By using the Y (gl(m|n)) super Yangian symmetry of the SU(m|n) supersymmetric Haldane-Shastry spin chain, we show that the partition function of this model satisfies a duality relation under the exchange of bosonic and fermionic spin degrees of freedom. As a byproduct of this study of the duality relation, we find a novel combinatorial formula for the super Schur polynomials associated with some irreducible representations of the Y (gl(m|n)) Yangian algebra. Finally, we reveal an intimate connection between the global SU(m|n) symmetry of a spin chain and the boson-fermion duality relation. *

By taking the freezing limit of a spin Calogero-Sutherland model containing 'anyon like' representation of the permutation algebra, we derive the exact partition function of SU(m|n) supersymmetric Haldane-Shastry (HS) spin chain. This partition function allows us to study global properties of the spectrum like level density distribution and nearest neighbour spacing distribution. It is found that, for supersymmetric HS spin chains with large number of lattice sites, continuous part of the energy level density obeys Gaussian distribution with a high degree of accuracy. The mean value and standard deviation of such Gaussian distribution can be calculated exactly. We also conjecture that the partition function of supersymmetric HS spin chain satisfies a duality relation under the exchange of bosonic and fermionic spin degrees of freedom.

We show that a topologically trivial zero bias conductance peak is produced in semiconductorsuperconductor hybrid structures due to a suppressed superconducting pair potential and/or an excess Zeeman field at the ends of the heterostructure, both of which can occur in experiments. The zero bias peak (ZBP) (a) appears above a threshold parallel bulk Zeeman field, (b) is stable for a range of bulk field before splitting, (c) disappears with rotation of the bulk Zeeman field, and, (d) is robust to weak disorder fluctuations. The topologically trivial ZBPs are also expected to produce splitting oscillations with the applied field similar to those from Majorana fermions. Because of such strong similarity with the phenomenology expected from Majorana fermions we find that the only unambiguous way to distinguish these trivial ZBPs (of height 4e 2 /h) from those arising from Majorana fermions (of height 2e 2 /h) is by comparing the (zero temperature) peak height and/or through an interference experiment.

The ground state and low energy excitations of the SU(m|n) supersymmetricHaldane-Shastry spin chain are analyzed. In the thermodynamic limit, it is found that the ground state degeneracy is finite only for the SU(m|0) and SU(m|1) spin chains, while the dispersion relation for the low energy and low momentum excitations is linear for all values of m and n. We show that the low energy excitations of the SU(m|1) spin chain are described by a conformal field theory of m non-interacting Dirac fermions which have only positive energies; the central charge of this theory is m/2. Finally, for n ≥ 1, the partition functions of the SU(m|n) Haldane-Shastry spin chain and the SU(m|n) Polychronakos spin chain are shown to be related in a simple way in the thermodynamic limit at low temperatures.

We construct polarized spin reversal operator (PSRO) which yields a class of representations for the BC N type of Weyl algebra, and subsequently use this PSRO to find out novel exactly solvable variants of the BC N type of spin Calogero model. The strong coupling limit of such spin Calogero models generates the BC N type of Polychronakos spin chains with PSRO. We derive the exact spectra of the BC N type of spin Calogero models with PSRO and compute the partition functions of the related spin chains by using the freezing trick. We also find out an interesting relation between the partition functions of the BC N type and A N −1 type of Polychronakos spin chains. Finally, we study spectral properties like level density and distribution of spacing between consecutive energy levels for BC N type of Polychronakos spin chains with PSRO.