Six-vertex model and Quantum XXZ Hamiltonian

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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


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 \([Q_n,Q_m]=0\)


R-matrix and Boltzmann weights

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

\[ \check{R}(x)= \left( \begin{array}{cccc} x-q^2 & 0 & 0 & 0 \\ 0 & 1-q^2 & q (x-1) & 0 \\ 0 & q (x-1) & \left(1-q^2\right) x & 0 \\ 0 & 0 & 0 & x-q^2 \\ \end{array} \right) \]

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

\[\hat H = -\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \Delta \sigma_j^z \sigma_{j+1}^z)=-\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \frac{1}{2} \sigma_j^z \sigma_{j+1}^z)\]

  • 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

\[\Delta=\frac{a^2+b^2-c^2}{2ab}=\cos \eta\]


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


related items


encyclopedia


books


expositions

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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Spacy 패턴 목록

  • [{'LOWER': 'ice'}, {'OP': '*'}, {'LOWER': 'type'}, {'LEMMA': 'model'}]
  • [{'LOWER': 'six'}, {'OP': '*'}, {'LOWER': 'vertex'}, {'LEMMA': 'model'}]
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