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We revisit the theory of dissipative mechanics in RLC circuits, allowing for circuit elements to have nonlinear constitutive relations, and for the circuit to have arbitrary topology. We systematically generalize the dissipationless Hamiltonian mechanics of an LC circuit to account for resistors and incorporate the physical postulate that the resulting RLC circuit thermalizes with its environment at a constant positive temperature. Our theory explains stochastic fluctuations, or Johnson noise, which are mandated by the fluctuation-dissipation theorem. Assuming Gaussian Markovian noise, we obtain exact expressions for multiplicative Johnson noise through nonlinear resistors in circuits with convenient (parasitic) capacitors and/or inductors. With linear resistors, our formalism is describable using a Kubo-Martin-Schwinger-invariant Lagrangian formalism for dissipative thermal systems. Generalizing our technique to quantum circuits could lead to an alternative way to study decoherence in nonlinear superconducting circuits without the Caldeira-Leggett formalism., Comment: 30 pages, 6 figures

The quantum mechanics of superconducting circuits is derived by starting from a classical Hamiltonian dynamical system describing a dissipationless circuit, usually made of capacitive and inductive elements. However, standard approaches to circuit quantization treat fluxes and charges, which end up as the canonically conjugate degrees of freedom on phase space, asymmetrically. By combining intuition from topological graph theory with a recent symplectic geometry approach to circuit quantization, we present a theory of circuit quantization that treats charges and fluxes on a manifestly symmetric footing. For planar circuits, known circuit dualities are a natural canonical transformation on the classical phase space. We discuss the extent to which such circuit dualities generalize to non-planar circuits., Comment: 29 pages, 6 figures

Circuit quantization is an extraordinarily successful theory that describes the behavior of quantum circuits with high precision. The most widely used approach of circuit quantization relies on introducing a classical Lagrangian whose degrees of freedom are either magnetic fluxes or electric charges in the circuit. By combining nonlinear circuit elements (such as Josephson junctions or quantum phase slips), it is possible to build circuits where a standard Lagrangian description (and thus the standard quantization method) does not exist. Inspired by the mathematics of symplectic geometry and graph theory, we address this challenge, and present a Hamiltonian formulation of non-dissipative electrodynamic circuits. The resulting procedure for circuit quantization is independent of whether circuit elements are linear or nonlinear, or if the circuit is driven by external biases. We explain how to re-derive known results from our formalism, and provide an efficient algorithm for quantizing circuits, including those that cannot be quantized using existing methods., Comment: 30 pages, 8 figures

Spatially resolved transport in two-dimensional quantum materials can reveal dynamics which is invisible in conventional bulk transport measurements. We predict striking patterns in spatially inhomogeneous transport just above the critical temperature in two-dimensional superconducting thin films, where electrical current will appear to flow as if it were a viscous fluid obeying the Navier-Stokes equations. Compared to viscous electron fluids in ultrapure metals such as graphene, this analogue viscous vortex fluid can exhibit a far more tunable crossover, as a function of temperature, from Ohmic to non-Ohmic transport, with the latter arising on increasingly large length scales close to the critical temperature. Experiments using nitrogen vacancy center magnetometry, or transport through patterned thin films, could reveal this analogue viscous flow in a wide variety of materials., Comment: 5+4 pages; 2+0 figures

When continuous rotational invariance of a two-dimensional fluid is broken to the discrete, dihedral subgroup $D_6$ - the point group of an equilateral triangle - the resulting anisotropic hydrodynamics breaks both spatial-inversion and time-reversal symmetries, while preserving their combination. In this work, we present the hydrodynamics of such $D_6$ fluids, identifying new symmetry-allowed dissipative terms in the hydrodynamic equations of motion. We propose two experiments - both involving high-purity solid-state materials with $D_6$-invariant Fermi surfaces - that are sensitive to these new coefficients in a $D_6$ fluid of electrons. In particular, we propose a local current imaging experiment (which is present-day realizable with nitrogen vacancy center magnetometry) in a hexagonal device, whose $D_6$-exploiting boundary conditions enable the unambiguous detection of these novel transport coefficients., Comment: 25+12 pages, 7+0 figures, 2+0 tables. v2: fixed typos. v3: revised version

We present the nonlinear fluctuating hydrodynamics which governs the late time dynamics of a chaotic many-body system with simultaneous charge/mass, dipole/center of mass, and momentum conservation. This hydrodynamic effective theory is unstable below four spatial dimensions: dipole-conserving fluids at rest become unstable to fluctuations, and are governed not by hydrodynamics, but by a fractonic generalization of the Kardar-Parisi-Zhang universality class. We numerically simulate many-body classical dynamics in one-dimensional models with dipole and momentum conservation, and find evidence for a breakdown of hydrodynamics, along with a new universality class of undriven yet non-equilbrium dynamics., Comment: 6+5 pages, 4 figures

When electrons flow as a viscous fluid in anisotropic metals, the reduced symmetry can lead to exotic viscosity tensors with many additional, nonstandard components. We present a viscometry technique that can, in principle, measure the multiple dissipative viscosities allowed in isotropic and anisotropic fluids alike. By applying representation theory to exploit the intrinsic symmetry of the fluid, our viscometry is also exceptionally robust to both boundary complications and ballistic effects. We present the technique via the illustrative example of dihedral symmetry, relevant in this context as the point symmetry of 2D crystals. Finally, we propose a present-day realizable experiment for detecting, in a metal, a novel hydrodynamic phenomenon: the presence of rotational dissipation in an otherwise isotropic fluid.

We develop the theory of hydrodynamics of an isotropic Fermi liquid of electrons coupled to isotropic acoustic phonons, assuming that umklapp processes may be neglected. At low temperatures, the fluid is approximately Galilean invariant; at high temperatures, the fluid is nearly relativistic; at intermediate temperatures, there are seven additional temperature regimes with unconventional thermodynamic properties and hydrodynamic transport coefficients in a three-dimensional system. We predict qualitative signatures of electron-phonon fluids in incoherent transport coefficients, shear and Hall viscosity, and plasmon dispersion relations. Our theory may be relevant for numerous quantum materials where strong electron-phonon scattering has been proposed to underlie a hydrodynamic regime, including $\mathrm{WTe}_2$, $\mathrm{WP}_2$, and $\mathrm{PtSn}_4$., Comment: 24+10 pages, 4 figures. v2: published version

Compressible electron flow through a narrow cavity is theoretically unstable, and the oscillations occurring during the instability have been proposed as a method of generating Terahertz radiation. We numerically demonstrate that the endpoint of this instability is a nonlinear hydrodynamic oscillator, consisting of an alternating shock wave and rarefaction-like relaxation flowing back and forth in the device. This qualitative physics is robust to cavity inhomogeneity and changes in the equation of state of the fluid. We discuss the frequency and amplitude dependence of the emitted radiation on physical parameters (viscosity, momentum relaxation rate, and bias current) beyond linear response theory, providing clear predictions for future experiments.

In sufficiently clean metals, it is possible for electrons to collectively flow as a viscous fluid at finite temperature. These viscous effects have been predicted to give a notable magnetoresistance, but whether the magnetoresistance is positive or negative has been debated. We argue that regardless of the strength of inhomogeneity, bulk magnetoresistance is always positive in the hydrodynamic regime. We also compute transport in weakly inhomogeneous metals across the ballistic-to-hydrodynamic crossover, where we also find positive magnetoresistance. The non-monotonic temperature dependence of resistivity in this regime (a bulk Gurzhi effect) rapidly disappears upon turning on any finite magnetic field, suggesting that magnetotransport is a simple test for viscous effects in bulk transport, including at the onset of the hydrodynamic regime., Comment: 14 pages, 5 figures; v2: published version

Recent experiments have observed hints of hydrodynamic electron flow in a number of materials, not all of which have an isotropic Fermi surface. We revisit these experiments in $\mathrm{PdCoO}_2$, a quasi-two-dimensional material whose Fermi surface is a rounded hexagon, and observe that the data appears quantitatively consistent with a non-hydrodynamic interpretation. Nevertheless, motivated by such experiments, we develop a simple model for the low temperature kinetics and hydrodynamics of a two-dimensional Fermi liquid with a polygonal Fermi surface. A geometric effect leads to a finite number of additional long-lived quasihydrodynamic "imbalance" modes and corresponding qualitative changes in transport at the ballistic-to-hydrodynamic crossover. In the hydrodynamic limit, we find incoherent diffusion and a new dissipative component of the viscosity tensor arising from the explicit breaking of rotational invariance by the Fermi surface. Finally, we compute the conductance of narrow channels across the ballistic-to-hydrodynamic crossover and demonstrate a modification of the Gurzhi effect that allows for non-monotonic temperature and width dependence in the channel conductance., Comment: 27 + 11 pages (main text + appendices/references); 8+1 figures; 1+2 tables. v2: published version

We present the theory of thermoelectric transport in metals with long-lived quasiparticles, carefully addressing the interplay of electron-electron scattering as well as electron-impurity scattering, but neglecting electron-phonon scattering. In Fermi liquids with a large Fermi surface and weak electron-impurity scattering, we provide universal and simple formulas for the behavior of the thermoelectric conductivities across the ballistic-to-hydrodynamic crossover. In this regime, the electrical conductivity is relatively unchanged by hydrodynamic effects. In contrast, the thermal conductivity can be parametrically smaller than predicted by the Wiedemann-Franz law. A less severe violation of the Mott law arises. We quantitatively compare the violations of the Wiedemann-Franz law arising from (i) momentum-conserving electron-electron scattering in the collision integral, (ii) hydrodynamic modifications of the electron-impurity scattering rate, and (iii) thermal broadening of the Fermi surface, and show that (i) is the largest. We present simple formulas for electrical and thermal magnetoconductivity across the ballistic-to-hydrodynamic limit, along with a more complicated formula for the thermoelectric magnetoconductivity. In a finite magnetic field, the Lorenz number may be smaller or larger than predicted by the Wiedemann-Franz law, and the crossover between these behaviors is a clear prediction for experiments. The arbitrarily strong violation of the Wiedemann-Franz law found in our work arises entirely from electron-electron interaction effects within the Fermi liquid paradigm, and does not imply any non-Fermi liquid behavior. We predict clear experimental signatures of bulk hydrodynamics in high-mobility 2D GaAs semiconductor structures, where failure of the Wiedemann-Franz law should persist down to very low temperatures in high-quality, low-density samples., Comment: 21+8 pages

We numerically solve semiclassical kinetic equations and compute the growth rate of the Dyakonov-Shur instability of a two-dimensional Fermi liquid in a finite length cavity. When electron-electron scattering is fast, we observe the well-understood hydrodynamic instability, and its disappearance due to viscous dissipation. When electron-electron scattering is negligible, we find that the instability re-emerges for certain boundary conditions, but not for others. We discuss the implications of these findings for experiments., Comment: 7+10 pages, 4+0 figures; v2: published version

Using an analytically tractable kinetic model of a two dimensional Fermi liquid of electrons, we characterize the crossovers between zero sound, first sound and plasmons. For experimentally realized Fermi liquids in a hydrodynamic limit, both zero and first sound waves are essentially replaced by plasmons. The plasmon dispersion relation is robust against hydrodynamic effects, up to acquiring the viscous-limited decay rate of a first sound wave in the hydrodynamic limit. We discuss implications for experiments in clean two dimensional electron gases., Comment: 14 pages. v2: published version

We present the theory of quasiparticle transport in perturbatively small inhomogeneous magnetic fields across the ballistic-to-hydrodynamic crossover. In the hydrodynamic limit, the resistivity $\rho$ generically grows proportionally to the rate of momentum-conserving electron-electron collisions at large enough temperatures $T$. In particular, the resulting flow of electrons provides a simple scenario where viscous effects suppress conductance below the ballistic value. This new mechanism for $\rho\propto T^2$ resistivity in a Fermi liquid may describe low $T$ transport in single-band $\mathrm{SrTiO}_3$., Comment: v2: published version. 7+8 pages, 1+0 figures

Generic interacting many-body quantum systems are believed to behave as classical fluids on long time and length scales. Due to rapid progress in growing exceptionally pure crystals, we are now able to experimentally observe this collective motion of electrons in solid-state systems, including graphene. We present a review of recent progress in understanding the hydrodynamic limit of electronic motion in graphene, written for physicists from diverse communities. We begin by discussing the "phase diagram" of graphene, and the inevitable presence of impurities and phonons in experimental systems. We derive hydrodynamics, both from a phenomenological perspective and using kinetic theory. We then describe how hydrodynamic electron flow is visible in electronic transport measurements. Although we focus on graphene in this review, the broader framework naturally generalizes to other materials. We assume only basic knowledge of condensed matter physics, and no prior knowledge of hydrodynamics., Comment: Review article. 79 pages, 21 figures. v2: published version

The interplay between electronic interactions and disorder is neglected in the conventional Boltzmann theory of transport, yet can play an essential role in determining the resistivity of unconventional metals. When quasiparticles are long-lived, one can account for these intertwined effects by solving spatially inhomogeneous Boltzmann equations. Assuming smooth disorder and neglecting umklapp scattering, we solve these inhomogeneous kinetic equations and compute the electrical resistivity across the ballistic-to-hydrodynamic transition. An important consequence of electron-electron interactions is the modification of the momentum relaxation time; this effect is ignored in the conventional theory. We characterize precisely when interactions enhance the momentum scattering rate, and when they decrease it. Our approach unifies existing semiclassical theories of transport and reveals novel transport mechanisms. In particular, we explain how the resistivity can be proportional to the rate of momentum-conserving collisions. We compare this result with existing transport mysteries, including the disorder-independent $T^2$ resistivity of many Fermi liquids, and the linear-in-$T$ "Planckian-limited" resistivity of many strange metals., Comment: 1+37+15 pages; 9 figures. v2: minor changes. v3: published version

The Stokes paradox is the statement that in a viscous two dimensional fluid, the "linear response" problem of fluid flow around an obstacle is ill-posed. We present a simple consequence of this paradox in the hydrodynamic regime of a Fermi liquid of electrons in two-dimensional metals. Using hydrodynamics and kinetic theory, we estimate the contribution of a single cylindrical obstacle to the global electrical resistance of a material, within linear response. Momentum relaxation, present in any realistic electron liquid, resolves the classical paradox. Nonetheless, this paradox imprints itself in the resistance, which can be parametrically larger than predicted by Ohmic transport theory. We find a remarkably rich set of behaviors, depending on whether or not the quasiparticle dynamics in the Fermi liquid should be treated as diffusive, hydrodynamic or ballistic on the length scale of the obstacle. We argue that all three types of behavior are observable in present day experiments., Comment: 23+3 pages, 7+1 figures. v2: published version with expanded introduction

We present a theory of thermoelectric transport in weakly disordered Weyl semimetals where the electron-electron scattering time is faster than the electron-impurity scattering time. Our hydrodynamic theory consists of relativistic fluids at each Weyl node, coupled together by perturbatively small inter-valley scattering, and long-range Coulomb interactions. The conductivity matrix of our theory is Onsager reciprocal and positive-semidefinite. In addition to the usual axial anomaly, we account for the effects of a distinct, axial-gravitational anomaly expected to be present in Weyl semimetals. Negative thermal magnetoresistance is a sharp, experimentally accessible signature of this axial-gravitational anomaly, even beyond the hydrodynamic limit., Comment: v3: 26 pages, published version

Inspired by the recent experimental signatures of relativistic hydrodynamics in graphene, we investigate theoretically the behavior of hydrodynamic sound modes in such quasi-relativistic fluids near charge neutrality, within linear response. Locally driving an electron fluid at a resonant frequency to such a sound mode can lead to large increases in the electrical response at the edges of the sample, a signature which cannot be explained using diffusive models of transport. We discuss the robustness of this signal to various effects, including electron-acoustic phonon coupling, disorder, and long-range Coulomb interactions. These long range interactions convert the sound mode into a collective plasmonic mode at low frequencies unless the fluid is charge neutral. At the smallest frequencies, the response in a disordered fluid is quantitatively what is predicted by a "momentum relaxation time" approximation. However, this approximation fails at higher frequencies (which can be parametrically small), where the classical localization of sound waves cannot be neglected. Experimental observation of such resonances is a clear signature of relativistic hydrodynamics, and provides an upper bound on the viscosity of the electron-hole plasma., Comment: 24+18 pages. 11+2 figures. v2: some small additions, published version

We develop a general hydrodynamic framework for computing direct current thermal and electric transport in a strongly interacting finite temperature quantum system near a Lorentz-invariant quantum critical point. Our framework is non-perturbative in the strength of long wavelength fluctuations in the background charge density of the electronic fluid, and requires the rate of electron-electron scattering to be faster than the rate of electron-impurity scattering. We use this formalism to compute transport coefficients in the Dirac fluid in clean samples of graphene near the charge neutrality point, and find results insensitive to long range Coulomb interactions. Numerical results are compared to recent experimental data on thermal and electrical conductivity in the Dirac fluid in graphene and substantially improved quantitative agreement over existing hydrodynamic theories is found. We comment on the interplay between the Dirac fluid and acoustic and optical phonons, and qualitatively explain experimentally observed effects. Our work paves the way for quantitative contact between experimentally realized condensed matter systems and the wide body of high energy inspired theories on transport in interacting many-body quantum systems., Comment: 19 + 12 pages; 8 + 3 figures; v2: very minor changes, published version

Interactions between particles in quantum many-body systems can lead to collective behavior described by hydrodynamics. One such system is the electron-hole plasma in graphene near the charge neutrality point which can form a strongly coupled Dirac fluid. This charge neutral plasma of quasi-relativistic fermions is expected to exhibit a substantial enhancement of the thermal conductivity, due to decoupling of charge and heat currents within hydrodynamics. Employing high sensitivity Johnson noise thermometry, we report the breakdown of the Wiedemann-Franz law in graphene, with a thermal conductivity an order of magnitude larger than the value predicted by Fermi liquid theory. This result is a signature of the Dirac fluid, and constitutes direct evidence of collective motion in a quantum electronic fluid., Comment: 5+5 pages, 3+7 figures

Parity-violating fluids in two spatial dimensions can appear in a variety of contexts such as liquid crystal films, anyon fluids, and quantum Hall fluids. Nonetheless, the consequences of parity-violation on the solutions to the equations of motion are largely unexplored. In this paper, we explore phenomenological consequences of parity-violation through simple, illustrative examples. Although incompressible velocity fields are essentially unchanged by parity violation, we discuss examples where parity violation plays a role at boundaries, or in the dynamics of temperature. We then discuss new types of compressible flows which only exist in a parity-violating fluid, including new sound waves, and solitons in the dissipationless limit. We conclude with a discussion of some curious features in Rayleigh-B\'enard convection of a parity-violating fluid., Comment: v2: expanded and published version. 33 pages, 9 figures