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The critical two-point correlation function shows that the field has scaling dimensionIn the ordered phase the correlation length is the only diverging scale, so the order parameter scales asTherefore

A continuum limit of a quantum field theory sends the ultraviolet cutoff ${\Lambda }_0$ to infinity while holding chosen physical masses and amplitudes fixed. In cutoff units this requires a diverging correlation length, so the bare couplings must be tuned onto the critical surface of an ultraviolet renormalization-group fixed point. Each relevant direction of a fixed point requires one coordinate fixed by measurement; predictivity requires only finitely many such directions, and irrelevant microscopic details disappear.

Several behaviors are possible. In an asymptotically free theory the trajectory approaches a Gaussian fixed point in the ultraviolet and interactions vanish logarithmically. In an asymptotically safe quantum field theory it approaches a non-Gaussian fixed point with finitely many relevant directions. A theory with a Landau pole may possess only a trivial quantum field theory as its continuum limit; retaining a nonzero interaction then requires a finite cutoff. Finally, if no ultraviolet-complete trajectory exists, the model remains an effective field theory valid only below a physical cutoff.

The mean-field approximation suppresses correlated order-parameter fluctuations. Near a continuous phase transition, the correlation length diverges and long-wavelength fluctuations become increasingly important. The Ginzburg criterion therefore fails in sufficiently low dimension, and the interacting renormalization-group fixed point changes the mean-field exponents. For the tricritical $m^6$ theory the upper critical dimension is three: below $d=3$ the exponents are generally non-mean-field, while at $d=3$ one expects logarithmic corrections to mean-field scaling.

Landau theory treats the order parameter as spatially uniform and expands the free-energy density in powers allowed by its symmetries. The Landau-Ginzburg theory promotes it to a field $\varphi (\mathbf{x})$ and adds gradient terms such as $|\nabla \varphi {|}^2$. It therefore describes spatial fluctuations, interfaces, defects, and correlation lengths, while reducing to Landau theory for uniform fields.

For the $O(2)$ model writeAt long distances the massive radial field $\rho$ can be neglected, leaving the Goldstone-mode effective free energyThereforeThe mode is massless, so its correlation length is infinite. For $d>2$ its large-distance Green function isFor $d=2$ it is $-(2\pi \gamma v^2{)}^{-1}\log (r/a)$ up to an infrared-dependent constant, and in $d=1$ it is $-|r|/(2\gamma v^2)$ up to such a constant.

The invariant diagnostic is the phase-difference varianceIt diverges linearly in $d=1$ and logarithmically in $d=2$, destroying true long-range order, but approaches a finite infrared limit for $d=3,4$. Thus the continuous-symmetry ordered phase exists for $d=3,4$ and not for $d=1,2$, in agreement with the Mermin-Wagner theorem. The lower critical dimension is $d_{\mathrm{lc}}=2$; the two-dimensional $O(2)$ model can instead show quasi-long-range order below a BKT transition.

To first order in the weak fluctuation, $n^2-1=2\mu W+O({\mu }^2)$, soForintroduce the signs $s=(1,-1,1,-1)$. The product rule gives

Assume that $W$ is a zero-mean stationary Gaussian random field, that its longitudinal correlation length is short compared with the envelope's evolution scale, and that the propagation distance is long compared with that correlation length. The forward Markov approximation then neglects diffraction during one correlation length. Applying the Furutsu–Novikov formula closes the last average at second order in $\mu$. Define the integrated longitudinal autocorrelation function of a random fieldThenwhereWriting $C_{ij}=C(z_i-z_j)$ and using the evenness of the covariance gives the equivalent expressionThe derivation also assumes paraxial propagation, weak scattering, sufficient regularity to interchange differentiation and expectation, and statistical homogeneity in both coordinates. Without the short-correlation approximation, the Gaussian identity produces a nonlocal longitudinal memory integral rather than this local closed equation.

When $T\ll \Delta$, the argument of the inverse hyperbolic sine is large. Using $\mathrm{arsinh}z=\log (2z)+1/(4z^2)+O(z^{-4})$ givesThe mass remains the zero-temperature gap, with an exponentially small correction from thermally activated excitations.

When $T\gg \Delta$, expand around $z=1/2$. Ifis the golden ratio, then $\mathrm{arsinh}(1/2)=\log \phi$, andThe leading value $m/T=2\log \phi$ is universal. This is the quantum-critical regime: temperature is the only leading energy scale and the correlation length is of order $v/T$.

In the exterior write $\varphi =-{\varphi }_B+\tilde{\varphi }$. Linearization at the bulk minimum givesFor a large droplet, variations occur on scale $R$ much larger than the interfacial correlation length, so the second term is smaller by $O(\kappa /(\alpha R^2))$ and $\mu \simeq \alpha \tilde{\varphi }$. The conserved order-parameter dynamics then becomesDiffusion relaxes the exterior profile much faster than the droplet radius changes. In this quasistatic limit ${\partial }_t\tilde{\varphi }\simeq 0$, and therefore

Choose the vacuum $\varphi =(v+\sigma ,{\pi }_1,\dots ,{\pi }_{N-1})$. The quadratic free energy contains a positive mass term for the radial field $\sigma$ but no mass term for any transverse field ${\pi }_a$:Thus the momentum-space correlation function is $⟨{\pi }_a(k){\pi }_b(-k)⟩={\delta }_{ab}/k^2$. Since a massive correlator has denominator $k^2+{\xi }^{-2}$, these Goldstone modes have ${\xi }^{-1}=0$ and hence infinite correlation length.

These critical exponents describe the leading singular powers of percolation probability, percolation susceptibility, correlation length and critical cluster distributions near the percolation critical probability. Their existence and exact values require theorems or scaling hypotheses for the particular model; they are not consequences of the definition of the threshold. For $\theta (p)=P_p(0\leftrightarrow \infty )$ and $\chi (p)={\mathbb{E}}_p|C(0)|$, the definitions are $\theta (p)=(p-p_c{)}^{\beta +o(1)}$ above $p_c$ and $\chi (p)=(p_c-p{)}^{-\gamma +o(1)}$ below $p_c$. The correlation length has $\xi (p)=|p-p_c{|}^{-\nu +o(1)}$. At criticality, the root-cluster tail is $P(|C(0)|\ge s)=s^{-1/\delta +o(1)}$; two-point connectivity has leading power $|x{|}^{-(d-2+\eta )}$; and the expected number per site of size-$s$ clusters has leading power $s^{-\tau }$. The last distribution differs from the size-weighted distribution seen at a specified site. Relations such as $\gamma =(2-\eta )\nu$ and $2\beta +\gamma =d\nu$ have specific scaling regimes, and the latter hyperscaling relation fails in ordinary mean-field regimes above the upper critical dimension of percolation.

A phase transition is a nonanalytic change in equilibrium behavior as a thermodynamic control parameter varies. A continuous phase transition has a diverging correlation length and no latent heat.

A pure thermodynamic phase is an extremal equilibrium Gibbs state rather than a probabilistic mixture of distinct macroscopic phases. On an ordered Ising branch, taking the thermodynamic limit before removing a selecting conjugate field gives a nonzero magnetization. A symmetric mixture can instead have zero mean but a nondecaying two-point contribution. The connected correlation function and correlation length describing ordinary critical fluctuations should be evaluated in a chosen pure phase when this distinction matters.

In a symmetric equal mixture of two ordered pure thermodynamic phases with magnetizations $\pm M_0$, the mean spin vanishes but the large-distance two-spin expectation tends to $M_0^2$. Thus the connected correlation function of the mixture has a nondecaying part. The correlation-function susceptibility sum rule gives an extensive contribution ${\beta }_{\mathrm{th}}N{M}_0^2$ to susceptibility per spin. This is phase-switching response, not the connected susceptibility within one pure phase. The latter uses the thermodynamic limit followed by a one-sided zero-field limit, and subtracts the nonzero pure-phase mean before defining the correlation length.

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.

At an ordinary thermodynamic critical point, two coexisting phases become indistinguishable: their order parameter difference vanishes, and the correlation length and appropriate response functions typically diverge. A continuous phase transition can be approached there by tuning the thermal and conjugate-field variables. A renormalization-group fixed point controls its singular long-distance behavior. A triple point with three separated minima is not an ordinary thermodynamic critical point, and a tricritical point requires an additional tuning and has a different scaling theory.

The thermodynamic limit describes a sequence of larger systems, not a single enormous specimen: particle number $N$ and volume $V$ tend to infinity with number density $n=N/V$ fixed. Hold the chosen intensive and extensive thermodynamic quantities such as temperature $T$ and an applied magnetic field fixed along that sequence. In the canonical ensemble the controls are $T,N,V$; in the grand canonical ensemble one instead fixes $T$ and chemical potential, and the limiting number density is a result, possibly with a jump at phase coexistence. These are different specifications of a limiting experiment.

The useful limit is usually an intensive and extensive thermodynamic quantities normalization of the Helmholtz free energy, not a finite limit of the total extensive quantity itself:Here $Z_{N,V}$ is the canonical partition function, $k_B$ is the Boltzmann constant, $f$ is free-energy density, and $a$ is Helmholtz free energy per particle. Thus $F_{N,V}=Vf+o(V)$ when the limit exists; finite systems need not be exactly extensive quantities because of the surface correction to thermodynamic free energy.

Specify the interactions and the growing containers too. For a conventional fluid, use a fixed stable pair potential with a tempered pair potential tail, and containers whose boundary layer of any fixed thickness occupies a vanishing fraction of the volume; uniformly enlarged cubes work, a tube of fixed cross-section does not give the same bulk limit. For a finite-range Ising model, keep the lattice spacing and bond coupling fixed while enlarging all intended spatial directions. Ordinary bounded wall interactions or periodic boundary conditions can then give the same bulk free-energy density, although boundary conditions can select different spontaneous symmetry breaking states. This is a set of hypotheses, not a theorem that every interaction has a boundary-independent limit.^[ref]

The Curie–Weiss model below deliberately uses a different limiting prescription: its all-to-all coupling is divided by $N$. Without that factor its interaction energy would grow as $N^2$, rather than as an extensive quantity. The model has no spatial wall or correlation length to compare with a container; it is an exactly summable illustration of finite-size rounding, not a quantitative model of the fluid experiment below.

The bulk tangent covariance is twice the inverse Fourier transform of $k_{B}T/(A{q}^2+f)$, one term for each transverse direction. Equivalently the inverse of $-A{\partial }_r^2+f$ is $e^{-|r|/\xi }/(2\sqrt{Af})$. This gives the displayed correlation and identifies a force-dependent correlation length, different from the zero-force persistence length. Finite chains require endpoint conditions or a periodic convention.

In the phase with quasi-long-range order, vortex defects of opposite winding remain bound in pairs. Their unbinding produces the disordered phase with a finite correlation length.