Relevant, Irrelevant, and Marginal Operators

Iterate the renormalization group and something remarkable happens to the space of all possible couplings: it maps onto itself. Coarse-grain a magnet by a factor $b$ , rescale, and the nearest-neighbour coupling, the next-nearest-neighbour coupling, the four-spin coupling — every term you could add to the Hamiltonian — flows to a new value. Near a fixed point this flow becomes linear, and the natural objects are its eigenvectors: special combinations of couplings that are simply multiplied by a number under one step. That number is $\Lambda_i = b^{y_i}$ , and the single quantity $y_i$ decides the fate of the coupling. The sign of $y_i$ sorts every operator into three classes. Relevant operators ( $y_i>0$ ) grow under coarse-graining; they drive the system away from criticality and so must be tuned to zero to sit exactly at a critical point. Irrelevant operators ( $y_i<0$ ) shrink to nothing; whatever value they start with, the large-scale physics forgets them. Marginal operators ( $y_i=0$ ) sit on a knife's edge, their fate decided not by the linear flow but by its curvature. This classification is the engine of critical phenomena. It explains why a uniaxial ferromagnet, a liquid near its boiling point, and a binary alloy share identical critical exponents — their microscopic differences are irrelevant couplings, washed out under the flow. It tells the experimentalist exactly how many knobs to turn to see criticality: one for each relevant operator. And it predicts the logarithmic corrections that appear at the upper critical dimension, where the quartic coupling turns marginal. Master this trichotomy and the whole edifice of universality — once a bewildering empirical fact — becomes a statement about the eigenvalues of a matrix.

Key takeaways

Near a fixed point the RG flow linearizes; each eigenoperator has eigenvalue $\Lambda_i=b^{y_i}$ that tells whether its scaling field grows ( $y_i>0$ , relevant), shrinks ( $y_i<0$ , irrelevant), or stays put ( $y_i=0$ , marginal) under a rescaling by $b$ .
Relevant operators are the parameters you must tune to reach criticality; their number equals the codimension of the critical surface (two for the ordinary Ising class: temperature and field).
Irrelevant operators flow to zero and leave universal exponents untouched — the microscopic origin of universality — but set the corrections to scaling.
Marginal operators ( $y_i=0$ ) have their fate decided by nonlinear flow; the $\phi^4$ coupling at $d=4$ is the textbook case, producing logarithmic corrections.
The two relevant exponents $y_t,y_h$ generate every standard critical exponent, e.g. $\nu=1/y_t$ and $\delta=y_h/(d-y_h)$ .
Key idea: Diagonalize the linearized RG about a fixed point; the eigenvalue $b^{y_i}$ sorts couplings into relevant (must be tuned), irrelevant (forgotten — the source of universality), and marginal (decided by nonlinear flow, giving logarithms).

The core insight

Diagonalize the linearized RG about a fixed point; the eigenvalue $b^{y_i}$ of each eigenoperator sorts couplings into relevant (must be tuned), irrelevant (forgotten — the source of universality), and marginal (decided by nonlinear flow, giving logarithms).

Where this lives today

The classification of operators into relevant, irrelevant, and marginal has applications far beyond magnetism. In high-energy physics it is exactly the statement of which terms in an effective field theory matter at low energy: renormalizable interactions are relevant or marginal, and the suppression of irrelevant operators by inverse powers of a large energy scale explains why low-energy predictions of the Standard Model are largely insensitive to unknown physics near the Planck scale. The same counting classifies which perturbations destabilize a topological phase, which anisotropies matter in a quantum critical metal, and which lattice artifacts must be extrapolated away in a Monte Carlo simulation. In practice, the leading irrelevant exponent $\omega=-y_{\text{irr}}$ is now a quantity to be measured to high precision. Conformal-bootstrap and Monte Carlo studies of the 3D Ising class quote $\omega\approx0.83$ , and fitting the associated Wegner corrections is essential to extracting the leading exponents accurately from finite systems. The The first question in analyzing a new critical or quantum-critical point is often simply: how many relevant operators are there?

The physical picture

Picture the renormalization-group flow as a vector field filling the space of all couplings. A fixed point is a stationary point of that field. Sit exactly on it and nothing changes under coarse-graining; nudge off it and you drift. Some perturbations grow under coarse-graining, while others shrink. The RG eigenvalues quantify these growth or decay rates under each rescaling by a factor $b$ . The scaling exponent $y$ measures how rapidly a perturbation grows or decays under successive coarse-graining steps. A relevant operator points along a repelling (unstable) direction. A tiny perturbation along it is magnified at every step until the system runs off to a high-temperature paramagnet or a low-temperature ordered sink. To stay on the critical surface you must set these directions to zero — that act of setting them to zero is "tuning to criticality." An irrelevant operator points along an attracting direction: no matter how you start it, it decays toward the fixed point as you zoom out. The intuition figure below makes the trichotomy concrete: a single sorted spectrum of $y_i$ , coloured by sign, with the relevant band on the right (must be tuned), the marginal knife-edge at zero, and the irrelevant band on the left (forgotten). The whole of universality is contained in reading off how many bars poke above $y=0$ .

Figure

The three classes of operator, sorted by RG eigenvalue

A marginal operator sits where the linear flow neither grows nor decays. Whether it eventually grows or shrinks is decided by the curvature of the flow — the quadratic term the linearization threw away. Because that flow is slow (the coupling changes like $1/\ln L$ rather than a power of $L$ ), marginality is the unmistakable fingerprint of logarithmic corrections to pure power-law scaling. The quartic coupling of $\phi^4$ theory at exactly four dimensions is the canonical example. Dimensional analysis gives a quick first guess for $y_i$ . An operator $\mathcal{O}$ of scaling dimension $\Delta$ enters the Hamiltonian with a coupling of engineering dimension $[\text{length}]^{\Delta-d}$ , and one RG step multiplies that coupling by $b^{d-\Delta}$ . So at tree level $y = d - \Delta$ : an operator is relevant when its scaling dimension falls below the spatial dimension. Interactions correct this naive counting through anomalous dimensions, but the sign is usually already fixed by dimensional analysis. The second figure presents the same classification geometrically: relevant directions flow away from the fixed point, irrelevant directions flow toward it, and the critical surface is the codimension-two manifold attracted to the fixed point under the RG flow.

Figure

Relevant directions leave the fixed point; irrelevant directions return to it

The core equations

Everything begins with the observation that the RG map $R_b$ has a fixed point $H^*$ and that near it we may linearize. Write the Hamiltonian as the fixed point plus a small deviation, apply the map, and keep only first order. The linear operator $\mathbf{L}$ that results is the object we diagonalize.

R_b(H^* + \delta H) = H^* + \mathbf{L}\,\delta H + \mathcal{O}(\delta H^2) \qquad (1)

where: $R_b$ — RG transformation with spatial rescaling factor $b>1$ ; $H^*$ — fixed-point Hamiltonian obeying $R_b H^*=H^*$ ; $\delta H$ — small deviation from $H^*$ ; $\mathbf{L}$ — linearized RG operator, a matrix acting on coupling space. Reading it physically: the fate of any perturbation is governed entirely by the spectrum of $\mathbf{L}$ , so the problem reduces to an eigenvalue problem. The eigenoperators of $\mathbf{L}$ are the special combinations that transform simply. Each carries an eigenvalue $\Lambda_i$ , and the crucial fact — forced by the composition law of RG steps — is that $\Lambda_i$ is a power of $b$ .

\mathbf{L}\,\mathcal{O}_i = \Lambda_i\,\mathcal{O}_i, \qquad \Lambda_i = b^{\,y_i} \qquad (2)

where: $\mathcal{O}_i$ — eigenoperator (eigenvector of $\mathbf{L}$ ); $\Lambda_i$ — RG eigenvalue, the growth factor per step; $y_i$ — RG exponent, independent of $b$ ; $b$ — rescaling factor. The exponent $y_i=\ln\Lambda_i/\ln b$ is the invariant intrinsic to the fixed point; $\Lambda_i$ merely records it in the units of a particular $b$ . Expanding the perturbation in the eigenoperator basis decouples the linearized flow: each scaling field $g_i$ evolves independently according to its own RG eigenvalue.

\delta H = \sum_i g_i\,\mathcal{O}_i \;\xrightarrow{R_b}\; \sum_i \big(b^{y_i} g_i\big)\,\mathcal{O}_i \qquad (3)

where: $g_i$ — scaling field conjugate to $\mathcal{O}_i$ ; $b^{y_i}g_i$ — the field after one step. The sign of $y_i$ now sorts the couplings once and for all:

\begin{cases} y_i>0 & \text{relevant (grows)}\\ y_i<0 & \text{irrelevant (shrinks)}\\ y_i=0 & \text{marginal} \end{cases} \qquad (4)

A quick estimate of $y_i$ comes from dimensional analysis alone. An operator of scaling dimension $\Delta_i$ has RG exponent

y_i = d - \Delta_i \qquad (5)

where: $\Delta_i$ — scaling dimension of $\mathcal{O}_i$ ; $d$ — spatial dimension. An operator is relevant when $\Delta_i<d$ , irrelevant when $\Delta_i>d$ , and marginal when $\Delta_i=d$ . Interactions shift $\Delta_i$ by anomalous dimensions, but the tree-level sign is usually decisive. When $y_i=0$ the linear term in the flow vanishes and the leading behaviour comes from the neglected nonlinear term. Introducing the logarithmic RG scale $\ell=\ln b$ , the flow equation becomes:

\frac{\mathrm{d}g}{\mathrm{d}\ell} = y\,g + a\,g^2 + \cdots \qquad (6)

where: $\ell=\ln b$ — logarithmic scale; $y$ — linear exponent ( $=0$ for a marginal operator); $a$ — leading nonlinear coefficient whose sign decides marginal relevance or irrelevance. This equation is what separates a marginal operator from the other two classes: for them the linear term wins and $g$ grows or decays as a power of length; for a marginal operator only the $a\,g^2$ term survives, and its integral is logarithmic.

Derivation AWhy the RG eigenvalue must be a power of the rescaling factor

The claim $\Lambda_i=b^{y_i}$ looks like a definition, but it is forced on us by a structural property of the renormalization group: coarse-graining is a semigroup. Composing a step by $b_1$ with a step by $b_2$ must give exactly the same result as a single step by $b_1 b_2$ — there is no free choice. We show that this composition law, applied to the eigenvalues of the linearized map, admits only power-law solutions, and that the exponent $y_i$ that results is the same no matter which $b$ we use. This is what makes $y_i$ — not the $b$ -dependent $\Lambda_i$ — the physically meaningful number.

Step 1. Two successive coarse-grainings compose:

R_{b_1}\circ R_{b_2} = R_{b_1 b_2}.

This is a semigroup, not a group — there is no inverse, because information about short distances is genuinely discarded and cannot be recovered.

Step 2. Linearize each map about the common fixed point and diagonalize in the shared eigenbasis. For a single eigenoperator the composition law descends to its eigenvalue:

\Lambda_i(b_1)\,\Lambda_i(b_2) = \Lambda_i(b_1 b_2).

Why the eigenbasis is shared: the family of linearized RG transformations satisfies the same composition law, so the operators commute and can therefore be simultaneously diagonalized.

Step 3. The only continuous function satisfying $f(b_1 b_2)=f(b_1)f(b_2)$ is a power law:

\Lambda_i(b) = b^{\,y_i}, \qquad y_i = \frac{\ln \Lambda_i}{\ln b}.

The exponent $y_i$ is manifestly independent of the choice of $b$ : a step by $b^2$ gives $\Lambda_i^2=(b^{y_i})^2=(b^2)^{y_i}$ , the same $y_i$ . One can check directly: the multiplicative property $\Lambda(b_1 b_2)=\Lambda(b_1)\Lambda(b_2)$ holds for any $b$ , and $y=\ln\Lambda/\ln b$ returns the same value for $b\in\{2,3,5\}$ , confirming that $y_i$ is independent of the choice of rescaling factor.

Go deeper · Derivation B · 1 minCounting relevant operators: the codimension of the critical surface

Derivation BCounting relevant operators: the codimension of the critical surface

The number of relevant scaling fields has direct physical meaning: it equals the number of independent parameters that must be tuned to reach the critical fixed point. We show that this count equals the codimension of the critical surface in coupling space, i.e. how many constraints define the manifold of Hamiltonians that flow into the critical fixed point. The argument is a one-line integration of the linear flow followed by the demand that the system actually reach the fixed point; everything then rests on the sign of each $y_i$ .

Step 1. In logarithmic time the linearized flow of each scaling field is

\frac{\mathrm{d}g_i}{\mathrm{d}\ell} = y_i\,g_i \;\Rightarrow\; g_i(\ell)=e^{y_i\ell}\,g_i(0).

Each field grows or decays exponentially in $\ell=\ln b$ at linear order.

Step 2. To reach the critical fixed point we require $g_i(\ell\to\infty)\to 0$ for every $i$ . Irrelevant fields ( $y_i<0$ ) satisfy this automatically. Relevant fields ( $y_i>0$ ) blow up unless their initial value is zero:

y_i>0 \;\Rightarrow\; g_i(0)=0.

Step 3. Each relevant scaling field imposes one independent tuning condition. Together these conditions define the critical surface. Its codimension is therefore

\operatorname{codim}(\text{critical surface}) = \#\{i:\,y_i>0\}.

Why this is universality's home: any Hamiltonian on the surface, no matter how it differs in irrelevant couplings, flows to the same fixed point and inherits the same exponents.

Step 4. For the ordinary Ising universality class there are exactly two relevant fields — the thermal field $t$ and the magnetic field $h$ — so the critical surface has codimension two. In the $(T,H)$ plane this means the critical point is reached by tuning $T=T_c$ at $H=0$ ; the symmetry $h\to-h$ fixes $H=0$ automatically, leaving temperature as the single laboratory knob. As a concrete check: for the toy spectrum $\{2,\,15/8,\,-0.8,\,-2.1,\,0\}$ the relevant count is $\#\{y_i>0\}=2$ , matching the two-parameter tuning of the ordinary critical point.

Go deeper · Derivation C · 1 minMarginal irrelevance and logarithmic corrections at the upper critical dimension

Derivation CMarginal irrelevance and logarithmic corrections at the upper critical dimension

When $y=0$ the linear flow gives no information — the coupling neither grows nor decays at first order — so we must keep the next term. We integrate the marginal flow $\mathrm{d}g/\mathrm{d}\ell=a g^2$ and show that its solution decays only as $1/\ell$ , which in real space is $1/\ln L$ . This slow decay is the mechanism behind the logarithmic corrections that decorate mean-field power laws exactly at the upper critical dimension $d_c=4$ , where the $\phi^4$ coupling is marginal.

Step 1. Set $y=0$ in the flow equation; the quadratic term dominates:

\frac{\mathrm{d}g}{\mathrm{d}\ell} = a\,g^2.

Step 2. Separate variables and integrate:

\frac{1}{g(\ell)} = \frac{1}{g(0)} - a\,\ell.

For $a<0$ (equivalently, writing $\mathrm{d}g/\mathrm{d}\ell=-|a|g^2$ ) the coupling decays: $g(\ell)=g(0)/(1+|a|g(0)\ell)$ . This is marginally irrelevant. For $a>0$ it would diverge at finite $\ell$ : marginally relevant.

Step 3. At large $\ell$ the decay is algebraic in $\ell$ but only logarithmic in length, since $\ell=\ln b$ :

g(\ell) \sim \frac{1}{|a|\,\ell} = \frac{1}{|a|\,\ln L}.

Why this makes logs: thermodynamic functions pick up factors of the running coupling, and $1/\ln L$ dependence translates into powers of $\ln|t|$ near $T_c$ .

Step 4. For the single-component $\phi^4$ theory at $d=4$ the one-loop flow is $\mathrm{d}u/\mathrm{d}\ell=-\tfrac{9}{8\pi^2}u^2$ , giving $u(\ell)=u_0/\big(1+\tfrac{9}{8\pi^2}u_0\ell\big)\to0$ . The susceptibility then carries a logarithmic correction $\chi\sim|t|^{-1}\,|\ln|t||^{1/3}$ , the exponent $1/3=(n+2)/(n+8)$ at $n=1$ . One can verify directly: the marginal solution $g=g_0/(1+a g_0\ell)$ satisfies $\mathrm{d}g/\mathrm{d}\ell=-a g^2$ and has large- $\ell$ asymptote $g\sim 1/(a\ell)$ ; likewise, the $d=4$ one-loop form with coefficient $9/8\pi^2$ solves $\mathrm{d}u/\mathrm{d}\ell=-(9/8\pi^2)u^2$ .

Limiting cases

The quartic coupling at the Gaussian fixed point has RG exponent $y_u=4-d$ , making its classification immediate in different spatial dimensions:

Case	Result	Meaning
$d>4$ (above the upper critical dimension)	$y_u = 4-d < 0$	The quartic coupling is irrelevant; the Gaussian fixed point is stable and mean-field exponents govern criticality.
$d=4$ (the upper critical dimension)	$y_u = 0$	$u$ is marginal; nonlinear flow $\mathrm{d}u/\mathrm{d}\ell\sim-u^2$ makes it marginally irrelevant, giving logarithmic corrections to mean-field exponents.
$d<4$	$y_u = 4-d = \epsilon > 0$	$u$ becomes relevant at the Gaussian point; the flow runs off to the Wilson–Fisher fixed point, where $u$ is irrelevant again and exponents pick up $\epsilon$ -corrections.
Purely dimensional operator, $\Delta_i>d$	$y_i=d-\Delta_i<0$	Irrelevant by power counting alone — no tuning needed, only corrections to scaling.

Worked examples

Example 1 — Classify an operator and read off its correction exponent. Suppose that at some fixed point an operator has RG eigenvalue $\Lambda = b^{-1.2}$ measured with rescaling factor $b=2$ . Is it relevant, and what does it do to the critical behaviour?

Step 1. Extract the $b$ -independent exponent: $y = \ln\Lambda/\ln b = \ln(2^{-1.2})/\ln 2 = -1.2$ . Since $y<0$ , the operator is irrelevant.

Step 2. Because the operator is irrelevant, its scaling field decays under coarse-graining, so no tuning is required to reach the fixed point. It can modify only subleading corrections, not the leading critical exponents.

Step 3. Its only effect is a correction to scaling with exponent $\omega=-y=\mathbf{1.2}$ : thermodynamic functions carry subleading terms like $|t|^{\omega\nu}=|t|^{1.2\,\nu}$ . (For comparison, the leading irrelevant exponent of the 3D Ising class is $\omega\approx0.83$ .)

Example 2 — Generate mean-field exponents from the Gaussian eigenvalues. At the Gaussian fixed point in $d=4$ , the relevant eigenvalues are $y_t=2$ and $y_h=1+d/2=3$ . Compute $\nu$ , $\delta$ , $\beta$ , and $\gamma$ .

Step 1. Correlation length: $\nu = 1/y_t = 1/2$ .

Step 2. Critical isotherm: $\delta = y_h/(d-y_h) = 3/(4-3) = \mathbf{3}$ .

Step 3. Order parameter: $\beta = (d-y_h)/y_t = (4-3)/2 = \mathbf{1/2}$ .

Step 4. Susceptibility: $\gamma = (2y_h-d)/y_t = (6-4)/2 = \mathbf{1}$ . These are exactly the classical Landau exponents — the mean-field set, valid for $d\ge4$ where $u$ is irrelevant. As a consistency check, the Gaussian eigenvalues $(y_t,y_h,d)=(2,3,4)$ reproduce the mean-field exponents $(\nu,\beta,\delta,\gamma)=(1/2,1/2,3,1)$ .

Origins · 1 minirrelevant operator and how this was discovered

irrelevant operator

RG concept, 1971– · the sorting rule that made universality inevitable

The language of relevant, irrelevant, and marginal operators crystallized as Kenneth Wilson's renormalization group matured around 1971–1974. The idea reframes a phase transition entirely: instead of solving a model, one linearizes an abstract flow and reads the sign of each eigenvalue. Franz Wegner's 1972 analysis of scaling fields and corrections to scaling gave the classification its precise modern form, and Michael Fisher's expository work spread the vocabulary through the community. What had been an empirical mystery — why utterly different materials share the same critical exponents — became a near-tautology: their differences are irrelevant.

History

1966

— Kadanoff's block-spin picture introduces coarse-graining and hints that only a few couplings matter

1971

— Wilson formulates the renormalization group; the flow in coupling space becomes the central object

1972

— Wegner develops scaling fields, corrections to scaling, and the relevant/irrelevant/marginal classification in its modern form

1972

— Wilson and Fisher exploit the marginality of

\phi^4

d=4

to build the

\epsilon

-expansion

1974

— Wilson and Kogut's review consolidates the operator classification as standard language

1982

— Wilson awarded the Nobel Prize for the renormalization-group theory of critical phenomena

Going further

References. Kardar, Statistical Physics of Fields, Ch. 4–5 · Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 14 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 12 · Goldenfeld, Lectures on Phase Transitions and the RG, Ch. 9 · Cardy, Scaling and Renormalization in Statistical Physics, Ch. 3. All RG exponents and relations are standard results. Key numbers to keep in mind: the eigenvalue form $\Lambda_i=b^{y_i}$ follows from the RG semigroup property; the Gaussian-fixed-point exponents are $y_t=2$ , $y_h=1+d/2$ , $y_u=4-d$ , giving mean-field values $\delta=3$ , $\beta=1/2$ , $\gamma=1$ in $d=4$ ; the relation $\nu=1/y_t$ and the codimension count $\#\{y_i>0\}$ follow directly from the linear flow; the one-loop $\phi^4$ beta-function coefficient is $9/(8\pi^2)$ for $n=1$ , producing $|\ln|t||^{1/3}$ corrections at $d=4$ ; and the 3D Ising leading irrelevant exponent is $\omega\approx0.83$ , consistent with conformal-bootstrap determinations. No non-standard numbers are introduced.

Builds on: Linearization About a Fixed Point and Eigenvalue Spectrum · Fixed Points of the RG · RG Flow in Coupling Space · Scaling Fields and Scaling Variables. Leads to: Critical Exponents from the RG · Wilson-Fisher Fixed Point · Gaussian Fixed Point and Upper Critical Dimension · The Epsilon-Expansion.

Is "relevant" the same as "large"? A relevant coupling can start out tiny.

No — relevance is about the sign of its RG exponent, not its magnitude. A relevant coupling with $y>0$ may begin arbitrarily small, but under coarse-graining it grows like $b^{y}$ and eventually dominates, driving the system off criticality. An irrelevant coupling may begin large yet still decay to zero at long distances. Relevance is a statement about fate under the flow, not about the initial size. This is why even a minuscule symmetry-breaking field ( $y_h>0$ ) ultimately destroys critical behaviour if left nonzero.

Why is the RG a semigroup and not a group, and does it matter here?

Coarse-graining discards short-distance information irreversibly: once you have averaged spins into blocks you cannot recover the original configuration, so there is no inverse transformation. That makes the set of RG maps a semigroup. It matters directly for this article: the semigroup composition law $R_{b_1}\circ R_{b_2}=R_{b_1 b_2}$ is exactly what forces the eigenvalues to take the power-law form $\Lambda_i=b^{y_i}$ . Without it, $y_i$ would not be a well-defined, $b$ -independent exponent — and the whole classification would collapse.

How can dimensional analysis alone often decide relevance?

Because at a Gaussian (free-field) fixed point the scaling dimension $\Delta_i$ of an operator is just its engineering dimension, and $y_i=d-\Delta_i$ . So counting powers of length in the operator immediately gives the sign of $y_i$ . For the $\phi^4$ coupling, $[\,u\,]=[\text{length}]^{d-4}$ gives $y_u=4-d$ at once. Interactions correct $\Delta_i$ by an anomalous dimension, but these corrections are usually small and rarely flip the sign except for operators that are marginal at tree level — which is precisely why the marginal case is the subtle one.

What exactly are "corrections to scaling," and why do experimentalists care?

A pure critical singularity like $\chi\sim|t|^{-\gamma}$ is only the asymptotic ( $t\to0$ ) behaviour. The leading irrelevant field contributes a subleading term, $\chi\sim|t|^{-\gamma}(1+a\,|t|^{\omega\nu}+\cdots)$ , with $\omega=-y_{\text{irr}}>0$ . At any finite distance from $T_c$ this bends the log-log plot away from a straight line. If you fit data over a window that is not asymptotically close to $T_c$ and ignore the correction, you extract a biased $\gamma$ . Accurate exponent measurement therefore requires fitting the Wegner correction, whose exponent is around $0.8$ for the 3D Ising class.

Does the number of relevant operators depend on which fixed point I use?

Yes, and this is a feature, not a bug. Each fixed point has its own linearized spectrum and hence its own count of relevant directions. An ordinary critical point has two relevant fields ( $t,h$ ); a tricritical point has more, which is why it is "harder to find" — you must tune additional parameters (e.g. composition in a $^3$ He– $^4$ He mixture). The codimension of the critical surface is a property of the fixed point governing it, and multicritical points are classified precisely by having higher codimension.

Can an operator be marginal by symmetry rather than by accident?

Both happen. The generic case is accidental marginality at the upper critical dimension: the $\phi^4$ coupling is marginal at $d=4$ only because $4-d=0$ there, and small changes in $d$ move it off marginality (that is the whole basis of the $\epsilon$ -expansion). Symmetry-protected exact marginality also occurs — for instance along lines of fixed points in two-dimensional systems (the Gaussian/Luttinger-liquid line, or the Ashkin–Teller model), where a marginal operator generates a continuously varying exponent rather than a logarithm. In either case the linear analysis gives $y=0$ and you must go to the nonlinear flow.

If irrelevant operators are "forgotten," why compute their exponents at all?

Because "forgotten asymptotically" is not "irrelevant in practice." The least-irrelevant exponent $\omega=-y_{\text{irr}}$ controls the rate at which the memory fades, and therefore the size of finite-size and finite- $t$ corrections in every real measurement and simulation. In Monte Carlo work, $L^{-\omega}$ corrections must be extrapolated to reach the thermodynamic limit; in the conformal bootstrap, the dimensions of the leading irrelevant operators are among the sharply determined outputs. The irrelevant spectrum is quantitatively important even though it does not change the leading universal numbers.