arXiv subjects

Yao-Zhong Zhang

Publications and source records attributed to Yao-Zhong Zhang.

At least 37 records · Page 2Linked to original sources

arXiv2023

Bethe ansatz solutions and hidden $sl(2)$ algebraic structure for a class of quasi-exactly solvable systems

The construction of analytic solutions for quasi-exactly solvable systems is an interesting problem. We revisit a class of models for which the odd solutions were largely missed previously in the literature: the anharmonic oscillator, the singular anharmonic oscillator, the generalized quantum isotonic oscillator, non-polynomially deformed oscillator, and the Schrödinger system from the kink stability analysis of $ϕ^6$-type field theory. We present a systematic and unified treatment for the odd and even sectors of these models. We find generic closed-form expressions for constraints to the allowed model parameters for quasi-exact solvability, the corresponding energies and wavefunctions. We also make progress in the analysis of solutions to the Bethe ansatz equations in the spaces of model parameters and provide insight into the curves/surfaces of the allowed parameters in the parameter spaces. Most previous analyses in this aspect were on a case-by-case basis and restricted to the first excited states. We present analysis of the solutions (i.e. roots) of the Bethe ansatz equations for higher excited states (up to levels $n$=30 or 50). The shapes of the root distributions change drastically across different regions of model parameters, illustrating phenomena analogous to phase transition in context of integrable models. Furthermore, we also obtain the $sl(2)$ algebraization for the class of models in their respective even and odd sectors in a unified way.

math-ph↗

arXiv2023

On polynomial symmetry algebras underlying superintegrable systems in Darboux spaces

We review three different approaches to polynomial symmetry algebras underlying superintegrable systems in Darboux spaces. The first method consists of using deformed oscillator algebra to obtain finite-dimensional representations of quadratic algebras. This allow one to gain information on the spectrum of the superintegrable systems. The second method has similarities with the induced module construction approach in the context of Lie algebras and can be used to construct infinite dimensional representations of the symmetry algebras. Explicit construction of these representations is a non-trivial task due to the non-linearity of the polynomial algebras. This method allows the construction of states of the superintegrable systems beyond the reach of separation of variables. As a result, we are able to construct a large number of states in terms of Airy, Bessel and Whittaker functions which would be difficult to obtain in other ways. We also discuss the third approach which is based on the notion of commutants of subalgebras in the enveloping algebra of a Poisson algebra or a Lie algebra. This allows us to discover new superintegrable models in the Darboux spaces and to construct their integrals and symmetry algebras via polynomials in the enveloping algebras.

math-ph↗

arXiv2023

Polynomial algebras from commutants: Classical and Quantum aspects of $\mathcal{A}_3$

We review some aspects of the Racah algebra $R(n)$, including the closure relations, pointing out their role in superintegrability, as well as in the description of the symmetry algebra for several models with coalgebra symmetry. The connection includes the generic model on the $(n-1)$ sphere. We discuss an algebraic scheme of constructing Hamiltonians, integrals of the motion and symmetry algebras. This scheme reduces to the Racah algebra $R(n)$ and the model on the $(n-1)$ sphere only for the case of specific differential operator realizations. We review the method, which allows us to obtain the commutant defined in the enveloping algebra of $\mathfrak{sl}(n)$ in the classical setting. The related $\mathcal{A}_3$ polynomial algebra is presented for the case $\mathfrak{sl}(3)$. An explicit construction of the quantization of the scheme for $\mathcal{A}_3$ by symmetrization of the polynomial and the replacement of the Berezin bracket by commutator and symmetrization of the polynomial relations is presented. We obtain the additional quantum terms. These explicit relations are of interest not only for superintegrability, but also for other applications in mathematical physics.

math-ph↗

arXiv2023

Infinite dimensional representations of cubic and quintic algebras and special functions

Finite and Infinite-dimensional representations of symmetry algebras play a significant role in determining the spectral properties of physical Hamiltonians. In this paper, we introduce and apply a practical method to construct infinite dimensional representations of certain polynomial algebras which appear in the context of quantum superintegrable systems. Explicit construction of these representations is a non-trivial task due to the non-linearity of the polynomial algebras. Our method has similarities with the induced module construction approach in the context of Lie algebras and allows the construction of states of the superintegrable systems beyond the reach of separation of variables. Our main focus is the representations of the polynomial algebras underlying superintegrable systems in 2D Darboux spaces. We are able to construct a large number of states in terms of complicated expressions of Airy, Bessel and Whittaker functions which would be difficult to obtain in other ways.

math-ph↗

arXiv2023

Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces

Superintegrable systems in 2D Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces. In this paper, we obtain exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems. We also introduce generic cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and cubic integrals, and obtain their Casimir operators and deformed oscillator realizations. As examples of applications, we present three classes of new superintegrable systems with cubic symmetry algebras in 2D Darboux spaces.

nlin.SI↗

arXiv2022

Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras

Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial symmetry algebra is proposed. The case of the special linear Lie algebra $\mathfrak{sl}(n)$ is discussed in detail, where an explicit basis for the commutant with respect to the Cartan subalgebra is obtained, and the order of the polynomial algebra is computed. It is further shown that, with an appropriate realization of $\mathfrak{sl}(n)$, this provides an explicit connection with the generic superintegrable model on the $(n-1)$-dimensional sphere $\mathbb{S}^{n-1}$ and the related Racah algebra $R(n)$. In particular, we show explicitly how the models on the $2$-sphere and $3$-sphere and the associated symmetry algebras can be obtained from the quadratic and cubic polynomial algebras generated by the commutants defined in the enveloping algebra of $\mathfrak{sl}(3)$ and $\mathfrak{sl}(4)$, respectively. The construction is performed in the classical (or Poisson-Lie) context, where the Berezin bracket replaces the commutator.

math-ph↗

arXiv2022

Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras

We propose a systematic procedure to construct polynomial algebras from intermediate Casimir invariants arising from (semisimple or non-semisimple) Lie algebras $\mathfrak{g}$. In this approach, we deal with explicit polynomials in the enveloping algebra of $\mathfrak{g} \oplus \mathfrak{g} \oplus \mathfrak{g}$. We present explicit examples to show how these Lie algebras can display different behaviours and can lead to Abelian algebras, quadratic algebras or more complex structures involving higher order nested commutators. Within this framework, we also demonstrate how virtual copies of the Levi factor of a Levi decomposable Lie algebra can be used as a tool to construct "copies" of polynomial algebras. Different schemes to obtain polynomial algebras associated to algebraic Hamiltonians have been proposed in the literature, among them the use of commutants of various type. The present approach is different and relies on the construction of intermediate Casimir invariants in the enveloping algebra $\mathcal{U}(\mathfrak{g} \oplus \mathfrak{g} \oplus \mathfrak{g})$.

math-ph↗

arXiv2022

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining higher order ladder operators. One feature of these algebras is that they preserve by construction some aspects of the structure of the $\mathfrak{gl}(n)$ Lie algebra. Among the classes of Hamiltonians arising in this framework are various deformations of harmonic oscillator and singular oscillator related to exceptional orthogonal polynomials and even Painlevé and higher order Painlevé analogs. As an explicit example, we investigate a new three-dimensional superintegrable system related to Hermite exceptional orthogonal polynomials of type III. Among the main results is the determination of the degeneracies of the model in terms of the finite-dimensional irreducible representations of the polynomial algebra.

math-ph↗

arXiv2021

Exact solution of the two-axis two-spin Hamiltonian

Bethe ansatz solution of the two-axis two-spin Hamiltonian is derived based on the Jordan-Schwinger boson realization of the SU(2) algebra. It is shown that the solution of the Bethe ansatz equations can be obtained as zeros of the related extended Heine-Stieltjes polynomials. Symmetry properties of excited levels of the system and zeros of the related extended Heine-Stieltjes polynomials are discussed. As an example of an application of the theory, the two equal spin case is studied in detail, which demonstrates that the levels in each band are symmetric with respect to the zero energy plane perpendicular to the level diagram and that excited states are always well entangled.

quant-ph↗

arXiv2021

$N$-dimensional Smorodinsky-Winternitz model and related higher rank quadratic algebra ${\cal SW}(N)$

The $N$-dimensional Smorodinsky-Winternitz system is a maximally superintegrable and exactly solvable model, being subject of study from different approaches. The model has been demonstrated to be multiseparable with wavefunctions given by Laguerre and Jacobi polynomials. In this paper we present the complete symmetry algebra ${\cal SW}(N)$ of the system, which it is a higher-rank quadratic one containing the recently discovered Racah algebra ${\cal R}(N)$ as subalgebra. The substructures of distinct quadratic ${\cal Q}(3)$ algebras and their related Casimirs are also studied. In this way, from the constraints on the oscillator realizations of these substructures, the energy spectrum of the $N$-dimensional Smorodinsky-Winternitz system is obtained. We show that ${\cal SW}(N)$ allows different set of substructures based on the Racah algebra ${\cal R}({ N})$ which can be applied independently to algebraically derive the spectrum of the system.

math-ph↗

arXiv2021

Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures

The recent interest in the study of higher-rank polynomial algebras related to $n$-dimensional classical and quantum superintegrable systems with coalgebra symmetry and their connection with the generalised Racah algebra $R(n)$, a higher-rank generalisation of the rank one Racah algebra $R(3)$, raises the problem of understanding the role played by the $n - 2$ quadratic subalgebras generated by the left and right Casimir invariants (sometimes referred as universal quadratic substructures) from this new perspective. Such subalgebra structures play a signficant role in the algebraic derivation of spectrum of quantum superintegrable systems. In this work, we tackle this problem and show that the above quadratic subalgebra structures can be understood, at a fixed $n > 3$, as the images of $n - 2$ injective morphisms of $R(3)$ into $R(n)$. We show that each of the $n-2$ quadratic subalgebras is isomorphic to the rank one Racah algebra $R(3)$. As a byproduct, we also obtain an equivalent presentation for the universal quadratic subtructures generated by the partial Casimir invariants of the coalgebra. The construction, which relies on explicit (symplectic or differential) realisations of the generators, is performed in both the classical and the quantum cases.

math-ph↗

arXiv2020

Embedding of the Racah Algebra R($\boldsymbol{n}$) and Superintegrability

The rank-$1$ Racah algebra $R(3)$ plays a pivotal role in the theory of superintegrable systems. It appears as the symmetry algebra of the $3$-parameter system on the $2$-sphere from which all second-order conformally flat superintegrable models in $2$D can be obtained by means of suitable limits and contractions. A higher rank generalization of $R(3)$, the so-called rank $n-2$ Racah algebra $R(n)$, has been considered recently and showed to be the symmetry algebra of the general superintegrable model on the $(n-1)$-sphere. In the present work, we show that such an algebraic structure naturally arises as embedded inside a larger quadratic algebra characterizing $n$D superintegrable models with non-central terms. This is shown both in classical and quantum mechanics through suitable (symplectic or differential) realisations of the Racah and additional generators. Among the main results, we present an explicit construction of the complete symmetry algebras for two families of $n$-dimensional maximally superintegrable models, the Smorodinsky-Winternitz system and the generalized Kepler-Coulomb system. For both families, the underlying symmetry algebras are higher-rank quadratic algebras containing the Racah algebra $R(n)$ as subalgebra. High-order algebraic relations among the generators of the full quadratic algebras are also obtained both in the classical and quantum frameworks. These results should shed new light to the further understanding of the structures of quadratic algebras in the context of superintegrable systems.

math-ph↗

arXiv2020

Analytic solutions of the Teukolsky equation for massless perturbations of any spin in de Sitter background

We present analytic solutions to the Teukolsky equation for massless perturbations of any spin in the 4-dimensional de Sitter background. The angular part of the equation fixes the separation constant to a discrete set and its solution is given by hypergeometric polynomials. For the radial part, we derive analytic power series solution which is regular at the poles and determine a transcendental function whose zeros give the characteristic values of the wave frequency. We study the existence of explicit polynomial solutions to the radial equation and obtain two classes of singular closed-form solutions, one with discrete wave frequencies and the other with continuous frequency spectra.

gr-qc↗

arXiv2020

New R-matrices with non-additive spectral parameters and integrable models of strongly correlated fermions

We present a general formula for constructing R-matrices with non-additive spectral parameters associated with a type-I quantum superalgebra. The spectral parameters originate from two one-parameter families of inequivalent finite-dimensional irreducible representations of the quantum superalgebra upon which the R-matrix acts. Applying to the quantum superalgebra $U_q(gl(2|1))$, we obtain the explicit expression for the $U_q(gl(2|1))$-invariant R-matrix which is of non-difference form in spectral parameters. Using this R-matrix we derive a new two-parameter integrable model of strongly correlated electrons with pure imaginary pair hopping terms.

math-ph↗

arXiv2020

Remarks on Replica Method and Sachdev-Ye-Kitaev Model

Through tracing back to EA/SK models, we disentangle the construction logic of SYK model. During the construction, we recall the technical essence of replica method. Then we point out the redundance of the flavor group and the slave indices as well as the color group via the generalization from 2-index SY model to its 4-index version and the simplification of the latter in terms of Majorana fermions. Spectacularly, such a simplification reserves the system's self-consistency condition. Getting rid of the redundance, the model itself and its effective action have the same $O(N_s)$ structural symmetry, while it realizes $AdS_2/CFT_1$ holography by the emergent conformal symmetry. We also clarify the model transmutation to matrix model and argue that the disorder-averaged SYK model naturally preserves a holographic nature. Based on the transmutation and the correspondence with vector model, SYK model is formally a hybrid of these two models. Its quantum and semiclassical properties are also discussed respectively.

hep-th↗

arXiv2020

On analytic solutions of the driven, 2-photon and two-mode quantum Rabi models

By applying the Bogoliubov transformations and through the introduction of the Bargmann-Hilbert spaces, we obtain analytic representations of solutions to the driven Rabi model without ${\cal Z}_2$ symmetry, and the 2-photon and two-mode quantum Rabi models. In each case, transcendental function is analytically derived whose zeros give the energy spectrum of the model. The zeros can be numerically found by standard root-search techniques.

quant-ph↗

arXiv2019

Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras

We present a new method for constructing $D$-dimensional minimally superintegrable systems based on block coordinate separation of variables. We give two new families of superintegrable systems with $N$ ($N\leq D$) singular terms of the partitioned coordinates and involving arbitrary functions. These Hamiltonians generalize the singular oscillator and Kepler systems. We derive their exact energy spectra via separation of variables. We also obtain the quadratic algebras satisfied by the integrals of motion of these models. We show how the quadratic symmetry algebras can be constructed by novel application of the gauge transformations from those of the non-partitioned cases. We demonstrate that these quadratic algebraic structures display an universal nature to the extent that their forms are independent of the functions in the singular potentials.

math-ph↗

arXiv2019

Finiteness of Nichols Algebras and Nichols (Braided) Lie Algebras

It is shown that if $\mathfrak B(V) $ is connected Nichols algebra of diagonal type with $\dim V>1$, then $\dim (\mathfrak L^-(V)) = \infty$ $($resp. $ \dim (\mathfrak L(V)) = \infty $$)$ $($ resp. $ \dim (\mathfrak B(V)) = \infty $$)$ if and only if $Δ(\mathfrak B(V)) $ is an arithmetic root system and the quantum numbers (i.e. the fixed parameters) of generalized Dynkin diagrams of $V$ are of finite order. Sufficient and necessary conditions for $m$-fold adjoint action in $\mathfrak B(V)$ equal to zero, viz. $\overline{l}_{x_{i}}^{m}[x_{j}]^ -=0$ for $x_i,~x_j\in \mathfrak B(V)$, are given.

math.QA↗