# Hydrodynamic flows of non-Fermi liquids: magnetotransport and bilayer drag

###### Abstract

We consider a hydrodynamic description of transport for generic two dimensional electron systems that lack Galilean invariance and do not fall into the category of Fermi liquids. We study magnetoresistance and show that it is governed only by the electronic viscosity provided that the wavelength of the underlying disorder potential is large compared to the microscopic equilibration length. We also derive the Coulomb drag transresistance for double-layer non-Fermi liquid systems in the hydrodynamic regime. As an example, we consider frictional drag between two quantum Hall states with half-filled lowest Landau levels, each described by a Fermi surface of composite fermions coupled to a gauge field. We contrast our results to prior calculations of drag of Chern-Simons composite particles and place our findings in the context of available experimental data.

Introduction. Hydrodynamic flow of electrons can occur in solid state systems provided that the microscopic length scale of momentum-conserving electron-electron collisions is sufficiently short Gurzhi (1968). Under this condition the electron liquid attains local equilibrium and can be described in terms of slow variables associated with conserved quantities such as momentum and energy. However, this transport regime was hard to realize experimentally as typically electron-impurity scattering degrades electron momentum, whereas electron-phonon collisions violate both momentum and energy conservations of the electron liquid. Early evidence for the so-called hydrodynamic Gurzhi effect, related to the negative temperature derivative of resistivity, was reported in thin potassium wires Yu et al. (1984), and later in the electrostatically defined wires in the two dimensional electron gas of (Al,Ga)As heterostructures de Jong and Molenkamp (1995). The recent surge of experiments devoted to revealing hydrodynamic regimes of electronic transport is mainly focused on measurements conducted on graphene Bandurin et al. (2016); Crossno et al. (2016).

In the context of transport theories, a hydrodynamic description is powerful as it accurately describes most liquids. All microscopic details of the system at hand are then encoded into a handful of kinetic coefficients such as viscosities and thermal conductivity. In certain cases the latter can be controllably derived from the linearized Boltzmann kinetic equation by following the perturbative Chapman-Enskog procedure developed originally for gases. However, we have examples now where this kind of microscopic approach has to be substantially revisited. Deriving hydrodynamics for linearly dispersing electronic excitations in graphene represents an interesting example where this standard computation scheme had to be redone from scratch Müller and Sachdev (2008); Müller et al. (2009); Narozhny et al. (2015); Principi et al. (2016); Lucas et al. (2016a). An even more dramatic example is given by strongly correlated electron liquids Spivak et al. (2010), where the effects of interactions are nonperturbative, and thus a Boltzmann-like description may not be applicable. Yet the hydrodynamic picture still holds Andreev et al. (2011) and has to be viewed as a phenomenology that enables one to express various transport observables in terms of pristine kinetic coefficients of the electron liquid and certain thermodynamic quantities. This is our motivation to consider a hydrodynamic description of transport for strongly correlated electron liquids where we do not assume Fermi liquid-like behavior. We also do not assume Galilean invariance to be present. In this study we focus on magnetotransport and frictional drag transresistance in bilayers.

Hydrodynamic formalism. The general linearized set of equations that govern nonrelativistic magnetohydrodynamic transport in two dimensional charged fluids are given by Hartnoll et al. (2007); Foster and Aleiner (2009); Lucas and Sachdev (2015); Lucas (2015); Hartnoll et al. (2016) (i) the force equations (repeated indices imply summation throughout this work)

(1) |

which relate the rate of change of the momentum density to pressure, viscous, thermoelectric and Lorentz forces. serves as an effective “mass density” and is the effective charge density of the fluid. and respectively are the shear and bulk viscosities. and represent the electric field and thermal gradient. Fluctuations in the fluid pressure are given by , where is the entropy density and is the local screened chemical potential per unit charge. (ii) The equations for charge and heat currents read respectively as

(2) |

where , and are microscopic “incoherent” conductivities Davison et al. (2015), and (iii) the continuity equations are

(3) |

Onsager reciprocity requires . The incoherent conductivities, viscosities and thermodynamic properties are derived from correlation functions of the underlying microscopic field theory of the non-Fermi liquid Eberlein et al. (2016, 2017); Patel et al. (2015). This is a generalization of the usual theory of hydrodynamics to systems without Galilean invariance.

Magnetotransport in a single layer. We consider the steady state solutions of these equations in the presence of a disordered chemical potential . In the absence of applied electric fields and temperature gradients, we can apply a background electric field to nullify currents and fluid motion, assuming a uniform temperature. We then look for steady state solutions when this background is perturbed by an infinitesimal uniform electric field in linear response Lucas (2015); Lucas et al. (2016a). The difference between the unperturbed and perturbed set of equations gives

(4) |

where the delta-quantities represent deviations from the background values generated by the applied electric field (). We read off transport coefficients by looking at the change of the uniform components of their respective currents with respect to the applied electric field. For example, and . The equations (4) need to be given periodic boundary conditions in order to ensure a unique solution; otherwise one may shift by a constant and cancel the effects by appropriately shifting by a function that has a constant gradient Lucas (2015). We can consider the solution of (4) while treating disorder perturbatively Lucas (2015); Lucas et al. (2016a). Against a uniform background chemical potential, it is easy to see that the only response is a finite uniform velocity field , which implies , where is the uniform charge density in the absence of any disorder. Introducing a small parameter to parameterize the strength of the disorder, we expand . All responses, densities, viscosities and microscopic conductivities may also be expanded in powers of . For example and .

Order by order in , there are 4 unknowns , , and 4 equations in (4), so a unique solution is possible. This expansion in disorder strength while keeping finite implies the assumption that the magnetic field relaxes momentum faster than the disordered potential (see Lucas et al. (2016b) for when both relaxation rates are comparable). The expression for the uniform charge current at is (in momentum space)

(5) |

Thus, solving the equations at gives all the information needed to obtain the uniform conductivities up to .

In general the densities, viscosities and incoherent conductivities depend on , and their functional forms can be deduced from the underlying quantum critical theory, which is beyond the scope of hydrodynamics. However, for small values of these dependences can be neglected as the dominant effect on magnetoresistance arises from the long-range modulations of the equilibrium density (see Refs. Grozdanov et al. (2017); Hernandez and Kovtun (2017); Baumgartner et al. (2017) for other large effects). This contribution exceeds the one due to the -dependence of the kinetic coefficients of the liquid by a parametrically large factor controlled by the ratio of disorder wavelength to electron equilibration length. We hence set the off-diagonal components of the quantum critical transport to zero. We assume that and , so and . The solutions of (4) are provided in the supplementary material.

Using (5) to read off the uniform charge current, we see that are , whereas is . Hence the symmetrized electrical resistance is given by . This is in general a very complicated function, with a potentially complicated temperature dependence due to the temperature dependences of all the microscopic coefficients. However, if we assume that the disorder is very long wavelength, thus retaining only the leading contribution in the inverse disorder wavelength in the diagonal conductivity, we find a rather simple result

(6) |

which is consistent with Onsager reciprocity . All corrections from the microscopic incoherent conductivities appear at higher orders in the inverse disorder wavelength (for details see supplementary information). For the second term of (6) to be smaller than the first, so the perturbative structure is consistent, we must have , where is a characteristic wavelength of the disorder. To leading order in , one gets the symmetrized magnetoresistance at leading order in the inverse disorder wavelength

(7) |

The temperature dependence of the magnetoresistance is controlled only by the viscosity in this long-wavelength disorder limit, as was the case in Levchenko et al. (2017) for the special case of Galilean-invariant fluids (). However, there the magnetoresistance was controlled only by the viscosity regardless of the spectrum of the disorder. Since we do not expect most non-Fermi liquid metals to be Galilean-invariant, this is an important strengthening of the previous result. It can additionally be shown that the long-wavelength disorder result (7) is also insensitive to the Hall viscosity Avron et al. (1995) and vorticity susceptibility Jensen et al. (2012), which are new parity-odd microscopic transport coefficients that can appear in the presence of a magnetic field.

The above result could enable the extraction of the temperature dependence of the viscosity of the electron liquid as and thus allow for testing theoretical models of potential non-Fermi liquid states in the hydrodynamic regime. In the supplementary material we also provide results for the magnetothermal resistance.

Drag transport in bilayers. For drag type transport Narozhny and Levchenko (2016), we use our hydrodynamic equations for each layer of the bilayer system, with . Drag is generated by intrinsic hydrodynamic fluctuations encoded in fluctuating noise terms Landau and Lifshitz (1957); Kovtun (2012); Apostolov et al. (2014); Chen et al. (2015) added to that are uncorrelated between the layers (, , )

(8) | |||

with all other correlators of the sources being zero. These fluctuations induce fluctuations in the charge and entropy densities in the layers (). The fluctuations of chemical potential and temperature are expressed in terms of the charge and entropy fluctuations

(9) |

and likewise for . We must add to the pressure term in each layer the effects of intra and inter-layer Coulomb forces generated by the fluctuations in the charge densities (the layers are separated by a distance )

(10) |

The drag resistance measures the sensitivity of the electric field induced by the dragging force in the open-circuit passive layer to the current flowing in the driven layer. It is given by

(11) | |||

The derivation of these force and pressure relations only requires a straightforward application of Coulomb’s law. In additional to the noise sources, we also linearize in the velocity (the driven layer is driven by this uniform velocity field, not by an electric field). Note that is valid even for non-Galilean invariant fluids as the noise terms themselves cannot induce any uniform current flow due to averaged inversion and time-reversal symmetries. Thus must vanish when , and renormalizations of due to the noise terms are subleading.

We neglect the effects of thermal currents: they produce only subleading effects at large spatial separations (see supplementary information for further details). Switching to the basis defined by , , , the hydrodynamic equations can be reduced to the form

(12) |

where , and . The solutions to these equations are linearized in : . Since the -less configuration obeys averaged inversion and time-reversal symmetry and always appears as which is odd under inversion, is even under whereas is odd. The dragging force may be written as

(13) |

All other terms vanish upon momentum/frequency integration due to even/odd cancellations. Inserting the solutions of (12), we obtain , where is generated by the charge fluctuations and is generated by the viscous fluctuations :

(14) |

This yields a complicated integral expression for . We can however make simplifications in the regimes of “large” and “small” . The model of Fermi surface coupled to gauge field has roughly the following properties Halperin et al. (1993); Eberlein et al. (2016, 2017) for dynamical critical exponent ( is the effective fermion mass), corresponding to the case of short-ranged interactions of composite fermions Halperin et al. (1993):

(15) |

is said to be “large” when . This gives

(16) |

We have set the electrostatic permittivity so far in the paper but restored it in the last equation. We also demand , which is trivially achieved as is typically a very small distance scale ( m for ).

For we obtain the leading contributions

(17) |

and have the same temperature scaling up to logarithms. However, falls off faster with than . This results should be contrasted to that obtained earlier for Fermi liquids Apostolov et al. (2014). Note that even though the power dependence on temperature is , there is a correction, which will make the temperature dependence appear faster than but slower than , which is consistent with the data of Refs. Lilly et al. (1998); Jorger et al. (2000) at large separations.

At small separations , all contributions to scale as (see further details in supplementary materials). This is again consistent with Jorger et al. (2000), which shows an apparent crossover from positive to negative curvature in as a function of as is increased. In Fig. 1 we show obtained by numerically evaluating the integrals without the above approximations that confirm the qualitative behaviors we discussed. It should be carefully noted that in Fig. 1 the line corresponding to nm appears superficially above the line of nm plot which is due to the choice of the normalization factor . Drag is obviously a decaying function of inter-layer separation as is clear from (17).

Discussion. The most extensively studied example of transresistance in the case of non-Fermi liquids corresponds to inter-layer frictional Coulomb drag between bilayers of half-filled Landau levels Jorger et al. (2000); Lilly et al. (1998); Zelakiewicz et al. (2000); Price et al. (2010). The theoretical approach that has proved most useful for understanding the filling fraction state is the fermion Chern-Simons field theory, which is based in turn on the composite-fermion picture Halperin et al. (1993). Previous calculations Ussishkin and Stern (1997); Sakhi (1997); Kim and Millis (1999) showed that the dominant low-temperature behavior for scales with temperature as (see supplementary material for a brief summary of this result). This unique power exponent can be traced back to a special momentum dependence of the electronic longitudinal conductivity, as can be deduced from surface acoustic wave measurements. Indeed, in the composite-fermion picture, at , the density response at small frequencies and small wave-vectors is of the form , which can be viewed as slow diffusion with an effective diffusion constant that vanishes linearly with (where is the thermodynamic compressibility of the state). Since the typical frequency is set by temperature , the pole structure of long-wavelength density fluctuations sets a characteristic scale for momentum transfer between the layers that then carriers over to drag resistance . This should be contrasted the Fermi liquid prediction at lowest temperatures, and our prediction . In our current understanding, the results of Ussishkin and Stern (1997) correspond to the “collisionless” regime of transport with respect to intra-layer collisions, namely a long equilibration length as mediated by interactions with the gauge field. We considered the opposite collision-dominated regime where this length scale is assumed to be short. This should explain the difference between the power exponents and between two limiting cases. We hope that understanding different transport regimes and corresponding temperature dependencies will be of help for the interpretation of future experiments, as it also deepens our current understanding of the existing transport data and corresponding theories.

Acknowledgements. We thank A. Andreev, L. V. Delacretaz, B. Halperin, P. Kim, A. Lucas, and S. Sachdev for helpful discussions. A.A.P. acknowledges support by NSF Grant DMR-1360789. R.A.D. is supported by the Gordon and Betty Moore Foundation Grant GBMF-4306. The work of A.L. was financially supported in part by NSF Grant DMR-1653661, and by the Office of the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin Alumni Research Foundation.

## References

- Gurzhi (1968) R. N. Gurzhi, Usp. Fiz. Nauk 94, 689 (1968), URL http://ufn.ru/ru/articles/1968/4/d/.
- Yu et al. (1984) Z. Z. Yu, M. Haerle, J. W. Zwart, J. Bass, W. P. Pratt, and P. A. Schroeder, Phys. Rev. Lett. 52, 368 (1984), URL https://link.aps.org/doi/10.1103/PhysRevLett.52.368.
- de Jong and Molenkamp (1995) M. J. M. de Jong and L. W. Molenkamp, Phys. Rev. B 51, 13389 (1995), URL https://link.aps.org/doi/10.1103/PhysRevB.51.13389.
- Bandurin et al. (2016) D. A. Bandurin, I. Torre, R. K. Kumar, M. Ben Shalom, A. Tomadin, A. Principi, G. H. Auton, E. Khestanova, K. S. Novoselov, I. V. Grigorieva, et al., Science 351, 1055 (2016), ISSN 0036-8075, URL http://science.sciencemag.org/content/351/6277/1055.
- Crossno et al. (2016) J. Crossno, J. K. Shi, K. Wang, X. Liu, A. Harzheim, A. Lucas, S. Sachdev, P. Kim, T. Taniguchi, K. Watanabe, et al., Science 351, 1058 (2016), ISSN 0036-8075, URL http://science.sciencemag.org/content/351/6277/1058.
- Müller and Sachdev (2008) M. Müller and S. Sachdev, Phys. Rev. B 78, 115419 (2008), URL https://link.aps.org/doi/10.1103/PhysRevB.78.115419.
- Müller et al. (2009) M. Müller, J. Schmalian, and L. Fritz, Phys. Rev. Lett. 103, 025301 (2009), URL https://link.aps.org/doi/10.1103/PhysRevLett.103.025301.
- Narozhny et al. (2015) B. N. Narozhny, I. V. Gornyi, M. Titov, M. Schütt, and A. D. Mirlin, Phys. Rev. B 91, 035414 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.91.035414.
- Principi et al. (2016) A. Principi, G. Vignale, M. Carrega, and M. Polini, Phys. Rev. B 93, 125410 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.93.125410.
- Lucas et al. (2016a) A. Lucas, J. Crossno, K. C. Fong, P. Kim, and S. Sachdev, Phys. Rev. B 93, 075426 (2016a), URL https://link.aps.org/doi/10.1103/PhysRevB.93.075426.
- Spivak et al. (2010) B. Spivak, S. V. Kravchenko, S. A. Kivelson, and X. P. A. Gao, Rev. Mod. Phys. 82, 1743 (2010), URL https://link.aps.org/doi/10.1103/RevModPhys.82.1743.
- Andreev et al. (2011) A. V. Andreev, S. A. Kivelson, and B. Spivak, Phys. Rev. Lett. 106, 256804 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.106.256804.
- Hartnoll et al. (2007) S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76, 144502 (2007), URL https://link.aps.org/doi/10.1103/PhysRevB.76.144502.
- Foster and Aleiner (2009) M. S. Foster and I. L. Aleiner, Phys. Rev. B 79, 085415 (2009), URL https://link.aps.org/doi/10.1103/PhysRevB.79.085415.
- Lucas and Sachdev (2015) A. Lucas and S. Sachdev, Phys. Rev. B91, 195122 (2015), eprint 1502.04704, URL http://dx.doi.org/10.1103/PhysRevB.91.195122.
- Lucas (2015) A. Lucas, New Journal of Physics 17, 113007 (2015), URL http://stacks.iop.org/1367-2630/17/i=11/a=113007.
- Hartnoll et al. (2016) S. A. Hartnoll, A. Lucas, and S. Sachdev, arXiv preprint arXiv:1612.07324 (2016), URL https://arxiv.org/abs/1612.07324.
- Davison et al. (2015) R. A. Davison, B. Gouteraux, and S. A. Hartnoll, JHEP 10, 112 (2015), eprint 1507.07137, URL https://link.springer.com/article/10.1007/JHEP10(2015)112.
- Eberlein et al. (2016) A. Eberlein, I. Mandal, and S. Sachdev, Phys. Rev. B 94, 045133 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.94.045133.
- Eberlein et al. (2017) A. Eberlein, A. A. Patel, and S. Sachdev, Phys. Rev. B 95, 075127 (2017), URL https://link.aps.org/doi/10.1103/PhysRevB.95.075127.
- Patel et al. (2015) A. A. Patel, P. Strack, and S. Sachdev, Phys. Rev. B 92, 165105 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.165105.
- Lucas et al. (2016b) A. Lucas, R. A. Davison, and S. Sachdev, Proc. Nat. Acad. Sci. 113, 9463 (2016b), eprint 1604.08598, URL http://www.pnas.org/content/113/34/9463.full.
- Grozdanov et al. (2017) S. Grozdanov, D. M. Hofman, and N. Iqbal, Phys. Rev. D95, 096003 (2017), eprint 1610.07392, URL http://dx.doi.org/10.1103/PhysRevD.95.096003.
- Hernandez and Kovtun (2017) J. Hernandez and P. Kovtun, JHEP 05, 001 (2017), eprint 1703.08757, URL http://dx.doi.org/10.1007/JHEP05(2017)001.
- Baumgartner et al. (2017) A. Baumgartner, A. Karch, and A. Lucas (2017), eprint 1704.01592, URL https://arxiv.org/abs/1704.01592.
- Levchenko et al. (2017) A. Levchenko, H.-Y. Xie, and A. V. Andreev, Phys. Rev. B 95, 121301 (2017), URL https://link.aps.org/doi/10.1103/PhysRevB.95.121301.
- Avron et al. (1995) J. E. Avron, R. Seiler, and P. G. Zograf, Phys. Rev. Lett. 75, 697 (1995), URL https://link.aps.org/doi/10.1103/PhysRevLett.75.697.
- Jensen et al. (2012) K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz, and A. Yarom, Journal of High Energy Physics 2012, 102 (2012), ISSN 1029-8479, URL https://doi.org/10.1007/JHEP05(2012)102.
- Narozhny and Levchenko (2016) B. N. Narozhny and A. Levchenko, Rev. Mod. Phys. 88, 025003 (2016), URL https://link.aps.org/doi/10.1103/RevModPhys.88.025003.
- Landau and Lifshitz (1957) L. Landau and E. Lifshitz, SOVIET PHYSICS JETP-USSR 5, 512 (1957), URL http://www.jetp.ac.ru/cgi-bin/e/index/e/5/3/p512?a=list.
- Kovtun (2012) P. Kovtun, Journal of Physics A: Mathematical and Theoretical 45, 473001 (2012), URL http://stacks.iop.org/1751-8121/45/i=47/a=473001.
- Apostolov et al. (2014) S. S. Apostolov, A. Levchenko, and A. V. Andreev, Phys. Rev. B 89, 121104 (2014), URL https://link.aps.org/doi/10.1103/PhysRevB.89.121104.
- Chen et al. (2015) W. Chen, A. V. Andreev, and A. Levchenko, Phys. Rev. B 91, 245405 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.91.245405.
- Halperin et al. (1993) B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312 (1993), URL https://link.aps.org/doi/10.1103/PhysRevB.47.7312.
- Lilly et al. (1998) M. P. Lilly, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 80, 1714 (1998), URL https://link.aps.org/doi/10.1103/PhysRevLett.80.1714.
- Jorger et al. (2000) C. Jorger, W. Dietsche, W. Wegscheider, and K. von Klitzing, Physica E: Low-dimensional Systems and Nanostructures 6, 586 (2000), ISSN 1386-9477, URL http://www.sciencedirect.com/science/article/pii/S1386947799001204.
- Zelakiewicz et al. (2000) S. Zelakiewicz, H. Noh, T. J. Gramila, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 85, 1942 (2000), URL https://link.aps.org/doi/10.1103/PhysRevLett.85.1942.
- Price et al. (2010) A. S. Price, A. K. Savchenko, and D. A. Ritchie, Phys. Rev. B 81, 193303 (2010), URL https://link.aps.org/doi/10.1103/PhysRevB.81.193303.
- Ussishkin and Stern (1997) I. Ussishkin and A. Stern, Phys. Rev. B 56, 4013 (1997), URL https://link.aps.org/doi/10.1103/PhysRevB.56.4013.
- Sakhi (1997) S. Sakhi, Phys. Rev. B 56, 4098 (1997), URL https://link.aps.org/doi/10.1103/PhysRevB.56.4098.
- Kim and Millis (1999) Y. B. Kim and A. J. Millis, Physica E: Low-dimensional Systems and Nanostructures 4, 171 (1999), URL http://www.sciencedirect.com/science/article/pii/S1386947799000132.
- Nave and Lee (2007) C. P. Nave and P. A. Lee, Phys. Rev. B 76, 235124 (2007), URL https://link.aps.org/doi/10.1103/PhysRevB.76.235124.

## Appendix A Supplementary information

## Appendix B Solution to linearized disordered magneto-hydrodynamic equations

At we get the following equations in momentum space from (4) of the main text

(18) | |||

(19) | |||

(20) | |||

(21) |

The zero-momentum limit of the equations (21) implies that and have only finite momentum components, since the periodic boundary conditions ensure that . also has only finite momentum components and a solution to doesn’t exist. Hence we need to look at to obtain nontrivial uniform conductivities.

Using the first equation of (21) to read off , we obtain

(22) |

Note that it is impossible to set the right hand side of this equation to zero in the limit of vanishing shear viscosity . Thus, this perturbative expansion is valid only for finite background shear viscosities.

Using (5) of the main text, the expression for the uniform electrical conductivity retaining only the leading and next-to-leading contributions in the inverse disorder wavelength in the diagonal conductivity is given by

(26) |

Note that the leading disorder-induced contribution depends only upon the shear viscosity , and that all corrections coming from the microscopic incoherent conductivities occur at higher orders in the inverse disorder wavelength.

In the presence of a magnetic field, the stress tensor can contain the effects of new parity-odd microscopic transport coefficients. These are the Hall viscosity Avron et al. (1995) and the vorticity susceptibility Jensen et al. (2012), which are both proportional to . The stress tensor is modified to

(27) |

At , this becomes

(28) |

Then, repeating our solution, we find that the long-wavelength disorder result for the magnetoresistance given by (7) of the main text is unaffected by these terms.

## Appendix C Magneto-thermal transport in the clean system

To obtain the thermal resistance,

(29) |

we apply a temperature gradient , an electric field to block electric currents, and solve the hydrodynamic equations in linear response. The clean solution is

(30) | |||

(31) |

This choice gives but