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
4,873 answers , 20,701 comments
1,470 users with positive rep
502 active unimported users
More ...

Is there a reasonable lower bound for free energy per site of the 2D Ising model in the presence of an external field?

+ 3 like - 0 dislike
531 views

Given the standard Ising partition function: $$Z(\theta ,h) = \sum\limits_{\bf{x}} {\exp \left\{ {\theta \sum\limits_{(i,j) \in E} {{x_i}{x_j}} + h\sum\limits_{i \in V} {{x_i}} } \right\}}, $$

is there a closed form expression for the lower bound of the free-energy (Pressure) per-site, defined as,

$$\psi (\theta ,h) = \mathop {\lim }\limits_{n \to \infty } \frac{{\log \left( {Z(\theta ,h)} \right)}}{n}.$$

My hunch is that is $2\theta + h$ based on taking limits of an approximation suggested in this article

Any ideas on its suitability as a lower bound?

This post imported from StackExchange Physics at 2014-06-21 09:02 (UCT), posted by SE-user user48476
asked Jun 19, 2014 in Theoretical Physics by user48476 (15 points) [ no revision ]

Your definition of the free energy differs from the standard one by a minus sign. The answer can't be that simple (or is too weak to be of any use). Set $h=0$ and the exact solution (due to Onsager) is known and exhibits different behaviour for $\theta<\theta_c$ and $\theta>\theta_c$ for a critical value that I denote by $\theta_c$ (given by $(\sinh 2\theta_c)^2=1$ for the square lattice). A similar exact solution is not known for non-zero $h$. Now one expects a phase transition across a real codimension one "wall" in the $(\theta,h)$ plane that can be written as $\theta_c(h)$. I expect two different bounds on either side of the wall.

@Suresh1: the sharp phase-transition doesn't happen at any nonzero h, and at large h, you might as well use a single-site lower bound.

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$\hbar$ys$\varnothing$csOverflow
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
...