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
$$T\left(u\right):A\otimes {V}^{\otimes N}\to A\otimes {V}^{\otimes N}\phantom{\rule{2em}{0ex}}\text{given by}\phantom{\rule{2em}{0ex}}T\left(u\right)={L}_{01}\left(u\right){L}_{02}\left(u\right)\cdots {L}_{0N}\left(u\right).$$

The transfer matrix is
$$\mathcal{T}\left(u\right):{V}^{\otimes N}\to {V}^{\otimes N}\phantom{\rule{2em}{0ex}}\text{given by}\phantom{\rule{2em}{0ex}}\mathcal{T}\left(u\right)={\mathrm{Tr}}_{A}T\left(u\right).$$

The Hamiltonian is
$$H=\left(\mathrm{const}\right){\frac{d}{du}log\mathcal{T}\left(u\right)|}_{u=\mathrm{const}}+\left(\mathrm{const}\right).$$

The partition function on an
$N\times M$ lattice is
$$Z\left(u\right)={\mathrm{Tr}}_{{V}^{\otimes N}}\mathcal{T}\left(u\right).$$

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
$A\otimes A\otimes {V}^{\otimes N}$,

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

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.