ( x ) = φ − Cx − ν − ν , E x = Cx ν , (29)where ν is the critical anomalous exponent, which is re-lated to β q , the renormalization β function of the electriccoupling q , and the vacuum dielectric constant (cid:15) by ν = 2 β q qc (cid:126) (cid:15) . (30)If the presence of the boundary is within a dielectric ma-terial, Eq. (30) can be generalized with the dielectricof the material by substituting (cid:15) → (cid:15) . In Fig.2 thepredicted electric field corrected by the BMS analogousanomaly is displayed.As a simple experimental set-up can be envisaged totest the effect of the quantum anomaly of analogous BMSsymmetries arising from the boundary. We can proposeexactly the same concept of Refs. [7, 8], being sufficientto consider a Dirac or Weyl semi-metal material, apply-ing a difference of potential on the two sides of it. Inthis way, we can test the electrostatic profile φ ( x ), whichis affected by the quantum screening effect. Indeed, theconformal anomaly pointed out in Refs. [7, 8] is relatedto the BMS analogue in QED. However, there is a newprediction of BMS anomaly that cannot be captured bythe conformal anomaly. This new class of effects are re- lated to the electromagnetic radiation that may be de-tected from these materials, providing a new memoryeffect as for BMS and gravity. As a matter of fact, thesoft Bremsstrahlung effect is described by the QED softlimits, in turn related to the Large Asymptotic Symme-try. An anomaly of it percolates onto the modificationof the soft Bremsstrahlung effect. Indeed, being Eq. (15)anomalous, the asymptotic future and past charges inEqs. (13)-(14) are modified by the anomaly. This affectsany QED scattering process considered in the material.These aspects certainly deserve future extended experi-mental and theoretical analysis beyond the purposes ofthis paper. Conclusions.—
The infinite dimensional asymptoticgauge symmetry in QED is quantum anomalous in pres-ence of a magnetized boundary. This effect may be testedin Dirac or Weyl semimetals over the next future. Thus,it might provide a better understanding of the role ofinfinite dimensional symmetries in quantum physics, in-cluding the BMS symmetries in gravity, which allow amemory effect for quantum black hole physics.
ACKNOWLEDGMENTS
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