# Dissipative and Hall viscosity of a disordered 2D electron gas

###### Abstract

We study the dissipative and Hall viscosity of a disordered noninteracting 2D electrons, both analytically and numerically. Analytically, we employ the self-consistent Born approximation, explicitly taking into account the modification of the single-particle density of states and the elastic transport time due to the Landau quantization. The reported results interpolate smoothly between the limiting cases of weak (strong) magnetic field and strong (weak) disorder. In the regime of weak magnetic field, our results describes the quantum (de Haas-van Alphen type) oscillations of the dissipative and Hall viscosity. For strong magnetic field, we computed the dependence of the dissipative and Hall viscosity on disorder broadening of a Landau level. In particular, for the Hall viscosity the effect of the disorder broadening is weak. This theoretical conclusion is in agreement with our numerical results for a few lowest Landau levels, which show that Hall viscosity is robust to disorder.

Introduction. — The notion of viscosity is among cornerstones of hydrodynamics, in which it is responsible for dissipation. Under certain conditions charge transport in an electron system can be dominated by hydrodynamic viscous electron flow Gurzhi (); Molenkamp (). Appearance of graphene stimulated theoretical Muller2009 (); Andreev2011 (); Polini2015 (); Levitov2016 (); NGMS (); Levitov2017 (); Kashuba (); Lucas2018 (); Schmalian () and experimental Titov2013 (); Polini2016 (); Kim2016 (); Moll2016 (); Geim2017 (); Levitov2018 (); Kvon2018 () interest in the hydrodynamic description of charge transport.

In absence of the time-reversal symmetry the viscosity tensor has non-dissipative antisymmetric components. In the presence of a magnetic field , this non-dissipative Hall viscosity () was studied theoretically in the classical limit of high temperature plasmas CC (); Marshall (); Kaufman (); Thompson (); Braginskii () and for low temperature electron gas Steinberg (). Later, interest in the Hall viscosity was renewed in quantum systems with a gapped spectrum, due to connection of to the geometric response Avron (); Levay (); Avron1998 (); Read (); RR (); Haldane (); HLF (), and its expected quantization in the presence of translational and rotational symmetries RR (). It was understood that in addition to the Hall conductivity and Hall viscosity there are additional non-dissipative physical quantities which determines the combined electro-magnetic and geometrical responses of gapped quantum systems HS2012 (); Abanov2013 (); AG2014 (); GA2014 (); Hoyos (); Andreev (); Wiegmann2014 (); GA2015 (); Wiegmann2015 (); Gurarie (); Andreev2015 (); NG2017 (); NCG2017 (). Within the hydrodynamic description of electron transport non-zero Hall viscosity influences significantly the structure of an electron flow Alekseev (); PTP (); SNSMM (); DG2017 (); GA2017 (); Alekseev-2 () that allows one to access experimentally Bandurin2018 (). Also, it was argued that the dissipative and Hall viscosity affect the spectrum of edge magnetoplasmons ACG2018 (); SDVV (); CG ().

For noninteracting electrons in the absence of disorder each filled Landau level (LL) gives the contribution to the Hall viscosity equal Avron (), where denotes the LL index and stands for the magnetic length. As mentioned above, this result is stable against perturbations of the Hamiltonian which preserve translational and rotational invariance RR (). However, the fate of this result in the presence of disorder has not been studied yet. Therefore, it is not clear how the clean result obtained within the quantum treatment of an electron motion in a magnetic field connects to the result derived for a classical disordered electron gas Steinberg (). Here denotes the chemical potential, the second transport time, the cyclotron frequency, the density of states at , and the effective electron mass.

In this Letter we report the results of an analytical and numerical study of the dissipative and Hall viscosity of noninteracting 2D electrons in the presence of disorder. Contrary to previous studies we explicitly take into account the Landau quantization of the electron spectrum. Analytically, within the self-consistent Born approximation (SCBA) AFS () we derive expressions for the dissipative and Hall viscosities, which smoothly interpolates between the results known in the literature for classical magnetic field CC (); Marshall (); Kaufman (); Thompson (); Braginskii (); Alekseev () and for the strong magnetic field in the absence of disorder Avron (). Since the SCBA is rigorously justified for high LLs only, we perform numerical calculation of the Hall viscosity for a few lowest LLs. The obtained numerical results are in a perfect agreement with the expectations based on our theoretical findings within SCBA.

Model. — Noninteracting electrons confined to a 2D plane are described by the following single-particle Hamiltonian

(1) |

where stands for a random potential and for the vector potential corresponding to the static perpendicular magnetic field . In this paper we use the Landau gauge: and . We assume that the random potential has Gaussian distribution with a pair correlation function which decays with a typical length scale .

Kubo formula for the viscosity. — The viscosity tensor can be computed by means of the Kubo formula Resibois (); McLennan (); Bradlyn2012 ():

(2) |

Here denotes the Fermi distribution function, the retarded Green’s function, the stress tensor, the system area, and the internal compressibility Footnote1 (). Averaging over the random potential is denoted by an overbar.

Self-consistent Born approximation. — In order to compute the viscosity tensor from Eq. (2) we treat the disorder scattering using the SCBA AFS (). This approximation holds under the following conditions RS (); LA ():

(3) |

Here stands for the magnetic length, while and denote the Fermi momentum and velocity, respectively. is the total elastic relaxation time at zero magnetic field. It can be expressed in terms of the Fourier transform of the pair correlation function . Furthermore, it is convenient to generalize it ro

(4) |

where , and denotes the density of states at .

The average density of states at non-zero is determined by the average retarded Green’s function . In the LL representation the average density of states is given as , where the retarded Green’s function satisfies the SCBA equation () AFS (); RS (); LA ()

(5) |

There are two limiting cases in which the self-consistent Eq. (5) can be easily solved AFS (). In the regime of overlapping LLs, , one can use the Poisson formula for summation over LL index. Then the averaged density of states becomes . Here is the Dingle parameter. In the opposite case, when the LLs are well separated, one can restrict the summation in Eq. (5) to the single LL which is closest to the energy of interest, , where . Then the average density of states acquires the semi-circle profile: , where determines the the broadened LL width.

In the presence of long-range disorder correlations, , it is important to take into account the vertex corrections to the “bubble” contribution in the Kubo formula (2) (see Fig. 1). This implies that in addition to the averaged Green’s function one needs also to know the renormalized vertex, which is the stress tensor in the case of the viscosity. Within the SCBA can be approximated as a linear combination of operators which change the LL index by . Under conditions (3) one can show that an operator , which transfers an electron from the -th LL to the -th LL, is renormalized by the ladder resummation of the disorder lines as follows RS (); LA (); DMP () (see the Supplemental Material for details SM ()):

(6) |

Here is the contribution of the bubble without ladder insertions. Using Eq. (5) it can be rewritten as . Therefore, within the SCBA the vertex corrections are expressed in terms of the average density of states only.

Dissipative viscosity. — Disorder averaging restores 2D rotational symmetry. Hence, the viscosity tensor is characterized by only three parameters:

(7) |

where and denotes the bulk and shear viscosities, respectively. Within the SCBA the bulk viscosity vanishes, . Using Eqs. (5) and (6), we find the following result for the shear viscosity at SM ():

(8) |

where is the renormalized second transport time and is the second transport rate at . We note that for the second transport time becomes . We mention that Eq. (8) is analogous to the result for the dissipative conductivity DMP ().

In the regime of overlapping LLs, , the shear viscosity exhibits Shubnikov-de Haas-type oscillations:

(9) |

where and . The non-oscillatory term in reproduces the classical result for the shear viscosity of an electron gas Steinberg ().

In the regime of well separated LLs, , one finds from Eq. (8) that the shear viscosity is non-zero when the chemical potential is inside the -th broadened Landau level ():

(10) |

For chemical potential at the center of the LL, the shear viscosity is times larger when one naively expects on the basis of purely classical expression. The dependence of the shear viscosity on the chemical potential in comparison with the density of states is shown in Fig. 2.

Hall viscosity. — The Hall viscosity can be extracted from the viscosity tensor as . Similar to the Hall conductance, the evaluation of from the Kubo formula (2) is complicated due to contributions which come from all the states below the chemical potential. Therefore, it is convenient to proceed in a way pioneered by Smrčka and Středa SmSt (): Similarly to the Hall conductivity we split the Hall viscosity at into two parts, , where SM ()

(11) | ||||

(12) |

Here denotes the stress generators which are related with the stress tensor as Bradlyn2012 (). One can evaluate in a similar way to SM ():

(13) |

The evaluation of is more involved. Although one can write down the viscoelastic analog of the Smrčka and Středa formula for the Hall viscosity VE (), it does not provided a suitable way for evaluation of in the presence of disorder. In order to compute one needs to know the expressions for the stress generators. In the absence of disorder they can be easily written down explicitly Bradlyn2012 (), e.g., and . In the presence of a random potential the stress generators can be constructed as a series in spatial derivatives of a random potential SM (). This allows us to evaluate within the SCBA SM ():

(14) |

where stands for the energy density. Combining Eqs. (13) and (14), we obtain

(15) |

In the absence of disorder and for the chemical potential above the -th Landau level the energy density at can be computed as , which yields the known result Avron (). Also, we mention that in the Boltzmann limit, , the energy density is given by , where denotes the particle density, such that the Hall viscosity in the absence of disorder and at becomes , in agreement with Eq. (59.38) of Ref. LL10 () in which the Hall viscosity is denoted by . We note that the structure of Eq. (15) resembles the structure of the result for the Hall conductivity DMPZ ().

The appearance of the non-zero can be explained on a pure classical level Kaufman (). The Hall viscosity describes the response of the on a shear velocity profile . In the presence of a magnetic field this velocity can be considered as the result of a non-uniform electric field, . This electric field results not only in a drift of the cyclotron orbit but in its deformation into an ellipse. To linear order in the ratio between ellipse axes is equal to . This asymmetry between motion in the and direction yields the non-zero ratio in the limit . Hence, non-zero Hall viscosity arises, which is given by the first term in Eq. (15). We note that an electron moving along an ellipse conserves its energy to the first order in , in agreement with non-dissipative nature of . In the presence of impurity scattering an electron experiences a friction force corresponding to the electric field . This electric field leads to a velocity component in the direction, . This non-uniform velocity produces additional correction to the difference, . Thus there is an additional correction to the Hall viscosity which corresponds the second term in Eq. (15) in the classical regime.

In the case of overlapping LLs, , from Eq.(15) we obtain the Shubnikov-de Haas oscillations of the Hall viscosity:

(16) |

The non-oscillatory term in coincides with the classical result for the Hall viscosity of electron gas Steinberg ().

In the case of well-separated LLs, , one finds from Eq. (15) that the Hall viscosity is reduced from the quantized value if the chemical potential lies within the broadened LL, :

(17) |

We note that for the long-range-correlated random potential the Hall viscosity dominates the shear viscosity, (cf. Eqs. (10) and (17)).

The deviation of the Hall viscosity from the clean value is controlled by the small parameter . In the case of short range random potential correlations, , the deviation of from its clean value is very small. For the long-range-correlated random potential, , the difference is additionally suppressed (see Fig. 3).

Numerical results.– We would now like to explore the quantum Hall regime, where the number of filled LLs is of order unity. Here the SCBA cannot be used anymore, and we resort to a numerical calculation. For this we discretize the system and employ the Hofstadter model with uncorrelated random potential, uniformly distributed between at each lattice site. We calculate the Hall viscosity using retarded correlation function of discretized stress operators TuegelHueghes (), and take both the continuum and thermodynamic limits to extrapolate to the behavior of our model (1) SM (). In the presence of disorder we can take these limits while keeping constant . The results for the Hall viscosity are plotted in Fig. 4, together with the behavior of the Hall conductivity () at zero wavevector. One sees that, somewhat surprisingly, the Hall viscosity maintains its quantization to the same extent as the Hall conductivity.

Conclusions. — To summarize, we studied the dissipative and Hall viscosity of 2D electron system in the presence of a random potential. Within the self-consistent Born approximation we derived an expressions for both the dissipative and Hall viscosities, which takes into account the modification of the single-particle density of states and the elastic transport time due to the Landau quantization. Our results smoothly interpolate between the case of weak magnetic field and strong disorder, on the one hand, and the case of strong magnetic field and weak disorder, on the other hand. In the former regime, we derived the expressions for the quantum (Shubnikov-de Haas type) oscillations of the dissipative and Hall viscosity. In the case of strong magnetic field, we found that the disorder broadening of the Landau level does no lead to significant change of the Hall viscosity in comparison with the clean result. Our numerical results for a few filled LLs support this conclusion.

There are various ways to extend our work. In Galilean invariant systems it was proven Hoyos (); Bradlyn2012 () that the Hall viscosity can be extracted from the Hall conductivity at finite wave-vector . This allows one to extract the Hall viscosity from the non-local conductivity. In the absence of Galilean invariance there is no reason to expect that is related with Radzihovsky (); HollerRead (). Also, the relation between and can be affected by the presence of lattice TuegelHueghes () or disorder. However, if one treats disorder on the level of Drude model with classical magnetic field the relation of Ref. Bradlyn2012 () between and still holds HKO (). This fact is not surprising since the Drude model does not properly take into account the LLs, which result in the energy dependence of the density of states and elastic scattering transport time. This simplification can be dangerous since and have contributions coming from the states well below the Fermi energy. It would therefore be worthwhile to extend the presented analytical and numerical approaches to the conductivity at finite wave vector elsewhere (). Also we mention that our techniques can be applied to calculation of the dissipative and Hall viscosity in graphene, where only the result in the absence of disorder in known SPV ().

Acknowledgements. — We thank A. Abanov, O. Andreev, I. Gornyi, A. Gromov, A. Mirlin, P. Ostrovsky, D. Polyakov, and P. Wiegmann for useful discussions. Hospitality by Tel Aviv University, the Weizmann Institute of Science, the Landau Institute for Theoretical Physics, and the Karlsruhe Institute of Technology is gratefully acknowledged. The work was partially supported by the Russian Foundation for Basic Research under Grant No. 17-02-00541, the program “Contemporary problems of low-temperature physics” of Russian Academy of Science, the Alexander von Humboldt Foundation, the Israel Ministry of Science and Technology (Contract No. 3-12419), the Israel Science Foundation (Grant No. 227/15), the German Israeli Foundation (Grant No. I-1259-303.10), the US-Israel Binational Science Foundation (Grant No. 2016224), and a travel grant by the BASIS Foundation.

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