General topological features and instanton vacuum in quantum Hall and spin liquids

We introduce the concept of super universality in quantum Hall and spin liquids which has emerged from previous studies. It states that all the fundamental features of the quantum Hall effect are generically displayed as general topological features of the $\theta$ parameter in nonlinear sigma models in two dimensions. To establish super universality in spin liquids we revisit the mapping by Haldane who argued that the anti ferromagnetic Heisenberg spin $s$ chain is effectively described by the O(3) nonlinear sigma model with a $\theta$ term. By combining the path integral representation for the dimerized spin $s=1/2$ chain with renormalization group decimation techniques we generalise the Haldane approach to include a more complicated theory, the fermionic rotor chain, involving four different renormalization group parameters. We show how the renormalization group calculation technique can be used to lay the bridge between the fermionic rotor chain and the sigma model. As an integral and fundamental aspect of the mapping we establish the topological significance of the dangling spin at the edge of the chain which is in all respects identical to the massless chiral edge excitations in quantum Hall liquids. We consider various different geometries of the spin chain and show that for each of the different geometries correspond to a topologically equivalent quantum Hall liquid.Comment: Title changed, Section 2 and Appendix expanded, an error in the expression for theta correcte

Ground state fidelity and quantum phase transitions in free Fermi systems

We compute the fidelity between the ground states of general quadratic fermionic hamiltonians and analyze its connections with quantum phase transitions. Each of these systems is characterized by a $L\times L$ real matrix whose polar decomposition, into a non-negative $\Lambda$ and a unitary $T$, contains all the relevant ground state (GS) information. The boundaries between different regions in the GS phase diagram are given by the points of, possibly asymptotic, singularity of $\Lambda$. This latter in turn implies a critical drop of the fidelity function. We present general results as well as their exemplification by a model of fermions on a totally connected graph.Comment: 4 pages, 2 figure

New nonlinear coherent states and some of their nonclassical properties

We construct a displacement operator type nonlinear coherent state and examine some of its properties. In particular it is shown that this nonlinear coherent state exhibits nonclassical properties like squeezing and sub-Poissonian behaviour.Comment: 3 eps figures. to appear in J.Opt

Phase properties of a new nonlinear coherent state

We study phase properties of a displacement operator type nonlinear coherent state. In particular we evaluate the Pegg-Barnett phase distribution and compare it with phase distributions associated with the Husimi Q function and the Wigner function. We also study number- phase squeezing of this state.Comment: 8 eps figures. to appear in J.Opt

Coherent states of non-Hermitian quantum systems

We use the Gazeau-Klauder formalism to construct coherent states of non-Hermitian quantum systems. In particular we use this formalism to construct coherent state of a PT symmetric system. We also discuss construction of coherent states following Klauder's minimal prescription.Comment: to appear in Phys.Lett

General 2+1 Dimensional Effective Actions and Soliton Spin Fractionalization

We propose actions for non-linear sigma models on cosets $G/H$ in 2+1 dimensions that include the most general non-linear realizations of Chern-Simons terms. When $G$ is simply connected and $H$ contains $r$ commuting U(1) factors, there are $r$ different topologically conserved charges and generically $r$ different types of topological solitons. Soliton spin fractionalizes as a function of the Chern-Simons couplings, with independent spins associated to each species of soliton charge, as well as to pairs of different charges. This model of soliton spin fractionalization generalizes to arbitrary $G/H$ a model of Wilczek and Zee for one type of soliton.Comment: 10 pp., Plain Tex, no figure

The quantum state vector in phase space and Gabor's windowed Fourier transform

Representations of quantum state vectors by complex phase space amplitudes, complementing the description of the density operator by the Wigner function, have been defined by applying the Weyl-Wigner transform to dyadic operators, linear in the state vector and anti-linear in a fixed `window state vector'. Here aspects of this construction are explored, with emphasis on the connection with Gabor's `windowed Fourier transform'. The amplitudes that arise for simple quantum states from various choices of window are presented as illustrations. Generalized Bargmann representations of the state vector appear as special cases, associated with Gaussian windows. For every choice of window, amplitudes lie in a corresponding linear subspace of square-integrable functions on phase space. A generalized Born interpretation of amplitudes is described, with both the Wigner function and a generalized Husimi function appearing as quantities linear in an amplitude and anti-linear in its complex conjugate. Schr\"odinger's time-dependent and time-independent equations are represented on phase space amplitudes, and their solutions described in simple cases.Comment: 36 pages, 6 figures. Revised in light of referees' comments, and further references adde

Non-commutative Duality: High Spin Fields and CP1CP^1 Model with Hopf Term

We show that the non-commutative $CP^1$ model coupled with Hopf term in 3 dimensions is equivalent to an interacting spin-$s$ theory where the spin $s$ of the dual theory is related to the coefficient of the Hopf term. We use the Seiberg-Witten map in studying this non-commutative duality equivalence, keeping terms to order $\theta$ and show that the spin of the dual theory do not get any $\theta$ dependant corrections. The map between current correlators show that topological index of the solitons in the non-commutative $CP^1$ model is unaffected by $\theta$ where as the Noether charge of the corresponding dual particle do get a $\theta$ dependence. We also show that this dual theory smoothly goes to the limit $\theta\to 0$ giving dual theory in the commutative plane.Comment: 13 pages, more references and a footnote are added. Version to appear in Phys. Lett.

Scalar Field Theory on Fuzzy S^4

Scalar fields are studied on fuzzy $S^4$ and a solution is found for the elimination of the unwanted degrees of freedom that occur in the model. The resulting theory can be interpreted as a Kaluza-Klein reduction of CP^3 to S^4 in the fuzzy context.Comment: 16 pages, LaTe

Regularization of the Singular Inverse Square Potential in Quantum Mechanics with a Minimal length

We study the problem of the attractive inverse square potential in quantum mechanics with a generalized uncertainty relation. Using the momentum representation, we show that this potential is regular in this framework. We solve analytically the s-wave bound states equation in terms of Heun's functions. We discuss in detail the bound states spectrum for a specific form of the generalized uncertainty relation. The minimal length may be interpreted as characterizing the dimension of the system.Comment: 30 pages, 3 figure