"Six-vertex model and Quantum XXZ Hamiltonian"의 두 판 사이의 차이
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| 41번째 줄: | 41번째 줄: | ||
<h5>entropy of two-dimensional ice</h5> | <h5>entropy of two-dimensional ice</h5> | ||
| − | * entropy is given as<br><math>Mk\ln W</math> where M is the number of molecules and <math>W=(4/3)^{3/2}</math><br> | + | * entropy is given as<br><math>Mk\ln W</math> where M is the number of molecules and <math>W=(4/3)^{3/2}=1.53960\cdots</math><br> |
| 152번째 줄: | 152번째 줄: | ||
** V. Fridkin, Yu. Stroganov, D. Zagier, 2000 | ** V. Fridkin, Yu. Stroganov, D. Zagier, 2000 | ||
* [http://arxiv.org/abs/hep-th/9204064 Diagonalization of the XXZ Hamiltonian by Vertex Operators]<br> | * [http://arxiv.org/abs/hep-th/9204064 Diagonalization of the XXZ Hamiltonian by Vertex Operators]<br> | ||
| − | ** | + | ** Brian Davies, Omar Foda, Michio Jimbo, Tetsuji Miwa, Atsushi Nakayashiki, 1993 |
| − | + | * [http://dx.doi.org/10.1103/PhysRevLett.18.1046 Exact Solution of the F Model of An Antiferroelectric]<br> | |
| − | * | + | ** E.H. Lieb. <em style="">Phys. Rev.</em> '''18''' (1967), p. 1046. |
| − | ** E.H. Lieb. <em style="">Phys. Rev.</em> '''18''' (1967), p. 1046. | + | * [http://dx.doi.org/10.1103/PhysRevLett.19.108 Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric]<br> |
| − | + | ** E.H. Lieb. <em style="">Phys. Rev.</em> '''19''' (1967), p. 108. | |
| − | ** E.H. Lieb. <em style="">Phys. Rev.</em> '''19''' (1967), p. 108. | + | * [http://dx.doi.org/10.1103/PhysRevLett.19.103 Exact Solution of a Two-Dimensional Model for Hydrogen-Bonded Crystals]<br> |
| − | + | ** B. Sutherland. <em style="line-height: 2em;">Phys. Rev.</em> '''19''' (1967), p. 103. | |
| − | ** B. Sutherland. <em style="line-height: 2em;">Phys. Rev.</em> '''19''' (1967), p. 103. | ||
* [http://dx.doi.org/10.1103/PhysRevLett.18.692 Exact Solution of the Problem of the Entropy of Two-Dimensional Ice]<br> | * [http://dx.doi.org/10.1103/PhysRevLett.18.692 Exact Solution of the Problem of the Entropy of Two-Dimensional Ice]<br> | ||
** E. H. Lieb, [http://link.aps.org/doi/10.1103/PhysRevLett.18.692 Phys. Rev. Letters 18, 692 (1967)] | ** E. H. Lieb, [http://link.aps.org/doi/10.1103/PhysRevLett.18.692 Phys. Rev. Letters 18, 692 (1967)] | ||
| 167번째 줄: | 166번째 줄: | ||
** C. N. Yang, C. P. Yang, 1966 | ** C. N. Yang, C. P. Yang, 1966 | ||
| − | * http://dx.doi.org/10.1103/PhysRevLett.19. | + | * http://dx.doi.org/10.1103/PhysRevLett.19.108 |
2010년 8월 3일 (화) 10:19 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