Six-vertex model and Quantum XXZ Hamiltonian
http://bomber0.myid.net/ (토론)님의 2010년 8월 3일 (화) 11:46 판
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
- partition function is calculated in terms of the eigenvalues of the transfer matrix
- the below is from Yang-Baxter equation
- transfer matrix is builtup from matrices of Boltzmann weights
- 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
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
- \(F=-kT \ln Z\)
partition function
correlation functions
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
books
- 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
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