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Extended Z-Invariance for Integrable Vector and Face Models and Multi-component Integrable Quad Equations

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Abstract

In a previous paper (Kels in J Phys A 50(49):495202, 2017), the author has established an extension of the Z-invariance property for integrable edge-interaction models of statistical mechanics, that satisfy the star–triangle relation (STR) form of the Yang–Baxter equation (YBE). In the present paper, an analogous extended Z-invariance property is shown to also hold for integrable vector models and interaction-round-a-face (IRF) models of statistical mechanics respectively. As for the previous case of the STR, the Z-invariance property is shown through the use of local cubic-type deformations of a 2-dimensional surface associated to the models, which allow an extension of the models onto a subset of next nearest neighbour vertices of \(\mathbb {Z}^3\), while leaving the partition functions invariant. These deformations are permitted as a consequence of the respective YBE’s satisfied by the models. The quasi-classical limit is also considered, and it is shown that an analogous Z-invariance property holds for the variational formulation of classical discrete Laplace equations which arise in this limit. From this limit, new integrable 3D-consistent multi-component quad equations are proposed, which are constructed from a degeneration of the equations of motion for IRF Boltzmann weights.

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Notes

  1. Also referred to simply as a “box” [9, 11].

  2. For N-state discrete spin models the constant factor is simply given by a power of N. As was noted in [1], there is some subtlety for continuous spin models where this constant becomes infinite (proportional to \(\delta (0)\)). These cases then appear to require an appropriate regularisation in order to properly define the deformed partition functions. In any case, the observables of the deformed model will not be affected, since these are typically expressed in terms of derivatives of the partition function.

  3. Since these models are not formed from edge Boltzmann weights, these would simply be elementary squares, rather than four-squares.

  4. As a slight abuse of notation, both spin variables of the previous section, and classical variables that arise from the quasi-classical limit in this section, are referred to with the notation \({x}_i\).

  5. Note that the scalar case of (42) (or (51)), corresponds to \(n=2\), while the scalar case of (61) corresponds to \(n=1\). This is because the extra condition on the variables (44) (or (52)), has already been used in taking the limit to (55).

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Acknowledgements

The majority of this work was completed while the author was an overseas researcher under Postdoctoral Fellowship of Japan Society for the Promotion of Science (JSPS), at the University of Tokyo, Komaba. The author thanks Atsuo Kuniba and Masahito Yamazaki for helpful discussions. Some of the n-component equations of Sect. 3.4 were presented at the 13th Symmetries and Integrability of Difference Equations (SIDE) conference, in Fukuoka, Japan, on November, 2018. The author thanks the organisers for the opportunity to give a talk at this conference, and also thanks the audience for their interest and feedback.

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    Andrew P. Kels

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Appendix: Deformations for Vertex and IRF Formulations

Appendix: Deformations for Vertex and IRF Formulations

The cubic deformations in Figs. 20, 21, 22, 23, 24, and 25, are for the vertex and IRF models respectively, as defined in Sect. 2. These deformations are derived with use of the respective Yang–Baxter equations (12), and (8), and inversion relations (13), (9), and are used to show the extended Z-invariance property described in Sect. 2.3 for the vertex and IRF models.

Fig. 22
figure 22

A deformation in the vertex formulation that is equivalent to the Yang–Baxter equation (12)

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Fig. 23
figure 23

A deformation in the IRF formulation corresponding to (A.2). Note that the rapidity lines which are not shown for two hidden four-squares on the right hand side are assigned according to Fig. 12

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Fig. 24
figure 24

A deformation in the IRF formulation. Note that the rapidity lines which are not shown for the hidden four-square on the right hand side are assigned according to Fig. 12

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Fig. 25
figure 25

A deformation in the IRF formulation that is equivalent to the Yang–Baxter equation (8)

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For the case of the vertex formulation, the most complicated case is when changing one four-square into five four-squares, where two positively oriented rapidity lines r and \(r'\) are added/removed, as is shown in Fig. 20. Using the Yang–Baxter equation (12) and inversion relation (13), the contribution to the partition function of the right hand side of Fig. 20 is given by

$$\begin{aligned}&\displaystyle \sum _{x''_i,x'''_i,x''_j,x'''_j,x_k,x'_k,x''_k,x'''_k} \left\langle x''_i,x''_j\,|\,\mathbb {R}_{{\mathbf {{\mathsf {p}}}}{\mathbf {{\mathsf {q}}}}}\,|\,x'''_i,x'''_j \right\rangle \,\left\langle x_k,x_j\,|\,\mathbb {R}_{{\mathbf {{\mathsf {r}}}}{\mathbf {{\mathsf {q}}}}} \,|\,x'_k,x'_j\right\rangle \,\left\langle x'''_i,x'_k \,|\,\mathbb {R}_{{\mathbf {{\mathsf {p}}}}{\mathbf {{\mathsf {r}}}}}\,|\,x'_i,x''_k\right\rangle \nonumber \\&\qquad \displaystyle \times \left\langle x'''_j,x''_k \,|\,\mathbb {R}_{{\mathbf {{\mathsf {q}}}}{\mathbf {{\mathsf {r}}}}}\,|\,x'_j,x'''_k\right\rangle \, \left\langle x'''_k,x_i\,|\,\mathbb {R}_{{\mathbf {{\mathsf {r}}}}{\mathbf {{\mathsf {p}}}}} \,|\,x_k,x''_i\right\rangle \nonumber \\&\quad \displaystyle =\sum _{x''_i,x'''_j,x_k,x''_k,x'''_k} \;\sum _{\hat{x}_i,\hat{x}'_j,\hat{x}'''_k}\left\langle x'''_j,x''_k \,|\,\mathbb {R}_{{\mathbf {{\mathsf {q}}}}{\mathbf {{\mathsf {r}}}}}\,|\,x'_j,x'''_k\right\rangle \,\left\langle x'''_k,x_i\,|\,\mathbb {R}_{{\mathbf {{\mathsf {r}}}}{\mathbf {{\mathsf {p}}}}} \,|\,x_k,x''_i\right\rangle \nonumber \\&\qquad \displaystyle \times \left\langle \hat{x}'''_k,\hat{x}'_j \,|\,\mathbb {R}_{{\mathbf {{\mathsf {r}}}}{\mathbf {{\mathsf {q}}}}}\,|\,x''_k,x'''_j\right\rangle \,\left\langle x''_i,x_k\,|\,\mathbb {R}_{{\mathbf {{\mathsf {p}}}}{\mathbf {{\mathsf {r}}}}}\,|\,\hat{x}_i, \hat{x}'''_k\right\rangle \,\left\langle \hat{x}_i,x_j \,|\,\mathbb {R}_{{\mathbf {{\mathsf {p}}}}{\mathbf {{\mathsf {q}}}}}\,|\,x'_i,\hat{x}'_j\right\rangle \nonumber \\&\quad \displaystyle =\left\langle x_i,x_j\,|\,\mathbb {R}_{{\mathbf {{\mathsf {p}}}}{\mathbf {{\mathsf {q}}}}} \,|\,x'_i,x'_j\right\rangle \sum _{x'''_k}\delta _{x'''_k,x'''_k}. \end{aligned}$$
(A.1)

This shows that the contributions to the partition function of the left and right hand sides of Fig. 20 are equal, up to the constant factor \(\sum _{x'''_k}\delta _{x'''_k,x'''_k}\) (this is left here as a \(\delta \)-function, because when considering models with continuous valued spins, this becomes an infinite constant). A similar type of deformation not pictured here, that instead adds two negatively oriented rapidity lines r and \(r'\), may be shown to hold with an analogous calculation to (A.1). Similar calculations involving the Yang–Baxter equation (12) and inversion relation (13), can be used to show the equalities of Figs. 21 and 22, with the latter Figure only requiring a simple use of (12).

For the case of the IRF formulation, the most complicated case again involves changing one four-square into five four-squares, where the deformation adds two positively oriented rapidity lines r and \(r'\), as is depicted in Fig. 23. Using the Yang–Baxter equation (8) and inversion relation (9), the contribution to the partition function of the right hand side of Fig. 23 is given by

$$\begin{aligned}&\displaystyle \sum _{x'_a,x'_b,x'_c,x'_d}\mathbb {V}^{(1)}_{{\mathbf {{\mathsf {p}}}}{\mathbf {{\mathsf {q}}}}} (x'_a,x'_b,x'_c,x'_d)\,\mathbb {V}^{(1)}_{{\mathbf {{\mathsf {r}}}}{\mathbf {{\mathsf {q}}}}}(x'_c,x'_d,x_c,x_d) \,\mathbb {V}^{(1)}_{{\mathbf {{\mathsf {p}}}}{\mathbf {{\mathsf {r}}}}}(x'_b,x_b,x'_d,x_d)\,\mathbb {V}^{(2)}_{{\mathbf {{\mathsf {q}}}}{\mathbf {{\mathsf {r}}}}} (x'_a,x_a,x'_b,x_b)\nonumber \\&\qquad \displaystyle \times \mathbb {V}^{(2)}_{{\mathbf {{\mathsf {r}}}}{\mathbf {{\mathsf {p}}}}}(x'_a,x'_c,x_a,x_c) \, W_{qq'}(x_a,x'_a)\, W_{q'q}(x'_b,x_b)\, W_{q'q} (x'_c,x_c)\, W_{qq'} (x_d,x'_d)\nonumber \\&\quad \displaystyle =\sum _{x'_a,x'_b,x'_c}\sum _{\hat{x}_a} \mathbb {V}^{(2)}_{{\mathbf {{\mathsf {q}}}}{\mathbf {{\mathsf {r}}}}}(x'_a,x_a,x'_b,x_b)\,\mathbb {V}^{(2)}_{{\mathbf {{\mathsf {r}}}}{\mathbf {{\mathsf {p}}}}} (x'_a,x'_c,x_a,x_c)\, W_{qq'}(x_a,x'_a)\, W_{q'q}(x'_a,\hat{x}_a)\nonumber \\&\qquad \displaystyle \times \mathbb {V}^{(1)}_{{\mathbf {{\mathsf {r}}}}{\mathbf {{\mathsf {q}}}}}(x'_a,x'_b,\hat{x}_a,x_b) \,\mathbb {V}^{(1)}_{{\mathbf {{\mathsf {p}}}}{\mathbf {{\mathsf {r}}}}}(x'_a,\hat{x}_a,x'_c,x_c)\,\mathbb {V}^{(1)}_{{\mathbf {{\mathsf {p}}}}{\mathbf {{\mathsf {q}}}}} (\hat{x}_a,x_b,x_c,x_d)\nonumber \\&\quad \displaystyle =\mathbb {V}^{(1)}_{{\mathbf {{\mathsf {p}}}}{\mathbf {{\mathsf {q}}}}}(x_a,x_b,x_c,x_d) \sum _{x_a}\delta _{x_a,x_a}. \end{aligned}$$
(A.2)

This shows that the contributions to the partition function of the left and right hand sides of Fig. 23 are equal, up to the constant factor \(\left( \sum _{x_a}\delta _{x_a,x_a}\right) \). A similar type of deformation not pictured here, that instead adds two negatively oriented rapidity lines r and \(r'\), may be shown with an analogous calculation to (A.2). Similar calculations involving the Yang–Baxter equation (8) and inversion relation (9), can be used to show the equalities of Figs. 24 and 25, with the latter Figure only requiring a simple use of (8).

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Kels, A.P. Extended Z-Invariance for Integrable Vector and Face Models and Multi-component Integrable Quad Equations. J Stat Phys 176, 1375–1408 (2019). https://doi.org/10.1007/s10955-019-02346-9

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