"Six-vertex model and Quantum XXZ Hamiltonian"의 두 판 사이의 차이

2015년 1월 15일 (목) 20:04 판

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
- 틀:수학노트
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-matrix

$$ R(u,\eta)=\rho\left( \begin{array}{cccc} \sin (u+\eta ) & 0 & 0 & 0 \\ 0 & \sin (u) & \sin (\eta ) & 0 \\ 0 & \sin (\eta ) & \sin (u) & 0 \\ 0 & 0 & 0 & \sin (u+\eta ) \end{array} \right) $$

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

$$ G'(a)= \begin{cases} \frac{q^{\frac{a^2}{2}}}{\prod_{n=1}^{\infty}(1-q^{2n})}, & \text{if $a$ is even}\\ 0, & \text{if $a$ is odd} \\ \end{cases} $$

$$

thermodynamic properties

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

$F=-kT \ln Z=-\frac{1}{\beta} \ln Z$

partition function

correlation functions

computational resource

https://docs.google.com/file/d/0B8XXo8Tve1cxSG5zZm80QVROMGc/edit

related items

encyclopedia

books

R. J. Baxter, Exactly Solved Models in Statistical mechanics, 1982

expositions

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.
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

A Theory of the Structure of Ice

articles

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 $\rm Sl_2$ 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.

@@ 130번째 줄: / 130번째 줄: @@
 ===free energy===
-* <math>F=-kT \ln Z=-\beta \ln Z</math>
+* <math>F=-kT \ln Z=-\frac{1}{\beta} \ln Z</math>
 ===partition function===