342 Transfer matrix and diagonalisation of spin chains
We see that the correlation drops off exponentially, unlessλ 0 =λ 1 , that is, unless
the largest eigenvalue is degenerate. This implies that critical behaviour can only
occur when the largest eigenvalue is degenerate – recall that critical behaviour is
characterised by power-law decay of correlation functions.
In this regard Frobenius’ theorem is important. This states that the largest eigen-
value of a positive matrix (i.e. a matrix with all positive elements) of finite size is
nondegenerate [2]; that is, our correlation functions will never show critical beha-
viour. The only way to obtain critical behaviour is by having a matrix of infinite
size, which is only possible when the degrees of freedom on a site can assume
an infinite number of different values. For a one-dimensional model on a lattice
(chain) in which the degrees of freedom on a lattice site can assume a finite number
of different values, the correlation function is always exponential.
In Chapter 12 we shall see that the analysis applied here carries over to quantum
mechanics, the transfer matrix corresponding to the quantum mechanical time evol-
ution operator exp(−itH/), whereHis now the quantum Hamiltonian. If this
Hamiltonian has eigenvaluesEα,wehaveλα = exp(−itEα/). After an ana-
lytic continuation into imaginary timet →−it, the eigenvalues become real:
λα=exp(−tEα/)and we see that degeneracy of the largest eigenvalue of the
transfer matrix (or time-evolution operator) implies degeneracy of the quantum
ground state energy. As an example, consider the one-dimensional Ising model in
zero magnetic field. It is easy to see that for this model the quantum Hamiltonian
occurring in the exponent of the time-evolution operator can be written as
H=−J ̃σx, (11.17)where
σx=(
01
10
)
(11.18)
is the Pauli matrix, andJ ̃is related toJby
tanh(βJ ̃)=exp(− 2 βJ) (11.19)(see Problem 11.1).
In quantum field theory (seeChapter 15), the ground state is the vacuum (no
particles present) and the first excited state is the state with one particle at rest.
As the relativistic energy is given byE=
√
p^2 c^2 +m^2 c^4 , the gap between these
two states is given by the particle rest energymc^2. Degeneracy of the ground state
implies therefore that the particle mass is zero. A critical point is therefore often
identified with a vanishing mass.