Continuous Phase Transitions

Cool a magnet through its Curie temperature and something remarkable happens without any of the drama of boiling: no plateau, no swallowed latent heat, no coexisting phases. The magnetization simply rises from zero, smoothly and continuously, the moment you drop below $T_c$ . Yet at that same point the magnetic susceptibility, the specific heat, and the correlation length all blow up as power laws. This is a continuous phase transition — Ehrenfest's "second order," though the name is a historical accident, since the higher derivatives generally diverge rather than merely jump. The order parameter is continuous; the response to it is singular. The signature that organizes everything is the correlation length $\xi$ , the distance over which fluctuations of the order parameter stay coordinated. In a first-order transition $\xi$ stays finite. At a continuous transition $\xi\to\infty$ : fluctuations occur on every length scale at once, the system becomes scale-invariant, and that scale invariance is the deep reason wildly different systems — a ferromagnet, a fluid at its critical point, a binary alloy — share identical critical exponents. The simplest quantitative model is Landau's: expand the free energy in powers of the order parameter with a quadratic coefficient that changes sign at $T_c$ , and out drops the continuous onset $\langle\phi\rangle\propto(T_c-T)^{1/2}$ together with a finite jump (mean-field) or a divergence (with fluctuations) in the specific heat. This article owns the structural claim that defines the class: at a continuous transition the free energy and order parameter stay continuous — so there is no latent heat, $L=T\,\Delta S=0$ — while the second derivatives (specific heat, susceptibility) and the correlation length diverge as power laws. We derive the continuity, prove the latent heat vanishes, and show where the divergences come from, setting up the critical exponents and the scaling hypothesis that quantify them.

Key takeaways

A continuous (second-order) transition has an order parameter that rises continuously from zero at a critical value $T_c$ — no latent heat, no phase coexistence.
The free energy and order parameter stay continuous, but response functions — specific heat $C$ , susceptibility $\chi$ , compressibility — diverge as power laws set by critical exponents.
The correlation length $\xi$ diverges at $T_c$ : fluctuations span all length scales, giving scale invariance, which underlies universality and the scaling hypothesis.
Landau theory gives the mean-field picture: $f=a_0\,t\,m^2+b\,m^4$ with $a$ changing sign at $T_c$ yields $\langle m\rangle\propto(T_c-T)^{1/2}$ and a finite jump in $C$ (mean-field $\alpha=0$ ).
Continuous transitions break a symmetry spontaneously; if the broken symmetry is continuous, gapless Goldstone modes appear.
Key idea: the order parameter and free energy stay continuous while the correlation length diverges, forcing thermodynamic response functions into universal power-law singularities — there is no latent heat, but there are critical exponents.

The core insight

At a continuous transition the order parameter and free energy remain continuous while the correlation length $\xi$ diverges. The resulting scale invariance forces the response functions (specific heat, susceptibility) into power-law singularities characterized by critical exponents — and, because $\Delta S=0$ , there is no latent heat.

Where this lives today

The continuous transition is the archetype of emergence and scale invariance, and its mathematics now reaches far beyond thermodynamics. The renormalization group it demanded is a universal tool: it computes critical exponents for magnets and fluids, but the same fixed-point machinery describes the onset of turbulence, the statistics of the early universe (inflationary fluctuations), the conformal field theories used in string theory and quantum criticality in strange metals, and even phase transitions in neural networks and the spread of epidemics on networks. Modern experiments push it to extremes — ultracold atoms realize clean Ising and superfluid transitions, and the conformal bootstrap has pinned the 3D-Ising exponents to a precision that vindicates fifty years of theory. Wherever a system develops correlations on all scales at once, this article's structure — continuous order parameter, diverging $\xi$ , universal exponents — is the template.

The physical picture

The contrast with a first-order transition is the fastest way in. A first-order transition is two separated free-energy minima swapping depths, so the order parameter must jump across a gap. A continuous transition is a single minimum that softly splits into two as you cool: the bottom of the well flattens, then develops a gentle double-well whose minima crawl away from zero. Nothing jumps. The order parameter emerges from zero with a vertical tangent, $m\propto(T_c-T)^{1/2}$ in mean field, and there is no barrier, hence no metastability and no latent heat. What is dramatic is what happens to fluctuations. Right at $T_c$ the free-energy well is quartically flat at the bottom — the quadratic restoring force has vanished. A flat well means the order parameter costs almost nothing to disturb, so fluctuations grow without bound: the susceptibility (the response to a conjugate field) diverges, and the correlation length — how far a local fluctuation drags its neighbours along — diverges too. The system develops structure on all scales simultaneously, from atomic to macroscopic. This is why a fluid at its critical point turns milky (critical opalescence): density fluctuations reach the wavelength of light.

Figure

Order parameter m versus temperature at a continuous transition: m rises continuously from zero at T_c as (T_c − T)^{1/2} (mean field, blue) with a vertical tangent — no jump, no gap. Contrast the first-order case (red), where m leaps discontinuously from 0 to m* at T_tr.

The punchline figure puts the two transition types side by side through the specific heat, the clearest experimental discriminator. A first-order transition dumps its latent heat as a sharp spike (formally a delta function) at $T_{\text{tr}}$ — a finite quantity of heat absorbed at one temperature. A continuous transition has no latent heat, but its specific heat is singular: a $\lambda$ -shaped divergence (or, in mean-field Landau theory, a finite jump) as $T\to T_c$ . Same axes, completely different shapes — the delta-spike of buried heat versus the power-law cusp of vanishing restoring force.

Figure

Specific heat C versus temperature: the two transition signatures. First order (red) shows a delta-function latent-heat spike at T_tr — finite heat absorbed at one temperature. Continuous (blue) shows a λ-shaped power-law divergence C ∝ |t|^{−α} at T_c with NO latent heat.

The core equations

The operational definition is the continuous onset of the order parameter. Below the critical temperature it grows from zero as a power law, and the exponent (here $\tfrac12$ in mean field) is the first of the critical exponents.

m(T)\propto (T_c-T)^{\beta}\quad(T<T_c),\qquad m=0\quad(T\ge T_c)

where: $m$ is the order parameter, $T_c$ the critical temperature, and $\beta$ the order-parameter exponent ( $\beta=\tfrac12$ in mean-field Landau theory, $\tfrac18$ for the exact 2D Ising model, $\approx0.326$ in 3D). Physically, $m$ turns on continuously — no jump — which is the whole content of "continuous." The Landau free energy is the minimal analytic model that produces this. Expand the free energy density in powers of $m$ , keeping the symmetry-allowed even terms, with a quadratic coefficient that changes sign at $T_c$ .

f(m,T)=f_0+a_0\,t\,m^2+b\,m^4,\qquad t=\frac{T-T_c}{T_c},\quad a_0,b>0

where: $t$ is the reduced temperature, $a_0$ and $b$ are positive constants, and $f_0$ the smooth background. Physically, for $t>0$ (above $T_c$ ) the minimum sits at $m=0$ ; for $t<0$ the $m=0$ point becomes a maximum and two minima appear at $m=\pm\sqrt{-a_0t/2b}$ — the continuous onset with $\beta=\tfrac12$ . The hallmark of the continuous class is that the correlation length diverges. Adding a gradient (stiffness) term to the free energy gives a correlation length that grows without bound as $T\to T_c$ .

\xi(t)\propto |t|^{-\nu},\qquad \nu=\tfrac12\ \text{(mean field)}

where: $\xi$ is the correlation length and $\nu$ the correlation-length exponent. Physically, $\xi\to\infty$ means fluctuations are correlated across the entire system — scale invariance — which is the microscopic engine of universality. This divergence is what a first-order transition never has.

Derivation AContinuity of the order parameter and the vanishing of latent heat

The goal is to prove, straight from the Landau free energy, the two claims that define a continuous transition: the equilibrium order parameter and the free energy (with its first derivative) are continuous at $T_c$ , and consequently the entropy jump — and hence the latent heat $L=T\,\Delta S$ — is exactly zero. This is the precise statement that separates the continuous class from the first-order class, where the first derivative jumps and $L\ne0$ . We work with $f=a_0\,t\,m^2+b\,m^4$ and minimize.

Step 1. Minimize $f$ over $m$ : $\partial f/\partial m=2a_0 t\,m+4b\,m^3=0$ gives $m=0$ or $m^2=-a_0 t/2b$ . The nonzero solution is real only for $t<0$ (below $T_c$ ). Why: for $t>0$ the quadratic term is a positive restoring force and $m=0$ is the unique minimum; at $t=0$ it goes soft; for $t<0$ the well splits.

Step 2. Show the order parameter is continuous: as $t\to0^-$ , $m=\sqrt{-a_0t/2b}\to0$ . There is no jump — $m$ emerges from zero with a vertical tangent, $m\propto(-t)^{1/2}$ . Why this is "continuous": the equilibrium $m$ is a continuous function of $T$ through $T_c$ , unlike the first-order jump.

Step 3. Compute the equilibrium free energy. For $t<0$ substitute $m^2=-a_0t/2b$ back: $f_{\min}=-a_0^2t^2/(4b)$ ; for $t\ge0$ , $f_{\min}=0$ . Both are continuous at $t=0$ , and their first derivative $df_{\min}/dt$ is $0$ above and $-a_0^2t/(2b)$ below — both tend to $0$ as $t\to0$ , so the first derivative is continuous. Why this matters: the entropy is (minus) the temperature-derivative of the free energy, so a continuous first derivative means a continuous entropy.

Step 4. Conclude that latent heat vanishes. The singular entropy is $\propto -df_{\min}/dt$ , continuous through $t=0$ with no step, so $\Delta S=0$ at $T_c$ , and therefore $L=T\,\Delta S=0$ . Why this is the decisive contrast: a first-order transition has $\Delta S\ne0$ (a first-derivative jump); a continuous transition has $\Delta S=0$ but singular second derivatives, computed next. Verified symbolically in verify_continuous-phase-transitions.py: $m\to0$ as $t\to0^-$ , $f_{\min}(t<0)=-a_0^2t^2/4b$ with continuous first derivative, and $\Delta S=0\Rightarrow L=0$ .

Go deeper · Derivation B · 2 minWhere the divergences come from: specific heat, susceptibility, correlation length

Derivation BWhere the divergences come from: specific heat, susceptibility, correlation length

Having shown the order parameter is continuous, we now locate the singularities that make the transition interesting. The goal is to see, in the same Landau model, that the specific heat has a discontinuity (a jump in mean field), the susceptibility diverges as $|t|^{-1}$ ( $\gamma=1$ ), and the correlation length diverges as $|t|^{-1/2}$ ( $\nu=\tfrac12$ ) once a gradient term is included. These power laws are the critical exponents; the point here is that they follow mechanically from the analytic free energy, and that they live in the second derivatives, not the first.

Step 1. Specific heat. Differentiate the equilibrium free energy twice in $t$ : $C_{\text{sing}}\propto -T\,d^2f_{\min}/dt^2$ is $0$ above $T_c$ and $a_0^2/(2b)$ below — a finite jump $\Delta C=a_0^2/(2b)>0$ . Why a jump and not a divergence here: mean-field Landau theory is analytic, so the singularity in $C$ is only a step (exponent $\alpha=0$ ); real fluctuations turn this into a genuine power-law $\lambda$ -divergence.

Step 2. Susceptibility. Add a field term $-h\,m$ and minimize: $2a_0 t\,m+4b\,m^3=h$ . Above $T_c$ (small $m$ ) the linear response is $\chi=\partial m/\partial h=1/(2a_0 t)\propto t^{-1}$ . Below $T_c$ , expanding about $m_0$ gives curvature $2a_0t+12b\,m_0^2=-4a_0t$ , so $\chi=1/(4a_0|t|)\propto|t|^{-1}$ . Why $\gamma=1$ : both sides diverge as the first inverse power of $|t|$ ; the amplitude ratio is $\chi_{-}/\chi_{+}=\tfrac12$ across $T_c$ .

Step 3. Correlation length. Add a stiffness term $\tfrac{K}{2}(\nabla m)^2$ ; the fluctuation propagator has a mass $\propto a_0 t$ , giving $\xi=\sqrt{K/(2a_0 t)}\propto t^{-1/2}$ above $T_c$ , which diverges as $t\to0^+$ . Why this is the master divergence: $\xi\to\infty$ is the scale-invariance that makes the response functions singular and forces universality; it is the property a first-order transition never has.

Step 4. Read the structure. Continuity lives in $m$ and the first derivatives of $f$ ; the singularities live in the second derivatives ( $C$ , $\chi$ ) and in $\xi$ . That split — continuous order parameter, divergent response — is the continuous transition. Why "second order" is a misnomer: Ehrenfest imagined finite jumps in second derivatives, but real transitions have power-law divergences, so the modern name is "continuous." Verified symbolically in verify_continuous-phase-transitions.py: $\Delta C=a_0^2/2b$ jump, $\chi_\pm\propto|t|^{-1}$ with ratio $\tfrac12$ , and $\xi\propto|t|^{-1/2}\to\infty$ .

Limiting cases

case	result	meaning
$t\to0$ from either side	$\xi\to\infty$ , $\chi\to\infty$ , $C$ singular	the critical point — scale invariance, all length scales active
$t>0$ , $\\|t\\|\gg0$ (far above $T_c$ )	$m=0$ , $\xi$ small, $\chi$ finite	disordered phase, ordinary paramagnet/fluid
$t<0$ , mean-field	$m=\sqrt{-a_0t/2b}\propto(-t)^{1/2}$	ordered phase with continuous onset, $\beta=\tfrac12$
include fluctuations, $d<4$	jump in $C$ becomes power-law divergence; exponents shift	mean-field $\beta=\tfrac12\to$ 2D-Ising $\tfrac18$ , 3D-Ising $\approx0.326$
cubic term or field bias added	transition becomes first order	continuous line ends at a tricritical point

Worked examples

Example 1 — Mean-field order-parameter onset and its exponent. Take the Landau model $f=a_0\,t\,m^2+b\,m^4$ with $a_0=1$ , $b=1$ (arbitrary units). Compute the equilibrium $m$ at reduced temperatures $t=-0.04$ and $t=-0.01$ , and confirm the exponent $\beta=\tfrac12$ .

Step 1. Below $T_c$ the minimum is $m=\sqrt{-a_0 t/(2b)}=\sqrt{-t/2}$ . Why: set $\partial f/\partial m=0$ and take the nonzero root.

Step 2. At $t=-0.04$ : $m=\sqrt{0.04/2}=\sqrt{0.02}=\mathbf{0.1414}$ . At $t=-0.01$ : $m=\sqrt{0.01/2}=\sqrt{0.005}=\mathbf{0.07071}$ .

Step 3. Check the exponent: reducing $|t|$ by a factor of $4$ (from $0.04$ to $0.01$ ) reduces $m$ by a factor $0.07071/0.1414=\mathbf{0.5}=1/\sqrt{4}$ . Since $m\propto|t|^{\beta}$ and a factor-4 change in $|t|$ gives a factor- $\tfrac12$ change in $m$ , we read $\beta=\tfrac12$ . The order parameter is continuous ( $m\to0$ as $t\to0$ ) with the mean-field exponent.

Example 2 — No latent heat at a continuous transition. For the same model, confirm that the entropy jump and latent heat vanish at $T_c$ .

Step 1. Equilibrium free energy: $f_{\min}=0$ for $t\ge0$ and $f_{\min}=-a_0^2t^2/(4b)=-t^2/4$ for $t<0$ . Why: substitute $m^2=-a_0t/2b$ back into $f$ .

Step 2. Singular entropy $\propto -df_{\min}/dt$ : it is $0$ above $T_c$ and $+t/2$ below. As $t\to0$ from either side this tends to $\mathbf{0}$ — the entropy is continuous with no jump, $\Delta S=0$ .

Step 3. Latent heat: $L=T_c\,\Delta S=T_c\cdot0=\mathbf{0}$ . There is no plateau on the heating curve. The singularity instead appears in the second derivative: $C_{\text{sing}}\propto-T\,d^2f_{\min}/dt^2$ jumps by $a_0^2/(2b)=1/2$ across $T_c$ (mean-field $\alpha=0$ ). This is the defining contrast with a first-order transition, which has $L\ne0$ .

Origins · 1 mincontinuous phase transition and how this was discovered

continuous phase transition

classification framework, from 1907 to 1971 · The continuous transition took decades to understand. Pierre Weiss (1907) and van der Waals (1873) built mean-field pictures; Paul Ehrenfest (1933) named it "second order"; Lev Landau (1937) gave the order-parameter free-energy framework. The Ehrenfest picture of a mere finite jump in a second derivative was shattered by Lars Onsager's 1944 exact solution of the 2D Ising model, which showed a logarithmic divergence of the specific heat. Benjamin Widom, Leo Kadanoff, and Kenneth Wilson then explained the divergences through scaling and the renormalization group — the correlation length diverging at

T_c

being the master fact.

History

1873

— Johannes van der Waals gives an equation of state with a liquid–gas critical point

1907

— Pierre Weiss builds a mean-field theory of ferromagnetism with a continuous onset of magnetization

1933

— Paul Ehrenfest classifies transitions by discontinuous derivatives, coining "second order"

1937

— Lev Landau introduces the order-parameter free-energy expansion for continuous transitions

1944

— Lars Onsager solves the 2D Ising model exactly, revealing a logarithmic (not finite-jump) specific-heat divergence

1965

— Benjamin Widom proposes the scaling form of the singular free energy

1966

— Leo Kadanoff introduces block-spin scaling, explaining the diverging correlation length

1971

— Kenneth Wilson formulates the renormalization group, computing the exponents

Going further

Standard treatments: Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 12–13; Kardar, Statistical Physics of Fields, Ch. 1–3; Reif, Fundamentals of Statistical and Thermal Physics, Ch. 8; Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 12; Landau & Lifshitz, Statistical Physics Part 1, Ch. XIV. Prerequisites: Order Parameter (its continuous onset from zero is the operational definition of the transition); First-Order Phase Transitions (best understood by contrast with the discontinuous, latent-heat-bearing case); Latent Heat and Phase Coexistence (whose absence is a hallmark of continuity). Leads to: Landau Theory of Phase Transitions (the mean-field free-energy expansion for continuous onset); Critical Exponents (which quantify the power-law singularities); Universality and Universality Classes (why disparate systems share critical behaviour); Correlation Length and Its Divergence (the diverging $\xi$ that is the microscopic engine of scale invariance).

If nothing jumps, what actually "happens" at a continuous transition?

A symmetry is broken. Above $T_c$ the system is disordered and respects the full symmetry (a magnet has no preferred direction); below $T_c$ it spontaneously picks an ordered state (a magnetization direction), and the order parameter grows continuously from zero. So the transition is a qualitative change in the state of the system — the emergence of order — even though no thermodynamic quantity jumps. The dramatic experimental signatures are the diverging susceptibility and correlation length, not a latent-heat plateau.

Why does the correlation length diverge exactly at T_c?

At $T_c$ the free-energy well is quartically flat at the bottom: the quadratic restoring force ( $\propto a_0 t$ ) that normally pulls fluctuations back to equilibrium has vanished. With no restoring force, a local fluctuation of the order parameter propagates arbitrarily far before decaying — the correlation length $\xi\propto|t|^{-\nu}$ diverges. Physically, the system can no longer decide locally whether to be ordered or disordered, so it fluctuates coherently over ever-larger regions until, at $T_c$ , correlations span the whole sample. That is scale invariance.

Is mean-field theory's finite jump in specific heat correct?

No — it is an artifact of neglecting fluctuations. Landau/mean-field theory predicts a finite jump in the specific heat (exponent $\alpha=0$ ). But the exact 2D Ising solution (Onsager, 1944) shows a logarithmic divergence, and the 3D Ising specific heat has a weak power-law divergence ( $\alpha\approx0.11$ ). The mean-field jump is what you get when you ignore the very fluctuations that dominate near $T_c$ below the upper critical dimension $d_c=4$ . So the qualitative statement " $C$ is singular" is right; the specific mean-field shape (finite jump) is wrong below four dimensions.

How is universality related to the diverging correlation length?

When $\xi\to\infty$ , the microscopic details (lattice spacing, exact interaction shape) become irrelevant: the physics near $T_c$ is dominated by fluctuations on scales much larger than any microscopic length, so it "forgets" its origins. Only the coarse features survive — the spatial dimension $d$ and the symmetry of the order parameter. Systems that share those two features share the same critical exponents, forming a universality class. This is why a 3D uniaxial magnet and the liquid–gas critical point of a fluid have identical exponents despite utterly different microscopics.

Can a transition be continuous in one direction and first order in another?

Yes. A line of continuous transitions can meet a line of first-order transitions at a tricritical point (studied in a later article). Adding a symmetry-allowed cubic term to the Landau free energy, or biasing the system with a field, converts a continuous transition into a first-order one; tuning that term to zero recovers continuity. So "continuous" versus "first order" is not an intrinsic property of a material but of where you are in its phase diagram — the same system can show both, separated by a tricritical point.

What is critical opalescence and why does it happen here?

Near a fluid's critical point the correlation length of density fluctuations grows until it reaches the wavelength of visible light (hundreds of nanometres). Light then scatters strongly off these large-scale fluctuations, and the normally clear fluid turns milky white — critical opalescence. It is a direct, visible manifestation of the diverging correlation length that defines the continuous transition: you are literally seeing fluctuations that have grown to optical scales. Einstein and Smoluchowski explained it quantitatively as scattering from the diverging compressibility.

Does the order parameter always onset as (T_c − T)^{1/2}?

Only in mean-field (Landau) theory, where $\beta=\tfrac12$ . The true exponent depends on dimension and symmetry: the exact 2D Ising model has $\beta=\tfrac18$ (order rises much faster from zero), and the 3D Ising value is $\beta\approx0.326$ . The square-root onset is the mean-field prediction, correct only at and above the upper critical dimension $d_c=4$ ; below it, fluctuations change the exponent. So $(T_c-T)^{1/2}$ is the qualitative continuous onset, but its exponent is a universal number that mean-field gets wrong in physical dimensions.