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The partition function sums the Boltzmann weights of all statistical configurations. For discrete states $s$ of energy $E_s$, $Z={\sum }_s{e}^{-{\beta }_{\mathrm{th}}{E}_s}$; for a regulated scalar field it is a functional integral, $Z=\int \mathcal{D}\varphi e^{-{\beta }_{\mathrm{th}}H[\varphi ]}$. It normalizes probabilities and generates thermodynamic averages. Its Helmholtz free energy isA quantum version uses $Z=\mathrm{Tr}{e}^{-{\beta }_{\mathrm{th}}\hat{H}}$. In the following identification, $F(M)$ is a constrained free-energy density, so its units match those of a Hamiltonian density rather than a total extensive energy.

Fix a microscopic cutoff ${\Lambda }_0$. Split the field into retained modes ${\varphi }_{<}$ with momenta below $\Lambda$ and eliminated modes ${\varphi }_{>}$ between $\Lambda$ and ${\Lambda }_0$. Define the Wilsonian coarse-grained statistical Hamiltonian exactly byIts coefficients and additive constant depend on $\Lambda$ so that integrating the retained modes gives precisely the original $Z$. With sources and observables carried along, the same construction preserves long-wavelength responses. Thus changing this intermediate ultraviolet cutoff is a change of description, not a change of the physical system. It is advantageous because one can integrate out fluctuations irrelevant to the observable length scale, organize a local gradient expansion, and use renormalization-group flow toward simpler universal fixed-point descriptions. Exact invariance retains all generated operators; truncating them is an approximation.

The quadratic gradient coefficient generally becomes $K_{\Lambda }$, not a constant. Field renormalization sets a chosen kinetic normalization, for example ${\varphi }_R=\sqrt{K_{\Lambda }}\varphi$ when this makes the gradient coefficient one in the energy convention. Sources and observables must be transformed inversely so that their physical meaning is preserved. Under rescaled coordinates $x^{\prime }=x/b$, the critical field scaling isIts anomalous part is the anomalous dimension; it gives $G(r)\sim r^{-2x_{\varphi }}$ and the conjugate-field exponent $y_h=D-x_{\varphi }$. Different conventions for the wave-function factor require corresponding changes in its logarithmic derivative. A fixed physical magnetization must not be confused with a cutoff-dependent dimensionless renormalized field.

To justify the zero-cutoff identification of constrained free energy, work first in a finite periodic volume $V$. Define the physical spatial mean $M=V^{-1}\int \varphi (x)d^{D}x$ and constrain that mean in the microscopic integral:Retain the constant mode $M$ while eliminating every nonzero mode. When $\Lambda$ is below the smallest nonzero box momentum, the remaining energy is $H_{\Lambda }[M]=V{U}_{\Lambda }(M)$. Consequently $e^{-{\beta }_{\mathrm{th}}V{U}_{\Lambda }(M)}=Z_M$ with a consistently fixed, possibly field-independent measure normalization. This provesFor a uniform field, $U_{\Lambda }(M)$ is the Hamiltonian density $\mathcal{H}(\Lambda ,M)$ in the question. The statement assumes that the physical zero mode is retained, all other modes are integrated exactly, field-rescaling factors are undone, and density and additive-normalization conventions agree. Taking the cutoff limit in the finite box before its thermodynamic limit makes this derivation unambiguous. With an applied uniform source, $Z(h)=\int dM{e}^{-{\beta }_{\mathrm{th}}V[F_c(M)-hM]}$ up to the same measure convention; its large-volume saddle minimizes $F_c(M)-hM$.

The Landau approximation further assumes that this potential can be approximated by a regular local polynomial on the homogeneous branches, and that the remaining fluctuations are negligible for the observables sought. It is not an assertion that an exact zero-cutoff free energy is the bare polynomial. In a coexistence region the exact constrained density can be convexified by domains, while a nonconvex Landau branch potential describes local phases and their metastability.

For the breakdown criterion use a short-range scalar theory with positive quadratic gradient stiffness. At the Gaussian fixed point, invariance of $\int (\nabla \varphi {)}^2{d}^{D}x$ gives field dimension $(D-2)/2$. A coupling $g_{2n}$ multiplying ${\varphi }^{2n}$ therefore has engineering scaling exponentIt is irrelevant only above its upper critical dimension. For the ordinary critical point the leading stabilizing interaction is quartic, while at a tricritical point the renormalized quartic term is tuned away and the leading interaction is sextic. ThereforeOne can also see the failure through the Ginzburg criterion. Long-wavelength fluctuations averaged over a correlation volume obey $⟨(\delta M{)}^2{⟩}_{\xi }\sim {\xi }^{2-D}$, up to temperature and stiffness factors. The quartic saddle has $M^2\sim {\xi }^{-2}$, giving a fluctuation/mean-square ratio proportional to ${\xi }^{4-D}$. The tricritical sextic saddle has $M^2\sim {\xi }^{-1}$, giving a ratio proportional to ${\xi }^{3-D}$. Both ratios grow without bound below their respective critical dimension, invalidating the fluctuation-free Landau exponents sufficiently close to a transition.

At equality the ratios are marginal, not power-law divergent. The loop expansion resolves this boundary: the quartic vertex correction contains $\int d^{D}q/(q^2+{\xi }^{-2}{)}^2$, logarithmic at $D=4$; the sextic vertex correction contains a two-loop three-propagator integral whose overall degree is $2D-6$, logarithmic at $D=3$. Marginal running therefore generally adds logarithmic corrections at these dimensions. Below them, where the corresponding transition exists, interactions control the long-distance critical behavior. Above them, leading thermodynamic powers are mean-field, although stabilizing dangerously irrelevant couplings explain why naive hyperscaling need not hold.

A partition function sums the Boltzmann weights of all states: $Z={\sum }_s{e}^{-{\beta }_{T}H(s)}$ for discrete states, or the corresponding integral for continuous fields. It normalizes equilibrium probabilities. Its physical Helmholtz free energy is $\mathcal{F}=-k_{B}T\log Z$, and its free-energy density is $\mathcal{F}/V$. With a momentum ultraviolet cutoff ${\Lambda }_0$, a scalar-field example isThe normalization of the functional measure and field-independent energy terms must be tracked when discussing absolute free energy.

Choose a smaller cutoff $\Lambda$ and split the field into retained modes ${\varphi }_{<}$ with $|q|<\Lambda$ and eliminated modes ${\varphi }_{>}$ with $\Lambda <|q|<{\Lambda }_0$. Define the Wilsonian coarse-grained statistical Hamiltonian bySubstitution into the partition function proves $Z=\int \mathcal{D}{\varphi }_{<}{e}^{-{\beta }_T{H}_{\Lambda }[{\varphi }_{<}]}$ for every intermediate cutoff. A source retained in this definition similarly preserves long-wavelength connected correlation functions and response. Thus the coefficients in a Hamiltonian density must run with $\Lambda$: the dependence induced by eliminated modes compensates the dependence of the remaining field integral. Holding the original bare coefficients fixed while changing the cutoff would generally change the system. Exact integration generates all interactions allowed by symmetry, including higher powers, derivative terms and an additive constant; a local derivative expansion or a truncation to a few couplings is an extra approximation.

Cutoff independence makes arbitrary intermediate coarse-graining scales unphysical. It allows calculations at the scale of the observable, avoids repeatedly resolving irrelevant short-distance details and exposes the renormalization-group fixed points responsible for universality. A microscopic cutoff, such as a Bravais lattice spacing, can still label a physical microscopic model; taking it to infinity as a continuum limit requires tuning appropriate bare couplings. This is distinct from lowering an intermediate Wilsonian cutoff in a fixed model.

To interpret the zero-cutoff limit, define $M=V^{-1}\int \varphi (x)d^{D}x$, the spatially uniform mode of the order parameter. Keep this mode fixed and integrate the others, rather than integrating over $M$ too. Write $\varphi =M+\eta$, with $\int \eta =0$, and define the constrained order-parameter free energy byIn a finite periodic box, once $\Lambda$ is below the smallest nonzero momentum, all nonconstant modes have been eliminated. The remaining effective energy is $H_{\Lambda }[M]=V{U}_{\Lambda }(M)$, so its defining integral is exactly $Z_M$. Therefore, with consistent normalization,When $\mathcal{H}(\Lambda ,M)$ denotes this uniform effective Hamiltonian density, this is the stated identification of $F(M)$ with its zero-cutoff limit. It requires a free-energy density constrained at fixed $M$, not the unconstrained extensive free energy; all integrated fluctuations and additive constants are included, the zero mode in field theory is retained, and a density for the uniform configuration is well defined. The finite-volume argument avoids an interchange of the thermodynamic and cutoff limits without justification. On a homogeneous pure-phase branch one may use a local potential description, while a thermodynamic limit allowing phase separation can convexify the constrained potential. A bare nonconvex Landau polynomial is accordingly a homogeneous-branch approximation, not a claim that the exact equilibrium Legendre potential stays nonconvex.

In a field $h$, the remaining integral is $Z(h)=\int dM{e}^{-{\beta }_{T}V[F_c(M)-hM]}$, up to the consistently chosen zero-mode measure. The thermodynamic saddle then gives $f(h)={\inf }_M\{F_c(M)-hM\}$ and selects the equilibrium magnetization. Landau theory assumes that the relevant uniform potential can be approximated by a regular analytic expansion in $M$, with symmetry-constrained coefficients, and minimizes it. At tree level the integrated fluctuations are neglected, recovering the familiar bare polynomial. With fluctuations retained, the coefficients run and can become singular near criticality; the zero-cutoff identification of constrained free energy alone does not justify mean-field critical exponents.

For an ordinary short-range scalar thermodynamic critical point, the leading interaction is quartic. Absorb inverse temperature and the gradient coefficient into field normalization and writeScale $x=b{x}^{\prime }$ and $\varphi (x)=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$ to leave the gradient term invariant. The quartic term changes to $g{b}^{4-D}\int {\varphi }^{\prime 4}/4!$, so its engineering scaling exponent is $4-D$. It is irrelevant at the Gaussian fixed point for $D>4$, relevant for $D<4$, and marginal for $D=4$. HenceThe loop expansion makes the breakdown concrete. A quartic-vertex correction is proportional to $g^2{I}_2(r)$, whereFor $0<D<4$, rescaling $q=\sqrt{r}k$ gives $I_2(r)\sim C_D{r}^{(D-4)/2}$, diverging as $r\to 0$. At $D=4$, radial integration giveswhich is logarithmically divergent. Above four dimensions its infrared limit is finite; the leading cutoff-dependent constant can be absorbed into a coupling renormalization. The effective expansion parameter is thus of order $g{\xi }^{4-D}$ below four, or $g\log (\Lambda \xi )$ at four, since $\xi \sim r^{-1/2}$. It eventually becomes large for any fixed bare interaction in the lower-dimensional critical limit, defeating the uncorrected mean-field expansion. The Ginzburg criterion gives the same threshold by comparing correlation-volume fluctuations with the squared mean order parameter.

Below four dimensions an interacting renormalization-group fixed point can replace the Gaussian fixed point and change critical exponents, provided a continuous transition exists. At four dimensions the interaction is marginal and produces logarithmic corrections: the leading power exponents can remain their mean-field values, so “breakdown” there means that pure mean-field power laws omit those corrections, rather than necessarily that every power exponent changes. Above four the stabilizing quartic interaction is dangerously irrelevant for the ordered phase, explaining why mean-field thermodynamic exponents need not obey naive hyperscaling with the spatial dimension.