Six-vertex model and Quantum XXZ Hamiltonian
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
integrability of the model
- $T(u)$ 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 YBE must be satisfied
R-matrix and Yang-Baxter equation
$$ R(u,\eta)=\rho\left( \begin{array}{cccc} \sin (u+\eta ) & 0 & 0 & 0 \\ 0 & \sin (u) & \sin (\eta ) & 0 \\ 0 & \sin (\eta ) & \sin (u) & 0 \\ 0 & 0 & 0 & \sin (u+\eta ) \end{array} \right) $$
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 a Bethe ansatz for the eigenvectors of the transfer matrix
- \(f(x_1,\cdots,x_n)\) be the amplitude in an eigenvector of the state with up arrows at the sites \( x_ 1<x_ 2<\cdots<x_n\)
- obtain the equation for amplitudes \[f(x_ 1,\cdots,x_n)=\sum_{P}A (P)\exp(i\sum_{j=1}^{n}x_jk _{P_j})\]
- Bethe ansatz equation for wave numbers : there are n conditions
\[\exp(ik_jn)=\prod_{j \neq i}B(k_i,k_j)=\prod_{j=1}^{n}B(k_i,k_j)\]
where
\[B(k,q)=-\frac{1+e^{ik}e^{iq}-e^{ik}}{1+e^{ik}e^{iq}-e^{iq}}\]
- eigenvalue
\[\lambda=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{\prod_{j=1}^{n}1-e^{ik_{j}}}=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{(1-e^{ik_{j}})\cdots(1-e^{ik_{j}})}\]
anistropic one-dimensional Heisenberg model (XXZ model)
- Heisenberg spin chain model
- Hamiltonian of XXZ model or XXZ spin chain with anisotropic parameter \(\Delta=1/2\)
\[\hat H = -\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \Delta \sigma_j^z \sigma_{j+1}^z)=-\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \frac{1}{2} \sigma_j^z \sigma_{j+1}^z)\]
- 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
\[\Delta=\frac{a^2+b^2-c^2}{2ab}=\cos \eta\]
- $\delta$ = 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
- \(F=-kT \ln Z=-\beta \ln Z\)
partition function
correlation functions
computational resource
books
- Exactly Solved Models in Statistical mechanics
- R. J. Baxter, 1982
encyclopedia
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Ice-type_model
- http://en.wikipedia.org/wiki/Spin_ice
- http://en.wikipedia.org/wiki/Heisenberg_model_(quantum)
expositions
- 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
- 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
- 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, 와/과Miguel Tierz. 2011. “Two-dimensional Yang-Mills theory, Painleve equations and the six-vertex model”. arXiv:1102.3640 (2월 17). http://arxiv.org/abs/1102.3640
- 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. http://dx.doi.org/10.1016/0550-3213(85)90271-8
- Kazuhiko Minami, The free energies of six-vertex models and the n-equivalence relation
- 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)
- The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement
- L. Pauling, Journal of the American Chemical Society, Vol. 57, p. 2680 (1935).
- http://dx.doi.org/10.1063/1.2890671