Quantcast
  • 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.

News

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

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,853 answers , 20,624 comments
1,470 users with positive rep
501 active unimported users
More ...

What is rigorously known about critical points?

+ 3 like - 0 dislike
169 views

What is rigorously known about the existence and properties of critical points (in the thermodynamic/statistical mechanics sense) in classical and quantum mechanical models in 3 space dimensions? I'd be particularly interested in pointers to survey articles that allow me to form a complete picture.

asked May 2 in Resources and References by Arnold Neumaier (12,385 points) [ revision history ]
edited Sep 5 by Arnold Neumaier

What do you mean by critical points? That which is called equilibria ($dH=0$) by Arnold? For that, I found the textbook Mathematical aspects of classical and celestial mechanics  by Arnold and others to be at the right briefness/detail ratio. Depending on what you are actually looking for, Bifurcation Theory and Catastrophe Theory from this series might also be interesting.

No. I had meant it in the thermodynamic sense; I hadn't notice the ambiguity. I added a link in my question.

You probably already know this. I heard the proof for triviality of $\phi^4$ in 4D is almost rigorous, this implies the universality class for Ising model in 4D being Gaussian is at least almost rigorously proven (although I  don't know how difficult it is to have a rigorous justification for saying the two are indeed of the same universality). Saying mean field treatment for 4D Ising model critical exponent is exact is an equivalent statement, but I'm ignorant on how much rigor has been achieved from this perspective.

@JiaYiyang: Without a reference I don't buy the triviality proof. The arguments I know of depend on the assumption that you can get the field theory in a lattice limit, which is a questional assumption in the absence of asymptotic freedom. Thus what is to be proved is almost assumed from the start! - 

See the discussion in https://www.physicsoverflow.org/32756 . The discussion in the paper by Gallavotti and Rivasseau from 1984 mentioned there is still unsurpassed, as far as I know. Se also  https://www.physicsoverflow.org/21328/

Anyway, 4D Ising is unphysical since the physically relevant dimension is $d=3$. 

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:
p$\hbar$ysicsOverfl$\varnothing$w
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).
To avoid this verification in future, please log in or register.




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

Your rights
...