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

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==integrability of the model and the Yang-Baxter equation==
 
==integrability of the model and the Yang-Baxter equation==
−
* $T(u)$ transfer matrix
+
* <math>T(u)</math> transfer matrix
−
* $\log T(u)=\sum_{n=0}^{\infty}Q_{n}u^n$
+
* <math>\log T(u)=\sum_{n=0}^{\infty}Q_{n}u^n</math>
−
* here $Q_1$ plays the role of the Hamiltonian
+
* 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
−
$$[T(u), T(v)]=0$$
+
:<math>[T(u), T(v)]=0</math>
−
which implies $[Q_n,Q_m]=0$
+
which implies <math>[Q_n,Q_m]=0</math>
−
* in order to have $[T(u), T(v)]=0$, the [[Yang-Baxter equation (YBE)]] must be satisfied
+
* in order to have <math>[T(u), T(v)]=0</math>, the [[Yang-Baxter equation (YBE)]] must be satisfied
  
  
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===R-matrix and Boltzmann weights===
 
===R-matrix and Boltzmann weights===
 
* [[R-matrix]]  
 
* [[R-matrix]]  
−
$$
+
:<math>
 
R(u,\eta)=\rho\left(
 
R(u,\eta)=\rho\left(
 
\begin{array}{cccc}
 
\begin{array}{cccc}
46번째 줄: 46번째 줄:
 
\end{array}
 
\end{array}
 
\right)
 
\right)
−
$$
+
</math>
  
 
* multiplicative form of [[S-matrix of the quantum sine-Gordon model]]
 
* multiplicative form of [[S-matrix of the quantum sine-Gordon model]]
−
$$
+
:<math>
 
\check{R}(x)=
 
\check{R}(x)=
 
\left(
 
\left(
59번째 줄: 59번째 줄:
 
\end{array}
 
\end{array}
 
\right)
 
\right)
−
$$
+
</math>
  
 
==transfer matrix formalism and coordinate Bethe ansatz==
 
==transfer matrix formalism and coordinate Bethe ansatz==
66번째 줄: 66번째 줄:
 
* one can regard the up(or down) arrows in a row as 'particles'
 
* 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
 
* because of the ice rule, their number is conserved and one can try [[Bethe ansatz]] for the eigenvectors of the transfer matrix
−
* let <math>f(x_1,\cdots,x_n)</math> be the coefficient in an eigenvector $v$ of the state with up arrows at the sites <math>x_ 1<x_ 2<\cdots<x_n</math> so that
+
* 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>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 </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 $f$
+
* Bethe ansatz suggests the following form for <math>f</math>
 
:<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>f(x_ 1,\cdots,x_n)=\sum_{P\in S_n}A (P)\exp(i\sum_{j=1}^{n}x_jk _{P_j})</math>
 
* Bethe ansatz equation for wave numbers : there are n conditions
 
* 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>\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>B(k,q)=-\frac{1+e^{ik}e^{iq}-e^{ik}}{1+e^{ik}e^{iq}-e^{iq}}</math>
 
:<math>B(k,q)=-\frac{1+e^{ik}e^{iq}-e^{ik}}{1+e^{ik}e^{iq}-e^{iq}}</math>
−
* eigenvalue $\lambda$ of $v$ is given by
+
* 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>
 
:<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>
  
102번째 줄: 102번째 줄:
 
*  the eigenvectors of the transfer matrix depended on a,b,c only via the parameter
 
*  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=\frac{a^2+b^2-c^2}{2ab}=\cos \eta</math>
−
*  $\Delta$ = anisotropic parameter in [[Heisenberg spin chain model]]
+
*  <math>\Delta</math> = anisotropic parameter in [[Heisenberg spin chain model]]
  
  
 
==one-point function==
 
==one-point function==
 
* by Baxter's corner transfer matrix method, we get
 
* 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)))}
 
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
 
* one can evaluate the sum
−
$$
+
:<math>
 
G'(a)=
 
G'(a)=
 
\begin{cases}  
 
\begin{cases}  
−
  \frac{q^{\frac{a^2}{2}}}{\prod_{n=1}^{\infty}(1-q^{2n})}, & \text{if $a$ is even}\\  
+
  \frac{q^{\frac{a^2}{2}}}{\prod_{n=1}^{\infty}(1-q^{2n})}, & \text{if </math>a<math> is even}\\  
−
  0, & \text{if $a$ is odd} \\  
+
  0, & \text{if </math>a<math> is odd} \\  
 
\end{cases}
 
\end{cases}
−
$$
+
</math>
  
−
$$
+
:<math>
  
 
==thermodynamic properties==
 
==thermodynamic properties==
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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
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===blogs===
 
===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]
−
 
−
 
−
 
  
 
==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.
 
* 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.
 
* 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.
+
* 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.
 
* 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.
 
* 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.
 
* 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:[http://dx.doi.org/10.1007/s00023-006-0290- 10.1007/s00023-006-0290-8]
+
* 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].
 
* 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]
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[[분류:integrable systems]]
 
[[분류:integrable systems]]
 
[[분류:math and physics]]
 
[[분류:math and physics]]
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[[분류:migrate]]
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==메타데이터==
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q5985139 Q5985139]
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===Spacy 패턴 목록===
 +
* [{'LOWER': 'ice'}, {'OP': '*'}, {'LOWER': 'type'}, {'LEMMA': 'model'}]
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* [{'LOWER': 'six'}, {'OP': '*'}, {'LOWER': 'vertex'}, {'LEMMA': 'model'}]

2021년 2월 17일 (수) 02:11 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


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.
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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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