arXiv subjects
Publications and source records attributed to Yao-Zhong Zhang.
All finite dimensional Nichols algebras with diagonal type of connected finite dimensional Yetter-Drinfeld modules over finite cyclic group $\mathbb Z_n$ are found. It is proved that finite dimensional Nichols algebra over $\mathbb Z_2$ is a quantum linear space and Nichols algebra of connected Yetter-Drinfeld module $V$ over $\mathbb Z_n$ with $\dim V >3$ is infinite dimensional.
A recursive method for construction of symmetric irreducible representations of O(2l+1) in the O(2l + 1) supset O(3) basis for identical boson systems is proposed. The formalism is realized based on the group chain U(2l + 1) supset U(2l- 1) x U(2), of which the symmetric irreducible representations are simply reducible. The basis vectors of the O(2l+1) supset O(2l-1) x U(1) can easily be constructed from those of U(2l + 1) supset U(2l-1) x U(2) supset O(2l-1) x U(1) with no boson pairs, from which one can construct symmetric irreducible representations of O(2l+1) in the O(2l-1) x U(1) basis when all symmetric irreducible representations of O(2l-1) are known. As a starting point, basis vectors of symmetric irreducible representations of O(5) are constructed in the O1(3) x U(1) basis. Matrix representations of O(5) supset O1(3) x U(1), together with the elementary Wigner coefficients, are presented. After the angular momentum projection, a three-term relation in determining the expansion coefficients of the O(5) supset O(3) basis vectors in terms of those of the O1(3) x U(1) is derived. The eigenvectors of the projection matrix with zero eigenvalues constructed according to the three-term relation completely determine the basis vectors of O(5) supset O(3). Formulae for evaluating the elementary Wigner coefficients of O(5) supset O(3) are derived explicitly. Analytical expressions of some elementary Wigner coefficients of O(5) supset O(3) for the coupling (tau, 0) x (1,0) with resultant angular momentum quantum number L = 2 tau+ 2 - k for k = 0, 2, 3,...,6 with a multiplicity 2 case for k = 6 are presented.
A new angular momentum projection for systems of particles with arbitrary spins is formulated based on the Heine-Stieltjes correspondence, which can be regarded as the solutions of the mean-field plus pairing model in the strong pairing interaction G ->Infinity limit. Properties of the Stieltjes zeros of the extended Heine-Stieltjes polynomials, of which the roots determine the projected states, and the related Van Vleck zeros are discussed. The electrostatic interpretation of these zeros is presented. As examples, applications to n nonidentical particles of spin-1/2 and to identical bosons or fermions are made to elucidate the procedure and properties of the Stieltjes zeros and the related Van Vleck zeros. It is shown that the new angular momentum projection for n identical bosons or fermions can be simplified with the branching multiplicity formula of U(N) supset O(3) and the special choices of the parameters used in the projection. Especially, it is shown that the solutions for identical bosons can always be expressed in terms of zeros of Jacobi polynomials. However, unlike non-identical particle systems, the n-coupled states of identical particles are non-orthogonal with respect to the multiplicity label after the projection.
We demonstrate the occurrence of free quasi-particle excitations obeying generalised exclusion statistics in a BCS model with asymmetric pair scattering. The results are derived from an exact solution of the Hamiltonian, which was obtained via the algebraic Bethe ansatz utilising the representation theory of an underlying Yangian algebra. The free quasi-particle excitations are associated to highest-weight states of the Yangian algebra, corresponding to a class of analytic solutions of the Bethe ansatz equations.
We investigate the quantum teleportation between a conformal detector Alice and an inertial detector Bob in de Sitter space in two schemes, (i) one uses free scalar modes and (ii) one utilizes cavity to store qubit. We show that the fidelity of the teleportation is degraded for Bob in both cases. While the fidelity-loss is due to the Gibbons-Hawking effect associated with his cosmological horizon in the scheme (i), the entanglement decreases in the scheme (ii) because the ability to entangle the cavities is reduced by the spacetime curvature. With a cutoff at Planck-scale, comparing with the standard Bunch-Davies choice, we also show that the possible Planckian physics cause extra modifications to the fidelity of the teleportation protocol in both schemes.
We present a model for compositional distributional semantics related to the framework of Coecke et al. (2010), and emulating formal semantics by representing functions as tensors and arguments as vectors. We introduce a new learning method for tensors, generalising the approach of Baroni and Zamparelli (2010). We evaluate it on two benchmark data sets, and find it to outperform existing leading methods. We argue in our analysis that the nature of this learning method also renders it suitable for solving more subtle problems compositional distributional models might face.
We present new quasi-exactly solvable models with inverse quartic, sextic, octic and decatic power potentials, respectively. We solve these models exactly via the functional Bethe ansatz method. For each case, we give closed-form solutions for the energies and the wave functions as well as analytical expressions for the allowed potential parameters in terms of a set of algebraic equations.
We present a pairing Hamiltonian of the Bardeen-Cooper-Schrieffer form which exhibits two quantum critical lines of deconfined excitations. This conclusion is drawn using the exact Bethe ansatz equations of the model which admit a class of simple, analytic solutions. The deconfined excitations obey generalised exclusion statistics. A notable property of the Hamiltonian is that it is non-hermitian. Although it does not have a real spectrum for all choices of coupling parameters, we provide a rigorous argument to establish that real spectra occur on the critical lines. The critical lines are found to be invariant under a renormalisation group map.
The quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one.
We study the quantum correlations of free scalar field with vacuum ambiguity of de Sitter space. We show the occurrence of degradation of quantum entanglement and quantum discord between field modes for inertial observer in curved space due to the radiation associated with cosmological horizon. In particular, we find that quantum correlations can be used to encode infinite de Sitter invariant vacua, which correspond to infinite set of possible physical worlds. This may provide a superselection rule of physical vacuum via quantum information tasks. We also discuss the simulation of such quantum effects of vacuum ambiguity in ion trap experiments.
We obtain the determinant representations of the scalar products for the XXZ Gaudin model with generic non-diagonal boundary terms.
We present a unified treatment of exact solutions for a class of four quantum mechanical models, namely the singular anharmonic potential, the generalized quantum isotonic oscillator, the soft-core Coulomb potential, and the non-polynomially modified oscillator. We show that all four cases are reducible to the same basic ordinary differential equation, which is quasi-exactly solvable. A systematic and closed form solution to the basic equation is obtained via the Bethe ansatz method. Using the result, general exact expressions for the energies and the allowed potential parameters are given explicitly for each of the four cases in terms of the roots of a set of algebraic equations. A hidden $sl(2)$ algebraic structure is also discovered in these models.
We prove that Nichols algebras of irreducible Yetter-Drinfeld modules over classical Weyl groups $A \rtimes \mathbb S_n$ supported by $\mathbb S_n$ are infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter-Drinfeld modules over classical Weyl groups $A \rtimes \mathbb S_n$ supported by $A$ to be finite dimensional.
The quasi-Gaudin algebra was introduced to construct integrable systems which are only quasi-exactly solvable. Using a suitable representation of the quasi-Gaudin algebra, we obtain a class of bosonic models which exhibit this curious property. These models have the notable feature that they do not preserve U(1) symmetry, which is typically associated to a non-conservation of particle number. An exact solution for the eigenvalues within the quasi-exactly solvable sector is obtained via the algebraic Bethe ansatz formalism.
The free-field representations of the $D(2,1;\a)$ current superalgebra and the corresponding energy-momentum tensor are constructed. The related screening currents of the first kind are also presented.
With the help of the Drinfeld twist or factorizing F-matrix for the eight-vertex SOS model, we obtain the explicit determinant expression of the partition function of the eight-vertex model with a generic non-diagonal reflecting end and domain wall boundary condition. Our result shows that, contrary to the eight-vertex model without a reflection end, the partition function can be expressed as a single determinant.
With the help of the F-basis provided by the Drinfeld twist or factorizing F-matrix for the open XXZ spin chain with non-diagonal boundary terms, we obtain the determinant representation of the partition function of the six-vertex model with a non-diagonal reflecting end under domain wall boundary condition.
We find all polynomials $Z(z)$ such that the differential equation $${X(z)\frac{d^2}{dz^2}+Y(z)\frac{d}{dz}+Z(z)}S(z)=0,$$ where $X(z), Y(z), Z(z)$ are polynomials of degree at most 4, 3, 2 respectively, has polynomial solutions $S(z)=\prod_{i=1}^n(z-z_i)$ of degree $n$ with distinct roots $z_i$. We derive a set of $n$ algebraic equations which determine these roots. We also find all polynomials $Z(z)$ which give polynomial solutions to the differential equation when the coefficients of X(z) and Y(z) are algebraically dependent. As applications to our general results, we obtain the exact (closed-form) solutions of the Schrödinger type differential equations describing: 1) Two Coulombically repelling electrons on a sphere; 2) Schrödinger equation from kink stability analysis of $ϕ^6$-type field theory; 3) Static perturbations for the non-extremal Reissner-Nordström solution; 4) Planar Dirac electron in Coulomb and magnetic fields; and 5) O(N) invariant decatic anharmonic oscillator.