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

146 submissions , 123 unreviewed
3,961 questions , 1,408 unanswered
4,890 answers , 20,766 comments
1,470 users with positive rep
506 active unimported users
More ...

Peierls Argument for Absence of Long Range Order

+ 1 like - 0 dislike
68 views

I'm really confused about the argument in Cardy's book for why there can't be long range order in 1D for discrete models. Let me just copy it out, and hopefully someone can explain it to me.

He takes an Ising-like system as an example. We start with the ground state with all spins up, and we want to see if this state is stable against flipping the spins in some chain of length $l$. This chain has two domain walls at the endpoints, so we get an energy change of $4J$. Then the claim is that there is an entropy of $\log l$ associated with this chain, since "each wall may occupy $O(l)$ positions." If this were true, we would get a free energy change of $4J-\beta^{-1} \log l$, and this would imply that the ground state is unstable to flipping very long chains.

The only part I'm not on board with is the claim about the entropy. I would say that if $L$ is the length of the system, then we have $L$ places to put the chain, so we get an entropy of $\log L$. Certainly as $L\to \infty$ this gives no long range order, as expected.

So, is the entropy $\log l$ or $\log L$?

(Incidentally, I'm perfectly happy with his argument in 2D...)


This post imported from StackExchange Physics at 2015-11-08 10:09 (UTC), posted by SE-user Matthew

asked Aug 4, 2013 in Theoretical Physics by Matthew (320 points) [ revision history ]
edited Nov 8, 2015 by Dilaton
Because the spin chain itself is the entire system that we study, so the length of the chain = the length of the system, i.e. $l=L$, hence $\log l$ and $\log L$ make no difference. The argument is to put domain walls (kinks) on the spin chain, but not to put a spin chain in some 1D space.

This post imported from StackExchange Physics at 2015-11-08 10:09 (UTC), posted by SE-user Everett You

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$\hbar$ysicsOver$\varnothing$low
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
...