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

BACKGROUND Materials by design is an appealing idea that is very hard to realize in practice. Combining the best of different ingredients in one ultimate material is a task for which we currently have no general solution. However, we do have some successful examples to draw upon: Composite materials and III-V heterostructures have revolutionized many aspects of our lives. Still, we need a general strategy to solve the problem of mixing and matching crystals with different properties, creating combinations with predetermined attributes and functionalities. ADVANCES Two-dimensional (2D) materials offer a platform that allows creation of heterostructures with a variety of properties. One-atom-thick crystals now comprise a large family of these materials, collectively covering a very broad range of properties. The first material to be included was graphene, a zero-overlap semimetal. The family of 2D crystals has grown to includes metals (e.g., NbSe 2 ), semiconductors (e.g., MoS 2 ), and insulators [e.g., hexagonal boron nitride (hBN)]. Many of these materials are stable at ambient conditions, and we have come up with strategies for handling those that are not. Surprisingly, the properties of such 2D materials are often very different from those of their 3D counterparts. Furthermore, even the study of familiar phenomena (like superconductivity or ferromagnetism) in the 2D case, where there is no long-range order, raises many thought-provoking questions. A plethora of opportunities appear when we start to combine several 2D crystals in one vertical stack. Held together by van der Waals forces (the same forces that hold layered materials together), such heterostructures allow a far greater number of combinations than any traditional growth method. As the family of 2D crystals is expanding day by day, so too is the complexity of the heterostructures that could be created with atomic precision. When stacking different crystals together, the synergetic effects become very important. In the first-order approximation, charge redistribution might occur between the neighboring (and even more distant) crystals in the stack. Neighboring crystals can also induce structural changes in each other. Furthermore, such changes can be controlled by adjusting the relative orientation between the individual elements. Such heterostructures have already led to the observation of numerous exciting physical phenomena. Thus, spectrum reconstruction in graphene interacting with hBN allowed several groups to study the Hofstadter butterfly effect and topological currents in such a system. The possibility of positioning crystals in very close (but controlled) proximity to one another allows for the study of tunneling and drag effects. The use of semiconducting monolayers leads to the creation of optically active heterostructures. The extended range of functionalities of such heterostructures yields a range of possible applications. Now the highest-mobility graphene transistors are achieved by encapsulating graphene with hBN. Photovoltaic and light-emitting devices have been demonstrated by combining optically active semiconducting layers and graphene as transparent electrodes. OUTLOOK Currently, most 2D heterostructures are composed by direct stacking of individual monolayer flakes of different materials. Although this method allows ultimate flexibility, it is slow and cumbersome. Thus, techniques involving transfer of large-area crystals grown by chemical vapor deposition (CVD), direct growth of heterostructures by CVD or physical epitaxy, or one-step growth in solution are being developed. Currently, we are at the same level as we were with graphene 10 years ago: plenty of interesting science and unclear prospects for mass production. Given the fast progress of graphene technology over the past few years, we can expect similar advances in the production of the heterostructures, making the science and applications more achievable.

2D materials and van der Waals heterostructures

http://sces.phys.utk.edu/mostcited/mc1_1114.pdf

Colossal Magnetoresistant Materials: The Key Role of Phase Separation

The ternary iron arsenide ${\mathrm{BaFe}}_{2}{\mathrm{As}}_{2}$ becomes superconducting by hole doping, which was achieved by partial substitution of the barium site with potassium. We have discovered bulk superconductivity at ${T}_{c}=38\text{ }\text{ }\mathrm{K}$ in $({\mathrm{Ba}}_{1\ensuremath{-}x}{\mathrm{K}}_{x}){\mathrm{Fe}}_{2}{\mathrm{As}}_{2}$ with $x\ensuremath{\approx}0.4$. The parent compound ${\mathrm{BaFe}}_{2}{\mathrm{As}}_{2}$ crystallizes in the tetragonal ${\mathrm{ThCr}}_{2}{\mathrm{Si}}_{2}$-type structure, which consists of $(\mathrm{FeAs}{)}^{\ensuremath{\delta}\ensuremath{-}}$ iron arsenide layers separated by ${\mathrm{Ba}}^{2+}$ ions. ${\mathrm{BaFe}}_{2}{\mathrm{As}}_{2}$ is a poor metal and exhibits a spin density wave anomaly at 140 K. By substituting ${\mathrm{Ba}}^{2+}$ for ${\mathrm{K}}^{+}$ ions we have introduced holes in the $(\mathrm{FeAs}{)}^{\ensuremath{-}}$ layers, which suppress the anomaly and induce superconductivity. The ${T}_{c}$ of 38 K in $({\mathrm{Ba}}_{0.6}{\mathrm{K}}_{0.4}){\mathrm{Fe}}_{2}{\mathrm{As}}_{2}$ is the highest in hole doped iron arsenide superconductors so far. Therefore, we were able to expand this class of superconductors by oxygen-free compounds with the ${\mathrm{ThCr}}_{2}{\mathrm{Si}}_{2}$-type structure.

Superconductivity at 38 K in the iron arsenide (Ba1-xKx)Fe2As2.

FOLLOWING the discovery of superconductivity at ∼30 K in La2−xBaxCuO4 (ref. 1), a large number of related compounds have been found that are superconducting at relatively high temperatures. The feature common to all of these materials is a layered crystal structure based on a perovskite template and containing planar networks of copper and oxygen. This raises the question of whether superconductivity can occur in layered perovskites that do not contain copper. To the best of our knowledge, no such material has been found to date, despite nearly a decade of searching. We describe here the discovery of superconductivity in Sr2RuO4, a layered perovskite isostructural with La2−xBaxCuO4 (Fig. 1). Our results demonstrate that the presence of copper is not a prerequisite for the existence of superconductivity in a layered perovskite. But the low value of the superconducting transition temperature (Tc = 0.93 K) points towards a special role for copper in the high-temperature superconductors.

Superconductivity in a layered perovskite without copper

Substantial hole doping has been achieved in (La,Sr,Ca${)}_{14}$C${\mathrm{u}}_{24}$${\mathrm{O}}_{41}$ compounds with Cu${\mathrm{O}}_{2}$ chain and C${\mathrm{u}}_{2}$${\mathrm{O}}_{3}$ ladder building blocks. Holes enter the structural units with larger O/Cu ratios which are the Cu${\mathrm{O}}_{2}$ chains. The ladders remain undoped and magnetically inert due to a large spin gap. The moments in the undoped chains interact weakly, and every hole added creates a nonmagnetic Cu${\mathrm{O}}_{2}$ unit and decreases the resistivity. At the maximum hole concentration $(\ensuremath{\sim}0.6/{\mathrm{CuO}}_{2})$ the residual moments form a singlet ground state possibly due to dimerization with an excitation gap of $\ensuremath{\sim}140\mathrm{K}$ to the triplet state.

Hole Doping of the CuO2 Chains in (La, Sr, Ca)14Cu24O41

${\mathrm{Sr}}_{2}$${\mathrm{IrO}}_{4}$ and ${\mathrm{Sr}}_{2}$${\mathrm{RuO}}_{4}$ are both ${\mathrm{K}}_{2}$${\mathrm{NiF}}_{4}$-structure oxides. ${\mathrm{Sr}}_{2}$${\mathrm{IrO}}_{4}$ is a weakly ferromagnetic insulator and ${\mathrm{Sr}}_{2}$${\mathrm{RuO}}_{4}$ a paramagnetic metal. The ${\mathrm{Sr}}_{2}$${\mathrm{Ir}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ru}}_{\mathit{x}}$${\mathrm{O}}_{4}$ solid solution displays surprising physical properties. On the substitution of Ru to ${\mathrm{Sr}}_{2}$${\mathrm{IrO}}_{4}$, the Ru first exhibits its expected S=1 local magnetic moment, but at a critical Ru concentration of x\ensuremath{\simeq}0.7 the local moment disappears, the apparent density of states at ${\mathit{E}}_{\mathit{F}}$ increases, and an insulator-to-metal transition occurs. These changes in physical properties are accompanied by a change in the crystallographic cell parameters.

Localized-to-itinerant electron transition in Sr 2 Ir 1 − x Ru x O 4

The pressure dependence of the Raman peaks of the scheelite-type alkaline-earth tungstates CaWO 4 , SrWO 4 , and BaWO 4 have been investigated to 90 kbar hydrostatic pressure, with the use of the diamond anvil cell. The (dν/dP)'s are all positive, except in the case of the lowest-frequency external mode. In BaWO 4 , a new reversible first-order phase transition occurs near 65 kbar, where the Raman peaks shift abruptly to lower frequencies. From the observed decrease in the W—O bond-stretching mode frequencies and the increased number of W—O bond-bending modes it is concluded that the oxygen coordination for tungsten becomes octahedral, and the phase transition itself is postulated to be to the monoclinic HgWO 4 structure. The previously noted phase transitions in CaWO4 and SrWO 4 are not confirmed in the present study.

High-pressure Raman study of alkaline-earth tungstates and a new pressure-induced phase transition in BaWO 4

We report measurements of entropy transport due to vortex motion in single-cyrstal ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7}$. From these magnetothermal measurements we are able to determine thermodynamic properties, such as the transport line energy and superconducting phase boundary ${\mathit{T}}_{\mathit{c}}$(H), as well as transport parameters associated with flux motion. Below ${\mathit{T}}_{\mathit{c}}$, the data are compared with the microscopic theory of Caroli and Maki. Interestingly, we also find entropy transport in the fluctuation regime above ${\mathit{T}}_{\mathit{c}}$, which grows with applied field and for which no theory exists.

/pdf/transport-entropy-of-vortex-motion-in-yba2cu3o7-139m8xd0jb.pdf

Transport entropy of vortex motion in YBa2Cu3O7.

The formation of a heavy-electron state in a magnetically ordered material is established by measurements of the low-temperature specific heat of U${\mathrm{Cu}}_{5}$ and UAg${\mathrm{Cu}}_{4}$. In U${\mathrm{Cu}}_{5}$ this state undergoes a continuous but hysteretic phase transition which removes parts of the Fermi surface with a high density of electronic states and leads to a resistivity increase of almost an order of magnitude.

/pdf/low-temperature-state-of-ucu5-formation-of-heavy-electrons-2xfoaj55eb.pdf

Bertram Batlogg

Papers

Hole Doping of the CuO2 Chains in (La, Sr, Ca)14Cu24O41

Localized-to-itinerant electron transition in Sr 2 Ir 1 − x Ru x O 4

High-pressure Raman study of alkaline-earth tungstates and a new pressure-induced phase transition in BaWO 4

Transport entropy of vortex motion in YBa2Cu3O7.

Low-temperature state of UCu5: Formation of heavy electrons in a magnetically ordered material.