There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

Recent technical advances in the atomic-scale synthesis of oxide heterostructures have provided a fertile new ground for creating novel states at their interfaces. Different symmetry constraints can be used to design structures exhibiting phenomena not found in the bulk constituents. A characteristic feature is the reconstruction of the charge, spin and orbital states at interfaces on the nanometre scale. Examples such as interface superconductivity, magneto-electric coupling, and the quantum Hall effect in oxide heterostructures are representative of the scientific and technological opportunities in this rapidly emerging field.

/pdf/emergent-phenomena-at-oxide-interfaces-29wwyhjrec.pdf

Emergent phenomena at oxide interfaces

인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

We present a review of the basic ideas and techniques of the spectral density functional theory which are currently used in electronic structure calculations of strongly{correlated materials where the one{electron description breaks down. We illustrate the method with several examples where interactions play a dominant role: systems near metal{insulator transition, systems near volume collapse transition, and systems with local moments.

/pdf/electronic-structure-calculations-with-dynamical-mean-field-244tl5lhyg.pdf

Electronic structure calculations with dynamical mean-field theory

This article reviews quantum cluster theories, a set of approximations for infinite lattice models which treat correlations within the cluster explicitly, and correlations at longer length scales either perturbatively or within a mean-field approximation. These methods become exact when the cluster size diverges, and most recover the corresponding mean-field approximation when the cluster size becomes 1. Although quantum cluster theories were originally developed to treat disordered systems, they have more recently been applied to the study of ordered and disordered correlated systems, which will be the focus of this review. After a brief historical review, the authors provide detailed derivations of three cluster formalisms: the cluster perturbation theory, the dynamical cluster approximation, and the cellular dynamical mean-field theory. They compare their advantages and review their applications to common models of correlated electron systems.

/pdf/quantum-cluster-theories-2ek4yxki9k.pdf

Quantum cluster theories

Weak-coupling approaches to the pairing problem in the iron pnictide superconductors have predicted a wide variety of superconducting ground states. We argue here that this is due both to the inadequacy of certain approximations to the effective low-energy band structure, and to the natural near degeneracy of different pairing channels in superconductors with many distinct Fermi surface sheets. In particular, we review attempts to construct two-orbital effective band models, the argument for their fundamental inconsistency with the symmetry of these materials, and compare the dynamical susceptibilities of two- and five-orbital tight-binding models. We then present results for the magnetic properties, pairing interactions and pairing instabilities within a five-orbital tight-binding random phase approximation model. We discuss the robustness of these results for different dopings, interaction strengths and variations in band structures. Within the parameter space explored, an anisotropic, sign-changing s-wave (A1g) state and a (B1g) state are nearly degenerate, due to the near nesting of Fermi surface sheets.

http://iopscience.iop.org/article/10.1088/1367-2630/11/2/025016/pdf

Near-degeneracy of several pairing channels in multiorbital models for the Fe pnictides

http://absimage.aps.org/image/MAR09/MWS_MAR09-2008-003042.pdf

Near degeneracy of several pairing channels in a multi-orbital model for the Fe-pnictides

The cluster size dependence of superconductivity in the conventional two-dimensional Hubbard model, commonly believed to describe high-temperature superconductors, is systematically studied using the dynamical cluster approximation and quantum Monte Carlo simulations as a cluster solver. Because of the nonlocality of the $d$-wave superconducting order parameter, the results on small clusters show large size and geometry effects. In large enough clusters, the results are independent of the cluster size and display a finite temperature instability to $d$-wave superconductivity.

https://arxiv.org/pdf/cond-mat/0504529.pdf

Systematic study of d-wave superconductivity in the 2D repulsive Hubbard model.

Despite the wealth of experimental data on the Fe-pnictide compounds of the $K{\text{Fe}}_{2}{\text{As}}_{2}$ type, $K=\text{Ba}$, Ca, or Sr, the main theoretical work based on multiorbital tight-binding models has been restricted so far to the study of the related 1111 compounds. This can be ascribed to the more three-dimensional electronic structure found by ab initio calculations for the 122 materials, making this system less amenable to model development. In addition, the more complicated Brillouin zone (BZ) of the body-centered tetragonal symmetry does not allow a straightforward unfolding of the electronic band structure into an effective 1Fe/unit cell BZ. Here we present an effective five-orbital tight-binding fit of the full density functional theory band structure for ${\text{BaFe}}_{2}{\text{As}}_{2}$ including the ${k}_{z}$ dispersions. We compare the five-orbital spin fluctuation model to one previously studied for LaOFeAs and calculate the random-phase approximation enhanced susceptibility. Using the fluctuation exchange approximation to determine the leading pairing instability, we then examine the differences between a strictly two-dimensional model calculation over a single ${k}_{z}$ cut of the BZ and a completely three-dimensional approach. We find pairing states quite similar to the 1111 materials, with generic quasi-isotropic pairing on the hole sheets and nodal states on the electron sheets at ${k}_{z}=0$, which however are gapped as the system is hole doped. On the other hand, a substantial ${k}_{z}$ dependence of the order parameter remains, with most of the pairing strength deriving from processes near ${k}_{z}=\ensuremath{\pi}$. These states exhibit a tendency for an enhanced anisotropy on the hole sheets and a reduced anisotropy on the electron sheets near the top of the BZ.

Thomas Maier

Papers

Quantum cluster theories

Near-degeneracy of several pairing channels in multiorbital models for the Fe pnictides

Near degeneracy of several pairing channels in a multi-orbital model for the Fe-pnictides

Systematic study of d-wave superconductivity in the 2D repulsive Hubbard model.

Spin fluctuations and superconductivity in a three-dimensional tight-binding model for BaFe 2 As 2