Six-vertex model and Quantum XXZ Hamiltonian

introduction

• ice-type model, R model, Rys model
• XXZ spin chain and the six-vertex transfer matrix have the same eigenvectors

• Boltzmann weights
• monodromy matrix
• trace of monodromy matrix is the transfer matrix
• power of transfer matrix becomes the partition function

 

 

types of six vertex models

• on a square lattice with periodic boundary conditions
• on a square lattice with domain wall boundary conditions
  
  • this is related to the Alternating sign matrix theorem

 

 

transfer matrix

• transfer matrix in statistical mechanics
• transfer matrix is builtup from matrices of  Boltzmann weights
• finding eigenvalues and eigenvectors of transfer matrix is crucial
• Bethe ansatz equation is used to find the eigenvectors and eigenvalues of the transfer matrix
• partition function  is calculated in terms of the eigenvalues of the transfer matrix
   
  

• 
  
  
• we need the trasfer matrices coming from different set of Boltzman weights commute 
  
• partition function = trace of power of transfer matrices
  
• so the problem of solving the model is reduced to the computation of this trace
  

 

 

 

transfer matrix formalism and the role of Bethe ansatz

• one can regard the down arrows in a row as 'particles'
  
• because of the ice rule, their number is conserved and one can try a Bethe ansatz for the eigenvectors of the transfer matrix
  

 

 

entropy of two-dimensional ice

• entropy is given as
  \(Mk\ln W\) where M is the number of molecules and \(W=(4/3)^{3/2}=1.53960\cdots\)
  

 

 

 

free energy

• \(F=-kT \ln Z\)

 

 

 

 

 

 

 

 

 

 

 

Sutherland's observation

• the eigenvectors of the transfer matrix depended on a,b,c only via the parameter
  \(\Delta=\frac{a^2+b^2-c^2}{2ab}\)
  
• is the \delta = anistropic parameter in Heisenberg spin chain model ?
  

 

 

anistropic one-dimensional Heisenberg model (XXZ model)

• Heisenberg spin chain model
• XXZ model or XXZ spin chain
• first solved by Bethe 
• Yang and Yang
• ground state eigevector for Hamiltonian  is a common eigenvector

 

 

related items

• Bethe ansatz
• Heisenberg spin chain model

 

 

books

• 찾아볼 수학책

• Exactly Solved Models in Statistical mechanics
  
  • R. J. Baxter, 1982

• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Ice-type_model
• http://en.wikipedia.org/wiki/Heisenberg_model_(quantum)
• http://en.wikipedia.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=

 

 

STATISTICAL MECHANICS-A REVIEW OF

SELECTED RIGOROUS RESULTS1•2

By JOEL L. LEBOWITZ

 

 

Method for calculating finite size corrections in Bethe ansatz systems: Heisenberg chain and six-vertex model
de Vega, H. J.; Woynarovich, F.

 

 

articles

• Integrability of the Quantum XXZ Hamiltonian
  
  • T Miwa, 2009
• Introduction to solvable lattice models in statistical and mathematical physics
  
  • Tetsuo Deguchi, 2003
• Finite Size XXZ Spin Chain with Anisotropy Parameter $\Delta = {1/2}$
  
  • V. Fridkin, Yu. Stroganov, D. Zagier, 2000
• Diagonalization of the XXZ Hamiltonian by Vertex Operators
  
  • Brian Davies, Omar Foda, Michio Jimbo, Tetsuji Miwa, Atsushi Nakayashiki, 1993
• Exact Solution of the F Model of An Antiferroelectric
  
  • E.H. Lieb. Phys. Rev. 18 (1967), p. 1046.
• Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric
  
  • E.H. Lieb. Phys. Rev. 19 (1967), p. 108.
• Exact Solution of a Two-Dimensional Model for Hydrogen-Bonded Crystals
  
  • B. Sutherland. Phys. Rev. 19 (1967), p. 103.
• Exact Solution of the Problem of the Entropy of Two-Dimensional Ice
  
  • E. H. Lieb, Phys. Rev. Letters 18, 692 (1967)
• One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System
  
  • C. N. Yang, C. P. Yang, 1966

• One-dimensional chain of anisotropic spin-spin interactions
  
  • C. N. Yang, C. P. Yang, 1966
• http://dx.doi.org/10.1103/PhysRev.150.327

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/