Numerical Renormalization Group Calculations for Impurity ...

2.1. The scaling limit and universality 5
where the correlation length, ξ, is given by
1
ξ = 1 lncoth K. (2.9)
a
The notion of the correlation length ξ given above helps us write a universal critical
theory of the Ising chain H I in the scaling limit, where the detail informations
of the finite-size system (M, K and a) are absorbed into the macroscopic lengths
ξ and L τ with replacements of M = L τ /a and K = ln coth −1 (a/ξ) and, finally,
take the limit a → 0 at fixed τ, L τ and ξ.
We first describe the results for the free energy. The quantity with the finite
scaling limit should clearly be the free energy density, F:
F
= − ln Z/Ma
= E 0 − 1 L τ
ln
[
2 cosh L ]
τ
, (2.10)
2ξ
where E 0 = −K/a is the ground state energy per unit length of the chain.
In a similar manner, we can take the scaling limit of the correlation function in
Eq. (2.5). We obtain
〈σ z (τ)σ z (0)〉 = e−|τ|/ξ −(Lτ −|τ|)/ξ
+ e
. (2.11)
1 + e −Lτ/ξ
The assertion of universality is that the results of the scaling limit are not sensitive
to the microscopic details. This can be seen as the formal consequence
of the physically reasonable requirement that correlations at the scale of large ξ
should not depend upon the details of the interactions on the scale of the lattice
spacing,a.
We can make the assertion more precise by introducing the concept of a universal
scaling function. We write Eq. (2.10) in the form
F = E 0 + 1 L τ
Φ F ( L τ
ξ
), (2.12)
where Φ F is the universal scaling function, whose explicit value can be easily
deduced by comparing with Eq. (2.10). Notice that the argument of Φ F is simply
the dimensionless ratios that can be made out of the large(macroscopic) lengths
at our disposal: L τ and ξ. The prefactor, 1/L τ , in front of Φ F is necessary
because the free energy density has dimensions of inverse length.
In a similar manner, we can introduce a universal scaling function of the two-point
correlation function. We have
〈σ z (τ)σ z (0)〉 = Φ σ ( τ L τ
, L τ
ξ
), (2.13)
where Φ σ is again a function of all the independent dimensionless combinations of
large lengths and the exact form of Φ σ is obtained by comparison with Eq. (2.11).