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Consider two real scalar fields $\mathbf{\varphi }=({\varphi }_1,{\varphi }_2)$ with positive $\lambda ,v$, standard kinetic terms, and the displayed scalar potential. At $h=0$, the circle group rotates one field into the other: this is a global symmetry in field theory, with the same rotation at every point in space. The vacuum manifold is the circle ${\mathbf{\varphi }}^2=v^2$. Choose $(v,0)$ and differentiate the scalar potential twice:The scalar mass matrix assigns a restoring force to the radial displacement, whereas its tangent is one of the Goldstone directions in the scalar mass matrix. Moving around the circle costs no potential energy; moving between neighbouring points in space can still cost gradient energy. Flat angular potential energy therefore means a soft collective excitation, not that every spatial pattern is free.

A positive source $h$ explicitly selects the positive ${\varphi }_1$ direction. The minimum is $(w,0)$, where $w>v$ is the positive solution of $\lambda w(w^2-v^2)=h$. The two eigenvalues of the scalar mass matrix there areFor example, $\lambda =v=1$ and $h=0.15$ give $w\simeq 1.06792$, $m_{\mathrm{radial}}^2\simeq 2.42138$ and $m_{\mathrm{angular}}^2\simeq 0.140459$. The light angular excitation becomes a Goldstone boson as $h\downarrow 0$, while the radial mass stays finite. The source changes both the preferred direction and the radius; expanding about the old point $(v,0)$ at nonzero $h$ would miss that shift.

Use a dimensionless statistical Hamiltonian $\mathcal{H}=H/(k_{B}T)$, so that the partition function is $Z={\sum }_s{e}^{-\mathcal{H}(s)}$. Choose a normalized blocking kernel $B(S\mid s)$ with ${\sum }_{S}B(S\mid s)=1$. For the majority example it is one when every block has its prescribed majority, and zero otherwise. Define the coarse statistical Hamiltonian by the displayed sum. Then an elementary interchange of finite sums provesThe probability of a retained pattern is therefore its original marginal probability. For any observable $A(S)$ of the retained variables,This is the sense in which elimination preserves predictions for retained information. It does not reconstruct every microscopic observable from a block sign.

Next express positions in units of the enlarged spacing: $x^{\prime }=x/b$, where $b>1$. In a continuum version, rescale the retained scalar field as well to restore a chosen gradient energy coefficient; any change in the integration measure contributes to the field-independent term. The result is a new set of dimensionless couplings. Repeating the procedure defines a renormalization-group flow. Additional interactions and the additive constant generally appear, even if the starting statistical Hamiltonian was simple. Keeping only a few terms is an approximation; the equality of partition functions above applies to the complete coarse weight. See also the Wilsonian coarse-grained statistical Hamiltonian for integration over fast Fourier modes.

An order parameter distinguishes thermodynamic phases. For a ferromagnet it is the magnetization; for a fluid near its liquid-gas critical point it can be the density relative to its critical value. In a system with a scalar order parameter, introduce a slowly varying real scalar field $\varphi (x)$ whose spatial average represents that order parameter. The Landau-Ginzburg theory organizes the effective free energy as a local expansion consistent with the symmetry:Here $K>0$ is the stiffness in the gradient energy, and $H$ is the field conjugate to an order parameter. The even powers assume a symmetry $\varphi \mapsto -\varphi$ at $H=0$. Without that symmetry further terms, including a cubic term, may be allowed. The coefficients are smooth functions of temperature and other controls; near a mean-field transition one usually writes $r=a(T-T_0)$ with $a>0$. A positive highest retained even coefficient makes the local potential bounded below.

The partition function is obtained by weighting configurations with $\exp [-\mathcal{F}/(k_{B}T)]$. The Landau approximation replaces this functional average by minimization, taking a uniform $\varphi =M$ in a homogeneous phase. In a symmetric finite system the zero-field average can vanish even below the transition because it averages both signs of $M$. The spontaneous order parameter selects a pure thermodynamic phase by taking the thermodynamic limit before removing the conjugate field:Thus the order parameter measures the ordering of a selected thermodynamic phase, rather than the average over all symmetry-related phases.

Write ${\beta }_T=(k_{B}T{)}^{-1}$ for inverse temperature to distinguish it from the order-parameter critical exponent. A normalized blocking kernel $B_b({\sigma }^{\prime },\sigma )$ is a nonnegative conditional distribution of coarse spins given the fine spins, normalized byFor deterministic blocking it can impose a majority rule or block average, with a Dirac delta replacing the sum constraint for continuous coarse variables. Each block occupies $b^D$ original cells, so the new lattice spacing and number of sites areThe physical volume $N{a}^D$ is unchanged. After $p$ blockings, $a_p=b^{p}a$ and $N_p=b^{-pD}N$.

Define the coarse statistical Hamiltonian by the exact Boltzmann-weight identityThe operator family must include all generated interactions for this renormalization-group transformation to be exact; restricting it to a few couplings is an approximation. The scalar $g(u)$ has energy units and is the contribution per original site that is independent of ${\sigma }^{\prime }$. Restoring the original constant term $NC$ giveshenceThis convention measures $g$ per original site; defining a constant per new site would move a factor $b^D$ into its definition.

The identity-operator contribution to renormalization-group free energy $C_p$ records the configuration-independent normalization generated by the eliminated spins. It includes their contribution to free energy, including entropy and measure factors, and need not be independent of temperature. It has no effect on normalized probabilities of the retained spins, but is indispensable for the total partition function and its temperature derivatives. Iterating its recursion givesThus $N_p{C}_p$ is the original constant $N{C}_0$ plus the accumulated free-energy contributions of the eliminated degrees of freedom.

To obtain the additive free-energy recursion under blocking, use energy-valued free energy per current lattice site,The blocking kernel normalization impliesSubstituting the recursion for $C^{\prime }$ cancels $C$, givingA second step gives $f_0(u_0)=b^{-2D}{f}_0(u_2)+g(u_0)+b^{-D}g(u_1)$, and induction provesFor dimensionless free energy $-\log Z/N$, multiply $F,f_0,g,C$ throughout by ${\beta }_T$.

The singular free-energy density must be distinguished from the exact per-site representative $f_0$. If $a(u)$ is a regular background and $f_s=f_0-a$, thenThus the same displayed inhomogeneous recursion holds with $f_s,g_s$ after subtraction. When a suitable analytic background solves $a(u)=g(u)+b^{-D}a(Ru)$, it leaves a homogeneous singular recursion. Analytic subtraction of an inhomogeneous renormalization recursion explains this separation. A renormalization-group free-energy resonance can obstruct analytic subtraction and produce a logarithm. The original function $g$ originates in the short-distance spins that were summed out, rather than in an extra external interaction.

Near a renormalization-group fixed point $u_{\ast }$, linearize the transformation:An eigenvector of $\mathsf{A}$ with eigenvalue $b^{y_i}$ defines a scaling coordinate $t_i$ with $t_i^{\prime }=b^{y_i}{t}_i$ to leading order. A relevant operator has $y_i>0$: its coupling grows towards large distances, so it must be tuned to reach the fixed point. An irrelevant operator has $y_i<0$ and dies away; a marginal operator has $y_i=0$ and requires higher-order flow analysis. An operator of scaling dimension $x_i$ has $y_i=D-x_i$.

The coarse spins also require a multiplicative normalization. In continuum notation, writewhere ${\varphi }_{<}$ is the retained field and $\eta$ is its anomalous dimension. This restores the normalization of the critical long-distance connected correlation function. Since a uniform field conjugate to an order parameter enters through $\int d^{D}xH\varphi$, it has eigenvalue exponent $y_h=D-x_{\varphi }=(D+2-\eta )/2$.

Let $t$ be the thermal relevant coordinate with exponent $y_t>0$. After regular backgrounds have been removed, the leading singular free-energy density obeysfor a large total scale factor $B=b^p$. Choosing $B=|t{|}^{-1/y_t}$ gives the scaling hypothesis for critical phenomenaTwo temperature derivatives give the heat-capacity critical exponent; one conjugate-field derivative gives the order-parameter critical exponent; two such derivatives give the magnetic-susceptibility critical exponent. Scaling the correlation length gives $\xi (t)=B\xi (B^{y_t}t)$. Consequently,At $t=0$, instead choose $B=|H{|}^{-1/y_h}$; differentiating gives $\delta =y_h/(D-y_h)$. These scaling relations for critical exponents assume that irrelevant variables can be set to their fixed-point values without singular consequences. A dangerously irrelevant coupling or logarithmic correction requires keeping the corresponding dependence.

For the Gaussian field theory, integrate out the Fourier modes in $\Lambda /b<|q|<\Lambda$ and retain ${\varphi }_{<}$. The modes are independent at zero external field, so this Gaussian thinning transformation changes the normalization by a Gaussian determinant and leaves a quadratic statistical Hamiltonian for the retained modes. Restore the cutoff withIn the integrated statistical Hamiltonian, the volume element contributes $b^D$, each gradient contributes $b^{-1}$, and each field contributes $b^{-(D-2)/2}$. Thus the coefficient ${\kappa }^{-1}$ of the gradient energy is unchanged, whileIn particular $y_t=2$, $y_h=(D+2)/2$ and $\eta =0$. The Gaussian critical exponents and the Gaussian order-parameter scaling index therefore give the requested indicesThe sign convention $+h\varphi$ corresponds to the physical conjugate field $H=-h$ and does not affect these powers.

There are physical qualifications to these Gaussian indices. The strictly quadratic Gaussian field theory is stable for positive $m^2$ and has no stable ordered phase for negative $m^2$; its purported spontaneous $\beta$ is therefore the formal Gaussian order-parameter scaling index. A stabilizing quartic term gives an actual ordered phase. Below four dimensions that term is relevant and changes the exponents; above four dimensions it is a dangerously irrelevant coupling and gives the actual mean-field $\beta =1/2$, rather than the Gaussian hyperscaling value.

The heat capacity power can also be checked directly. Apart from constants, the zero-field Gaussian free-energy density isIf $m^2\propto t$, its singular second temperature derivative is controlled by $\int d^{D}q(q^2+\kappa m^2{)}^{-2}$. For $0<D<4$, rescaling $q=\sqrt{\kappa m^2}k$ gives a power $t^{D/2-2}$, as above. At $D=4$ there is a logarithm, and for $D>4$ an analytic background dominates the heat capacity. The negative formal Gaussian $\alpha$ then describes the subleading singular piece, with logarithmic resonances at appropriate even dimensions. This is the Gaussian specific-heat infrared threshold, and it prevents confusing singular indices with the entire observable.

An order parameter distinguishes equilibrium thermodynamic phases. For a scalar magnetic system, a slowly varying coarse-grained scalar field $\varphi (x)$ represents local magnetization, and its uniform expectation $M=⟨\varphi ⟩$ is the order parameter. Under spin inversion symmetry, $\varphi \mapsto -\varphi$ and $h\mapsto -h$. The disordered phase has $M=0$ at $h=0$, while a pure ordered phase has $M\ne 0$. In a finite symmetric system the average still vanishes at $h=0$; spontaneous symmetry breaking means taking the thermodynamic limit before $h\to 0^{+}$ or $h\to 0^{-}$ to select a pure thermodynamic phase.

Landau-Ginzburg theory assigns a local free-energy functionalSpin inversion symmetry forbids odd powers at zero conjugate field; locality and slow variation motivate the gradient expansion. The gradient energy penalizes interfaces and determines long-wavelength response. Integrating $e^{-\mathcal{F}/(k_{B}T)}$ over fields includes fluctuations, whereas the Landau approximation replaces that integral by its stable stationary configuration. For a uniform phase this reduces to minimizing $V(M)$. If $f$ is the equilibrium free-energy density, then $M=-\partial f/\partial h$, so the order parameter is also the thermodynamic response to its conjugate field.

In Landau-Ginzburg theory, a scalar order parameter $M$ distinguishes macroscopic phases: for an Ising model it is the magnetization per site, while near a liquid–gas endpoint it can be a shifted density. An ordered pure thermodynamic phase has a nonzero value even when the conjugate field is removed. For a symmetry-related pair, take the thermodynamic limit before $h\to 0^{\pm }$; a finite-volume symmetric ensemble instead has zero average.

Coarse-grain to a slowly varying real field and use a local, symmetry-constrained free-energy density. A useful convention isFor spin inversion symmetry, odd powers other than the externally imposed source are absent. Take $v>0$ when the quartic coefficient can be negative; for an ordinary transition $u>0$ already stabilizes the quartic truncation. The gradient energy penalizes spatial variations. In the Landau approximation, minimize this functional, usually first over constant fields. The corresponding statistical field theory also integrates fluctuations with weight $e^{-\mathcal{F}/(k_{B}T)}$. The order parameter identifies the phase; its spatial field describes both equilibrium order and its fluctuations.

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.

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 dimensionThe engineering dimensions of ${\varphi }^2,{\varphi }^4,{\varphi }^6$ are respectively $1,2,3$. Their coupling eigenvalues are $d-[{\varphi }^{2m}]$, 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:

Here $b=e^s>1$, so increasing $s$ means flowing toward longer distances. At leading order, the fast field has a Gaussian distribution with covarianceThis perturbative integration requires $k^2+{\mu }_0^2>0$ on the shell. By the Wick theorem, odd fast-field moments vanish and $⟨{\eta }^4⟩=3I_b^2$, $⟨{\eta }^6⟩=15I_b^3$. Therefore the first term of the cumulant expansion of a coarse-grained free energy isThis is the quartic interaction generated by a sextic interaction. Including the rescaling, the requested flow isIn spherical coordinates, $I_b=(2{\pi }^2{)}^{-1}{\int }_{\Lambda /b}^{\Lambda }{k}^2/(k^2+{\mu }_0^2)dk$. In particular, for ${\mu }_0^2=0$,The same integration shifts the quadratic coefficient by $90{\lambda }_0{I}_b^2$ 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 $V={\lambda }_0\int d^{3}x(\phi +\eta {)}^6$. The required cumulant expansion isThe subtraction removes disconnected contributions. The term $20{\lambda }_0{\phi }^3{\eta }^3$ at each vertex supplies the primitive sextic correction. The Wick contractions giveThe 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 thereforeExpanding the slow fields about a common point gives a negative local sextic correction proportional towhere ${\chi }_b$ 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 $C(z)=1/(4\pi |z|)$. The same three-line contraction over short separations $a<|z|<ba$ givesIts 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, givesThe 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-$ds$ phase space. The finite-shell cumulant and the logarithmic renormalized calculation above are distinct stages of the computation. Field normalization corrections at order ${\lambda }^2$ enter this renormalization-group beta function only at order ${\lambda }^3$.

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.

At the Gaussian fixed point, $[{\varphi }_i]=(d-2)/2$, so all three quartic couplings have engineering dimension $d-4[{\varphi }_i]=4-d$. ThusTake $A>0$ and $ϵ>0$, as in the perturbative epsilon expansion. Define $x=A{g}_1/ϵ$, $y=A{g}_2/ϵ$, $z=A\lambda /ϵ$. The renormalization-group fixed point equations becomeIf $z=0$, each of $x,y$ is independently $0$ or $1/36$, giving four renormalization-group fixed points. If $z\ne 0$, subtracting the first two equations givesThe branch $x+y=1/36$ forces $z=1/12$; inserting this in the first equation yields a repeated root $x=y=1/72$. Hence every nonzero-$z$ solution has $x=y$. Then $z=1/8-3x$, andhas roots $x=1/40$ and $x=1/72$. The complete list of six coupled Ising fixed points near four dimensions, in coordinates $(g_1,g_2,\lambda )$, isThe internal symmetries of these renormalization-group fixed points are:

The last point displays the field-rotation equivalence of decoupled Ising theories. With the orthogonal transformation ${\psi }_{\pm }=({\varphi }_1\pm {\varphi }_2)/\sqrt{2}$,It is the same pair of decoupled Ising models written in fields rotated by $45^{\circ }$, but remains a distinct coordinate solution of the stated renormalization-group beta functions. Here $D_4$ denotes the dihedral group of order eight; an alternative convention calls it $D_8$. We have described linear internal transformations preserving the gradient energy; free massless sectors also have constant scalar-field shift symmetries. At $ϵ=0$, the six coordinates coalesce at the Gaussian fixed point.

Surfactants accumulate at an interface and orient their two chemically distinct ends toward their preferred fluid components. The relaxation found in part c lowers the coefficient of the gradient energy, and hence generally lowers the surface tension. If the reduction makes $\tilde{\kappa }<0$, the positive fourth-gradient term proportional to $\gamma$ can stabilize structure at a nonzero wavevector, producing the modulated correlations characteristic of a microemulsion.

For $z=(e^{-i\varphi }\cos (\theta /2),\sin (\theta /2){)}^T$,Thus $i\kappa z^{†}\dot{z}=(\kappa /2)(1+\cos \theta )\dot{\varphi }$. Dropping the total derivative $(\kappa /2)\dot{\varphi }$ and writing $s=\kappa /2$ givesThe leading rotationally invariant gradient energy is $({\rho }_s/2)(\nabla \mathbf{n}{)}^2$, so an effective Lagrangian is