Fixed Points of the RG

Two magnets, a liquid boiling in a kettle, and a strip of alloy losing its order at high temperature look nothing alike at the atomic scale, yet at their critical points they obey the same power laws with the same exponents. The renormalization group explains this by treating a physical system not as a fixed Hamiltonian but as a moving point in an infinite-dimensional space of coupling constants, dragged along by repeated coarse-graining. Everything universal about criticality is encoded in the special points of that motion where the dragging stops — the fixed points. A fixed point $\mu^*$ satisfies $R_b\mu^*=\mu^*$ : a Hamiltonian that reproduces itself after you blur out short-distance detail and shrink the picture back by a factor $b$ . It is a theory with no preferred length scale. Since coarse-graining reports every length in units stretched by $b$ , the correlation length obeys $\xi\to\xi/b$ , and self-consistency at a fixed point forces $\xi^*=\xi^*/b$ . For $b>1$ this has only two solutions: $\xi^*=\infty$ , a critical fixed point that governs a continuous phase transition, or $\xi^*=0$ , a trivial fixed point describing a featureless bulk phase such as the fully ordered ferromagnet or the high-temperature paramagnet. This article defines fixed points, shows why their correlation length is quantized to zero or infinity, and classifies them as attractors (sinks) or saddles by whether nearby trajectories flow in or out. We solve the fixed points of an exactly tractable example — the one-dimensional Ising decimation map — by hand, finding the disordered sink and the zero-temperature critical point, and we read off why one dimension has no finite-temperature transition. We then survey the Gaussian fixed point and note where the Wilson–Fisher fixed point takes over below four dimensions. How the eigenvalues around a fixed point turn into critical exponents is the job of the following articles; here we build the object those articles stand on.

Key takeaways

A fixed point of the renormalization group is a scale-invariant Hamiltonian, $R_b\mu^*=\mu^*$ : it looks statistically identical at every magnification.
Because coarse-graining sends $\xi\to\xi/b$ , a fixed point must have $\xi^*=0$ (a trivial sink — a stable bulk phase) or $\xi^*=\infty$ (a critical fixed point — a continuous transition).
Fixed points organize the flow: trivial ones are sinks that attract whole regions (phases); critical ones are saddles, attracting along the critical surface and repelling along relevant directions.
The 1D Ising decimation map $\tanh K'=\tanh^2 K$ has a stable sink at $K^*=0$ and a critical fixed point only at $K^*=\infty$ ( $T=0$ ) — a hand-solvable proof that 1D has no finite- $T$ transition.
Above four dimensions the Gaussian fixed point is stable (mean-field behaviour); below four the quartic coupling turns relevant and the Wilson–Fisher fixed point takes over.
Key idea: a fixed point is where coarse-graining stops changing the theory; its correlation length is pinned to $0$ or $\infty$ , and the critical fixed point — a saddle with $\xi=\infty$ — is the single object every model in a universality class flows toward.

The core insight

A fixed point is a scale-invariant Hamiltonian $R_b\mu^*=\mu^*$ ; because $\xi\to\xi/b$ under coarse-graining it must have $\xi^*=0$ (trivial sink) or $\xi^*=\infty$ (critical), and the critical fixed point is the saddle that every microscopically distinct model in one universality class flows toward.

Where this lives today

Fixed points are now the organizing objects of an enormous swath of physics well beyond magnets. In high-energy physics, an ultraviolet fixed point at vanishing coupling is asymptotic freedom — the reason quarks behave almost freely at short distances — while infrared fixed points underlie the scale-invariant correlation functions of conformal field theories. In condensed matter, the conformal-bootstrap program pins down the 3D Ising fixed point's exponents to extraordinary precision ( $\nu=0.62999$ , $\eta=0.03631$ ) by demanding self-consistency of a scale-invariant theory, and quantum critical points — fixed points of a zero-temperature RG — are invoked to explain strange-metal behaviour in high- $T_c$ superconductors and heavy-fermion compounds. Whenever a system shows power laws that ignore its microscopic makeup, a fixed point is lurking behind them.

The physical picture

Photograph a spin configuration, blur out every detail finer than a block of size $b$ , then shrink the photo back to its original size. A fixed point is a configuration whose statistics are unchanged by this operation — it looks the same at every zoom level. Only two kinds of picture have no preferred scale: one where fluctuations of every size coexist ( $\xi=\infty$ , a critical system) and one that is perfectly uniform ( $\xi=0$ , a pure phase with no fluctuations at all). Any finite correlation length would single out a special size and get magnified away. Now picture the space of all couplings as a landscape and the coarse-graining operation as water flowing downhill. Trivial fixed points are sinks: they pull in every nearby trajectory, so they sit at the bottom of whole basins — the deep interior of the ordered phase, the deep interior of the disordered phase. A critical fixed point is different: it is a saddle. Along most directions (the irrelevant ones, spanning the critical surface) it attracts; along a few special directions (the relevant ones, like temperature offset and magnetic field) it repels.

Figure

The flow in coupling space: sinks, a saddle, and the critical surface

A real material tuned near its transition sits slightly off the critical surface. The flow carries it ever closer to the saddle, and it lingers there — because a saddle is slow — before the relevant directions finally fling it toward one phase or the other. The long time spent near the saddle is exactly what makes the correlation length enormous near $T_c$ . Universality is then almost a tautology: the saddle, not the starting material, governs the long-distance physics, so a uniaxial magnet and a boiling liquid that flow to the same saddle share identical exponents. Their microscopic differences were irrelevant directions that washed out on the way in.

Figure

Why the correlation length is pinned to 0 or ∞ at a fixed point

Counting the relevant directions tells you how much tuning criticality needs. The ordinary Ising critical point has two relevant fields — temperature offset $t$ and field $h$ — so you must set two knobs (cool to $T_c$ , zero the field) to sit on it. A tricritical point has an extra relevant direction and demands an extra tuning. The number of relevant eigenvalues is therefore the codimension of the special point in the phase diagram.

The core equations

A fixed point is defined by the single requirement that the RG map returns it unchanged. Because the map $R_b$ coarse-grains by a linear factor $b>1$ and then rescales, its output is a new set of couplings $\mu'=R_b\mu$ ; the fixed point is where output equals input.

R_b\,\mu^* = \mu^* \qquad (1)

where: $R_b$ — the RG transformation, coarse-grain by factor $b>1$ then rescale lengths; $\mu=(\mu_1,\mu_2,\dots)$ — a point in coupling space, the full list of symmetry-allowed couplings; $\mu^*$ — the fixed point, invariant under $R_b$ for all $b$ . Physically $\mu^*$ is a Hamiltonian identical to itself at every scale — a scale-invariant theory. The one macroscopic length in the problem is the correlation length $\xi$ , and it must be reported in the coarsened units. This is the constraint that pins a fixed point's correlation length.

\xi(R_b\mu) = \frac{\xi(\mu)}{b} \;\;\Longrightarrow\;\; \xi(\mu^*)=\frac{\xi(\mu^*)}{b} \qquad (2)

where: $\xi$ — correlation length in lattice units; $b$ — rescaling factor. Reading it physically: at a fixed point the second equation has, for $b>1$ , only the solutions $\xi^*=0$ and $\xi^*=\infty$ . A featureless phase ( $\xi=0$ ) and a critical system ( $\xi=\infty$ ) are the only scale-free states. Near a fixed point the nonlinear map becomes a matrix. Writing $\mu=\mu^*+\delta\mu$ , the leading response is the Jacobian evaluated at $\mu^*$ ; its full eigen-analysis is developed in the linearization article, but the object itself is defined here.

T_{ij} = \left.\frac{\partial \mu_i'}{\partial \mu_j}\right|_{\mu^*},\qquad \delta\mu'_i=\sum_j T_{ij}\,\delta\mu_j \qquad (3)

where: $T_{ij}$ — linearized RG map (Jacobian) at the fixed point; $\delta\mu=\mu-\mu^*$ — deviation. The eigenvalues of $T$ decide whether $\mu^*$ attracts or repels along each direction — the criterion that sorts fixed points into sinks and saddles. The classification of a fixed point is then geometric: it is a sink if every eigenvalue pulls inward, a saddle (critical) if at least one direction pushes outward. The stable directions span the critical surface; the unstable ones are the relevant fields.

\text{sink (trivial): all directions stable};\qquad \text{saddle (critical): } \ge 1 \text{ unstable direction} \qquad (4)

where: stable/unstable is decided by whether an eigenvalue shrinks or grows a deviation under $R_b$ . Reading it physically: the number of unstable (relevant) directions equals the number of parameters you must tune to place a real system exactly on the fixed point.

Derivation AWhy a fixed point's correlation length is quantized to 0 or ∞

We prove the single most distinctive property of any RG fixed point — that its correlation length can only be zero or infinity — directly from how coarse-graining rescales lengths. The strategy is short but rigorous: track the one macroscopic length, $\xi$ , through a coarse-graining step; it is reported in the new, expanded units, so it shrinks by $b$ ; imposing the fixed-point condition then turns this into an exact algebraic constraint on $\xi^*$ whose only solutions are the two scale-free values. This is what makes a fixed point a scale-invariant theory rather than one with some special finite length.

Step 1. After one RG step every physical length is measured in units stretched by $b$ , so the correlation length of the coarse-grained system is reported as

\xi(R_b\mu)=\frac{\xi(\mu)}{b}.

Nothing physical has changed about the correlation of fluctuations; only the ruler has been rescaled.

Step 2. At a fixed point the coarse-grained Hamiltonian equals the original, $R_b\mu^*=\mu^*$ , so it must have the same correlation length. Substituting $\mu=\mu^*$ into Step 1,

\xi(\mu^*)=\frac{\xi(\mu^*)}{b}.

Step 3. For $b>1$ the equation $\xi^*=\xi^*/b$ has only the solutions

\xi^*=0 \qquad\text{or}\qquad \xi^*=\infty.

No finite nonzero value survives: $\xi^*=10$ would become $\xi^*/b\neq10$ , contradicting invariance. Why the two solutions mean opposite things. A featureless bulk phase has $\xi^*=0$ (a trivial sink — no fluctuations to correlate), while a critical system has $\xi^*=\infty$ (fluctuations of every size, no preferred scale). Both are scale-free, which is exactly why only they can be fixed points; everything in between is dragged toward one of them by the flow. Verified: verify_rg-fixed-points.py solves $\xi^*=\xi^*/b$ symbolically and confirms $\xi^*=0$ is the only finite solution (with $\xi^*=\infty$ the scale-invariant critical value).

Go deeper · Derivation B · 2 minFixed points of the 1D Ising decimation map

Derivation BFixed points of the 1D Ising decimation map

We solve, exactly and by hand, the fixed points of a real RG map — the decimation of the one-dimensional Ising chain — to make the abstract $R_b\mu^*=\mu^*$ concrete. The strategy is to sum over every second spin, which produces an effective coupling $K'$ between the survivors, giving a closed recursion $K'=R(K)$ ; we then locate its fixed points, test their stability with $|R'(K^*)|$ , and read the physical meaning off the result. The recursion itself is derived in the decimation article; here we take it as given and mine it for its fixed-point structure. Watch for the subtlety that the ordered fixed point lives at $K=\infty$ , which is best examined in the variable $x=e^{-2K}$ .

Step 1. The decimation of the nearest-neighbour chain (zero field) yields the exact recursion, in two equivalent forms,

K' = \tfrac12\ln\cosh 2K \qquad\Longleftrightarrow\qquad \tanh K' = \tanh^2 K,

where $K=\beta J$ is the dimensionless coupling. Both forms describe the same map; the second is cleaner for the algebra.

Step 2. A fixed point satisfies $K^*=R(K^*)$ . Setting $\tanh K^*=\tanh^2 K^*$ gives $\tanh K^*\,(1-\tanh K^*)=0$ , so $\tanh K^*=0$ or $\tanh K^*=1$ , i.e.

K^*=0 \quad(\text{high-temperature disordered}) \qquad\text{and}\qquad K^*=\infty \quad(\text{zero-temperature ordered}).

Step 3. Test $K^*=0$ . Differentiating the first form, $R'(K)=\tanh 2K$ , so $R'(0)=0$ . Since $|R'(0)|=0<1$ , the disordered fixed point is attractive: it is a stable sink. Indeed $R(K)<K$ for every finite $K>0$ (because $\tfrac12\ln\cosh 2K<K$ ), so any starting temperature flows down to $K^*=0$ . Why this already settles the physics. Every finite-temperature chain drains to the disordered sink under coarse-graining. There is no finite- $K$ fixed point in between, so there is no finite-temperature phase transition in one dimension.

Step 4. Examine $K^*=\infty$ in the variable $x=e^{-2K}$ , which sends $x\to0$ as $K\to\infty$ . Using $\tanh K=(1-x)/(1+x)$ and $\tanh K'=\tanh^2K$ , expand for small $x$ :

x' = 2x + O(x^2).

The leading factor is $2$ , i.e. $\lambda=2=b$ , so this fixed point is unstable (relevant): a chain infinitesimally warmer than $T=0$ flows away from order. The only critical point of the 1D Ising model sits at zero temperature. Why the factor is $2$ , not $4$ . A careless expansion (keeping the wrong power) suggests $x'\approx4x$ ; the correct leading term of $\tanh K'=\tanh^2 K$ in $x=e^{-2K}$ is $x'=2x$ . The eigenvalue is $\lambda=2$ and the thermal exponent is $y_t=\ln\lambda/\ln b=\ln2/\ln2=1$ , consistent with the exact chain result $\xi=1/x=e^{2K}$ diverging exponentially (not as a power law) as $T\to0$ . Verified: verify_rg-fixed-points.py solves $\tanh K'=\tanh^2K$ , confirms the fixed points $K^*=0$ (with $R'=\tanh2K=0$ , attractive) and $K^*=\infty$ (with $x'=2x$ , repulsive), and iterates $K=1\to0$ ; it flags the $x'=2x$ (not $4x$ ) correction.

The general lesson of that example — a stable sink at high temperature, an unstable critical fixed point at the transition — repeats in higher dimensions, except that there the critical fixed point moves to a finite temperature and acquires an infinite correlation length. Two canonical higher-dimensional fixed points bracket the interesting range: the Gaussian fixed point of the free field, and the Wilson–Fisher fixed point that inherits its stability below four dimensions.

\text{Gaussian FP: quartic eigenvalue } y_u = 4-d; \qquad \begin{cases} d>4: & y_u<0 \text{ (stable, mean-field)}\\ d=4: & y_u=0 \text{ (marginal, logs)}\\ d<4: & y_u>0 \text{ (unstable)} \end{cases} \qquad (5)

where: $y_u$ — RG eigenvalue of the quartic coupling at the Gaussian fixed point; $d$ — spatial dimension. Reading it physically: above four dimensions fluctuations are too weak to matter, the Gaussian (free-field) fixed point is stable, and mean-field exponents hold. At $d=4$ the quartic coupling is marginal, producing logarithmic corrections. Below four dimensions it turns relevant, the Gaussian point destabilizes, and a new stable critical fixed point — the Wilson–Fisher fixed point at $u^*\propto\epsilon$ with $\epsilon=4-d$ — branches off to govern the true critical behaviour. The construction of that fixed point and its exponents belongs to the Wilson–Fisher and $\epsilon$ -expansion articles; here we only place it in the classification.

Limiting cases

The character of a fixed point is read off from its correlation length and the signs of its eigenvalues.

Case	Result	Meaning
Critical fixed point	$\xi^*=\infty$ , at least one $y_i>0$	Scale-invariant theory of a continuous transition; a saddle in coupling space.
Trivial (sink) fixed point	$\xi^*=0$ , all $y_i<0$	Featureless stable bulk phase (fully ordered or disordered); attracts a whole basin.
1D Ising, disordered	$K^*=0$ , $R'(0)=0$	Stable sink; every finite- $T$ chain flows here — no finite- $T$ transition.
1D Ising, ordered	$K^*=\infty$ , $x'=2x$ ( $\lambda=2$ )	Unstable critical point at $T=0$ only; $\xi\sim e^{2K}$ diverges exponentially.
Gaussian FP, $d>4$	$y_u=4-d<0$	Quartic coupling irrelevant; Gaussian point stable, mean-field exponents.
Gaussian FP, $d<4$	$y_u=4-d>0$	Quartic relevant; Gaussian destabilizes, Wilson–Fisher fixed point takes over.

Worked examples

Example 1 — Decimating a 1D Ising chain toward its sink. Start a nearest-neighbour Ising chain at coupling $K_0=0.8$ (a moderate temperature) and apply the decimation map $K'=\tfrac12\ln\cosh 2K$ three times. Track the coupling and confirm the exact halving of the correlation length.

Step 1. Apply the map once: $K_1=\tfrac12\ln\cosh(1.6)=\tfrac12\ln(2.5775)=\mathbf{0.4734}$ .

Step 2. Again: $K_2=\tfrac12\ln\cosh(0.9468)=\mathbf{0.1969}$ ; and once more $K_3=\tfrac12\ln\cosh(0.3939)=\mathbf{0.03782}$ . The coupling is collapsing toward the sink $K^*=0$ — the chain is coarse-graining into a disordered paramagnet, as it must in one dimension.

Step 3. Check the correlation length. Using $\xi(K)=-1/\ln\tanh K$ (the exact 1D result), $\xi(K_0)=-1/\ln\tanh0.8=1.797$ and $\xi(K_1)=-1/\ln\tanh0.4734=0.898$ . The ratio is $\xi(K_1)/\xi(K_0)=\mathbf{0.500}$ exactly: one decimation step ( $b=2$ ) halves the correlation length, the defining property of a fixed-point flow. The correlation length only diverges as $K\to\infty$ , confirming the transition is at $T=0$ .

Example 2 — Where does the Gaussian fixed point lose control? A field theory just below the upper critical dimension has $d=3.5$ . Its quartic coupling starts at a small seed $u_0=0.01$ (in the natural units where the Gaussian eigenvalue is $y_u=4-d$ ). Using coarse-graining steps of $b=2$ , how many steps does it take for the quartic coupling to grow by a factor of ten, signalling that the Gaussian description has broken down?

Step 1. The quartic eigenvalue is $y_u=4-d=4-3.5=\mathbf{0.5}>0$ : the coupling is relevant here, so it grows. Under one step $u\to b^{y_u}u=2^{0.5}u=1.414\,u$ .

Step 2. Require $2^{y_u\,n}=10$ , i.e. $n=\ln10/(y_u\ln2)=2.3026/(0.5\times0.6931)=6.64$ . Rounding up, $\mathbf{n=7}$ coarse-graining steps.

Step 3. Check: after 7 steps $u_7=2^{0.5\times7}\,u_0=2^{3.5}\times0.01=11.31\times0.01=0.113$ , indeed just over ten times the seed. The growing quartic coupling drives the flow off the Gaussian fixed point toward Wilson–Fisher — the concrete meaning of "the Gaussian fixed point is unstable below $d=4$ ."

Recall the tools we lean on

The decimation recursion $\tanh K'=\tanh^2K$ used above is derived in Decimation and the One-Dimensional Ising RG; the exact 1D correlation length $\xi=-1/\ln\tanh K$ comes from One-Dimensional Ising Model and Transfer Matrix Method. How the eigenvalues of the linearized map $T$ become critical exponents is worked out in Linearization About a Fixed Point and Eigenvalue Spectrum and Critical Exponents from the RG; the full construction of the Wilson–Fisher fixed point lives in its own article. This one supplies the object those results act on.

Origins · 1 minfixed point and how this was discovered

fixed point

scale-invariant Hamiltonians in the RG · The notion of an RG fixed point as a scale-invariant theory crystallized in the late 1960s and early 1970s. Leo Kadanoff's 1966 block-spin picture supplied the intuition that coarse-graining maps one coupling to another; Kenneth Wilson's 1971 formulation made the transformation an explicit flow in coupling space whose fixed points classify critical behaviour, work recognized with the 1982 Nobel Prize in Physics. A fixed point is not a person but the central object of the theory — the point where the flow, and with it the change of scale, comes to rest.

History

1937

— Landau frames continuous transitions through an order parameter and a symmetry, setting the stage for a scale-based theory

1966

— Kadanoff's block-spin argument suggests coarse-graining maps couplings to new couplings, hinting at fixed points

1971

— Wilson formulates the renormalization group as an explicit flow in coupling space, with fixed points governing criticality

1972

— Wilson and Fisher find the nontrivial fixed point below four dimensions, opening the ε-expansion

1974

— Wilson and Kogut's review consolidates the fixed-point picture of critical phenomena

1982

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

Going further

References. Kardar, Statistical Physics of Fields, Ch. 4–5 · Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Ch. 9 · Cardy, Scaling and Renormalization in Statistical Physics, Ch. 3 · Wilson & Kogut, Phys. Rep. 12, 75 (1974) · Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 14 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 12.

Provenance. The fixed-point condition, the correlation-length constraint $\xi^*=\xi^*/b\Rightarrow\xi^*\in\{0,\infty\}$ , and the sink/saddle classification follow Kardar (Ch. 4–5) and Goldenfeld (Ch. 9). The 1D Ising decimation fixed points are worked exactly here; the recursion $\tanh K'=\tanh^2K$ and $\xi=-1/\ln\tanh K$ are standard (Pathria Ch. 13, Reif). The Gaussian eigenvalue $y_u=4-d$ and the Wilson–Fisher branch at $u^*\propto\epsilon$ are stated for classification only; their derivations are cited to the momentum-shell and Wilson–Fisher articles. Quoted conformal-bootstrap 3D Ising values ( $\nu=0.62999$ , $\eta=0.03631$ ) are from Kos–Poland–Simmons-Duffin–Vichi (2016). A base-error correction is recorded: the 1D decimation eigenvalue in $x=e^{-2K}$ is $x'=2x$ ( $\lambda=2$ , $y_t=1$ ), not $x'=4x$ .

Builds on: The Renormalization Group Idea: Coarse-Graining and Rescaling · RG Flow in Coupling Space · Block-Spin (Real-Space) Renormalization Group · Decimation and the One-Dimensional Ising RG. Leads to: Linearization About a Fixed Point and Eigenvalue Spectrum · Relevant, Irrelevant, and Marginal Operators · Critical Exponents from the RG · Wilson–Fisher Fixed Point · Gaussian Fixed Point and Upper Critical Dimension.

Is a fixed point a specific temperature, or something else?

Neither a temperature nor a single material — it is a point in the abstract space of all couplings, invariant under coarse-graining. A given material traces a curve through that space as you vary temperature, and it crosses the critical surface at exactly one temperature ( $T_c$ ). But the fixed point itself sits at the far end of the flow, common to every material that shares the same dimension and symmetry. That is why one fixed point can govern a magnet and a boiling liquid at once.

Why must the correlation length be exactly zero or infinity, not just "very small" or "very large"?

Because the fixed-point equation $\xi^*=\xi^*/b$ is exact, not approximate. For any $b>1$ , the only real numbers unchanged by division by $b$ are $0$ and $\infty$ . A finite value like $\xi^*=10$ would become $5$ after one step — different from $10$ — so it cannot be a fixed point. The quantization is a rigorous consequence of scale invariance, not a modelling choice.

How can a saddle point "attract" and "repel" at the same time?

A saddle is like a mountain pass: uphill in some directions, downhill in others. Along the irrelevant directions (which span the critical surface) trajectories flow in toward the fixed point; along the relevant directions they flow out. A system placed exactly on the critical surface is captured; one placed slightly off is first drawn near, then flung away toward a phase. The competition between these behaviours is what makes the correlation length large but finite near, though not at, $T_c$ .

The 1D Ising model has a fixed point at

T=0

— does that count as a phase transition?

It is a zero-temperature critical point, not a finite-temperature transition. The correlation length diverges only as $T\to0$ , and it diverges exponentially ( $\xi\sim e^{2J/k_BT}$ ) rather than as a power law. So one dimension orders only at absolute zero, and there is no finite- $T$ transition — exactly what the decimation flow shows by draining every finite-temperature chain to the disordered sink. Genuine finite- $T$ critical fixed points appear only in two or more dimensions.

Why does the Gaussian fixed point stop working below four dimensions?

The quartic (interaction) coupling has RG eigenvalue $y_u=4-d$ . Above four dimensions this is negative, so the interaction is irrelevant: it decays under coarse-graining, the free-field (Gaussian) description survives, and you get mean-field exponents. Below four dimensions $y_u$ turns positive — the interaction grows — so the Gaussian fixed point becomes unstable. The flow then funnels critical trajectories into a new, stable fixed point (Wilson–Fisher) whose exponents differ from mean-field. Four is therefore the upper critical dimension for this class.

If exponents come from the eigenvalues, why isn't that derivation in this article?

Scope. This article establishes what a fixed point is — the invariance condition, the $\xi\in\{0,\infty\}$ quantization, and the sink/saddle classification — and identifies the relevant directions as the eigenvalues that repel. Turning those eigenvalues $y_t,y_h$ into the numbers $\nu,\beta,\gamma,\delta,\alpha,\eta$ requires linearizing the map (next article) and feeding the two relevant eigenvalues into the free-energy homogeneity relation, which is the dedicated job of Critical Exponents from the RG. Keeping the derivation there avoids repeating it in every RG article.

How many parameters do I have to tune to reach a given fixed point?

Exactly the number of relevant directions (positive eigenvalues) at that fixed point. The ordinary Ising critical point has two — temperature offset and magnetic field — so you tune two knobs. A tricritical point has three, requiring an extra thermodynamic variable (like a vacancy concentration). This count is the codimension of the special point in the phase diagram, and it is one of the most useful things a fixed point tells you before you compute a single exponent.