Renormalization group

A renormalization group compares models at different observation lengths. A Wilsonian coarse-grained statistical Hamiltonian sums over short-distance fluctuations; a renormalization-group transformation rescales coordinates and fields for comparison. Kenneth Geddes Wilson's 1982 lecture treats fluctuations across successive length scales to explain critical phenomena.^[ref]

The experimental motivation is Andrews's carbon dioxide experiment. Dried carbon dioxide occupied a capillary tube above mercury; a screw compressed the transmitting water. A second air-filled tube gauged pressure. Flowing water controlled temperature, and a cathetometer read column positions for volume. The liquid meniscus vanished near the liquid-gas critical point. Residual air and imperfect air-gauge corrections limited accuracy.^[ref] These observations motivate critical-point theory without determining a correlation-length critical exponent. The renormalization group explains how critical measurements can become insensitive to many molecular details.

In 1972, Kenneth Geddes Wilson and Michael Ellis Fisher used an epsilon expansion about $d=4$ to compute nonclassical critical exponents. Their paper distinguishes a Gaussian fixed point from an interacting renormalization-group fixed point, tunes the thermal parameter, and extracts powers from a linearized recursion. Its Ising-like result is $\gamma =1+ϵ/6+O({ϵ}^2)$ for $d=4-ϵ$.^[ref] The worked model below isolates this mechanism with chosen coefficients; it is not a precision calculation for the three-dimensional Ising model.

Consider Ising spins $s_i=\pm 1$ on a square lattice of spacing $a$. Divide it into disjoint $3\times 3$ blocks $B$ and retain the majority sign $S_B$. Each block contains nine spins, so a tie is impossible. For example, five positive and four negative spins produce $S_B=+1$. All $2^9$ configurations are assigned a retained sign; the individual pattern within a block is lost. A coarse-grained variable has fewer possible values than the microscopic configuration it summarizes.

The spacing between block centres is $3a$. In coordinates $x^{\prime }=x/3$ it is again $a$, so the retained pattern can be compared with the original lattice at the same numerical cutoff. This coordinate change does not physically shrink the sample. A majority picture alone does not specify the new interactions: the blocking kernel and original Boltzmann weights determine their probabilities. The figure is a constructed configuration, not a simulated equilibrium state or a claim that majority blocking closes the nearest-neighbour Ising model.

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.

Take the following bounded analytic example for a dimensionless positive coupling $g$, with fixed $ϵ>0$ and $g(0)=g_0>0$:Increasing $\ell$ means increasing observation length $L$, hence motion toward the infrared fixed point. This convention matters: if the inverse-length scale is $\mu ={\mu }_0{e}^{-\ell }$, the same trajectory has renormalization-group beta function ${\beta }_{\mu }(g)=\mu dg/d\mu =-ϵg+g^2$. The two derivatives have opposite signs.

To solve the equation, set $v=1/g$. It obeys $dv/d\ell =-ϵv+1$, a linear ordinary differential equation. ThereforeFor $\ell \ge 0$ its denominator is positive, and $g$ stays between $g_0$ and $ϵ$. Thus the trajectory is bounded on this domain. The renormalization-group fixed points are $g=0$ and $g_{\ast }=ϵ$. The derivative of the length-flow right-hand side is $ϵ-2g$: zero is repulsive from positive $g$, whereas $g_{\ast }$ has eigenvalue $-ϵ$ and attracts positive trajectories. Linearization gives $g-g_{\ast }\propto e^{-ϵ\ell }=b^{-ϵ}$ near $g_{\ast }$. The inverse-length beta function has positive derivative at the same infrared-attractive point.

For the chosen $ϵ=1/5$, take two starting models with $g_0=1/20$ and $g_0=3/5$. At $b=1024$, $e^{-ϵ\ell }=1/4$, so their couplings are respectively $4/35$ and $6/25$. Both move toward $1/5$, from opposite sides. They are still visibly different: slow decay of an irrelevant operator can leave substantial corrections at accessible lengths. The case $g_0=0$ stays exactly zero and is outside this positive-coupling basin.

Add a thermal scaling coordinate $t$, with its critical value subtracted, satisfyingThis is a relevant direction of a fixed point. A thermal scaling coordinate agrees to leading order with a nonzero multiple of reduced temperature; in a general model it can mix several bare parameters. An ordering conjugate field is held at its critical value. The combined stability matrix at $(g,t)=(ϵ,0)$ has eigenvalues $-ϵ$ and $y_t$. Attraction along $g$ therefore does not make the critical point attractive against thermal detuning. The critical surface in this example is $t=0$.

Let $X=\xi /a$ denote the correlation length in microscopic lattice units. A renormalization-group transformation of length factor $b$ givesIndeed, a tail with exponential envelope $\exp (-r/\xi )$ becomes $\exp [-r^{\prime }/(\xi /b)]$ under $r^{\prime }=r/b$. Choose the stopping scale $b_{\ast }=|t_0{|}^{-1/y_t}$ so that the transformed thermal coordinate is $+1$ or $-1$. If the stopped correlation length tends to a finite nonzero value $C_{\pm }$ as the irrelevant coupling approaches $g_{\ast }$, thenThe limit requires $|t_0|\ll 1$ and enough room between $a$, $\xi$, and the sample size. Singular dependence on a dangerously irrelevant coupling or a marginal operator needs separate analysis. On an ordered branch, use the connected correlation function in a pure thermodynamic phase.

For a completely specified illustrative scaling model choose $y_t=3/2$ and $X=C_{+}{t}^{-2/3}$ for $t>0$. Then $t=10^{-3}$ gives $X=100C_{+}$, whereas $t=10^{-6}$ gives $X=10^4{C}_{+}$. With $b=4$, $t^{\prime }=8t$ and $X(t^{\prime })=X(t)/4$. Models with $C_{+}=1$ and $C_{+}=2$ have different lengths but identical curves after plotting $X/C_{+}$ against $t$. This collapse checks the chosen scaling law; agreement of real data would test that law, rather than follow from this example. Neither $ϵ=1/5$ nor $y_t=3/2$ is being fitted to a three-dimensional Ising model.

Near a renormalization-group fixed point, expand a difference between two nearby models in eigenvectors of the stability matrix. An irrelevant operator direction has $y_i<0$, so its dimensionless coefficient shrinks under increasing length factor $b$. A relevant operator direction grows and must be tuned to study the same critical behavior. This is a mechanism for losing microscopic detail, not an assertion that every parameter disappears. Critical temperature, individual amplitudes and regular contributions to free energy can remain material dependent.

For the solvable example, write $A=ϵ/g_0-1$. Its exact solution gives $g-ϵ=-ϵA{b}^{-ϵ}+O(b^{-2ϵ})$. At the thermal stopping scale $b_{\ast }=|t_0{|}^{-1/y_t}$ this is of order $|t_0{|}^{ϵ/y_t}$. If the stopped correlation length is smooth in $g$, the leading correction to its power law has that order. Thus two microscopic couplings can yield the same limiting critical exponent, yet approach it at different rates.

This mechanism makes universality useful: one tractable representative can predict critical exponents and normalized scaling forms for many systems in its universality class. A liquid-gas critical point and a scalar magnetic transition can share a long-distance description even though their microscopic constituents differ. The scope must be checked: spatial dimension, order parameter symmetry, interaction range, and extra soft variables can change the renormalization-group fixed point.

In his lecture, Kenneth Geddes Wilson connects limiting renormalization-group fixed points with common critical behavior and states the limits of his elementary truncation.^[ref]

A zero of the beta function approached by the running coupling as the energy scale increases. A simple one-coupling fixed point with ${\beta }^{\prime }(g_{\ast })<0$ attracts nearby flows. Such a point can control two-point scaling at an ultraviolet fixed point.

If $g(t)\to g_{\ast }$, $\gamma (g(t))\to {\gamma }_{\ast }$ and the reference correlator has a finite nonzero fixed-point limit, the characteristic solution of the multiplicative Callan-Symanzik equation gives $\log f(t)=2{\gamma }_{\ast }t+o(t)$. A simple attractive fixed point and smooth anomalous dimension give a finite prefactor multiplying $e^{2{\gamma }_{\ast }t}$. A nonsimple fixed point can retain slower corrections.

A renormalization-group transformation can scale different coordinate directions by different factors to keep distinct gradient terms fixed. If momentum component $p_i$ scales to $b^{w_i}{p}_i$, its coordinate scales by $b^{-w_i}$. The momentum/volume measure has total scaling weight ${\sum }_i{w}_i$, motivating the anisotropic effective dimension.

For a Gaussian kernel $K{q}_{\parallel }^2+L{q}_{\perp }^4+t$, keeping both gradient coefficients fixed gives $q_{\parallel }^{\prime }=b{q}_{\parallel }$ and $q_{\perp }^{\prime }=b^{1/2}{q}_{\perp }$. The volume rescales by $b^{(d+1)/2}$. Fourier-field normalization is $m^{\prime }(q^{\prime })=m(q)/b^{(d+5)/4}$; the real-space normalization instead divides by $b^{(3-d)/4}$. The thermal coefficient scales by $b^2$ and the uniform source by $b^{(d+5)/4}$. Thus ${\xi }_{\parallel }\sim t^{-1/2}$, ${\xi }_{\perp }\sim t^{-1/4}$ and the formal Gaussian free-energy powers are $\alpha =(7-d)/4$, $\Delta =(d+5)/8$.

The effective dimension for a specified anisotropic rescaling is the total scaling weight of its coordinate measure. With $D_{\perp }$ directions of weight 1 and $D_{\parallel }$ directions of weight $\theta$, $d_{\mathrm{eff}}=D_{\perp }+\theta D_{\parallel }$. The uniaxial Gaussian Lifshitz point has $\theta =1/2$, so $d_{\mathrm{eff}}=D-1/2$. This is not a topological or Hausdorff dimension.

A renormalization-group transformation integrates out selected short-distance degrees of freedom and restores chosen cutoff and field-normalization conventions. It defines a map $R_b$ on couplings at a length factor $b$. Its discrete Jacobian matrix multipliers differ from eigenvalues of the generator of a continuous renormalization-group flow; the relation for a scaling direction is $\lambda =b^y$.

For dimensionless free energy per site $F(u,C)=f(u)+\beta C$, an exact blocking kernel preserves the partition function and gives $F(u,C)=b^{-D}F(u^{\prime },C^{\prime })$. Thus $f(u)=b^{-D}f(u^{\prime })+g(u)$, where $g(u)=\beta [b^{-D}{C}^{\prime }-C]$ is the additive contribution from the eliminated degrees of freedom. Repeating this equality gives the displayed sum. After subtracting analytic backgrounds, the singular free-energy density obeys the homogeneous scaling hypothesis for critical phenomena, subject to marginal or dangerous-variable qualifications.

A blocking kernel assigns a conditional probability to coarse configurations given microscopic configurations. Multiplying each microscopic Boltzmann weight by the kernel and summing over microscopic configurations defines the coarse Hamiltonian. The normalization preserves the partition function exactly when the coarse operator space is not truncated. In $D$ dimensions a blocking factor $b$ changes the lattice spacing to $ba$ and the number of sites to $N/b^D$. Generated interactions and additive constants must be retained for the transformation to remain exact.

A real-space renormalization group replaces groups of microscopic degrees of freedom by retained coarse variables, summing over the eliminated variables to define a coarse Hamiltonian. The effective Boltzmann weights must preserve the partition function up to a tracked normalization. Spin decimation retains selected spins rather than forming an average block variable. Coarse-graining generally generates additional interactions, so closure of a chosen finite coupling family requires justification.

A real-space approximation that moves or combines lattice bonds so that selected intermediate spins can be summed through a chain recursion. On a one-dimensional nearest-neighbour Ising model, no transverse bond moving is needed and exact zero-field Ising-chain decimation closes exactly. In higher dimensions, neglected geometric correlations generally make the transformed model approximate on the original lattice. The order of bond moving and decimation must be stated; the two operations need not commute. The bond-moving-first convention is described in Position Space Renormalization Group by Mehran Kardar lecture 14, section V.D.

In the bond-moving-first hypercubic construction with blocking length $b$, combine $b^{d-1}$ transverse-equivalent bonds into each link of a surviving $b$-link chain. Summing that chain gives the displayed Migdal-Kadanoff approximation. For $b=2$, it becomes $K^{\prime }=\frac{1}{2}\log \cosh (2^{d}K)$. The $d=1$ specialization is exact; the higher-dimensional formula is a convention-dependent approximation, not an exact closure of the original hypercubic model.

A normalized blocking kernel assigns probabilities or delta constraints for coarse-grained variables given a microscopic configuration. Multiplying the microscopic Boltzmann weight by this kernel and summing over eliminated variables defines the blocked statistical Hamiltonian. Normalization preserves the partition function exactly when all generated operators and the field-independent constant are retained. A finite-coupling truncation may lose that exactness.

If an energy per site $C$ multiplies the identity operator, blocking sends $C$ to $b^{D}C+c(u)$. For free energy per site $F=C+f(u)$, exact partition function invariance gives the displayed inhomogeneous equation with $g=b^{-D}c$. It records eliminated-mode entropy, vacuum contributions and measure normalization. Removing a suitable regular background leaves homogeneous singular scaling; additive logarithmic resonances can obstruct a strictly analytic subtraction.

For the renormalization-group transformation $f(u)=b^{-D}f(R_{b}u)+g(u)$, a background solving $a(u)=b^{-D}a(R_{b}u)+g(u)$ leaves a homogeneous remainder. In linear scaling coordinates $t^{\prime }=b^{{\lambda }_t}t$ and $h^{\prime }=b^{{\lambda }_h}h$, an analytic source term $g_{mn}{t}^m{h}^n$ is removed by $a_{mn}=g_{mn}/[1-b^{m{\lambda }_t+n{\lambda }_h-D}]$ unless the denominator vanishes. Such a resonance may produce a logarithm and invalidate a pure homogeneous power law. The scaling hypothesis for critical phenomena concerns the singular remainder. A regular field-dependent background cannot generally be represented by a function of temperature alone; an arbitrary analytic identity term proportional to $h^2$ makes this explicit.

An analytic source monomial $g_{mn}{t}^m{h}^n$ in an additive renormalization-group transformation of free-energy density is resonant when its scaling degree equals the spatial dimension. The analytic background coefficient would require division by $1-b^{m{\lambda }_t+n{\lambda }_h-D}=0$. On a branch with $t\ne 0$ and ${\lambda }_t>0$, a particular nonanalytic background is $-g_{mn}{t}^m{h}^n\log |t|/({\lambda }_t\log b)$: substituting $t^{\prime }=b^{{\lambda }_t}t$, $h^{\prime }=b^{{\lambda }_h}h$ verifies the additive recursion exactly. Thus logarithms can be required even when the eliminated-mode source is analytic. This is distinct from marginal interaction running, which can create additional logarithmic corrections.

An infinite positive coupling coordinate $v$ can be studied through $w=1/v$, with $v=\infty$ represented by $w=0$. For example, $v^{\prime }=v^4/A(u)$ with $A(u)>0$ becomes $w^{\prime }=A(u)w^4$, a smooth map at zero. This gives a well-defined boundary renormalization-group fixed point whose derivative in the reciprocal direction is zero. One should use this chart rather than substituting infinity into an ordinary finite-coordinate Jacobian.

Spin decimation sums over spins at selected lattice sites while retaining the others. For alternate-site elimination in a nearest-neighbour periodic chain of even length, the coarse spin-chain transfer matrix is $W^2$, because its $(s,t)$ entry sums $W_{sa}{W}_{at}$ over the eliminated middle spin. Exact preservation $\mathrm{tr}(W^2{)}^{N/2}=\mathrm{tr}{W}^N$ requires keeping the scalar normalization as well as the entry ratios.

Keep every $b$th spin in a ferromagnetic nearest-neighbour Ising model at zero field, with coupling $K=J/(k_{B}T)$. Multiplying its two-state transfer matrix $b$ times yields the displayed recursion. For $b=2$, direct summation of one intermediate spin gives $2\cosh [K(s+s^{\prime })]=A{e}^{K^{\prime }s{s}^{\prime }}$, with $K^{\prime }=\frac{1}{2}\log \cosh 2K$. The new spacing is $ba$ before coordinate rescaling. All finite positive couplings flow toward the disordered fixed point $K=0$; the ordered zero-temperature endpoint is unstable to a nonzero domain-wall number per bond.

Use the symmetric transfer matrix $T_{s{s}^{\prime }}=\exp [Ks{s}^{\prime }+h(s+s^{\prime })/2]$ with dimensionless field $h$. For blocking length two set $W=T^2$, then write $W_{s{s}^{\prime }}=A\exp [K^{\prime }s{s}^{\prime }+h^{\prime }(s+s^{\prime })/2]$. Matching its three independent entries proves the title relations and gives $A=(W_{++}{W}_{--}{W}_{+-}^2{)}^{1/4}$. Explicitly $W_{\pm \pm }=e^{2K\pm 2h}+e^{-2K}$ and $W_{+-}=2\cosh h$. At zero field the exact chain recursion is recovered; at $K=0$ the independent-spin field stays unchanged. Splitting the field equally between neighboring bonds is essential for this normalization.

A reversed bond costs $2J$, so its Boltzmann factor $w=e^{-2J/(k_{B}T)}$ is a natural thermal coordinate near the zero-temperature chain fixed point. Exact length-two decimation gives $w^{\prime }\simeq 2w$, and the One-dimensional Ising correlation function gives $\xi /a=-1/\log \tanh K\sim 1/(2w)$. This is a length exponent one with respect to $w$. The correlation length instead diverges exponentially in inverse physical temperature; claiming a conventional finite power $T^{-1}$ would confuse scaling coordinates. Choosing $e^{-K}$ gives multiplier $\sqrt{2}$ and power two in that alternative coordinate, the same physical length.

For a periodic even chain, eliminating alternate spins gives $Z_N(K)=A^{N/2}{Z}_{N/2}(K^{\prime })$. Thus dimensionless free energy per original spin obeys $f(K)=-\frac{1}{2}\log A+\frac{1}{2}f(K^{\prime })$. The exact solution $f(K)=-\log (2\cosh K)$ satisfies this relation. The multiplicative constant cancels from normalized retained-spin probabilities but records eliminated-spin entropy; discarding it loses thermodynamics even if the interaction recursion is correct.

Checkerboard spin decimation retains one parity sublattice of a square-lattice Ising model and sums over the other. The retained primitive vectors are $a(1,1)$ and $a(1,-1)$, so the coarse lattice is again square with spacing $\sqrt{2}a$. A rotation and length rescaling restore the original geometry. Every original nearest-neighbour bond joins the two parity classes, whereas a diagonal next-nearest-neighbour bond stays within one class.

For dimensionless nearest-neighbour coupling $K$ and diagonal coupling $L$, the second cumulant in checkerboard decimation of the square-lattice Ising model generates $K^2$ for each pair of retained neighbours of an eliminated centre. A retained diagonal pair shares two centres, and an axial pair at distance $2a$ shares one. The displayed two-coupling recursion neglects $K^4$, $K^{2}L$, and $L^2$, treating $L$ as order $K^2$. Its finite nontrivial renormalization-group fixed point is $(K,L)=(1/3,1/9)$. Higher orders generate additional operators, so the closure is a property of the truncation.

For the spin inversion symmetry parameterization $W=c\left(\begin{array}{ccc}1& x& y\\ x& z& x\\ y& x& 1\end{array}\right)$ with positive entries, spin decimation gives the same form with $x^{\prime }=x(1+y+z)/(1+x^2+y^2)$, $y^{\prime }=(x^2+2y)/(1+x^2+y^2)$ and $z^{\prime }=(z^2+2x^2)/(1+x^2+y^2)$. Multiply $W$ by itself and divide by its $(+,+)$ entry to prove these ratios. The leftover scalar is an additive free-energy coupling; dropping it leaves normalized expectations unchanged but loses the full free energy.

The momentum-shell renormalization group splits a field into slow modes below $\Lambda /\zeta$ and fast modes in $\Lambda /\zeta <|k|<\Lambda$, integrates out the fast modes, then rescales momenta and fields to restore the cutoff and kinetic normalization.

Integrate field modes with momenta between a fixed microscopic ultraviolet cutoff ${\Lambda }_0$ and a lower cutoff $\Lambda$, retaining the modes below $\Lambda$ as backgrounds. The exact statistical Hamiltonian includes all generated interactions and field-independent constants. Integrating its retained modes reproduces the original partition function, so long-distance observables do not depend on the arbitrary intermediate cutoff. With sources retained, the corresponding coarse observables reproduce long-distance response and connected correlation functions. A local gradient expansion or a finite-coupling truncation is an additional approximation, rather than a consequence of cutoff independence. The momentum-shell construction is also explained in the Cambridge statistical field theory lecture notes.

For centered fast Fourier modes and $V=u\int {\varphi }^4/4!$, the two-fast-field vertex is $(u/4)\int {\phi }^2{\eta }^2$. Its connected second cumulant uses $⟨{\eta }^2(x){\eta }^2(y){⟩}_c=2G_{>}(x-y{)}^2$, producing $-u^2\int {\phi }^2(x){\phi }^2(y)G_{>}(x-y{)}^2/16$. Extracting the local zero-external-momentum quartic coefficient gives $u^{\prime }=b^{4-D}(u-3u^2{I}_2/2)+\cdots$, where $I_2={\int }_{\mathrm{shell}}{d}^{D}p(2\pi {)}^{-D}K(p{)}^{-2}>0$. At $D=4$ its sign makes weak positive quartic coupling marginally irrelevant for increasing length scale, after tuning the critical mass. This local truncation omits generated derivative and higher-order operators; it is not the exact complete finite-shell effective action.

For interaction $V=u\int {\varphi }^4/4!$, split $\varphi =\phi +\eta$ into slow and fast Fourier modes, with a centered fast Gaussian measure. The first cumulant expansion term is $⟨V⟩=(u/24)\int [{\phi }^4+6G_{>}(0){\phi }^2+3G_{>}(0{)}^2]$ by Wick theorem. The constant shifts the free energy, the quadratic term gives $\delta r=u{G}_{>}(0)/2$, and there is no gradient correction at this order. Rescaling gives $r^{\prime }=b^2[r+u{G}_{>}(0)/2]+O(u^2)$ and $u^{\prime }=b^{4-D}u+O(u^2)$. The mass shift mixes thermal and quartic perturbations; the interacting critical tuning need not have zero bare mass.

For a centered shell Gaussian measure of a real scalar field with positive shell kernel $K(p)=\kappa p^2+r$, its Gaussian functional integral gives $⟨\tilde{\varphi }(p)\tilde{\varphi }(q)⟩=(2\pi {)}^D{\delta }^{(D)}(p+q)/K(p)$ on the shell. Inverse Fourier transforms therefore yield $G_{>}(x-y)={\int }_{\Lambda /b<|p|\le \Lambda }{d}^{D}p(2\pi {)}^{-D}{e}^{ip\cdot (x-y)}/K(p)$. The kernel must be positive throughout the shell. Translation invariance makes the separation essential; a numerator involving $x$ alone is valid only with $y=0$ or if $x$ is redefined as the separation.

Splitting a scalar field into slow and fast components $\varphi =\phi +\eta$, the Wick contractions in $\lambda ⟨(\phi +\eta {)}^6⟩$ generate $15\lambda ⟨{\eta }^2⟩{\phi }^4$. A vanishing bare quartic coupling is therefore not generally preserved by renormalization-group flow.

Writing the interaction as $V$ and averaging over the fast modes, which have a Gaussian distribution, gives the cumulant expansion $F_{\mathrm{eff}}=F_0+⟨V⟩-\frac{1}{2}(⟨V^2⟩-⟨V{⟩}^2)+\cdots$. The second cumulant selects connected Feynman diagrams.

The engineering dimension of a field, parameter or operator is its dimension under the rescaling that leaves the quadratic theory invariant. Interactions can change it into a full scaling dimension.

An anomalous dimension is the interaction-generated difference between a full scaling dimension and its engineering dimension. For a scalar field, momentum-dependent self-energy corrections require wave-function renormalization and produce ${\Delta }_{\varphi }=(d-2+\eta )/2$.

If ${\varphi }_B=Z_{\varphi }^{1/2}{\varphi }_R$, differentiating a renormalized $n$-point function at fixed bare parameters gives $(\mu {\partial }_{\mu }+\beta {\partial }_g+n{\gamma }_{\varphi })G_{n,R}=0$. With this convention the field scaling dimension at a fixed point is its engineering dimension plus ${\gamma }_{\varphi }$. The critical exponent $\eta$ for scalar two-point decay is $2{\gamma }_{\varphi }$.

Under a length blocking factor $b$, a scalar scaling field transforms as ${\varphi }^{\prime }(x^{\prime })=b^{x_{\varphi }}{\varphi }_{<}(b{x}^{\prime })$. Its critical connected correlation function consequently decays as $r^{-2x_{\varphi }}$. Comparing with $r^{-(D-2+\eta )}$ identifies the anomalous part of the field dimension. The uniform conjugate field has scaling exponent $y_h=D-x_{\varphi }$. Wave-function factors must be defined explicitly before assigning a sign to their logarithmic derivatives.

A renormalization-group flow is the trajectory of effective couplings as the observation scale changes.

A repulsive renormalization-group trajectory leaves a renormalization-group fixed point along its unstable manifold as the coarse-graining length increases. Its tangent directions are relevant operators. The critical surface is instead the stable manifold, whose irrelevant coordinates approach the critical fixed point after relevant fields are tuned. For a discrete step, relevance is determined by multipliers larger than one in magnitude, not merely by their being positive.

A renormalization-group fixed point is a set of couplings left unchanged by coarse-graining. It describes a scale-invariant theory.

An infrared fixed point attracts renormalization-group flows toward low energies or long distances. If it is interacting, it defines a nontrivial scale-invariant quantum or statistical field theory.

The stability matrix is the Jacobian of the beta functions at a renormalization-group fixed point. Its eigenvalues classify perturbations as relevant, irrelevant, or marginal.

The critical surface is the stable manifold of a critical renormalization-group fixed point. Tuning every relevant direction places a theory on this surface.

At the Gaussian fixed point all interaction couplings vanish and scaling dimensions follow from the quadratic theory.

For quadratic dispersion $p^2+t$ with no interacting anomalous dimension, the thermal and uniform-source scaling eigenvalues are $y_t=2$ and $y_h=(d+2)/2$. The generic homogeneous singular free-energy density has $2-\alpha =d/2$ and gap exponent $\Delta =(d+2)/4$. These Gaussian-model values below the upper critical dimension are not the exponents of an interacting scalar critical point.

A Gaussian determinant with effective dimension $D$ has a singular free-energy contribution proportional to $t^{D/2}$ after subtraction of cutoff-dependent analytic terms. When $D/2$ is a positive integer, a logarithm replaces the pure power. Differentiating the determinant sufficiently often gives a marginal momentum integral proportional to $\log t$. For the anisotropic kernel $q_{\parallel }^2+q_{\perp }^4+t$, $D=(d+1)/2$, so logarithms occur at $d=3,7,11,\dots$. These qualify simple homogeneous-power formulas rather than changing the nominal scaling dimensions.

The Gaussian momentum-shell scaling has thermal eigenvalue $2$ and source eigenvalue $(D+2)/2$. The formal homogeneous magnetization index is therefore $[D-(D+2)/2]/2=(D-2)/4$. A free quadratic Hamiltonian alone does not possess a stable negative-mass ordered phase, so this index is not a claim of spontaneous magnetization in an unstabilized Gaussian integral. In $2<D<4$, a quartic interaction is relevant at the Gaussian fixed point and changes the generic scalar critical behavior.

Two thermal derivatives of a regulated Gaussian determinant produce this integral. For $0<D<4$, rescaling $p=\sqrt{t}q$ yields a divergent power $t^{D/2-2}$ and heat-capacity critical exponent $\alpha =(4-D)/2$. At four dimensions the singularity is logarithmic, with power exponent zero. Above four the leading heat capacity is a finite cutoff-dependent background; its leading bounded-power convention has exponent zero, although the background-subtracted singular Gaussian contribution retains the negative index $2-D/2$. In a stable quartic theory the mean-field ordered free energy proportional to $-t^2/u$ instead produces a heat-capacity jump. This is why a dangerously irrelevant coupling qualifies naive hyperscaling relation.

After integrating a free-field momentum shell, restore the cutoff with $x^{\prime }=x/b$ and ${\varphi }^{\prime }(x^{\prime })=b^{(D-2)/2}{\varphi }_{<}(b{x}^{\prime })$. The gradient coefficient is invariant, the mass changes as $r^{\prime }=b^{2}r$, and a uniform conjugate field changes as $h^{\prime }=b^{(D+2)/2}h$. This follows by counting measure, derivative and field factors in the quadratic Hamiltonian. A nonuniform source instead obeys $h^{\prime }(x^{\prime })=b^{(D+2)/2}h(b{x}^{\prime })$, with the source on the right projected onto the retained Fourier modes. The field has engineering dimension $(D-2)/2$ and zero anomalous dimension.

For a stable massive Gaussian field theory, a finite blocking step yields $F(t,h)=F_{\mathrm{shell}}(t)+b^{-d}F(b^{2}t,b^{(d+2)/2}h)$ for free-energy density. The finite-step shell and measure terms are analytic backgrounds. Generic-dimension singular scaling is homogeneous after subtracting those backgrounds; resonances such as Gaussian free-energy logarithm at effective dimension two need additive logarithmic terms.

For a quadratic determinant in two effective momentum dimensions, differentiating the free-energy density with respect to the squared mass $R$ gives a term proportional to $\log ({\Lambda }^2/R)$. Integration gives $F_S\propto R\log R$ after subtracting analytic terms. Its thermal power index is still 1, but a strictly homogeneous pure-power expression misses this logarithm. It is a Gaussian determinant resonance, distinct from interaction-generated logarithms at an upper critical dimension.

Below four dimensions, the scalar quartic theory has an interacting Wilson-Fisher fixed point that controls the Ising universality class.

This is a particular fixed point of the renormalization group, governing the long-distance behavior of a critical phenomenon.

The scalar Wilson-Fisher fixed point is attractive within its critical surface, but has a relevant thermal direction. With $r=m^2{\Lambda }^{-2}$, $\lambda =g{\Lambda }^{-ϵ}$, and $K_D={\Omega }_D/(2\pi {)}^D$, its one-loop equations are $\dot{r}=2r+K_D\lambda /[2(1+r)]$ and $\dot{\lambda }=ϵ\lambda -3K_D{\lambda }^2/[2(1+r{)}^2]$. Their stability matrix at $r_{\ast }=-ϵ/6+O({ϵ}^2)$ and ${\lambda }_{\ast }=16{\pi }^2ϵ/3+O({ϵ}^2)$ has the displayed eigenvalues. Thus an unrestricted claim of infrared attraction is false: the bare mass must be tuned to the critical surface.

The $O(N)$ model has an $N$-component real field and an orthogonal group symmetry. Its standard Landau-Ginzburg interaction is $g(\mathbf{\varphi }\cdot \mathbf{\varphi }{)}^2$; below the upper critical dimension its continuous transition is governed by an $O(N)$ Wilson-Fisher fixed point.

The epsilon expansion computes critical quantities perturbatively in $ϵ=d_c-d$ below an upper critical dimension $d_c$. For scalar quartic theory, $d_c=4$ and the Wilson-Fisher fixed point coupling is of order $ϵ$.

The quartic interaction $g_1{\varphi }_1^4+g_2{\varphi }_2^4+\lambda {\varphi }_1^2{\varphi }_2^2$ includes the Gaussian fixed point, single and double Ising model Wilson-Fisher fixed points, and an O(N) model fixed point with $N=2$. A rotation of two decoupled identical Ising models produces a second coordinate representation of their renormalization-group fixed point.

The orthogonal transformation ${\psi }_{\pm }=({\varphi }_1\pm {\varphi }_2)/\sqrt{2}$ takes $g({\varphi }_1^4+{\varphi }_2^4+6{\varphi }_1^2{\varphi }_2^2)$ to $2g({\psi }_{+}^4+{\psi }_{-}^4)$. These are the same two decoupled Ising models expressed in rotated fields.

The relevance of a perturbation is determined by the eigenvalue of its dimensionless coupling under linearized renormalization-group flow. Positive, negative and zero eigenvalues define relevant, irrelevant and marginal perturbations, respectively.

An operator is relevant at a fixed point when its coupling has positive renormalization-group eigenvalue. Its dimensionless coupling grows under coarse-graining and drives the theory away from the fixed point unless tuned.

A perturbation of a renormalization-group fixed point is relevant if it grows as the coarse-graining length increases. For a continuous flow $d\delta g/d\ell =M\delta g$, an eigenvector of the linearized matrix with eigenvalue $\sigma$ grows as $e^{\sigma \ell }$; a positive real part is relevant, a negative real part is irrelevant and a zero real part requires nonlinear analysis. For a discrete transformation with length factor $b>1$, its Jacobian matrix multiplier $\lambda$ instead gives $\delta g_n={\lambda }^n\delta g_0$: $|\lambda |>1$ is relevant, $|\lambda |<1$ is irrelevant and $|\lambda |=1$ requires nonlinear analysis. For a positive multiplier the scaling exponent is $y=\log \lambda /\log b$. Thus a positive discrete eigenvalue smaller than one is an irrelevant operator direction, while the sign criterion belongs to the continuous generator.

A discrete renormalization group with length factor $b>1$ and positive relevant thermal multiplier ${\lambda }_t>1$ sends $t$ to ${\lambda }_{t}t$ and the correlation length to $\xi /b$. Combining this with $\xi \sim |t{|}^{-\nu }$ gives the displayed correlation-length critical exponent. The corresponding scaling dimension is $y_t=\log {\lambda }_t/\log b=1/\nu$. Relevance of discrete multipliers is determined by magnitude relative to one, unlike the sign criterion for continuous-flow eigenvalues.

An operator is irrelevant at a fixed point when its coupling has negative renormalization-group eigenvalue. Its dimensionless coupling decreases under coarse-graining.

An operator is marginal at linear order when its coupling has zero renormalization-group eigenvalue. Nonlinear terms in its renormalization-group beta function determine whether it is marginally relevant, marginally irrelevant, or exactly marginal.

Systems belong to one universality class when their long-distance renormalization-group flows approach the same fixed point.

Under the scaling hypothesis for critical phenomena, $F_s=A_f|a_{t}t{|}^{D/y_t}{\Phi }_{\pm }(0)$ at zero field. The common free-energy and thermal metric factors cancel from $A_{+}/A_{-}$, leaving a property of the universality class. This concerns the singular part at equal $|t|$, not the ratio of full free energies including arbitrary analytic identity terms. Matching constants in ${\Phi }_{\pm }$ belong to the same scaling solution and cannot be assigned independently.

When the singular free-energy density has $F_s=A_f|a_{t}t{|}^{2-\alpha }{\Phi }_{\pm }(0)$, the same thermal and free-energy metric factors multiply the singular heat capacity above and below the critical point. Their ratio cancels these factors and depends only on the normalized scaling functions of the universality class. This statement concerns singular amplitudes, not regular heat-capacity backgrounds, and needs separate treatment at logarithmic resonances or zero amplitudes. Branch matching constants belong to the fixed-point scaling functions and cannot be chosen as independent nonuniversal parameters.

Ordinary short-range independent bond percolation and site percolation on regular lattices of the same spatial dimension are expected to share their percolation critical exponents and large-scale limiting behavior. Their percolation critical probabilities can differ. Long-range or correlated models can lie in a different universality class. A theorem for one lattice alone does not establish this transfer.

An irrelevant coupling is dangerously irrelevant when setting it to zero makes a scaling function singular or removes the term needed to stabilize the ordered phase.