"Six-vertex model and Quantum XXZ Hamiltonian"의 두 판 사이의 차이
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| 42번째 줄: | 42번째 줄: | ||
<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">transfer matrix formalism and the role of Bethe ansatz</h5> | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">transfer matrix formalism and the role of Bethe ansatz</h5> | ||
| − | * one can regard the | + | * one can regard the up(or down) arrows in a row as 'particles'<br> |
| − | * because of the ice rule, their number is conserved and one can try a Bethe ansatz for the eigenvectors of the transfer matrix<br> | + | * because of the ice rule, their number is conserved and one can try a [[Bethe ansatz]] for the eigenvectors of the transfer matrix<br> |
* <br> | * <br> | ||
| 57번째 줄: | 57번째 줄: | ||
* Hamiltonian of XXZ model or XXZ spin chain with anisotropic parameter <math>\Delta=1/2</math><br><math>\hat H = -\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \Delta \sigma_j^z \sigma_{j+1}^z)</math><br> | * Hamiltonian of XXZ model or XXZ spin chain with anisotropic parameter <math>\Delta=1/2</math><br><math>\hat H = -\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \Delta \sigma_j^z \sigma_{j+1}^z)</math><br> | ||
* ground state eigevector for Hamiltonian is a common eigenvector although the eigenvalues are different | * ground state eigevector for Hamiltonian is a common eigenvector although the eigenvalues are different | ||
| − | * | + | * see '''[YY1966-2]''' |
| 187번째 줄: | 187번째 줄: | ||
* [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, Phys. Rev. Letters 18, 692 (1967) | ** E. H. Lieb, Phys. Rev. Letters 18, 692 (1967) | ||
| − | * [http://dx.doi.org/10.1103/PhysRev.150.327 One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System]<br> | + | * '''[YY1966-2]'''[http://dx.doi.org/10.1103/PhysRev.150.327 One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System]<br> |
| − | ** C. N. Yang, C. P. Yang, 1966 | + | ** C. N. Yang, C. P. Yang, Phys. Rev. 150, 327 (1966) |
* [http://dx.doi.org/10.1016/0031-9163(66)91024-9 One-dimensional chain of anisotropic spin-spin interactions]<br> | * [http://dx.doi.org/10.1016/0031-9163(66)91024-9 One-dimensional chain of anisotropic spin-spin interactions]<br> | ||
| − | ** C. N. Yang, C. P. Yang, Phys. Rev. 150, 321 (1966) | + | ** C. N. Yang, C. P. Yang, Phys. Rev. 150, 321 (1966) |
* http://dx.doi.org/10.1103/PhysRev.150.327 | * http://dx.doi.org/10.1103/PhysRev.150.327 | ||
2010년 8월 3일 (화) 12:01 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
- borrowed from 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 = trace of power of transfer matrices
- so the partition function is calculated in terms of the eigenvalues of the transfer matrix
- then 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 up(or 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
-
anistropic one-dimensional Heisenberg model (XXZ model)
- Heisenberg spin chain model
- Hamiltonian of XXZ model or XXZ spin chain with anisotropic parameter \(\Delta=1/2\)
\(\hat H = -\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \Delta \sigma_j^z \sigma_{j+1}^z)\) - ground state eigevector for Hamiltonian is a common eigenvector although the eigenvalues are different
- see [YY1966-2]
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 ?
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
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
- 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)
- [YY1966-2]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, Phys. Rev. 150, 327 (1966)
- One-dimensional chain of anisotropic spin-spin interactions
- C. N. Yang, C. P. Yang, Phys. Rev. 150, 321 (1966)
- http://dx.doi.org/10.1103/PhysRev.150.327