Gapless Fermions - from Fermi liquids to strange metals

I will review the basic physics of interacting Dirac fermions in condensed matter settings. Graphene's honeycomb lattice will be used as a playground, with the universal features of the problem extracted and highlighted. Plan of lectures: 1) Symmetries of two-dimensional massless Dirac fermions: emergent Lorentz symmetry and Clifford algebra, "chiral" vs. time reversal symmetry, inversion. 2) Importance of masses; breaking of chiral and time reversal symmetries. Two types of insulators. 3) Electron-electron interactions: from lattice Hamiltonians to field theory. Mass as order. Scaling near critical point. Death of Fermi liquid. 4) Gross-Neveu and Gross-Neveu-Yukawa field theories of metal-insulator transitions of Dirac fermions. Neel transition in the Hubbard model, and comparison with the Monte Carlo. Time permitting some open problems in the field will be discussed towards the end. Useful references: 1) I. H., Phys. Rev. Lett. vol. 97, 146401, (2006), 2) I. H., V. Juricic, and B. Roy, Phys. Rev. B, vol. 79, 085116 (2009), 3) I. H., V. Juricic, O. Vafek, Phys. Rev. B vol. 80, 065432 (2009), 4) N. Zerf, L. Mihaila, P. Marquard, I. H., M. Scherer Phys. Rev. D vol. 96, 096010 (2017), 5) Y. Otsuka, S. Yunoki, S. Sorella, Phys. Rev. X vol. 6, 011029 (2016).

I will review the basic physics of interacting Dirac fermions in condensed matter settings. Graphene's honeycomb lattice will be used as a playground, with the universal features of the problem extracted and highlighted. Plan of lectures: 1) Symmetries of two-dimensional massless Dirac fermions: emergent Lorentz symmetry and Clifford algebra, "chiral" vs. time reversal symmetry, inversion. 2) Importance of masses; breaking of chiral and time reversal symmetries. Two types of insulators. 3) Electron-electron interactions: from lattice Hamiltonians to field theory. Mass as order. Scaling near critical point. Death of Fermi liquid. 4) Gross-Neveu and Gross-Neveu-Yukawa field theories of metal-insulator transitions of Dirac fermions. Neel transition in the Hubbard model, and comparison with the Monte Carlo. Time permitting some open problems in the field will be discussed towards the end. Useful references: 1) I. H., Phys. Rev. Lett. vol. 97, 146401, (2006), 2) I. H., V. Juricic, and B. Roy, Phys. Rev. B, vol. 79, 085116 (2009), 3) I. H., V. Juricic, O. Vafek, Phys. Rev. B vol. 80, 065432 (2009), 4) N. Zerf, L. Mihaila, P. Marquard, I. H., M. Scherer Phys. Rev. D vol. 96, 096010 (2017), 5) Y. Otsuka, S. Yunoki, S. Sorella, Phys. Rev. X vol. 6, 011029 (2016).

TBD

Connecting theoretical models for exotic quantum states to real physical systems is a key goal in the study of quantum materials. Among such theoretical models, a “toy model” is one made deliberately simplistic in order to demonstrate new physical concepts and their underlying mechanisms. Such models have proven to be tremendously successful in offering insight in to new condensed matter phenomena including those involving electronic topology and correlation. We describe here our recent progress in experimentally realizing “toy model” quantum materials which, in analogy to their theoretical counterparts, are designed to capture simple model systems by lattice and superlattice design. We detail developments in synthesizing and studying magnetic and superconducting materials that allow for new connections to long-standing predictions for unusual topological electronic phases. We close with a perspective for realizing further toy model systems in complex material structures.

TBD

I will review the basic physics of interacting Dirac fermions in condensed matter settings. Graphene's honeycomb lattice will be used as a playground, with the universal features of the problem extracted and highlighted. Plan of lectures: 1) Symmetries of two-dimensional massless Dirac fermions: emergent Lorentz symmetry and Clifford algebra, "chiral" vs. time reversal symmetry, inversion. 2) Importance of masses; breaking of chiral and time reversal symmetries. Two types of insulators. 3) Electron-electron interactions: from lattice Hamiltonians to field theory. Mass as order. Scaling near critical point. Death of Fermi liquid. 4) Gross-Neveu and Gross-Neveu-Yukawa field theories of metal-insulator transitions of Dirac fermions. Neel transition in the Hubbard model, and comparison with the Monte Carlo. Time permitting some open problems in the field will be discussed towards the end. Useful references: 1) I. H., Phys. Rev. Lett. vol. 97, 146401, (2006), 2) I. H., V. Juricic, and B. Roy, Phys. Rev. B, vol. 79, 085116 (2009), 3) I. H., V. Juricic, O. Vafek, Phys. Rev. B vol. 80, 065432 (2009), 4) N. Zerf, L. Mihaila, P. Marquard, I. H., M. Scherer Phys. Rev. D vol. 96, 096010 (2017), 5) Y. Otsuka, S. Yunoki, S. Sorella, Phys. Rev. X vol. 6, 011029 (2016).

In this talk I will present the electronic signatures of different quantum materials from experimental studies in high magnetic fields. I will describe in detail quantum oscillations and the quantities extracted from these studies. I will also discuss magnetotransport properties of various quantum materials, such as iron-based superconductors and materials with Dirac dispersions.

We present here a brief overview of experiments in strongly correlated metals focusing on the role of electronic correlation, topology, and combinations thereof to realize exotic behavior in metal systems. We discuss the role of both lattice and compositional design to promote correlation. We also touch on the utility of the growing materials databases for quantum material design.

These four lectures will introduce the students to basics of fermionic path integrals, RG philosophy in general, some initial schematic calculations of flow, RG in d=1 and lessons learned, RG in 2+1 dimensions, tree and loop levels, fixed points, flow equations for coupling functions, BCS and CDW instabilities, generalizations to codimension>1.

Topology, a mathematical concept, recently became a hot and truly transdisciplinary topic in condensed matter physics, solid state chemistry and materials science. Since there is a direct connection between real space: atoms, valence electrons, bonds and orbitals, and reciprocal space: bands, Fermi surfaces and Berry curvature, a simple classification of topological materials in a single particle picture should be possible [1]. Binary phosphides are an ideal material class for a systematic study of Dirac, Weyl and new Fermion physics, since these compounds can be grown as high-quality single crystals. A new class of topological phases that have Weyl points was also predicted in the family that includes NbP, NbAs. TaP, MoP and WP2. [3-5]. Beyond Weyl and Dirac, new fermions can be identified in compounds that have linear and quadratic 3-, 6- and 8- band crossings that are stabilized by space group symmetries [2]. Crystals of chiral topological materials CoSi, AlPt and RhSi were investigated by angle resolved photoemission and show giant unusual helicoid Fermi arcs with topological charges of ±2 [6,7]. In agreement with the chiral crystal structure two different chiral surface states are observed. In magnetic materials the Berry curvature and the classical anomalous Hall (AHE) and spin Hall effect (SHE) helps to identify potentially interesting candidates. As a consequence, the magnetic Heusler compounds have already been identified as Weyl semimetals: for example, Co2YZ [8-11], and Co3Sn2S2 [12-15]. The Anomalous Hall angle also helps to identify materials in which a QAHE should be possible in thin films. Heusler compounds with non-collinear magnetic structures also possess real-space topological states in the form of magnetic antiskyrmions, which have not yet been observed in other materials [16]. [1] Bradlyn et al., Nature 547 298, (2017), Vergniory, et al., Nature 566 480 (2019), [2] Bradlyn, et al., Science 353, aaf5037A (2016). [3] Shekhar, et al., Nat. Phys. 11, 645 (2015) [4] Liu, et al., Nat. Mat. 15, 27 (2016) [5] Gooth et al., Nature 547, 324 (2017) [6] Schröter, et al., Nature Physics 15, 759 (2019) preprint arXiv: 1812.03310 [7] Sanchez, et al., Nature 567, 500 (2019) [8] Kübler and Felser, EPL 114, 47005 (2016) [9] Wang, et al. Phys. Rev. Lett. 117, 236401 (2016) [10] Chang et al., Scientific Reports 6, 38839 (2016) [11] Belopolski, et al., Science 365, 1278 (2019) [12] Liu, et al. Nature Physics 14, 1125 (2018) [13] Liu, et al. Nat. Phys. Nature Physics 14, 1125 (2018) [14] Liu, et al., Science 365, 1282 (2019) [15] Morali, et al., Science 365, 1286 (2019) [16] Nayak, et al., Nature 548, 561 (2017)

Electron quasiparticles in all materials are collective excitations of electrons, arising from the complex interplay of atomic bonds and electron interactions. In many cases, the behavior of these quasiparticles can be approximated by the model of a massive free electron gas. However, special conditions and combinations of atoms in so-called quantum materials can fundamentally alter the quasiparticles’ properties. For example, their effective mass may vanish, their electric charge can be effectively fractionalized, or entirely new electron quasiparticles may arise. In this talk, I present recent experiments on electron quasiparticle physics in three-dimensional topological semimetals and show how electron-atom, electron-electron and electron-phonon interaction give rise to a rich variety of quasi-particle states in these systems. Such states include Weyl fermions with quantum anomalies, viscous electron fluids, particles with fractional electric charge and axions.

Delafossite oxides are layered compounds, which can be thought of as natural heterostructures of triangularly coordinated metallic sheets and transition metal oxide blocks. A fascinating range of electronic states can be found both in their bulk and on their surfaces, including extremely high conductivity [1] in PtCoO$_{2}$ and PdCoO$_{2}$, maximal Rashba-like spin-splitting [2] on the transition metal terminated surfaces of PtCoO$_{2}$, PdCoO$_{2}$ and PdRhO$_{2}$, Stoner ferromagnetism [3] on the Pd-terminated surface of PdCoO$_{2}$ and, perhaps most remarkably, an intertwined spin-charge response due to a Kondo coupling between metallic and Mott insulating layers[4] in PdCrO$_{2}$. Our group has investigated these states experimentally with transport measurements and angle resolved photoemission, and theoretically with first principles calculations and model Hamiltonians, where applicable. I will show how in a number of cases the simplicity and cleanliness of the materials allows us to pinpoint to the underlying cause for the remarkable electronic behaviour, and in turn to use delafossites as model systems to understand complex phenomena. [1] Kushwaha, P., Sunko, V., Moll, P.J.W., Bawden, L., Riley, J.M., Nandi, N., Rosner, H., Schmidt, M.P., Arnold, F., Hassinger, E., Kim, T.K., Hoesch, M., Mackenzie, A.P., King, P.D.C., 2015, Science Advances 1, e1500692. [2] Sunko, V., Rosner, H., Kushwaha, P., Khim, S., Mazzola, F., Bawden, L., Clark, O.J., Riley, J.M., Kasinathan, D., Haverkort, M.W., Kim, T.K., Hoesch, M., Fujii, J., Vobornik, I., Mackenzie, A.P., King, P.D.C., 2017, Nature 549, 492 [3] Mazzola, F., Sunko, V., Khim, S., Rosner, H., Kushwaha, P., Clark, O.J., Bawden, L., Marković, I., Kim, T.K., Hoesch, M., Mackenzie, A.P., King, P.D.C., 2017., PNAS 51, 12956 [4] Sunko, V., Mazzola, F., Kitamura, S., Khim, S., Kushwaha, P., Clark, O. J. , Watson, M., Marković, I., Biswas, D., Pourovskii, L., Kim, T. K., Lee, T.-L. ,
Thakur, P. K., Rosner, H., Georges, A., Moessner, R., Oka, T. , Mackenzie, A. P., and King, P. D. C., 2018, arXiv:1809.08972

Delafossite oxides are layered compounds, which can be thought of as natural heterostructures of triangularly coordinated metallic sheets and transition metal oxide blocks. A fascinating range of electronic states can be found both in their bulk and on their surfaces, including extremely high conductivity [1] in PtCoO$_{2}$ and PdCoO$_{2}$, maximal Rashba-like spin-splitting [2] on the transition metal terminated surfaces of PtCoO$_{2}$, PdCoO$_{2}$ and PdRhO$_{2}$, Stoner ferromagnetism [3] on the Pd-terminated surface of PdCoO$_{2}$ and, perhaps most remarkably, an intertwined spin-charge response due to a Kondo coupling between metallic and Mott insulating layers[4] in PdCrO$_{2}$. Our group has investigated these states experimentally with transport measurements and angle resolved photoemission, and theoretically with first principles calculations and model Hamiltonians, where applicable. I will show how in a number of cases the simplicity and cleanliness of the materials allows us to pinpoint to the underlying cause for the remarkable electronic behaviour, and in turn to use delafossites as model systems to understand complex phenomena. [1] Kushwaha, P., Sunko, V., Moll, P.J.W., Bawden, L., Riley, J.M., Nandi, N., Rosner, H., Schmidt, M.P., Arnold, F., Hassinger, E., Kim, T.K., Hoesch, M., Mackenzie, A.P., King, P.D.C., 2015, Science Advances 1, e1500692. [2] Sunko, V., Rosner, H., Kushwaha, P., Khim, S., Mazzola, F., Bawden, L., Clark, O.J., Riley, J.M., Kasinathan, D., Haverkort, M.W., Kim, T.K., Hoesch, M., Fujii, J., Vobornik, I., Mackenzie, A.P., King, P.D.C., 2017, Nature 549, 492 [3] Mazzola, F., Sunko, V., Khim, S., Rosner, H., Kushwaha, P., Clark, O.J., Bawden, L., Marković, I., Kim, T.K., Hoesch, M., Mackenzie, A.P., King, P.D.C., 2017., PNAS 51, 12956 [4] Sunko, V., Mazzola, F., Kitamura, S., Khim, S., Kushwaha, P., Clark, O. J. , Watson, M., Marković, I., Biswas, D., Pourovskii, L., Kim, T. K., Lee, T.-L. ,
Thakur, P. K., Rosner, H., Georges, A., Moessner, R., Oka, T. , Mackenzie, A. P., and King, P. D. C., 2018, arXiv:1809.08972

These four lectures will introduce the students to basics of fermionic path integrals, RG philosophy in general, some initial schematic calculations of flow, RG in d=1 and lessons learned, RG in 2+1 dimensions, tree and loop levels, fixed points, flow equations for coupling functions, BCS and CDW instabilities, generalizations to codimension>1.

These four lectures will introduce the students to basics of fermionic path integrals, RG philosophy in general, some initial schematic calculations of flow, RG in d=1 and lessons learned, RG in 2+1 dimensions, tree and loop levels, fixed points, flow equations for coupling functions, BCS and CDW instabilities, generalizations to codimension>1.

Non-Fermi liquids are unconventional metallic states which do not support well-defined quasiparticles. Due to strong quantum fluctuations and the presence of extensive gapless modes near the Fermi surface, it has been difficult to understand the universal low energy properties of non-Fermi liquids in 2+1 dimensions. In these lectures, I will review recent progress made on non-Fermi liquid theories based on a dimensional regularization scheme and a non-perturbative interaction driven scaling.

Non-Fermi liquids are unconventional metallic states which do not support well-defined quasiparticles. Due to strong quantum fluctuations and the presence of extensive gapless modes near the Fermi surface, it has been difficult to understand the universal low energy properties of non-Fermi liquids in 2+1 dimensions. In these lectures, I will review recent progress made on non-Fermi liquid theories based on a dimensional regularization scheme and a non-perturbative interaction driven scaling.

These four lectures will introduce the students to basics of fermionic path integrals, RG philosophy in general, some initial schematic calculations of flow, RG in d=1 and lessons learned, RG in 2+1 dimensions, tree and loop levels, fixed points, flow equations for coupling functions, BCS and CDW instabilities, generalizations to codimension>1.

Non-Fermi liquids are unconventional metallic states which do not support well-defined quasiparticles. Due to strong quantum fluctuations and the presence of extensive gapless modes near the Fermi surface, it has been difficult to understand the universal low energy properties of non-Fermi liquids in 2+1 dimensions. In these lectures, I will review recent progress made on non-Fermi liquid theories based on a dimensional regularization scheme and a non-perturbative interaction driven scaling.

Non-Fermi liquids are unconventional metallic states which do not support well-defined quasiparticles. Due to strong quantum fluctuations and the presence of extensive gapless modes near the Fermi surface, it has been difficult to understand the universal low energy properties of non-Fermi liquids in 2+1 dimensions. In these lectures, I will review recent progress made on non-Fermi liquid theories based on a dimensional regularization scheme and a non-perturbative interaction driven scaling.

The lecture covers the Kondo effect, its lattice version in heavy-fermion metals, their quantum phase transitions, and possibly emerging non-Fermi-liquid metallic states. We will start with the traditional Kondo problem, describing a magnetic moment coupled to conduction electrons of a metal, and discuss its phenomenology. We then turn to variations of the Kondo model, such as a magnetic moments in semimetals or unconventional superconductors. These models possess zero-temperature phase transitions associated with the breakdown of Kondo screening. Moving from a single magnetic moment to a lattice, we will explain key aspects of Kondo lattices and associated heavy-fermion phenomena. A significant part of the lecture will discuss scenarios for quantum phase transitions in heavy-fermion systems. The breakdown of lattice Kondo screening opens the way to stable non-Fermi liquid phases with topological properties, and we will in particular discuss fractionalized Fermi liquids. Finally, we will highlight connections to the celebrated pseudogap regime of underdoped cuprate superconductors.

The auxiliary field Quantum Monte Carlo approach has recently been very successful at describing phases and phase transitions where fermonic degrees of freedom are gapless and hence cannot be integrated out. In these lectures, I will review the basic formulation, symmetry properties leading to an absence of the negative sign problem, as well as generic implementations of the method.

The auxiliary field Quantum Monte Carlo approach has recently been very successful at describing phases and phase transitions where fermonic degrees of freedom are gapless and hence cannot be integrated out. In these lectures, I will review the basic formulation, symmetry properties leading to an absence of the negative sign problem, as well as generic implementations of the method.

The lecture covers the Kondo effect, its lattice version in heavy-fermion metals, their quantum phase transitions, and possibly emerging non-Fermi-liquid metallic states. We will start with the traditional Kondo problem, describing a magnetic moment coupled to conduction electrons of a metal, and discuss its phenomenology. We then turn to variations of the Kondo model, such as a magnetic moments in semimetals or unconventional superconductors. These models possess zero-temperature phase transitions associated with the breakdown of Kondo screening. Moving from a single magnetic moment to a lattice, we will explain key aspects of Kondo lattices and associated heavy-fermion phenomena. A significant part of the lecture will discuss scenarios for quantum phase transitions in heavy-fermion systems. The breakdown of lattice Kondo screening opens the way to stable non-Fermi liquid phases with topological properties, and we will in particular discuss fractionalized Fermi liquids. Finally, we will highlight connections to the celebrated pseudogap regime of underdoped cuprate superconductors.

The lecture covers the Kondo effect, its lattice version in heavy-fermion metals, their quantum phase transitions, and possibly emerging non-Fermi-liquid metallic states. We will start with the traditional Kondo problem, describing a magnetic moment coupled to conduction electrons of a metal, and discuss its phenomenology. We then turn to variations of the Kondo model, such as a magnetic moments in semimetals or unconventional superconductors. These models possess zero-temperature phase transitions associated with the breakdown of Kondo screening. Moving from a single magnetic moment to a lattice, we will explain key aspects of Kondo lattices and associated heavy-fermion phenomena. A significant part of the lecture will discuss scenarios for quantum phase transitions in heavy-fermion systems. The breakdown of lattice Kondo screening opens the way to stable non-Fermi liquid phases with topological properties, and we will in particular discuss fractionalized Fermi liquids. Finally, we will highlight connections to the celebrated pseudogap regime of underdoped cuprate superconductors.

The functional renormalization group (RG) is an ideal tool for dealing with the diversity of energy scales and competition of instabilities in interacting fermion systems. Starting point is an exact flow equation which yields the gradual evolution from a microscopic model action to the effective low-energy action as a function of a continuously decreasing energy scale. Expanding in powers of the fields yields an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes [1]. Applications reviewed in the lectures include: (i) d-wave superconductivity and other instabilities in the two-dimensional Hubbard model, and (ii) transport through a barrier and resonant tunneling in a one-dimensional Luttinger liquid metal. Recently, the functional RG has been upgraded from a weak-coupling method to a computational tool for strongly interacting fermion systems [2,3]. [1] W. Metzner et al., Rev. Mod. Phys. 84, 299 (2012). [2] C. Taranto et al., Phys. Rev. Lett. 112, 196402 (2014). [3] D. Vilardi et al., Phys. Rev. B 99, 104501 (2019).

The functional renormalization group (RG) is an ideal tool for dealing with the diversity of energy scales and competition of instabilities in interacting fermion systems. Starting point is an exact flow equation which yields the gradual evolution from a microscopic model action to the effective low-energy action as a function of a continuously decreasing energy scale. Expanding in powers of the fields yields an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes [1]. Applications reviewed in the lectures include: (i) d-wave superconductivity and other instabilities in the two-dimensional Hubbard model, and (ii) transport through a barrier and resonant tunneling in a one-dimensional Luttinger liquid metal. Recently, the functional RG has been upgraded from a weak-coupling method to a computational tool for strongly interacting fermion systems [2,3]. [1] W. Metzner et al., Rev. Mod. Phys. 84, 299 (2012). [2] C. Taranto et al., Phys. Rev. Lett. 112, 196402 (2014). [3] D. Vilardi et al., Phys. Rev. B 99, 104501 (2019).

This course will be a general introduction to the physics of 1D quantum systems, as well as to the techniques to study them.

This course will be a general introduction to the physics of 1D quantum systems, as well as to the techniques to study them.

The functional renormalization group (RG) is an ideal tool for dealing with the diversity of energy scales and competition of instabilities in interacting fermion systems. Starting point is an exact flow equation which yields the gradual evolution from a microscopic model action to the effective low-energy action as a function of a continuously decreasing energy scale. Expanding in powers of the fields yields an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes [1]. Applications reviewed in the lectures include: (i) d-wave superconductivity and other instabilities in the two-dimensional Hubbard model, and (ii) transport through a barrier and resonant tunneling in a one-dimensional Luttinger liquid metal. Recently, the functional RG has been upgraded from a weak-coupling method to a computational tool for strongly interacting fermion systems [2,3]. [1] W. Metzner et al., Rev. Mod. Phys. 84, 299 (2012). [2] C. Taranto et al., Phys. Rev. Lett. 112, 196402 (2014). [3] D. Vilardi et al., Phys. Rev. B 99, 104501 (2019).

I will give an overview over a number of theoretical methods that determine transport properties of gapless electronic systems. We will cover the Kubo, memory function, and the kinetic Boltzmann theories. We will also touch upon holographic methods to describe ellectron transport in systems without well-defined quasiparticles The focus will however be on applications. Topics will include the Bloch-Grüneisen theory, phonon drag, scattering due to electron-electron interactions, the distinction between Fermi and Dirac liquids, and aspects of hydrodynamic transport, and non-dissipative transport in systems with nontrivial topological properties.

I will give an overview over a number of theoretical methods that determine transport properties of gapless electronic systems. We will cover the Kubo, memory function, and the kinetic Boltzmann theories. We will also touch upon holographic methods to describe ellectron transport in systems without well-defined quasiparticles The focus will however be on applications. Topics will include the Bloch-Grüneisen theory, phonon drag, scattering due to electron-electron interactions, the distinction between Fermi and Dirac liquids, and aspects of hydrodynamic transport, and non-dissipative transport in systems with nontrivial topological properties.

In this lecture, I will provide a self-contained introduction to Dynamical Mean Field Theory - both as a theoretical and computational framework and as a physical point of view on materials with strong electronic correlations.

In this lecture, I will provide a self-contained introduction to Dynamical Mean Field Theory - both as a theoretical and computational framework and as a physical point of view on materials with strong electronic correlations.

TBA

I will give an overview over a number of theoretical methods that determine transport properties of gapless electronic systems. We will cover the Kubo, memory function, and the kinetic Boltzmann theories. We will also touch upon holographic methods to describe ellectron transport in systems without well-defined quasiparticles The focus will however be on applications. Topics will include the Bloch-Grüneisen theory, phonon drag, scattering due to electron-electron interactions, the distinction between Fermi and Dirac liquids, and aspects of hydrodynamic transport, and non-dissipative transport in systems with nontrivial topological properties.

I will give an overview over a number of theoretical methods that determine transport properties of gapless electronic systems. We will cover the Kubo, memory function, and the kinetic Boltzmann theories. We will also touch upon holographic methods to describe ellectron transport in systems without well-defined quasiparticles The focus will however be on applications. Topics will include the Bloch-Grüneisen theory, phonon drag, scattering due to electron-electron interactions, the distinction between Fermi and Dirac liquids, and aspects of hydrodynamic transport, and non-dissipative transport in systems with nontrivial topological properties.