Residual bulk viscosity of a disordered 2D electron gas
RResidual bulk viscosity of a disordered 2D electron gas
V. A. Zakharov and I. S. Burmistrov
2, 3 Skolkovo Institute of Science and Technology, Moscow 121205, Russia L. D. Landau Institute for Theoretical Physics, acad. Semenova av. 1-a, 142432 Chernogolovka, Russia Laboratory for Condensed Matter Physics, HSE University, 101000 Moscow, Russia (Dated: February 20, 2021)The nonzero bulk viscosity signals breaking of the scale invariance. We demonstrate that adisorder in two-dimensional noninteracting electron gas in a perpendicular magnetic field results inthe nonzero disorder–averaged bulk viscosity. We derive analytic expression for the bulk viscositywithin the self-consistent Born approximation. This residual bulk viscosity provides the lower boundfor the bulk viscosity of 2D interacting electrons at low enough temperatures.
I. INTRODUCTION
Hydrodynamic description of a viscous electron flowhas long history [1]. The progress in this field was de-tained by a lack of experiments (see, however, Ref. [2]).After experimental realization of graphene there wererevival in the theoretical [3–13] and experimental [14–24] research on the hydrodynamic description of electrontransport in two spatial dimensions.In the presence of the rotational symmetry the vis-cosity tensor of a two dimensional system can beparametrized by three parameters only, 𝜂 𝑗𝑘,𝑙𝑠 = (︀ 𝜁 − 𝜂 𝑠 )︀ 𝛿 𝑗𝑘 𝛿 𝑙𝑠 + 𝜂 𝑠 (︀ 𝛿 𝑗𝑙 𝛿 𝑘𝑠 + 𝛿 𝑗𝑠 𝛿 𝑘𝑙 )︀ + ( 𝜂 𝐻 / (︀ 𝜖 𝑗𝑙 𝛿 𝑘𝑠 + 𝜖 𝑗𝑠 𝛿 𝑘𝑙 + 𝜖 𝑘𝑙 𝛿 𝑗𝑠 + 𝜖 𝑘𝑠 𝛿 𝑗𝑙 )︀ . (1)Here 𝜁 stands for the bulk viscosity. The shear viscosityis denoted as 𝜂 𝑠 . The second line of Eq. (1) appears if thetime reversal symmetry is broken, e.g. by the perpendic-ular magnetic field 𝐵 . Similarly to the Hall conductivity,the Hall viscosity, 𝜂 𝐻 , describes the non–dissipative partof the viscosity tensor. We mention that the existenceof the Hall viscosity has been well appreciated long timeago in the field of high temperature magnetized plasma[25–29].Although it is frequently said that viscosity exists onlyin a context of hydrodynamics, in fact, it has implica-tion on its own: as a linear response that characterizes achange of the stress tensor under a time–dependent defor-mations [30]. In electron systems microscopic calculationof the viscosity tensor has been traditionally performedfor the shear and Hall components only [30–45]. The lat-ter attracted much interest due to its relation to the ge-ometrical response [32–38] and quantization for transla-tionally and rotationally invariant gapped quantum sys-tems [36].It is well-known that for a monoatomic gas the Boltz-mann kinetic equation predicts zero value for the bulkviscosity [46, 47]. Zero bulk viscosity implies that thesystem is scale invariant and can expand isotropicallywithout dissipation. One more example of such systemis the unitary Fermi gas [48]. However, generically, in-teraction breaks scale invariance and results in nonzerobulk viscosity. The canonical example is the Fermi liquid with nonzero albeit small bulk viscosity [49, 50]. Re-cently, breaking of scale invariance has been extensivelystudied in the context of strongly interacting Fermi gas,both theoretical [51–57] and experimentally [58–60], aswell as in the quantum chromodynamics [61–64].Typically, a condensed matter electron system containsa quenched disorder. A presence of a random potentialin the Hamiltonian inevitably breaks the scale invariance.Therefore, one may expect a nonzero value of the bulkviscosity even in the absence of electron-electron interac-tions.In this paper, to unravel this issue, we consider a two-dimensional (2D) noninteracting electron gas in the pres-ence of a perpendicular static magnetic field and a ran-dom potential. Based on the Kubo formula for the bulkviscosity we demonstrate explicitly how a nonzero mag-nitude of the disorder–averaged bulk viscosity appearsdue to the presence of a random potential in the Hamil-tonian. We find that the real part of the bulk viscosity asa function of frequency contains two contributions: (i) adelta-function peak with the weight which is determinedby such thermodynamic quantities as pressure, energydensity, and isentropic compressibility; and (ii) a smoothpart depending on the total elastic scattering time 𝜏 .Within the self-consistent Born approximation (SCBA)we derive expression for the smooth contribution to thereal part of the bulk viscosity at a finite frequency. Inthe absence of the magnetic field it acquires a remarkablysimple form for all frequencies, 𝜔 , and temperatures, 𝑇 ,much smaller than the chemical potential, 𝜇 ,Re 𝜁 ( 𝜔 ) = 𝜈 / (2 𝜏 ) , | 𝜔 | , 𝑇 ≪ 𝜇, (2)where 𝜈 denotes the density of states at the Fermi level.We emphasize that 𝜁 ( 𝜔 →
0) is proportional to the elasticscattering rate in contrast to the shear viscosity which,as standard transport quantities, is proportional to theelastic scattering time. The result (2) indicates that inorder to derive nonzero bulk viscosity within the kineticequation approach one needs to take into account higherorder corrections due to scattering (see discussion in Sec.VI). The results reported in this paper together with theresults of Ref. [42] provide the full answer for the viscos-ity tensor of 2D noninteracting electrons subjected to aperpendicular magnetic field. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b The outline of the paper is as follows. In Sec. II weformulate the problem and write down the Kubo-type ex-pression for the bulk viscosity in which the delta-functionpeak at zero frequency is single out. The weight of thedelta-function peak is analyzed in Sec. III. In Sec. IVthe SCBA is reviewed. We present calculation of thebulk viscosity within SCBA in Sec. V. We end the paperwith summary and conclusions (Sec. VI). Some detailsof calculations are given in Appendices.
II. FORMALISM
A 2D electron gas in the presence of an external staticperpendicular magnetic field 𝐵 and a random potential 𝑉 ( 𝑟 ) is described by the following Hamiltonian, 𝐻 = (︀ − 𝑖 ∇ − 𝑒 𝐴 )︀ / (2 𝑚 𝑒 ) + 𝑉 ( 𝑟 ) . (3)Here 𝑚 𝑒 denotes the electron mass. The vector potential 𝐴 ( 𝑟 ) corresponds to the static magnetic field 𝐵 , ∇× 𝐴 = 𝐵 𝑒 𝑧 . We shall work in the Landau gauge: 𝐴 𝑦 = − 𝐵𝑥 and 𝐴 𝑥 = 𝐴 𝑧 = 0. We assume the Gaussian distributionfor a random potential with zero mean and characterizedby the pair correlation function 𝑉 ( 𝑟 ) 𝑉 ( 𝑟 ′ ) = 𝑊 ( | 𝑟 − 𝑟 ′ | ). The function 𝑊 ( 𝑟 ) is assumed to decay at a typicallength scale 𝑑 𝑊 . The magnetic field 𝐵 is assumed to bestrong enough to polarize the electron spins. Throughoutthe paper we shall use units with 𝑐 = (cid:126) = 𝑘 𝐵 = 1.In the microscopic theory the disorder-averaged viscos-ity tensor can be computed from the Kubo formula (seeEqs. (3.4), (3.11), and (3.14) of Ref. [30]): 𝜂 𝑗𝑘,𝑙𝑠 ( 𝜔 ) = 𝛿 𝑗𝑘 𝛿 𝑙𝑠 𝑖𝜔 + ( 𝜅 − − 𝑃 ) − ∫︁ 𝑑𝜀𝑓 𝜀 𝜋 𝒜 𝜔 + Tr[ 𝑇 𝑗𝑘 , 𝐽 𝑙𝑠 ] Im 𝐺 𝑅𝜀 + ∫︁ 𝑑𝜀𝑑 Ω 𝜋 𝒜 (︀ 𝑓 𝜀 − 𝑓 𝜀 +Ω )︀ 𝑖 (Ω − 𝜔 + ) 𝜔 + Tr 𝑇 𝑗𝑘 Im 𝐺 𝑅𝜀 +Ω 𝑇 𝑙𝑠 Im 𝐺 𝑅𝜀 . (4)Here 𝑃 stands for the pressure of the electron gas, 𝜅 − denotes the inverse isentropic compressibility at constantparticle number, 𝒜 is the system area, and 𝜔 + = 𝜔 + 𝑖 𝐺 𝑅𝜀 = 1 / ( 𝜀 − 𝐻 + 𝑖 𝑓 𝜀 = 1 / [1 + exp(( 𝜀 − 𝜇 ) /𝑇 )]denotes the Fermi distribution function with the chemicalpotential 𝜇 and temperature 𝑇 . The stress tensor opera-tor 𝑇 𝑗𝑘 = 𝑚 𝑒 ( 𝑣 𝑗 𝑣 𝑘 + 𝑣 𝑘 𝑣 𝑗 ) / 𝑣 = ( − 𝑖 ∇ − 𝑒 𝐴 ) /𝑚 𝑒 isthe velocity operator [30, 42]. The strain generator op-erator 𝐽 𝑗𝑘 is related with the stress tensor operator as 𝑇 𝑗𝑘 = − 𝑖 [ 𝐻, 𝐽 𝑗𝑘 ]. We note that contrary to the stresstensor operator, the expression for 𝐽 𝑗𝑘 is sensitive to thepresence of a random potential.Bulk viscosity 𝜁 can be derived from the viscosity ten-sor by tracing the spatial indices, 𝜁 = 𝜂 𝑗𝑗,𝑙𝑙 /𝑑 , where 𝑑 = 2 is the spatial dimension. Using Eq.(4), we find 𝜁 ( 𝜔 ) = 𝜅 − − 𝑃 − 𝑋𝑖𝜔 + + ∫︁ 𝑑𝜀𝑑 Ω( 𝜋𝑑 ) 𝒜 (︀ 𝑓 𝜀 − 𝑓 𝜀 +Ω )︀ 𝑖 (Ω − 𝜔 + ) 𝜔 + × Tr 𝑇 Σ Im 𝐺 𝑅𝜀 +Ω 𝑇 Σ Im 𝐺 𝑅𝜀 , (5) where 𝑇 Σ = 𝑇 𝑗𝑗 and the frequency independent quantity 𝑋 is defined as 𝑋 = 𝑖 ∫︁ 𝑑𝜀𝑓 𝜀 𝜋𝑑 𝒜 Tr[ 𝑇 Σ , 𝐽 Σ ] Im 𝐺 𝑅𝜀 . (6)Here we introduce 𝐽 Σ = 𝐽 𝑗𝑗 . Using the relation 𝑇 Σ =2( 𝐻 − 𝑉 ), we can rewrite Eq. (5) as follows 𝜁 ( 𝜔 ) = 𝜅 − − 𝑃 − 𝑋𝑖𝜔 + + 4 ∫︁ 𝑑𝜀𝑑 Ω( 𝜋𝑑 ) 𝒜 (︀ 𝑓 𝜀 − 𝑓 𝜀 +Ω )︀ 𝑖 (Ω − 𝜔 + ) 𝜔 + × Tr 𝑉 Im 𝐺 𝑅𝜀 +Ω 𝑉 Im 𝐺 𝑅𝜀 . (7)It is worthwhile to emphasize that the last term in theright hand side of the above expression represents themany-body two-point correlation function of a randompotential. Thus the structure of Eq. (7) resembles thestructure of the Kubo formula for the interacting cleanFermi gas (see Ref. [65] and references therein). In ourcase a random potential plays a role of the contact oper-ator [66–68].The expression (7) suggests the following sum rule forthe disorder averaged bulk viscosity, ∞ ∫︁ −∞ 𝑑𝜔𝜋 𝜁 ( 𝜔 ) = 𝑃 + 𝑋 − 𝜅 − . (8)This expression is analogous to the sum rule found forthe interacting clean Fermi gas [51, 65].Using Eq. (7), we obtain the following Kubo formulafor the real part of the bulk viscosity,Re 𝜁 ( 𝜔 ) = 4 𝑑 ∫︁ 𝑑𝜀𝜋 𝒜 𝑓 𝜀 − 𝑓 𝜀 + 𝜔 𝜔 Tr 𝑉 Im 𝐺 𝑅𝜀 + 𝜔 𝑉 Im 𝐺 𝑅𝜀 + 𝜋 𝒟 𝛿 ( 𝜔 ) , (9)where the weight of the delta-function peak at 𝜔 = 0 isgiven as 𝒟 = 𝑃 + 𝑋 − 𝜅 − − 𝑑 ∫︁ 𝑑𝜀𝑑 Ω 𝜋 𝒜 𝑓 𝜀 − 𝑓 𝜀 +Ω Ω × Tr 𝑉 Im 𝐺 𝑅𝜀 +Ω 𝑉 Im 𝐺 𝑅𝜀 . (10)We emphasize that the appearance of a random potential 𝑉 as vertices in Eq. (9) reflects the fact that the bulkviscosity vanishes in the clean case. III. THE WEIGHT OF THE ZEROFREQUENCY DELTA-FUNCTION PEAK
The definition of the pressure 𝑃 in terms of the stresstensor and the relation 𝑇 Σ = 2( 𝐻 − 𝑉 ) yield 𝑃 = ⟨ 𝑇 Σ ⟩ /𝑑 = − ∫︁ 𝑑𝜀𝜋𝑑 𝒜 𝑓 𝜀 Tr 𝑇 Σ Im 𝐺 𝑅𝜀 = 2 𝑑 ℰ + 2 𝑑 ∫︁ 𝑑𝜀𝜋 𝒜 𝑓 𝜀 Tr 𝑉 Im 𝐺 𝑅𝜀 . (11)Here ℰ = ∫︀ 𝑑𝜀𝜈 ( 𝜀 ) 𝜀𝑓 𝜀 denotes the energy density where 𝜈 ( 𝜀 ) stands for the disorder-averaged density of states.We mention that the relation (11) is analogous to theTan’s relation for the pressure of an interacting Fermigas [68]. In our case the random potential plays a role ofthe contact operator.In a similar way as above, we obtain 𝑋 = 2 𝑑 𝑃 − 𝑖𝑑 ∫︁ 𝑑𝜀𝜋 𝒜 𝑓 𝜀 Tr[
𝑉, 𝐽 Σ ] Im 𝐺 𝑅𝜀 . (12)Using Eq. (11) and the relation Tr[ 𝑉, 𝐽 Σ ] Im 𝐺 𝑅𝜀 =Tr[ 𝑇 Σ , 𝐽 𝑉 ] Im 𝐺 𝑅𝜀 , where 𝐽 𝑉 is the generator operatorcorresponding to 𝑉 , i.e. 𝑉 = − 𝑖 [ 𝐻, 𝐽 𝑉 ], we find 𝑋 = 4 𝑑 𝑃 − 𝑑 ℰ + 4 𝑖𝑑 ∫︁ 𝑑𝜀𝜋 𝒜 𝑓 𝜀 Tr[
𝑉, 𝐽 𝑉 ] Im 𝐺 𝑅𝜀 . (13)With the help of identity (see Appendix A), ∫︁ 𝑑𝜀𝑑 Ω 𝜋 𝒜 𝑓 𝜀 − 𝑓 𝜀 +Ω Ω Tr 𝑉 Im 𝐺 𝑅𝜀 +Ω 𝑉 Im 𝐺 𝑅𝜀 = 𝑖 ∫︁ 𝑑𝜀𝜋 𝒜 𝑓 𝜀 Tr[
𝑉, 𝐽 𝑉 ] Im 𝐺 𝑅𝜀 , (14)the expression (10) becomes 𝒟 = 𝑑 + 4 𝑑 𝑃 − 𝑑 ℰ − 𝜅 − . (15)We emphasize that the weight 𝒟 of the delta-functionpeak is expressed in terms of the thermodynamic quan-tities only.With the help of Eq. (14) the sum rule (8) can berewritten as ∞ ∫︁ −∞ 𝑑𝜔𝜋 𝜁 ( 𝜔 ) = 𝒟 + 4 𝑖𝑑 ∫︁ 𝑑𝜀𝜋 𝒜 𝑓 𝜀 Tr[
𝑉, 𝐽 𝑉 ] Im 𝐺 𝑅𝜀 . (16)We note that the right hand side of Eq.(16) is purely real.In the absence of the magnetic field, the in-verse isentropic compressibility is defined as 𝜅 − = −𝒜 (︀ 𝜕𝑃/𝜕 𝒜 )︀ 𝑠 𝒜 ,𝑛 𝑒 𝒜 , where 𝑠 and 𝑛 𝑒 denote the entropyand electron densities, respectively. Using the thermo-dynamic relation 𝑇 𝑠 = ℰ + 𝑃 − 𝜇𝑛 𝑒 we find that a vari-ation of the area 𝛿 𝒜 under conditions 𝑠 𝒜 = const and 𝑛 𝑒 𝒜 = const results in the following variation of the en-ergy density, 𝛿 ℰ = − ( ℰ + 𝑃 ) 𝛿 𝒜 / 𝒜 . Also, a variation ofthe area leads to the variation of the electron density, 𝛿𝑛 𝑒 = − 𝑛 𝑒 𝛿 𝒜 / 𝒜 . Hence, we obtain [65] 𝜅 − = ( ℰ + 𝑃 ) (︂ 𝜕𝑃𝜕 ℰ )︂ 𝑛 𝑒 + 𝑛 𝑒 (︂ 𝜕𝑃𝜕𝑛 𝑒 )︂ ℰ . (17)We note that 𝜅 − is related with the sound velocity, 𝑐 𝑠 =1 / √ 𝜅𝑚 𝑒 𝑛 𝑒 .In the absence of the disorder, 𝑉 = 0, and the mag-netic field, 𝐵 = 0, the energy density and the pressure of the ideal Fermi gas are related as 𝑃 = 2 ℰ /𝑑 [69]. Thisrelation implies that the pressure is fixed if the energydensity is fixed, i.e ( 𝜕𝑃/𝜕𝑛 𝑒 ) ℰ ≡
0. Then, from Eq.(17), we find 𝜅 − = ( 𝑑 + 2) 𝑃/𝑑 . As a result, we ob-tain that the weight of the delta-function peak is zero, 𝒟 = 0. Therefore, Eq. (7) implies that the bulk viscosityvanishes identically, 𝜁 ( 𝜔 ) = 0, for the ideal Fermi gas inagreement with its scale invariance.In the presence of magnetic field, the trace of the stresstensor yields the so-called, internal pressure, 𝑃 int = 𝑃 − 𝑚𝐵 , where 𝑚 stands for the magnetization density [70].Repeating similar steps as in the case of zero magneticfield, we find that the weight of the zero frequency delta-function peak in a 2D electron gas in the perpendicularmagnetic fields becomes 𝒟 = 3 𝑃 int − ℰ − 𝜅 − , (18)where 𝜅 − = −𝒜 (︀ 𝜕𝑃 int /𝜕 𝒜 )︀ 𝑠 𝒜 ,𝑛 𝑒 𝒜 ,𝐵 𝒜 is the inverseisentropic compressibility at the constant particle num-ber and the magnetic flux. We note that the expression(18) for the weight of the delta-function peak in the mag-netic field can be obtained from Eq. (15) by substitutionof 𝑃 int instead of 𝑃 and setting 𝑑 = 2.Using the thermodynamic relation 𝑇 𝑠 = ℰ + 𝑃 int + 𝑚𝐵 − 𝜇𝑛 𝑒 , we find that a variation of the area 𝛿 𝒜 underconditions 𝑠 𝒜 = const, 𝑛 𝑒 𝒜 = const, and 𝐵 𝒜 = const re-sults in the following variation of the energy density, 𝛿 ℰ = − ( ℰ + 𝑃 int ) 𝛿 𝒜 / 𝒜 . Also, a variation of the area yields thevariations of the electron density, 𝛿𝑛 𝑒 = − 𝑛 𝑒 𝛿 𝒜 / 𝒜 andthe magnetic field, 𝛿𝐵 = − 𝐵𝛿 𝒜 / 𝒜 . Hence, we obtain 𝜅 − = ( ℰ + 𝑃 int ) (︂ 𝜕𝑃 int 𝜕 ℰ )︂ 𝑛 𝑒 ,𝐵 + 𝑛 𝑒 (︂ 𝜕𝑃 int 𝜕𝑛 𝑒 )︂ ℰ ,𝐵 + 𝐵 (︂ 𝜕𝑃 int 𝜕𝐵 )︂ ℰ ,𝑛 𝑒 . (19)Again, in the absence of a random potential, the weight ofthe delta-function peak vanishes. It is easy to check thisstatement at zero temperature. Then for 𝑁 filled Landaulevels we find 𝑃 int = ℰ = 𝑚𝜔 𝑐 𝑁 / (4 𝜋 ) and 𝜅 − = 2 ℰ .Hence Eq. (18) leads to 𝒟 = 0.We note that in the presence of a magnetic field thesum rule (16) still holds but with the weight of the zero-frequency delta-function peak given by Eq. (18). IV. SELF-CONSISTENT BORNAPPROXIMATION
In order to take into account a random potential weemploy a self-consistent Born approximation [71]. Thisapproximation is justified under the following condi-tions [72–74],1 /𝑘 𝐹 , 𝑑 𝑊 ≪ 𝑙 𝐵 , 𝑑 𝑊 ≪ 𝑣 𝐹 𝜏 . (20)Here 𝑙 𝐵 = 1 / √ 𝑒𝐵 stands for the magnetic length and 𝑘 𝐹 = 𝑚 𝑒 𝑣 𝐹 stands for the Fermi momentum with theFermi velocity denoted as 𝑣 𝐹 . The total elastic relaxationtime in the absence of a magnetic field is defined by thefollowing relation1 𝜏 𝑛 = 𝜈 𝜋 ∫︁ 𝑑𝜑 𝜋 ˜ 𝑊 (︀ 𝑘 𝐹 sin( 𝜑/ )︀ cos( 𝑛𝜑 ) , 𝑛 = 0 , , , . . . (21)with 𝑛 = 0. Here ˜ 𝑊 ( 𝑞 ) stands for the Fourier transformof 𝑊 ( 𝑟 ). We note that the condition 𝑘 𝐹 𝑙 𝐵 ≫ 𝑁 ≫ 𝑁 is the number offilled Landau levels.Within the SCBA the physical quantities of interestare usually fully expressed in terms of the disorder aver-aged retarded Green’s function 𝒢 𝑅𝜀 . It satisfies the self-consistency equation (see Fig. 1(a)), 𝒢 𝑅𝑛 = ( 𝜀 − 𝜖 𝑛 − Σ 𝑅𝜀 ) − , Σ 𝑅𝜀 = 𝜔 𝑐 𝜋𝜏 ∑︁ 𝑛 𝒢 𝑅𝑛 , (22)where 𝜖 𝑛 = 𝜔 𝑐 ( 𝑛 + 1 /
2) denotes the energy of the 𝑛 -thLandau level (LL) and Σ 𝑅𝜀 stands for the disorder aver-aged self energy. Here 𝜔 𝑐 = 𝑒𝐵/𝑚 𝑒 is the cyclotron fre-quency. The self-consistency relation (22) can be solvedanalytically for Σ 𝑅𝜀 in two limiting cases [71]. In theregime of weak magnetic field, 𝜔 𝑐 𝜏 ≪
1, when LLs over-lap, one can perform summation over LL index 𝑛 withthe help of the Poisson formula and find [71]Σ 𝑅𝜀 = − 𝑖 𝜏 (︁ − 𝛿𝑒 𝜋𝑖𝜀/𝜔 𝑐 )︁ , (23)where 𝛿 = exp( − 𝜋/𝜔 𝑐 𝜏 ) ≪ 𝜔 𝑐 𝜏 ≫
1, onecan restrict the summation over LL index 𝑛 in Eq. (22)to 𝑛 = 𝑁 only, where 𝜖 𝑁 is the closest LL energy to theenergy of interest: | 𝜀 − 𝜖 𝑁 | < 𝜔 𝑐 /
2. Then one obtains [71]Σ 𝑅𝜀 = 12 (︁ 𝜀 − 𝜖 𝑁 − 𝑖 √︀ Γ − ( 𝜀 − 𝜖 𝑁 ) )︁ . (24)Here the LL broadening is controlled by the energy scaleΓ = √︀ 𝜔 𝑐 / ( 𝜋𝜏 ).The disorder–averaged density of states can be ex-pressed in terms of the disorder–averaged Green’s func-tion as 𝜈 𝜀 = − 𝜋 𝑙 𝐵 ∑︁ 𝑛 Im 𝒢 𝑅𝑛 ( 𝜀 ) = − 𝜏 𝜈 Im Σ 𝑅𝜀 . (25)Using Eqs. (23) and (24), we find the disorder–averageddensity of states [71] 𝜈 𝜀 = 𝜈 {︃ − 𝛿 cos(2 𝜋𝜀/𝜔 𝑐 ) , 𝜔 𝑐 𝜏 ≪ ,𝜏 ∑︀ 𝑛 Re √︀ Γ − ( 𝜀 − 𝜖 𝑛 ) , 𝜔 𝑐 𝜏 ≫ . (26) FIG. 1. Diagrams used in SCBA. (a) The self-energy dia-gram; (b) and (c) Diagrams corresponding to the bulk vis-cosity within SCBA. Bold solid lines denote the disorder av-eraged Green’s function 𝒢 𝜀 , dashed lines stand for the paircorrelation function 𝑊 ( 𝑟 ). V. BULK VISCOSITY WITHIN SCBA
The bulk viscosity at nonzero frequency, 𝜔 ̸ = 0, isgiven by the first term in the right hand side of Eq. (9).We assume that frequency and temperature are muchsmaller than the chemical potential, | 𝜔 | , 𝑇 ≪ 𝜇 . Underthis assumption, the integral over energy 𝜀 is dominatedby the vicinity of the chemical potential. The unusualfeature of the Kubo formula for the real part of the bulkviscosity, Eq. (9), is that vertex is a random potential.The diagrams contributing to Re 𝜁 ( 𝜔 ) within SCBA areshown in Fig. 1(b) and (c).We start from computation of the diagram of Fig. 1(b).Using Eq. (22), we can rewrite this contribution asRe 𝜁 ( 𝑏 ) = ∫︁ 𝑑𝜀𝜋 𝑓 𝜀 − 𝑓 𝜀 + 𝜔 𝜔 𝜋𝑙 𝐵 ∑︁ 𝑛 Im 𝒢 𝑅𝑛 ( 𝜀 ) Im Σ 𝑅𝜀 + 𝜔 = ∫︁ 𝑑𝜀 𝑓 𝜀 − 𝑓 𝜀 + 𝜔 𝜔 𝜈 𝜀 𝜈 𝜀 + 𝜔 𝜏 𝜈 . (27)We note that the contribution to Re 𝜁 from the diagramof Fig. 1(b) can be expressed solely in terms of the densityof states, 𝜈 𝜀 , computed within SCBA.In addition to the diagram in Fig. 1(b) within SCBAone needs to take into account a set of diagrams shownin Fig. 1(c). They correspond to the impurity ladder in-sertion and describe vertex renormalization. As we shallsee below, in spite of the scalar nature of the vertex (arandom potential), the diagrams of Fig. 1(c) provide asignificant contribution to the real part of the bulk vis-cosity in the case of strong magnetic field. Evaluation ofthe four diagrams in Fig. 1(c) yields (see Appendix B)Re 𝜁 ( 𝑐 ) = 𝜈 ∫︁ 𝑑𝜀 𝑓 𝜀 − 𝑓 𝜀 + 𝜔 𝜔 Re [︃ (︀ Σ 𝑅𝜀 + Σ 𝐴𝜀 + 𝜔 )︀ Π 𝑅𝐴 ( 𝜔 )1 − Π 𝑅𝐴 ( 𝜔 ) /𝜏 − (︀ Σ 𝑅𝜀 + Σ 𝑅𝜀 + 𝜔 )︀ Π 𝑅𝑅 ( 𝜔 )1 − Π 𝑅𝑅 ( 𝜔 ) /𝜏 ]︃ . (28)Here the polarization operator,Π 𝑅𝐴 ( 𝜔 ) = 𝜔 𝑐 𝜋 ∑︁ 𝑛 𝒢 𝑅𝑛 ( 𝜀 + 𝜔 ) 𝒢 𝐴𝑛 ( 𝜀 ) = 𝜏 (Σ 𝐴𝜀 − Σ 𝑅𝜀 + 𝜔 ) 𝜔 + Σ 𝐴𝜀 − Σ 𝑅𝜀 + 𝜔 , (29)provides the contribution to the “bubble” without theimpurity ladder insertion. The expression for Π 𝑅𝑅 ( 𝜔 )can be obtained from Eq. (29) by changing superscript 𝐴 to 𝑅 . Combining contributions (27) and (28), we findthe following expression for the disorder–averaged bulkviscosity of a 2D electron gas,Re 𝜁 ( 𝜔 ) = ∫︁ 𝑑𝜀 𝑓 𝜀 − 𝑓 𝜀 + 𝜔 𝜔 𝜈 𝜀 𝜈 𝜀 + 𝜔 𝜏 𝜈 Re [︃ − Σ 𝑅𝜀 + 𝜔 − Σ 𝑅𝜀 𝜔 − 𝜈 𝜀 + 𝜔 − 𝜈 𝜀 𝜔𝜈 𝜀 𝜈 𝜀 + 𝜔 (︀ Σ 𝑅𝜀 + 𝜔 + Σ 𝑅𝜀 )︀]︃ . (30)We emphasize that the above expression involves notonly the density of states, 𝜈 𝜀 , computed within SCBAbut also the real part of the SCBA self energy. In thelimit of zero frequency the expression (30) becomesRe 𝜁 ( 𝜔 →
0) = ∫︁ 𝑑𝜀 𝜏 𝜈 (︀ − 𝑓 ′ 𝜀 )︀ 𝜈 𝜀 [︁ − 𝜕 𝜀 (︀ 𝜈 𝜀 Re Σ 𝑅𝜀 )︀ 𝜈 𝜀 ]︁ . (31)In the absence of magnetic field the density of states andself-energies are independent of energy. Therefore, Eq.(30) transforms into remarkably simple result (2). It isinstructive to compare the bulk viscosity and the shearviscosity in the absence of magnetic field [31] and in thelimit of zero frequency,Re 𝜁 ( 𝜔 → 𝜂 𝑠 = 1 𝜇 𝜏 tr , 𝜏 ≪ /𝜏 tr , = 1 /𝜏 − /𝜏 denotes the inverse secondtransport time.In the case of a weak magnetic field, 𝜔 𝑐 𝜏 ≪
1, thegeneral expression (30) can be drastically simplified. Tothe first order in the Dingle parameter 𝛿 we findRe 𝜁 ( 𝜔 ) = 𝜈 𝜏 [︃ − 𝛿 sin Ω 𝜔 (︂ − 𝜋𝜔 𝑐 𝜏 tan(Ω / )︂ ×ℱ 𝑇 cos 2 𝜋𝜇𝜔 𝑐 ]︃ . (33)Here ℱ 𝑇 = (2 𝜋 𝑇 /𝜔 𝑐 ) / sinh (︀ 𝜋 𝑇 /𝜔 𝑐 )︀ and Ω = 2 𝜋𝜔/𝜔 𝑐 describe the temperature and frequency dependence of the Shubnikov–de Haas-type oscillations of the bulk vis-cosity, respectively. The real part of the bulk viscosity asa function of the frequency and the chemical potential inthe case of a weak magnetic field is shown on Fig. 2(c)and (d). We mention that the amplitude of oscillationsof Re 𝜁 ( 𝜔 ) decays with the frequency as ∼ 𝜔 − , while atzero frequency the amplitude of oscillations of Re 𝜁 withthe chemical potential is independent of 𝜇 . This ampli-tude is enhanced by the factor ∼ / ( 𝜔 𝑐 𝜏 ) in comparisonwith oscillations of the density of states. Therefore, theShubnikov–de Haas-type oscillations in the bulk viscosityare stronger than in the longitudinal conductivity [74, 75]and the shear viscosity [42].Now we consider the case of a strong magnetic field, 𝜔 𝑐 𝜏 ≫
1, in which Landau levels are well separated.Then in the limit of zero frequency the general result(31) can be reduced to the following expression,Re 𝜁 ( 𝜔 →
0) = 2 𝜋 ( 𝜇 − 𝜖 𝑁 ) 𝑙 𝐵 Γ , | 𝜇 − 𝜖 𝑁 | (cid:54) Γ . (34)This result is a bit counterintuitive. The real part of thebulk viscosity vanishes when the chemical potential is atthe center of the 𝑁 -th Landau level. With deviation of 𝜇 from the center of the Landau level Re 𝜁 ( 𝜔 →
0) increasesand reaches the magnitude 2 / ( 𝜋𝑙 𝐵 ) at the boundary ofthe disorder broadened Landau level. This dependenceon chemical potential is shown on Fig.2(b). Such un-usual behavior of the real part of the bulk viscosity occurssince it is proportional to the derivative of the density ofstates with respect to the energy. The maximum valueof Re 𝜁 ( 𝜔 →
0) is the factor 𝑁 𝜏 /𝜏 tr , smaller than themaximal value of the shear viscosity and the factor 𝑁 smaller than the maximal value of the Hall viscosity [42].The result (30) suggests that the bulk viscosity oscil-lates as a function of frequency with the period 𝜔 𝑐 . Nearharmonics of the cyclotron resonance, | 𝜔 − 𝑘𝜔 𝑐 | = | ∆ 𝜔 | ≪ 𝜔 𝑐 , 𝑘 = 1 , , . . . , Eq. (30) transforms into the followingexpressionRe 𝜁 ( 𝜔 ) ≈ ∫︁ 𝑑𝜀 𝑓 𝜀 − 𝑓 𝜀 + 𝑘𝜔 𝑐 +Δ 𝜔 𝑘𝜔 𝑐 𝜈 𝜀 𝜈 𝜀 +Δ 𝜔 𝜏 𝜈 . (35)We note that at frequencies 𝜔 (cid:38) 𝜔 𝑐 one can neglect theterms with the self-energies in the right hand side of Eq.(30).At zero temperature, 𝑇 = 0, and under assumptionthat 𝑘 is much smaller than the number of filled Landaulevels, 𝑁 , we obtain that the bulk viscosity near the 𝑘 -th harmonics of the cyclotron resonance, 𝑘 = 1 , , . . . , isgiven byRe 𝜁 ( 𝜔 ) = ΓΘ (︀ − | ∆ 𝜔 | )︀ 𝜋 𝑙 𝐵 𝜔 𝑐 [︃ ℱ (︂ | ∆ 𝜔 | Γ )︂ + sgn(∆ 𝜔 ) 𝑘 × Θ (︀ Γ − | 𝜇 − 𝜖 𝑁 | )︀ ℱ (︂ | ∆ 𝜔 | Γ , 𝜇 − 𝜖 𝑁 Γ )︂]︃ . (36)Here Θ( 𝑥 ) stands for the Heaviside theta function andsgn(∆ 𝜔 ) at ∆ 𝜔 = 0 is equal to zero. The functions ℱ , FIG. 2. The real part of the bulk viscosity in the regimes of strong ( 𝜔 𝑐 𝜏 = 100, panels (a) and (b)) and weak ( 𝜔 𝑐 𝜏 = 0 . 𝑇 = 0. The dependence of Re 𝜁 on frequency is shown on panels (a)and (c). Black dashed line at the panel (a) correspond to the value of the bulk viscosity at the 𝜔 = 𝑘𝜔 𝑐 for well-separatedLandau levels, cf. Eq. (38). Panels (b) and (d) show Re 𝜁 ( 𝜔 →
0) as a function of the chemical potential for strong and weakmagnetic fields. Black dashed line at the panel (b) corresponds to the limiting value of Re 𝜁 ( 𝜔 →
0) in the strong magneticfield for 𝜇 = 𝜖 𝑁 ± Γ, cf. Eq. (34). are defined as (0 (cid:54) 𝑥 <
2, 0 (cid:54) | 𝑦 | (cid:54) ℱ ( 𝑥 ) = − 𝑥 ∫︁ − 𝑑𝑡 √︀ − 𝑡 √︀ − ( 𝑡 + 𝑥 ) , ℱ ( 𝑥, 𝑦 ) = min { 𝑦, − 𝑥 } ∫︁ max { 𝑦 − 𝑥, − } 𝑑𝑡 √︀ − 𝑡 √︀ − ( 𝑡 + 𝑥 ) . (37)We mention that this result suggests that the magnitudeof the bulk viscosity at the harmonics of the cyclotronresonance is independent of the harmonics number andthe chemical potential,Re 𝜁 ( 𝜔 = 𝑘𝜔 𝑐 ) = 2Γ3 𝜋 𝑙 𝐵 𝜔 𝑐 , 𝑘 = 1 , , . . . . (38)As one can see, the magnitude of the bulk viscosity atthe harmonics of the cyclotron resonance are the factorΓ /𝜔 𝑐 smaller than the maximal value of the bulk viscosityat small frequencies, | 𝜔 | ≪ Γ. Dependence of Re 𝜁 onfrequency 𝜔 is shown on Fig. 2(a). We note that thebulk viscosity decays relatively fast with detuning fromthe cyclotron resonance harmonics. VI. SUMMARY AND CONCLUSIONS
To summarize, we have developed the theory of thedisorder-averaged bulk viscosity of the disordered 2Delectron gas in the presence of a perpendicular magneticfield within the self-consistent Born approximation. Wedemonstrated that the real part of the bulk viscosity hastwo contributions: delta-function peak at zero frequency,see Eq. (18), and the smooth part, see Eq. (30). Thelatter is explicitly computed in the case of weak, see Eq.(33) and strong magnetic fields, see Eq. (34). Also weanalyzed the harmonics of the cyclotron resonance in thecase of strong magnetic fields, see Eq. (38).The zero field result (2) indicates that the method ofthe kinetic equation is not convenient for computation ofthe bulk viscosity. This statement is well enough illus-trated by Ref. [50] where the bulk viscosity in the cleanFermi liquid was derived from the kinetic equation. Onemore example is calculations of the bulk viscosity of aclean interacting Fermi gas near the unitary limit withinthe kinetic equation approach [55, 56, 65]. However, it isworthwhile to explain for a reader how the kinetic equa-tion can lead to the bulk viscosity which is proportionalto the scattering rate, 1 /𝜏 but not to the scattering time(as standard dissipative coefficients, e.g. the dissipativeconductivity, the shear viscosity, etc.). We start fromexpansion of the left hand side of the kinetic equationinto formal series in 1 /𝜏 . Such an expansion can besymbolically written as ℒ ( 𝑛 (0) 𝑞 + 𝛿𝑛 𝑞 ) + ℒ 𝑛 (0) 𝑞 + . . . .Here 𝑛 (0) 𝑞 denotes the equilibrium distribution functionand 𝛿𝑛 𝑞 stands for the out-of-equilibrium perturbationof the distribution function induced by a bulk flow of theelectron gas. The operator ℒ coincides with the oper-ator in the kinetic equation for the clean noninteractingelectron gas [46]. As a consequence, it vanishes actingon both 𝑛 (0) 𝑞 and 𝛿𝑛 𝑞 . The operator ℒ appears due torenormalization of the electron spectrum by scattering offa random potential, i.e., in other words, due to Re Σ 𝑅𝜀 .Therefore, the term ℒ 𝑛 (0) 𝑞 is proportional to 1 /𝜏 . Sincethe collision integral is also proportional to 𝛿𝑛 𝑞 /𝜏 , wefind that the kinetic equation yields 𝛿𝑛 𝑞 ∝ (1 /𝜏 ) . Thisshould be contrasted with a standard situation for which 𝛿𝑛 𝑞 ∝ ℒ 𝑛 (0) 𝑞 (1 /𝜏 ) − . Next, the bulk viscosity can becomputed as 𝜁 ∝ ∫︀ 𝑑 𝑞 𝒞 𝑞 𝛿𝑛 𝑞 [50, 55, 56]. However, thefunction 𝒞 𝑞 becomes nonzero only due the renormaliza-tion of the electron spectrum by scattering off a randompotential, i.e. 𝒞 𝑞 ∝ /𝜏 (see similar cancellation forclean interacting problem [55, 56]). Again, we remindthat in a standard case 𝒞 𝑞 is independent of 𝜏 . Combin-ing the estimates for 𝛿𝑛 𝑞 and 𝒞 𝑞 , we find that the kineticequation results in 𝜁 ∝ /𝜏 . We emphasize that actualcomputation of ℒ and 𝒞 𝑞 , especially, in the presence ofmagnetic field is much more complicated task than thediagrammatic approach developed in this work.We mention that the viscosity tensor affects the spec-trum of bulk and edge magnetoplasmons [76]. Our resultfor the bulk viscosity in a weak magnetic field impliesthat the contribution to the magnetoplasmon spectrumdue to the bulk viscosity can be neglected for wave vec-tors 𝑞 ≪ 𝑘 𝐹 in comparison with the contribution due tothe shear viscosity.It is worthwhile to compare our result for the bulk vis-cosity due to a random potential with the result for thebulk viscosity in a clean weakly degenerate interactingFermi gas. The interaction contribution to the bulk vis-cosity decreases with the temperature as a power law, ∝ 𝑇 (see Ref. [50] for the three-dimensional Fermi liq-uid). This implies that the contribution to the bulk vis-cosity due to disorder dominates at low enough tempera-tures. Therefore, we expect that our results provide thelower bound for the residual bulk viscosity in 2D inter-acting disordered electron system at low temperatures.The bulk viscosity can be estimated from measure-ments in interacting Fermi gases [58–60]. It is an ex-perimental challenge to extract the bulk viscosity fromexperiments in 2D electron systems. ACKNOWLEDGMENTS
The authors are grateful to M. Goldstein and I. Gornyifor very useful discussions. The research was partiallysupported by the Russian Ministry of Science and HigherEducation, the Russian Foundation for Basic Research(grant No. 20-52-12013) - Deutsche Forschungsgemein- schaft (grant No. EV 30/14-1) cooperation, and by theBasic Research Program of HSE.
Appendix A: Derivation of the identity (14)
In this Appendix we present a brief derivation of theidentity (14). It worth to mention that this identity sat-isfied for arbitrary 𝑉 and, therefore, we shall omit thebar which denotes averaging over a random potential.It is convenient to work in the eigen basis of the Hamil-tonian 𝐻 , 𝐻 | 𝑎 ⟩ = 𝐸 𝑎 | 𝑎 ⟩ . In this basis the imaginarypart of the Green function is a sum of the delta-functionpeaks, Im 𝐺 𝑅 ( 𝜀 ) = − 𝜋 ∑︀ 𝑎 | 𝑎 ⟩⟨ 𝑎 | 𝛿 ( 𝜀 − 𝐸 𝑎 ). Substituting 𝑉 = − 𝑖 [ 𝐻, 𝐽 𝑉 ] for one of two operators of the randompotential, we obtainTr 𝑉 Im 𝐺 𝑅𝜀 +Ω 𝑉 Im 𝐺 𝑅𝜀 = − 𝑖𝜋 ∑︁ 𝑎,𝑏 ( 𝐸 𝑎 − 𝐸 𝑏 ) ⟨ 𝑎 | 𝐽 𝑉 | 𝑏 ⟩×⟨ 𝑏 | 𝑉 | 𝑎 ⟩ 𝛿 ( 𝜀 + Ω − 𝐸 𝑏 ) 𝛿 ( 𝜀 − 𝐸 𝑎 ) . (A1)With the help of the above result, the left hand side ofEq. (14) becomes ∫︁ 𝑑𝜀𝑑 Ω 𝜋 𝒜 𝑓 𝜀 − 𝑓 𝜀 +Ω Ω Tr 𝑉 Im 𝐺 𝑅𝜀 +Ω 𝑉 Im 𝐺 𝑅𝜀 = 𝑖 𝒜 ∑︁ 𝑎,𝑏 (︀ 𝑓 𝐸 𝑎 − 𝑓 𝐸 𝑏 )︀ ⟨ 𝑎 | 𝐽 𝑉 | 𝑏 ⟩⟨ 𝑏 | 𝑉 | 𝑎 ⟩ . (A2)The right hand side of Eq. (14) can be written in theeigen basis of 𝐻 as 𝑖 ∫︁ 𝑑𝜀𝜋 𝒜 𝑓 𝜀 Tr[
𝑉, 𝐽 𝑉 ] Im 𝐺 𝑅𝜀 = 𝑖 𝒜 ∑︁ 𝑎,𝑏 𝑓 𝐸 𝑎 (︀ ⟨ 𝑎 | 𝐽 𝑉 | 𝑏 ⟩⟨ 𝑏 | 𝑉 | 𝑎 ⟩−⟨ 𝑎 | 𝑉 | 𝑏 ⟩⟨ 𝑏 | 𝐽 𝑉 | 𝑎 ⟩ )︀ . (A3)As one can see, Eqs. (A2) and (A3) are identical. Appendix B: Ladder contribution to the bulkviscosity
In this Appendix we present a brief derivation of Eq.(28). Diagrams corresponding to 𝜁 ( 𝑐 ) are shown in Fig.1(c). We findRe 𝜁 ( 𝑐 ) = 1 𝜋 𝒜 ∫︁ 𝑑𝜀 𝑓 𝜀 − 𝑓 𝜀 + 𝜔 𝜔 [︃ 𝑋 (1) 𝜀,𝜔 + 𝑋 (2) 𝜀,𝜔 + 𝑋 (3) 𝜀,𝜔 + 𝑋 (4) 𝜀,𝜔 ]︃ , (B1)Here 𝑋 (1) stands for the diagram in the upper left panelin Fig. 1c, 𝑋 (2) for the upper right panel, 𝑋 (3) for thebottom left panel and 𝑋 (4) for the bottom right panel,respectively. Each of the four diagram 𝑋 𝑖 consists ofthree blocks: two self-energies at the vertices and thediffuson ladder in the middle. This ladder represents aninfinite sum of diagrams with the Green’s functions atthe top and bottom, and arbitrary number of verticaldashed scattering lines. For computation of such dia-grams it is convenient to rewrite Tr 𝑉 Im 𝐺 𝑅𝜀 + 𝜔 𝑉 Im 𝐺 𝑅𝜀 as − (1 / 𝑉 (︀ 𝐺 𝑅𝜀 + 𝜔 − 𝐺 𝐴𝜀 + 𝜔 )︀ 𝑉 ( 𝐺 𝑅𝜀 − 𝐺 𝐴𝜀 ). After suchtransformation each contribution 𝑋 ( 𝑖 ) has four differentterms with particular combination of the Green’s func-tions. For the first contribution we obtain 𝑋 (1) 𝜀,𝜔 = − 𝜋𝜈 𝒜 𝑅𝜀 Σ 𝑅𝜀 + 𝜔 Π 𝑅𝑅 ( 𝜔 ) ∞ ∑︁ 𝑛 =0 (︂ Π 𝑅𝑅 ( 𝜔 ) 𝜏 )︂ 𝑛 − 𝜋𝜈 𝒜 𝐴𝜀 Σ 𝐴𝜀 + 𝜔 Π 𝐴𝐴 ( 𝜔 ) ∞ ∑︁ 𝑛 =0 (︂ Π 𝐴𝐴 ( 𝜔 ) 𝜏 )︂ 𝑛 + 𝜋𝜈 𝒜 𝑅𝜀 Σ 𝐴𝜀 + 𝜔 Π 𝑅𝐴 ( 𝜔 ) ∞ ∑︁ 𝑛 =0 (︂ Π 𝑅𝐴 ( 𝜔 ) 𝜏 )︂ 𝑛 + 𝜋𝜈 𝒜 𝐴𝜀 Σ 𝑅𝜀 + 𝜔 Π 𝐴𝑅 ( 𝜔 ) ∞ ∑︁ 𝑛 =0 (︂ Π 𝐴𝑅 ( 𝜔 ) 𝜏 )︂ 𝑛 = − 𝜋𝜈 𝒜 [︃ Σ 𝑅𝜀 Σ 𝑅𝜀 + 𝜔 Π 𝑅𝑅 ( 𝜔 )1 − Π 𝑅𝑅 ( 𝜔 ) /𝜏 + Σ 𝐴𝜀 Σ 𝐴𝜀 + 𝜔 Π 𝐴𝐴 ( 𝜔 )1 − Π 𝐴𝐴 ( 𝜔 ) /𝜏 − Σ 𝑅𝜀 Σ 𝐴𝜀 + 𝜔 Π 𝑅𝐴 ( 𝜔 )1 − Π 𝑅𝐴 ( 𝜔 ) /𝜏 − Σ 𝐴𝜀 Σ 𝑅𝜀 + 𝜔 Π 𝐴𝑅 ( 𝜔 )1 − Π 𝐴𝑅 ( 𝜔 ) /𝜏 ]︃ . (B2) As one can check, 𝑋 (2) 𝜀,𝜔 = 𝑋 (1) 𝜀,𝜔 . Next, in a similar way,we find 𝑋 (3) 𝜀,𝜔 = − 𝜋𝜈 𝒜 [︃ (︀ Σ 𝑅𝜀 )︀ Π 𝑅𝑅 ( 𝜔 )1 − Π 𝑅𝑅 ( 𝜔 ) /𝜏 + (︀ Σ 𝐴𝜀 )︀ Π 𝐴𝐴 ( 𝜔 )1 − Π 𝐴𝐴 ( 𝜔 ) /𝜏 − (︀ Σ 𝑅𝜀 )︀ Π 𝑅𝐴 ( 𝜔 )1 − Π 𝑅𝐴 ( 𝜔 ) /𝜏 − (︀ Σ 𝐴𝜀 )︀ Π 𝐴𝑅 ( 𝜔 )1 − Π 𝐴𝑅 ( 𝜔 ) /𝜏 ]︃ (B3)and 𝑋 (4) 𝜀,𝜔 = − 𝜋𝜈 𝒜 [︃ (︀ Σ 𝑅𝜀 + 𝜔 )︀ Π 𝑅𝑅 ( 𝜔 )1 − Π 𝑅𝑅 ( 𝜔 ) /𝜏 + (︀ Σ 𝐴𝜀 + 𝜔 )︀ Π 𝐴𝐴 ( 𝜔 )1 − Π 𝐴𝐴 ( 𝜔 ) /𝜏 − (︀ Σ 𝐴𝜀 + 𝜔 )︀ Π 𝑅𝐴 ( 𝜔 )1 − Π 𝑅𝐴 ( 𝜔 ) /𝜏 − (︀ Σ 𝑅𝜀 + 𝜔 )︀ Π 𝐴𝑅 ( 𝜔 )1 − Π 𝐴𝑅 ( 𝜔 ) /𝜏 ]︃ . 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