## Transfer matrices

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 $T(u): A⊗V⊗N → A⊗V⊗N given by T(u) =L01(u) L02(u) ⋯ L0N(u) .$
• The transfer matrix is $𝒯(u) :V⊗N →V⊗N given by 𝒯(u) =TrAT(u) .$
• The Hamiltonian is $H=(const) ddu log𝒯(u) | u=const + (const) .$
• The partition function on an $N×M$ lattice is $Z(u) = Tr V⊗N 𝒯(u) .$

As operators on $A\otimes A\otimes V$, $R12(u,v) L13(v) L23(u) = L23(u) L13(v) R12(u,v)$ and this implies that, as operators on $A\otimes A\otimes {V}^{\otimes N}$,

 $R12(u,v) T13(v) T23(u) = T23(u) T13(v) R12(u,v)$ (1.1)
Multiplying on the left by ${{R}_{12}\left(u,v\right)}^{-1}$ and taking ${\mathrm{Tr}}_{A\otimes A}$ gives $𝒯(u) 𝒯(v) = 𝒯(v) 𝒯(u).$

Remark: In many applications $R(u,v) = R(u-v)$ and the quantum Yang-Baxter equation (QYBE) gives $R12(u-v) R13(v) R23(u) = R23(u) R13(v) R12(u-v) ,$ 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.