arXiv subjects

Yao-Zhong Zhang

Publications and source records attributed to Yao-Zhong Zhang.

At least 19 recordsLinked to original sources

arXiv2026

Superintegrable systems in curved spaces via coalgebras and deformed Racah algebras

We determine the Poisson algebras generated by partial Casimirs of undeformed and non-standard $q$-deformed $\mathfrak{sl}(2)$ coalgebras, respectively. The derivation is performed on the abstract algebraic level and does not depend on any particular symplectic realization. For the 3- and 4-site cases the constructions associated with the undeformed $\mathfrak{sl}(2)$ produce the rank-one and rank-two Racah Poisson algebras, respectively, while those with the non-standard deformed $\mathfrak{sl}$ yield finite quadratic Poisson algebras generated by connected partial Casimirs. At both the three and four sites, the final closure coefficients do not explicitly depend on the deformation parameter. The results hold at the Poisson coalgebra level and hence apply to any realizations which preserve the defining brackets and coproducts. Canonical symplectic realizations give rise to Hamiltonians in spaces of non-constant curvature. For the Stäckel $q$-deformed Smorodinsky-Winternitz model we show explicitly how the universal coalgebra sector is completed to maximal superintegrability. An exact two-block identity, proved directly in the localized Poisson algebra without appealing to a symplectic realization, controls the additional Stäckel integrals. In three dimensions, it generates a quadratic Poisson block with its own Casimir relation; in four dimensions, its coupling to the full left/right coalgebra family is organized into three nested quadratic blocks, and the same identity extends this nested coupling to arbitrary $N$. Explicit higher-site partial Casimirs and factorized bracket generators are collected in the appendices.

math-ph↗

arXiv2026

Quasi-exactly solvable deformations of quantum systems associated with exceptional orthogonal polynomials

Quasi-exactly solvable (QES) deformations of harmonic and singular oscillators have been widely studied via a range of approaches. In this paper, we obtain families of new QES deformations of solvable quantum systems associated with exceptional orthogonal polynomials (EOPs). The construction builds upon the theory of Darboux-Crum transformations and the classification of exactly solvable quantum systems associated with EOPs of Hermite type. It is shown that the deformations break the exact solvability of the undeformed systems and introduce model parameters into the deformed systems that permit the existence of a finite number of polynomial solutions whose roots satisfy systems of algebraic equations. The new families presented and studied consist of deformations of quantum systems related to Hermite EOPs of type III with arbitrary codimensions. We present polynomial and rational deformations and analyze, in each case, the conditions for quasi-exact solvability in terms of Bethe ansatz equations and parameter constraints. In general, the structures of the underlying polynomial solutions are no longer associated with well-known classical orthogonal polynomials. So for the general cases, we mainly focus on presenting the new approach as well as the Bethe ansatz equations and the constraints for model parameters. As applications, we construct particular families of QES deformations related to solvable models allowing one, two, and up to four gaps, and obtain the closed form expressions for their wavefunctions and spectra. We analyze the existence of QES solutions in the spaces of model parameters, providing information on the number of solutions for given parameters.

math-ph↗

arXiv2026

Geometric construction of superintegrable Poisson projection chains via Poisson centralizers

We introduce a geometric framework for constructing superintegrable systems from Poisson centralizers (commutants) in the Lie-Poisson algebra $S(\mathfrak{g})$ of a complex semisimple Lie algebra. Starting from a chain of reductive subgroups, we study the corresponding invariant Poisson subalgebras and their Poisson centers, and formulate superintegrability in terms of a \emph{Poisson projection chain} of affine Poisson varieties. For a maximal torus $T\subset G$, we prove that the inclusions $S(\mathfrak{g})^G\subset S(\mathfrak{g})^T\subset S(\mathfrak{g})$ determine a superintegrable chain and identify the associated quotient maps $\mathfrak{g}\xrightarrow{χ_T}\mathfrak{g}//T\xrightarrowρ\mathfrak{g}//G$. The rank (transcendence degree) computations yield the expected dimension split between commuting Hamiltonians and first integrals, and we describe the corresponding symplectic leaves in the intermediate space. Several examples illustrate how the centralizer generators organize into explicit superintegrable Poisson chains.

math-ph↗

arXiv2026

Poisson Centralisers and Polynomial Superintegrability for Magnetic Geodesic Flows on Reductive Homogeneous Spaces

We provide a method for formulating superintegrable magnetic geodesic flows on reductive homogeneous spaces $M=G/A$, with $G$ a compact semisimple Lie group and $A$ a closed subgroup of $G$. In the twisted cotangent bundle $(T^*M,ω_\varepsilon)$, with $ω_\varepsilon=ω_{\mathrm{can}}+\varepsilon\,π^*ω_{\mathrm{KKS}}$ being the canonical plus Kirillov-Kostant-Souriau (KKS) forms, we build two canonical and commuting families of polynomial first integrals: one pulled back from the Lie algebra $\mathfrak{g}$ of $G$ via the magnetic moment map $P$, and one pulled back from a $\mathrm{Ad}(A)$-invariant affine slice of $\mathfrak{m} \cong T_{eA}M$, where $eA$ is the identity of $G/A$. Their common image generates a reduced Poisson algebra obtained from a fiber tensor product, and the natural multiplication map into a Poisson subalgebra of polynomial functions $\mathcal{O}(T^*M) \subset C^\infty(T^*M)$ is Poisson and injective. The center of this fiber tensor product is contained in the Poisson center of the symmetric algebra of $\mathfrak{g}$. In a dense regular locus, the resulting projection chain realises a superintegrable system. As examples, two $\mathrm{SU}(3)$ cases are studied (regular torus and irregular $\mathrm{S}(\mathrm{U}(2)\times \mathrm{U}(1))$ quotients), which illustrate the construction and produce explicit action-angle coordinates.

math-ph↗

arXiv2025

Polynomial Poisson Algebras and Superintegrable Systems from Cartan centralisers of Types $B_3$, $C_3$ and $D_3$

In this work, we construct explicit formulas for the generators of the Cartan centralisers of complex semisimple Lie algebras $B_n,C_n$ and $D_n$, the case $A_n$ being already known \cite{campoamor2023algebraic}. The precise structures for the cases of rank-three simple Lie algebras ($B_3,C_3$ and $D_3$) are provided, and the inclusion relations between the corresponding polynomial Poisson algebras (finitely generated Poisson algebras over $\mathbb{C}[\mathfrak{h}^*]$) are illustrated. We develop the idea of constructing algebraic superintegrable systems and their integrals from the generators of these polynomial Poisson algebras. In particular, we explicitly present the algebraic superintegrable systems corresponding to the Cartan reduction chains $\mathfrak{h} \subset \mathfrak{so}(6,\mathbb{C})$, $\mathfrak{h} \subset \mathfrak{so}(7,\mathbb{C})$, and $\mathfrak{h} \subset \mathfrak{sp}(6,\mathbb{C})$.

math-ph↗

arXiv2025

New quasi-exactly solvable systems from SUSYQM and Bethe Ansatz

We give a systematic construction of new quasi-exactly solvable systems via Bethe ansatz and supersymmetric quantum mechanics (SUSYQM). Methods based on the intertwining of supercharges have been extensively used in the literature for exactly solvable systems. We generalize the state-deleting (Krein-Adler) supersymmetric transformations to quasi-exactly exactly solvable (QES) systems building on the Bethe ansatz method and related Bethe roots. This enables us to construct superpartners for a wide class of known QES systems classified previously through a hidden $sl(2)$ algebra. We present our constructions of factorizations and intertwining relations related to 1st-order SUSYQM and the $n=1$ state for 10 nonequivalent types, denoted I,...,X. In order to have a unified treatment we rely on their ODE standard form as this is also the appropriate setting to obtain the Bethe ansatz equations which constrain the polynomial solutions. This setting also allows one to deal with systems with $n$ states in a unified manner, using analysis based on the Bethe ansatz equations to build the supersymmetric transformations in terms of the Bethe ansatz roots. We derive the Schrödinger potentials for the $n=1$ superpartners of the 10 QES cases and give closed-form solutions for the spectra and wavefunctions of the corresponding QES SUSYQM systems. Furthermore, we present numerical results for higher excited states up to the $n=10$ level. The results obtained may have wider applicability as our framework is built on ODEs with polynomial coefficients.

math-ph↗

arXiv2025

Subalgebra chains and nuclear physics: Commutant approach and construction of polynomial algebras

In this paper, we review a new approach to study subalgebra chains $\mathfrak{g} \supset \mathfrak{g}'$ in the context of nuclear physics. This approach does not rely on explicit realizations as bosons or differential operators. We rely on the enveloping algebra, the notion of commutant $C_{U(\mathfrak{g})}(\mathfrak{g}^{\prime})$ and $\mathfrak{g}^{\prime}$-invariant polynomials. This approach builds on those $\mathfrak{g}^{\prime}$-invariant polynomials and finding the underlying finitely generated polynomial algebras. Those algebraic structures can then provide further information on sets of labeling operators. Another aspect of this method consists in exploiting the dual space and the symmetric algebra. Being independent of explicit realizations, it endows the algebraic relations with a universal character. We review the chains associated with $\mathfrak{su}(3) \supset \mathfrak{so}(3)$, $\mathfrak{so}(5) \supset \mathfrak{su}(2) \times \mathfrak{u}(1)$, $\mathfrak{su}(4) \supset \mathfrak{su}(2) \times \mathfrak{su}(2)$. Those chains are known as the Elliott, Seniority and Supermultiplet. We also provide new results and insights into the subalgebra chain $\mathfrak{so}(5) \supset \mathfrak{so}(3)$ of the Surfon model. For all chains, we present the related commutant, $\mathfrak{g}^{\prime}$-invariant polynomials and Poisson algebras.

math-ph↗

arXiv2025

Exact surface energies and boundary excitations of the Izergin-Korepin model with generic boundary fields

The Izergin-Korepin model is an integrable model with the simplest twisted quantum affine algebra $U_q(A_2^{(2)})$ symmetry. Applying the $t-W$ method, we derive the homogeneous zero roots Bethe ansatz equations and the corresponding zero root patterns of the Izergin-Korepin model with generic integrable boundaries. Based on these results, we analytically compute the surface energies and boundary excitations in different regimes of boundary parameters of the model. It is shown that in some regimes, correlation effect appears between two boundary fields.

math-ph↗

arXiv2025

Exact physical quantities of the $D_2^{(2)}$ spin chain model with generic open boundary conditions

We study the quantum integrable spin chain model associated with the twisted $D_2^{(2)}$ algebra (or simply the $D_2^{(2)}$ model) under generic open boundary conditions. The Hamiltonian of this model can be factorized into the sum of two staggered XXZ spin chains. Applying the $t$-$W$ method, we derive the homogeneous Bethe ansatz equations for the zeros of the transfer matrix eigenvalues and the patterns of the corresponding zeros of the staggered XXZ spin chain with generic integrable boundaries. Based on these results, we analytically compute the surface energies and excitation energies of the $D_2^{(2)}$ model in different regimes of boundary parameters.

math-ph↗

arXiv2025

Polynomial deformations of $sl(2)$ and unified algebraic framework for solutions of a class of spin models

We introduce novel polynomial deformations of the Lie algebra $sl(2)$. We construct their finite-dimensional irreducible representations and the corresponding differential operator realizations. We apply our results to a class of spin models with hidden polynomial algebra symmetry and obtain the closed-form expressions for their energies and wave functions by means of the Bethe ansatz method. The general framework enables us to give an unified algebraic and analytic treatment for three interesting spin models with hidden cubic algebra symmetry: the Lipkin-Meshkov-Glick (LMG) model, the molecular asymmetric rigid rotor, and the two-axis countertwisting squeezing model. We provide analytic and numerical insights into the structures of the roots of the Bethe ansatz equations (i.e. the so-called Bethe roots) of these models. We give descriptions of the roots on the spheres using the inverse stereographic projection. The changes in nature and pattern of the Bethe roots on the spheres indicate the existence of different phases of the models. We also present the fidelity and derivatives of the ground-state energies (with respect to model parameters) of the models. The results indicate the presence of critical points and phase transitions of the models. In the appendix, we show that, unlike the so-called Bender-Dunne polynomials, the set of polynomials in the energy $E$, $P_\ell(E)$, corresponding to each of the three spin models has two critical polynomials whose zeros give the quasi-exact energy eigenvalues of the model. Such types of polynomials seem new.

math-ph↗

arXiv2025

Algebraic structures and Hamiltonians from the equivalence classes of 2D conformal algebras

The construction of superintegrable systems based on Lie algebras and their universal enveloping algebras has been widely studied over the past decades. However, most constructions rely on explicit differential operator realisations and Marsden-Weinstein reductions. In this paper, we develop an algebraic approach based on the subalgebras of the 2D conformal algebra $\mathfrak{c}(2)$. This allows us to classify the centralisers of the enveloping algebra of the conformal algebra and construct the corresponding Hamiltonians with integrals in algebraic form. It is found that the symmetry algebras underlying these algebraic Hamiltonians are six-dimensional quadratic algebras. The Berezin brackets and commutation relations of the quadratic algebraic structures are closed without relying on explicit realisations or representations. We also give the Casimir invariants of the symmetry algebras. Our approach provides algebraic perspectives for the recent work by Fordy and Huang on the construction of superintegrable systems in the Darboux spaces.

math-ph↗

arXiv2025

Polynomial algebra from the Lie algebra reduction chain $\mathfrak{su}(4) \supset \mathfrak{su}(2) \times \mathfrak{su}(2)$: The supermultiplet model

The supermultiplet model, based on the reduction chain $\mathfrak{su}(4) \supset \mathfrak{su}(2) \times \mathfrak{su}(2)$, is revisited through the lens of commutants within universal enveloping algebras of Lie algebras. From this analysis, a collection of twenty polynomials up to degree nine emerges from the commutant associated with the $\mathfrak{su}(2) \times \mathfrak{su}(2)$ subalgebra. This study is conducted in the Poisson (commutative) framework using the Lie-Poisson bracket associated with the dual of the Lie algebra under consideration. As the main result, we obtain the polynomial Poisson algebra generated by these twenty linearly independent and indecomposable polynomials, with five elements being central. This incorporates polynomial expansions up to degree seventeen in the Lie algebra generators. We further discuss additional algebraic relations among these polynomials, explicitly detailing some of the lower-order ones. As a byproduct of these results, we also show that the recently introduced 'grading method' turns out to be essential for deriving the Poisson bracket relations when the degree of the expansions becomes so high that standard approaches are no longer applicable, due to computational limitations. These findings represent a further step toward the systematic exploration of polynomial algebras relevant to nuclear models.

math-ph↗

arXiv2025

On the construction of polynomial Poisson algebras: a novel grading approach

In this work, we refine recent results on the explicit construction of polynomial algebras associated with commutants of subalgebras in enveloping algebras of Lie algebras by considering an additional grading with respect to the subalgebra. It is shown that such an approach simplifies and systematizes the explicit derivation of the Lie--Poisson brackets of elements in the commutant, and several fundamental properties of the grading are given. The procedure is illustrated by revisiting three relevant reduction chains associated with the rank-two complex simple Lie algebra $\mathfrak{sl}(3,\mathbb{C})$. Specifically, we analyze the reduction chains $\mathfrak{so}(3) \subset \mathfrak{su}(3)$, corresponding to the Elliott model in nuclear physics, the chain $\mathfrak{o}(3) \subset \mathfrak{sl}(3,\mathbb{C})$ associated with the decomposition of the enveloping algebra of $\mathfrak{sl}(3,\mathbb{C})$ as a sum of modules, and the reduction chain $\mathfrak{h} \subset \mathfrak{sl}(3,\mathbb{C})$ connected to the Racah algebra $R(3)$. In addition, a description of the classification of the centralizer with respect to the Cartan subalgebra $\mathfrak{h}$ associated with the classical series $A_n$ in connection with its root system is reconsidered. As an illustration of the procedure, the case of $S(A_3)^\mathfrak{h}$ is considered in detail, which is connected with the rank-two Racah algebra for specific realizations of the generators as vector fields. This case has attracted interest with regard to orthogonal polynomials.

math-ph↗

arXiv2025

Stäckel transform, coupling constant metamorphosis and algebraization of quasi-exactly solvable systems

We generalize the notions of the Stäckel transform and the coupling constant metamorphosis to quasi-exactly solvable systems. We discover that for a variety of one-dimensional and separable multidimensional quasi-exactly solvable systems, their $sl(2)$ algebraizations can only be achieved via coupling constant metamorphosis after appropriate Stäckel transformations. This discovery has interesting applications, allowing us to derive algebraizations and energies for a wide class of quasi-exactly solvable systems, such as Hooke's atoms in magnetic fields and Newtonian cosmology. The approach of coupling constant metamorphosis was successfully applied previously in the context of exactly solvable, integrable and superintegrable systems. To our knowledge, the present work is the first to apply the idea and approach in the context of quasi-exactly solvable systems.

math-ph↗

arXiv2024

Charged massless scalar fields in a charged $C$-metric black hole: Exact solutions, Hawking radiation, and scattering of scalar waves

We study Hawking radiation and wave scattering of charged scalar fields in a charged $C$-metric black hole background. The conformally invariant wave equation for charged scalar fields can be separated into radial and angular parts, each with five singularities. We first show that the radial and angular equations can be respectively transformed into the general Heun equation, and then we explore exact solutions of the radial Heun equation in terms of the local Heun functions and connection coefficients. Exact behaviors of the asymptotic wave functions are determined without approximations. We further apply the exact results to derive Hawking radiation, quasinormal modes and superradiance.

gr-qc↗

arXiv2024

T-W relation and free energy of the antiperiodic XXZ chain with η=iπ/3 at a finite temperature

We study the thermodynamics of the antiperiodic XXZ chain with anisotropy parameter η=iπ/3 by means of the t-W method. We parameterize the eigenvalues of both the transfer matrix and the corresponding fused transfer matrix by their zero points instead of Bethe roots. Based on the patterns of the zero points distribution and the reconstructed entropy, we obtain the nonlinear integral equations (NLIEs) describing the thermodynamics of the model and compute its free energy at a finite temperature.

cond-mat.stat-mech↗

arXiv2024

Root patterns and exact surface energy of the spin-1 Heisenberg model with generic open boundaries

We investigate the thermodynamic limit and exact surface energy of the isotropic spin-1 Heisenberg chain with integrable generic open boundary conditions by a novel Bethe ansatz method. We obtain the homogeneous Bethe ansatz equations for the zero roots of the transfer matrix. Based on the patterns of the zero roots, we analytical calculate the densities of zero roots and the surface energies of the model in all regimes of the boundary parameters.

math-ph↗

arXiv2023

Polynomial algebras from Lie algebra reduction chains $\mathfrak{g} \supset \mathfrak{g}'$

We reexamine different examples of reduction chains $\mathfrak{g} \supset \mathfrak{g}'$ of Lie algebras in order to show how the polynomials determining the commutant with respect to the subalgebra $\mathfrak{g}'$ leads to polynomial deformations of Lie algebras. These polynomial algebras have already been observed in various contexts, such as in the framework of superintegrable systems. Two relevant chains extensively studied in Nuclear Physics, namely the Elliott chain $\mathfrak{su}(3) \supset \mathfrak{so}(3)$ and the chain $\mathfrak{so}(5) \supset \mathfrak{su}(2) \times \mathfrak{u}(1)$ related to the Seniority model, are analyzed in detail from this perspective. We show that these two chains both lead to three-generator cubic polynomial algebras, a result that paves the way for a more systematic investigation of nuclear models in relation to polynomial structures arising from reduction chains. In order to show that the procedure is not restricted to semisimple algebras, we also study the chain $\hat{S}(3) \supset \mathfrak{sl}(2,\mathbb{R}) \times \mathfrak{so}(2)$ involving the centrally-extended Schr{ö}dinger algebra in $(3+1)$-dimensional space-time. The approach chosen to construct these polynomial algebras is based on the use of the Lie-Poisson bracket, so that all the results are presented in the Poisson (commutative) setting. The advantage of considering this approach {\it vs.} the enveloping algebras is emphasized, commenting on the main formal differences between the polynomial Poisson algebras and their noncommutative analogue. As an illustrative example of the latter, the three-generator cubic algebra associated to the Elliott chain is reformulated in the Lie algebraic (noncommutative) setting.

math-ph↗