Superintegrable models are very special dynamical systems: they possess more conservation laws than what is necessary for integrability. This severely constrains their dynamical processes, and it often leads to their exact solvability, even in non-equilibrium situations. In this paper we consider special Hamiltonian deformations of superintegrable quantum circuits. The deformations break superintegrability, but they preserve integrability. We focus on a selection of concrete models and show that for each model there is an (at least) one parameter family of integrable deformations. Our most interesting example is the so-called Rule54 model. We show that the model is compatible with a one parameter family of Yang-Baxter integrable spin chains with six-site interaction. Therefore, the Rule54 model does not have a unique integrability structure, instead it lies at the intersection of a family of quantum integrable models.Superintegrable models are very special dynamical systems: they possess more conservation laws than what is necessary for integrability. This severely constrains their dynamical processes, and it often leads to their exact solvability, even in non-equilibrium situations. In this paper we consider special Hamiltonian deformations of superintegrable quantum circuits. The deformations break superintegrability, but they preserve integrability. We focus on a selection of concrete models and show that for each model there is an (at least) one parameter family of integrable deformations. Our most interesting example is the so-called Rule54 model. We show that the model is compatible with a one parameter family of Yang-Baxter integrable spin chains with six-site interaction. Therefore, the Rule54 model does not have a unique integrability structure, instead it lies at the intersection of a family of quantum integrable models.

arXiv:2205.02038SciPost PhysicsApr 26, 202417 citations

Dynamical symmetry of integrable quantum systems

Alexander A. Belavin

Nuclear Physics BMar 1, 1981255 citations

Parametric-time coherent states in a superintegrable Coulomb-type system

Nalan Kandırmaz

Abstract In this work, we constructed parametric-time coherent states (CS) for a superintegrable Coulomb-type system using the Feynman path integral formalism. This system corresponds to a deformation-dependent extension of the Kepler–Coulomb potential and exhibits maximal superintegrability governed by an indexing parameter. Using the Duru–Kleinert transformation, we mapped the problem onto an effective four-dimensional isotropic harmonic oscillator in parametric time. This transformation allowed the explicit derivation of the propagator, energy spectrum, and coherent-state wave functions while preserving the underlying symmetry structure. The resulting parametric-time CS satisfied the minimum uncertainty relation in the unsqueezed limit and exhibited non-trivial time-dependent squeezing behaviour. We showed that the deformation parameter controlled both the effective spectral scale and the redistribution of quantum fluctuations, leading to deformation-dependent noise suppression.Abstract In this work, we constructed parametric-time coherent states (CS) for a superintegrable Coulomb-type system using the Feynman path integral formalism. This system corresponds to a deformation-dependent extension of the Kepler–Coulomb potential and exhibits maximal superintegrability governed by an indexing parameter. Using the Duru–Kleinert transformation, we mapped the problem onto an effective four-dimensional isotropic harmonic oscillator in parametric time. This transformation allowed the explicit derivation of the propagator, energy spectrum, and coherent-state wave functions while preserving the underlying symmetry structure. The resulting parametric-time CS satisfied the minimum uncertainty relation in the unsqueezed limit and exhibited non-trivial time-dependent squeezing behaviour. We showed that the deformation parameter controlled both the effective spectral scale and the redistribution of quantum fluctuations, leading to deformation-dependent noise suppression.

Physica ScriptaSep 10, 20260 citations

Heisenberg-Integrable Spin Systems

Robin Steinigeweg, Heinz–Jürgen Schmidt

arXiv:math-ph/0504009Mathematical Physics Analysis and GeometryNov 20, 200815 citations

Quadratic symmetry algebras and Sturm algebraization of superintegrable systems with magnetic fields

Shams Ara, Md Fazlul Hoque, Ian Marquette

We study two three-dimensional quantum superintegrable systems in nonvanishing axially symmetric magnetic fields. For each model the integrals generate a quadratic algebra with a Casimir, and finite-dimensional deformed-oscillator representations yield algebraic spectra. The same spectra are recovered by separation of variables. In circular parabolic coordinates the separated equations are sextic biconfluent-Heun equations with finite polynomial sectors. We formulate them as Sturm eigenvalue problems with the physical energy as a parameter and the separation constant diagonalized. After gauge transformation, each equation admits an $sl(2)$ algebraization and reduces to a finite tridiagonal matrix. We also construct a pre-separation Sturm representation of each complete Hamiltonian, taking the inverse-square coupling as the Sturm eigenvalue. The transformed integrals remain commuting and, after fixing the axial integral, changing variables and gauge rotating, the two-variable Sturm Hamiltonian and its non-central integrals belong to the universal enveloping algebra $U(sl(3))$ . We thus obtain three complementary algebraic descriptions: the quadratic-algebra/deformed-oscillator representation of the original Hamiltonian, the $sl(2)$ algebraization after parabolic separation, and the $sl(3)$ algebraization of the complete Sturm problem before separation. Although the two magnetic Hamiltonians are distinct in the original classification, fixing the axial integral reduces them to the same singular 2:1 oscillator normal form with different effective parameter embeddings. This explains the common $sl(3)$ structure without implying equivalence of the original magnetic systems. These results show how distinct hidden algebraic structures and exact or quasi-exact solvability can emerge at different stages of a multiseparable superintegrable system.We study two three-dimensional quantum superintegrable systems in nonvanishing axially symmetric magnetic fields. For each model the integrals generate a quadratic algebra with a Casimir, and finite-dimensional deformed-oscillator representations yield algebraic spectra. The same spectra are recovered by separation of variables. In circular parabolic coordinates the separated equations are sextic biconfluent-Heun equations with finite polynomial sectors. We formulate them as Sturm eigenvalue problems with the physical energy as a parameter and the separation constant diagonalized. After gauge transformation, each equation admits an $sl(2)$ algebraization and reduces to a finite tridiagonal matrix. We also construct a pre-separation Sturm representation of each complete Hamiltonian, taking the inverse-square coupling as the Sturm eigenvalue. The transformed integrals remain commuting and, after fixing the axial integral, changing variables and gauge rotating, the two-variable Sturm Hamiltonian and its non-central integrals belong to the universal enveloping algebra $U(sl(3))$. We thus obtain three complementary algebraic descriptions: the quadratic-algebra/deformed-oscillator representation of the original Hamiltonian, the $sl(2)$ algebraization after parabolic separation, and the $sl(3)$ algebraization of the complete Sturm problem before separation. Although the two magnetic Hamiltonians are distinct in the original classification, fixing the axial integral reduces them to the same singular 2:1 oscillator normal form with different effective parameter embeddings. This explains the common $sl(3)$ structure without implying equivalence of the original magnetic systems. These results show how distinct hidden algebraic structures and exact or quasi-exact solvability can emerge at different stages of a multiseparable superintegrable system.

math-phCross-listedarXiv:2610.01869Oct 2, 2026

Quantum-algebraic description of quantum superintegrable systems in two dimensions

Dennis Bonatsos A, C. Daskaloyannis, Kostas D. Kokkotas

An alternative method for the description of quantum superintegrable systems in two dimensions through the use of quantum algebraic techniques is introduced. It is suggested that such systems can be described in terms of a generalized deformed oscillator, characterized by a structure function specific to the system. The energy eigenvalues corresponding to a state with finite-dimensional degeneracy can then be determined directly from the properties of the relevant structure function. The validity of the method is demonstrated in the case of the isotropic harmonic oscillator in a space with constant curvature. The method can be used for constructing the quantum versions of several classical superintegrable systems, the Holt potential being given as an example.An alternative method for the description of quantum superintegrable systems in two dimensions through the use of quantum algebraic techniques is introduced. It is suggested that such systems can be described in terms of a generalized deformed oscillator, characterized by a structure function specific to the system. The energy eigenvalues corresponding to a state with finite-dimensional degeneracy can then be determined directly from the properties of the relevant structure function. The validity of the method is demonstrated in the case of the isotropic harmonic oscillator in a space with constant curvature. The method can be used for constructing the quantum versions of several classical superintegrable systems, the Holt potential being given as an example.

Physical Review ANov 1, 199350 citations

sl(2)

algebraization and reduces to a finite tridiagonal matrix. We also construct a pre-separation Sturm representation of each complete Hamiltonian, taking the inverse-square coupling as the Sturm eigenvalue. The transformed integrals remain commuting and, after fixing the axial integral, changing variables and gauge rotating, the two-variable Sturm Hamiltonian and its non-central integrals belong to the universal enveloping algebra

U(sl(3))

. We thus obtain three complementary algebraic descriptions: the quadratic-algebra/deformed-oscillator representation of the original Hamiltonian, the

sl(2)

algebraization after parabolic separation, and the

sl(3)

algebraization of the complete Sturm problem before separation. Although the two magnetic Hamiltonians are distinct in the original classification, fixing the axial integral reduces them to the same singular 2:1 oscillator normal form with different effective parameter embeddings. This explains the common

sl(3)

structure without implying equivalence of the original magnetic systems. These results show how distinct hidden algebraic structures and exact or quasi-exact solvability can emerge at different stages of a multiseparable superintegrable system.