3-states Potts model

introduction

• 3-states Potts model = M(5,6) minimal model
• two modular invariant partition functions
• c=4/5, effective central charge=4/5
• having the Z_3 symmetry W_3 algebra (W-algebra)

ferromagnetic three-state Potts spin chain

• [DKMM93]

conformal field theory

• discrete series unitary representations and GKO coset construction

\(m= 5, c = \frac{4}{5}\) \[h_{r,s}(c) = {((m+1)r-ms)^2-1 \over 4m(m+1)}\]\(r = 1, 2, 3,4\) and \(s= 1, 2, 3,\cdots, r\) i.e. \(1\leq s\leq r< 5\) ( or \(1\leq r< s\leq m\) condition is also used)

• 10 irreducible representations
• conformal dimensions

\[\Delta_{r,s}^{(p,p')}=h_{r,s}^{(p,p')}=\frac{(p'r-ps)^2-(p'-p)^2}{4pp'}\]\(1\leq r \leq p-1\) and \(1\leq s \leq p'-1\)

• note that \(\Delta_{r,s}^{(p,p')}=\Delta_{p-r,p'-s}^{(p,p')}\) r=1,2,3,4 s=1,2,3,4,5

related items

• parafermion theory

• table of minimal models
• rank 2 case
• W-algebra
• Schramm–Loewner evolution (SLE)
• Ising models
• hard hexagon model
• eight-vertex model and quantum XYZ model

articles

• Schramm–Loewner evolution in the three-state Potts model—a numerical study
  • Adam Gamsa and John Cardy J. Stat. Mech. (2007) P08020
• On thermodynamic approaches to conformal field theory
  • Jose Gaite, Nuclear Physics B Volume 525, Issue 3, 17 August 1998, Pages 627-640
• Critical exponents of the chiral Potts model from conformal field theory
  • John L. Cardy, 1993
• [DKMM93]Virasoro Characters from Bethe Equations for the Critical Ferromagnetic Three-State Potts Model
  • Srinandan Dasmahapatra, Rinat Kedem, Barry M. McCoy and Ezer Melzer, 1933
• Thermodynamics of the 3-state Potts Spin chain
  • Rinat Kedem, 1992
• Construction of Modular Branching Functions from Bethe’s Equations in the 3-State Potts Chain
  • Rinat Kedem, Barry M. McCoy, 1992
• Thermodynamic Bethe ansatz in relativistic models: Scaling 3-state potts and Lee-Yang models
  • Al. B. Zamolodchikov, 1990
• Integrals of Motion in Scaling 3-STATE Potts Model Field Theory.
  • Zamolodchikov, A, Volume: 3, Issue: 3(1988) pp. 743-750
• Conformal quantum field theory models in two dimensions having Z3 symmetry
  • V. A. Fateev and A. B. Zamolodchikov, Nuclear Phys. B 280 (1987), no. 4, 644–660
• Virasoro Algebra, von Neumann Algebra and Critical Eight-Vertex SOS Models
  • Atsuo Kuniba, Yasuhiro Akutsu1 and Miki Wadati, 1986
• Critical behavior of the three-state Potts model: Monte Carlo renormalization group
  • Phys. Rev. B 28, 3897–3903 (1983)
• Critical exponents of the three-state potts model
  • Bambi Hu, 1980
• Hard hexagons: exact solution
  • R J Baxter 1980 J. Phys. A: Math. Gen.