Nonlinear Quantum Electrodynamics in Dirac materials
NNonlinear Quantum Electrodynamics in Dirac materials
Aydın Cem Keser,
1, 2
Yuli Lyanda-Geller, and Oleg P. Sushkov
1, 2 School of Physics, University of New South Wales, Sydney, NSW 2052, Australia Australian Research Council Centre of Excellence in Low-Energy Electronics Technologies,The University of New South Wales, Sydney 2052, Australia Department of Physics and Astronomy, Purdue University, West Lafayette, IN, 47907, USA (Dated: January 26, 2021)Classical electromagnetism is linear. However, fields can polarize the vacuum Dirac sea, causingquantum nonlinear electromagnetic phenomena, e.g., scattering and splitting of photons that occuronly in very strong fields found in neutron stars or heavy ion colliders. We show that strongnonlinearity arises in Dirac materials at much lower fields ∼ Classical electromagnetism is linear and hence sup-ports the principle of superposition. It has been pointedout by Heisenberg and Euler in 1936 that due to quantummechanics and the presence of the Dirac sea, linearityand the principle of superposition do not hold in strongfields [1]. Quantum electrodynamics (QED) is thereforenonlinear owing to the fact that the electromagnetic fieldpolarises the Dirac sea as though it is a material medium.This effect becomes significant at electric and magneticfields E S (cid:39) . × V / cm, B S (cid:39) . × T respec-tively. These are the so-called Schwinger scale values [2]and they are enormous on the laboratory scale. Such elec-tric and magnetic fields exist in heavy ion collisions [3]and in neutron stars [4], respectively. Nevertheless, loworder nonlinear QED effects have been observed in thelaboratory, such as (i) the elastic scattering of a photonfrom the Coulomb field of a nucleus and (ii) the photonsplitting into two photons in the Coulomb field [5]. Forthe discussion of light by light scattering see Ref. [6]Dirac materials are known for decades, [7–9] neverthe-less, their recently understood topological properties andsurface excitations have led to an enormous activity [10–15]. There has been a recent surge of interest in the non-linear electromagnetic response of materials [15–17] dueto their transport properties (e.g. rectification) and pos-sible applications in photovoltaics. In this letter, ratherthan transport, we study dielectric and magnetizationproperties of three dimensional (3D) Dirac materials. Weinclude nonlinear corrections to all orders in perturbationtheory by analysing the Heisenberg-Euler action [1, 18–20], that goes beyond the general framework in condensedmatter physics [21].In known Dirac materials, we find the typical valuesof Schwinger fields, E S ∼ V / cm, B S ∼ FIG. 1. Diagrams for the Heisenberg-Euler action. Thedashed line corresponds to the external electromagnetic fields B , E and the solid line is the Green function of a Dirac seaelectron. muth [22] and ii) the nonlinear quantum magnetisationfor Weyl semimetal TaAs [23]. Hence, Heisenberg-Eulertheory not only provides a unifying framework for the ex-perimentally known effects but also enables us to predictnew strongly nonlinear phenomena. Most significant isthe magnetic field tunable, very large dielectric enhance-ment reaching up to δ(cid:15) r ∼
10 per every 1 T of appliedmagnetic field, an effect which does not saturate up toextremely high fields. In addition, we also predict an in-duced magnetization, quadratic in applied electric field,which can be easily detected with squid magnetometryand lock-in techniques. Remarkably, both of these effectsare highly anisotropic, since they require that the electricand magnetic fields are collinear.To adopt results from QED, we work with the La-grangian description of electromagnetic field. In freespace, using CGS units [24] L = E − B π (1)Heisenberg-Euler polarisation corrections to the La-grangian can be represented by the infinite chain of dia-grams shown in Fig.1, that arise from the Hamiltonian ofthe Dirac material coupled to the electromagnetic field, H = v α · (cid:16) p − ec A (cid:17) + β ∆ + eϕ. (2)Here α and β are Dirac matrices, A and ϕ are the vectorand the scalar potentials. For typical systems under con- a r X i v : . [ c ond - m a t . o t h e r] J a n sideration, such as, eg., PbTe, the gap is 2∆ ∼
100 meVinstead of 2 m e c = 1 MeV in the QED vacuum, and theFermi-Dirac velocity v ∼ c/
400 is significantly smallerthan the speed of light c. To make specific, for estimateswe will take ∆ = 50meV and v = c/ v (cid:28)
1, the QED fine structure constant α = e (cid:126) c = 1 /
137 is replaced by α D = e (cid:126) v = 2 .
9. Diagramsin Fig.1 do not contain the electron-electron interaction,therefore the dielectric constant is not included in α D .Note also that, compared to the electric field, E = − ∇ ϕ ,the interaction with magnetic field, B = ∇ × A , containsan additional factor vc per each power of B .In Fig.1 the first diagram quadratic in external fieldsis ultraviolet divergent, and all other diagrams are con-vergent. In QED this diagram describes corrections toelectric and magnetic permeability of vacuum. The cor-responding correction to the Lagrangian is [18] δL = α D π ln (cid:18) Λ∆ (cid:19) (cid:18) E − v c B (cid:19) . (3)Here the subscript ‘1’ indicates contribution from the firstdiagram in Fig.1, Λ ∼ v (cid:126) πa ∼ a is the lattice spacing. Comparing L + δL with the classical Lagrangian L cl = (cid:15) e E − B /µ π (4)we find the electronic dielectric constant (cid:15) e = 1 + 4 πχ ( e ) ( χ ( e ) is the electric susceptibility) and the diamagneticsusceptibility χ , where µ = 1 + 4 πχ . (cid:15) e = 1 + 2 α D π ln (cid:18) Λ∆ (cid:19) ∼ χ = − α D π v c ln (cid:18) Λ∆ (cid:19) ∼ − − (6)The dielectric constant in most materials have also a con-tribution that is due to the polarization of ions, (cid:15) i , andthe total relative permittivity is (cid:15) ≈ (cid:15) i + (cid:15) e − e = e q =0 . However, in con-densed matter, the charge is normalised at the latticespacing scale, e = e q , q ∼ (cid:126) π/a . The dielectric constantin Eq. (5) relates the two as e = e/ √ (cid:15) e . In the con-densed matter context this is called the RPA dielectric screening, whereas in QED it is referred to as the run-ning electric charge [20]. Definitions of the electric fieldare also different. In condensed matter, the Lagrangianof the free field is given by the first term in Eq.(4) whilein QED it is E / (8 π ) where E = √ (cid:15) e E . Importantly,the coupling term appears as a product of the chargeand electric field, that is independent of the definition, e E = e E . Hence we can literally use the QED resultsfor diagrams of Fig.1 beyond the first one. With this un-derstanding we can map a Dirac material to QED. Bohrmagneton, µ B = e (cid:126) m e c , is replaced by “Dirac magneton” µ D = e (cid:126) v × vc = 64 × µ B = 3 . / T, and the Comptonwavelength λ C = (cid:126) m e c is replaced by “Dirac wavelength” λ D = (cid:126) v ∆ = 2 . × λ C = 9 . λ D is equal to the rest energy.2 µ D B S = ∆ → B S = ∆ ce (cid:126) v ≡ . λ D eE S = ∆ → E S = ∆ e (cid:126) v = 0 . × V / cm (7)It is convenient to represent the nonlinear Heisenberg-Euler action in terms of dimensionless fields e = E E S , b = B B S (8)Considering, first, the weak coupling limit, | e | , | b | (cid:28) δL = α D λ D π ∆ (cid:34)(cid:18) E − v c B (cid:19) + 7 v c ( E · B ) (cid:35) . (9)At higher fields, all the diagrams in Fig.1, beyond thefirst one, add up to [19] δL (cid:48) = ∆8 π λ D (cid:90) ∞ dηe − η η × (cid:20) ( − ηA cot( ηA ) ηC coth( ηC ) + 1 − η ( A − C ) (cid:21) A = − i (cid:104)(cid:112) ( b + i e ) − (cid:112) ( b − i e ) (cid:105) C = 12 (cid:104)(cid:112) ( b + i e ) + (cid:112) ( b − i e ) (cid:105) (10)To avoid electric breakdown we consider only weakelectric fields, | e | <
1. However, the magnetic field can bemuch larger than B S . In a strong magnetic field, | b | (cid:29) δL (cid:48) → α D π (cid:20) (ˆ b · E ) | b | + v c B ln | b | (cid:21) (11) FIG. 2. Dimensionless functions F,G,D of Eqs.(12),(13),(15)versus the dimensionless magnetic field. Functions G and Ddepend on the relative orientation of electric and magneticfield. The left (right) panel corresponds to B (cid:107) E ( B ⊥ E ).Note that the vertical axes in two panels differ significantly. Here ˆ b = B /B . Although Eqs.(10,9,11) are derived forDC fields, they are valid for AC fields as long as thefrequency is sufficiently small, ω (cid:28) E = 0.Using (10), (9),(11) we find the nonlinear paramagneticcorrection to the magnetic susceptibility, χ = ∂ L∂B , δχ (cid:48) = α D π v c F ( | b | ) | b | (cid:28) F ( | b | ) = 25 | b | | b | (cid:29) F ( | b | ) = ln | b | (12)The dimensionless function F ( | b | ) obtained by numeri-cal integration of (10) is shown in Fig.2. The total mag-netic susceptibility is the sum of the linear susceptibility,Eq.(6) and the nonlinear correction, χ = χ + δχ (cid:48) .When | b | (cid:29) χ = − α D π v c ln (cid:16) c Λ eB (cid:126) v (cid:17) is indepen-dent of ∆ but depends on B . This effect was observed inthe Weyl semimetal TaAs where B S ∝ ∆ → δ(cid:15) e = α D π G i ( | b | ) (13) | b | (cid:28) G || ( | b | ) = 13 | b | , G ⊥ ( | b | ) = − | b | | b | (cid:29) G || ( | b | ) = | b | , G ⊥ ( | b | ) = − ln( | b | )Here the index i = || , ⊥ shows the relative orientationof electric and magnetic fields. Dimensionless functions G i ( | b | ) obtained by numerical integration of (10) areshown in Fig.2. For B (cid:107) E the correction δ(cid:15) e is posi-tive and can be very large, see below. For B ⊥ E thecorrection δ(cid:15) e is negative. Furthermore, according to (4),(9) the dielectric constant receives a nonlinear correction quadratic in the electric field, so that δ(cid:15) e = 2 α D π | e | , | b | = 0 (14)When | b | (cid:29)
1, this effect is suppresed by | e | / | b | , whereas for | e | , | b | (cid:28)
1, the contributions from (13) and (14)are additive.Another manifestation of the Dirac nonlinearity is thedependence of magnetic properties on the applied elec-tric field. The electric field dependent magnetisation M ( e ) = ∂δL∂ B , reads4 π M ( e ) = ˆ b µ D πλ D | e | D i ( | b | ) (15) | b | (cid:28) D || ( | b | ) = 23 | b | , D ⊥ ( | b | ) = − | b || b | (cid:29) D || ( | b | ) = 1 , D ⊥ ( | b | ) = − | b | Here the magnetization is presented in units of “Diracmagnetons” per “Dirac volume”. Dimensionless func-tions D i ( | b | ) obtained by numerical integration of (10)are shown in Fig.2. For B (cid:107) E the magnetization is largeand paramagnetic, while for B ⊥ E the magnetization isdiamagnetic. The electric field induced magnetization isquadratic in the applied electric field and as a function ofmagnetic field, saturates when | b | (cid:29)
1. In a weak mag-netic field, | b | (cid:28)
1, the same physics can be articulatedas electric field dependent magnetic susceptibility. δχ || = α D π v c | e | , δχ ⊥ = − α D π v c | e | (16)To enhance the nonlinear effects given by Eqs. (15) and(16), one needs the electric field as strong as possible.However, the field is limited by the dielectric strength, E d of the material, beyond which dielectric breakdownoccurs. We can estimate E d in the following way. Therate per unit volume of the particle-hole generation byelectric field (Zener tunnelling) is [19] P = ∆4 π (cid:126) λ D | e | e − π/ | e | . (17)The most important here is the exponential dependence,which universally applies to both Dirac and quadraticdispersions. Therefore, one expects that the dielectricstrength of a material, E d , is proportional to E S . Takingtwo well known band insulators, diamond (2∆ ≈ . E d ≈ V/cm), and silicon (2∆ ≈ .
14 eV, E d ≈ × V/cm), as reference materials, we observe that the di-electric strength scales as E d ∝ ∆ . Therefore E d is afixed fraction of the Schwinger field E S , see Eq.(7), corre-sponding to | e | ≈ .
1. Of course, there are other factorslike the dielectric strength dependence on impurities, thesize of the sample, etc. Nevertheless, the values of theelectric field up to | e | = 0 . . E -dependent magnetic effects (15) and (16) can besignificant. Furthermore, as usual in solids, setups withhuge built-in electric fields can be explored [25].So far we considered the isotropic crystals, with thevelocity independent of the orientation with respect tothe crystal axes, yet the nonlinear response is inherentlyanisotropic due to the mutual alignment of E and B .(This holds for physical vacuum too.) Crystal anisotropyadds an additional source of dependence on orientation.In real crystals the velocity is a tensor that can be diag-onalized in particular axes, with three different diagonalcomponents v , v , v . In the supplement A we summa-rize a well-known procedure [26] to employ the formulaein Eqs. (5),(12)-(16) in a general anisotropic crystal.We now discuss various experimental settings. Bis-muth is a semimetal with a small electron Fermi surfacecentered near the L-points and a small hole Fermi surfacecentered near the T-points of the Brillouine zone [27].The chemical potential is slightly above the bandgap2∆ ≈ . T = 370meV at T-points [27]. Because ofthe presence of free electrons, any applied voltage leadsto electric conductivity, so only the magnetic effects arerelevant for Bismuth. The contribution of conductionelectrons and holes to the magnetic susceptibility is neg-ligible, hence we can apply the our theory which consid-ers a full valence band only. First, we disregard the T-points, as 2∆ (cid:28) ∆ T . At the three L-points the electrondispersion is described by the Dirac Eq. with c/v ≈ α D ≈ .
5, and Λ ∼ χ ≈ − × − in the binary-bisectrix plane and wasfound to be consistent with the experimentally observedvalue, χ exp ≈ − . × − [29]. However, there is alsoa paramagnetic Heisenberg-Euler contribution that hasbeen missing in the previous work [22, 28]. Due to thesmall gap, the Schwinger field for Bismuth is very low, B S ≈ B = 300 mT which corresponds to b = B/B S =60 (cid:29)
1. In this situation the nonlinear paramagneticeffect (12) is very significant. We have calculated thiscontribution, δχ (cid:48) (0 . ≈ +1 × − . To investigatethe compensation of diamagnetic and paramagnetic con-tributions, we re-evaluated the diamagnetic effect (5) bytaking in to account that there are three Dirac points andarrived to a the value χ ≈ − × − . Thus, the total χ + δχ (cid:48) remains in good agreement with the experiment[29]. More importantly, the paramagnetic contributiongrows with the field, and we predict a significant non-linearity of susceptibility as function of magnetic field.For example [ χ (500 mT) − χ (100 mT)] /χ (300 mT) = 0 . − x Sb x is a Dirac semimetal at x < .
07, at x = 0 .
07 the gap is closed and at x > . x > .
07 we get anarray of Dirac insulators with a varying band gap. If wetake x ≈ .
14, the bandgap of is approximately the sameas that in Bi [31], hence for this substitution ratio, wepredict magnetic properties similar to those in pure Bi.However in this case, given the temperature is small com-pared to the gap 2∆ ∼
200 K the sample is an insulatorand hence a sufficiently stong electric field can be applied.The Schwinger field is very low E S ≈ × V / cm.By Eq. (13), the enhancement of of the dielectric con-stant (measured at E (cid:28) E S ) due to applied magneticfield is enormous, e.g. at B = 1 T, δ(cid:15) e ≈
12 andit continues to grow linearly with magnetic field up to B ∼ B S × Λ / ∆ ∼ πM ( e ) = 10 µ B / cm =10 − G = 10 − T at | e | = 0 . E = 40 V / cm). Theexternal field B ∼ − T / √ Hz [32]. We suggest thelock-in experiment, where if the AC applied electric fieldhas frequency f , assuming that f < ∼ f .Similar to the Bi and Sb alloys, a variety of com-pounds with the neighbouring elements Pb and Sn andtheir chalcogenides such as with S, Se and Te containDirac fermions [11, 13, 14]. In particular topological in-sulators are categorically bulk Dirac insulators with in-verted band gap. [12]. In lead chalcogenides (PbS, PbSeand PbTe), there is a single Dirac point with the bandgap about 2∆ = 0 . ∼ c/
400 [33, 34]. Therefore the estimates for susceptibili-ties presented earlier in the text in Eqs. (5)- (7) are basedon these compounds. The Schwinger fields are B S ≈ E S ≈ . × V / cm. The linear diamagnetic susceptibil-ity (5) is χ ≈ − − . The nonlinear, B-dependent, para-magnetic susceptibility (12) at B = 20T is δχ (cid:48) ≈ +10 − .The nonlinearity in χ is particularly strong, for example[ χ (35 T) − χ (7 T)] /χ (20 T) = 1 .
2. The B-dependent partof the dielectric constant (13) at B = 20 T is δ(cid:15) e ≈ . πM ( e ) ≈ × µ B / cm = 3 × − G = 3 × − Tat | e | = 0 . E ≈ V/cm). For the saturation of theE-field contribution to magnetization, a magnetic field B = 10-20 T is needed. Thanks to the relatively large α D = 2 .
9, the nonlinear, E-dependent, relative permit-tivity in (14) is measurable. Large electric fields up toabove 10 V/cm are feasible, bringing the correction upto δ(cid:15) e = 0 .
5, in particular, in an experimental configura-tion of a tunnel junction [25] with the reverse bias lessthan the current threshold. Then the system is in theinsulating regime, the electric field has a built-in charac-ter. An applied bias can be used to vary the field andthe change in dielectric constant can be determined by acapacitance measurement (See for example [35]). Theseestimates are also valid for the topological systems suchas the ternary compounds Pb(Bi − x Sb x ) Te [13, 36]and Dirac materials such as Bi Se , Bi Te , Sb Te .A closely related alloy structure to the above, is theseries Pb − x Sn x Se. The Dirac velocity is v ≈ c/ x =0 .
23 and varying between −
100 meV to 100 meV [37],that allows to cover all the regimes that we consideredin the previous paragraphs. Another possible route toa tunable gap Dirac material is to apply a pressure of ∼ In conclusion , we predict and calculate the nonlinearelectromagnetic effects in three dimensional Dirac mate-rials. The nonlinearity is due to the polarization of theDirac valence band. The nonlinear effects are large andcan be observed in laboratory experiments. We discussseveral experimental settings where the nonlinear elec-tromagnetism can be observed.
Acknowledgements
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Here, for the sake of completeness, we provide a prescription to generalize of our results to anisotropic crystals,adopted from Ref. [26] of the main text. In real crystals the velocity is a tensor that can be diagonalized to have threedifferent diagonal components v , v , v , in a particular choice of coordinate axes. All equations below are written forthese axes. The Dirac velocity is v = ( v v v ) / , and it is convenient to introduce the rescaling parameters u = v /v, u = v /v, u = v /v. (18)with which calculate the rescaled fields ˜E , ˜B by ˜E = ( u E , u E , u E ) ˜B = ( B /u , B /u , B /u ) . (19)By using these rescaled fields, we can still use the susceptibility formulae derived for isotropic crystals in the main textwhich produce the scaled susceptibilities ˜ χ and ˜ χ ( e ) . Finally, the physical susceptibilities χ, χ ( e ) in the unisotropiccrystal as a function of applied fields E , B are X ij ( E , B ) = ˜ χ ij ( ˜E , ˜B ) / ( u i u j ) X ( e ) ij ( E , B ) = u i u j ˜ χ ( e ) ij ( ˜E , ˜B ) (20)Here, there is no summation over repeated indices. We re-iterate that even when vv
Here, for the sake of completeness, we provide a prescription to generalize of our results to anisotropic crystals,adopted from Ref. [26] of the main text. In real crystals the velocity is a tensor that can be diagonalized to have threedifferent diagonal components v , v , v , in a particular choice of coordinate axes. All equations below are written forthese axes. The Dirac velocity is v = ( v v v ) / , and it is convenient to introduce the rescaling parameters u = v /v, u = v /v, u = v /v. (18)with which calculate the rescaled fields ˜E , ˜B by ˜E = ( u E , u E , u E ) ˜B = ( B /u , B /u , B /u ) . (19)By using these rescaled fields, we can still use the susceptibility formulae derived for isotropic crystals in the main textwhich produce the scaled susceptibilities ˜ χ and ˜ χ ( e ) . Finally, the physical susceptibilities χ, χ ( e ) in the unisotropiccrystal as a function of applied fields E , B are X ij ( E , B ) = ˜ χ ij ( ˜E , ˜B ) / ( u i u j ) X ( e ) ij ( E , B ) = u i u j ˜ χ ( e ) ij ( ˜E , ˜B ) (20)Here, there is no summation over repeated indices. We re-iterate that even when vv = vv
Here, for the sake of completeness, we provide a prescription to generalize of our results to anisotropic crystals,adopted from Ref. [26] of the main text. In real crystals the velocity is a tensor that can be diagonalized to have threedifferent diagonal components v , v , v , in a particular choice of coordinate axes. All equations below are written forthese axes. The Dirac velocity is v = ( v v v ) / , and it is convenient to introduce the rescaling parameters u = v /v, u = v /v, u = v /v. (18)with which calculate the rescaled fields ˜E , ˜B by ˜E = ( u E , u E , u E ) ˜B = ( B /u , B /u , B /u ) . (19)By using these rescaled fields, we can still use the susceptibility formulae derived for isotropic crystals in the main textwhich produce the scaled susceptibilities ˜ χ and ˜ χ ( e ) . Finally, the physical susceptibilities χ, χ ( e ) in the unisotropiccrystal as a function of applied fields E , B are X ij ( E , B ) = ˜ χ ij ( ˜E , ˜B ) / ( u i u j ) X ( e ) ij ( E , B ) = u i u j ˜ χ ( e ) ij ( ˜E , ˜B ) (20)Here, there is no summation over repeated indices. We re-iterate that even when vv = vv = vv
Here, for the sake of completeness, we provide a prescription to generalize of our results to anisotropic crystals,adopted from Ref. [26] of the main text. In real crystals the velocity is a tensor that can be diagonalized to have threedifferent diagonal components v , v , v , in a particular choice of coordinate axes. All equations below are written forthese axes. The Dirac velocity is v = ( v v v ) / , and it is convenient to introduce the rescaling parameters u = v /v, u = v /v, u = v /v. (18)with which calculate the rescaled fields ˜E , ˜B by ˜E = ( u E , u E , u E ) ˜B = ( B /u , B /u , B /u ) . (19)By using these rescaled fields, we can still use the susceptibility formulae derived for isotropic crystals in the main textwhich produce the scaled susceptibilities ˜ χ and ˜ χ ( e ) . Finally, the physical susceptibilities χ, χ ( e ) in the unisotropiccrystal as a function of applied fields E , B are X ij ( E , B ) = ˜ χ ij ( ˜E , ˜B ) / ( u i u j ) X ( e ) ij ( E , B ) = u i u j ˜ χ ( e ) ij ( ˜E , ˜B ) (20)Here, there is no summation over repeated indices. We re-iterate that even when vv = vv = vv = vv