This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.

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The electronic properties of graphene

This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

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Many-Body Physics with Ultracold Gases

A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

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Pattern formation outside of equilibrium

The behaviour of strongly correlated materials, and in particular unconventional superconductors, has been studied extensively for decades, but is still not well understood. This lack of theoretical understanding has motivated the development of experimental techniques for studying such behaviour, such as using ultracold atom lattices to simulate quantum materials. Here we report the realization of intrinsic unconventional superconductivity-which cannot be explained by weak electron-phonon interactions-in a two-dimensional superlattice created by stacking two sheets of graphene that are twisted relative to each other by a small angle. For twist angles of about 1.1°-the first 'magic' angle-the electronic band structure of this 'twisted bilayer graphene' exhibits flat bands near zero Fermi energy, resulting in correlated insulating states at half-filling. Upon electrostatic doping of the material away from these correlated insulating states, we observe tunable zero-resistance states with a critical temperature of up to 1.7 kelvin. The temperature-carrier-density phase diagram of twisted bilayer graphene is similar to that of copper oxides (or cuprates), and includes dome-shaped regions that correspond to superconductivity. Moreover, quantum oscillations in the longitudinal resistance of the material indicate the presence of small Fermi surfaces near the correlated insulating states, in analogy with underdoped cuprates. The relatively high superconducting critical temperature of twisted bilayer graphene, given such a small Fermi surface (which corresponds to a carrier density of about 1011 per square centimetre), puts it among the superconductors with the strongest pairing strength between electrons. Twisted bilayer graphene is a precisely tunable, purely carbon-based, two-dimensional superconductor. It is therefore an ideal material for investigations of strongly correlated phenomena, which could lead to insights into the physics of high-critical-temperature superconductors and quantum spin liquids.

Unconventional superconductivity in magic-angle graphene superlattices

We review the dynamical mean-field theory of strongly correlated electron systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact for models of correlated electrons in the limit of large lattice coordination (or infinite spatial dimensions). It extends the standard mean-field construction from classical statistical mechanics to quantum problems. We discuss the physical ideas underlying this theory and its mathematical derivation. Various analytic and numerical techniques that have been developed recently in order to analyze and solve the dynamical mean-field equations are reviewed and compared to each other. The method can be used for the determination of phase diagrams (by comparing the stability of various types of long-range order), and the calculation of thermodynamic properties, one-particle Green's functions, and response functions. We review in detail the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to experiments on three-dimensional transition-metal oxides. We present an overview of the rapidly developing field of applications of this method to other systems. The present limitations of the approach, and possible extensions of the formalism are finally discussed. Computer programs for the numerical implementation of this method are also provided with this article.

Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions

The authors develop and explore a bilocal quantum critical theory, which arises from the coupling of a Fermi surface to SU(2) charged bosons at criticality. A strongly-coupled fixed point is identified, with a dynamic critical exponent z g 1 and a finite enhancement of the specific heat near the critical point.

https://link.aps.org/pdf/10.1103/PhysRevResearch.2.033390

Bilocal quantum criticality

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Dual vortex theory of doped antiferromagnets

M agic-angle graphene captured the attention of condensed-matter physicists in 2018 when it was discovered that this material—made of two sheets of graphene with slightly misaligned lattice orientations (Fig. 1)—is a superconductor. Moreover, observations showed that the phase diagram of magicangle graphene is similar to that of copper oxide hightemperature superconductors, with an insulating region next to a dome-shaped superconducting region (see Trend: Bilayer Graphene’s Wicked, Twisted Road). Now, Pablo Jarillo-Herrero from the Massachusetts Institute of Technology, Cambridge, and his colleagues report that magic-angle graphene has another remarkable feature of the cuprate phase diagram: a “strange metal” phase in which the resistivity scales linearly with temperature down to very low temperatures [1]. This temperature dependence is unlike that of most metals and has proved difficult to explain with traditional models of electron conduction. The one clue that theorists have is that the strange metal phase is associated

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Graphene Reveals Its Strange Side

Research on high-temperature superconducting cuprates is at present focused on identifying the relationship between the classic ‘pseudogap’ phenomenon1,2 and the more recently investigated density wave state3–13. This state is generally characterized by a wavevector Q parallel to the planar Cu–O–Cu bonds4–13 along with a predominantly d-symmetry form factor14–16 (dFF-DW). To identify the microscopic mechanism giving rise to this state17–29, one must identify the momentum-space states contributing to the dFF-DW spectral weight, determine their particle–hole phase relationship about the Fermi energy, establish whether they exhibit a characteristic energy gap, and understand the evolution of all these phenomena throughout the phase diagram. Here we use energy-resolved sublattice visualization14 of electronic structure and reveal that the characteristic energy of the dFF-DW modulations is actually the ‘pseudogap’ energy ∆1. Moreover, we demonstrate that the dFF-DW modulations at E=−∆1 (filled states) occur with relative phase π compared to those at E=∆1 (empty states). Finally, we show that the conventionally defined dFF-DW Q corresponds to scattering between the ‘hot frontier’ regions of momentum-space beyond which Bogoliubov quasiparticles cease to exist30–32. These data indicate that the cuprate dFF-DW state involves particle–hole interactions focused at the pseudogap energy scale and between the four pairs of ‘hot frontier’ regions in momentum space where the pseudogap opens. A conventional ‘Peierls’ charge density wave (CDW) in a metal results from particle–hole interactions which open an energy gap at specific regions of k-space that are connected by a common wavevector Q. This generates a modulation in the density of free charge at Q along with an associated modulation of the crystal lattice parameters. Such CDW states are now very well known33. In principle, a density wave modulating at Q can also exhibit a ‘form factor’ (FF) with different possible symmetries34,35 (see Supplementary Section 1). This is relevant to the high-temperature superconducting cuprates because numerous researchers have recently proposed that the ‘pseudogap’ regime1,2 (PG in Fig. 1a) contains an unconventional density wave with a d-symmetry form factor17–29. The basic phenomenology of such a state is that intraunit-cell (IUC) symmetry breaking renders the Ox and Oy sites within each CuO2 unit-cell electronically inequivalent, and that this inequivalence is then modulated periodically at wavevector Q parallel to (1,0);(0,1). The real-space (r-space) schematic of such a d-symmetry FF density wave (dFF-DW) atQx , as shown in Fig. 1b, exemplifies periodic modulations at the Ox sites that are π out of phase with those at the Oy sites. Such a state is then described by A(r)=D(r)cos(φ(r)+φ0(r)), where A(r) represents whatever is the modulating electronic degree of freedom, φ(r)=Qx ·r is the DW spatial phase at location r, φ0(r) represents disorder related spatial phase shifts, and D(r) is the magnitude of the d-symmetry form factor14,21,23. To distinguish between the various microscopic mechanisms proposed for the Q=(Q, 0);(0,Q) dFF-DW state of cuprates17–29, it is essential to establish its atomic-scale phenomenology, including the momentum space (k-space) eigenstates contributing to its spectral weight, the relationship (if any) between modulations occurring above and below the Fermi energy, whether the modulating states in the DW are associated with a characteristic energy gap, and how the dFF-DW evolves with doping. To visualize such phenomena directly as in Fig. 1c, we use sublattice-phase-resolved imaging of the electronic structure14 of the CuO2 plane. Both the scanning tunnelling microscope (STM) tip–sample differential tunnelling conductance dI/dV (r,E=eV)≡g (r,E) and the tunnel-current I(r, E) are measured at bias voltage V = E/e and with sub-unitcell spatial resolution. Because the density of electronic states N (r, E) is related to the differential conductance as g (r, E) ∝ [eIs/ ∫ eVs 0 N (r,E )dE ]N (r, E), with Is and Vs being arbitrary parameters and the denominator ∫ eVs 0 N (r,E )dE ′ unknown, valid imaging of N (r,E) is challenging (Supplementary Section 2). However, one can suppress these serious systematic ‘set-point’ errors by using R(r, E) = I(r, E)/I(r, −E) or Z(r,E)=g (r,E)/g (r,−E); this allows distances, wavelengths and spatial phases of electronic structure to be measured accurately. The unprocessed g (r, E) acquired for and analysed in this paper are measured over very large fields of view (to achieve high phase precision in Fourier analysis), simultaneously maintain deeply sub-unit-cell precision measurements in r (to achieve high precision in sublattice segregation), and are taken over a wide range of energies E with fine energy spacing, so that energy dependences of d-symmetry FF modulations may be accurately determined. We then calculate each sublattice-phase-resolved Z(r,E) image and separate it into three: the first, Cu(r), contains

Underpinning Data: Atomic-scale Electronic Structure of the Cuprate d-Symmetry Form Factor Density Wave State

A colloquium style review of the connections between the Sachdev-Ye-Kitaev model and strange metals without quasiparticles, and between the SYK model and the quantum properties of black holes. To appear in ICTS News , newsletter of the International Centre for Theoretical Sciences, Tata Institute of Fundamental Research.

Subir Sachdev

Papers

Bilocal quantum criticality

Dual vortex theory of doped antiferromagnets

Graphene Reveals Its Strange Side

Underpinning Data: Atomic-scale Electronic Structure of the Cuprate d-Symmetry Form Factor Density Wave State

Statistical mechanics of strange metals and black holes