The gap probabilities at the hard and soft edges of scaled random matrix ensembles with orthogonal symmetry are known in terms of τ -functions. Extending recent work relating to the soft edge, it is shown that these τ -functions, and their generalizations to contain a generating function like parameter, can be expressed as Fredholm determinants. These same Fredholm determinants also occur in ex...

Funding information UK-EPSRC Grant ‘Attosecond Electron Dynamics in Molecular and Condensed Phase Systems’, Grant/Award Number: EP/I032517/1 Abstract We present an implementation of the spin-dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry-adapted functions. Using S eigenfunctions helps to reduce the size of conf...

We study the symmetry classes of graphene quantum dots, both open and closed, through the conductance and energy level statistics. For abrupt termination of the lattice, these properties are well described by the standard orthogonal and unitary ensembles. However, for smooth mass confinement, special time-reversal symmetries associated with the sublattice and valley degrees of freedom are criti...

Our interest is in the cumulative probabilities Pr(L(t) ≤ l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry r...

P.J. Forrester and N.S. Witte Department of Mathematics and Statistics (and School of Physics), University of Melbourne, Victoria 3010, Australia; Email: [email protected]; [email protected] It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact τ -functions for certain Painlevé systems. We sho...

In quantum mechanics, symmetry groups can be realized by projective, as well as by ordinary unitary, representations. For the permutation symmetry relevant to quantum statistics of N indistinguishable particles, the simplest properly projective representation is highly non-trivial, of dimension 2 N−1 2 , and is most easily realized starting with spinor geometry. Quasiparticles in the Pfaffian q...

We study a few classes of Hilbert space operators whose matrix representations are complex symmetric with respect to a preferred orthonormal basis. The existence of this additional symmetry has notable implications and, in particular, it explains from a unifying point of view some classical results. We explore applications of this symmetry to Jordan canonical models, selfadjoint extensions of s...

Particle systems admit a variety of tensor product structures (TPSs) depending on the complete system of commuting observables chosen for the analysis. Different notions of entanglement are associated with these different TPSs. Global symmetry transformations and dynamical transformations factor into products of local unitary operators with respect to certain TPSs and not with respect to others...

Our interest is in the cumulative probabilities Pr(L(t) l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry res...

We construct second-quantized (field) theories on coset spaces of pseudo-unitary groups U(N+, N−). The existence of degenerated quantum vacua (coherent states of zero modes) leads to a breakdown of the original pseudo-unitary symmetry. The action of some spontaneously broken symmetry transformations destabilize the vacuum and make it to radiate. We study the structure of this thermal radiation ...