Questions tagged [integrable-systems]

Liouville integrability means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian vector fields associated to the invariants of the foliation span the tangent distribution: there exists a maximal set of Poisson-commuting invariants in phase space. May be used more broadly for systems possessing simple analytic solutions.

128 questions
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Integrability condition of perturbations of Wess-Zumino-Witten (WZW) models

When one tries to analyze the renormalization group of marginal perturbations of Wess-Zumino-Witten (WZW) model in 1+1d, only those "integrable perturbations" can be computed analytically. I wonder ...
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Question about notation of a Jimbo's paper

I am reading Jimbo's Introduction to Yang-Baxter Equations. And I am confused by the notation he used in the definition: Here he uses $u\in C$ without previously mentioning what is $C$. I guess ...
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Couple of non-interacting, non-integrable Hamiltonian systems

I have two Hamiltonians, $H(p_1, q_1, \dots, p_n, q_n)$ and $H'(p_{n+1}, q_{n+1}, \dots, p_m, q_m)$. They are non-interacting, indeed they have different variables. Moreover, I know that they are both ...
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Liouville's integrability theorem: action-angle variables

For classical dynamical systems, let $I_{\alpha}$ stand for independent constants of motion which commute with each other. Pg 323 of Jose-Saletan and also 'Remark 11.12' on pg 443 of Fasano-Marmi's '...
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Troubles with Haldane Shastry Spin Chain

I'm actually reading the article of Shastry "Exact solution of an S= 1/2 Heisenberg Antiferromagnetic Chain with Long-rnaged interactions", Phys. Rev. Lett. 60, 639 (1988)" The articles ...
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Physical intuition behind Poincaré–Bendixson theorem

The Poincaré–Bendixson theorem states that: In continuous systems, chaotic behaviour can only arise in systems that have 3 or more dimensions. What is the best way to understand this criteria ...
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Is there any approximation in which the double pendulum has an exact solution?

We know that the double pendulum isn't an integrable system, since only energy is conserved versus two degrees of freedom. My question is, does a physical approximation exist, in which the total time ...
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Link between integrability and soliton solutions

I have been doing some research on the properties and dynamics of solitons (in particular, solitons in superfluids) and several works and papers mention the link between solitonic solutions and ...
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How to deduce the commutativity of the transfer matrix in algebraic bethe ansatz (and integrable systems)?

This is supposed to be a one-line derivation, but how to deduce the commutativity of the transfer matrix (in algebraic bethe ansatz) from the RTT relation? Thank you!
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Meaning of Yang-Baxter equation for classical $r$-matrix

I'm reading this [math/9802054] paper on the structure of the phase space of Chern-Simons TQFT. I'm stuck at the definition of the classical $r$-matrix, which goes as follows: This might sound dumb, ...
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What is the relationship between the integrability of a quantum many-body system and thermalization?

If a quantum many-body system is integrable, does it imply the system would always thermalized or many-body localized?
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R-matrix for spin chains

In algebraic Bethe ansatz procedure, one of the central objects is the R-matrix satisfying the Yang-Baxter equation, but all the papers/books give directly its expression without deriving it, so my ...
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Elliptic R-matrix and Yang Baxter solution for XYZ model [duplicate]

in the framework of QISM, How can i derive the R-matrix for XYZ Heisenberg model?
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$R$ matrix for XYZ spin chain [duplicate]

Trying to understand how the Algebraic Bethe Ansatz works, I'm actually reading some papers and trying to apply for XXZ or XYZ model. But my problem is that I don't know how to find the R-matrix ...
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Question about a system with all bounded orbits closed and maximal integrable

Given Hamiltonian system with $2n$-dim phase space, if there exist $k\ge n$ independent integrals of motions then we call it integrable Hamiltonian system. The largest number of independent integrals ...
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What does a non-integrable Hamiltonian system mean? [duplicate]

Is it something that cant be solved exactly? Are there any examples of such system? Is high-dimension lattice models like Ising model or Hubbard models considered as non-integrable systems?
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How are action variables linked to first integrals of a Hamiltonian?

Suppose I have an integrable Hamiltonian system $H(q_{1}, p_{1},..., q_{n}, p_{n})$, with first integrals $F_{1} = H, F_{2},..., F_{n}$. Excluding certain singular level sets (i.e. separatrices), one ...
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What is the point of using statistical mechanics on an integrable system?

On one hand, the ergodic hypothesis is usually justified by chaotic dynamics. On the other hand, it seems necessary to consider an integrable system in order to compute a nice, closed-form partition ...
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Reference for Bethe Ansatz solution of 1D spinless Hubbard model

I want to numerically solve 1D spinless Hubbard model using Bethe Ansatz. Can you provide me some online references for that.
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Proof of constructing Action-Angle Coordinates on Hamiltonian System

By Liouville-Arnold Theorem, we know we can construct action-angle coordinates such that the Hamiltonian system, when described in these coordinates, will have a form that is integrable by quadratures....
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Any model integrable but not separable?

In textbooks on classical mechanics, the exactly solvable models are all separable. Is there any model integrable but not separable?
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I am studying L. Fadeev's "How Algebraic Bethe Ansatz works for integrable model". He takes Heisenberg $XXX_{1/2}$ model as an example. After obtaining the Bethe Ansatz Equations (BAE) for the roots {$... 1answer 157 views Quantum Integrable models and string theory I’ve seen some recent papers on integrability by string theorists like Ed Witten or Kevin Costello but I don’t know why quantum integrable models are of interest to the string community. I would ... 1answer 161 views Conditions for periodic motions in classical mechanics What are the conditions for classical motion to be periodic? In one dimension, if the motion is bounded, then it is also periodic. However, I don't think this generalizes to higher dimensions. I am ... 0answers 64 views Correlation functions of exactly solvable 1D quantum models Quantum 1d spin-1/2 transverse Ising and XY models are both related to 2d classical Ising model. Are there any known simple explicit relations between correlation functions of this models? Something ... 2answers 69 views Maximal number of conserved quantities (classical integrability) In these notes on page 4 the author says that if a$2d$-dimensional phase space has$d$conserved quantities$F_{\mu}$that Poisson commute, then$H$can be written as a function of the$F_{\mu}$. Why ... 1answer 315 views Are there necessary and sufficient conditions for ergodicity? What are the necessary and sufficient conditions (if any) for ergodicity (or non-ergodicity)? I see for instance that some integrable systems are not ergodic. For instance a linear chain of harmonic ... 0answers 33 views Conserved Quantities And Lagrangian Mechanics [duplicate] By Noether's Theorem we know that any conserved quantity can be associated with a continuous symmetry. I was reading Kepler's Problem and I saw that there can be conserved quantities like Laplace-... 1answer 77 views XXh model for$1/2$-spin chain I'm looking for correlation function (i.e.$\langle g \rvert \hat{S}_i^z \hat{S}_{i+n}^z\lvert g \rangle$, where$g$stands for ground state) for given Hamiltonian($\hat{S}^z_i = \hat{c}^+_{j} \hat{c}...
The KdV equation $$v_t+\frac{1}{4}v_{xxx}-\frac{3}{2}vv_x=0$$ was originally invented to model waves in shallow water. However, it is well known that it also has applications in quantum mechanics. ...