Timeline for Elementary proof of lack of phase transition in Ising models with external fields
Current License: CC BY-SA 3.0
10 events
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| Nov 3, 2021 at 20:39 | comment | added | Aernout van Enter | Simple answer: phase transitions in an external field are possible on trees ( and more generally on non-amenable graphs). See for example Georgii’s book. | |
| Oct 17, 2016 at 11:28 | comment | added | Ben | The updated link seems to be ie.technion.ac.il/~ieioffe/Papers/LNM.pdf. | |
| Jan 13, 2014 at 12:45 | comment | added | Yvan Velenik | I haven't really thought about it, but one should be able to do it using Aizenman's random current representation. For example, it is quite easy to prove exponential decay of the truncated 2-point function at non-zero field using this representation, and a variant should be enough to establish uniqueness. The argument for the decay of correlation can be found, for example, here (see p.13) : iew3.technion.ac.il/~ieioffe/Papers/LNM.pdf . | |
| Dec 21, 2013 at 17:53 | comment | added | Anthony Quas | Thanks for the suggestion. I haven't yet looked in Baxter's book, but will see if has anything for me. | |
| Dec 21, 2013 at 17:46 | comment | added | Matthew Titsworth | Onsager's solution is for the case with zero field though, which falls outside of your question. My apologies. Baxter's book may still be useful though. | |
| Dec 21, 2013 at 17:39 | comment | added | Matthew Titsworth | On a general graph it is possible to have a phase transition in the Ising model. Lars Onsager found one in the square lattice in 1944 (en.wikipedia.org/wiki/Square-lattice_Ising_model). Separately, the usual reference for these things is Rodney Baxter's book "Exactly solved models in statistical mechanics" which (if I remember correctly) contains Onsager's solution and may be more helpful to you. | |
| Dec 21, 2013 at 17:26 | comment | added | Anthony Quas | I mean the Ising model on a general graph, but the question is specifically about the Ising model on $\mathbb Z^2$. By phase transition, I mean the existence of more than one Gibbs measure. | |
| Dec 21, 2013 at 17:21 | comment | added | Matthew Titsworth | By "the Ising model" I assume you mean the one dimensional Ising model? Also, (classical) phase transitions are defined (at least physically) as a discontinuity in the order parameter (or it's first derivative) in the system being described. For the Ising model, the order parameter is the magnetization, which is dependent upon the magnetic field strength. At finite temperature in the 1-D Ising model the magnetization and its derivative vary continuously with respect to magnetic field strength, however a discontinuity can be introduced at T=0. | |
| Dec 20, 2013 at 16:14 | history | edited | Anthony Quas |
++tag;
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| Dec 20, 2013 at 16:04 | history | asked | Anthony Quas | CC BY-SA 3.0 |