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

Quantum electrodynamics (resulting from the merger of quantum mechanics and relativity theory) has provided a clear understanding of phenomena ranging from particle physics to cosmology and from astrophysics to quantum chemistry. The ideas underlying quantum electrodynamics also influence the theory of condensed matter, but quantum relativistic effects are usually minute in the known experimental systems that can be described accurately by the non-relativistic Schrodinger equation. Here we report an experimental study of a condensed-matter system (graphene, a single atomic layer of carbon) in which electron transport is essentially governed by Dirac's (relativistic) equation. The charge carriers in graphene mimic relativistic particles with zero rest mass and have an effective 'speed of light' c* approximately 10(6) m s(-1). Our study reveals a variety of unusual phenomena that are characteristic of two-dimensional Dirac fermions. In particular we have observed the following: first, graphene's conductivity never falls below a minimum value corresponding to the quantum unit of conductance, even when concentrations of charge carriers tend to zero; second, the integer quantum Hall effect in graphene is anomalous in that it occurs at half-integer filling factors; and third, the cyclotron mass m(c) of massless carriers in graphene is described by E = m(c)c*2. This two-dimensional system is not only interesting in itself but also allows access to the subtle and rich physics of quantum electrodynamics in a bench-top experiment.

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Two-dimensional gas of massless Dirac fermions in graphene

Topological insulators are new states of quantum matter which cannot be adiabatically connected to conventional insulators and semiconductors. They are characterized by a full insulating gap in the bulk and gapless edge or surface states which are protected by time-reversal symmetry. These topological materials have been theoretically predicted and experimentally observed in a variety of systems, including HgTe quantum wells, BiSb alloys, and Bi2Te3 and Bi2Se3 crystals. Theoretical models, materials properties, and experimental results on two-dimensional and three-dimensional topological insulators are reviewed, and both the topological band theory and the topological field theory are discussed. Topological superconductors have a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions. The theory of topological superconductors is reviewed, in close analogy to the theory of topological insulators.

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Topological insulators and superconductors

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

For a system at a temperature of absolute zero, all thermal fluctuations are frozen out, while quantum fluctuations prevail. These microscopic quantum fluctuations can induce a macroscopic phase transition in the ground state of a many-body system when the relative strength of two competing energy terms is varied across a critical value. Here we observe such a quantum phase transition in a Bose-Einstein condensate with repulsive interactions, held in a three-dimensional optical lattice potential. As the potential depth of the lattice is increased, a transition is observed from a superfluid to a Mott insulator phase. In the superfluid phase, each atom is spread out over the entire lattice, with long-range phase coherence. But in the insulating phase, exact numbers of atoms are localized at individual lattice sites, with no phase coherence across the lattice; this phase is characterized by a gap in the excitation spectrum. We can induce reversible changes between the two ground states of the system.

Quantum Phase Transition From a Superfluid to a Mott Insulator in a Gas of Ultracold Atoms

Arguments are presented that the $T=0$ conductance $G$ of a disordered electronic system depends on its length scale $L$ in a universal manner. Asymptotic forms are obtained for the scaling function $\ensuremath{\beta}(G)=\frac{d\mathrm{ln}G}{d\mathrm{ln}L}$, valid for both $G\ensuremath{\ll}{G}_{c}\ensuremath{\simeq}\frac{{e}^{2}}{\ensuremath{\hbar}}$ and $G\ensuremath{\gg}{G}_{c}$. In three dimensions, ${G}_{c}$ is an unstable fixed point. In two dimensions, there is no true metallic behavior; the conductance crosses over smoothly from logarithmic or slower to exponential decrease with $L$.

Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions

This paper reviews the progress made in the last several years in understanding the properties of disordered electronic systems. Even in the metallic limit, serious deviations from the Boltzmann transport theory and Fermi-liquid theory have been predicted and observed experimentally. There are two important ingredients in this new understanding: the concept of Anderson localization and the effects of interaction between electrons in a disordered medium. This paper emphasizes the theoretical aspect, even though some of the relevant experiments are also examined. The bulk of the paper focuses on the metallic side, but the authors also discuss the metal-to-insulator transition and comment on problems associated with the insulator.

Disordered electronic systems

A first-principles order-parameter theory of the fluid-solid transition is presented in this paper. The thermodynamic potential $\ensuremath{\Omega}$ of the system is computed as a function of order parameters ${\ensuremath{\lambda}}_{i}(={\ensuremath{\lambda}}_{{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{i}})$ proportional to the lattice periodic components of the one-particle density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}}_{i}$'s being the reciprocal-lattice vectors (RLV) of the crystal. Computation of $\ensuremath{\Omega}({{\ensuremath{\lambda}}_{i}})$ is shown to require knowing $\ensuremath{\Omega}$ for a fluid placed in lattice periodic potentials with amplitudes depending on ${\ensuremath{\lambda}}_{i}$. Using systematic nonperturbative functional methods for calculating the response of the fluid to such potentials, we find $\ensuremath{\Omega}({{\ensuremath{\lambda}}_{i}})$. The fluid properties (response functions) determining it are the Fourier coefficients ${c}_{i}(={c}_{{\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}}_{i}})$ and ${c}_{0}(={c}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}=0})$ of the direct correlation function $c(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$. The system freezes when at constant chemical potential $\ensuremath{\mu}$ and pressure $P$, locally stable fluid and solid phases [i.e., minima of $\ensuremath{\Omega}({{\ensuremath{\lambda}}_{i}})$ with ${{\ensuremath{\lambda}}_{i}}=0$ and ${{\ensuremath{\lambda}}_{i}}\ensuremath{\ne}0$, respectively] have the same $\ensuremath{\Omega}$. The order-parameter mode most effective in reducing $\ensuremath{\Omega}({{\ensuremath{\lambda}}_{i}})$ corresponds to ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}}_{j}$ being of the smallest-length RLV set (${\mathrm{c}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}$ is largest for $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}|\ensuremath{\simeq}|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}}_{j}|$). In some cases one has to consider a second order parameter ${\ensuremath{\lambda}}_{n}$ with a RLV ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}}_{n}$ lying near the second peak in ${c}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}$. The effect of further order-parameter modes on $\ensuremath{\Omega}$ is shown to be small. The theory can be viewed as one of a strongly first-order density-wave phase transition in a dense classical system. The transition is a purely structural one, occurring when the fluid-phase structural correlations (measured by ${c}_{j}$, etc.) are strong enough. This fact has been brought out clearly by computer experiments but had not been theoretically understood so far. Calculations are presented for freezing into some simple crystal structures, i.e., fcc, bcc, and two-dimensional hcp. The input information is only the crystal structure and the fluid compressibility (related to ${c}_{0}$). We obtain as output the freezing criterion stated as a condition on ${c}_{j}$ or as a relation between ${c}_{j}$ and ${c}_{n}$, the volume change $V$, the entropy change $\ensuremath{\Delta}s$, and the Debye-Waller factor at freezing for various RLV values. The numbers are all in very good agreement with those available experimentally.

First-principles order-parameter theory of freezing

Results of a detailed investigation of the structure and electron-transport properties of $La_{1-x}A_xMnO_3$ (A =Ca, Sr) over a wide range of compositions are presented along with those of parent $LaMnO_3$ containing different percentages of $Mn^{4+}$. The electrical resistivity $(\rho)$ and magnetoresistance (MR) of polycrystalline pellets have been measured in the 4.2–400 K range in magnetic fields up to 6 T and the Seebeck coefficient (S) from 100 to 400 K. The electrical measurements were supplemented by ac susceptibility and magnetization measurements. MR is large and negative over a substantial range of compositions and peaks around temperatures close to the ferromagnetic transition temperatures $(T_c)$. An insulator to metal-like transition occurs near the $T_c$ and the temperature dependence of $\rho$ below $T_c$ is related to the magnetization although $\rho$ in the metallic state is generally much larger than the Mott’s maximum metallic resistivity. The occurrence of giant magnetoresistance is linked to the presence of an optimal proportion of $Mn^{4+}$ ions and is found in the rhombohedral and the cubic structures where the Mn-O distance is less than 1.97 \AA and the Mn-O-Mn angle is $170^o\pm10^o$. The field dependence of MR shows the presence of two distinct regimes. The thermopower S shows a positive peak in the composition range at a temperature where MR also peaks; S becomes more negative with increase in $Mn^{4+}$.

Structure, electron-transport properties, and giant magnetoresistance of hole-doped LaMnO 3 systems

We introduce the idea of an incoherence length ${L}_{2}$ at which inelastic collision broadening equals electronic energy-level separation. The two-dimensional conductance crosses over at ${L}_{2}$ from a $\mathrm{ln}L$ dependence suggested recently by Abrahams et al. to Ohmic behavior. These ideas and plausible relaxation and heating models are used to explain the nonlinear conductivity observations of Dolan and Osheroff.

T. V. Ramakrishnan

Papers

Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions

Disordered electronic systems

First-principles order-parameter theory of freezing

Structure, electron-transport properties, and giant magnetoresistance of hole-doped LaMnO 3 systems

Possible explanation of nonlinear conductivity in thin-film metal wires