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

• finding eigenvalues and eigenvectors of transfer matrix is crucial
• Bethe ansatz equation is used to find the eigenvectors and eigenvalues of the transfer matrix
•  
•  

 

 

entropy of two-dimensional ice

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

 

 

free energy

• \(F=-kT \ln Z\)

 

 

 

 

 

 

 

anistropic one-dimensional Heisenberg model

• Heisenberg 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

 

 

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
  
  • Authors: Brian Davies, Omar Foda, Michio Jimbo, Tetsuji Miwa, Atsushi Nakayashiki, 1993
• E. H. Lieb, Phys. Rev. Letters18, 692 (1967)
• Exact Solution of the F Model of An Antiferroelectric
  
  • E.H. Lieb. Phys. Rev. 18 (1967), p. 1046. Full Text via CrossRef
• Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric
  
  • E.H. Lieb. Phys. Rev. 19 (1967), p. 108. Full Text via CrossRef
• Exact Solution of a Two-Dimensional Model for Hydrogen-Bonded Crystals
  
  • B. Sutherland. Phys. Rev. 19 (1967), p. 103. Full Text via CrossRef
• 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
• 논문정리
• http://www.ams.org/mathscinet/search/publications.html?pg4=ALLF&s4=
• http://www.zentralblatt-math.org/zmath/en/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html

• http://www.ams.org/mathscinet
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
• http://dx.doi.org/