Transfer matrices

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 6 December 2011

Lattice models and Transfer matrices

Let $A$ be a vector space with basis $\left\{f_{\mu }\right\}$ and let $V$ be a vector space with basis $\left\{e_{\alpha }\right\}$.

• The Boltzmann weights are the matrix entries
  
  of an operator $L\left(u\right):A\otimes V\to A\otimes V$.
• The monodromy matrix is $$T\left(u\right):A\otimes V^{\otimes N}\to A\otimes V^{\otimes N}\text{given by}T\left(u\right)=L_{01}\left(u\right)L_{02}\left(u\right)\cdots L_{0N}\left(u\right).$$
• The transfer matrix is $$𝒯\left(u\right):V^{\otimes N}\to V^{\otimes N}\text{given by}𝒯\left(u\right)={\mathrm{Tr}}_{A}T\left(u\right).$$
• The Hamiltonian is $$H=\left(\mathrm{const}\right){\frac{d}{du}log𝒯\left(u\right)|}_{u=\mathrm{const}}+\left(\mathrm{const}\right).$$
• The partition function on an $N\times M$ lattice is $$Z\left(u\right)={\mathrm{Tr}}_{V^{\otimes N}}𝒯\left(u\right).$$

As operators on $A\otimes A\otimes V$, $$R_{12}(u,v)L_{13}\left(v\right)L_{23}\left(u\right)=L_{23}\left(u\right)L_{13}\left(v\right)R_{12}(u,v)$$ and this implies that, as operators on $A\otimes A\otimes V^{\otimes N}$,

$$R_{12}(u,v)T_{13}\left(v\right)T_{23}\left(u\right)=T_{23}\left(u\right)T_{13}\left(v\right)R_{12}(u,v)$$

(1.1)

Multiplying on the left by ${R_{12}(u,v)}^{-1}$ and taking ${\mathrm{Tr}}_{A\otimes A}$ gives $$𝒯\left(u\right)𝒯\left(v\right)=𝒯\left(v\right)𝒯\left(u\right).$$

Remark: In many applications $$R(u,v)=R(u-v)$$ and the quantum Yang-Baxter equation (QYBE) gives $$R_{12}(u-v)R_{13}\left(v\right)R_{23}\left(u\right)=R_{23}\left(u\right)R_{13}\left(v\right)R_{12}(u-v),$$ with $L\left(u\right)=R\left(u\right)$ gives (1.1).

Notes and References

This page is based on section 5.2 of [dG]. The formula ??? appears as ??? in [TF] who quote [Bax].

References

[Bax] R. Baxter, One-dimensional anisotropic Heisenberg chain, Ann. Phys. 70 (1972), 323-337.

[dG] J. de Gier, Random Tilings and Solvable Lattice Models, Ph.D Thesis, University of Amsterdam, 1998.

[TF] L.A. Takhtajan and L.D. Faddeev, The quantum method of the inverse problem and the Heisenberg XYZ model, Russian Math Surveys 34 5 (1979), 11-68.

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