Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1008.1054 · Journal of Mathematical Physics

Higher charges and regularized quantum trace identities in su(1,1) Landau–Lifshitz model

A. Melikyan, Aleksandr Pinzul, G. Weber

Published Dec 1, 2010• 8 citations
arXivHTMLPDF

Abstract

We solve the operator ordering problem for the quantum continuous integrable su(1,1) Landau–Lifshitz model, and give a prescription to obtain the quantum trace identities, and the spectrum for the higher-order local charges. We also show that this method, based on operator regularization and renormalization, which guarantees quantum integrability, as well as the construction of self-adjoint extensions, can be used as an alternative to the discretization procedure, and unlike the latter, is based only on integrable representations.

DOI
10.1063/1.3509374

Related papers

On quantum integrability of the Landau–Lifshitz model

A. Melikyan, Aleksandr Pinzul

We investigate the quantum integrability of the Landau–Lifshitz (LL) model and solve the long-standing problem of finding the local quantum Hamiltonian for the arbitrary n-particle sector. The particular difficulty of the LL model quantization, which arises due to the ill-defined operator product, is dealt with by simultaneously regularizing the operator product and constructing the self-adjoint extensions of a very particular structure. The diagonalizibility difficulties of the Hamiltonian of the LL model, due to the highly singular nature of the quantum-mechanical Hamiltonian, are also resolved in our method for the arbitrary n-particle sector. We explicitly demonstrate the consistency of our construction with the quantum inverse scattering method due to Sklyanin [Lett. Math. Phys. 15, 357 (1988)] and give a prescription to systematically construct the general solution, which explains and generalizes the puzzling results of Sklyanin for the particular two-particle sector case. Moreover, we demonstrate the S-matrix factorization and show that it is a consequence of the discontinuity conditions on the functions involved in the construction of the self-adjoint extensions.

arXiv:0812.0188•Journal of Mathematical Physics•Oct 1, 2009•14 citations

Thermodynamics of the quantumsu(1, 1) Landau–Lifshitz model

A Melikyan, A Pinzul

We present the thermodynamics of the quantum su (1, 1) Landau–Lifshitz model, following our earlier exposition of the quantum integrability of the theory, which is based on the construction of self-adjoint extensions, leading to a regularized quantum Hamiltonian for an arbitrary n -particle sector. Starting from general discontinuity properties of the functions used to construct the self-adjoint extensions, we derive the thermodynamic Bethe ansatz equations. We show that due to the non-symmetric and singular kernel, the self-consistency implies that only negative chemical potential values are allowed, which leads to the conclusion that, unlike its su (2) counterpart, the su (1, 1) LL theory at T = 0 has no instabilities.

arXiv:1001.5450•Journal of Statistical Mechanics Theory and Experiment•Jun 3, 2010•7 citations

Classical Integrable Spin Chains of Landau–Lifshitz Type from R-Matrix Identities

D. Domanevsky, Andrei Vladimirovich Zotov

We describe a family of 1+1 classical integrable space-discrete models of the Landau–Lifshitz type through the usage of ansatz for U–V (Lax) pair with spectral parameter satisfying the semi-discrete Zakharov–Shabat equation. The ansatz for U–V pair is based on $$R$$ -matrices satisfying the associative Yang–Baxter equation and certain additional properties. Equations of motion are obtained using a set of $$R$$ -matrix identities. In the continuous limit we reproduce the previously known family of the higher rank Landau–Lifshitz equations.

arXiv:2505.09918•Journal of Experimental and Theoretical Physics Letters•Jun 1, 2025•2 citations

Quantization of models with non-compact quantum group symmetry: modular XXZ magnet and lattice sinh-Gordon model

Andrei Bytsko, Jörg Teschner

We define and study certain integrable lattice models with non-compact quantum group symmetry (the modular double of ) including an integrable lattice regularization of the sinh-Gordon model and a non-compact version of the XXZ model. Their fundamental R -matrices are constructed in terms of the non-compact quantum dilogarithm. Our choice of the quantum group representations naturally ensures self-adjointness of the Hamiltonian and the higher integrals of motion. These models are studied with the help of the separation of variables method. We show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations (Baxter equation and quantum Wronskian equation) which is characterized by suitable analytic and asymptotic properties. A key technical tool is the so-called -operator, for which we give an explicit construction. Our results allow us to establish some connections to related results and conjectures on the sinh-Gordon theory in continuous spacetime. Our approach also sheds some light on the relations between massive and massless models (in particular, the sinh-Gordon and Liouville theories) from the point of view of their integrable structures.

arXiv:hep-th/0602093•Journal of Physics A Mathematical and General•Sep 27, 2006•167 citations

Coherent states withSU(2) andSU(3) charges

Manu Mathur, Samir K Paul

We define coherent states carrying SU (2) charge by exploiting the Schwinger boson representation of the SU (2) Lie algebra. These coherent states satisfy the continuity property and provide resolution of identity on S 3 . We further generalize these techniques to construct the corresponding SU (3) charge coherent states. The SU ( N ) extension is also discussed.

arXiv:quant-ph/0303146•Journal of Physics A Mathematical and General•Aug 23, 2005•1 citation

Coherent-state path integral operator ordering, and regularization: A simple example

L. C. Lew Yan Voon

For a restricted class of potential problems, the Feynman path integral in the coordinate representation circumvents operator-ordering ambiguities. We will show that, in the coherent-state representation, the traditional continuum time path integral cannot resolve this issue. Using a simple solvable model, we carry out a careful analysis of the continuum limit with regularization. The latter is found to be necessary to address the operator-ordering issue.

Canadian Journal of Physics•Oct 1, 1991•0 citations
We solve the operator ordering problem for the quantum continuous integrable su(1,1) Landau–Lifshitz model, and give a prescription to obtain the quantum trace identities, and the spectrum for the higher-order local charges. We also show that this method, based on operator regularization and renormalization, which guarantees quantum integrability, as well as the construction of self-adjoint extensions, can be used as an alternative to the discretization procedure, and unlike the latter, is based only on integrable representations.
We investigate the quantum integrability of the Landau–Lifshitz (LL) model and solve the long-standing problem of finding the local quantum Hamiltonian for the arbitrary n-particle sector. The particular difficulty of the LL model quantization, which arises due to the ill-defined operator product, is dealt with by simultaneously regularizing the operator product and constructing the self-adjoint extensions of a very particular structure. The diagonalizibility difficulties of the Hamiltonian of the LL model, due to the highly singular nature of the quantum-mechanical Hamiltonian, are also resolved in our method for the arbitrary n-particle sector. We explicitly demonstrate the consistency of our construction with the quantum inverse scattering method due to Sklyanin [Lett. Math. Phys. 15, 357 (1988)] and give a prescription to systematically construct the general solution, which explains and generalizes the puzzling results of Sklyanin for the particular two-particle sector case. Moreover, we demonstrate the S-matrix factorization and show that it is a consequence of the discontinuity conditions on the functions involved in the construction of the self-adjoint extensions.
We present the thermodynamics of the quantum su (1, 1) Landau–Lifshitz model, following our earlier exposition of the quantum integrability of the theory, which is based on the construction of self-adjoint extensions, leading to a regularized quantum Hamiltonian for an arbitrary n -particle sector. Starting from general discontinuity properties of the functions used to construct the self-adjoint extensions, we derive the thermodynamic Bethe ansatz equations. We show that due to the non-symmetric and singular kernel, the self-consistency implies that only negative chemical potential values are allowed, which leads to the conclusion that, unlike its su (2) counterpart, the su (1, 1) LL theory at T = 0 has no instabilities.
We describe a family of 1+1 classical integrable space-discrete models of the Landau–Lifshitz type through the usage of ansatz for U–V (Lax) pair with spectral parameter satisfying the semi-discrete Zakharov–Shabat equation. The ansatz for U–V pair is based on RRR -matrices satisfying the associative Yang–Baxter equation and certain additional properties. Equations of motion are obtained using a set of RRR -matrix identities. In the continuous limit we reproduce the previously known family of the higher rank Landau–Lifshitz equations.
We define and study certain integrable lattice models with non-compact quantum group symmetry (the modular double of ) including an integrable lattice regularization of the sinh-Gordon model and a non-compact version of the XXZ model. Their fundamental R -matrices are constructed in terms of the non-compact quantum dilogarithm. Our choice of the quantum group representations naturally ensures self-adjointness of the Hamiltonian and the higher integrals of motion. These models are studied with the help of the separation of variables method. We show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations (Baxter equation and quantum Wronskian equation) which is characterized by suitable analytic and asymptotic properties. A key technical tool is the so-called -operator, for which we give an explicit construction. Our results allow us to establish some connections to related results and conjectures on the sinh-Gordon theory in continuous spacetime. Our approach also sheds some light on the relations between massive and massless models (in particular, the sinh-Gordon and Liouville theories) from the point of view of their integrable structures.