The one-dimensional Hubbard model is integrable in the sense that it has an infinite family of conserved currents. We explicitly construct a ladder operator which can be used to iteratively generate all of the conserved current operators. This construction is different from that used for Lorentz invariant systems such as the Heisenberg model. The Hubbard model is not Lorentz invariant, due to the separation of spin and charge excitations. The ladder operator is obtained by a very general formalism which is applicable to any model that can be derived from a solution of the Yang-Baxter equation.The one-dimensional Hubbard model is integrable in the sense that it has an infinite family of conserved currents. We explicitly construct a ladder operator which can be used to iteratively generate all of the conserved current operators. This construction is different from that used for Lorentz invariant systems such as the Heisenberg model. The Hubbard model is not Lorentz invariant, due to the separation of spin and charge excitations. The ladder operator is obtained by a very general formalism which is applicable to any model that can be derived from a solution of the Yang-Baxter equation.

arXiv:cond-mat/0011368Physical Review LettersMay 28, 200135 citations

Current density in heisenberg representation

Michele Cini, S. Fubini

Il Nuovo CimentoJul 1, 19551 citation

On some mathematical aspects of the Heisenberg relation

Liu Zhe

Science China MathematicsAug 4, 201112 citations

Closed hierarchy of Heisenberg equations in integrable models with Onsager algebra

Oleg Lychkovskiy

Dynamics of a quantum system can be described by coupled Heisenberg equations. In a generic many-body system these equations form an exponentially large hierarchy that is intractable without approximations. In contrast, in an integrable system a small subset of operators can be closed with respect to commutation with the Hamiltonian. As a result, the Heisenberg equations for these operators can form a smaller closed system amenable to an analytical treatment. We demonstrate that this indeed happens in a class of integrable models where the Hamiltonian is an element of the Onsager algebra. We explicitly solve the system of Heisenberg equations for operators from this algebra. Two specific models are considered as examples: the transverse field Ising model and the superintegrable chiral 3-state Potts model.Dynamics of a quantum system can be described by coupled Heisenberg equations. In a generic many-body system these equations form an exponentially large hierarchy that is intractable without approximations. In contrast, in an integrable system a small subset of operators can be closed with respect to commutation with the Hamiltonian. As a result, the Heisenberg equations for these operators can form a smaller closed system amenable to an analytical treatment. We demonstrate that this indeed happens in a class of integrable models where the Hamiltonian is an element of the Onsager algebra. We explicitly solve the system of Heisenberg equations for operators from this algebra. Two specific models are considered as examples: the transverse field Ising model and the superintegrable chiral 3-state Potts model.

arXiv:2012.00388SciPost PhysicsJun 1, 202127 citations

Integration of the Heisenberg equations for inverse power-law potentials

M. Hron, Mohsen Razavy

The direct method of integration of the operator Heisenberg equations of motion is extended to the solution of quantum tunneling when the central potential is a sum of inverse powers of the radial distance. By obtaining the equation of motion for the Weyl-ordered basis set { ${\mathit{S}}_{\mathit{m},\mathit{n}}$ (t)} formed from r(t) and ${\mathit{p}}_{\mathit{r}}$ (t), one can express the time evolution of any member of the set as an infinite sum involving the operators { ${\mathit{S}}_{\mathit{m},\mathit{n}}$ (0)}. The direct integration enables one to find the expectation values of the radial position operator and its conjugate momentum and higher moments of these operators as functions of time. \textcopyright{} 1996 The American Physical Society.The direct method of integration of the operator Heisenberg equations of motion is extended to the solution of quantum tunneling when the central potential is a sum of inverse powers of the radial distance. By obtaining the equation of motion for the Weyl-ordered basis set {${\mathit{S}}_{\mathit{m},\mathit{n}}$(t)} formed from r(t) and ${\mathit{p}}_{\mathit{r}}$(t), one can express the time evolution of any member of the set as an infinite sum involving the operators {${\mathit{S}}_{\mathit{m},\mathit{n}}$(0)}. The direct integration enables one to find the expectation values of the radial position operator and its conjugate momentum and higher moments of these operators as functions of time. \textcopyright{} 1996 The American Physical Society.

Physical Review ANov 1, 19964 citations

On the Heisenberg commutation relation. I

Konrad Schmüdgen

Journal of Functional AnalysisJan 1, 198370 citations

{\mathit{S}}_{\mathit{m},\mathit{n}}

(t)} formed from r(t) and

{\mathit{p}}_{\mathit{r}}

(t), one can express the time evolution of any member of the set as an infinite sum involving the operators {

{\mathit{S}}_{\mathit{m},\mathit{n}}

(0)}. The direct integration enables one to find the expectation values of the radial position operator and its conjugate momentum and higher moments of these operators as functions of time. \textcopyright{} 1996 The American Physical Society.