Crossover Phenomena and Crossover Exponents

Bring a nearly isotropic magnet toward its Curie point and measure its critical exponents at, say, one part in a hundred of $T_c$ . You will read off the exponents of a Heisenberg magnet. Push ten times closer and the same sample quietly changes its mind: the exponents drift, and deep enough in you find Ising values instead. Nothing about the material changed — what changed is which fixed point of the renormalization-group flow the system's coarse-grained description was sitting near. This is crossover, and it is the reason a single substance can appear to belong to two different universality classes depending only on how closely you look. The engine is the renormalization group's habit of possessing several fixed points at once. A trajectory launched near an unstable (usually more symmetric) fixed point lingers there for many decades of length scale, so measurements at those scales report the unstable point's exponents. Eventually a relevant symmetry-breaking field — uniaxial anisotropy, a cubic term, a dipolar coupling, even the inverse system size — grows large enough to sweep the flow away to the stable fixed point that truly governs the asymptotics. The rate of that escape relative to the rate at which temperature itself grows under coarse-graining is captured by one number, the crossover exponent $\phi = y_g/y_t$ , the ratio of the perturbation's RG eigenvalue to the thermal eigenvalue. Because two scaling fields are simultaneously relevant, the singular free energy is a homogeneous function of two arguments, and eliminating the arbitrary rescaling factor collapses it to a one-variable scaling function of the crossover combination $g\,|t|^{-\phi}$ . Its two limits are the two fixed points; the interpolation between them, seen as a continuous drift of effective exponents across a crossover window of width $t_\times \sim g^{1/\phi}$ , is the experimental fingerprint. The very same mathematics governs why mean-field exponents give way to Wilson-Fisher ones near $T_c$ , and — with $1/L$ as the relevant field — underpins finite-size scaling.

Key takeaways

When more than one fixed point can govern a critical region, a system exhibits crossover: at long wavelengths its coarse-grained description flows from an unstable (often more symmetric) fixed point toward the stable one that fixes the true asymptotic exponents.
A symmetry-breaking perturbation $g$ attached to the unstable fixed point grows under the RG with eigenvalue $y_g$ ; the crossover exponent is $\phi = y_g/y_t$ , the ratio of its RG eigenvalue to the thermal eigenvalue $y_t=1/\nu$ .
The crossover occurs when the scaled variable $g\,|t|^{-\phi}$ reaches order one, defining a crossover line $g \sim |t|^{\phi}$ and a temperature width $t_\times \sim g^{1/\phi}$ .
Singular thermodynamics obeys the two-variable scaling form $f_s \sim |t|^{2-\alpha}\,\Phi(g\,|t|^{-\phi})$ ; effective exponents drift continuously through the window and reach their asymptotic values only past the crossover scale.
Classic realizations: Heisenberg to Ising under uniaxial anisotropy, Gaussian to Wilson-Fisher near the upper critical dimension (Ginzburg crossover), and finite-size crossover with $\phi=\nu$ .
Key idea: a relevant symmetry-breaking field destabilizes a fixed point; the crossover exponent $\phi = y_g/y_t$ measures how fast it grows relative to temperature, and the system crosses from the unstable to the stable fixed point's critical behavior along the line $g \sim |t|^{\phi}$ .

The core insight

A relevant symmetry-breaking field destabilizes a fixed point; the crossover exponent $\phi = y_g/y_t$ is the ratio of its RG eigenvalue to the thermal eigenvalue, and the system crosses over from the unstable to the stable fixed point's critical behavior along the line $g \sim |t|^{\phi}$ .

Where this lives today

Crossover is the working vocabulary of anyone extracting critical exponents from real data or simulations. In Monte-Carlo studies the inverse system size $1/L$ is the relevant crossover field with $\phi=\nu$ , so every finite-lattice measurement is literally a crossover between bulk criticality and finite-size rounding — the basis of the finite-size scaling used to pin down exponents. Experimentally, the mean-field-to-Wilson-Fisher (Ginzburg) crossover explains why superconductors and conventional magnets look classical over wide temperature ranges while high- $T_c$ cuprates and superfluid helium reveal genuine three-dimensional critical exponents within an accessible window. The concept has migrated well beyond magnets. Dimensional crossover in thin films and layered materials ( $d\to d-1$ as $\xi$ exceeds the film thickness), quantum-to-classical crossover near quantum critical points (where the inverse correlation time plays the role of $g$ ), and crossovers between long-range and short-range universality in systems with power-law interactions all use the same $\phi=y_g/y_t$ machinery. Wherever two fixed points compete for control of the same critical region, the crossover exponent is the number that says which one wins, and how quickly.

The physical picture

Picture the RG flow as a landscape of rivers. The unstable fixed point is a hilltop spring; the stable fixed point is the basin where every river eventually collects. A trajectory that starts very near the critical surface but slightly off the unstable fixed point runs almost into the spring, loiters at the summit for a long stretch, then is swept downhill to the basin. An experiment that measures exponents at a given length scale samples whichever fixed point the trajectory is nearest at that moment — so the apparent, or effective, exponents change continuously as you probe larger scales. Nothing is discontinuous: you watch one set of exponents morph into another. The crossover exponent adjudicates a race. Under coarse-graining by a factor $b$ , the temperature field grows as $b^{y_t}$ and the symmetry-breaking field as $b^{y_g}$ . Whichever reaches order one first seizes control of the physics. They tie when $b^{y_t}|t|\sim 1$ and $b^{y_g}g\sim 1$ hold together, which forces $g \sim |t|^{y_g/y_t} = |t|^{\phi}$ . That tie-line is the crossover boundary in the $(t,g)$ plane. Read $\phi$ as a lever ratio: if $\phi>1$ the symmetry-breaking field grows slowly relative to temperature, so you must come very close to $T_c$ before crossover sets in and the window is narrow; if $\phi<1$ the perturbation takes over quickly and the new asymptotic behavior appears well away from $T_c$ . The knife-edge is a marginal perturbation, $y_g=0$ and hence $\phi=0$ . Then the power-law crossover variable degenerates and the crossover happens only logarithmically slowly — which is precisely the origin of the logarithmic corrections seen at the upper critical dimension and in marginal cases like uniaxial dipolar magnets. The three RG eigenvalue-signs (relevant, irrelevant, marginal) established by the article on relevant, irrelevant, and marginal operators are exactly what decide whether crossover happens, is absent, or is logarithmic.

Figure

Crossover in the RG flow: lingering near the unstable fixed point, then escaping to the stable one

Figure

Punchline — data at different g collapse onto one crossover scaling function of g|t|^(−φ)

The core equations

Two scaling fields are relevant here, so the whole apparatus follows from writing the singular free energy as a function of both and then removing the arbitrary RG scale. Before the definition, note what $\phi$ compares: the temperature field carries the thermal eigenvalue $y_t=1/\nu$ , while the symmetry-breaking field $g$ carries its own eigenvalue $y_g$ at the unstable fixed point. The crossover exponent is their ratio.

\phi \equiv \frac{y_g}{y_t} \qquad (1)

where: $\phi$ — crossover exponent, fixing both the crossover line and the argument of the crossover scaling function; $y_g$ — RG eigenvalue of the symmetry-breaking scaling field $g$ at the unstable fixed point; $y_t$ — thermal RG eigenvalue $y_t=1/\nu$ of the temperature field at the same point. Physically $\phi$ answers "how many decades of length scale does $g$ gain for each decade $t$ gains?" — a large $\phi$ means $g$ grows fast and crossover comes early (far from $T_c$ ); a small $\phi$ means it comes late. Since $y_t=1/\nu$ , an equivalent reading is $\phi=\nu\,y_g$ . Because both fields are relevant, the singular free-energy density is a generalized homogeneous function of the two rescaled arguments. This is the two-variable extension of the single-field scaling that the article on the scaling form of the free energy from the RG derives; here we simply carry a second relevant direction along.

f_s(t,g) = b^{-d}\, f_s\!\left(b^{y_t} t,\; b^{y_g} g\right) \qquad (2)

where: $f_s$ — singular part of the free-energy density; $b>1$ — RG length-rescaling factor; $d$ — spatial dimension; $t=(T-T_c)/T_c$ and $g$ — the two relevant scaling fields. The equation says coarse-graining by $b$ shrinks the free-energy density by $b^{-d}$ (a volume factor) while stretching both scaling fields; it holds for any $b$ , which is exactly the freedom we now exploit. Eliminating $b$ by fixing the temperature argument collapses the two-variable form to a one-variable scaling function of the crossover combination. Choosing $b=|t|^{-1/y_t}$ makes $b^{y_t}|t|=1$ , and then the free energy becomes a prefactor times a function of a single argument.

f_s(t,g) = |t|^{2-\alpha}\,\Phi\!\left(g\,|t|^{-\phi}\right),\qquad 2-\alpha=\frac{d}{y_t} \qquad (3)

where: $\Phi$ — universal crossover scaling function of one variable; $2-\alpha=d/y_t$ — the free-energy exponent of the unstable fixed point (Josephson hyperscaling); $g\,|t|^{-\phi}$ — the crossover scaling variable, of order one on the crossover boundary. All the crossover physics is compressed into $\Phi$ : its small-argument limit reproduces the unstable fixed point, its large-argument limit the stable one. Setting the scaling variable to order one gives the crossover boundary in the phase plane and the temperature width of the crossover window.

g \sim |t|^{\phi} \quad\Longleftrightarrow\quad t_\times \sim g^{1/\phi} \qquad (4)

where: $t_\times$ — reduced-temperature width of the crossover region for a given $g$ . For $|t|\gg t_\times$ the unstable fixed point's exponents are seen; for $|t|\ll t_\times$ the stable point's. A small anisotropy $g$ and a large $\phi$ push $t_\times$ to tiny values, so the asymptotic regime hides very close to $T_c$ . The correlation length obeys its own crossover form, differing only in the length dimension of $\xi$ (a single power of $b$ instead of $b^{-d}$ ).

\xi(t,g) = |t|^{-\nu_1}\,X\!\left(g\,|t|^{-\phi}\right) \qquad (5)

where: $\nu_1=1/y_t$ — correlation-length exponent of the unstable fixed point; $X$ — crossover scaling function for $\xi$ . The same argument $g\,|t|^{-\phi}$ appears, so $\xi$ and $f_s$ cross over together. Finally, for the concrete Gaussian-to-Wilson-Fisher (Ginzburg) crossover, the quartic coupling $u$ is the relevant field of the Gaussian fixed point below four dimensions. Its eigenvalue is the engineering dimension, so its crossover exponent and the Ginzburg temperature follow immediately.

y_t^{G}=2,\quad y_u^{G}=4-d=\epsilon \;\Rightarrow\; \phi_G=\frac{\epsilon}{2},\quad t_{\mathrm{Gi}}\sim u^{2/\epsilon} \qquad (6)

where: $\phi_G=\epsilon/2$ — crossover exponent at the Gaussian fixed point; $t_{\mathrm{Gi}}$ — Ginzburg temperature below which mean field fails and Wilson-Fisher behavior takes over; $u$ — dimensionless quartic coupling. For $|t|\gg t_{\mathrm{Gi}}$ mean-field exponents are observed; for $|t|\ll t_{\mathrm{Gi}}$ the true fluctuation-dominated exponents emerge.

Derivation ACrossover exponent from the linearized RG eigenvalues

This derivation extracts $\phi=y_g/y_t$ from first principles by linearizing the RG map about the unstable fixed point, showing that the crossover boundary is not an assumption but a direct consequence of two relevant eigenvalues competing. The strategy is to diagonalize the RG recursion near the fixed point so each coupling combination transforms multiplicatively, identify the two relevant directions, iterate the flow, and read off the combination of $t$ and $g$ that stays invariant as the trajectory escapes. Watch that every eigenvalue is evaluated at the unstable fixed point about which the flow lingers — this is the subtle point the whole result turns on.

Step 1. Linearize the RG recursion $K_i' = R_i(\{K_j\})$ about the fixed point $K^\ast$ , writing $\delta K_i' = \sum_j M_{ij}\,\delta K_j$ with $M_{ij}=\partial K_i'/\partial K_j|_{K^\ast}$ the Jacobian. Why: near a fixed point the nonlinear map is controlled by its linear part, and the eigenstructure of $M$ organizes the flow into independent directions. This linearized RG and its eigenvalue spectrum are exactly what the companion article on linearization about a fixed point establishes; we take that spectrum as given.

Step 2. Diagonalize $M$ : its left eigenvectors define scaling fields $\mu_a$ that transform multiplicatively, $\mu_a' = \Lambda_a\,\mu_a = b^{y_a}\mu_a$ , with $y_a=\ln\Lambda_a/\ln b$ . Why: multiplicative transformation is what makes an eigen-combination a "scaling field" — it grows or shrinks by a pure power of the rescaling factor, with no mixing.

Step 3. Identify the two relevant scaling fields. Temperature supplies $\mu_t\equiv t$ with $y_t>0$ ; the symmetry-breaking perturbation supplies $\mu_g\equiv g$ with $y_g>0$ . Why: it is the second positive eigenvalue $y_g$ that makes $K^\ast$ unstable — with only $y_t$ relevant the point would fully control criticality and there would be no crossover.

Step 4. Iterate $\ell$ times with $b=e^{\ell}$ , so $t(\ell)=e^{y_t\ell}t$ and $g(\ell)=e^{y_g\ell}g$ . The trajectory leaves the fixed-point neighborhood when the growing fields reach order one. Why: linearization is valid only while the fields are small; the escape is signalled by the first field to reach $O(1)$ .

Step 5. Impose the escape condition on both fields at the same RG time $\ell_\times$ : $e^{y_t\ell_\times}|t|=1$ fixes $\ell_\times$ , and substituting into $e^{y_g\ell_\times}g$ gives $g\,|t|^{-y_g/y_t}=O(1)$ . Why: eliminating the common $\ell_\times$ leaves an invariant combination of $t$ and $g$ — the crossover scaling variable — whose exponent is $\phi=y_g/y_t$ . Since $y_t=1/\nu$ this is equivalently $\phi=\nu\,y_g$ . Verified: symbolic collapse of the two-variable form under $b=|t|^{-1/y_t}$ and the identity $\phi=\nu y_g$ checked in verify_aq-sm-crossover-phenomena-exponents.py (checks 1–2).

Go deeper · Derivation B · 2 minEffective exponents drift across the crossover region

Derivation BEffective exponents drift across the crossover region

This derivation shows that a measured exponent is not a constant but a smooth function of the crossover variable, gliding from the unstable fixed point's value to the stable one's as you approach $T_c$ . The plan is to write any singular response — take the susceptibility — in its crossover scaling form, take the logarithmic temperature derivative that experimentalists actually fit, and use the chain rule through the crossover variable to expose how the drift is controlled by $\phi$ and the shape of the scaling function. The payoff is a formula that reduces to $\gamma_1$ at small argument and to a shifted $\gamma_2$ at large argument, making "effective exponent" a precise object rather than a hand-wave.

Step 1. Write the susceptibility in crossover form, $\chi(t,g)=|t|^{-\gamma_1}\,\Psi(g\,|t|^{-\phi})$ , where $\gamma_1$ is the susceptibility exponent of the unstable fixed point and $\Psi$ the crossover scaling function. Why: near the unstable fixed point $\chi$ diverges with $\gamma_1$ ; the function $\Psi$ carries all dependence on how far the perturbation has grown.

Step 2. Define the effective exponent as the local logarithmic slope, $\gamma_{\mathrm{eff}}\equiv -\,\mathrm{d}\ln\chi/\mathrm{d}\ln|t|$ . Why: this is exactly the quantity a log-log plot of $\chi$ versus $|t|$ measures over any finite window, so it is the operationally meaningful exponent.

Step 3. Apply the chain rule through the scaling variable $x=g\,|t|^{-\phi}$ . Because $\ln x=\ln g-\phi\ln|t|$ , we have $\mathrm{d}\ln x/\mathrm{d}\ln|t|=-\phi$ , so $\gamma_{\mathrm{eff}}=\gamma_1+\phi\,x\,\dfrac{\mathrm{d}\ln\Psi}{\mathrm{d}x}$ . Why: the crossover variable is the only channel through which $\Psi$ depends on temperature, and its slope with respect to $\ln|t|$ is exactly $-\phi$ .

Step 4. Take the small-argument limit $x\to0$ : $\Psi\to\mathrm{const}$ , its derivative vanishes, and $\gamma_{\mathrm{eff}}\to\gamma_1$ . Why: far outside the crossover window the perturbation is negligible and the unstable fixed point rules.

Step 5. Take the large-argument limit $x\to\infty$ , where matching to the stable fixed point requires $\Psi\sim x^{p}$ with $\gamma_1+\phi\,p=\gamma_2$ , i.e. $p=(\gamma_2-\gamma_1)/\phi$ . Why: deep past the window the stable fixed point must reappear with its own $\gamma_2$ , which fixes the asymptotic power of $\Psi$ and closes the interpolation. *Verified: the large-argument power $p=(\gamma_2-\gamma_1)/\phi$ solved symbolically in verify_aq-sm-crossover-phenomena-exponents.py (check 8).*

Limiting cases

The crossover scaling function $\Phi(x)$ with $x=g\,|t|^{-\phi}$ interpolates between regimes; its limits recover the individual fixed points and the marginal knife-edge.

Case	Result	Meaning
Unstable fixed point ( $x\to 0$ )	$f_s \sim \vert t\vert^{2-\alpha_1}\,\Phi(0),\; \xi\sim\vert t\vert^{-\nu_1}$	For $g\,\vert t\vert^{-\phi}\ll1$ the perturbation has not grown; exponents are those of the unstable, more symmetric fixed point.
Stable fixed point ( $x\to\infty$ )	$f_s \to \vert t'\vert^{2-\alpha_2}$	For $g\,\vert t\vert^{-\phi}\gg1$ the scaling function restores the stable point's exponents in a shifted reduced temperature $t'$ ; matching fixes the power of $x$ .
Marginal perturbation ( $\phi\to0$ )	$\vert t\vert^{-\phi}=1-\phi\ln\vert t\vert+\cdots$	With $y_g\to0$ the power-law crossover degenerates into logarithmic dependence — the origin of log corrections at $d=d_c$ .
Finite-size crossover ( $g\to L^{-1}$ )	$f_s(t,L)=L^{-d}\,\mathcal F(t\,L^{1/\nu}),\; \phi_L=\nu$	Inverse size is relevant with $y_L=1$ , so $\phi_L=y_L/y_t=\nu$ ; bulk criticality crosses over to finite-size rounding when $\xi\sim L$ .

Worked examples

Example 1 — Where does a Heisenberg magnet stop looking Heisenberg? A nearly isotropic Heisenberg ferromagnet ( $n=3$ ) carries a weak uniaxial anisotropy $g=0.02$ . Taking the accepted crossover exponent $\phi\approx1.25$ for the Heisenberg-to-Ising crossover, find the crossover temperature width and decide whether a measurement at $t=0.05$ still reports Heisenberg exponents.

Step 1. Compute the crossover temperature from equation (4): $t_\times \sim g^{1/\phi} = 0.02^{1/1.25} = 0.02^{0.8}$ . Evaluating, $t_\times = \exp(0.8\ln 0.02) = \mathbf{0.0437}$ .

Step 2. Form the crossover scaling variable at the measurement point, $x=g\,t^{-\phi}=0.02\times 0.05^{-1.25}$ . Numerically $0.05^{-1.25}=\exp(1.25\times 2.996)=42.3$ , so $x=0.02\times42.3=\mathbf{0.846}$ .

Step 3. Interpret. Since $x=0.846<1$ — equivalently $t=0.05>t_\times=0.0437$ — the perturbation has not yet grown to order one, so the measurement sits on the Heisenberg side of the crossover. To reach Ising exponents one must cool to $|t|\ll 0.044$ , i.e. within about four percent of $T_c$ . The answer: at $t=0.05$ the sample still shows Heisenberg critical exponents; Ising behavior emerges only for $|t|\lesssim 0.044$ .

Example 2 — Why cuprates show a critical region and low- $T_c$ superconductors do not. The Ginzburg temperature is $t_{\mathrm{Gi}}\sim u^{2/\epsilon}$ (equation 6) with $\epsilon=1$ in $d=3$ , so $t_{\mathrm{Gi}}\sim u^{2}$ in terms of the dimensionless quartic coupling $u$ . Estimate the critical window for a conventional low- $T_c$ superconductor and for a high- $T_c$ cuprate.

Step 1. Low- $T_c$ superconductor: the effective coupling is minute, $u\sim10^{-6}$ , because the coherence length is enormous. Then $t_{\mathrm{Gi}}\sim(10^{-6})^{2}=\mathbf{10^{-12}}$ . The fluctuation window is $10^{-12}$ of $T_c$ — utterly unobservable, so mean-field exponents hold everywhere accessible.

Step 2. High- $T_c$ cuprate: the short coherence length gives a much larger coupling, $u\sim10^{-1}$ , so $t_{\mathrm{Gi}}\sim(10^{-1})^{2}=\mathbf{10^{-2}}$ . Now the fluctuation window is one percent of $T_c$ — easily reached in experiment.

Step 3. Conclusion: the seven-to-fourteen orders of magnitude separating the two Ginzburg numbers is why three-dimensional-XY critical exponents were first cleanly seen in cuprates and not in classic superconductors. The crossover mathematics is identical; only the size of $u$ differs. Cuprate window $\sim10^{-2}$ (accessible), low- $T_c$ window $\sim10^{-12}$ (inaccessible).

Origins · 1 mincrossover and how this was discovered

crossover

concept — a phenomenon named collectively, not for one person · The idea that a critical system can pass between two fixed points crystallized in the early-1970s renormalization-group work of Riedel and Wegner, Fisher, and Pfeuty–Toulouse, rather than in a single discovery.

The crossover exponent entered the literature as the renormalization group matured in the early 1970s. Eberhard Riedel and Franz Wegner (1969–1974) formulated crossover scaling and the crossover exponent for anisotropic and multicritical systems; Michael Fisher and collaborators framed the anisotropy crossover of Heisenberg magnets; and Pierre Pfeuty and Gérard Toulouse's influential textbook systematized the picture for a generation of students. Unlike an eponymous fixed point, "crossover" honors no single author — it names a universal phenomenon that many hands made precise.

History

1966

— Kadanoff's block-spin picture makes scaling fields and their eigenvalues conceptual, setting the stage for crossover

1969

— Riedel and Wegner introduce crossover scaling and the crossover exponent for anisotropic critical systems

1972

— Wilson and Fisher's ε-expansion supplies the one-loop O(n) anisotropy crossover exponent φ = 1 + (n−1)/[2(n+8)] ε

1974

— Wegner's review formalizes scaling fields, corrections to scaling, and crossover in the RG framework

1977

— Pfeuty and Toulouse's textbook systematizes crossover and multicritical phenomena for students

1990

— high-Tc cuprates make the Ginzburg (mean-field to 3D-XY) crossover experimentally accessible

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 14 (RG and crossover scaling) · Kardar, Statistical Physics of Fields, Ch. 4–5 (perturbative RG, crossover, anisotropy) · Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Ch. 9–10 · Cardy, Scaling and Renormalization in Statistical Physics, Ch. 3–5 · Aharony, in Phase Transitions and Critical Phenomena Vol. 6 (ed. Domb & Green) — crossover and multicritical phenomena · Pfeuty & Toulouse, Introduction to the Renormalization Group and to Critical Phenomena.

Provenance. Definitions and one-loop results cross-checked against Pathria & Beale Ch. 14, Kardar SPF Ch. 4–5, Goldenfeld Ch. 9–10, and Cardy. Recomputed here from scratch: (1) $\phi=y_g/y_t$ from the linearized RG eigenvalues; (2) the one-variable collapse of the two-variable scaling form under $b=|t|^{-1/y_t}$ ; (3) the Gaussian eigenvalues $y_t=2$ , $y_u=4-d$ giving $\phi_G=\epsilon/2$ and $t_{\mathrm{Gi}}\sim u^{2/\epsilon}$ ; (4) the O( $n$ ) one-loop anisotropy crossover exponent $\phi=1+(n-1)/[2(n+8)]\,\epsilon$ . Note: the one-loop value at $n=3,\epsilon=1$ is $12/11\approx1.09$ ; the accepted Heisenberg-to-Ising $\phi\approx1.25$ used in Example 1 is a higher-order/experimental value, stated as such.

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

Is crossover a real thermodynamic transition, or just an apparent change of exponents?

It is not a transition — no thermodynamic singularity occurs at the crossover line. What changes is which fixed point governs the coarse-grained physics at the length scale you are probing. The free energy is perfectly smooth across the crossover; only the effective exponents, defined as local logarithmic slopes, drift. You cross a true singularity only at $T_c$ itself, where the correlation length diverges.

Why must both eigenvalues come from the unstable fixed point?

Because the trajectory spends its measurable lifetime near the unstable point before escaping. The perturbation $g$ is a scaling field of that point's linearized RG, and its eigenvalue $y_g>0$ there is precisely what makes the point unstable and forces the eventual crossover. Using the stable point's spectrum would describe a different operator — usually irrelevant there — and give the wrong crossover line and the wrong $\phi$ .

What if the symmetry-breaking field is irrelevant at the unstable fixed point?

Then no asymptotic crossover occurs. An irrelevant $g$ ( $y_g<0$ ) decays under the RG, so the unstable point with $g=0$ controls the asymptotics and $g$ produces only a correction-to-scaling term with exponent $|y_g|/y_t$ . Crossover in the strict sense requires a relevant perturbation, $y_g>0$ ; that is the whole point of relevance.

How is the crossover exponent related to ν and the thermal eigenvalue?

Directly: $\phi=y_g/y_t$ and $y_t=1/\nu$ , so equivalently $\phi=\nu\,y_g$ . The thermal eigenvalue sets the "clock" of the flow — how fast temperature grows under coarse-graining — and $\phi$ measures the perturbation's growth against that clock. This is why finite-size scaling, where $y_g=y_L=1$ , has $\phi=\nu$ .

Where do the logarithmic corrections at the upper critical dimension come from?

From the marginal limit $\phi\to0$ . When $y_g=0$ the crossover variable $g\,|t|^{-\phi}$ loses its power-law form; expanding $|t|^{-\phi}=1-\phi\ln|t|+\cdots$ around $\phi=0$ turns power-law dependence into logarithmic dependence on reduced temperature. The quartic coupling at exactly $d=4$ is marginal in this sense, which is why mean-field exponents at the upper critical dimension come dressed with logarithmic corrections rather than clean power laws.

Can I measure the crossover exponent directly, or only infer it?

Both. The cleanest route is the crossover line itself: mapping where the effective exponents change gives $g\sim|t|^{\phi}$ , so a log-log plot of the crossover temperature against the anisotropy (or other field) yields $\phi$ as a slope. Alternatively, collapsing thermodynamic data onto a single crossover scaling function $\Phi(g\,|t|^{-\phi})$ is a two-parameter fit in $\phi$ and the fixed-point exponents; the value of $\phi$ that gives the best collapse is the measurement, exactly as in ordinary data collapse.