Brezis–Gallouet inequality: Difference between revisions

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The '''Chiral Potts model''' is a spin model on a planar lattice in [[statistical mechanics]]. As with the [[Potts model]], each spin can take n=0,...N-1 values. To each pair of nearest neighbor of spins n and n', a Boltzmann weight W(n-n') ([[Boltzmann factor]]) is assigned. The model is chiral, meaning W(n-n')≠ W(n'-n).  When its weights satisfy the [[Yang-Baxter equation]], (or the [[star-triangle relation]]), it is integrable. For the integrable Chiral Potts model, its weights are parameterized by a high genus curve-- [[Chiral Potts curve]].
<ref name=AMPTY>Au-Yang H, McCoy B M, Perk J H H, Tang S and Yan M-L ( 1987),
"Commuting transfer matrices in the chiral Potts models: Solutions of the star-triangle equations with genus > 1",
[[Physics Letters A]] '''123''' 219--23.</ref>
<ref name=BPA>Baxter R J, Perk J H H and Au-Yang H (1988),
"New solutions of the star-triangle relations for the chiral Potts model", [[Physics Letters A]]  '''128''' 138--42.</ref>
Unlike the other solvable models,<ref name=Baxterbk>[[R. J. Baxter]],"Exactly Solved Models in Statistical Mechanics", [[Academic Press]], [[ISBN]]13: 978-0-486-46271-4; [[ISBN]]  978-0-12-083180-7.</ref><ref name=McCoybk>B.M.McCoy,"Advanced  Statistical Mechanics", '''146''' International Series of Monographs on Physics,  (Oxford, England), [[ISBN]] 0199556636, 9780199556632</ref>
whose weights are parametrized by curves of genus less or equal to one, so that they can be expressed in term of trigonometric, or rational function (genus=0) or by [[theta functions]] (genus=1), this model involves high genus theta functions, which are not yet well developed. Therefore, it was thought that no progress could be made for such a difficult problem. Yet, many breakthroughs have been made since the 1990s. It must be stressed again that the Chiral Potts model was not invented because it was integrable but the integrable case was found, after it was introduced to explain experimental data. In a very profound way physics is here far ahead of mathematics. The history and its development will be presented here briefly.


== The Model ==


This model is out of the class of all previously known models and raises a host of unsolved questions which are related to some of the most intractable problems of algebraic geometry which have been with us for 150 years. The chiral Potts models are used to understand the commensurate-incommensurate phase transitions.<ref name=HKN83>S. Howes, L.P. Kadanoff and M. den Nijs (1983), [[Nuclear Physics]] B
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'''215''', 169.</ref>
<ref>Julia M. Yeomans, Michael E. Fisher (1984), "Analysis of the multiphase region in the three-state chiral clock model", [[Physica A]]
'''127''', 1-37.</ref> For N = 3 and 4, the integrable case was discovered in 1986 in Stony Brook and published the following year. <ref name=AMPTY/><ref name=MPTS87>McCoy B M, Perk J H H, Tang S and Sah C H (1987),
"Commuting transfer matrices for the 4 state self-dual chiral Potts model with a genus 3 uniformizing Fermat curve", [[Physics Letters A]],  '''125''' , 9--14.</ref>
 
=== Self-Dual Case ===
The model is called self-dual, if the Fourier transform of the weight is equal to the weight. A special (genus 1) case had been solved in 1982 by Fateev and Zamolodchikov.<ref name=FZ82>V.A. Fateev and A.B. Zamolodchikov, (1982) [[Physics Letters]] A  '''92 '''.</ref>
By removing certain restrictions of the work of Alcaraz and Santos
,<ref name=Alcaraz>E.C. Alcaraz and A. Lima Santos, [[Nuclear Physics]] B 275.</ref> a more general self-dual case of the integrable chiral Potts model was discovered.<ref name=AMPTY/> The weight are given in product form <ref name=AMPS88> H. Au-Yang, B.M. McCoy, J.H.H. Perk, and S. Tang, (1988),``Solvable models in statistical mechanics and Riemann surfaces of genus greater than one, in Algebraic Analysis", Vol. 1, M. Kashiwara and T. Kawai, eds., [[Academic Press]], pp. 29-40.</ref><ref name=Perk87>
J.H.H. Perk, ( 1987 ),``Star-triangle equations, quantum Lax pairs, and higher genus curves", in Proc. 1987 Summer Research Institute on Theta Functions, Proc. Symp. Pure Math., Vol. 49, part 1, (Am. Math. Soc., Providence, R.I., 1989), pp. 341-354.</ref>and the parameters in the weight are shown to be on the [[Fermat curve]], with genus greater than 1.
 
=== General Case ===
 
In Canberra, the general solution for all ''k'' ( the temperature variable) was found.<ref name=BPA/> The weights were also given in product form and it was tested by Fortran that they satisfy the [[star-triangle relation]]. The proof was published later.
<ref name=AP89Tani>Au-Yang H and Perk J H H (1989).
"Onsager's star-triangle equation: Master key to
integrability", Proc. Taniguchi Symposium, Kyoto, October 1988
Advanced Studies in Pure Mathematics vol 19
(Tokyo: Kinokuniya--Academic) pp 57--94</ref>
 
== Results ==
 
=== Order Parameter ===
 
From the series <ref name=HKN83/><ref>M. Henkel and J. Lack, preprint Bonn-He- 85-22</ref>  the order parameter is conjectured <ref name=AMPT89>Albertini G., McCoy B.M., Perk J.H.H. and Tang S.
(1989), "Excitation spectrum and order parameter for the integrable
$N$-state chiral Potts model", [[Nuclear Physics]] B
  '''314''',741--763</ref> to have the simple form
:<math>\langle \sigma^n\rangle=(1-k'^2)^\beta,\quad \beta=n(N-n)/2N^2.</math>
It took many years to prove this conjecture, as the usual corner transfer matrix technique could not be used, because of the higher genus curve. This conjecture was finally proven by Baxter in 2005 <ref>Baxter R J (2005)
"Derivation of the order parameter of the chiral Potts model",
[[Physics Review Letters]] , '''94''' 130602 (3 pp)
[[arXiv:cond-mat/0501227]],</ref>
<ref>Baxter R J (2005)
"The order parameter of the chiral Potts model",
[[Journal of Statistical Physics]]  '''120''', 1--36:
[[arXiv:cond-mat/0501226]].</ref> using functional equations and the "broken rapidity line" technique of
Jimbo {\it et al.} <ref>Jimbo M, Miwa T and Nakayashiki A (1993),
"Difference equations for the correlation functions of the eight-vertex model",
[[Journal of Physics A]]: Math. Gen.  '''26''', 2199--210:
[[arXiv:hep-th/9211066]].</ref> assuming two rather mild analyticity
conditions of the type commonly used in the field of Yang—Baxter
integrable models. Most recently, in a series of papers
<ref>
Baxter R J (2008)
"Algebraic reduction of the Ising model",
[[Journal of Statistical Physics]]  '''132''',959--82,
[[arXiv:0803.4036]];</ref>
<ref>Baxter R J (2008)
"A conjecture for the superintegrable chiral Potts model",
[[Journal of Statistical Physics]] '''132''', 983--1000,
[[arXiv:0803.4037]];</ref>
<ref>Baxter R J (2009)
"Some remarks on a generalization
of the superintegrable chiral Potts model}",
[[Journal of Statistical Physics]]  '''137''', 798--813,
[[arXiv:0906.3551]];</ref>
<ref>Baxter R J (2010)
"Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant $D_{PQ}$",
[[Journal of Physics A]]:  '''43''' 145002 (16pp)
[[arXiv:0912.4549]];</ref>
<ref>Baxter R J (2010)
"Proof of the determinantal form of the spontaneous magnetization of the superintegrable chiral Potts model",
Australian & New Zealand Industrial and Applied Mathematics Journal, '''51'''
[[arXiv:1001.0281]].</ref><ref>
Iorgov N, Pakuliak S, Shadura V, Tykhyy Yu and von Gehlen G (2009)
"Spin operator matrix elements in the superintegrable chiral Potts quantum chain", [[Journal of Statistical Physics]]  '''139'', 743--68 [[arXiv:0912.5027]];</ref><ref>
Au-Yang H and Perk J H H,  (2011),"Spontaneous magnetization of the integrable chiral Potts model",
[[Journal of Physics A]]:  '''44''', 445005 (20pp),[[arXiv:1003.4805]].
</ref>
an algebraic (Ising-like) way of obtaining order parameter has been given, giving more insight into the algebraic structure.
 
=== Connection to 6-[[Vertex model]] ===
 
In 1990 Bazhanov and Stroganov<ref>V.V. Bazhanov and Yu.G. Stroganov (1990),"Chiral Potts model as a descendant of the six-vertex model",[[Journal of Statistical Physics]],'''59''', pp 799-817.</ref> show that there exist an 2 × N '''''L'''''-operators, which satisfy the [[Yang-Baxter equation]]
:<math> L_{i_1\alpha}^{j_1\beta}(x)L_{i_2\beta}^{j_2\gamma} (y)R_{j_1j_2}^{k_1k_2}(y/x)= R_{i_1i_2}^{j_1j_2} (y/x)L_{j_2\alpha}^{k_2\beta} (y)L_{j_1\beta}^{k_1\gamma}(x),\quad 0<i,j,k\le1,\quad 0\le \alpha, \beta,\gamma\le N-1.</math>
where 2 × 2 '''''R'''''-operator is the 6-vertex  '''''R'''''-matrix (see [[Vertex model]]).  
The product of four chiral Potts weight '''''S''''' was shown to intertwine two '''''L'''''-operators as
:<math> L_{i_1\alpha_1}^{i_2\alpha_2} {\hat L}_{i_2\beta_1}^{i_3\beta_2} S_{\alpha_2\beta_2}^{\alpha_3\beta_3}= S_{\alpha_1\beta_1}^{\alpha_2\beta_2} {\hat L}_{i_1\beta_2}^{i_2\beta_3} L_{i_2\alpha_2}^{i_3\alpha_3},\quad 0<i_i\le1,\quad 0\le \alpha_i, \beta_i\le N-1.</math>
This inspired a most important breakthrough, namely the functional relations for the transfer matrices of the chiral Potts models are discovered.<ref>Baxter R J, Bazhanov V V and Perk J H H
(1990),"Functional relations for transfer matrices of the chiral Potts model",
International Journal of Modern Physics, B '''4''', 803--70.</ref>
 
=== Free Energy and Interfacial tension ===
 
Using these functional relation, Baxter was able to calculate the eigenvalues of the transfer matrix of the Chiral Potts model,
<ref>Baxter R J (1991),"Calculation of the eigenvalues of the transfer matrix of the chiral Potts model", Proceeding of Fourth Asia Pacific Physics Conference,(Singapore: World Scientific) pp 42--58.</ref> and obtained the critical exponent for the specific heat α=1-2/N, which was also conjectured in reference 12. The interfacial tensions are also calculated by him with the exponent μ=1/2+1/N.<ref>Baxter R J (1993),"Chiral Potts model with skewed boundary conditions", [[Journal of Statistical Physics]],'''73''', 461--95.</ref><ref>Baxter R J (1994),"Interfacial tension of the chiral Potts model", [[Journal of Physics A]]: '''27''', pp 1837--49.</ref>
 
==References ==
{{reflist}}
 
[[Category:Spin models]]
[[Category:Statistical mechanics]]
[[Category:Lattice models]]

Latest revision as of 17:12, 25 November 2014


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