Abstract:
An interacting spin-fermion model is exactly solved on an open chain. In a certain representation, it is the nearest-neighbor Hubbard model in the limit of infinite $U$ (local interaction). Exact solution of its complete energy eigen-spectrum is accomplished by introducing a unitary transformation which maps the original problem to a tight-binding model of the fermions only. Physically, the exact solution implies the absence of Nagaoka ferromagnetism in the ground state for arbitrary electron densities. The present method solves a class of very general models exactly. Few more problems are discussed as an application of this unitary transform method.

Abstract:
The mean-field triplon analysis is developed for spin-S quantum antiferromagnets with dimerized ground states. For the spin-1/2 case, it reduces to the well known bond-operator mean-field theory. It is applied to a coupled-dimer model on square lattice, and to a model on honeycomb lattice with spontaneous dimerization in the ground state. Different phases in the ground state are investigated as a function of spin. It is found that under suitable conditions (such as strong frustration) a quantum ground state (dimerized singlet phase in the present study) can survive even in the limit $S\rightarrow \infty$. Two quick extensions of this representation are also presented. In one case, it is extended to include the quintet states. In another, a similar representation is worked out on a square plaquette. A convenient procedure for evaluating the total-spin eigenstates for a pair of quantum spins is presented in the appendix.

Abstract:
A two-level atom interacting with a single mode of quantized electromagnetic radiation is discussed using a representation in which the atom and the radiation are unified into a {\em new} canonical radiation. At the {\em twice-resonance}, when the frequency of the original radiation is twice the atomic transition frequency ($\omega=2\epsilon$), the {\em emergent} unified field in the non-interacting atom-field system resembles a free radiation of frequency $\epsilon$. This free emergent radiation is further shown to exist in the presence of an interaction which looks similar to the atom-field interaction in the dipole approximation. The one-photon correlation and the population inversion are discussed as the possible means of observing the emergent radiation. The entanglement properties of the emergent radiation are also discussed.

Abstract:
A new representation for electrons is introduced, in which the electron operators are written in terms of a spinless fermion and the Pauli operators. This representation is canonical, invertible and constraint-free. Importantly, it simplifies the Hubbard interaction. On a bipartite lattice, the Hubbard model is reduced to a form in which the exchange interaction emerges simply by decoupling the Pauli subsystem from the spinless fermion bath. This exchange correctly reproduces the large $U$ superexchange. Also derived, for $U=\pm\infty$, is the Hamiltonian to study Nagaoka ferromagnetism. In this representation, the infinite-$U$ Hubbard problem becomes elegant and easier to handle. Interestingly, the ferromagnetism in Hubbard model is found to be related to the gauge invariance of the spinless fermions. Generalization of this representation for the multicomponent fermions, a new representation for bosons, the notion of a `soft-core' fermion, and some interesting unitary transformations are introduced and discussed in the appendices.

Abstract:
A one-dimensional model of coupled spin-1/2 spins and pseudospin-1/2 orbitals with nearest-neighbor interaction is rigorously shown to exhibit spin-orbital separation by means of a non-local unitary transformation. On an open chain, this transformation completely decouples the spins from the orbitals in such a way that the spins become paramagnetic while the orbitals form the soluble XXZ Heisenberg model. The nature of various correlations is discussed. The more general cases, which allow spin-orbital separation by the same method, are pointed out. A generalization for the orbital pseudospin greater than 1/2 is also discussed. Some qualitative connections are drawn with the recently observed spin-orbital separation in Sr2CuO3.

Abstract:
Two quantum spin models with bilinear-biquadratic exchange interactions are constructed on the checkerboard lattice. It is proved that, under certain sufficient conditions on the exchange parameters, their ground states consist of two degenerate Shastry-Sutherland singlet configurations. The constructions are studied for arbitrary spin-S. The sufficient conditions for the existence of ferromagnetic ground state are also found exactly. The approximate quantum phase diagrams are presented using the exact results, together with a variational estimate for the N\'eel antiferromagnetic phase. A two-leg spin-1/2 ladder model, based on one of the above constructions, is considered which admits exact solution for a large number of eigenstates. The ladder model is shown to have exact level-crossing between the rung-singlet state and the AKLT state in the singlet ground state. Also introduced is the notion of perpendicularity for quantum spin vectors, which appears in the discussion on one of the two checkerboard models, and is discussed in the Appendix.

Abstract:
We present survey results obtained from the UBVRI optical photometric follow-up of 19 bright core-collapse SNe during 2002-2012 using 1-m class optical telescopes operated by the Aryabhatta Research Institute of Observational Science (acronym ARIES), Nainital India. This homogeneous set of data have been used to study behavior of optical light/color curve, and to gain insight into object-to-object peculiarity. We derive integrated luminosities for types IIP, Ibc and luminous SNe. Two peculiar type IIP events having photometric properties similar to normal IIP and spectroscopic properties similar to sub-luminous IIP have been identified.

Abstract:
Inspired by the exact solution of the Majumdar-Ghosh model, a family of one-dimensional, translationally invariant spin hamiltonians is constructed. The exchange coupling in these models is antiferromagnetic, and decreases linearly with the separation between the spins. The coupling becomes identically zero beyond a certain distance. It is rigorously proved that the dimer configuration is an exact, superstable ground state configuration of all the members of the family on a periodic chain. The ground state is two-fold degenerate, and there exists an energy gap above the ground state. The Majumdar-Ghosh hamiltonian with two-fold degenerate dimer ground state is just the first member of the family. The scheme of construction is generalized to two and three dimensions, and illustrated with the help of some concrete examples. The first member in two dimensions is the Shastry-Sutherland model. Many of these models have exponentially degenerate, exact dimer ground states.

Abstract:
We construct and study two frustrated quantum spin-1/2 models on square lattice, which are like the antiferromagnetic $J_1$-$J_2$ model with some additional four-spin exchange interactions. These models admit an exactly solvable case in which the ground state consists of four degenerate columnar-dimer singlet (CDS) configurations. Away from the exact case, we employ bond-operator mean-field theory to investigate the evolution of the ground state by varying the interaction parameters. The mean-field calculation reveals a quantum phase diagram in which the CDS phase undergoes a continuous phase transition to either N\'eel or collinear ordered antiferromagnetic phases.

Abstract:
In a recent work [Phys. Rev. B {\bf 77}, 014419 (2008)], a quantum spin-1/2 model having an exact fourfold degenerate Shastry-Sutherland ground state was constructed, and studied using finite-size numerical exact diagonalization. There, a schematic quantum phase diagram suggesting many competing phases was also proposed. In the present work, an extended version of this model is investigated using dimer and plaquette triplon mean-field theories. The quantum phase diagram of the extended model (in the thermodynamic limit) is found to have many interesting phases: commensurate and incommensurate magnetically ordered phases, and columnar dimer, sublattice columnar dimer, Shastry-Sutherland and plaquette spin-gapped phases. Among themselves, these phases undergo a continuous or a level-crossing transition.