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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 a constraint when $x$ and $y$ depend on $t$ [closed]

I'm working through problem 6 in chapter 1 in Goldstein's classical mechanics book. I've reduced it to asking, if $x$ and $y$ are coordinates and function of time $t$, whether the differential ...
CasualPhysicsEnjoyer's user avatar
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Does the functional form of a perturbative Hamiltonian indicate how many nonzero terms are in energy corrections?

Suppose we add a perturbative Hamiltonian to a quantum system. In principle, we can compute high order corrections to the energy levels with perturbation theory. However, I've come across some systems ...
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Help with computations of first return time in an integrable system

I have example of an integrable system $(J,H):M\rightarrow \mathbb{R}^2$ where $J$ generates an $S^1$-action for which I'm trying to compute the action-angle coordinates. I have done all the steps but ...
Someone's user avatar
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Gap of the XXZ model in fixed magnetisation sectors

I am wondering whether it is known, or whether it can easily seen from the Bethe ansatz solution, what the gap of the spin-1/2 XXZ model of finite size $N$ with periodic boundary conditions ($H=\sum\...
lm1909's user avatar
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Why do colliding blocks compute to $\pi$?

If you take two blocks on a frictionless plane (each block is 1kg) with one wall and perfectly elastic collisions, if you collided the block on the right with the one on the left (wall is on the left) ...
Kellan Heerdegen's user avatar
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Applications of Schrodinger's to dark solitons [closed]

The Schrodinger equation (SE) admits dark solitons as particular solutions. The SE and the The Korteweg-de Vries (KdV) equations can be used to model them. Questions: What are the applications of ...
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Conserved charges and quantum integrability

In classical mechanics it is known that a certain model is solvable exactly (integrable) if it posses a sufficient amount of "well behaved" conserved charges. On the other hand in quantum ...
Truth and Beauty and Hatred's user avatar
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Mathematical meaning for Algebraic Bethe Ansatz

I'm a mathematician who's trying to understand the meaning of Algebraic Bethe Ansatz. What I understood is that when dealing with quantum integrable models (like XXZ Heisenberg spin chain), one is ...
BlueCharlie's user avatar
2 votes
1 answer
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Computing the Free Parafermion Spectrum

In Fendley's paper on Free Parafermion (https://arxiv.org/abs/1310.6049), Fendley used some operator techniques to show that $$Q_{2 L}\left(\epsilon_k^n\right)=0$$ which is the formal derivation of ...
ZHENGYAO HUANG's user avatar
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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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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{...
Matías Cerioni's user avatar
2 votes
1 answer
161 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 (...
Ad infinitum's user avatar
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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/...
purestate's user avatar
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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)}\...
Young Kindaichi's user avatar
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1 answer
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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. ...
Ting-Kai Hsu's user avatar
3 votes
2 answers
109 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 ...
symanzik138's user avatar
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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, ...
Robby Ram's user avatar
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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 ...
sakata's user avatar
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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/...
Physics Cat's user avatar
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1 answer
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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{...
raskolnikov's user avatar
3 votes
0 answers
70 views

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. ...
gab-ert's user avatar
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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 ...
miggle's user avatar
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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 ...
Ahsan Hayat's user avatar
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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 ...
chrispy's user avatar
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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-...
islam's user avatar
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1 answer
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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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1 answer
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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 ...
Clara Díaz Sanchez's user avatar
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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 ...
LorP's user avatar
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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} + ...
Ad infinitum's user avatar
1 vote
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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 ...
unifymchn_MCR's user avatar
4 votes
1 answer
131 views

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)...
Mr. Egg's user avatar
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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(...
Vivek Panchal 's user avatar
3 votes
1 answer
81 views

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 ...
Rebour's user avatar
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1 answer
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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 ...
ElementSegment's user avatar
0 votes
2 answers
66 views

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'' ...
mEXsACHINE's user avatar
3 votes
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81 views

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 ...
Clara Díaz Sanchez's user avatar
2 votes
1 answer
224 views

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 ...
Erithacus Rubecula's user avatar
2 votes
1 answer
82 views

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 ...
Souparna Nath's user avatar
3 votes
1 answer
135 views

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)- ...
3 votes
0 answers
117 views

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} [...
Spida's user avatar
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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?
J. H's user avatar
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2 votes
1 answer
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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 ...
Zarathustra's user avatar
3 votes
1 answer
411 views

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: $$\...
TaeNyFan's user avatar
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1 vote
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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 ...
SultanDeGranada's user avatar
2 votes
1 answer
230 views

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)...
John's user avatar
  • 1,028
1 vote
1 answer
104 views

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}^{...
Mtheorist's user avatar
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