This thesis focuses on several examples of strongly correlated electronic quantum fluids. In chapter 2 we use a kinetic theory approach to describe the transport properties of bilayer graphene near charge neutrality. After solving the semiclassical Boltzmann equation, we find that the results are well approximated by a hydrodynamic two-fluid model. A fluid of electrons and a fluid of holes interact with each other via Coulomb drag. Chapter 3 looks at quantum Hall edges. We consider a quantum Hall edge coupled to a driven quantum dot and compute the current on the edge downstream from the contact. We show that Coulomb interactions between the dot and the edge renormalize the charge in the current pulses. Chapter 4 focuses on the bilayer quantum Hall system. Two layers of quantum Hall fluid are separated by a variable distance. At large interlayer separation, we have two well-separated composite Fermi liquids and we consider the pairing between the two layers as the interlayer separation is decreased. We propose a new trial state for this paired phase of matter and use exact diagonalization numerics to investigate this proposal.
This thesis focuses on several examples of strongly correlated electronic quantum fluids. In chapter 2 we use a kinetic theory approach to describe the transport properties of bilayer graphene near charge neutrality. After solving the semiclassical Boltzmann equation, we find that the results are well approximated by a hydrodynamic two-fluid model. A fluid of electrons and a fluid of holes interact with each other via Coulomb drag. Chapter 3 looks at quantum Hall edges. We consider a quantum Hall edge coupled to a driven quantum dot and compute the current on the edge downstream from the contact. We show that Coulomb interactions between the dot and the edge renormalize the charge in the current pulses. Chapter 4 focuses on the bilayer quantum Hall system. Two layers of quantum Hall fluid are separated by a variable distance. At large interlayer separation, we have two well-separated composite Fermi liquids and we consider the pairing between the two layers as the interlayer separation is decreased. We propose a new trial state for this paired phase of matter and use exact diagonalization numerics to investigate this proposal.
We develop a theory of Coulomb drag in ultraclean double layers with strongly correlated carriers. In the regime where the equilibration length of the electron liquid is shorter than the interlayer spacing the main contribution to the Coulomb drag arises from hydrodynamic density fluctuations. The latter consist of plasmons driven by fluctuating longitudinal stresses, and diffusive modes caused by temperature fluctuations and thermal expansion of the electron liquid. We express the drag resistivity in terms of the kinetic coefficients of the electron fluid. Our results are nonperturbative in interaction strength and do not assume Fermi-liquid behavior of the electron liquid.
After its first isolation by Geim and Novoselov in 2004, graphene has grown quickly into an independent subfield of contemporary condensed matter physics, counting thousands of publications. This material has attracted so much attention not only because it is truly two-dimensional -- a feature long thought to be impossible to be realized in nature -- but also because it is the prime representative of a new class of condensed matter systems, the Dirac materials, whose electronic spectrum disperses linearly around isolated points in the Brillouin zone. An outstanding consequence of this linear energy-momentum dispersion is the collapse of the Fermi surface to isolated points at charge neutrality, accompanied by a vanishing single-particle density of states. While this feature of the noninteracting model is what separates graphene from most other, more conventional condensed matter systems, giving rise to many of its astounding physical properties, it leads to severe computational complications when extending the model to contain two-body interactions and/or disorder. In this thesis we consider both extensions separately. After rigorously deriving the low-energy quantum field theory from a tight-binding model and establishing the one-particle irreducible (1PI) vertex functions as the fundamental building blocks of correlation functions, we construct a nonperturbative formalism for the calculation of these vertex functions using the well-established functional renormalization group (fRG). By combining the fRG with the Keldysh formalism, accounting for possible external electromagnetic fields and nontrivial initial correlations, this framework is capable of handling thermal equilibrium and true nonequilibrium alike. To circumvent some of the technical problems of the standard fRG formalism when treating fermionic systems at finite density, we also explore a variant of the fRG, where the chemical potential is interpreted as a flow parameter. We obtain hierarchical sets of flow equations, which describe the change of the 1PI vertex functions upon varying an artificial cutoff scale or the chemical potential, respectively. These nonperturbative formalisms have been used to calculate the renormalization of the Fermi velocity and the static dielectric function at finite temperature and density in the strong coupling regime as an explicit demonstration of their capabilities. Since the Keldysh fRG is an exact formulation of quantum field theory, in particular when external magnetic fields are present, it should be possible to describe the fractional quantum Hall effect in this framework. However, the typical truncation schemes that are currently available to approximately solve the fRG flow equation fail to account for the nontrivial correlation effects necessary to access the fractional quantum Hall regime. To circumvent this problem we consider a modified field theory, where these correlations are implemented ``by hand'' via a Chern-Simons gauge field, coupling to the fermions in addition to the Coulomb interaction term. This modified field theory is analyzed in a stationary phase approximation including Gaussian fluctuations. The electromagnetic response tensor is calculated in the random phase approximation, yielding the Hall conductivities. In the second part of this thesis we consider disordered Dirac fermions in the absence of two-particle interactions. After sketching how the most general disorder potential for Dirac fermions arises from a tight-binding model, we first consider an explicit physical scenario, a disordered graphene pn junction in the presence of a quantizing magnetic field perpendicular to the graphene sheet. We derive an effective one-dimensional theory for the chiral states that propagate along the junction interface and calculate the full conductance distribution in the crossover between the clean and the strong-disorder limit, via an exact solution of a Fokker-Planck equation. Lastly, we develop a nonperturbative approach to calculate the disorder induced self-energy that is based on exact Schwinger-Dyson equations and Ward identities, not only for graphene, but also for other paradigmatic semimetals with a nodal point dispersion.
Quantum Hall bilayer systems at filling fractions near $\ensuremath{\nu}=\frac{1}{2}+\frac{1}{2}$ undergo a transition from a compressible phase with strong intralayer correlation to an incompressible phase with strong interlayer correlations as the layer separation d is reduced below some critical value. Deep in the intralayer phase (large separation) the system can be interpreted as a fluid of composite fermions (CFs), whereas deep in the interlayer phase (small separation) the system can be interpreted as a fluid of composite bosons (CBs). The focus of this paper is to understand the states that occur for intermediate layer separation by using trial variational wave functions. We consider two main classes of wave functions. In the first class, previously introduced in M\"oller et al. [Phys. Rev. Lett. 101, 176803 (2008)], we consider interlayer BCS pairing of two independent CF liquids. We find that these wave functions are exceedingly good for $d\ensuremath{\gtrsim}{\ensuremath{\ell}}_{0}$ with ${\ensuremath{\ell}}_{0}$ as the magnetic length. The second class of wave functions naturally follows the reasoning of Simon et al. [Phys. Rev. Lett. 91, 046803 (2003)] and generalizes the idea of pairing wave functions by allowing the CFs also to be replaced continuously by CBs. This generalization allows us to construct exceedingly good wave functions for interlayer spacings of $d\ensuremath{\lesssim}{\ensuremath{\ell}}_{0}$ as well. The accuracy of the wave functions discussed in this work, compared with exact diagonalization, approaches that of the celebrated Laughlin wave function.
Modern quantum materials host a wide range of electronic phases in which transport properties are governed by strong interactions and collective effects. This thesis investigates charge, heat, and momentum transport in interacting electronic systems across different theoretical frameworks, ranging from Fermi liquid theory to strongly coupled strange metals.The first part of the thesis focuses on two-dimensional Fermi liquids at finite temperature, where rapid electron-electron scattering can give rise to hydrodynamic behavior. A kinetic-theory framework allowing an exact treatment of quasiparticle distribution functions beyond the asymptotic low-temperature regime is used to compute the full spectrum of collective modes, providing a detailed characterization of long-lived odd-parity excitations and the intermediate transport regime between ballistic and hydrodynamic flow. The same framework is employed to determine the shear viscosity of interacting electron liquids beyond conventional perturbative limits.The second part of the thesis concerns strongly interacting quantum systems where the quasiparticle picture breaks down.Holographic duality is used to study transport and thermoelectric response in quantum critical systems subject to periodic potentials, providing a controlled framework for momentum relaxation and magnetotransport in non-Fermi liquids. The thesis further develops holographic models of collective charge dynamics in systems with dynamical electromagnetism, including bulk plasmons and surface plasmon polaritons, and explores connections between Fermi surface physics and holographic gauge-field dynamics.
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 U(1) 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.
In this thesis, we investigated the evolution of low-dimensional quantum many-body systems through the lenses of their dynamics. The main tools we used are unitary dynamics and kinetic theory. The interest in this class of systems derives from the fact that they are dominated by the effects of interactions between fundamental constituents. In one dimension, the situation is even more dramatic: even the smallest interaction leads to a description that cannot be cast into a small perturbation of the initial model. Even though, a particular set of systems still allows for exact solutions; hence, they are deemed integrable and they allow for an infinite number of conserved currents. A simple way, that can also be performed experimentally, to drive many-body systems outside of the equilibrium is called quantum quench. It consists in switching instantaneously the parameters of the theory after the system is at the equilibrium and measuring an observable after the driving has occurred. Our contributions to the field amount to providing a full characterization of the evolution of the observables of an integrable system after a global quantum quench. We were also able to extend these results, valid in principle for relativistic theories, to the Lieb-Liniger model, prominent for the description of experimental setups, by the means of non-relativistic limit. For two-dimensional models, interactions are not changing the system from its foundations, although there are unexpected effects. A paradigmatic example is graphene: in a two-dimensional system where carriers are charged and massless, namely their dispersion relation does not have any gap and particles and holes can be freely excited from the thermal state, it is predicted that they are interacting via Coulomb force hundreds of times more than normal electrons in void. Hence, the system behaves in a completely different way from electrons in quantum electrodynamics: for instance, there are additional modes which are called plasmons that are also massless but neutral. They represent collective excitations of the model, as oscillations in the local density of charge. This system is not integrable but still strongly interacting; solving it as we do for one-dimensional systems is not possible. The role of conservation laws is a common theme of this thesis. In the example above exposed, total charge, momentum and energy are conserved in the pristine model, but in presence of impurities and defects they dissipate. The evolution of such quantities can though be derived exactly by the means of hydrodynamics. We were able to derive the equations that rule their evolution in time for any electronic system with two bands touching at the Dirac point. They are generalizations of the celebrated Navier-Stokes equations, and they differ from them only outside the compressible regime. We are able to show that the transport properties in the hydrodynamic regime strongly differ from the usual ballistic picture popularized by Drude and his model; this is captured for instance by violations of the Wiedemann-Franz ratio. Moreover, we prove that the role of interaction is prominent in the viscosity of these fluids.