Quantum Transfer-Matrix Method and Its Application to Quantum Spin Systems

Abstract

It is well known that the transfer-matrix method is very useful in the statistical mechanics of classical systems [1,2]. For example, the partition function of the three-dimensional (3D) Ising model is formulated in terms of the transfer matrix. If the size of the lattice is N=l×m×n, the partition function can be written as

$$\text{Z=Tr }{{T}^{n}}$$

(1.1)

where the transfer matrix T acts on the spin states of the 2D layer of l×m sites sliced from the 3D lattice. The transfer matrix can be pictured as an operator which governs the evolution of spin states from layer to layer. Since each layer has 2^lm spin states, the size of the matrix T is 2^lm×2^lm. In the limit n →∞ the partition function is dominated by the maximum eigenvalue Λ of T:

$$\text{Z}\sim {{\Lambda }^{n}}$$

(1.2)