# 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.

231 questions
Filter by
Sorted by
Tagged with
30 views

### Integrability of long range Heisenberg chain

Is the long range heisenberg spin 1/2 chain integrable? More generally, is the long range version of famous spin chain models integrable?
198 views

### Particularity of symmetries generated by the action variables of a classically integrable system

Background I was reading this article on the unviersal $SO(4)$ and $SU(3)$ symmetries in all central potential problem. Turns out every bounded planar motion in any smooth central potential will all ...
20 views

### $\check{R}$ matrix acting on $2$ out of $m$ vector spaces [migrated]

We know that $\check{R} \in \operatorname{End}(V \otimes V)$ is a solution of Yang-Baxter equation: $$\check{R}_{23} \check{R}_{12} \check{R}_{23}=\check{R}_{12}\check{R}_{23} \check{R}_{12}$$ where ...
3k views

### What exactly are the 12 conserved quantities in the Two-Body Problem?

The Two-Body problem consists of 6 2nd-order differential equations \begin{equation} \ddot{\mathbf{r}}_1 = \frac{1}{m_1}\ \mathbf{F_g} \\ \ddot{\mathbf{r}}_2 = -\ \frac{1}{m_2}\ \mathbf{F_g} \end{...
39 views

### Definition of conserved quantities in integrable system

This question is about the definition of conserved quantities integrable systems. Using Algebraic Bethe ansatz,a family of commuting operators $F(\lambda)$ can be contructed by taking a partial trace (...
46 views

### Explicit construction of integrals of motion in 1d XXZ model for few sites

I was studying the algebraic Bethe ansatz for the spin-1/2 XXZ model. In the end one ends up with $2^L$ integrals of motion $Q_k$ that commute with the Hamiltonian, (https://doi.org/10.1103/...
69 views

### Is there a generic behavior of Spectral Form Factor for Integrable models?

The spectral form factor is defined as (usually taken at $\beta = 0$ by definition along with disorder average) \begin{equation}\label{eq:SFF1} g(\beta,t) = \left| \frac{Z(\beta,t)}{Z(\beta)}\...
70 views

### Necessity and Sufficiency of Yang-Baxter Equation for Integrability

Yang-Baxter Equation (YBE) seems to be a sufficient condition for integrability, i.e. if you have an $R$-matrix satisfying YBE, then the model is integrable. But how about the reverse? More ...
224 views

### Conserved Quantities in Kepler Problem?

In our classical mechanics class, professor said that Kepler's problem is a kind of Integrable System such that the number of conserved quantities would be equal to the number of degrees of freedom. ...
25 views

### Integrable many-body system and complete set of conserved charges

In an integrable quantum system (say XXZ model), where there is an extensive number of conserved charges, does the set of local conserved charges obtained from expanding the log of the transfer matrix ...
186 views

1 vote
107 views

### Can the conservation law be extended to the 2d Burgers equation?

I know that for the 1d inviscid Burgers' equation of the form $$\frac {\partial u}{\partial t} + u\frac {\partial u}{\partial x} = 0$$ the conservation law converts $u(u)_x$ to $(u^2/2)_x$. However, ...
1 vote
28 views

### Spatial component of energy-mometum tensor for the 2D infinite cylinder [duplicate]

I am reading Zamolodchikov's paper and a question arises, so I would like to ask it. In this paper, he considers QFT on a 2D infinite cylinder where spatial direction is compactified on a circle of ...
121 views

### Deriving Burger's equation for energy eigenvalues in $T\bar T$-deformed theories

When doing $T \bar T$-deformation to 2d CFTs, it is interesting to ask how the original energy spectrum is shifted throughout the procedure. This is done as follows. As mentioned in several papers/...
93 views

58 views

### Semi-classical spinning strings and AdS-CFT

I'm trying to understand how the AdS/CFT correspondence is precisely formulated when on the bulk side people are working with the string theory as a sigma model on the worldsheet expanded about some ...