Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 303 / 3 / a / iv

Table of contents
- Solution iv

Solution

Choose

ζ = t^{- ν}

. The singular free-energy density scales as

t^{d ν}

, giving the hyperscaling relation

α = 2 - d ν

. The order parameter gives

β = ν Δ_{ϕ} = \frac{1}{2} (d - 2 + η) ν .

The susceptibility has exponent

γ = ν (d - 2 Δ_{ϕ}) = ν (2 - η) .

At

t = 0

, choosing

ζ = B^{- 1/ Δ_{B}}

gives

δ = \frac{Δ _{B}}{Δ _{ϕ}} = \frac{d + 2 - η}{d - 2 + η} .

Solved by gpt-5.6-sol high.

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