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The renormalization group organizes how a statistical system changes when observed at progressively larger length scales. Start with its partition function and a microscopic cutoff

Λ

. Split the Fourier transform field into modes below

Λ/ b

and modes in the shell

Λ/ b < ∣ q ∣ < Λ

. Integrate the shell modes, rescale coordinates

x^{'} = x / b

to restore the cutoff, and rescale the field to keep a chosen gradient normalization. This is a renormalization-group transformation. It preserves the partition function when the generated interactions and additive free energy terms are retained; truncating the resulting operator expansion is an approximation.

A renormalization-group fixed point has an invariant dimensionless interaction structure. The continuous flow of a nearby scaling field

v_{i}

takes the linearized form

\frac{d v _{i}}{d ℓ} = y_{i} v_{i} + O (v^{2}), ℓ = lo g b .

A positive

y_{i}

defines a relevant operator, a negative

y_{i}

an irrelevant operator, and a zero

y_{i}

a marginal operator requiring nonlinear analysis. The continuous-flow eigenvalues

y_{i}

should not be confused with the discrete multipliers

b^{y_{i}}

. To reach a critical point, tune the relevant thermal variable and set the relevant symmetry-breaking source to zero. The resulting flow stays on the critical surface.

Microscopically different systems whose remaining couplings approach the same fixed point share a universality class. Irrelevant perturbations lose their influence on long-distance exponents. Relevant directions explain why varying the temperature or field moves the system away from criticality. The thermal scaling field obeys

t^{'} = b^{y_{t}} t

, while the rescaled correlation length is

ξ^{'} = ξ / b

. Taking

b = ∣ t ∣^{- 1/ y_{t}}

therefore gives

ν = 1/ y_{t}

If the field has scaling dimension

x_{ϕ} = (d - 2 + η) /2

, the source term

\int h ϕ

has eigenvalue

y_{h} = d - x_{ϕ} = (d + 2 - η) /2

. The singular free-energy density then obeys

f_{s} (t, h) = b^{- d} f_{s} (b^{y_{t}} t, b^{y_{h}} h), 2 - α = d / y_{t}, Δ = y_{h} / y_{t},

when the hyperscaling relation is valid. This derives the scaling hypothesis for critical phenomena and the exponent relations of Question 2 from the fixed-point picture.

At the Gaussian fixed point, keeping

\int (\nabla ϕ)^{2}

invariant gives

x_{ϕ} = (d - 2) /2

. The quadratic coupling has

y_{r} = 2

and the quartic coupling has

y_{u} = 4 - d

. Thus quartic interactions are relevant below four dimensions, marginal at four, and irrelevant above four. Below four, in dimensions where an ordinary continuous short-range scalar transition exists, an interacting Wilson-Fisher fixed point governs that transition. Above four, the Gaussian fixed point gives mean-field powers, but the quartic term remains necessary to stabilize the ordered state: it is a dangerously irrelevant coupling, explaining the failure of naive hyperscaling. The renormalization group connects scale invariance, universality, relevant tuning parameters and fluctuation corrections in one framework.

Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 64 / 3 / Solution Created 2026-10-03 Updated 2026-10-14

The correlation length is the distance over which fluctuations remain appreciably correlated. More precisely, in a massive phase the large-distance connected correlation function

G_{c} (x) = ⟨ ϕ (x) ϕ (0)⟩ - ⟨ ϕ ⟩^{2}

has an exponential factor

e^{- ∣ x ∣/ ξ}

, possibly multiplied by a power. Equivalently,

ξ^{- 1}

is the lowest inverse length in the long-wavelength fluctuation spectrum. In the Gaussian field theory with unit gradient coefficient,

G_{c} (q) \propto (q^{2} + m_{R}^{2})^{- 1}

gives

ξ = 1/ m_{R}

for positive renormalized mass.

At an ordinary continuous phase transition, the restoring force for long-wavelength order-parameter fluctuations vanishes and there is no finite correlation scale. The critical connected correlation function decays as a power rather than exponentially, so

ξ

diverges. The correlation-function susceptibility sum rule relates magnetic susceptibility to the integral of this connected correlation function; its divergence is consistent with correlations extending over increasingly large distances. Interactions change the powers, so the renormalized mass and the bare mass need not vanish at the same temperature.

Absorb the thermal normalization into the action and set

r = u^{2} = \frac{m ^{2} ( Λ , T )}{Λ ^{2}}, λ = g (Λ, T) Λ^{- ϵ}, d = 4 - ϵ, K_{d} = \frac{Ω _{d}}{( 2 π ) ^{d}} .

Here

r

, although denoted

u^{2}

in the question, is a signed mass-squared coordinate, not a nonnegative square. This matters because its interacting fixed-point value is negative. Increasing

b = lo g (Λ_{0} /Λ)

eliminates high-momentum modes and moves towards the infrared. The initial values at

b = 0

are

r_{0} = m^{2} (Λ_{0}, T) / Λ_{0}^{2}

and

λ_{0} = g (Λ_{0}, T) Λ_{0}^{- ϵ}

To derive the flow, use the momentum-shell decomposition of a scalar field

ϕ = φ + χ

, where

χ

has momenta

Λ e^{- d b} < ∣ q ∣ < Λ

. Its Gaussian shell covariance is

(q^{2} + m^{2})^{- 1}

. Integrating out

χ

while preserving low-momentum observables gives

S_{eff} [φ] = S_{0} [φ] - lo g ⟨ e^{- V [φ + χ]} ⟩_{>}, V = \frac{g}{4 !} \int d^{d} x (φ + χ)^{4} .

The cumulant expansion of a coarse-grained free energy yields

⟨ V ⟩_{>} - \frac{1}{2} ⟨ V^{2} ⟩_{>, c} + \dots

. By Wick theorem,

⟨ V ⟩_{>} = \frac{g}{24} \int d^{d} x [φ^{4} + 6 φ^{2} G_{>} (0) + 3 G_{>} (0)^{2}] .

The last term is field independent, while the quadratic term gives the one-loop shell mass renormalization in scalar quartic theory

δ m^{2} = \frac{g}{2} I_{1}, I_{1} = \int_{shell} \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{q ^{2} + m ^{2}} .

This tadpole is independent of the external momentum, so no gradient renormalization occurs at this order.

For the quartic correction, the mixed vertex is

(g /4) \int φ^{2} χ^{2}

. Its connected second cumulant uses

⟨ χ^{2} (x) χ^{2} (y) ⟩_{c} = 2 G_{>} (x - y)^{2}

and gives

- \frac{g ^{2}}{16} \int d^{d} x d^{d} y φ^{2} (x) φ^{2} (y) G_{>} (x - y)^{2} .

Extracting the local zero-external-momentum quartic interaction, the integral over

x - y

I_{2} = \int_{shell} d^{d} q (2 π)^{- d} (q^{2} + m^{2})^{- 2}

. Comparing its coefficient with

g φ^{4} /4!

gives the one-loop shell quartic renormalization

δ g = - \frac{3}{2} g^{2} I_{2} .

The negative sign comes from the second cumulant; the numerical factor incorporates the three quartic channels.

For the thin spherical shell, radial integration gives

I_{1} = K_{d} \frac{Λ ^{d - 2}}{1 + r} d b + O (d b^{2}), I_{2} = K_{d} \frac{Λ ^{d - 4}}{( 1 + r ) ^{2}} d b + O (d b^{2}) .

The shell is stable when

1 + r > 0

, including a small negative critical mass. Because

d Λ/ d b = - Λ

, differentiating the dimensionless couplings contributes their engineering dimensions

2

and

ϵ

. Together with the shell corrections this yields the renormalization-group flow

\frac{d r}{d b} = 2 r + \frac{K _{d}}{2} \frac{λ}{1 + r}, \frac{d λ}{d b} = ϵ λ - \frac{3 K _{d}}{2} \frac{λ ^{2}}{( 1 + r ) ^{2}} .

These are the requested equations with

r = u^{2}

. They express independence of the long-distance theory from the arbitrarily chosen ultraviolet cutoff, to the retained perturbative order; higher interactions and gradient corrections enter at higher orders.

Besides the Gaussian fixed point

(r, λ) = (0, 0)

, a nonzero fixed point satisfies

λ_{*} = \frac{2 ϵ}{3 K _{d}} (1 + r_{*})^{2}, 0 = 2 r_{*} + \frac{ϵ}{3} (1 + r_{*}) .

Thus, within these truncated equations,

r_{*} = - \frac{ϵ}{6 + ϵ}, λ_{*} = \frac{2 ϵ}{3 K _{d}} (\frac{6}{6 + ϵ})^{2} .

Since

Ω_{4} = 2 π^{2}

and

K_{4} = 1/ (8 π^{2})

, their leading epsilon expansion is

r_{*} = - \frac{ϵ}{6} + O (ϵ^{2}), λ_{*} = \frac{16 π ^{2} ϵ}{3} + O (ϵ^{2}) .

The rational expressions from the truncated flow are not a calculation of the genuine higher-order coefficients.

To assess stability, linearize in

δr, δ λ

about this Wilson-Fisher fixed point. The stability matrix of a renormalization-group fixed point is

M = 2 - \frac{K _{d} λ _{*}}{2 ( 1 + r _{*} ) ^{2}} \frac{3 K _{d} λ _{*}^{2}}{( 1 + r _{*} ) ^{3}} \frac{K _{d}}{2 ( 1 + r _{*} )} ϵ - \frac{3 K _{d} λ _{*}}{( 1 + r _{*} ) ^{2}} .

Using the fixed-point equation, its diagonal entries are

2 - ϵ /3

and

- ϵ

, respectively. Its upper off-diagonal entry is

1/ (16 π^{2}) + O (ϵ)

and its lower off-diagonal entry is

O (ϵ^{2})

. Hence its eigenvalues are

y_{t} = 2 - \frac{ϵ}{3} + O (ϵ^{2}), y_{irr} = - ϵ + O (ϵ^{2}) .

(14)

The interaction perturbation along the critical surface decays towards the infrared for small positive

ϵ

, whereas the thermal perturbation grows. Therefore infrared attraction requires tuning to the critical surface. The source's unqualified wording cannot mean attraction in both coupling directions: the positive thermal eigenvalue is precisely what produces the divergence of the correlation length. This is the thermal relevant direction at the Wilson-Fisher fixed point.

Let

τ

be the thermal scaling coordinate, a linear combination of

δr

and

δ λ

corresponding to the relevant left eigenvector of

M

. To leading order,

τ = δr + [1/ (32 π^{2}) + O (ϵ)] δ λ

is possible. Its initial value is proportional to

T - T_{c}

for a generic thermal path crossing the critical surface; the interaction also shifts the nonuniversal value of

T_{c}

. In the linear regime,

τ (b) ≃ e^{y_{t} b} τ (0) .

(15)

Choose

b_{*}

so that

∣ τ (b_{*}) ∣

is of order one. At this scale the dimensionless correlation length is finite. Restoring the eliminated length scale gives

ξ ≍ Λ_{0}^{- 1} e^{b_{*}} ≍ ∣ τ (0) ∣^{- 1/ y_{t}} ≍ ∣ T - T_{c} ∣^{- ν} .

(16)

Consequently the correlation-length critical exponent is

ν = \frac{1}{2 - ϵ /3 + O ( ϵ ^{2} )} = \frac{1}{2} + \frac{ϵ}{12} + O (ϵ^{2}) .

(17)

There is no field anomalous dimension at one loop in this scalar quartic theory, since the one-loop self-energy is momentum independent; this is consistent with

η = O (ϵ^{2})

Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 46 / 2 / Solution Created 2026-10-03 Updated 2026-10-14

Write

β_{T} = (k_{B} T)^{- 1}

to distinguish inverse temperature from an order-parameter critical exponent. For the Ising model, define the full two-point correlation function

G (r) = ⟨ σ_{0} σ_{r} ⟩

and its connected correlation function

G_{c} (r) = G (r) - M^{2}

in a translation-invariant phase. Away from a thermodynamic critical point, the large-distance envelope decays exponentially, possibly multiplied by a power of distance:

ξ^{- 1} = - lim_{∣ r ∣ \to \infty} \frac{l o g ∣ G _{c} ( r ) ∣}{∣ r ∣} .

This defines the correlation length in physical distance units when the limit exists. At a continuous phase transition,

ξ \to \infty

and the critical connected correlation function typically has algebraic decay

G_{c} (r) \sim ∣ r ∣^{- (D - 2 + η)}

, where

η

is the anomalous dimension. Below the transition the full

G (r)

tends to

M^{2}

, so the decay definition must use

G_{c}

Let

S = \sum_{r} σ_{r}

and

F = - β_{T}^{- 1} lo g Z

be the extensive Helmholtz free energy. Holding the field-independent constant

C

and other physical couplings fixed,

M = - \frac{1}{N} \frac{\partial F}{\partial h}, χ = - \frac{1}{N} \frac{\partial ^{2} F}{\partial h ^{2}} .

Because the Hamiltonian contains

- h S

\partial_{h} lo g Z = β_{T} ⟨ S ⟩

and

\partial_{h}^{2} lo g Z = β_{T}^{2} (⟨ S^{2} ⟩ - ⟨ S ⟩^{2})

. Translation invariance then proves the correlation-function susceptibility sum rule:

χ = \frac{β _{T}}{N} Var S = \frac{β _{T}}{N} r, r^{'} \sum ⟨ σ_{r} σ_{r^{'}} ⟩_{c} = β_{T} r \sum G_{c} (r) .

A normalized blocking kernel

In the ordered zero-field phase these statements refer to a selected pure thermodynamic phase; averaging equally over opposite magnetizations introduces the separate spin-mixture contribution to zero-field susceptibility.

K_{b} (σ^{'}, σ) \geq 0

satisfies

\sum_{σ^{'}} K_{b} (σ^{'}, σ) = 1

. For example, assign one block spin to each block of

b^{D}

microscopic spins using majority sign, with equal probabilities for a tie. Define the blocked Hamiltonian, retaining the entire generated operator space, by

\sum_{σ} K_{b} (σ^{'}, σ) e^{- β_{T} H (u, σ)} = e^{- β_{T} H (R_{b} u, σ^{'}) - β_{T} N g_{E} (u)} .

The field-independent contribution

N g_{E}

belongs to the identity operator. With

N^{'} = N b^{- D}

and

u^{'} = R_{b} u, C^{'} = b^{D} [C + g_{E} (u)],

summing over

σ^{'}

proves $Z (u, C, N) = Z (u^{'}, C^{'}, N^{'})$ exactly. Iterating gives

N_{p} = N b^{- p D}

and the same partition function with

(u_{p}, C_{p}, N_{p})

. Observables at large scales are reproduced by appropriately blocked observables and sources. Truncating the generated interactions makes this an approximation; dropping the additive constant already destroys the equality of free energies even when normalized spin expectations remain correct.

The extensive Helmholtz free energy is unchanged:

F (u_{0}, C_{0}, N) = F (u_{p}, C_{p}, N_{p})

. In contrast, its value per site changes because the number of sites changes. Define the reduced zero-constant free-energy density

f_{0} (u) = β_{T} F (u, 0, N) / N

in the thermodynamic limit. The preceding equality yields

f_{0} (u) = b^{- D} f_{0} (R_{b} u) + g_{0} (u), g_{0} (u) = β_{T} g_{E} (u),

and iteration proves the additive free-energy recursion under blocking:

f_{0} (u_{0}) = b^{- p D} f_{0} (u_{p}) + j = 0 \sum p - 1 b^{- j D} g_{0} (u_{j}) .

Equivalently

C_{p} = b^{p D} [C_{0} + \sum_{j = 0}^{p - 1} b^{- j D} g_{E} (u_{j})]

. Thus the inhomogeneous source records the identity-operator contribution to renormalization-group free energy from eliminated short-distance degrees of freedom.

To express this recursion for the singular part, write

f_{0} = a + f_{s}

, where

a

is the regular analytic background. Then the same displayed recursion holds with

f_{s}

and

g_{s} (u) = g_{0} (u) + b^{- D} a (R_{b} u) - a (u) .

This is the analytic subtraction of an inhomogeneous renormalization recursion. A purely analytic source can often be absorbed into

a

, after which it does not control the leading singular powers. It is not legitimate simply to discard an analytic source in the full free energy, where it can dominate the regular background. Moreover an analytic monomial

c t^{m} h^{n}

contributes

c t^{m} h^{n} \sum_{j = 0}^{p - 1} b^{j (m l_{t} + n l_{h} - D)} .

m l_{t} + n l_{h} = D

, the sum is

p

and hence generates a logarithm when

p \sim - lo g ∣ t ∣/ (l_{t} lo g b)

: a renormalization-group free-energy resonance. Such a term cannot be discarded when describing that logarithmic singularity. Neglect of the inhomogeneous term in the leading power-law scaling therefore assumes that its regular part has been subtracted, that any remaining source only changes finite scaling amplitudes, and that there are no relevant resonances or marginal logarithms. Omitting an irrelevant operator further requires absence of a dangerously irrelevant coupling, whose vanishing limit can make a scaling function singular.

A renormalization-group fixed point obeys

R_{b} u_{*} = u_{*}

. Linearization gives

δ u^{'} = A δ u

; in eigenvector coordinates

w_{i}^{'} = b^{l_{i}} w_{i}

. Positive

l_{i}

label relevant operators, negative

l_{i}

label irrelevant operators, and zero

l_{i}

labels a marginal operator requiring nonlinear analysis. The critical surface is the stable manifold obtained by tuning all relevant scaling fields to zero. A repulsive renormalization-group trajectory leaves the fixed point along a relevant direction as the observation length increases. Locally a typical zero-field section has

d w / d ℓ = - ω w

d t / d ℓ = l_{t} t

ℓ = lo g b

, so the critical surface

t = 0

attracts along

w

, while trajectories with either sign of

t

depart from it. A magnetic scaling field adds another relevant direction.

Figure 1.
Renormalization-group saddle flow in a zero-field section and two relevant directions transverse to the critical surface
. **Details**.

Under the stated homogeneous-scaling assumptions and after fixing irrelevant fields at their limiting values, two relevant scaling fields obey

t^{'} = b^{l_{t}} t

h^{'} = b^{l_{h}} h

. These fields are analytic coordinates near the renormalization-group fixed point, proportional to the physical reduced temperature and conjugate field to leading order. The singular free-energy density consequently obeys

F_{s} (t, h) = b^{- D} F_{s} (b^{l_{t}} t, b^{l_{h}} h) .

Here

F_{s}

denotes the singular free energy per original site or unit volume, with regular dimensional factors absorbed; the extensive singular part is

N F_{s}

. Iterate and choose

b^{p} = ∣ t ∣^{- 1/ l_{t}}

to obtain the scaling hypothesis for critical phenomena:

F_{s} (t, h) = ∣ t ∣^{D / l_{t}} f_{\pm} (h /∣ t ∣^{l_{h} / l_{t}}) .

The sign labels the two sides of the transition:

f_{+}

describes

t > 0

, the disordered side, while

f_{-}

describes

t < 0

, the ordered side. The two functions need not coincide;

f_{-}

has the zero-field one-sided derivative appropriate to spontaneous magnetization. The numbers

l_{t}, l_{h}

are scaling eigenvalues in logarithmic form: the corresponding discrete RG multipliers are

b^{l_{t}}, b^{l_{h}}

The correlation length obeys

ξ (t, h) = b ξ (b^{l_{t}} t, b^{l_{h}} h)

. The same scale choice gives

ν = 1/ l_{t}

. Differentiating

F_{s}

once and twice with respect to the conjugate field, and twice with respect to temperature, gives

β = \frac{D - l _{h}}{l _{t}}, γ = \frac{2 l _{h} - D}{l _{t}}, α = 2 - \frac{D}{l _{t}} .

t = 0

, instead choose the scale

b^{p} = ∣ h ∣^{- 1/ l_{h}}

. Then

F_{s} (0, h) \sim ∣ h ∣^{D / l_{h}}

and

M \sim sgn (h) ∣ h ∣^{D / l_{h} - 1}

, so

δ = l_{h} / (D - l_{h})

. Thus differentiation and scale matching establish

β δ = \frac{l _{h}}{l _{t}} = β + γ, α = 2 - D ν .

The first is the Widom scaling relation and the second the hyperscaling relation. The hyperscaling relation requires precisely the regular homogeneous scaling used here. Above the ordinary upper critical dimension, the quartic coupling is dangerously irrelevant in the ordered phase and this derivation cannot discard it: the mean-field critical exponents

ν = 1/2

α = 0

do not obey

α = 2 - D ν

for

D > 4

. At marginal dimensions logarithmic factors likewise require care beyond the displayed pure powers.

Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 46 / 2 / b / Solution Created 2026-10-03 Updated 2026-10-14

R_{b} (u_{*}) = u_{*}

. A critical fixed point is scale invariant and has divergent correlation length. The critical surface consists of couplings flowing toward it after relevant variables have been tuned; noncritical flows ultimately leave its neighborhood toward finite-correlation-length phases.

Linearize the renormalization-group transformation near the fixed point and choose scaling coordinates. If the thermal and magnetic coordinates transform as

t^{'} = b^{y_{t}} t

and

h^{'} = b^{y_{h}} h

, the singular free-energy density obeys

F_{s} (t, h) = b^{- D} F_{s} (b^{y_{t}} t, b^{y_{h}} h) .

Choose

b = ∣ t ∣^{- 1/ y_{t}}

to stop at an order-one thermal coordinate. This gives the scaling hypothesis for critical phenomena

F_{s} (t, h) = ∣ t ∣^{D / y_{t}} Φ_{\pm} (h /∣ t ∣^{y_{h} / y_{t}}), ξ (t, 0) \sim ∣ t ∣^{- 1/ y_{t}} .

Consequently

ν = 1/ y_{t}

and

Δ = y_{h} / y_{t}

, with

2 - α = D / y_{t}

when hyperscaling applies. The fixed point determines universal powers, while microscopic couplings set metric factors and amplitudes. Dangerously irrelevant variables and marginal logarithms can qualify the simple homogeneous form.

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 51 / 2 / Solution Created 2026-10-03 Updated 2026-10-14

Use

β_{T} = 1/ (k_{B} T)

for inverse temperature, to distinguish it from the order-parameter critical exponent

β

. In a translation-invariant pure thermodynamic phase, the connected two-point correlation function is

G (r) = ⟨ σ_{0} σ_{r} ⟩ - ⟨ σ_{0} ⟩ ⟨ σ_{r} ⟩ = ⟨ σ_{0} σ_{r} ⟩ - M^{2} .

For finite correlation length, at

∣ r ∣ ≫ ξ

it decays exponentially, possibly with an algebraic prefactor;

ξ^{- 1}

is the asymptotic exponential decay rate. Below an ordered transition the pure phase is selected by taking the thermodynamic limit before

h \to 0^{+}

h \to 0^{-}

. A zero-field symmetric mixture of the two ordered phases has a nondecaying connected term and does not describe the usual pure-phase magnetic susceptibility.

Let

F = - k_{B} T lo g Z

denote extensive physical free energy and

S = \sum_{r} σ_{r}

. Since the field contributes

- h S

to the statistical Hamiltonian, differentiation gives

M = - \frac{1}{N} \frac{\partial F}{\partial h} = \frac{⟨ S ⟩}{N}, χ = \frac{\partial M}{\partial h} = - \frac{1}{N} \frac{\partial ^{2} F}{\partial h ^{2}} = \frac{β _{T}}{N} Var (S) .

Expanding the variance and using translation invariance proves the correlation-function susceptibility sum rule:

χ = β_{T} r \sum G (r) .

The sum includes

r = 0

. If the source is instead defined as the dimensionless quantity

β_{T} h

, the corresponding derivative omits the displayed inverse-temperature factor.

A real-space renormalization group step of length factor

b > 1

uses a local blocking kernel

B_{b} (σ^{'}, σ) \geq 0

satisfying

\sum_{σ^{'}} B_{b} (σ^{'}, σ) = 1

. For example, a deterministic majority-spin rule assigns one coarse Ising spin to each block; tied blocks can be resolved with equal probabilities. Define the blocked statistical Hamiltonian by

e^{- β_{T} [H (u^{'}, σ^{'}) + N^{'} C^{'}]} = \sum_{σ} B_{b} (σ^{'}, σ) e^{- β_{T} [H (u, σ) + NC]}, N^{'} = b^{- D} N .

The spacing becomes

a^{'} = ba

. All generated interactions, including multispin terms and the field-independent constant, must be retained for an exact transformation. Summing over

σ^{'}

uses normalization to prove

Z (u^{'}, C^{'}, N^{'}) = Z (u, C, N)

. Iteration gives

N_{p} = b^{- p D} N

and

Z (u_{p}, C_{p}, N_{p}) = Z (u_{0}, C_{0}, N)

The coarse probability law is the exact marginal law of these block variables. Indeed, for any coarse observable

A

, its expected value is the microscopic expected value of

\sum_{σ^{'}} B_{b} (σ^{'}, σ) A (σ^{'})

. A blocking rule that retains the long-wavelength order-parameter mode, with source and observable transformations tracked, therefore retains long-distance response and correlation functions. Their field normalization may change, while their physical length scales and critical singularities do not. Partition-function equality alone would not justify this assertion for an arbitrary kernel that discards all slow variables. Truncating the generated operator family is likewise an approximation.

For the free-energy recursion use dimensionless free energy per site,

F (u, C) = - \frac{1}{N} lo g Z (u, C, N) = f (u) + β_{T} C,

in the thermodynamic limit. Exact partition-function invariance gives

F (u_{0}, C_{0}) = b^{- p D} F (u_{p}, C_{p})

. If one step sends

C

C^{'} = b^{D} C + c (u)

, the coefficient

c (u)

records the identity-operator contribution from eliminated fluctuations and normalization. Consequently

f (u) = b^{- D} f (u^{'}) + g (u), g (u) = β_{T} b^{- D} c (u) = β_{T} (b^{- D} C^{'} - C) .

Repeated substitution derives the additive free-energy recursion under blocking:

f (u_{0}) = b^{- p D} f (u_{p}) + j = 0 \sum p - 1 b^{- j D} g (u_{j}) .

In physical energy units, divide this convention by

β_{T}

. The additive term is essential for absolute free energies even when it does not affect the leading critical powers.

R_{b} (u_{*}) = u_{*}

. Linearize in scaling coordinates

v_{i}

so that

v_{i}^{'} = b^{y_{i}} v_{i} + O (v^{2})

. A relevant operator has

y_{i} > 0

, an irrelevant operator has

y_{i} < 0

, and a marginal operator has

y_{i} = 0

with behaviour decided by nonlinear terms. Thus the eigenvalues of the discrete linear map are

b^{y_{i}}

; the sign criterion applies to the generator exponents

y_{i}

, not to the discrete multipliers. The critical surface is the stable manifold on which every relevant scaling field has been tuned away. Irrelevant coordinates flow to the renormalization-group fixed point along it. A repulsive renormalization-group trajectory leaves the renormalization-group fixed point along an unstable relevant direction, ultimately entering a noncritical phase or another scaling regime.

Figure 1.
Local renormalization-group flow in the zero-field slice: attraction along the critical surface and repulsion in the thermal direction
. **Details**.

To separate the singular free-energy density from its regular background, suppose the eliminated-mode contribution

g

is analytic near the renormalization-group fixed point. Look for an analytic background

a (u)

satisfying

a (u) = b^{- D} a (R_{b} u) + g (u)

. Then

F_{s} = f - a

satisfies

F_{s} (u) = b^{- D} F_{s} (R_{b} u)

. In linear thermal and field coordinates, an analytic monomial

g_{mn} t^{m} h^{n}

has background coefficient

a_{mn} = \frac{g _{mn}}{1 - b ^{m λ_{t} + n λ_{h} - D}},

provided the denominator is nonzero. This explicitly explains when analytic subtraction of an inhomogeneous renormalization recursion works. The printed inhomogeneous equation may describe the free-energy function before this subtraction, or a singular part defined only up to analytic terms. The inhomogeneous sum need not be small compared with the full free energy: an analytic constant can dominate a vanishing singular contribution. It can be omitted in deriving leading singular powers after regular subtraction, provided no renormalization-group free-energy resonance, marginal logarithm or dangerously irrelevant coupling obstructs the homogeneous critical asymptotics. At a renormalization-group free-energy resonance

m λ_{t} + n λ_{h} = D

, additive logarithms can remain, so simply dropping the term would miss them.

Suppose there are exactly two relevant scaling fields, thermal

t

and magnetic

h

, with positive exponents

λ_{t}, λ_{h}

, and the remaining fields approach a regular limit. Physical reduced temperature and magnetic field agree with these coordinates to leading order, up to nonuniversal metric factors. The homogeneous scaling hypothesis for critical phenomena is

F_{s} (t, h) = ℓ^{- D} F_{s} (ℓ^{λ_{t}} t, ℓ^{λ_{h}} h), ℓ > 0,

as an asymptotic scaling equation. Choose

ℓ = ∣ t ∣^{- 1/ λ_{t}}

to make the thermal argument

+ 1

- 1

. Defining

f_{\pm} (z) = F_{s} (\pm 1, z)

gives

F_{s} (t, h) = ∣ t ∣^{D / λ_{t}} f_{\pm} (\frac{h}{∣ t ∣ ^{λ_{h} / λ_{t}}}) .

The two functions describe respectively

t > 0

and

t < 0

: approaching the transition from disordered and ordered sides need not give the same function. In particular the ordered zero-field function has one-sided derivatives associated with spontaneous magnetization. Metric factors multiply this formula in different microscopic normalizations.

The correlation length in Bravais lattice units obeys

ξ (t, h) = ℓ ξ (ℓ^{λ_{t}} t, ℓ^{λ_{h}} h)

. At

h = 0

, the same choice of

ℓ

gives

ν = 1/ λ_{t}

. Two thermal derivatives of

F_{s} (t, 0)

give

2 - α = D / λ_{t}

. One and two field derivatives give, respectively,

β = (D - λ_{h}) / λ_{t}

and

γ = (2 λ_{h} - D) / λ_{t}

, assuming the relevant branch amplitudes are nonzero. At

t = 0

, choose

ℓ = ∣ h ∣^{- 1/ λ_{h}}

; then

M \propto sgn (h) ∣ h ∣^{(D - λ_{h}) / λ_{h}}

, so

δ = λ_{h} / (D - λ_{h})

. Hence

ν = \frac{1}{λ _{t}}, α = 2 - \frac{D}{λ _{t}}, β = \frac{D - λ _{h}}{λ _{t}}, γ = \frac{2 λ _{h} - D}{λ _{t}}, δ = \frac{λ _{h}}{D - λ _{h}} .

Simple algebra now gives the Rushbrooke scaling relation

α + 2 β + γ = 2

and the hyperscaling relation

α = 2 - D ν

. Also

β δ = λ_{h} / λ_{t} = β + γ

, giving the Widom scaling relation. These statements refer to leading singular contributions with the homogeneous two-field scaling assumptions. Above the upper critical dimension, the quartic coupling is dangerously irrelevant because it is still needed to stabilize the ordered phase; ordinary mean-field critical exponents then satisfy Rushbrooke and Widom but generally not hyperscaling. Marginal logarithms also require qualifications at the upper critical dimension.

For the connected-correlation scaling form

G (r) = ∣ r ∣^{- (D - 2 + η)} f_{G} (∣ r ∣/ ξ)

, the function

f_{G} (x)

tends to a finite nonzero value as

x \to 0

and decays exponentially at large

x

, up to powers of

x

, in a massive short-range phase. The scaling ansatz alone does not fix that prefactor. If the usual massive Gaussian field correlation tail

G (r) \propto r^{- (D - 1) /2} e^{- r / ξ}

applies, matching the powers gives

f_{G} (x) \sim B x^{(D - 3) /2 + η} e^{- x}

, with a correlation-length-dependent amplitude in

G

fixed by matching the critical scaling form.

In physical coordinates the lattice sum has site-density factor

a^{- D}

. Its long-distance singular part therefore gives

χ_{s} ≃ β_{T} a^{- D} Ω_{D - 1} \int_{a}^{\infty} r^{1 - η} f_{G} (r / ξ) d r = β_{T} a^{- D} Ω_{D - 1} ξ^{2 - η} \int_{a / ξ}^{\infty} x^{1 - η} f_{G} (x) d x,

where

Ω_{D - 1}

is the area of the unit sphere in

D

dimensions. For

η < 2

, finite

f_{G} (0)

, an integrable exponential tail and a nonzero integral, the last factor tends to a finite nonzero constant as

ξ \to \infty

. Microscopic distances contribute a regular background. Thus

χ_{s} \propto ξ^{2 - η} \propto ∣ t ∣^{- (2 - η) ν}

, proving the Fisher scaling relation

γ = (2 - η) ν .

Equivalently the field has scaling dimension

(D - 2 + η) /2

, so its conjugate field has

λ_{h} = (D + 2 - η) /2

, which gives the same identity from the exponent formulas above.

Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 50 / 2 / Solution Created 2026-10-03 Updated 2026-10-14

Use

β_{th} = 1/ (k_{B} T)

for inverse temperature, to distinguish it from the order-parameter exponent. In a translation-invariant pure thermodynamic phase, the connected correlation function is

G (r) = ⟨ σ_{r} σ_{0} ⟩ - ⟨ σ_{r} ⟩ ⟨ σ_{0} ⟩ .

The subtraction is essential below the transition. Its exponential correlation length is defined by

ξ^{- 1} = - lim_{r \to \infty} r^{- 1} lo g ∣ G (r) ∣

, with direction specified if the lattice is anisotropic. Away from criticality,

G (r)

has a tail proportional to

e^{- r / ξ}

times an algebraic prefactor. In the massive continuum scalar approximation that prefactor is

r^{- (D - 1) /2}

; at criticality the exponential cutoff disappears and

G (r) ≍ r^{- (D - 2 + η)}

instead. The large-distance limit here is

r ≫ ξ

, as the PDF shows.

Let

F_{tot} = - β_{th}^{- 1} lo g Z

. With the energy convention

- h \sum_{r} σ_{r}

, the magnetization per site and magnetic susceptibility are

M = - \frac{1}{N} \frac{\partial F _{tot}}{\partial h}, χ = \frac{\partial M}{\partial h} = - \frac{1}{N} \frac{\partial ^{2} F _{tot}}{\partial h ^{2}} .

Writing

S = \sum_{r} σ_{r}

, direct differentiation of the partition function gives

\partial_{h} ⟨ S ⟩ = β_{th} (⟨ S^{2} ⟩ - ⟨ S ⟩^{2})

. Thus the correlation-function susceptibility sum rule is

χ = \frac{β _{th}}{N} r, r^{'} \sum ⟨ σ_{r} σ_{r^{'}} ⟩_{c} = β_{th} r \sum G (r) .

In physical continuum coordinates the lattice sum becomes

a^{- D} \int d^{D} r G (r)

, so omitting the site-density factor would change the normalization. A derivative with respect to the dimensionless source

β_{th} h

instead absorbs the explicit inverse-temperature factor.

An exact renormalization-group transformation with length factor

b > 1

can be defined by a nonnegative normalized blocking kernel

K_{b} (σ^{'}, σ)

satisfying

\sum_{σ^{'}} K_{b} (σ^{'}, σ) = 1

. A deterministic majority rule is one example; randomized tie-breaking preserves its normalization. Define the blocked energy, including its constant, by

e^{- β_{th} H (u^{'}, σ^{'}) - β_{th} N^{'} C^{'}} = \sum_{σ} K_{b} (σ^{'}, σ) e^{- β_{th} H (u, σ) - β_{th} NC}, N^{'} = N / b^{D} .

Summing over

σ^{'}

proves

Z (u^{'}, C^{'}, N^{'}) = Z (u, C, N)

. All generated operators must be retained for this identity to remain exact; a finite-coupling truncation is an approximation. After

p

iterations,

N_{p} = N / b^{p D}

and the spacing is

a_{p} = b^{p} a

. Long-distance physics is preserved because this is a marginalization of microscopic probabilities, with observables and sources transformed along with the energy. It integrates short-distance variables rather than changing their effects on retained observables. Physical lengths remain fixed even though the correlation length in units of the new spacing is reduced by

b^{p}

For clarity, now use dimensionless free energy per site

F (u, C) = - N^{- 1} lo g Z = f (u) + β_{th} C

. Exact equality of the partition functions gives

F (u_{0}, C_{0}) = b^{- p D} F (u_{p}, C_{p}) .

A constant microscopic energy shifts every blocked weight by the same factor, so

C^{'} = b^{D} C + c (u)

. Substitution at one step yields the additive free-energy recursion under blocking

f (u) = b^{- D} f (R_{b} u) + g (u), g (u) = β_{th} b^{- D} c (u) .

Induction now gives the requested relation

f (u_{0}) = b^{- p D} f (u_{p}) + j = 0 \sum p - 1 b^{- j D} g (u_{j}) .

The function

g

is the identity-operator contribution to renormalization-group free energy: it records eliminated-mode entropy, integration constants and normalization factors. A free-energy function satisfying this inhomogeneous equation need not be purely singular until its analytic background has been subtracted.

At a renormalization-group fixed point,

R_{b} u_{*} = u_{*}

. Linearize in scaling coordinates

v_{i}

v_{i}^{'} = b^{λ_{i}} v_{i}

. Here

λ_{i}

is a scaling exponent, not the discrete multiplier itself. A positive

λ_{i}

gives a relevant operator, a negative one an irrelevant operator, and zero a marginal operator. The critical surface is the stable manifold: tune every relevant scaling field to zero and the irrelevant coordinates flow toward

u_{*}

. A repulsive renormalization-group trajectory leaves along a relevant direction; it can still have irrelevant coordinates contracting toward that manifold. The plot shows the

h = 0

section when temperature and field are relevant:

t = 0

is the critical line in this section, while an independent nonzero field is another direction away from criticality.

Figure 1.
Renormalization-group trajectories near a critical fixed point, showing thermal repulsion and irrelevant contraction in the zero-field section
. **Details**.

The inhomogeneous term cannot simply be discarded from the full free energy. If an analytic background

A (u)

solves

A (u) = b^{- D} A (R_{b} u) + g (u)

, its subtraction leaves

F_{s} = f - A

obeying homogeneous scaling. For example, in linear thermal and field coordinates, a source monomial

g_{mn} t^{m} h^{n}

is removed by

A_{mn} = \frac{g _{mn}}{1 - b ^{m λ_{t} + n λ_{h} - D}} .

When the denominator vanishes, the source is resonant and can instead produce a logarithm. Thus the homogeneous singular equation is justified after analytic subtraction when there is no relevant resonance affecting the leading singularity, no marginal logarithm that has been omitted, and no singular dependence on a dangerously irrelevant coupling. Merely saying that

g

is regular is insufficient to discard the whole weighted sum without this analysis.

Under these assumptions, with two relevant scaling fields

t, h

, choose

p

so that

b^{p λ_{t}} ∣ t ∣

is of order one. The scaling hypothesis for critical phenomena then gives

F_{s} (t, h) = ∣ t ∣^{D / λ_{t}} Φ_{\pm} (\frac{h}{∣ t ∣ ^{λ_{h} / λ_{t}}}) .

Writing

Φ_{\pm} = f_{\pm} + I_{\pm}

recovers the PDF's notation:

F_{\pm} = ∣ t ∣^{D / λ_{t}} [f_{\pm} (h /∣ t ∣^{λ_{h} / λ_{t}}) + I_{\pm}] .

The sign labels the two matching branches

t > 0

and

t < 0

, which enter different phases when the flow leaves the critical neighborhood. The constants

I_{\pm}

retain a branch-dependent identity contribution or a chosen split of the scaling function. They are determined by the same matching problem, not independently adjustable nonuniversal free-energy offsets. An arbitrary analytic offset from

C_{0}

belongs to the subtracted background. The formula describes the singular part; adding that background recovers the full physical free energy.

h = 0

, restore the common metric factors as

F_{s, \pm} = A_{f} ∣ a_{t} t ∣^{D / λ_{t}} Φ_{\pm} (0)

. Their ratio at equal

∣ t ∣

Φ_{+} (0) / Φ_{-} (0)

, proving the expected universal singular free-energy amplitude ratio because

A_{f}

and

a_{t}

cancel. The ratio of full free energies including an arbitrary regular

C_{0}

is not universal; zero amplitudes and logarithmic resonances also require separate treatment.

The critical exponents are obtained by rescaling and differentiating, not by identifying them with the discrete multipliers. The correlation length obeys

ξ (t, h) = b^{p} ξ (b^{p λ_{t}} t, b^{p λ_{h}} h)

, hence

ν = 1/ λ_{t}

. Two temperature derivatives of

F_{s} (t, 0)

give

C_{V, s} ≍ ∣ t ∣^{D / λ_{t} - 2}

, while one and two field derivatives give the ordered magnetization and susceptibility powers. Thus

ν = \frac{1}{λ _{t}}, α = 2 - \frac{D}{λ _{t}}, β = \frac{D - λ _{h}}{λ _{t}}, γ = \frac{2 λ _{h} - D}{λ _{t}}, δ = \frac{λ _{h}}{D - λ _{h}} .

For

δ

, instead set

t = 0

and choose the blocking scale by

h

, obtaining

F_{s} (0, h) ≍ ∣ h ∣^{D / λ_{h}}

and then differentiate. A nonzero spontaneous ordered-branch derivative is assumed in the formula for

β

. In particular the hyperscaling relation is

α = 2 - D ν

. Its physical assumption is an order-one singular free energy per correlation volume, with a single diverging length and no dangerous variable changing its volume scaling. Above the upper critical dimension, a stabilizing quartic coupling can invalidate that assumption even though the correlation length still scales.

For the last model, take the quadratic Hamiltonian exactly as visually printed: its source term is

+ h ϕ

, outside the half multiplying the gradient and mass terms. With

κ > 0

and

m^{2} > 0

, its connected quadratic covariance is proportional to

G (q) = \frac{1}{κ ^{- 1} q ^{2} + m ^{2}} = \frac{κ}{q ^{2} + κ m ^{2}} .

The pole scale, or the massive Gaussian field correlation tail obtained by its Fourier transform, gives

ξ = (κ m^{2})^{- 1/2} = \frac{1}{κ m} (m > 0) .

The uniform source shifts the mean to

⟨ ϕ ⟩ = - h / m^{2}

and does not change the connected covariance. Replacing

h

- h

matches the earlier magnetic source convention without changing any scaling index.

Define a Gaussian thinning transformation by retaining momenta

∣ q ∣ < Λ/ b

, integrating the Gaussian shell, and rescaling

x^{'} = x / b, ϕ^{'} (x^{'}) = b^{(D - 2) /2} ϕ_{<} (b x^{'}) .

(14)

The gradient coefficient is unchanged. Counting volume and field powers in the remaining action gives

κ^{'} = κ, (m^{2})^{'} = b^{2} m^{2}, h^{'} = b^{(D + 2) /2} h .

(15)

Shell integration contributes only the additive determinant term. Thus

λ_{t} = 2

when the thermal variable is proportional to

m^{2}

, and

λ_{h} = (D + 2) /2

. Inserting these into the homogeneous index formulas gives the requested Gaussian critical exponents

α_{G} = (4 - D) /2, β_{G} = (D - 2) /4.

(16)

The specific-heat power can also be verified without hyperscaling: two mass-squared derivatives of

\frac{1}{2} \int d^{D} q (2 π)^{- D} lo g (κ^{- 1} q^{2} + m^{2})

produce an integral proportional to

\int d^{D} q (q^{2} + κ m^{2})^{- 2}

. For

0 < D < 4

, rescaling

q = κ m p

gives

(m^{2})^{D /2 - 2}

; at

D = 4

the divergence is logarithmic, with power index zero. At

D = 2

the determinant itself has an additive

m^{2} lo g m^{2}

resonance, although its heat-capacity power remains one.

There is a genuine limitation to the last printed request. The Gaussian order-parameter scaling index

β_{G}

is formal: a pure quadratic theory has

⟨ ϕ ⟩ = 0

h = 0, m^{2} > 0

, and at

m^{2} < 0

the uniform energy

m^{2} ϕ^{2} /2

is unbounded below, so its partition function diverges. It therefore has no stable ordered branch from which a spontaneous-magnetization exponent can be measured. The thinning derivation establishes the index requested by the PDF, but not a nonexistent ordered phase. A stabilizing quartic interaction is relevant for

D < 4

and generally changes these Gaussian indices to the interacting critical ones.

Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 50 / 4 / b / Solution Created 2026-10-03 Updated 2026-10-14

Besides the Gaussian fixed point, solve the nonzero-coupling fixed-point equations. The quartic equation gives

λ_{*} = 2 ϵ (1 + ρ_{*})^{2} / (3 K_{D})

. Substitute into the mass equation to get

2 ρ_{*} + ϵ (1 + ρ_{*}) /3 = 0

, hence

ρ_{*} = - ϵ / (6 + ϵ)

within the displayed one-loop truncation. Since

K_{4} = 2 π^{2} / (2 π)^{4} = 1/ (8 π^{2})

ρ_{*} = - ϵ /6 + O (ϵ^{2}), λ_{*} = 16 π^{2} ϵ /3 + O (ϵ^{2}) .

These are the Wilson-Fisher fixed point coordinates in this cutoff scheme. The higher terms obtained by solving the truncated equations exactly should not be regarded as a full higher-order epsilon calculation.

The attraction claim in the PDF needs a thermal qualification. The stability matrix of a renormalization-group fixed point at this solution has entries

J = (2 - ϵ /3 3 K_{D} λ_{*}^{2} / (1 + ρ_{*})^{3} K_{D} / [2 (1 + ρ_{*})] - ϵ) .

The lower-left entry is

O (ϵ^{2})

, so its eigenvalues are

y_{t} = 2 - ϵ /3 + O (ϵ^{2}) > 0, y_{g} = - ϵ + O (ϵ^{2}) < 0.

Increasing

ℓ

means approaching the infrared. The quartic scaling direction is attractive, but a perturbation in the thermal scaling direction grows as

e^{y_{t} ℓ}

. Therefore the Wilson-Fisher fixed point is an infrared attractor on its tuned critical surface, not an attractive fixed point of the entire two-dimensional coupling plane. In the plane it is a saddle; an arbitrary small thermal perturbation is a concrete counterexample to unconditional attraction. This is the thermal relevant direction at the Wilson-Fisher fixed point.

Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 43 / 2 / Solution Created 2026-10-03 Updated 2026-10-14

Take the connected correlation function

G (r) = ⟨ σ_{0} σ_{r} ⟩ - ⟨ σ_{0} ⟩ ⟨ σ_{r} ⟩ .

For a massive phase its large-distance form is an algebraic prefactor times

e^{- ∣ r ∣/ ξ}

, defining the exponential correlation length

ξ

. To specify the normalization of magnetic susceptibility, add

- h \sum_{r} σ_{r}

to the energy and differentiate the mean spin. Translation invariance gives the correlation-function susceptibility sum rule

χ = \frac{\partial M}{\partial h} = \frac{β _{th}}{N} Var (r \sum σ_{r}) = β_{th} r \sum G (r), β_{th} = (k_{B} T)^{- 1} .

Using the dimensionless source

β_{th} h

removes the prefactor

β_{th}

. Below the transition, use a selected pure phase rather than a mixture of the two magnetizations, which would have a macroscopic disconnected fluctuation.

Choose a nonnegative blocking kernel

K_{b} (σ^{'}, σ)

normalized by

\sum_{σ^{'}} K_{b} (σ^{'}, σ) = 1

. It may enforce a block average, a majority rule or a chosen rescaled coarse spin. Define the transformed weight by

e^{- β_{th} [H (u^{'}, σ^{'}) + N^{'} C^{'}]} = \sum_{σ} K_{b} (σ^{'}, σ) e^{- β_{th} [H (u, σ) + NC]} .

Summing this identity over coarse spins shows exact preservation of the partition function. The geometrical changes are $a^{'} = ba$ and $N^{'} = N / b^{D}$ . A coordinate rescaling subsequently restores the reference cutoff if desired. The transformation is exact only if all generated operators are retained; a finite-coupling truncation is an approximation.

Define the coupling-independent term generated when

C = 0

\sum_{σ} K_{b} (σ^{'}, σ) e^{- β_{th} H (u, σ)} = e^{- β_{th} [H (R_{b} u, σ^{'}) + N^{'} c (u)]} .

Restoring the original

C

gives the identity-operator contribution to renormalization-group free energy:

u_{p + 1} = R_{b} (u_{p}), C_{p + 1} = b^{D} C_{p} + c (u_{p}) .

The parameter

C_{p}

is an accumulated additive energy per coarse site. It multiplies the identity operator and leaves normalized spin probabilities unchanged, but it must be kept when calculating the absolute free energy. It records the eliminated-mode entropy and the normalization of their statistical measure.

Use an energy-per-site convention:

F (u, C) = - lo g Z (u, C, N) / (β_{th} N)

and

f (u) = F (u, 0)

, so

F = f + C

. Partition-function invariance gives

F (u_{p}, C_{p}) = b^{- D} F (u_{p + 1}, C_{p + 1}) .

Consequently, with

g (u) = b^{- D} c (u)

f (u) = b^{- D} f (R_{b} u) + g (u), f (u_{0}) = b^{- p D} f (u_{p}) + j = 0 \sum p - 1 b^{- j D} g (u_{j}) .

This is the additive free-energy recursion under blocking. In a dimensionless free-energy convention multiply

f, C, c, g

β_{th}

; the same recursion holds. The inhomogeneous term comes from the generated identity operator, not from a change in the spin-dependent couplings alone.

The affine equation naturally applies to the unsubtracted free energy. To extract its singular part, write

f = f_{reg} + F_{s}

and choose a regular background satisfying

f_{reg} (u) = b^{- D} f_{reg} (R_{b} u) + g (u)

. Subtraction then gives

F_{s} (u) = b^{- D} F_{s} (R_{b} u)

. This analytic subtraction of an inhomogeneous renormalization recursion is justified when the eliminated finite-shell contribution is analytic near the fixed point and admits a regular subtraction. It can fail in a resonant direction, producing additive logarithms. The simple homogeneous formulas also assume the remaining irrelevant operators are not dangerously irrelevant couplings, and that marginal operators do not produce unaccounted logarithmic corrections. Calling the quantity in the affine equation “singular” does not make

g

vanish automatically.

Near a renormalization-group fixed point

u_{*}

, linearize the coupling map. In scaling coordinates

v_{i}

, write

v_{i}^{'} = b^{λ_{i}} v_{i}

to leading order. Here

b^{λ_{i}}

is a discrete multiplier, whereas

λ_{i}

is its logarithmic scaling exponent. A relevant operator has

λ_{i} > 0

and grows under coarse-graining; an irrelevant operator has

λ_{i} < 0

and decays. A marginal operator needs nonlinear analysis. Tuning the relevant fields to zero places the system on the critical surface. At the fixed point the spin field is rescaled as

σ^{'} (x^{'}) = b^{Δ_{σ}} σ_{<} (b x^{'})

with

Δ_{σ} = (D - 2 + η) /2

. Its conjugate source therefore has

λ_{h} = D - Δ_{σ} = (D + 2 - η) /2

. This accounts for both coupling and spin rescaling; the field normalization is part of the transformation, not a change in the physical microscopic spin length.

Assume the thermal field

t = (T - T_{c}) / T_{c}

and magnetic field

h

are the only remaining relevant directions. The homogeneous scaling hypothesis for critical phenomena gives

F_{s} (t, h) = b^{- p D} F_{s} (b^{p λ_{t}} t, b^{p λ_{h}} h) .

Choose the stopping scale with

b^{p λ_{t}} ∣ t ∣ = 1

. Then

F_{s} (t, h) = ∣ t ∣^{D / λ_{t}} f_{\pm} (h /∣ t ∣^{λ_{h} / λ_{t}}),

where $f_{+}$ is the branch for $t > 0$ and $f_{-}$ the branch for $t < 0$ . Nonuniversal metric factors can be absorbed into the definitions of

t, h

and these functions. Correlation lengths transform as

ξ (t) = b^{p} ξ (b^{p λ_{t}} t)

, so

ν = 1/ λ_{t}

. Differentiating the singular free energy twice with respect to temperature gives

C_{s} \propto ∣ t ∣^{D / λ_{t} - 2}

, and hence the hyperscaling relation

α = 2 - D ν .

Two source derivatives similarly give

γ = (2 λ_{h} - D) / λ_{t}

. A single source derivative gives the ordered-branch exponent

β = (D - λ_{h}) / λ_{t}

, and setting

t = 0

and scaling

h

to one gives

δ = λ_{h} / (D - λ_{h})

when the corresponding magnetization branch exists. These show explicitly how the critical exponents follow from the fixed-point scaling eigenvalues.

The independent correlation derivation uses

G (r) = A r^{- (D - 2 + η)} f_{G} (r / ξ)

with a single crossover scaling function. For lattice spacing

a

, the singular continuum contribution to the magnetic susceptibility is

χ_{s} ≃ β_{th} a^{- D} A S_{D} \int_{a}^{\infty} r^{1 - η} f_{G} (r / ξ) d r \sim β_{th} a^{- D} A S_{D} ξ^{2 - η} \int_{0}^{\infty} s^{1 - η} f_{G} (s) d s .

Here

S_{D}

is the area of the unit

(D - 1)

-sphere. The final integral is finite and nonzero when

η < 2

f_{G} (0)

is finite and nonzero, and the tail decays exponentially. Microscopic distances contribute a regular background. With

ξ \sim ∣ t ∣^{- ν}

this proves the Fisher scaling relation

χ_{s} \propto ξ^{2 - η}, γ = (2 - η) ν .

There is an amplitude qualification in the printed large-distance formula: its tail has the Gaussian shape and, taken literally with amplitude independent of

ξ

, integrates to a term proportional to

ξ^{2}

. For a nonzero anomalous exponent it must instead be matched to the critical short-distance normalization. The matching the amplitude of a massive critical correlation tail gives

G (r) \sim A_{\infty} ξ^{- η} \frac{ξ}{( ξ r ) ^{(D - 1) /2}} e^{- r / ξ}, r ≫ ξ .

r = s ξ

this has precisely the same overall

ξ^{- (D - 2 + η)}

scaling as the critical form. Thus the requested identity uses the full scaling hypothesis, or equivalently this implicit amplitude factor. The printed tail without it is consistent as written in the Gaussian case

η = 0

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 41 / 2 / Solution Created 2026-10-03 Updated 2026-10-14

Use the connected correlation function

G (r) = ⟨ σ_{0} σ_{r} ⟩ - ⟨ σ_{0} ⟩ ⟨ σ_{r} ⟩

. Away from criticality its large-distance envelope decays exponentially on the correlation length

ξ

, often with an algebraic prefactor. At criticality

ξ

diverges and a short-range scalar model instead has the power law

G (r) \sim r^{- (D - 2 + η)}

, with anomalous dimension

η

A blocking kernel

K_{b} (σ^{'}, σ)

assigns coarse spins to each block of side

b

lattice spacings, with

\sum_{σ^{'}} K_{b} (σ^{'}, σ) = 1

. Define the coarse Hamiltonian by

e^{- β H (u^{'}, σ^{'}) - β N^{'} C^{'}} = \sum_{σ} K_{b} (σ^{'}, σ) e^{- β H (u, σ) - βNC} .

The operator basis must be sufficiently complete to include interactions generated by blocking. Summing over coarse spins preserves the partition function, while the physical lattice spacing and site count change to

a^{'} = ba

N^{'} = b^{- D} N

. After

p

steps they are

a_{p} = b^{p} a

and

N_{p} = b^{- p D} N

. Exact blocking preserves the long-distance predictions; truncating the generated interaction space introduces an approximation.

Define the dimensionless free energy per site

F (u, C) = - N^{- 1} lo g Z (u, C, N) = f (u) + βC

. Preservation of

Z

gives

F (u_{0}, C_{0}) = b^{- p D} F (u_{p}, C_{p})

. For one step, removing the explicitly additive constant gives

f (u_{j}) = b^{- D} f (u_{j + 1}) + g (u_{j}), g (u_{j}) = β [b^{- D} C_{j + 1} - C_{j}] .

The function

g

records the additive free-energy density generated by eliminated degrees of freedom; it is usually regular near a finite blocking step's critical point. Iteration proves

f (u_{0}) = b^{- p D} f (u_{p}) + j = 0 \sum p - 1 b^{- j D} g (u_{j}) .

This is the additive free-energy recursion under blocking. After removing regular backgrounds, its singular part obeys homogeneous scaling.

R_{b} (u^{*}) = u^{*}

. Linearizing the map gives scaling coordinates

v_{i}^{'} = b^{λ_{i}} v_{i}

. A relevant operator has

λ_{i} > 0

, an irrelevant operator has

λ_{i} < 0

, and a marginal operator has

λ_{i} = 0

and needs nonlinear analysis. The critical surface consists of points flowing into the critical fixed point after all relevant directions are tuned. An outgoing or repulsive trajectory lies in the unstable manifold and describes perturbations that grow under coarse-graining.

Figure 1.
Renormalization-group flow near a critical surface
. **Details**.

With relevant thermal and magnetic coordinates

t, h

, the singular free-energy density obeys

F_{s} (t, h) = b^{- D} F_{s} (b^{λ_{t}} t, b^{λ_{h}} h)

. Choose

b = ∣ t ∣^{- 1/ λ_{t}}

to obtain

F_{s} (t, h) = ∣ t ∣^{D / λ_{t}} f_{\pm} (\frac{h}{∣ t ∣ ^{λ_{h} / λ_{t}}}) .

Here

λ_{t}, λ_{h}

are the positive scaling eigenvalues, and

f_{\pm}

refer to the signs of

t

. Since

ξ (t, h) = b ξ (b^{λ_{t}} t, b^{λ_{h}} h)

, the correlation-length exponent is

ν = 1/ λ_{t}

. Two temperature derivatives of

F_{s} (t, 0)

give

α = 2 - D / λ_{t}

, establishing the hyperscaling relation

α = 2 - D ν .

The magnetization exponent similarly is

β = (D - λ_{h}) / λ_{t}

. These expressions assume no additional dangerous scaling variables.

For the Gaussian field theory, decompose the field into low and high Fourier modes, integrate out

Λ/ b < ∣ p ∣ < Λ

, and restore the cutoff using

x = b x^{'}

and

ϕ (x) = b^{- (D - 2) /2} ϕ^{'} (x^{'})

. The gradient coefficient remains fixed, the mass coefficient transforms as

m^{'2} = b^{2} m^{2}

, and the uniform source as

h^{'} = b^{(D + 2) /2} h

. Hence

λ_{t} = 2

λ_{h} = (D + 2) /2

, and the Gaussian momentum-shell scaling gives

α = \frac{4 - D}{2}, β = \frac{D - 2}{4} .

The sign convention

+ h ϕ

reverses the magnetization's sign but not these exponents. For

2 < D < 4

, these are Gaussian scaling predictions, not the exponents of a generic interacting scalar critical point: the quartic interaction is relevant there. The free quadratic Hamiltonian also has no stable ordered phase at negative

m^{2}

, so the quoted

β

is its formal order-parameter scaling index, not a spontaneously magnetized state of the unstabilized Gaussian integral. The relevance distinction is also explained in Tong's renormalization-group lecture notes.

Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 49 / 2 / Solution Created 2026-10-03 Updated 2026-10-14

Use a selected translationally invariant pure thermodynamic phase and define the connected lattice correlation function

G (r) = ⟨ σ_{0} σ_{n} ⟩ - ⟨ σ_{0} ⟩ ⟨ σ_{n} ⟩, r = a n .

Away from criticality its long-distance tail has an exponential factor

e^{- r / ξ}

, possibly multiplied by a power. Equivalently

ξ^{- 1} = - lim_{r \to \infty} r^{- 1} lo g ∣ G (r) ∣

when this limit exists; different directions can have different lengths on an anisotropic lattice. At a thermodynamic critical point the correlation length diverges and the tail becomes algebraic. Differentiate

M = N^{- 1} ⟨ \sum_{n} σ_{n} ⟩

with respect to the physical field appearing in the energy Statistical Hamiltonian. With

β_{th} = 1/ (k_{B} T)

, the correlation-function susceptibility sum rule is

χ = \frac{β _{th}}{N} Var (n \sum σ_{n}) = β_{th} n \sum G (a n) .

A normalized blocking kernel

In physical continuum coordinates the sum becomes

a^{- D} \int d^{D} r G (r)

. Differentiating with respect to the dimensionless source

β_{th} h

instead removes the explicit inverse-temperature factor. Below the transition, take a one-sided pure-phase limit rather than a macroscopic symmetric mixture.

B_{b} (σ^{'}, σ)

assigns a probability or a delta constraint to a coarse configuration for each microscopic one, with

\sum_{σ^{'}} B_{b} (σ^{'}, σ) = 1

. Define the blocked Boltzmann weight by

e^{- β_{th} [H (u_{1}, σ^{'}) + N_{1} C_{1}]} = \sum_{σ} B_{b} (σ^{'}, σ) e^{- β_{th} [H (u_{0}, σ) + N C_{0}]} .

Summing over

σ^{'}

proves exact equality of the partition functions. Repeated real-space renormalization group steps compose the kernels and integrate out shorter-scale variables. To retain equality one must keep all generated interactions and the field-independent constant; a finite truncation is an approximation. Long-distance observables are mapped through the corresponding coarse observables, not identified blindly with microscopic spins. In physical units,

a_{p} = b^{p} a, N_{p} = b^{- p D} N, N_{p} a_{p}^{D} = N a^{D} .

A subsequent coordinate rescaling restores the reference lattice spacing while expressing couplings in cutoff units.

Let

F (u, C) = - (β_{th} N)^{- 1} lo g Z = C + f (u)

be free energy per microscopic site. Invariance of

Z

gives

F (u_{0}, C_{0}) = b^{- D} F (u_{1}, C_{1}) .

The constant transforms as

C_{1} = b^{D} C_{0} + c (u_{0})

. Therefore, with

g (u) = b^{- D} c (u)

, the identity-operator contribution to renormalization-group free energy gives

f (u_{0}) = b^{- D} f (u_{1}) + g (u_{0}), f (u_{0}) = b^{- p D} f (u_{p}) + j = 0 \sum p - 1 b^{- j D} g (u_{j}) .

The second equality follows by substitution and induction. The inhomogeneous term records eliminated-mode entropy, determinants, and normalization factors, all multiplying the identity operator; it is not an interaction with the retained spins. Strictly, this

f

can contain regular background terms. After an analytic subtraction of an inhomogeneous renormalization recursion, the genuinely singular part satisfies the homogeneous scaling equation, except for possible logarithmic resonances.

R_{b} (u_{*}) = u_{*}

. Linearize in scaling coordinates

s_{i}

s_{i}^{'} = b^{λ_{i}} s_{i}

. A relevant operator has

λ_{i} > 0

, an irrelevant operator has

λ_{i} < 0

, and a marginal operator has

λ_{i} = 0

and needs nonlinear analysis. These

λ_{i}

are logarithmic scaling eigenvalues; the discrete Jacobian eigenvalues are

b^{λ_{i}}

, not

λ_{i}

. The critical surface is the stable manifold obtained by tuning all relevant coordinates to zero. A repulsive renormalization-group trajectory leaves the fixed point along a relevant direction as the coarse-graining length grows. Stable and unstable manifolds become curved away from linear order.

Figure 1.
Linear renormalization-group flows: attraction along a critical surface and repulsion in two relevant directions
. **Details**.

The left section has

h = 0

, one irrelevant coordinate

w

and a relevant thermal coordinate

t

. Its critical surface is

t = 0

in that section. With two relevant variables, the full critical surface has codimension two: the second section shows repulsion in the

(t, h)

plane at

w = 0

Assume an isolated fixed point with two relevant scaling fields, analytic changes from physical controls to

(t, h)

, short-range isotropic scaling, no dangerously irrelevant variable in the thermodynamic scaling function, and no marginal or resonant logarithm. Let

λ_{t}, λ_{h} > 0

denote their scaling eigenvalues. The homogeneous singular free-energy density obeys

F_{s} (t, h) = b^{- p D} F_{s} (b^{p λ_{t}} t, b^{p λ_{h}} h) .

Choose a stopping scale

L = b^{p} ≃ ∣ t ∣^{- 1/ λ_{t}}

so the renormalized thermal field is order one. Irrelevant fields vanish at this scale. With

s = D / λ_{t}

and

Δ = λ_{h} / λ_{t}

F_{s} (t, h) = ∣ t ∣^{s} Φ_{\pm} (h /∣ t ∣^{Δ}) .

Continuous scale interpolation absorbs the immaterial integer stopping-scale convention. Nonuniversal thermal, field and free-energy metric factors can be included explicitly; the normalized functions depend only on the universality class. The

+

and

-

labels refer to

t > 0

and

t < 0

, which flow to different sides of the critical surface.

For clarity about the additive terms, let

a (t, h)

solve

a (t, h) = b^{- D} a (b^{λ_{t}} t, b^{λ_{h}} h) + g (t, h)

. If

g = \sum g_{mn} t^{m} h^{n}

, its analytic coefficients are

a_{mn} = \frac{g _{mn}}{1 - b ^{m λ_{t} + n λ_{h} - D}},

provided the denominators do not vanish. Subtracting

a

makes the recursion homogeneous. A vanishing denominator is a resonance and generically produces logarithms, which the simple power-law hypothesis excludes. In a convention where the regular field-dependent background has been subtracted, write

Φ_{\pm} = f_{\pm} + C_{\pm}

and

I_{\pm} (t) = C_{0} + a (t, 0)

. This yields the requested form

F (t, h) = ∣ t ∣^{D / λ_{t}} [f_{\pm} (\frac{h}{∣ t ∣ ^{λ_{h} / λ_{t}}}) + C_{\pm}] + I_{\pm} (t) .

The

C_{\pm}

contain branch-dependent matching contributions; only

f_{\pm} + C_{\pm}

is physically determined. The

I_{\pm}

are nonsingular functions analytic through the thermodynamic critical point and represent the same regular Taylor background on the two sides, though either branch may be used to describe them. For the full free energy at nonzero field, the regular background is generally $I (t, h)$ , not just $I (t)$ . Adding an analytic identity-operator contribution proportional to

h^{2}

provides a direct counterexample to an exact unrestricted

I (t)

formula; it changes no singular exponents. The printed expression is justified for the leading singular scaling form with that analytic field background removed. Confluent corrections from irrelevant fields are also omitted in this leading form.

Differentiate the singular scaling form, with one-sided derivatives in the ordered phase. It gives

α = 2 - \frac{D}{λ _{t}}, β = \frac{D - λ _{h}}{λ _{t}}, γ = \frac{2 λ _{h} - D}{λ _{t}}, δ = \frac{λ _{h}}{D - λ _{h}} .

The length transforms as

ξ (t, h) = L ξ (L^{λ_{t}} t, L^{λ_{h}} h)

, so

ν = 1/ λ_{t}

. Thus the Rushbrooke scaling relation and hyperscaling relation follow by direct substitution:

α + 2 β + γ = 2, α = 2 - D ν .

The same reasoning also gives

γ = β (δ - 1)

. Hyperscaling relation relies on the assumptions above and can fail above an upper critical dimension because of a stabilizing dangerously irrelevant coupling.

For the universal specific-heat amplitude ratio, write

F_{s} = A_{f} ∣ a_{t} t ∣^{s} Φ_{\pm} (0)

h = 0

. Two temperature derivatives multiply both branches by the same dimensional and thermal metric factors. Away from logarithmic resonances and exceptional zero amplitudes,

A_{+} / A_{-} = Φ_{+} (0) / Φ_{-} (0)

with the same heat-capacity sign convention on both sides. Hence

A_{+} / A_{-} is universal within a fixed universality class .

The functions and their matched branch constants are tied to the fixed-point theory; the constants are not independent arbitrary experimental parameters. If

α < 0

, this means the amplitudes of the subleading singular contribution after subtracting the analytic specific-heat background. A finite background ratio itself need not be universal.

Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 42 / 2 / Solution Created 2026-10-03 Updated 2026-10-14

Work in a translation-invariant pure thermodynamic phase, and write

β_{th} = 1/ (k_{B} T)

to distinguish inverse temperature from the order-parameter exponent. The connected correlation function is

G (r) = ⟨ σ_{0} σ_{n} ⟩ - ⟨ σ ⟩^{2}, r = a n .

Away from criticality in the massive scalar order channel, it decays exponentially, possibly multiplied by an algebraic factor. An exponential correlation length is defined by

ξ^{- 1} = - lim_{r \to \infty} r^{- 1} lo g ∣ G (r) ∣

when that limit exists. At a continuous transition

ξ

diverges and the critical decay is algebraic. In an ordered symmetric mixture, a nondecaying contribution can remain; selecting a pure branch prevents confusing it with connected critical fluctuations.

Let

M = \sum_{n} σ_{n}

and

m = ⟨ M ⟩ / N

. Differentiating the partition function with the energy term

- h M

gives the correlation-function susceptibility sum rule

χ = \frac{\partial m}{\partial h} = \frac{β _{th}}{N} (⟨ M^{2} ⟩ - ⟨ M ⟩^{2}) = β_{th} r \sum G (r) .

If the source is instead the dimensionless

β_{th} h

, the explicit inverse-temperature factor is absent. In continuum physical coordinates the lattice sum is approximately

a^{- D} \int d^{D} r G (r)

Choose a normalized blocking kernel

B_{b} (σ^{'}, σ) \geq 0

such that

\sum_{σ^{'}} B_{b} (σ^{'}, σ) = 1

, replacing the sum by an integral for continuous variables. A deterministic example is a product of delta functions imposing that each coarse variable equals the average of spins in a block of

b^{D}

sites, with its field normalization included consistently. Define the effective Boltzmann weight by

e^{- β_{th} [H (u^{'}, σ^{'}) + N^{'} C^{'}]} = \sum_{σ} B_{b} (σ^{'}, σ) e^{- β_{th} [H (u, σ) + NC]} .

Kernel normalization makes the blocked partition function exactly equal to the original. Its coarse-grained variables retain the long-distance observables through their defining relation to the original fields. An exact RG step generally generates all symmetry-allowed operators; keeping only a small coupling set is an approximation, not an exact closure assumption. The geometrical coarse lattice has

a_{p} = b^{p} a, N_{p} = b^{- p D} N,

so its physical volume is unchanged. A subsequent coordinate rescaling may restore the numerical lattice spacing to its initial value, which is the equivalent rescaled-coordinate convention.

Here define

F (u, C)

as free energy per lattice site,

F = - (β_{th} N)^{- 1} lo g Z = C + f (u)

. Exact partition-function invariance gives

F (u_{0}, C_{0}) = b^{- p D} F (u_{p}, C_{p}) .

Because

C

multiplies the identity operator, a blocking step has

C^{'} = b^{D} C + c (u)

, where

c

is the generated constant per blocked site. With

g (u) = b^{- D} c (u)

this implies

f (u) = b^{- D} f (u^{'}) + g (u), f (u_{0}) = b^{- p D} f (u_{p}) + j = 0 \sum p - 1 b^{- j D} g (u_{j}) .

The identity-operator contribution to renormalization-group free energy includes the eliminated modes' entropy, connected vacuum terms and field-measure normalization. It cannot be discarded when calculating an absolute free energy, even though it cancels out of normalized correlation functions.

There is a useful convention behind the word “singular” in this inhomogeneous equation. After removing

C

f (u)

still contains a regular coupling-dependent background. If

f = f_{reg} + F_{s}

and

f_{reg} (u) = b^{- D} f_{reg} (u^{'}) + g (u)

, then the genuinely nonanalytic part obeys

F_{s} (u) = b^{- D} F_{s} (u^{'})

. Thus the printed inhomogeneous representative and homogeneous singular scaling below are consistent after background subtraction. At resonances or marginal points this subtraction may leave additive or multiplicative logarithms; a strictly homogeneous pure-power form is then qualified accordingly.

R_{b} (u_{*}) = u_{*}

. Diagonalizing the linearized map gives scaling coordinates

s_{i}^{'} = b^{λ_{i}} s_{i}

. For increasing coarse-graining length,

λ_{i} > 0

defines a relevant operator,

λ_{i} < 0

an irrelevant operator, and

λ_{i} = 0

a marginal operator, whose fate needs nonlinear analysis. Here the

λ_{i}

are logarithmic scaling exponents: the eigenvalues of the discrete Jacobian matrix are

b^{λ_{i}}

, not the numbers

λ_{i}

themselves. The critical surface is the stable manifold flowing into the critical fixed point after every relevant scaling field is tuned to zero. A repulsive renormalization-group trajectory leaves the fixed point along relevant directions and flows toward a different long-distance phase.

Figure 1.
Linearized renormalization-group flows in a zero-field slice and the relevant-field plane
. **Details**.

For the requested two relevant fields, take

t^{'} = b^{λ_{t}} t

and

h^{'} = b^{λ_{h}} h

, with

λ_{t}, λ_{h} > 0

. Ignoring nonsingular irrelevant corrections and choosing a total blocking scale

ℓ = b^{p}

, the homogeneous singular free-energy density obeys

F_{s} (t, h) = ℓ^{- D} F_{s} (ℓ^{λ_{t}} t, ℓ^{λ_{h}} h) .

Choose

ℓ = ∣ t ∣^{- 1/ λ_{t}}

so the renormalized thermal field is

\pm 1

. This proves

F_{s} (t, h) = ∣ t ∣^{D / λ_{t}} f_{\pm} (\frac{h}{∣ t ∣ ^{λ_{h} / λ_{t}}}) .

Metric factors and regular analytic redefinitions of the scaling fields only change amplitudes. Likewise

ξ (t, h) = ℓ ξ (ℓ^{λ_{t}} t, ℓ^{λ_{h}} h)

, so at

h = 0

the correlation-length critical exponent is

ν = 1/ λ_{t}

. Two thermal derivatives of

F_{s}

give the heat-capacity critical exponent

α = 2 - D / λ_{t}

, proving the hyperscaling relation

α = 2 - D ν .

Two field derivatives also give

γ = (2 λ_{h} - D) / λ_{t}

; one field derivative gives

β = (D - λ_{h}) / λ_{t}

, and using

h

to set the blocking scale on the critical isotherm gives

δ = λ_{h} / (D - λ_{h})

. These statements assume that no dangerously irrelevant coupling makes the scaling function singular as it is removed. In particular the naive hyperscaling form need not describe the quartic ordered phase above four dimensions, where its stabilizing quartic coupling is dangerously irrelevant. That restriction reconciles this derivation with the mean-field exponents and Question 3.

The scaling form of the connected correlation function is

G (r) = r^{- (D - 2 + η)} f_{G} (r / ξ)

in the continuum long-distance regime. For

r ≪ ξ

f_{G}

tends to a finite nonzero constant. For

r ≫ ξ

, it decays exponentially with a possible algebraic prefactor; an isolated massive pole gives the familiar Ornstein--Zernike correlation function tail

G (r) \propto r^{- (D - 1) /2} e^{- r / ξ}

with a temperature-dependent amplitude. Exponential decay is the essential feature; the same critical algebraic power need not remain the exact large-distance prefactor.

Using the susceptibility sum rule and spherical integration gives

χ ≃ β_{th} a^{- D} S_{D - 1} ξ^{2 - η} \int_{a / ξ}^{\infty} d y y^{1 - η} f_{G} (y) .

For

η < 2

and an ordinary exponentially decaying scaling function the integral tends to a finite constant: it converges at zero because

f_{G} (0)

is finite, and at infinity because of the exponential decay. Microscopic distances add a regular background. Thus

χ_{s} \propto ξ^{2 - η} \propto ∣ t ∣^{- (2 - η) ν}

, establishing the Fisher scaling relation

γ = (2 - η) ν .

For a continuous field, field scaling and anomalous dimension give

x_{σ} = \frac{D - 2 + η}{2}, σ^{'} (x^{'}) = b^{x_{σ}} σ_{<} (b x^{'}) .

This field renormalization is additional to the canonical engineering factor

b^{(D - 2) /2}

. It makes

G (r) = b^{- 2 x_{σ}} G^{'} (r / b)

. Invariance of the uniform source coupling

\int hσ

then gives

λ_{h} = D - x_{σ} = (D + 2 - η) /2

, agreeing with the susceptibility exponent above. With a convention

σ^{'} = b^{(D - 2) /2} Z_{σ}^{- 1/2} σ_{<}

, the fixed-point choice is

Z_{σ} (b) \propto b^{- η}

; the sign of the logarithmic derivative depends on this stated convention.

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 303 / 3 / b / Solution Created 2026-10-03 Updated 2026-10-05

Compute the coupling flow at zero magnetic field, or assume the source has support only in retained Fourier modes, so that the shell Gaussian measure is centered. First justify the shell propagator. For a positive quadratic kernel on the shell, the Gaussian functional integral gives covariance equal to the inverse kernel,

⟨ ϕ_{>} (p) ϕ_{>} (q) ⟩_{0}^{shell} = (2 π)^{D} δ^{(D)} (p + q) \frac{α}{p ^{2} + α r _{0}},

with both momenta restricted to

Λ/ b < ∣ p ∣ \leq Λ

. Inserting the Fourier transforms and using the Dirac delta function yields the Gaussian shell covariance

G_{>} (x - y) = \int_{Λ/ b < ∣ p ∣ \leq Λ} \frac{d ^{D} p}{( 2 π ) ^{D}} \frac{α e ^{i p \cdot (x - y)}}{p ^{2} + α r _{0}} .

Changing

p

- p

gives the negative-exponent convention as well. The dependence is on the separation, as required by translation invariance. The printed numerator uses

x

alone: it is correct only if that symbol means the separation or if

y = 0

. For example at

x = y \neq = 0

, the covariance must equal the constant coincident value

G_{>} (0)

, while the literal printed integral generally depends on

x

. Thus the missing separation is a real source qualification, not a change of Fourier-sign convention.

Integrating out the shell gives

H_{eff} [ϕ_{<}] = H_{0} [ϕ_{<}] - lo g ⟨ e^{- V [ϕ_{<} + ϕ_{>}]} ⟩_{0}^{shell}

, up to a field-independent constant. At first order in

u_{0}

, the cumulant expansion of a coarse-grained free energy retains

⟨ V ⟩

. Odd moments of the centered Gaussian measure vanish and Wick theorem gives

⟨ ϕ_{>}^{4} ⟩ = 3 G_{>} (0)^{2}

. Writing

I_{1} = G_{>} (0)

⟨ V ⟩ = \frac{u _{0}}{24} \int d^{D} x [ϕ_{<}^{4} + 6 I_{1} ϕ_{<}^{2} + 3 I_{1}^{2}] .

The last term affects only the constant; the second is a momentum-independent tadpole diagram correction. Before rescaling it changes

r_{0}

r_{0} + u_{0} I_{1} /2

, leaves the quartic coefficient at

u_{0}

, and produces no gradient correction. Combining with part (a) gives the leading coupling recursion

r_{0}^{'} = b^{2} (r_{0} + \frac{u _{0}}{2} I_{1}) + O (u_{0}^{2}), u_{0}^{'} = b^{4 - D} u_{0} + O (u_{0}^{2}), α^{- 1'} = α^{- 1} + O (u_{0}^{2}) .

This is one-loop shell mass renormalization in scalar quartic theory. Here

I_{1} = \int_{shell} d^{D} p (2 π)^{- D} α / (p^{2} + α r_{0})

. The shell Gaussian is well defined if

p^{2} + α r_{0} > 0

there; near

r_{0} = 0

this follows from

α > 0

. A negative slow-mode mass may be stabilized by the retained quartic term and does not require pretending that the full unconstrained quadratic measure at negative mass is normalizable.

Linearizing about

(r_{0}, u_{0}) = (0, 0)

gives

(r_{0}^{'} u_{0}^{'}) = (b^{2} 0 \frac{1}{2} b^{2} I_{1} (0) b^{4 - D}) (r_{0} u_{0}) + O (r_{0} u_{0}, u_{0}^{2}) .

Its mass and quartic eigenvalues are

b^{2}

and

b^{4 - D}

. The off-diagonal term is an additive critical-mass shift: setting the bare

r_{0} = 0

is not generally the critical tuning when

u_{0} > 0

. The mass remains relevant for every

D

; at

D = 2

the two linear eigenvalues coincide and the matrix can have a Jordan block, but both perturbations are still relevant.

For

D > 4

, the quartic interaction is an irrelevant operator, and the Gaussian fixed point attracts weak quartic perturbations after the mass and magnetic field have been tuned. The quartic can nevertheless be a dangerously irrelevant coupling: its positive value stabilizes the ordered phase. For

D < 4

, it is a relevant operator, so the Gaussian description is unstable toward interactions. At

D = 4

it is a marginal operator; first-order perturbation theory alone does not decide its fate.

To settle that borderline case, retain the leading quartic contribution from the second cumulant, at zero external momentum in the local expansion. The two-fast-field part of

V

(u_{0} /4) \int ϕ_{<}^{2} ϕ_{>}^{2}

. Since

⟨ ϕ_{>}^{2} (x) ϕ_{>}^{2} (y) ⟩_{c} = 2 G_{>} (x - y)^{2}

, its connected second cumulant contributes

- \frac{u _{0}^{2}}{16} \int d^{D} x d^{D} y ϕ_{<}^{2} (x) ϕ_{<}^{2} (y) G_{>} (x - y)^{2} .

Keeping the local quartic term and using Parseval identity gives

\int d^{D} y G_{>} (x - y)^{2} = I_{2}

, where

I_{2} = \int_{shell} d^{D} p (2 π)^{- D} [α / (p^{2} + α r_{0})]^{2}

. Thus the one-loop shell quartic renormalization is

u_{0}^{'} = b^{4 - D} [u_{0} - \frac{3}{2} u_{0}^{2} I_{2}] + higher-order and derivative terms .

The coefficient follows from multiplying the quartic density correction

- u_{0}^{2} I_{2} /16

4! = 24

. This local, zero-external-momentum coupling extraction is not a claim that the exact finite-shell effective action retains only its original polynomial; further operators are generated.

More explicitly, for a thin shell

b = e^{d ℓ}

, let

c_{D} = area (S^{D - 1}) / (2 π)^{D}

R = α r_{0} / Λ^{2}

and

g = c_{D} α^{2} Λ^{D - 4} u_{0}

. With increasing length scale

ℓ

, the leading flow is

\frac{d R}{d ℓ} = 2 R + \frac{g}{2 ( 1 + R )} + O (g^{2}), \frac{d g}{d ℓ} = (4 - D) g - \frac{3 g ^{2}}{2 ( 1 + R ) ^{2}} + O (g^{3}) .

This convention has the opposite scale direction to a renormalization-group beta function defined using increasing momentum. On the tuned critical surface at

D = 4

R = O (g)

and

d g / d ℓ = - 3 g^{2} /2 + O (g^{3})

, so weak positive

g

is marginally irrelevant, tending to zero with logarithmic corrections rather than remaining an exactly marginal parameter. Just below four dimensions,

D = 4 - ε

with

0 < ε ≪ 1

, the same calculation yields the Wilson-Fisher fixed point

g_{*} = 2 ε /3 + O (ε^{2})

R_{*} = - ε /6 + O (ε^{2})

. It does not justify extrapolating a small-

ε

expansion to every lower dimension. The ordinary scalar quartic upper critical dimension is therefore

D_{c} = 4.

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 303 / 2 / Solution Created 2026-10-03 Updated 2026-10-05

Absorb

k_{B} T

into the free energy, so the statistical weight is

e^{- F}

. Under the Gaussian fixed point rescaling, invariance of the gradient energy gives the engineering dimension

[ϕ] = \frac{d - 2}{2} = \frac{1}{2} (d = 3) .

The engineering dimensions of

ϕ^{2}, ϕ^{4}, ϕ^{6}

are respectively

1, 2, 3

. Their coupling eigenvalues are

d - [ϕ^{2 m}]

, hence

2, 1, 0

. The quadratic and quartic interactions are relevant operators; the sextic interaction is a marginal operator at this order.

The three steps of the momentum-shell renormalization group are:

Split $ϕ = φ + η$ into slow modes $∣ k ∣ < Λ/ b$ and fast modes $Λ/ b < ∣ k ∣ < Λ$ , and integrate over $η$ .
Rescale $x^{'} = x / b$ to restore the ultraviolet cutoff to $Λ$ .
Rescale the field as $ϕ^{'} (x^{'}) = b^{(d - 2) /2} φ (b x^{'})$ to normalize the gradient energy.

Here

b = e^{s} > 1

, so increasing

s

means flowing toward longer distances. At leading order, the fast field has a Gaussian distribution with covariance

C_{b} (x - y) = \int_{Λ/ b < ∣ k ∣ < Λ} \frac{d ^{3} k}{( 2 π ) ^{3}} \frac{e ^{ik \cdot (x - y)}}{k ^{2} + μ _{0}^{2}}, I_{b} = C_{b} (0) .

This perturbative integration requires

k^{2} + μ_{0}^{2} > 0

on the shell. By the Wick theorem, odd fast-field moments vanish and

⟨ η^{4} ⟩ = 3 I_{b}^{2}

⟨ η^{6} ⟩ = 15 I_{b}^{3}

. Therefore the first term of the cumulant expansion of a coarse-grained free energy is

λ_{0} ⟨(φ + η)^{6} ⟩ = λ_{0} (φ^{6} + 15 I_{b} φ^{4} + 45 I_{b}^{2} φ^{2} + 15 I_{b}^{3}) .

This is the quartic interaction generated by a sextic interaction. Including the rescaling, the requested flow is

g (b) = 15 b λ_{0} \int_{Λ/ b < ∣ k ∣ < Λ} \frac{d ^{3} k}{( 2 π ) ^{3}} \frac{1}{k ^{2} + μ _{0}^{2}} + O (λ_{0}^{2}) .

In spherical coordinates,

I_{b} = (2 π^{2})^{- 1} \int_{Λ/ b}^{Λ} k^{2} / (k^{2} + μ_{0}^{2}) d k

. In particular, for

μ_{0}^{2} = 0

g (b) = \frac{15 λ _{0} Λ}{2 π ^{2}} (b - 1) + O (λ_{0}^{2}), \frac{d g}{d s} = g + \frac{15Λ}{2 π ^{2}} λ + \dots .

The same integration shifts the quadratic coefficient by

90 λ_{0} I_{b}^{2}

before rescaling and adds an irrelevant constant to the free energy. Thus setting the bare quartic coupling to zero does not place the theory on the tricritical critical surface.

To calculate the sextic renormalization-group beta function to second order, let

V = λ_{0} \int d^{3} x (φ + η)^{6}

. The required cumulant expansion is

F_{eff} = F_{0} + ⟨ V ⟩ - \frac{1}{2} (⟨ V^{2} ⟩ - ⟨ V ⟩^{2}) + O (λ_{0}^{3}) .

The subtraction removes disconnected contributions. The term

20 λ_{0} φ^{3} η^{3}

at each vertex supplies the primitive sextic correction. The Wick contractions give

⟨ η (x)^{3} η (y)^{3} ⟩ = 6 C_{b} (x - y)^{3} + 9 I_{b}^{2} C_{b} (x - y) .

The first term contains three propagators connecting the two vertices; its factor

6

is the number of bijections between the three fields at each vertex. The second contains a local tadpole diagram at each vertex and one connecting propagator. The primitive contribution to the free energy is therefore

Δ F_{6} = - 1200 λ_{0}^{2} \int d^{3} x d^{3} y φ (x)^{3} φ (y)^{3} C_{b} (x - y)^{3} .

Expanding the slow fields about a common point gives a negative local sextic correction proportional to

J_{b} = \int d^{3} z C_{b} (z)^{3} = \int \frac{d ^{3} p d ^{3} q}{( 2 π ) ^{6}} \frac{χ _{b} ( p ) χ _{b} ( q ) χ _{b} ( p + q )}{( p ^{2} + μ _{0}^{2} ) ( q ^{2} + μ _{0}^{2} ) (( p + q ) ^{2} + μ _{0}^{2} )} \geq 0,

where

χ_{b}

restricts a momentum to the shell. This connected Feynman diagram has two independent loop momenta. Other Wick contractions also generate lower interactions and tadpole diagrams; they must be included or subtracted consistently when fixing the tricritical critical surface.

To extract the logarithmic tricritical sextic beta function, use normal ordering to remove local tadpoles and tune the quadratic and quartic relevant operators. The massless three-dimensional propagator is

C (z) = 1/ (4 π ∣ z ∣)

. The same three-line contraction over short separations

a < ∣ z ∣ < ba

gives

\int_{a < ∣ z ∣ < ba} d^{3} z C (z)^{3} = \frac{1}{16 π ^{2}} lo g b .

Its positive coefficient and the negative sign of the second cumulant expansion term establish the sign of the flow. Matching this logarithm to a renormalized coupling, or performing consistent iterated shell integration, gives

\frac{d λ}{d s} = - C λ^{2} + O (λ^{3}), C > 0, λ (s) = \frac{λ ( 0 )}{1 + C λ ( 0 ) s} + \dots .

The precise normalization of

C

is not needed. In a sharp momentum cutoff scheme, simply forcing all three internal momenta into one infinitesimal shell does not extract this two-loop logarithm: the shell restrictions remove its linear-in-

d s

phase space. The finite-shell cumulant and the logarithmic renormalized calculation above are distinct stages of the computation. Field normalization corrections at order

λ^{2}

enter this renormalization-group beta function only at order

λ^{3}

A positive, stabilizing sextic coupling is marginally irrelevant in three dimensions on the tricritical critical surface. Reversing the convention for

s

reverses the sign of the renormalization-group beta function. A negative sextic coupling without higher stabilizing powers makes the potential unbounded; moreover, the untuned bare choice

g = 0

generally flows away through its generated relevant operator.

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 304 / 2 / d / Solution 2026-10-03

A continuum limit of a quantum field theory sends the ultraviolet cutoff

Λ_{0}

to infinity while holding chosen physical masses and amplitudes fixed. In cutoff units this requires a diverging correlation length, so the bare couplings must be tuned onto the critical surface of an ultraviolet renormalization-group fixed point. Each relevant direction of a fixed point requires one coordinate fixed by measurement; predictivity requires only finitely many such directions, and irrelevant microscopic details disappear.

Several behaviors are possible. In an asymptotically free theory the trajectory approaches a Gaussian fixed point in the ultraviolet and interactions vanish logarithmically. In an asymptotically safe quantum field theory it approaches a non-Gaussian fixed point with finitely many relevant directions. A theory with a Landau pole may possess only a trivial quantum field theory as its continuum limit; retaining a nonzero interaction then requires a finite cutoff. Finally, if no ultraviolet-complete trajectory exists, the model remains an effective field theory valid only below a physical cutoff.