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.
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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?
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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 ...
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$\check{R}$ 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 ...
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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{...
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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 (...
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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/...
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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)}\...
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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 ...
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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. ...
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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 ...
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Calculate partition function of 1D quantum Heisenberg models?
For the 1D Quantum Heisenberg Spin Model:
$\displaystyle {\hat H = -\frac{1}{2} \sum_{j=1}^{N} (J_x \sigma_j^x \sigma_{j+1}^x + J_y \sigma_j^y \sigma_{j+1}^y + J_z \sigma_j^z \sigma_{j+1}^z + h\...
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Perturbations of an integrable system with no resonant tori
Suppose I have a Hamiltonian $H_0$ which is just a collection of $N$ non-interacting harmonic oscillators. Written in action-angle coordinates $(J_i, \theta_i)$ we have $H_0 = \sum_{i=1}^N \omega_i ...
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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, ...
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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 ...
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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/...
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Integrability of spin central model
I have a central model of this form $$H = \sum_{i=1}^{N} S^z_0\otimes S^z_i$$ where the $S^z_i$ acts on the $i$th element of the environment, i.e. the Hilbert space is of the following form $\mathcal{...
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Generalised hydrodynamics and the Dirac delta potential
Generalised Hydrodynamics is a theory of hydrodynamics for Quantum integrable systems. Those system are integrable in the sense that one can find an infinite number of conserved charges i.e. ...
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Solubility of integrable systems and the classical XXZ model
I've been learning about integrability in the Hamiltonian sense, and trying to wrap my mind around the analytic power afforded by integrability, both in quantum and classical systems. My goal with ...
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Does Poisson Distribution means the system is chaotic?
The Berry-Tabor Conjecture says that for classically integrable systems, the corresponding quantum systems obey the Poisson distribution for their energy-level spacing. But generally, the integrable ...
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Is a free rigid body in 3D space an integrable system? [duplicate]
I am trying to find three integrable systems with 6 degrees of freedom using the Liouville–Arnold theorem. That means that a set of integrals of motion that correspond to a conserved quantity for ...
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Are Hamilton equations for coordinate difference $q_i-q_j$ equivalent to ordinary ones?
From Lax pair equations:
$$\dot{L}=[L,M];\quad L_{i j}=p_i \delta_{i j}+\nu\left(1-\delta_{i j}\right) \frac{1}{q_i-q_j};$$
$$M_{i j}=d_i \delta_{i j}-\nu\left(1-\delta_{i j}\right) \frac{1}{\left(q_i-...
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Could you explain how the one soliton solution for the KdV equation was mathematically derived?
I’m confused as to how the solution $u(x,t)$ was attained from the KdV equation, I understand there has to be some hyperbolic integral substitution however I’m not too sure how this was attained.
If ...
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Understanding Constants of Motion and the dimension of the effective Space available
I'm currently learning about Integrable Hamiltonian Systems and Constants of Motion. In my notes, there is this passage:
"With each Constant of Motion, the dimension of the effective space ...
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Is the attractive Fermi-Hubbard model solvable by Bethe Ansatz?
I know that the one-dimension Fermi-Hubbard model is solvable by using the Bethe Ansatz method. The results I have seen, however, seem only to treat the repulsive case, i.e. $U > 0$, and I have not ...
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Sine-gordon mass term
Simple question: are there some notes or explicit calculations of the mass term from the paper of Zamolodchikov - Mass scale in the sine-gordon model and its reduction (1994)? I need to justify this ...
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Connection between diffusion and non-integrable 1D spin chains
My question concerns non-integrable (à la Bethe) 1D spin chains.
Consider, for example, the 1D non-integrable Ising model
\begin{equation}
H = \sum_{i \in \mathbb{Z}}\sigma_{i}^{z} \sigma_{i+1}^{z} + ...
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Effective action for chiral - anti-chiral interaction in 4d Chern-Simons
So the question I have is regarding the derivation of eqn (2.9) of 'Gauge theory and Integrability III' by Costello and Yamazaki - https://arxiv.org/abs/1908.02289.
Beginning with 4d Chern-Simons, we ...
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Almost all Liouville torus is preserved for small oscillation problems even if we don't use second-order approximation to potential energy, right?
In small oscillation problems, we use a second-order approximation to the potential energy function (suppose the oscillation is around the point $(0,\cdots, 0)$),
$$
V(x) = V(0) + \frac{\partial^2 V(0)...
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Which potentials in real life are separable in variables?
We usually see in 3d potential problems, that we consider potential to be separable as a sum of three independent one dimensional like potential, for all three variables, i.e
$$V(x, y, z)=V(x)+V(y)+V(...
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Proof that infinite set of conservation laws imply no pair production
In QFT in 1+1 dimensions it is known that the presence of an infinite number of conservation laws, specifically in integrable systems like Sine-Gordon, implies that there is no pair production, and ...
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Is there any exact solution to a multivariable problem in physics not using separation of variables?
Related question (The system is not limited to integrable model, so I think this question is different)
As far as I know in quantum mechanics, exact solutions for multivariable systems (from partial ...
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Trying to prove chaotic motion from the equation of a nonlinear oscillation [closed]
So I'm given the equation of a nonlinear oscillation:
$x''+ω_0^2x=λx^3$
Assume that $x_1$ and $x_2$ are solutions to the differential equation above.
Therefore;
$x = αx_1+βx_2$
$x' = αx_1'+βx_2'$
$x'' ...
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Spectral statistics and integrability
It is commonly believed that the energy level spacings of integrable systems follow a Poisson distribution, while those of classically chaotic systems follow Wigner-Dyson statistics instead. Someone ...
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Importance of the Yang-Baxter equation for integrable models
The Yang-Baxter equation, or rather one given solution of the Yang-Baxter equation corresponding to some model is often described as the fundamental relation defining an integrable model. However in ...
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How can non-chaotic curves fill a (hyper)torus and chaotic curves fill the entire energy hypersphere?
What I already know
Before I ask my question, I would prefer to briefly explain what I already know, so that any gap in my understanding could be rectified.
Note: I consider only bounded phase space ...
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Source for Feynman quote on the Bethe ansatz
There is a well-known quote of Feynman on the Bethe ansatz that appeared in an article in Asia-Pacific Physics News,volume 3, 22 (June/July 1988):
"I got really fascinated by these (1 + 1)-
...
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Why is the Lax-Jacobi identity useful in integrability?
I'm studying integrabilty and there is the so-called Lax-Jacobi identity, which is an implication of the classical Jacobi identity of the Poisson brackets of the Lax-Poisson structure:
$$Cycl_{123} [...
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Do all members of the affine Toda field theories have fermion duals?
I know that the sine-Gordon model is $S$-dual to the massive Thirring model. Since sine-Gordon is a special case of affine Toda theory I was wondering if this extends to the generality of the models?
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Yang-Baxter equation for an $S$ matrix depending on total momentum
I have a system where the two-particle scattering matrix $S_{12}(p_1,p_2)$ depends on the momentum difference $p_1-p_2$, and also on the total momentum $P=p_1+p_2$ in some non-trivial way. One can use ...
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Separability of Hamiltonian and Factorization of Wavefunction
In Shankar's QM book Chapter 10 pg. 274, it was said that quantum mechanically, the separability of the hamiltonian $$H=H_1(x_1, p_1)+H_2(x_2,p_2)$$ leads to the factorization of the wave function:
$$\...
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Integrability of one-dimensional system of motion?
How can I prove that every one-dimensional system is integrable (meaning that there is a constant of motion)?
It is clear that if $H$ does not depend explicitly on time then $H$ is indeed a constant ...
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Confusion about the action variable definition
Suppose we have an integrable system consisting of a $2n$-dimensional phase space $M$ together with $n$ independent functions $f_{1\leq j \leq n }$ in involution. Suppose the level set
$$M_f = \{ (p,q)...
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Discretization of derivative of delta function and affine Kac-Moody algebra
In equation (4.16) of https://arxiv.org/abs/1506.06601, a discretization of the (classical) affine Kac-Moody algebra is presented:
$$
\frac{1}{\gamma}\left\{J_{m}^{1}, J^2_{n}\right\}=J_{m}^{1} J_{n}^{...
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Thermodynamic Bethe ansatz validity
The thermodynamic Bethe ansatz (TBA) involves introducing a chemical potential $h$ conjugate to some charge $Q$, and considering the hamiltonian $H-hQ$. Particles with the smallest value of the mass ...
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Is the Hamiltonian of interacting systems integrable if the interaction is linear?
Suppose we allow two integrable systems with Hamiltonians $H_1$ and $H_2$ to interact. Then their combined dynamics can be described by a joint Hamiltonian, $$H = H_1(\mathbf{q}_1,\mathbf{p}_1) + H_2(\...
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Numerical solution of Bethe ansatz equation
In this paper https://arxiv.org/abs/2003.14202, the authors plot the solution of coupled non-linear algebraic equation (Bethe Ansatz Equations):
\begin{equation}
k_j L=2\pi I_j -\sum_{\beta=1}^M \...
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Differentiation of an operator equation in paper by Chen, Lee, Pereira 1979
This 1979 paper by Chen, Lee, and Pereira gives an operator $L$ satisfying
$$\dot L = [A, L],\tag{1}$$
where $A$ is another operator, and the dot denotes time differentiation. They then define $I_n = \...
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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 ...
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Hilbert transform in soliton paper
I asked this question over at the Mathematics SE, see here, but have not gotten any responses, so I figured I might as well try here as well. While the question is mathematical, it does appear in a ...
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Correspondence between quantum and classical integrability
I'm looking for connections between quantum and classical integrability. I know quantum integrability is not well-defined, but let us just take one of the popular definitions by promoting the Poisson ...
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