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:2006.14624 · Nuclear Physics B

Exchange interactions, Yang-Baxter relations and transparent particles

Alexios P. Polychronakos

Published Nov 16, 2020• 3 citations
arXivHTMLPDF

Abstract

We introduce a class of particle models in one dimension involving exchange interactions that have scattering properties satisfying the Yang-Baxter consistency condition. A subclass of these models exhibits reflectionless scattering, in which particles are “transparent” to each other, generalizing a property hitherto only known for the exchange Calogero model. The thermodynamics of these systems can be derived using the asymptotic Bethe-Ansatz method.

DOI
10.1016/j.nuclphysb.2020.115243

Related papers

A Note on Statistical Interactions and the Thermodynamic Bethe Ansatz

Denis Bernard, Yong-Shi Wu

We show that the thermodynamic Bethe ansatz equations for one-dimensional integrable many-body systems can be reinterpreted in such a way that they only code the statistical interactions, in the sense of Haldane, between particles of identical or different momenta. Thus, the thermodynamic properties of these systems can be characterized by the generalized ideal gases recently proposed by one of us. For example, the Yang-Yang $δ$-function gas is a gas with specific statistical interactions between particles of different momenta, while the Calogero-Sutherland system provides a model for an ideal gas of particles with a fractional statistics.

arXiv:cond-mat/9404025•Apr 12, 1994•15 citations

Completeness of Interacting Particles

Pavel Abramski

This thesis concerns the completeness of scattering states of n _-interacting particles in one dimension. Only the repulsive case is treated, where thereare no bound states and the spectrum is entirely absolutely continuous, so the scattering Hilbert space is the whole of L2(Rn). The thesis consists of 4 chapters: The first chapter describes the model, the scattering states as given by the Bethe Ansatz, and the main completeness problem. The second chapter contains the proof of the completeness relation in the case of two particles: n = 2. This case had in fact already been treated by B. Smit (1997), [17], but it is useful to include this case as it clarifies the more general case. In particular, the more algebraic approach used for the n-particle case is illustrated in this simple example. In Chapter 3 the case n = 3 is examined. This is useful for illustrative purposes as the scattering states can still be written explicitly term by term and it is not yet necessary to introduce the complicated notation used in the general case. On the other hand, this case shows up certain technical difficulties to do with the non-commutativity of the permutation group (S3) which do not occur in the 2-particle case. Finally, Chapter 4 contains the proof of the completeness relation in the general n-particle case. The method used is the same as in the 3-particle case, but the algebra is much more complicated. In particular a number of interesting lemmas and one theorem is proved. The first lemma for 3-particle case and its generalisation - theorem for n-particle case essentially concerns the Yang-Baxter relation for this model, as first written by Yang. Indeed, Yang proposed his version of these relations as a consistency condition for the Bethe Ansatz solution of the model but never actually gave a complete proof of the consistency given these relations. Here a complete inductive proof is given. Some algebraic manipulation reduces the left-hand side of the completeness relation to a simpler form. Another lemma, which seems to be new, then shows that this expression does not contain divergent terms and consists of a sum of integrals similar to those encountered in the 2- and 3-particle cases. Evaluation of these integrals then leads to the required _-relation.

•Jan 1, 2011•0 citations

Integrability of two-species partially asymmetric exclusion processes

Ivan Lobaskin, M. R. Evans, Kirone Mallick

Abstract We work towards the classification of all one-dimensional exclusion processes with two species of particles that can be solved by a nested coordinate Bethe ansatz (BA). Using the Yang–Baxter equations, we obtain conditions on the model parameters that ensure that the underlying system is integrable. Three classes of integrable models are thus found. Of these, two classes are well known in literature, but the third has not been studied until recently, and never in the context of the BA. The Bethe equations are derived for the latter model as well as for the associated dynamics encoding the large deviation of the currents.

arXiv:2212.11646•Journal of Physics A Mathematical and Theoretical•Mar 17, 2023•6 citations

Integrability in three dimensions: Algebraic Bethe ansatz for anyonic models

Sh. Khachatryan, A. Ferraz, Andreas Klümper +1 more

We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang–Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson) scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.

Nuclear Physics B•Aug 13, 2015•5 citations

Aspects of Integrable Models

Pranav Diwakar

The objective of this thesis is to study the isotropic XXX-1/2 spin chain model using the Algebraic Bethe Ansatz. To this end, we discuss the concept of integrability as well as the Lax operator and R-matrix, which help generate as many commuting operators in involution as there are degrees of freedom. We establish that the spin chain Hamiltonian belongs to this set and provide a definition of a state vector whose parameters, the Bethe roots, are constrained by a set of equations called the Bethe Ansatz Equations. We show that there is a one-to-one correspondence between the Bethe roots and the eigenfunctions of the system. Next, we proceed to study the nature of the low-lying excitations of both the ferromagnetic and antiferromagnetic model in the thermodynamic limit N → ∞ and show that the Bethe roots can be grouped into complexes or strings, which behave like bound states. We see that integrability is directly related to diffractionless scattering, which is obeyed by systems whose scattering matrices satisfy the Yang-Baxter Equation. In order to provide a more physical interpretation, we calculate the scattering matrix of the two-body problem for a system that satisfies the Yang-Baxter Equation and obtain exchange relations that are identical to those obtained using the Algebraic Bethe Ansatz for the XXX-1/2 spin chain model. Finally, we calculate the scattering matrix for a two-body problem interacting with a delta potential and show that this is the same as what we derived using the Coordinate Bethe Ansatz.

•Oct 18, 2018•0 citations

Diffractive scattering of three particles in one dimension: A simple result for weak violations of the Yang-Baxter equation

Austen Lamacraft

We study scattering of three equal-mass particles in one dimension. Integrable interactions are synonymous with nondiffractive scattering, meaning that the set of incoming momenta for any scattering event coincides with the set of outgoing momenta. A system is integrable if the two-particle scattering matrix obeys the Yang-Baxter equation. Nonintegrable interactions correspond to diffractive scattering, where the set of outgoing momenta may take on all values consistent with energy and momentum conservation. Such processes play a vital role in the kinetics of one-dimensional gases, where binary collisions are unable to alter the distribution function. When integrability is broken weakly, the result is a small diffractive scattering amplitude. Our main result is a simple formula for the diffractive part of the scattering amplitude, when the violation of the Yang-Baxter equation is small. Although the derivation is given for $\ensuremath{\delta}$-function interactions, the result depends only on the two-particle scattering matrix, and should therefore also apply to finite-range interactions close to integrable.

arXiv:1211.4110•Physical Review A•Jan 16, 2013•26 citations
We introduce a class of particle models in one dimension involving exchange interactions that have scattering properties satisfying the Yang-Baxter consistency condition. A subclass of these models exhibits reflectionless scattering, in which particles are “transparent” to each other, generalizing a property hitherto only known for the exchange Calogero model. The thermodynamics of these systems can be derived using the asymptotic Bethe-Ansatz method.
We show that the thermodynamic Bethe ansatz equations for one-dimensional integrable many-body systems can be reinterpreted in such a way that they only code the statistical interactions, in the sense of Haldane, between particles of identical or different momenta. Thus, the thermodynamic properties of these systems can be characterized by the generalized ideal gases recently proposed by one of us. For example, the Yang-Yang δδδ-function gas is a gas with specific statistical interactions between particles of different momenta, while the Calogero-Sutherland system provides a model for an ideal gas of particles with a fractional statistics.
This thesis concerns the completeness of scattering states of n _-interacting particles in one dimension. Only the repulsive case is treated, where thereare no bound states and the spectrum is entirely absolutely continuous, so the scattering Hilbert space is the whole of L2(Rn). The thesis consists of 4 chapters: The first chapter describes the model, the scattering states as given by the Bethe Ansatz, and the main completeness problem. The second chapter contains the proof of the completeness relation in the case of two particles: n = 2. This case had in fact already been treated by B. Smit (1997), [17], but it is useful to include this case as it clarifies the more general case. In particular, the more algebraic approach used for the n-particle case is illustrated in this simple example. In Chapter 3 the case n = 3 is examined. This is useful for illustrative purposes as the scattering states can still be written explicitly term by term and it is not yet necessary to introduce the complicated notation used in the general case. On the other hand, this case shows up certain technical difficulties to do with the non-commutativity of the permutation group (S3) which do not occur in the 2-particle case. Finally, Chapter 4 contains the proof of the completeness relation in the general n-particle case. The method used is the same as in the 3-particle case, but the algebra is much more complicated. In particular a number of interesting lemmas and one theorem is proved. The first lemma for 3-particle case and its generalisation - theorem for n-particle case essentially concerns the Yang-Baxter relation for this model, as first written by Yang. Indeed, Yang proposed his version of these relations as a consistency condition for the Bethe Ansatz solution of the model but never actually gave a complete proof of the consistency given these relations. Here a complete inductive proof is given. Some algebraic manipulation reduces the left-hand side of the completeness relation to a simpler form. Another lemma, which seems to be new, then shows that this expression does not contain divergent terms and consists of a sum of integrals similar to those encountered in the 2- and 3-particle cases. Evaluation of these integrals then leads to the required _-relation.
Abstract We work towards the classification of all one-dimensional exclusion processes with two species of particles that can be solved by a nested coordinate Bethe ansatz (BA). Using the Yang–Baxter equations, we obtain conditions on the model parameters that ensure that the underlying system is integrable. Three classes of integrable models are thus found. Of these, two classes are well known in literature, but the third has not been studied until recently, and never in the context of the BA. The Bethe equations are derived for the latter model as well as for the associated dynamics encoding the large deviation of the currents.
We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang–Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson) scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.
The objective of this thesis is to study the isotropic XXX-1/2 spin chain model using the Algebraic Bethe Ansatz. To this end, we discuss the concept of integrability as well as the Lax operator and R-matrix, which help generate as many commuting operators in involution as there are degrees of freedom. We establish that the spin chain Hamiltonian belongs to this set and provide a definition of a state vector whose parameters, the Bethe roots, are constrained by a set of equations called the Bethe Ansatz Equations. We show that there is a one-to-one correspondence between the Bethe roots and the eigenfunctions of the system. Next, we proceed to study the nature of the low-lying excitations of both the ferromagnetic and antiferromagnetic model in the thermodynamic limit N → ∞ and show that the Bethe roots can be grouped into complexes or strings, which behave like bound states. We see that integrability is directly related to diffractionless scattering, which is obeyed by systems whose scattering matrices satisfy the Yang-Baxter Equation. In order to provide a more physical interpretation, we calculate the scattering matrix of the two-body problem for a system that satisfies the Yang-Baxter Equation and obtain exchange relations that are identical to those obtained using the Algebraic Bethe Ansatz for the XXX-1/2 spin chain model. Finally, we calculate the scattering matrix for a two-body problem interacting with a delta potential and show that this is the same as what we derived using the Coordinate Bethe Ansatz.
We study scattering of three equal-mass particles in one dimension. Integrable interactions are synonymous with nondiffractive scattering, meaning that the set of incoming momenta for any scattering event coincides with the set of outgoing momenta. A system is integrable if the two-particle scattering matrix obeys the Yang-Baxter equation. Nonintegrable interactions correspond to diffractive scattering, where the set of outgoing momenta may take on all values consistent with energy and momentum conservation. Such processes play a vital role in the kinetics of one-dimensional gases, where binary collisions are unable to alter the distribution function. When integrability is broken weakly, the result is a small diffractive scattering amplitude. Our main result is a simple formula for the diffractive part of the scattering amplitude, when the violation of the Yang-Baxter equation is small. Although the derivation is given for $\ensuremath{\delta}$-function interactions, the result depends only on the two-particle scattering matrix, and should therefore also apply to finite-range interactions close to integrable.