• 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
5,075 questions , 2,226 unanswered
5,347 answers , 22,749 comments
1,470 users with positive rep
818 active unimported users
More ...

  lattice model with compact boson CFT fixed point

+ 4 like - 0 dislike

The free compact boson at some radius $R\neq \sqrt{p/q}$ is probably the simplest example of an irrational CFT. Does this theory (or any other irrational CFT) arise as the thermodynamic limit of any 1+1d quantum lattice model with finite dimensional Hilbert space?

asked Apr 23, 2016 in Theoretical Physics by Ryan Thorngren (1,925 points) [ no revision ]

3 Answers

+ 4 like - 0 dislike

A one-dimensional Luttinger liquid should be an example. Concrete lattice realizations include XXZ model ($H=-\sum_i (S_i^+ S_{i+1}^- + S_i^- S_{i+1}^+ + \Delta S_i^z S_{i+1}^z)$ where $|\Delta|<1$ or Hubbard model. For XXZ, the Luttinger parameter, which determines the radius $R$ and all conformal dimensions, continuously varies with $\Delta$.

answered May 4, 2016 by Meng (550 points) [ no revision ]

Thanks Meng, this is exactly what I was looking for (:

+ 3 like - 0 dislike

Can you not just discretize the target space? I mean, the compact boson is just a free field $\phi$ with the identification $\phi \sim \phi + \beta$. You could have a lattice with $L$ sites $x$ and at every site $\phi(x)$ can take values $0, \beta/N, \ldots, \beta$. You are truncating the spectrum of the theory, obviously: you only probe $\sim N$ vertex operators with low enough winding numbers and throw away the rest. The Hamiltonian would just be the kinetic term $(\partial_\mu \phi)^2$, and of course you need to take the identification $\beta \sim 0$ into account when computing its value. A nice feature is that this construction is obviously invariant under shifts $\phi \to \phi + \text{const}$. 

answered Apr 24, 2016 by Very Anonymous User [ no revision ]
+ 3 like - 0 dislike

Quite a lot of the logCFT literature is concerned with such examples.  The spin chains studied by Saleur's group, the loop models studied by Pearce, the sandpiles studied by Ruelle.  JPhysA had an entire special issue in 2013 devoted to logCFT with reviews from these folks.

answered Apr 27, 2016 by anonymous [ no revision ]

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