Hydrodynamics in solid state transport, from microscopic to mesoscopic scales

The thesis is devoted to some aspects of the solid-state electronic transport in the so-called viscous or hydrodynamic regime. Hydrodynamic regime in this context means that due to the large carrier density and non-negligible carrier-carrier interactions, the transport properties follow from collective, rather than single-particle phenomena. To capture the dynamics of such a system one may use description based on the conserved quantities, i.e. momentum, energy or charge. If the interactions between the constituents of the system are strong enough, such a description is provided by the hydrodynamic equations which for conserved momentum and energy are the Navier-Stokes equations or their relativistic counterparts. This thesis focuses on such a situation: when the equations governing transport properties follow from conservation of the momentum or, at most, can be treated as a modification of such equations due to weak momentum relaxation. Presented here are two lines of investigation. The first one focuses on the mesoscopic effects, i.e. on the dependence of the outcome of the transport measurements on the physical parameters of the sample such as size and shape. Here also the effects of the weak momentum relaxation are studied. In the second one, the issue of parity and time reversal symmetry breaking, occurring in a 2 dimensional system due to the presence of an external magnetic field, is investigated. An effective model of a strongly coupled quantum system is introduced and used to compute the odd (Hall) viscosity -- a transport coefficient allowed once the discrete symmetries are broken -- as a function of magnetic field, temperature and chemical potential. The first part of results concerns the behaviour of the electronic fluid in a typical AC measurement -- modeled by an elongated channel in which the fluid is subject to a periodically time dependent electric potential. Assuming standard, no-slip boundary conditions, the spatial distribution of the current density is found to be much different to the one known for Ohmic conduction. For small frequency the current distribution has a parabolic profile across the channel, while for high frequency the current in the bulk of the channel becomes flat (position-independent), while two maxima terminating a so-called boundary layer develop. In these boundary layers large gradients of current can be found, contributing to high local entropy production due to the viscous force. Despite this differences in the local current density profile, when the global conductance is measured as a function of the frequency, the result much resembles the well known Drude curve, with a distinct maximum visible in the imaginary part of the AC conductance. There is, however, a global signature of the boundary layer formation -- the scaling of the conductance with the channel width, that changes from quadratic (for parabolic flow) to asymptotically constant (for a flow with boundary layers). Moreover, in the hydrodynamic regime, the position of the Drude peak is not only determined by microscopic parameter but again by a combination of microscopic (viscosity) and mesoscopic (width) parameters. Since the Drude peak occurs for experimentally feasible values of parameters, the mentioned mesoscopic dependence may be used to measure the value of viscosity coefficient. The results discussed above are obtained assuming, as is traditional for hydrodynamics on everyday length-scales, a no-slip boundary condition which forces the fluid to be immobile at the boundary. This boundary condition was also assumed in most of the previous works on the electronic hydrodynamics. However, this is not the only possibility. There exists a one-parameter family of consistent boundary conditions involving velocity and its derivative on the boundary, parametrized by a coefficient called the slip length. Recent theoretical and experimental publications suggest that it may be dependent on the state parameters of the system (i.e temperature, chemical potential) and its value may be relatively large for some experimental situations. One of the consequences of the slip length being large is that hydrodynamic effects are obscured in the simple AC set-up discussed before. In this work it is shown that by an appropriate micro-structuring of the boundary, the effects of slip can be suppressed. Once the array of defects is introduced on the edges of the sample, the no-slip behavior is restored for all the values of the microscopic slip length. Furthermore, the interplay between the microscopic slip length and the sample geometry is investigated and used to propose a simple device for measuring the dependence of the microscopic slip length on the state parameters such as the temperature or the chemical potential. The final part of this thesis is devoted to a different aspect of the hydrodynamic transport -- a computation of the value of hydrodynamic transport coefficients using a microscopic theory. The physical situation of interest is one in which time reversal and parity invariance of a 2-dimensional system are broken, due to the presence of an external magnetic field. In such a situation an unusual class of transport coefficients is allowed in the hydrodynamic description, so-called odd coefficients. The term comes from the fact that they encode response that is transverse to the applied perturbation. These odd coefficients for 2 dimensions were previously studied mostly at weak coupling, i.e. using descriptions based on quasi-particles. This work, however, presents the way of calculating them for strongly coupled model system. To achieve this a high-energy-physics-inspired framework of holographic duality (AdS/CFT) is used. In that approach, an effective model involving magnetically-sourced parity-breaking interactions is constructed for the system at finite temperature and chemical potential. Performing a linear response analysis around the thermal states in that model allows one to read off the transport coefficients, especially the odd (Hall) viscosity coefficient that is of central interest in this study. The mentioned Hall viscosity is found to be non-zero whenever the magnetic field is present, even for zero chemical potential. This is unusual, as odd viscosity is expected to only be non-zero for non-zero charge density states. The mechanism responsible for the presence of Hall viscosity in the discussed case turns out to be the following: charge density in the model is induced by either the chemical potential or the magnetic field, i.e. for non-zero magnetic field even at zero chemical potential some density of charge is present. This charge contributes to the Hall viscosity in the usual way. The odd viscosity coefficient is found to have different scaling behaviors for weak and strong magnetic field. Interestingly, it turns out that the computations of the Hall (and shear) viscosities are relatively straightforward and analytically tractable in the proposed model. This means that the results could be generalized to the zero-temperature case, which however is yet to be done. It also suggests that the model may capture some universal mechanisms of generating the odd viscosity due to the presence of the magnetic field. That intuition is backed by the fact that some of the effective models of quantum Hall states also predict similar mechanism in which charge density is induced by the presence of the magnetic field. Despite these similarities, further studies are needed to establish a solid connection between these systems. In particular, in the model under consideration no mechanism of quantization of the Hall viscosity is found, while the mentioned models of quantum Hall states predict quantization of that transport coefficient