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A free-energy functional assigns a free energy to an entire spatial order parameter configuration rather than only to one uniform value. In Landau-Ginzburg theory, a local derivative expansion gives $\mathcal{A}[m]=\int d^{D}x\{\kappa (\nabla m{)}^2/2+V(m)\}$, with appropriate boundary terms and regular backgrounds. The Landau approximation minimizes this functional; a fluctuating statistical field theory instead integrates its Boltzmann weight over configurations. These procedures need not give identical critical exponents.

Changing the ultraviolet cutoff changes which fluctuations remain explicit. Integrating out a shell of Fourier modes shifts the coefficients of all symmetry-allowed operators and a field-independent constant. Hence the cutoff-dependent coefficients are chosen so that the remaining partition function and long-distance observables reproduce the same microscopic system. This is the Wilsonian coarse-grained statistical Hamiltonian; retaining only the printed operators is a local derivative expansion, not an exact closure.

In a massive symmetric phase, the quadratic long-wavelength inverse two-point function has the form ${\alpha }^{-1}{p}^2+m^2$. Its Landau scalar correlation length satisfies ${\xi }^{-2}=\alpha m^2$ in this approximation. For canonical normalization of a scalar gradient term, set $\varphi =\sqrt{\alpha }\psi$; then the mass and quartic coefficients become $r=\alpha m^2$ and $g_c={\alpha }^{2}g$. With this normalization the infrared mass is the inverse correlation length, $m_R={\xi }^{-1}$, within the massive quadratic or one-loop approximation. More generally the decay rate is a pole mass, whereas the zero-momentum inverse magnetic susceptibility is a curvature mass; an anomalous dimension or nontrivial momentum dependence can distinguish them. In the ordered phase one must first expand about a selected stable equilibrium magnetization, rather than identify a negative quadratic coefficient at the unstable origin with a squared inverse correlation length. Thus the intended mass identification includes a field normalization and a stable-phase qualification.

For momentum-shell renormalization group, split $\varphi ={\varphi }_{<}+{\varphi }_{>}$, retaining $|p|<\Lambda /b$ and eliminating $\Lambda /b<|p|<\Lambda$. DefineTaking a logarithm gives the cumulant expansion of a coarse-grained free energy. For a centered fast Gaussian measure with $G_{>}(0)=⟨{\varphi }_{>}^2⟩$, the first quartic cumulant isThe Wick contractions give a mass shift $\Delta r=g{G}_{>}(0)/2$ and an identity-operator contribution, while leaving the quartic term at this order. The second cumulant produces higher interactions and a quartic correction. After the length and field rescalings $x=b{x}^{\prime }$ and ${\varphi }_{<}(x)=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$, the gradient coefficient remains one, andThe one-loop shell mass renormalization in scalar quartic theory explicitly shows why the critical bare mass is shifted by fluctuations.

Assume short-range interactions, an analytic even local potential, a positive gradient coefficient, and a weak-coupling trajectory in the basin of the Gaussian fixed point. Above four dimensions the quartic interaction has negative engineering dimension and its dimensionless strength decreases under coarse graining; higher local powers and higher derivative terms also decrease. After absorbing analytic short-distance shifts into the coefficients, long-wavelength fluctuations are small compared with the ordered order parameter. The remaining coarse Landau free energy can therefore be evaluated at its stable saddle:The quartic stabilizer must still be retained for the ordered minimum even though it is dangerously irrelevant. Stationarity, $r_{R}M+g_R{M}^3/6=h$, gives the ordinary Landau-Ginzburg theory and its mean-field critical exponents. This derives the saddle approximation from decreasing fluctuation strength under the stated RG assumptions; it does not assume that every exact coarse-grained functional becomes a quartic polynomial. Below four dimensions the quartic coupling grows, long-wavelength loop corrections cease to be small, and the argument fails. At four dimensions the interaction is marginal and generates logarithmic corrections. The tuned sextic theory has the analogous boundary at three dimensions.

Use a consistent Fourier transform pair for the connected correlation function:The printed forward-transform expression integrates over $p$ while leaving $x$ free; that integration variable is an error. The above convention resolves it and produces the loop measure used below. At zero magnetization, $G_c(x)=⟨\varphi (0)\varphi (x)⟩$; otherwise the disconnected product of expectations must be subtracted.

In the inverse-propagator identity used here, the truncated two-point function means the full one-particle-irreducible two-point vertex, including its free quadratic part. This convention differs from using “truncated” for a connected cumulant. To derive the inverse relation, introduce $W[J]=\log Z[J]$, $\phi =\delta W/\delta J$ and the effective action $\Gamma [\phi ]=\int J\phi -W[J]$. Then $\delta \Gamma /\delta \phi =J$ and the chain rule gives ${\Gamma }^{(2)}{W}^{(2)}=1$. Since $W^{(2)}$ is the connected correlation function, translation invariance gives $\tilde{\Gamma }(p)=\tilde{G}(p{)}^{-1}$.

Now use canonically normalized coefficients, denoting the cutoff mass by $m_b^2=m^2(\Lambda ,T)$ and the renormalized infrared mass by $m_R^2=m^2(0,T)$. Choose the free propagator ${\tilde{G}}_0(p)=(p^2+m_R^2{)}^{-1}$ and put $\delta m^2=m_b^2-m_R^2$ in the interaction as a quadratic counterterm. With the Euclidean self-energy convention in which an insertion changes the connected propagator by $-G_0\Sigma G_0$, the perturbative series beginsThe self-energy sums one-particle-irreducible loop insertions; the explicit mass counterterm is kept separate. Repeated insertions generate the geometric Dyson resummation, which explains why reducible chains occur in $G$ but not as independent terms in its inverse. Beyond one loop a momentum-dependent self-energy and wave-function renormalization must also be included.

At one loop the only quartic two-point graph is the tadpole diagram. At a $g{\varphi }^4/4!$ vertex, attaching the two labeled external fields uses $4\cdot 3=12$ choices and contracts the remaining two fields together. The factor is $12/4!=1/2$. Equivalently this is the same quadratic term in the shell cumulant expansion above. ThusIt is positive for $g>0$ and independent of external momentum, so there is no one-loop gradient renormalization. The condition $\tilde{\Gamma }(0)=m_R^2$ implies $\delta m^2+\Sigma (0)=0$, henceThe cutoff is necessary: the unregulated integral is ultraviolet divergent for $D\ge 2$. Using the renormalized mass internally is a self-consistent one-loop organization, not an exact resummation of all critical fluctuations. With nonunit $\alpha$, this formula instead applies to $r=\alpha m^2$, $g_c={\alpha }^{2}g$ after canonical normalization of a scalar gradient term.

To test a linear thermal mass, approach the ordinary transition from the symmetric side and set $R=m_R^2>0$. For $D>2$ the massless tadpole is infrared finite. Subtract its critical value, absorb regular temperature dependence of the coupling into a nonzero thermal coefficient $A$, and write $m_b^2(T)-m_b^2(T_C)=A(T-T_C)+\cdots$. The one-loop critical-mass subtraction giveswhereFor $D>4$, $J_D(0)=c_D{\Lambda }^{D-4}/(D-4)$ is finite, so the bracket tends to a finite constant and $R\propto T-T_C$ is self-consistent. At $D=4$,and the growing logarithm obstructs an asymptotically constant linear coefficient at nonzero coupling. For $2<D<4$, put $k=\sqrt{R}q$ to obtainThe divergent bracket then invalidates the assumed linear thermal mass. The radial critical-mass subtraction in dimensions three to five gives explicit checks of all three behaviors. For $D\le 2$, the massless tadpole subtraction itself is infrared divergent; that failure of the Gaussian expansion does not prove absence of an interacting transition. Therefore the ordinary upper critical dimension is $D_C=4$, with the unmodified Landau-Ginzburg theory asymptotically justified by this test only for $D>4$. Exponents inferred by solving the self-consistent one-loop equation below four dimensions would not be exact interacting exponents.

For a tricritical point the renormalized quartic coefficient must be tuned to zero, and a positive sextic interaction stabilizes the order parameter. The gradient term gives the scalar field engineering dimension $(D-2)/2$. A coefficient of ${\varphi }^6$ therefore has dimension $D-6(D-2)/2=6-2D$, which becomes marginal at $D=3$. This gives $D_C^{\mathrm{tri}}=3$. A direct Ginzburg criterion reaches the same result: long-wavelength variance in a correlation volume iswhile the tricritical saddle has $M^2\sim |r{|}^{1/2}$. Their ratio scales as $|r{|}^{(D-3)/2}$, vanishing only above three dimensions. At three dimensions it is marginal; below three it grows. In contrast the ordinary saddle $M^2\sim |r|/g$ gives a ratio $g|r{|}^{(D-4)/2}$. Both calculations assume the thermal mass is measured relative to its shifted critical value. A sextic interaction generates quartic terms under coarse graining, so tricriticality requires tuning the renormalized quartic scaling field as well as the mass.

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.

The ultraviolet cutoff specifies which fluctuations remain explicit. Changing it integrates out some modes; to preserve long-distance predictions, their effect must be absorbed into the mass, interactions, and other couplings. Thus bare coefficients depend on $\Lambda$. The long-distance inverse connected correlation function has the form $\tilde{\Gamma }(p)\simeq Z^{-1}(p^2+{\xi }^{-2})$. After fixing the gradient normalization, the infrared mass is an inverse correlation length. With the question's unrescaled gradient coefficient, ${\xi }^{-2}=\alpha m^2$ at Gaussian level, so proportionality rather than literal equality is appropriate until that normalization is chosen.

As a concrete momentum-shell renormalization group step, split $\varphi ={\varphi }_{<}+{\varphi }_{>}$ into modes below $\Lambda /b$ and in the shell $\Lambda /b<|p|<\Lambda$, and defineThen rescale coordinates and the field to restore the cutoff and gradient coefficient. A local derivative expansion of $H_{\mathrm{eff}}$ gives new quadratic, quartic, and higher even couplings. If the system has short-range interactions, the field varies slowly, the effective potential is analytic, and remaining fluctuations are small, retaining the leading gradient and lowest stabilizing powers and minimizing the resulting functional gives Landau-Ginzburg theory. Suppressing the omitted fluctuations is the substantive Landau approximation; it is justified sufficiently far from the narrow fluctuation region or above the appropriate upper critical dimension.

The truncated two-point function in this question is the one-particle-irreducible two-point vertex, $\tilde{\Gamma }(p)=\tilde{G}(p{)}^{-1}$. It is the Hessian of the Legendre effective action, not merely the connected two-point function. In perturbation theory,Here ${\tilde{G}}_0$ is the Gaussian propagator about the chosen reference mass, $\delta m^2$ is the mass counterterm or bare-to-reference mass shift, and $\Sigma$ is the interaction self-energy, computed from proper diagrams. Fix the gradient normalization so ${\tilde{G}}_0(p)=(p^2+M^2{)}^{-1}$, and choose the infrared reference mass $M^2=m^2(0,T)$. Then $\delta m^2=m^2(\Lambda ,T)-M^2$ at this order. The mass renormalization condition is $\tilde{\Gamma }(0)=M^2$.

At one loop the only two-point diagram is the quartic tadpole diagram. There are $4\cdot 3$ choices for attaching its two external lines, leaving the other pair contracted; dividing by $4!$ gives its factor $1/2$. It is independent of external momentum, soThe renormalization condition therefore gives the self-consistent one-loop equationUsing $M$ inside the loop is a self-consistent resummation; replacing it by the bare mass gives the same formal first-order expansion away from infrared problems.

Subtract the equation at $T_c$, where $M=0$, and absorb regular temperature dependence of $g$ into the thermal coefficient. Let $\tau =m^2(\Lambda ,T)-m^2(\Lambda ,T_c)\propto T-T_c$ and $I(M)={\int }_{|q|<\Lambda }{d}^{D}q/(2\pi {)}^D(q^2+M^2{)}^{-1}$. ThenFor $D>4$, the integral divided by $M^2$ tends to a finite cutoff-dependent constant, since its small-momentum radial integrand is $q^{D-5}$. Thus $\tau$ is proportional to $M^2$, consistent with the Landau prediction. For $2<D<4$, set $q=Mz$: the correction instead scales as $M^{D-2}$ and dominates $M^2$. At $D=4$ it is proportional to $M^2\log (\Lambda /M)$. Hence a pure linear Landau mass law without logarithmic modification is consistent only aboveFor $D\le 2$, the massless integral itself is infrared divergent, so the subtraction already needs additional care. The sub-four-dimensional self-consistent approximation locates the breakdown of mean-field behavior; it is not an exact calculation of interacting critical exponents.

At a tricritical point the quartic coupling is tuned away and the leading stabilizing interaction is $g_6{\varphi }^6$. At the Gaussian fixed point, the field has engineering dimension $(D-2)/2$, so the sextic coefficient rescales as $g_6^{\prime }=b^{D-3(D-2)}{g}_6=b^{6-2D}{g}_6$. It changes from irrelevant to relevant atThis is the sextic case of the upper critical dimension of an even scalar interaction; marginality at three dimensions allows logarithmic corrections.

Use the usual homogeneous, local, analytic Landau-Ginzburg theory; rotational symmetry and Z2 symmetry alone do not specify locality, analyticity, or the signs of the coefficients. The local derivative expansion contains every rotational scalar with an even total number of fields:The omitted terms include all allowed higher field powers and contracted spatial derivatives. Terms differing by a total derivative are equivalent for bulk behavior with suitable boundary conditions. The displayed gradient energy assumes spatial homogeneity. If only rotations about an origin are imposed, rotationally invariant position-dependent coefficients are also possible.

In the mean-field approximation, take a spatially uniform order parameter and minimize its Landau free energy. For $\gamma >0$, $u>0$ and $r=a(T-T_c)$ with $a>0$, keeping the quadratic and quartic terms givesThe single minimum splits into two minima as the temperature passes below the critical temperature. Choosing one minimum gives spontaneous symmetry breaking of the Z2 symmetry and a continuous phase transition. The order parameter is the expectation in a selected ordered state; averaging equally over both states gives zero. A Z2 symmetry does not guarantee a continuous phase transition: a negative quartic coefficient stabilized by a positive sextic coefficient can instead give a first-order phase transition.

Relaxing the Z2 symmetry permits odd terms. A nonzero linear source generally rounds this ordinary continuous phase transition. A cubic term can instead give a first-order phase transition: for $V(\varphi )=r{\varphi }^2/2+w{\varphi }^3/3+u{\varphi }^4/4$, $u>0$ and $w\ne 0$, equal minima occur atThe order parameter jumps from zero to ${\varphi }_{\ast }$. A continuous phase transition without an exact Z2 symmetry is still possible with additional tuning; odd terms do not logically exclude it.

For the multicritical even Landau potential, set $r={\mu }^2=a(T-T_c)$ and assume an integer $n\ge 2$, ${\lambda }_n>0$ and $\gamma >0$. For $n>2$, all lower even interactions must be absent or tuned to zero to obtain these mean-field critical exponents. The stationary equation isIts nonzero solutions are minima because $V^{\prime \prime }({\varphi }_0)=-(2n-2)r>0$. Thus the order-parameter critical exponent isSubstituting into the minimum free energy density givesThe singular heat capacity per unit volume is $c_s=-T{\partial }_T^2{f}_{\min }$. Smooth variation of the other coefficients does not change the leading power. ConsequentlyFor $n=2$, this means a finite heat capacity jump, with ${\alpha }_{\mathrm{MF}}=0$; for $n=3$, the tricritical point has ${\beta }_{\mathrm{MF}}=1/4$ and ${\alpha }_{\mathrm{MF}}=1/2$. The case $n=1$ would contain only a quadratic potential and would not stabilize an ordered phase.

The Gaussian fluctuation correction near a critical point is more singular than the mean-field approximation contribution whenThis is the upper critical dimension of an even scalar interaction. Equivalently, its coupling has engineering dimension $2n-(n-1)d$, positive below $d_c$. Dominant fluctuations signal breakdown of the mean-field approximation; the Gaussian expression is not then a calculation of the exact interacting heat-capacity critical exponent. At $d=d_c$, equality of the powers allows logarithmic corrections.