• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,719 comments
1,470 users with positive rep
818 active unimported users
More ...

  What does it mean when a spin chain is integrable or chaotic?

+ 4 like - 0 dislike

I am looking for the definition of integrable and the definition of chaotic in the context of spin chains. In particular, consider the NN Ising model with both transverse and longitudinal fields:

\[H=\sum_j Z_j Z_{j+1}+h\sum_j X +g \sum_j Z\]

It is said that for \(g=0\), the spin chain is integrable for all \(h\). Whereas for general values of \(h\) and \(g\) the system is chaotic. What makes the system integrable at \(g=0\), besides being exactly solvable via Jordan-Wigner transformation? Are all chaotic spin chains nonintegrable?


asked Nov 7, 2016 in Theoretical Physics by anonymous [ no revision ]
recategorized Nov 7, 2016 by Dilaton

1 Answer

+ 4 like - 0 dislike

In practice, exactly solvable = integrable. In theory one requires for the latter a set of as many independent commuting variables (classically: Poisson-commuting) as there are degrees of freedom. (Without these one cannot usually find an explicit solution.) See https://en.wikipedia.org/wiki/Integrable_system

Chaoticity excludes integrability, as a chaotic system cannot be solved exactly. However, a nonintegrable Hamiltonian system may have nonchaotic behavior at low energies. This is related to the so-called KAM theorem.

answered Nov 8, 2016 by Arnold Neumaier (15,787 points) [ no revision ]

Thanks for your answer. Can you elaborate a little more in the context of spin chains? If a spin chain is chaotic so what? What are some observable signatures in chaotic chains that make them fundamentally different from integrable spin chains?

Integrable is a very special case. Chaotic means that there are some initial conditions where you cannot compute the detailed long-term behavior since it depends exponentially sensitive on the integration time. But in practice one is interested only in a few macro variables of spin chains, which are far less sensitive. Monte Carlo studies typically have far more uncertainty than is introduced by the chaos.

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights