Six-vertex model and Quantum XXZ Hamiltonian

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

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

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

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

• \(F=-kT \ln Z=-\beta \ln Z\)

computational resource

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

related items

• Bethe ansatz
• Heisenberg spin chain model
• Alternating sign matrix theorem
• 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

• 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

• 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.