In statistical mechanics, the transfer-matrix method is a mathematical technique which is used to write the partition function into a simpler form.
[1][2] In many one dimensional lattice models, the partition function is first written as an n-fold summation over each possible microstate, and also contains an additional summation of each component's contribution to the energy of the system within each microstate.
For systems with more than a few particles, such expressions can quickly become too complex to work out directly, even by computer.
In some cases, particularly for systems with periodic boundary conditions, the partition function may be written more simply as where "tr" denotes the matrix trace.
The dimension p of the p × p transfer matrix equals the number of states the subsystem may have; the transfer matrix itself Wk encodes the statistical weight associated with a particular state of subsystem k − 1 being next to another state of subsystem k. Importantly, transfer matrix methods allow to tackle probabilistic lattice models from an algebraic perspective, allowing for instance the use of results from representation theory.
Transfer-matrix methods have been critical for many exact solutions of problems in statistical mechanics, including the Zimm–Bragg and Lifson–Roig models of the helix-coil transition, transfer matrix models for protein-DNA binding, as well as the famous exact solution of the two-dimensional Ising model by Lars Onsager.