Transfer matrix renormalization

The finite-size scaling theory combined with transfer matrix calculations had been, since the development of the phenomenological renormalization in 1976 by Nightingale [70], one of the most powerful tools to study critical phenomena in two-dimensional lattice models. For these models the partition function and all the physical quantities of the system (free energy, correlation length, response functions, etc) can be written as a function of the eigenvalues of the transfer matrix [71]. In particular, the free energy takes the form... 

A good model for this phenomenon is a SAW on some lattice with an absorbing surface (boundary). Every site on surface visited by the polymer contributes an energy —E,. This model has widely been studied on various lattices and via a number of techniques that include exact enumeration [48,49], Monte Carlo [50], transfer matrix [51], renormalization group [52] etc. For a 2-d Euclidean lattice, exact value of found from conformal field theory is 1/2 [53]. 

Here n = (n, a) labels the molecules in the crystal, where n is the lattice vector and a enumerates the molecules in a unit cell and / are the creation and annihilation operators of an exciton on molecule n, obeying Pauli commutation relations. Eq is the renormalized excitation energy in the monomer (see Section 3.2), and Mnm is the matrix element of the excitation energy transfer from the molecule m to the molecule n. 

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