Abstract:
The intriguing coexistence of the charge density wave (CDW) and superconductivity in SrPt$_2$As$_2$ and LaPt$_2$Si$_2$ has been investigated based on the {\it ab initio} density functional theory band structure and phonon calculations. We have found that the local split distortions Pt atoms in the [As-Pt-As] layers play an essential role in driving the five-fold supercell CDW instability as well as the phonon softening instability in SrPt$_2$As$_2$. By contrast, the CDW and phonon softening instabilities in LaPt$_2$Si$_2$ occur without split distortions of Pt atoms, indicating that the driving mechanisms of the CDW in SrPt$_2$As$_2$ and LaPt$_2$Si$_2$ are different. We have found that the CDW instability for the former arises from the Fermi surface nesting, while, for the latter, from the saddle point scattering. The phonon calculations, however, suggest that the CDW and the superconductivity coexist in [{\it X}-Pt-{\it X}] layers ({\it X} = As or Si) for both cases.

Abstract:
In order to explore the driving mechanism of the concomitant metal-insulator and structural transitions in quasi-one-dimensional hollandite K$_{2}$Cr$_{8}$O$_{16}$, electronic structures and phonon properties are investigated by employing the {\it ab initio} density functional theory (DFT) calculations. We have found that the imaginary phonon frequency reflecting the structural instability appears only in the DFT+$U$ ($U$: Coulomb correlation) calculation, which indicates that the Coulomb correlation plays an essential role in the structural transition. The lattice displacements of the softened phonon at X explain the observed lattice distortions in K$_{2}$Cr$_{8}$O$_{16}$ perfectly well, suggesting the Peierls distortion vector {\bf Q} of X (0, 0, 1/2). The combined study of electronic and phonon properties reveals that half-metallic K$_{2}$Cr$_{8}$O$_{16}$, upon cooling, undergoes the correlation-assisted Peierls transition to become a Mott-Peierls ferromagnetic insulator at low temperature.

Abstract:
To investigate the pressure-induced structural transitions of chromium dioxide (CrO$_{2}$), phonon dispersions and total energy band structures are calculated as a function of pressure. The first structural transition has been confirmed at P$\approx$ 10 GPa from the ground state tetragonal CrO$_{2}$ (t-CrO$_{2}$) of rutile type to orthorhombic CrO$_{2}$ (o-CrO$_{2}$) of CaCl$_{2}$ type. The half-metallic property is found to be preserved in o-CrO$_{2}$. The softening of Raman-active B$_{1g}$ phonon mode, which is responsible for this structural transition, is demonstrated. The second structural transition is found to occur for P$\geq$ 61.1 GPa from ferromagnetic (FM) o-CrO$_{2}$ to nonmagnetic (NM) monoclinic CrO$_{2}$ (m-CrO$_{2}$) of MoO$_{2}$ type, which is related to the softening mode at {\bf q} = R(1/2,0,1/2). The third structural transition has been newly identified at P= 88.8 GPa from m-CrO$_{2}$ to cubic CrO$_{2}$ of CaF$_{2}$ type that is a FM insulator.

Abstract:
To explore the driving mechanisms of the metal-insulator transition (MIT) and the structural transition in VO2, we have investigated phonon dispersions of rutile VO2 (R-VO2) in the DFT and the DFT+U (U : Coulomb correlation) band calculations. We have found that the phonon softening instabilities occur in both cases, but the softened phonon mode only in the DFT+U describes properly both the MIT and the structural transition from R-VO2 to monoclinic VO2 (M1-VO2). This feature demonstrates that the Coulomb correlation effect plays an essential role of assisting the Peierls transition in R-VO2. We have also found from the phonon dispersion of M1-VO2 that M1 structure becomes unstable under high pressure. We have predicted a new phase of VO2 at high pressure that has a monoclinic CaCl2-type structure with metallic nature.

Abstract:
In this paper, we discuss the Yang-Mills functional and a certain family of its critical points on quantum Heisenberg manifolds using noncommutative geometrical methods developed by A. Connes and M. Rieffel. In our main result, we construct a certain family of connections on a projective module over a quantum Heisenberg manifold that give rise to critical points of the Yang-Mills functional. Moreover, we show that this set of solutions can be described as a set of solutions to Laplace's equation on quantum Heisenberg manifolds.

Abstract:
We have investigated the surface states of a potential mixed-valent topological insulator SmB$_6$ based on the first principles density functional theory slab band structure analysis. We have found that metallic surface states are formed in the bulk band gap region, providing evidence for the topological insulating nature of SmB$_6$. The obtained surface in-gap states are quite different from those in existing reports in that they are formed differently depending on the Sm or B$_6$ surface termination, and are composed of mainly Sm $4f$ state indicating the essentiality of including $f$ electrons in describing the surface states. We have obtained the spin chiral structures of the Fermi surfaces, which are also in accordance with the topological insulating nature of SmB$_6$.

Abstract:
We construct a representation of each finitely aligned aperiodic k-graph \Lambda\ on the Hilbert space H^{ap} with basis indexed by aperiodic boundary paths in \Lambda. We show that the canonical expectation on B(H^{ap}) restricts to an expectation of the image of this representation onto the subalgebra spanned by the final projections of the generating partial isometries. We then show that every quotient of the Toeplitz algebra of the k-graph admits an expectation compatible with this one. Using this, we prove that the image of our representation, which is canonically isomorphic to the Cuntz-Krieger algebra, is co-universal for Toeplitz-Cuntz-Krieger families consisting of nonzero partial isometries.

Abstract:
We describe the primitive ideal space of the $C^{\ast}$-algebra of a row-finite $k$-graph with no sources when every ideal is gauge invariant. We characterize which spectral spaces can occur, and compute the primitive ideal space of two examples. In order to do this we prove some new results on aperiodicity. Our computations indicate that when every ideal is gauge invariant, the primitive ideal space only depends on the 1-skeleton of the $k$-graph in question.

Abstract:
The quantum Heisenberg manifolds are noncommutive manifolds constructed by M. Rieffel as strict deformation quantizations of Heisenberg manifolds and have been studied by various authors. Rieffel constructed the quantum Heisenberg manifolds as the generalized fixed-point algebras of certain crossed product $C^*$-algebras, and they also can be realized as crossed products of $C(\mathbb{T}^2)$ by Hilbert $C^*$-bimodules in the sense of Abadie et al. In this paper, we describe how the quantum Heisenberg manifolds can also be realized as twisted groupoid $C^*$-algebras.

Abstract:
Several authors have recently been studying the equilibrium or KMS states on the Toeplitz algebras of finite higher-rank graphs. For graphs of rank one (that is, for ordinary directed graphs), there is a natural dynamics obtained by lifting the gauge action of the circle to an action of the real line. The algebras of higher-rank graphs carry a gauge action of a higher-dimensional torus, and there are many potential dynamics arising from different embeddings of the real line in the torus. Previous results show that there is nonetheless a "preferred dynamics" for which the system exhibits a particularly satisfactory phase transition, and that the unique KMS state at the critical inverse temperature can then be implemented by intregrating vector states against a measure on the infinite path space of the graph. Here we obtain a similar description of the KMS state at the critical inverse temperature for other dynamics. Our spatial implementation is given by integrating against a measure on a space of paths which are infinite in some directions but finite in others. Our results are sharpest for the algebras of rank-two graphs.