"Six-vertex model and Quantum XXZ Hamiltonian"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
| + | * six-vertex model, also called ice-type model, R model, Rys model | ||
| + | * The Hamiltonian of Hisenberg XXZ spin chain and the six-vertex transfer matrix have the same eigenvectors | ||
| + | ** {{수학노트|url=하이젠베르크_스핀_1/2_XXZ_모형}} | ||
| + | * [[Bethe ansatz]] can be applied to solve the model | ||
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==types of six vertex models== | ==types of six vertex models== | ||
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* on a square lattice with periodic boundary conditions | * on a square lattice with periodic boundary conditions | ||
| − | * on a square lattice with domain wall boundary conditions | + | * on a square lattice with domain wall boundary conditions |
** this is related to the [[alternating sign matrix theorem|Alternating sign matrix theorem]] | ** this is related to the [[alternating sign matrix theorem|Alternating sign matrix theorem]] | ||
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==transfer matrix== | ==transfer matrix== | ||
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* borrowed from [[transfer matrix in statistical mechanics]] | * borrowed from [[transfer matrix in statistical mechanics]] | ||
| − | * transfer matrix is builtup from matrices of | + | * transfer matrix is builtup from matrices of Boltzmann weights |
* finding eigenvalues and eigenvectors of transfer matrix is crucial | * finding eigenvalues and eigenvectors of transfer matrix is crucial | ||
* Bethe ansatz equation is used to find the eigenvectors and eigenvalues of the transfer matrix | * Bethe ansatz equation is used to find the eigenvectors and eigenvalues of the transfer matrix | ||
* partition function = trace of power of transfer matrices | * partition function = trace of power of transfer matrices | ||
* so the partition function is calculated in terms of the eigenvalues of the transfer matrix | * 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 | + | * then the problem of solving the model is reduced to the computation of this trace |
| − | ==integrability of the model== | + | ==integrability of the model and the Yang-Baxter equation== |
| − | * | + | * <math>T(u)</math> transfer matrix |
| − | * | + | * <math>\log T(u)=\sum_{n=0}^{\infty}Q_{n}u^n</math> |
| − | * here | + | * here <math>Q_1</math> plays the role of the Hamiltonian |
* necessary and sufficient codntion to have infinitely many conserved quantities | * necessary and sufficient codntion to have infinitely many conserved quantities | ||
| − | + | :<math>[T(u), T(v)]=0</math> | |
| − | which implies | + | which implies <math>[Q_n,Q_m]=0</math> |
| − | * in order to have | + | * in order to have <math>[T(u), T(v)]=0</math>, the [[Yang-Baxter equation (YBE)]] must be satisfied |
| − | ==R-matrix and | + | ===R-matrix and Boltzmann weights=== |
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* [[R-matrix]] | * [[R-matrix]] | ||
| − | + | :<math> | |
R(u,\eta)=\rho\left( | R(u,\eta)=\rho\left( | ||
\begin{array}{cccc} | \begin{array}{cccc} | ||
| 58번째 줄: | 46번째 줄: | ||
\end{array} | \end{array} | ||
\right) | \right) | ||
| − | + | </math> | |
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| + | * multiplicative form of [[S-matrix of the quantum sine-Gordon model]] | ||
| + | :<math> | ||
| + | \check{R}(x)= | ||
| + | \left( | ||
| + | \begin{array}{cccc} | ||
| + | x-q^2 & 0 & 0 & 0 \\ | ||
| + | 0 & 1-q^2 & q (x-1) & 0 \\ | ||
| + | 0 & q (x-1) & \left(1-q^2\right) x & 0 \\ | ||
| + | 0 & 0 & 0 & x-q^2 \\ | ||
| + | \end{array} | ||
| + | \right) | ||
| + | </math> | ||
==transfer matrix formalism and coordinate Bethe ansatz== | ==transfer matrix formalism and coordinate Bethe ansatz== | ||
| − | * <math>M=N^{2}</math> number of molecules | + | * <math>M=N^{2}</math> number of molecules |
| − | * | + | * 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 [[Bethe ansatz]] for the eigenvectors of the transfer matrix |
| − | * <math>f(x_1,\cdots,x_n)</math> be the | + | * let <math>f(x_1,\cdots,x_n)</math> be the coefficient in an eigenvector <math>v</math> of the state with up arrows at the sites <math>x_ 1<x_ 2<\cdots<x_n</math> so that |
| − | * | + | :<math>v(k_1,\cdots,k_n)= |
| − | <math>f(x_ 1,\cdots,x_n)=\sum_{P}A (P)\exp(i\sum_{j=1}^{n}x_jk _{P_j})</math> | + | \sum_{\substack{\mathbf{x}=(x_ 1,x_ 2,\cdots,x_n) \\ x_ 1<x_ 2<\cdots<x_n}} f(x_1,\cdots,x_n|k_1,\cdots,k_n)\sigma_{-}^{(x_1)}\cdots\sigma_{-}^{(x_n)}|0\rangle </math> |
| − | * | + | * Bethe ansatz suggests the following form for <math>f</math> |
| − | :<math>\exp(ik_jn)=\prod_{ | + | :<math>f(x_ 1,\cdots,x_n)=\sum_{P\in S_n}A (P)\exp(i\sum_{j=1}^{n}x_jk _{P_j})</math> |
| − | :<math>B(k,q)=-\frac{1+e^{ik}e^{iq}-e^{ik}}{1+e^{ik}e^{iq}-e^{iq}}</math | + | * Bethe ansatz equation for wave numbers : there are n conditions |
| − | * | + | :<math>\exp(ik_jn)=\prod_{\ell \neq j}B(k_j,k_\ell)=\prod_{\ell=1}^{n}B(k_j,k_\ell),\quad \forall j=1,\cdots, n</math> where |
| − | :<math>\lambda=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{\prod_{j=1}^{n}1-e^{ik_{j}}}=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{(1-e^{ik_{j}})\cdots(1-e^{ik_{j}})}</math | + | :<math>B(k,q)=-\frac{1+e^{ik}e^{iq}-e^{ik}}{1+e^{ik}e^{iq}-e^{iq}}</math> |
| + | * eigenvalue <math>\lambda</math> of <math>v</math> is given by | ||
| + | :<math>\lambda=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{\prod_{j=1}^{n}1-e^{ik_{j}}}=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{(1-e^{ik_{j}})\cdots(1-e^{ik_{j}})}</math> | ||
| − | ==anistropic one-dimensional Heisenberg | + | ==anistropic one-dimensional Heisenberg XXZ model== |
* [[Heisenberg spin chain model]] | * [[Heisenberg spin chain model]] | ||
| − | * Hamiltonian of XXZ model | + | * Hamiltonian of XXZ model with anisotropic parameter <math>\Delta=1/2</math> |
| − | :<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)=-\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \frac{1}{2} \sigma_j^z \sigma_{j+1}^z)</math | + | :<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)=-\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \frac{1}{2} \sigma_j^z \sigma_{j+1}^z)</math> |
* two body scattering term | * two body scattering term | ||
:<math>s_{jl}=1-2\Delta e^{ik_l}+ e^{ik_l+ik_j}=1-e^{ik_l}+ e^{ik_l+ik_j}</math> | :<math>s_{jl}=1-2\Delta e^{ik_l}+ e^{ik_l+ik_j}=1-e^{ik_l}+ e^{ik_l+ik_j}</math> | ||
| 85번째 줄: | 88번째 줄: | ||
:<math>\exp(ik_jN)=(-1)^{N-1}\prod_{l=1}^{N}\exp(-i\theta(k_j,k_l))</math> | :<math>\exp(ik_jN)=(-1)^{N-1}\prod_{l=1}^{N}\exp(-i\theta(k_j,k_l))</math> | ||
where <math>\theta(p,q)</math> is defined as | where <math>\theta(p,q)</math> is defined as | ||
| − | :<math>\exp(-i\theta(p,q))=\frac{1-2\Delta e^{ip}+e^{i(p+q)}}{1-2\Delta e^{iq}+e^{i(p+q)}}=\frac{1-e^{ip}+e^{i(p+q)}}{1- e^{iq}+e^{i(p+q)}}</math | + | :<math>\exp(-i\theta(p,q))=\frac{1-2\Delta e^{ip}+e^{i(p+q)}}{1-2\Delta e^{iq}+e^{i(p+q)}}=\frac{1-e^{ip}+e^{i(p+q)}}{1- e^{iq}+e^{i(p+q)}}</math> |
* fundamental equation | * fundamental equation | ||
| − | :<math>k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)</math | + | :<math>k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)</math> |
| − | * eigenvalue | + | * eigenvalue |
* ground state eigenvector for Hamiltonian is a common eigenvector although the eigenvalues are different | * ground state eigenvector for Hamiltonian is a common eigenvector although the eigenvalues are different | ||
* the maximum eigenstate of the transfer matrix and the ground state of the above Hamiltonian are identical because both are characterized by the fact that <math>f(x_ 1,\cdots,x_n)>0</math> | * the maximum eigenstate of the transfer matrix and the ground state of the above Hamiltonian are identical because both are characterized by the fact that <math>f(x_ 1,\cdots,x_n)>0</math> | ||
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| − | + | ==Sutherland's observation== | |
| − | + | * the eigenvectors of the transfer matrix depended on a,b,c only via the parameter | |
| + | :<math>\Delta=\frac{a^2+b^2-c^2}{2ab}=\cos \eta</math> | ||
| + | * <math>\Delta</math> = anisotropic parameter in [[Heisenberg spin chain model]] | ||
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| − | * | + | ==one-point function== |
| − | :<math>\ | + | * by Baxter's corner transfer matrix method, we get |
| − | + | :<math> | |
| + | G'(a)=\sum_{{\mathbb{p}\in \mathcal{P}(\Lambda_0)}\atop {W(0,\mathbb{p})=a}}q^{2\sum_{k=0}^{\infty}(k+1)(H(\mathbb{p}(k+1),\mathbb{p}(k))-H(\mathbb{p}_{\Lambda_0}(k+1),\mathbb{p}_{\Lambda_0}(k)))} | ||
| + | </math> | ||
| + | * one can evaluate the sum | ||
| + | :<math> | ||
| + | G'(a)= | ||
| + | \begin{cases} | ||
| + | \frac{q^{\frac{a^2}{2}}}{\prod_{n=1}^{\infty}(1-q^{2n})}, & \text{if </math>a<math> is even}\\ | ||
| + | 0, & \text{if </math>a<math> is odd} \\ | ||
| + | \end{cases} | ||
| + | </math> | ||
| + | :<math> | ||
| − | == | + | ==thermodynamic properties== |
| − | + | ===entropy of two-dimensional ice=== | |
| − | + | * entropy is given as <math>Mk\ln W</math> where M is the number of molecules and <math>W=(4/3)^{3/2}=1.53960\cdots</math> | |
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| − | ==free energy== | + | ===free energy=== |
| − | * <math>F=-kT \ln Z=-\beta \ln Z</math> | + | * <math>F=-kT \ln Z=-\frac{1}{\beta} \ln Z</math> |
| − | ==partition function== | + | ===partition function=== |
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| − | ==correlation functions== | + | ===correlation functions=== |
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==computational resource== | ==computational resource== | ||
* https://docs.google.com/file/d/0B8XXo8Tve1cxSG5zZm80QVROMGc/edit | * https://docs.google.com/file/d/0B8XXo8Tve1cxSG5zZm80QVROMGc/edit | ||
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==related items== | ==related items== | ||
| 135번째 줄: | 148번째 줄: | ||
* [[Bethe ansatz]] | * [[Bethe ansatz]] | ||
* [[Heisenberg spin chain model]] | * [[Heisenberg spin chain model]] | ||
| + | * [[Alternating sign matrix theorem]] | ||
| + | * [[Proofs and Confirmation]] | ||
* [[2D Yang-Mills gauge theory]] | * [[2D Yang-Mills gauge theory]] | ||
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==encyclopedia== | ==encyclopedia== | ||
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* http://en.wikipedia.org/wiki/Ice-type_model | * http://en.wikipedia.org/wiki/Ice-type_model | ||
* http://en.wikipedia.org/wiki/Spin_ice | * http://en.wikipedia.org/wiki/Spin_ice | ||
| 161번째 줄: | 161번째 줄: | ||
| − | + | ==books== | |
| − | + | * R. J. Baxter, [http://tpsrv.anu.edu.au/Members/baxter/book Exactly Solved Models in Statistical mechanics], 1982 | |
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==expositions== | ==expositions== | ||
| − | * | + | * http://arxiv.org/abs/1512.07955 |
| + | * Lamers, J. “A Pedagogical Introduction to Quantum Integrability, with a View towards Theoretical High-Energy Physics.” arXiv:1501.06805 [hep-Th, Physics:math-Ph, Physics:nlin], January 27, 2015. http://arxiv.org/abs/1501.06805. | ||
* Reshetikhin, N. 2010. “Lectures on the Integrability of the Six-vertex Model.” In Exact Methods in Low-dimensional Statistical Physics and Quantum Computing, 197–266. Oxford: Oxford Univ. Press. http://www.ams.org/mathscinet-getitem?mr=2668647. | * Reshetikhin, N. 2010. “Lectures on the Integrability of the Six-vertex Model.” In Exact Methods in Low-dimensional Statistical Physics and Quantum Computing, 197–266. Oxford: Oxford Univ. Press. http://www.ams.org/mathscinet-getitem?mr=2668647. | ||
| + | ** Reshetikhin, N. “Lectures on the Integrability of the 6-Vertex Model.” arXiv:1010.5031 [cond-Mat, Physics:math-Ph], October 24, 2010. http://arxiv.org/abs/1010.5031. | ||
* T Miwa [http://www.springerlink.com/content/f9961j132852j27q/ Integrability of the Quantum XXZ Hamiltonian], 2009 | * T Miwa [http://www.springerlink.com/content/f9961j132852j27q/ Integrability of the Quantum XXZ Hamiltonian], 2009 | ||
* Tetsuo Deguchi [http://arxiv.org/abs/cond-mat/0304309 Introduction to solvable lattice models in statistical and mathematical physics], 2003 | * Tetsuo Deguchi [http://arxiv.org/abs/cond-mat/0304309 Introduction to solvable lattice models in statistical and mathematical physics], 2003 | ||
| + | * De Vega, H. J. 1993. “Bethe Ansatz and Quantum Groups.” arXiv:hep-th/9308008, August. http://arxiv.org/abs/hep-th/9308008. | ||
| + | * Karowski, M. 1990. “Yang-Baxter Algebra — Bethe Ansatz — Conformal Quantum Field Theories — Quantum Groups.” In Quantum Groups, edited by H.-D. Doebner and J.-D. Hennig, 183–218. Lecture Notes in Physics 370. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/3-540-53503-9_47. | ||
| + | * LeBowitz, J L. 1968. “Statistical Mechanics-A Review of Selected Rigorous Results.” Annual Review of Physical Chemistry 19 (1): 389–418. doi:10.1146/annurev.pc.19.100168.002133. http://www.annualreviews.org/doi/abs/10.1146/annurev.pc.19.100168.002133 | ||
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| + | ===blogs=== | ||
* [http://paulingblog.wordpress.com/2010/08/18/a-theory-of-the-structure-of-ice/ A Theory of the Structure of Ice] | * [http://paulingblog.wordpress.com/2010/08/18/a-theory-of-the-structure-of-ice/ A Theory of the Structure of Ice] | ||
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==articles== | ==articles== | ||
| + | * Reshetikhin, Nicolai, and Ananth Sridhar. “Integrability of Limit Shapes of the Six Vertex Model.” arXiv:1510.01053 [cond-Mat, Physics:hep-Th, Physics:math-Ph], October 5, 2015. http://arxiv.org/abs/1510.01053. | ||
| + | * Kozlowski, K. K. “On Condensation Properties of Bethe Roots Associated with the XXZ Chain.” arXiv:1508.05741 [math-Ph, Physics:nlin], August 24, 2015. http://arxiv.org/abs/1508.05741. | ||
| + | * Martins, M. J. ‘The Symmetric Six-Vertex Model and the Segre Cubic Threefold’. arXiv:1505.07418 [math-Ph], 27 May 2015. http://arxiv.org/abs/1505.07418. | ||
| + | * Morin-Duchesne, Alexi, Jorgen Rasmussen, Philippe Ruelle, and Yvan Saint-Aubin. ‘On the Reality of Spectra of </math>\boldsymbol{U_q(sl_2)}<math>-Invariant XXZ Hamiltonians’. arXiv:1502.01859 [cond-Mat, Physics:hep-Th, Physics:math-Ph], 6 February 2015. http://arxiv.org/abs/1502.01859. | ||
| + | * Vieira, R. S., and A. Lima-Santos. “Where Are the Roots of the Bethe Ansatz Equations?” arXiv:1502.05316 [cond-Mat, Physics:math-Ph, Physics:nlin], February 18, 2015. http://arxiv.org/abs/1502.05316. | ||
| + | * Hamel, Angèle M., and Ronald C. King. “Tokuyama’s Identity for Factorial Schur Functions.” arXiv:1501.03561 [math], January 14, 2015. http://arxiv.org/abs/1501.03561. | ||
| + | * Tavares, T. S., G. A. P. Ribeiro, and V. E. Korepin. “The Entropy of the Six-Vertex Model with Variety of Different Boundary Conditions.” arXiv:1501.02818 [cond-Mat, Physics:math-Ph, Physics:nlin], January 12, 2015. http://arxiv.org/abs/1501.02818. | ||
| + | * Garbali, Alexander. ‘The Scalar Product of XXZ Spin Chain Revisited. Application to the Ground State at </math>\Delta=-1/2<math>’. arXiv:1411.2938 [math-Ph], 11 November 2014. http://arxiv.org/abs/1411.2938. | ||
| + | * Ribeiro, G. A. P., and V. E. Korepin. “Thermodynamic Limit of the Six-Vertex Model with Reflecting End.” arXiv:1409.1212 [cond-Mat, Physics:hep-Th, Physics:math-Ph, Physics:nlin], September 3, 2014. http://arxiv.org/abs/1409.1212. | ||
| + | * Mangazeev, Vladimir V. “Q-Operators in the Six-Vertex Model.” arXiv:1406.0662 [hep-Th, Physics:math-Ph], June 3, 2014. http://arxiv.org/abs/1406.0662. | ||
* António, N. Cirilo, N. Manojlović, and Z. Nagy. 2013. “Trigonometric Sl(2) Gaudin Model with Boundary Terms.” arXiv:1303.2481 (March 11). http://arxiv.org/abs/1303.2481. | * António, N. Cirilo, N. Manojlović, and Z. Nagy. 2013. “Trigonometric Sl(2) Gaudin Model with Boundary Terms.” arXiv:1303.2481 (March 11). http://arxiv.org/abs/1303.2481. | ||
| − | * | + | * Szabo, Richard J., and Miguel Tierz. 2011. “Two-Dimensional Yang-Mills Theory, Painleve Equations and the Six-Vertex Model”. ArXiv e-print 1102.3640. http://arxiv.org/abs/1102.3640. |
| + | * Deguchi, Tetsuo. 2006. “The Six-vertex Model at Roots of Unity and Some Highest Weight Representations of the </math>\rm Sl_2<math> Loop Algebra.” Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics 7 (7-8): 1531–1540. doi:[http://dx.doi.org/10.1007/s00023-006-0290- 10.1007/s00023-006-0290-8] | ||
| + | * De Vega, H.J., and F. Woynarovich. 1985. “Method for Calculating Finite Size Corrections in Bethe Ansatz Systems: Heisenberg Chain and Six-vertex Model.” Nuclear Physics B 251: 439–456. doi:[http://dx.doi.org/10.1016/0550-3213(85)90271-8 10.1016/0550-3213(85)90271-8]. | ||
* Kazuhiko Minami, [http://dx.doi.org/10.1063/1.2890671 The free energies of six-vertex models and the n-equivalence relation] | * Kazuhiko Minami, [http://dx.doi.org/10.1063/1.2890671 The free energies of six-vertex models and the n-equivalence relation] | ||
| − | * | + | * Lieb, Elliott H. 1967. “Exact Solution of the F Model of An Antiferroelectric.” Physical Review Letters 18 (24): 1046–48. doi:[http://dx.doi.org/10.1103/PhysRevLett.18.1046 10.1103/PhysRevLett.18.1046]. |
| − | + | * Lieb, Elliott H. 1967. “Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric.” Physical Review Letters 19 (3): 108–10. doi:[http://dx.doi.org/10.1103/PhysRevLett.19.108 10.1103/PhysRevLett.19.108]. | |
| − | + | * Sutherland, Bill. “Exact Solution of a Two-Dimensional Model for Hydrogen-Bonded Crystals.” Physical Review Letters 19, no. 3 (July 17, 1967): 103–4. doi:10.1103/PhysRevLett.19.103. | |
| − | + | * Lieb, Elliott H. “Exact Solution of the Problem of the Entropy of Two-Dimensional Ice.” Physical Review Letters 18, no. 17 (April 24, 1967): 692–94. doi:[http://dx.doi.org/10.1103/PhysRevLett.18.692 10.1103/PhysRevLett.18.692]. | |
| − | + | * '''[YY1966-2]''' Yang, C. N., and C. P. Yang.“One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System.” Physical Review 150, no. 1 (October 7, 1966): 327–39. doi:[http://dx.doi.org/10.1103/PhysRev.150.327 10.1103/PhysRev.150.327]. | |
| − | + | * Yang, C. N., and C. P. Yang. “One-Dimensional Chain of Anisotropic Spin-Spin Interactions.” Physics Letters 20, no. 1 (January 15, 1966): 9–10. doi:[http://dx.doi.org/10.1016/0031-9163%2866%2991024-9 10.1016/0031-9163(66)91024-9]. | |
| − | * [http://dx.doi.org/10.1103/PhysRevLett.18.692 | + | * Pauling, Linus. 1935. “The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement.” Journal of the American Chemical Society 57 (12): 2680–84. doi:[http://dx.doi.org/10.1021/ja01315a102 10.1021/ja01315a102]. |
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| − | * '''[YY1966-2]''' | ||
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[[분류:개인노트]] | [[분류:개인노트]] | ||
[[분류:integrable systems]] | [[분류:integrable systems]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
| + | [[분류:migrate]] | ||
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| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q5985139 Q5985139] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'ice'}, {'OP': '*'}, {'LOWER': 'type'}, {'LEMMA': 'model'}] | ||
| + | * [{'LOWER': 'six'}, {'OP': '*'}, {'LOWER': 'vertex'}, {'LEMMA': 'model'}] | ||
2021년 2월 17일 (수) 01:11 기준 최신판
introduction
- six-vertex model, also called ice-type model, R model, Rys model
- The Hamiltonian of Hisenberg XXZ spin chain and the six-vertex transfer matrix have the same eigenvectors
- Bethe ansatz can be applied to solve the model
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
integrability of the model and the Yang-Baxter equation
- \(T(u)\) transfer matrix
- \(\log T(u)=\sum_{n=0}^{\infty}Q_{n}u^n\)
- here \(Q_1\) plays the role of the Hamiltonian
- necessary and sufficient codntion to have infinitely many conserved quantities
\[[T(u), T(v)]=0\] which implies \([Q_n,Q_m]=0\)
- in order to have \([T(u), T(v)]=0\), the Yang-Baxter equation (YBE) must be satisfied
R-matrix and Boltzmann weights
\[ R(u,\eta)=\rho\left( \begin{array}{cccc} \sin (u+\eta ) & 0 & 0 & 0 \\ 0 & \sin (u) & \sin (\eta ) & 0 \\ 0 & \sin (\eta ) & \sin (u) & 0 \\ 0 & 0 & 0 & \sin (u+\eta ) \end{array} \right) \]
- multiplicative form of S-matrix of the quantum sine-Gordon model
\[ \check{R}(x)= \left( \begin{array}{cccc} x-q^2 & 0 & 0 & 0 \\ 0 & 1-q^2 & q (x-1) & 0 \\ 0 & q (x-1) & \left(1-q^2\right) x & 0 \\ 0 & 0 & 0 & x-q^2 \\ \end{array} \right) \]
transfer matrix formalism and coordinate Bethe ansatz
- \(M=N^{2}\) number of molecules
- 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 Bethe ansatz for the eigenvectors of the transfer matrix
- let \(f(x_1,\cdots,x_n)\) be the coefficient in an eigenvector \(v\) of the state with up arrows at the sites \(x_ 1<x_ 2<\cdots<x_n\) so that
\[v(k_1,\cdots,k_n)= \sum_{\substack{\mathbf{x}=(x_ 1,x_ 2,\cdots,x_n) \\ x_ 1<x_ 2<\cdots<x_n}} f(x_1,\cdots,x_n|k_1,\cdots,k_n)\sigma_{-}^{(x_1)}\cdots\sigma_{-}^{(x_n)}|0\rangle \]
- Bethe ansatz suggests the following form for \(f\)
\[f(x_ 1,\cdots,x_n)=\sum_{P\in S_n}A (P)\exp(i\sum_{j=1}^{n}x_jk _{P_j})\]
- Bethe ansatz equation for wave numbers : there are n conditions
\[\exp(ik_jn)=\prod_{\ell \neq j}B(k_j,k_\ell)=\prod_{\ell=1}^{n}B(k_j,k_\ell),\quad \forall j=1,\cdots, n\] where \[B(k,q)=-\frac{1+e^{ik}e^{iq}-e^{ik}}{1+e^{ik}e^{iq}-e^{iq}}\]
- eigenvalue \(\lambda\) of \(v\) is given by
\[\lambda=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{\prod_{j=1}^{n}1-e^{ik_{j}}}=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{(1-e^{ik_{j}})\cdots(1-e^{ik_{j}})}\]
anistropic one-dimensional Heisenberg XXZ model
- Heisenberg spin chain model
- Hamiltonian of XXZ model with anisotropic parameter \(\Delta=1/2\)
\[\hat H = -\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \Delta \sigma_j^z \sigma_{j+1}^z)=-\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \frac{1}{2} \sigma_j^z \sigma_{j+1}^z)\]
- two body scattering term
\[s_{jl}=1-2\Delta e^{ik_l}+ e^{ik_l+ik_j}=1-e^{ik_l}+ e^{ik_l+ik_j}\]
- equation satisfied by wave numbers
\[\exp(ik_jN)=(-1)^{N-1}\prod_{l=1}^{N}\exp(-i\theta(k_j,k_l))\] where \(\theta(p,q)\) is defined as \[\exp(-i\theta(p,q))=\frac{1-2\Delta e^{ip}+e^{i(p+q)}}{1-2\Delta e^{iq}+e^{i(p+q)}}=\frac{1-e^{ip}+e^{i(p+q)}}{1- e^{iq}+e^{i(p+q)}}\]
- fundamental equation
\[k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)\]
- eigenvalue
- ground state eigenvector for Hamiltonian is a common eigenvector although the eigenvalues are different
- the maximum eigenstate of the transfer matrix and the ground state of the above Hamiltonian are identical because both are characterized by the fact that \(f(x_ 1,\cdots,x_n)>0\)
- see [YY1966-2]
Sutherland's observation
- the eigenvectors of the transfer matrix depended on a,b,c only via the parameter
\[\Delta=\frac{a^2+b^2-c^2}{2ab}=\cos \eta\]
- \(\Delta\) = anisotropic parameter in Heisenberg spin chain model
one-point function
- by Baxter's corner transfer matrix method, we get
\[ G'(a)=\sum_{{\mathbb{p}\in \mathcal{P}(\Lambda_0)}\atop {W(0,\mathbb{p})=a}}q^{2\sum_{k=0}^{\infty}(k+1)(H(\mathbb{p}(k+1),\mathbb{p}(k))-H(\mathbb{p}_{\Lambda_0}(k+1),\mathbb{p}_{\Lambda_0}(k)))} \]
- one can evaluate the sum
\[ G'(a)= \begin{cases} \frac{q^{\frac{a^2}{2}}}{\prod_{n=1}^{\infty}(1-q^{2n})}, & \text{if \]a\( is even}\\ 0, & \text{if \)a\( is odd} \\ \end{cases} \)
\[ =='"`UNIQ--h-9--QINU`"'thermodynamic properties== ==='"`UNIQ--h-10--QINU`"'entropy of two-dimensional ice=== * entropy is given as \(Mk\ln W\] where M is the number of molecules and <math>W=(4/3)^{3/2}=1.53960\cdots\)
free energy
- \(F=-kT \ln Z=-\frac{1}{\beta} \ln Z\)
partition function
correlation functions
computational resource
- Bethe ansatz
- Heisenberg spin chain model
- Alternating sign matrix theorem
- Proofs and Confirmation
- 2D Yang-Mills gauge theory
encyclopedia
- http://en.wikipedia.org/wiki/Ice-type_model
- http://en.wikipedia.org/wiki/Spin_ice
- http://en.wikipedia.org/wiki/Heisenberg_model_(quantum)
books
- R. J. Baxter, Exactly Solved Models in Statistical mechanics, 1982
expositions
- http://arxiv.org/abs/1512.07955
- Lamers, J. “A Pedagogical Introduction to Quantum Integrability, with a View towards Theoretical High-Energy Physics.” arXiv:1501.06805 [hep-Th, Physics:math-Ph, Physics:nlin], January 27, 2015. http://arxiv.org/abs/1501.06805.
- Reshetikhin, N. 2010. “Lectures on the Integrability of the Six-vertex Model.” In Exact Methods in Low-dimensional Statistical Physics and Quantum Computing, 197–266. Oxford: Oxford Univ. Press. http://www.ams.org/mathscinet-getitem?mr=2668647.
- Reshetikhin, N. “Lectures on the Integrability of the 6-Vertex Model.” arXiv:1010.5031 [cond-Mat, Physics:math-Ph], October 24, 2010. http://arxiv.org/abs/1010.5031.
- T Miwa Integrability of the Quantum XXZ Hamiltonian, 2009
- Tetsuo Deguchi Introduction to solvable lattice models in statistical and mathematical physics, 2003
- De Vega, H. J. 1993. “Bethe Ansatz and Quantum Groups.” arXiv:hep-th/9308008, August. http://arxiv.org/abs/hep-th/9308008.
- Karowski, M. 1990. “Yang-Baxter Algebra — Bethe Ansatz — Conformal Quantum Field Theories — Quantum Groups.” In Quantum Groups, edited by H.-D. Doebner and J.-D. Hennig, 183–218. Lecture Notes in Physics 370. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/3-540-53503-9_47.
- LeBowitz, J L. 1968. “Statistical Mechanics-A Review of Selected Rigorous Results.” Annual Review of Physical Chemistry 19 (1): 389–418. doi:10.1146/annurev.pc.19.100168.002133. http://www.annualreviews.org/doi/abs/10.1146/annurev.pc.19.100168.002133
blogs
articles
- Reshetikhin, Nicolai, and Ananth Sridhar. “Integrability of Limit Shapes of the Six Vertex Model.” arXiv:1510.01053 [cond-Mat, Physics:hep-Th, Physics:math-Ph], October 5, 2015. http://arxiv.org/abs/1510.01053.
- Kozlowski, K. K. “On Condensation Properties of Bethe Roots Associated with the XXZ Chain.” arXiv:1508.05741 [math-Ph, Physics:nlin], August 24, 2015. http://arxiv.org/abs/1508.05741.
- Martins, M. J. ‘The Symmetric Six-Vertex Model and the Segre Cubic Threefold’. arXiv:1505.07418 [math-Ph], 27 May 2015. http://arxiv.org/abs/1505.07418.
- Morin-Duchesne, Alexi, Jorgen Rasmussen, Philippe Ruelle, and Yvan Saint-Aubin. ‘On the Reality of Spectra of </math>\boldsymbol{U_q(sl_2)}\(-Invariant XXZ Hamiltonians’. arXiv:1502.01859 [cond-Mat, Physics:hep-Th, Physics:math-Ph], 6 February 2015. http://arxiv.org/abs/1502.01859. * Vieira, R. S., and A. Lima-Santos. “Where Are the Roots of the Bethe Ansatz Equations?” arXiv:1502.05316 [cond-Mat, Physics:math-Ph, Physics:nlin], February 18, 2015. http://arxiv.org/abs/1502.05316. * Hamel, Angèle M., and Ronald C. King. “Tokuyama’s Identity for Factorial Schur Functions.” arXiv:1501.03561 [math], January 14, 2015. http://arxiv.org/abs/1501.03561. * Tavares, T. S., G. A. P. Ribeiro, and V. E. Korepin. “The Entropy of the Six-Vertex Model with Variety of Different Boundary Conditions.” arXiv:1501.02818 [cond-Mat, Physics:math-Ph, Physics:nlin], January 12, 2015. http://arxiv.org/abs/1501.02818. * Garbali, Alexander. ‘The Scalar Product of XXZ Spin Chain Revisited. Application to the Ground State at \)\Delta=-1/2\(’. arXiv:1411.2938 [math-Ph], 11 November 2014. http://arxiv.org/abs/1411.2938. * Ribeiro, G. A. P., and V. E. Korepin. “Thermodynamic Limit of the Six-Vertex Model with Reflecting End.” arXiv:1409.1212 [cond-Mat, Physics:hep-Th, Physics:math-Ph, Physics:nlin], September 3, 2014. http://arxiv.org/abs/1409.1212. * Mangazeev, Vladimir V. “Q-Operators in the Six-Vertex Model.” arXiv:1406.0662 [hep-Th, Physics:math-Ph], June 3, 2014. http://arxiv.org/abs/1406.0662. * António, N. Cirilo, N. Manojlović, and Z. Nagy. 2013. “Trigonometric Sl(2) Gaudin Model with Boundary Terms.” arXiv:1303.2481 (March 11). http://arxiv.org/abs/1303.2481. * Szabo, Richard J., and Miguel Tierz. 2011. “Two-Dimensional Yang-Mills Theory, Painleve Equations and the Six-Vertex Model”. ArXiv e-print 1102.3640. http://arxiv.org/abs/1102.3640. * Deguchi, Tetsuo. 2006. “The Six-vertex Model at Roots of Unity and Some Highest Weight Representations of the \)\rm Sl_2<math> Loop Algebra.” Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics 7 (7-8): 1531–1540. doi:10.1007/s00023-006-0290-8
- De Vega, H.J., and F. Woynarovich. 1985. “Method for Calculating Finite Size Corrections in Bethe Ansatz Systems: Heisenberg Chain and Six-vertex Model.” Nuclear Physics B 251: 439–456. doi:10.1016/0550-3213(85)90271-8.
- Kazuhiko Minami, The free energies of six-vertex models and the n-equivalence relation
- Lieb, Elliott H. 1967. “Exact Solution of the F Model of An Antiferroelectric.” Physical Review Letters 18 (24): 1046–48. doi:10.1103/PhysRevLett.18.1046.
- Lieb, Elliott H. 1967. “Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric.” Physical Review Letters 19 (3): 108–10. doi:10.1103/PhysRevLett.19.108.
- Sutherland, Bill. “Exact Solution of a Two-Dimensional Model for Hydrogen-Bonded Crystals.” Physical Review Letters 19, no. 3 (July 17, 1967): 103–4. doi:10.1103/PhysRevLett.19.103.
- Lieb, Elliott H. “Exact Solution of the Problem of the Entropy of Two-Dimensional Ice.” Physical Review Letters 18, no. 17 (April 24, 1967): 692–94. doi:10.1103/PhysRevLett.18.692.
- [YY1966-2] Yang, C. N., and C. P. Yang.“One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System.” Physical Review 150, no. 1 (October 7, 1966): 327–39. doi:10.1103/PhysRev.150.327.
- Yang, C. N., and C. P. Yang. “One-Dimensional Chain of Anisotropic Spin-Spin Interactions.” Physics Letters 20, no. 1 (January 15, 1966): 9–10. doi:10.1016/0031-9163(66)91024-9.
- Pauling, Linus. 1935. “The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement.” Journal of the American Chemical Society 57 (12): 2680–84. doi:10.1021/ja01315a102.
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위키데이터
- ID : Q5985139
Spacy 패턴 목록
- [{'LOWER': 'ice'}, {'OP': '*'}, {'LOWER': 'type'}, {'LEMMA': 'model'}]
- [{'LOWER': 'six'}, {'OP': '*'}, {'LOWER': 'vertex'}, {'LEMMA': 'model'}]