Phase Transitions and Critical Phenomena
A phase transition is a singularity in a thermodynamic potential, and therefore something the finite-volume statistical mechanics of Statistical Mechanics cannot produce: partition functions of finitely many degrees of freedom are analytic, so every transition is a statement about the thermodynamic limit. That is the organizing fact of this chapter. It begins with phase diagrams, latent heats and the liquid–gas critical point, whose discovery [Andrews:1869] and first theory [vanderWaals:1873] set the vocabulary; it introduces the Ising paradigm [Ising:1925] and its exact two-dimensional solution [Onsager:1944], the one non-trivial case in which the singularity can be watched forming; and it then treats order parameters, critical exponents and universality — the discovery that a fluid near its critical point and a uniaxial magnet near its Curie point obey the same power laws with the same exponents, so that microscopic detail is irrelevant to the singular behaviour. The measured exponents that vindicate that picture are collected here, against the renormalization-group explanation developed in The Renormalization Group.
The chapter sits after the equilibrium ensembles of Statistical Mechanics and the quantum statistics of Quantum Statistics, both of which supply the free energies it differentiates, and before the specific ordered phases of the rest of the part: the magnetic transitions of Magnetism in Matter, the superfluid and superconducting transitions of Superconductivity and Superfluidity, and the condensation of Experiment: Bose–Einstein Condensation. Its last section treats liquid crystals, whose mesophases are the cleanest laboratory realization of an orientational order parameter. Standard monographs are [Stanley:1971] [Goldenfeld:1992].
Phase Transitions and Critical Phenomena: all derivations of this chapter are pending.
Phases and phase diagrams
Coexistence, triple and critical points
[Reserved: the phase diagram in the pressure–temperature plane, its coexistence curves, triple point and critical point; Gibbs's phase rule \(f=c-p+2\) derived from equality of the chemical potentials [Gibbs:1876]; the triple point of water as a former defining fixture of the kelvin, now superseded (Measurement, SI Units, and the Theory of Errors); metastability, superheating and nucleation as the reason a first-order transition can be delayed but not avoided.]
Latent heat and the Clausius–Clapeyron relation
[Reserved: latent heat as the discontinuity in entropy across a first-order line, and the slope relation \(\dd P/\dd T=L/(T\,\Delta v)\) [Clapeyron:1834], whose thermodynamic derivation belongs to Classical Thermodynamics; measured latent heats of fusion and vaporization in SI, with the anomalous negative slope of the ice–water line and its consequences; the vanishing of \(L\) at the critical point as the definition of second order.]
Along a first-order phase boundary two phases coexist at one pressure and one temperature, and converting a mole of the first into the second absorbs a finite latent heat \(L\) at constant temperature while the molar volume jumps discontinuously by \(\Delta v\). The boundary in the pressure–temperature plane is then not free: its slope is fixed by those two measured discontinuities,
a relation of Clapeyron [Clapeyron:1834] that has held wherever it has been tested. Its most conspicuous instance is an anomaly: water contracts on melting, so \(\Delta v<0\) for the ice–water line, whose slope is therefore negative — pressure lowers the melting point of ice, which is true of very few substances.
Derivation. Two phases in equilibrium at \((T,P)\) have equal molar Gibbs functions, \(\mu_{1}(T,P)=\mu_{2}(T,P)\) [Gibbs:1876]. Displace along the coexistence curve by \((\dd T,\dd P)\); the equality must persist, so \(\dd\mu_{1}=\dd\mu_{2}\). The Gibbs–Duhem relation for each phase gives \(\dd\mu=-s\,\dd T+v\,\dd P\) with \(s\) and \(v\) the molar entropy and volume, whence
The conversion is reversible and isothermal, so the heat absorbed is \(L=T\Delta s\); substituting gives Equation (123.1). Nothing in the argument uses a model of either phase, which is exactly what makes Equation (123.1) a test rather than a fit: \(L\), \(\Delta v\) and the slope are three independently measurable quantities constrained by one identity. The sign of the slope is the sign of \(\Delta v\), since \(L>0\) on melting and on boiling; the ice anomaly is therefore a statement about the density of ice and nothing else.
∎The Ehrenfest classification and its limits
[Reserved: Ehrenfest's ordering of transitions by the lowest discontinuous derivative of the free energy [Ehrenfest:1933]; why the scheme fails in practice — the specific heat at the helium lambda point and at the Curie point diverges rather than jumping, so there is no “second derivative discontinuity” to classify; the modern two-way division into first-order (latent heat, coexistence) and continuous (diverging correlation length) transitions, adopted for the rest of the chapter.]
The liquid–gas transition
Andrews and the critical point
[Reserved: Andrews's isotherms of carbon dioxide, showing the coexistence region shrinking to a point at \(T_{c}=304.1\,\mathrm{K}\) and \(P_{c}=7.38\,\mathrm{MPa}\), above which liquid and gas are not distinguishable [Andrews:1869]; the continuity of state that follows — one can pass from liquid to vapour without ever crossing a phase boundary; the flat critical isotherm and the diverging compressibility as the first measured critical exponents.]
Compressing a fluid along isotherms of rising temperature, the flat portion of the isotherm — the interval over which liquid and vapour coexist, the volume changes and the pressure does not — grows shorter and finally closes at a single point. Andrews located that point for carbon dioxide at \(T_{c}=304.1\,\mathrm{K}\) and \(P_{c}=7.38\,\mathrm{MPa}\) [Andrews:1869]. Above \(T_{c}\) no compression produces a meniscus, and one may pass from a dense liquid to a dilute vapour along a path that circles the critical point without ever crossing a phase boundary: liquid and gas are not two kinds of matter but two regions of one surface. At the critical point the isotherm is flat, so the isothermal compressibility diverges, and the fluid becomes strongly scattering (Section 123.2.3).
Derivation. That such a point must exist in any equation of state combining a short-range repulsion with a long-range attraction is van der Waals' result [vanderWaals:1873]. Take
in which \(b\) is the excluded molar volume and \(a\) measures the mean attraction. A critical point is an inflection of the isotherm with horizontal tangent, so
Dividing the first of Equation (123.4) by the second eliminates \(RT\) and gives \(v-b=\tfrac{2}{3}v\), hence \(v_{c}=3b\); substituting back yields
The critical point exists for every \(a>0\), \(b>0\), and the divergence of the compressibility there is immediate from the first of Equation (123.4). Eliminating \(a\) and \(b\) from Equation (123.5) gives the dimensionless combination \(P_{c}v_{c}/RT_{c}=3/8\) for every fluid, and rewriting Equation (123.3) in the reduced variables \(P/P_{c}\), \(v/v_{c}\), \(T/T_{c}\) removes \(a\) and \(b\) altogether: this is the law of corresponding states, the first statement of universality in physics. Two caveats belong with the result and are taken up later in this chapter. Below \(T_{c}\), Equation (123.3) produces a loop along which \((\pp P/\pp v)_{T}>0\), which is mechanically unstable and must be replaced by the horizontal tie line fixed by Maxwell's equal-area rule [Maxwell:1875]. And the exponents that Equation (123.4) implies near the critical point are the mean-field ones, which measurement contradicts (Section 123.5.2).
∎The van der Waals equation and the Maxwell construction
[Reserved: the equation \((P+a/v^{2})(v-b)=RT\) derived from excluded volume and a mean attraction [vanderWaals:1873], the first theory with a critical point built in; the unphysical loop below \(T_{c}\) and Maxwell's equal-area rule that replaces it with the coexistence tie line [Maxwell:1875]; the law of corresponding states in reduced variables; the mean-field exponents \(\beta=1/2\), \(\gamma=1\), \(\delta=3\) it predicts, and the measurements that contradict them (Section 123.5.2).]
Critical opalescence
[Reserved: the milky scattering of a fluid at its critical point, explained as light scattering from density fluctuations whose size grows with the correlation length [Einstein:1910]; the Ornstein–Zernike analysis of the pair correlation function and its Lorentzian structure factor [Ornstein:1914]; opalescence as the first direct evidence that a diverging correlation length, not a diverging response alone, is what characterizes criticality.]
A transparent fluid held within a fraction of a kelvin of its critical point turns milky white and scatters light so strongly that a centimetre of it is opaque; on warming or cooling by a degree the turbidity vanishes again. The scattered intensity is strongly peaked in the forward direction and its angular width narrows as the critical point is approached. The effect is not a mist of droplets — there is one phase present — but scattering from density fluctuations whose spatial extent has grown to the wavelength of visible light [Einstein:1910] [Ornstein:1914]. It is the first direct evidence that what diverges at a critical point is a length, and that the divergence of the compressibility is a consequence rather than the essence.
The scattered intensity as the Fourier transform of the density–density correlation function; the Ornstein–Zernike form with its Lorentzian structure factor and correlation length; and the fluctuation–response identity that ties the forward scattering to the isothermal compressibility, so that a diverging compressibility and a diverging correlation length are one statement
The Ising model
Lenz, Ising and the one-dimensional failure
[Reserved: the model of spins on a lattice with nearest-neighbour coupling, proposed by Lenz [Lenz:1920] and solved in one dimension by Ising, who found no transition at any positive temperature and wrongly concluded that none occurs in higher dimensions either [Ising:1925]; the transfer-matrix solution and the domain-wall (entropy versus energy) argument that explains the one-dimensional result; the honest note that the model's name records its first solver, not its inventor.]
Existence of a transition in two dimensions
[Reserved: Peierls's contour argument, the first proof that the two-dimensional Ising model has spontaneous magnetization below some finite temperature [Peierls:1936]; the counting of domain boundaries that makes the same entropy–energy comparison come out the other way in two dimensions; the general lesson that the lower critical dimension depends on the symmetry of the order parameter, taken up in Section 123.6.1.]
Onsager's exact solution
[Reserved: the exact free energy of the two-dimensional Ising model on a square lattice in zero field, with \(\sinh(2J/k_{B}T_{c})=1\) and a logarithmically divergent specific heat [Onsager:1944]; the spontaneous magnetization \(m=(1-\sinh^{-4}2\beta J)^{1/8}\), hence \(\beta=1/8\), announced by Onsager and first published in full by Yang [Yang:1952a]; why an exponent of \(1/8\) rather than \(1/2\) was the decisive evidence against mean-field theory; the absence of any exact three-dimensional solution.]
Lattice gas and the Yang–Lee theory
[Reserved: the exact mapping of the Ising magnet onto a lattice gas, which makes the ferromagnetic and liquid–gas transitions the same problem [Lee:1952]; the theory of condensation locating the singularity in the accumulation of zeros of the grand partition function in complex fugacity [Yang:1952], and — in the companion paper, which is where the result actually stands — the circle theorem placing those zeros on the unit circle for the ferromagnetic Ising model [Lee:1952]; why this is the sharpest available statement that a phase transition exists only in the thermodynamic limit.]
Mean-field theory and order parameters
The Weiss molecular field
[Reserved: the Curie law \(\chi\propto1/T\) measured across a wide temperature range [Curie:1895]; Weiss's hypothesis of an internal field proportional to the magnetization, giving the Curie–Weiss law \(\chi\propto1/(T-T_{c})\) and a spontaneous moment below \(T_{c}\) [Weiss:1907]; the same approximation for alloy ordering [Bragg:1934]; the self-consistency equation \(m=\tanh(T_{c}m/T)\) and its universality across mean-field models; magnetic instances in Magnetism in Matter.]
Landau's expansion of the free energy
[Reserved: Landau's postulate that the free energy is analytic in the order parameter near \(T_{c}\) and constrained only by symmetry, \(f=f_{0}+a(T-T_{c})m^{2}+bm^{4}\) [Landau:1937]; the classification of transitions by which terms symmetry permits, with a cubic term forcing first order; the resulting exponents \(\alpha=0\), \(\beta=1/2\), \(\gamma=1\), \(\delta=3\), identical to van der Waals and to Weiss; the gradient term that turns it into Landau–Ginzburg theory, used for superconductors in Superconductivity and Superfluidity.]
Symmetry breaking and the order parameter
[Reserved: the order parameter as the quantity that is zero in the symmetric phase and non-zero below the transition, with its symmetry group and number of components as the only data that matter for universality; the catalogue — magnetization (\(\Z_{2}\) or \(\SO(3)\)), density difference, condensate wavefunction (\(\U(1)\)), nematic director; spontaneous symmetry breaking as the selection of one of degenerate minima, the classical counterpart of the mechanism in Electroweak Unification and the Higgs Boson; Goldstone modes for continuous symmetries.]
Where mean field fails
[Reserved: the Ginzburg criterion comparing the fluctuation of the order parameter within a correlation volume to its mean value, and the resulting estimate of the reduced-temperature window in which mean-field theory is wrong [Ginzburg:1960]; the upper critical dimension \(d_{c}=4\) above which mean field is exact; why superconductors have a Ginzburg window too narrow to observe while fluids and magnets do not — the quantitative reason mean field survived so long.]
Critical exponents, scaling and universality
The exponents and the inequalities they obey
[Reserved: definitions of \(\alpha,\beta,\gamma,\delta,\nu,\eta\) in terms of the reduced temperature \(t=(T-T_{c})/T_{c}\); the thermodynamic inequality \(\alpha+2\beta+\gamma\geq2\) [Rushbrooke:1963] and its companion \(\gamma\geq\beta(\delta-1)\) [Griffiths:1965], proved from convexity alone; the empirical fact that both are satisfied as equalities, which is the clue scaling explains; the review that organized the field [Fisher:1967].]
Thermodynamics alone, using only the convexity of the free energy, bounds the measured critical exponents by \(\alpha+2\beta+\gamma\geq2\) [Rushbrooke:1963] and \(\gamma\geq\beta(\delta-1)\) [Griffiths:1965]. What is observed, in fluids and in magnets alike and across every universality class measured, is that both hold as equalities to within the experimental uncertainties [Fisher:1967] [Sengers:1986]. Of the six exponents \(\alpha,\beta,\gamma,\delta,\nu,\eta\) only two are independent. That is a fact about nature, not about thermodynamics, and it is the clue from which the scaling picture was read.
Derivation. Widom's hypothesis [Widom:1965] is that the singular part of the free energy per unit volume is a generalized homogeneous function of the reduced temperature \(t=(T-T_{c})/T_{c}\) and the ordering field \(h\),
with \(\Phi\) a scaling function finite and non-zero at the origin. Two differentiations extract the exponents. The specific heat is \(c\sim-T\pp^{2}f_{s}/\pp T^{2}\sim\abs{t}^{-\alpha}\) at \(h=0\), which is what fixes the prefactor in Equation (123.6). The order parameter is \(m=-\pp f_{s}/\pp h\sim\abs{t}^{2-\alpha-\beta\delta}\) at \(h=0\), and identifying this with \(\abs{t}^{\beta}\) gives
The susceptibility is \(\chi=-\pp^{2}f_{s}/\pp h^{2}\sim\abs{t}^{2-\alpha-2\beta\delta}\), and identifying it with \(\abs{t}^{-\gamma}\) gives, using Equation (123.7),
which is the Griffiths relation as an equality. Adding \(\alpha+2\beta\) to Equation (123.8) and using Equation (123.7) once more,
the Rushbrooke relation as an equality. Both are consequences of the single homogeneity assumption Equation (123.6), which also leaves exactly two free exponents — the scale \(2-\alpha\) and the ratio \(\beta\delta\) — as observed. The hypothesis itself is not derived here; it is explained by the renormalization group of The Renormalization Group, in which the homogeneity is the statement that the coarse-graining flow has a fixed point.
∎Corresponding states and universality
[Reserved: Guggenheim's collapse of the coexistence curves of eight different fluids onto a single cubic curve with \(\beta\approx1/3\), not the mean-field \(1/2\) [Guggenheim:1945]; the parallel measurement of the same exponent in uniaxial magnets; universality classes labelled by dimensionality and order-parameter symmetry alone; the modern precision values for the three-dimensional Ising and XY classes [Pelissetto:2002] and the fluid data [Sengers:1986].]
Plotted in reduced variables, the liquid–vapour coexistence curves of eight chemically unrelated substances — among them neon, argon, krypton, xenon, nitrogen, oxygen, carbon monoxide and methane — fall on one and the same curve, and that curve is a cube root rather than the square root of mean-field theory: \(\beta\approx1/3\), not \(\beta=1/2\) [Guggenheim:1945]. The same exponent, within uncertainty, is measured for the spontaneous magnetization of a uniaxial ferromagnet near its Curie point, a system with nothing microscopically in common with a fluid [Fisher:1967]. Modern determinations for the three-dimensional Ising class [Pelissetto:2002] and for fluids [Sengers:1986] agree to several decimal places. What survives at a critical point is therefore only the dimensionality of space and the symmetry and number of components of the order parameter; the chemistry drops out entirely.
Universality as the basin of attraction of a fixed point of the coarse-graining flow: that the couplings distinguishing one microscopic system from another are irrelevant in the technical sense, so that the singular behaviour is governed by the fixed point alone; and the accompanying calculation of the exponents, developed in the renormalization-group chapter
Scaling hypotheses
[Reserved: Widom's homogeneity assumption for the singular part of the free energy, from which every exponent relation follows and only two exponents remain independent [Widom:1965]; Kadanoff's block-spin construction, which explains the homogeneity by coarse graining and introduces the flow of couplings under a change of scale [Kadanoff:1966]; the data collapse of the equation of state onto a single scaling function as the experimental form of the hypothesis; hyperscaling \(2-\alpha=d\nu\) and where it fails.]
Correlation functions and the correlation length
[Reserved: the pair correlation \(G(r)\sim r^{-(d-2+\eta)}\ee^{-r/\xi}\) and the divergence \(\xi\sim\abs{t}^{-\nu}\) as the physical content of criticality [Ornstein:1914] [Fisher:1967]; scale invariance at \(T_{c}\), where \(\xi\) is infinite and the system looks the same at every magnification; measurement of \(\xi\) and \(\eta\) by neutron and light scattering; the fluctuation–response identity linking \(G\) to the susceptibility, an instance of Nonequilibrium Thermodynamics and Transport.]
The renormalization group and measured exponents
[Reserved: Wilson's realization that Kadanoff's flow can be made a calculation, with fixed points, relevant and irrelevant operators, and universality as the basin of attraction of a fixed point [Wilson:1971a] — developed in full in The Renormalization Group; the epsilon expansion about four dimensions and the first controlled exponent calculation [Wilson:1972]; the sharpest test on record, the specific-heat exponent of the helium lambda transition measured in microgravity as \(\alpha=-0.0127(3)\) [Lipa:2003], against theory at the same precision.]
The specific heat of liquid helium-4 at its lambda transition can be followed to within nanokelvin of \(T_{c}\) only in free fall, because on the ground the hydrostatic pressure head across a sample of any usable height smears the transition temperature. Measured aboard an orbiting laboratory, the singular specific heat yields \(\alpha=-0.0127(3)\) [Lipa:2003]: the exponent is small and negative, so the specific heat approaches a finite cusp rather than diverging. This is the most precisely measured critical exponent in physics, and it is a quantitative test of a calculated number rather than a qualitative confirmation of a picture.
The superfluid transition as the three-dimensional XY universality class, and the calculation of its specific-heat exponent to a precision comparable with the measurement, by renormalization-group and high-order perturbative methods; together with the hyperscaling relation that ties the negative sign of the exponent to the correlation-length exponent in three dimensions
Lower critical dimension and topological order
The Mermin–Wagner–Hohenberg theorem
[Reserved: the proof that a continuous symmetry cannot break spontaneously in one or two dimensions at any positive temperature, because Goldstone fluctuations diverge logarithmically [Mermin:1966], extended to superfluid and crystalline order [Hohenberg:1967]; the contrast with the discrete Ising symmetry, which does order in two dimensions (Section 123.3.2); why the theorem does not forbid a transition of a different kind, which is Section 123.6.2.]
The Berezinskii–Kosterlitz–Thouless transition
[Reserved: the two-dimensional XY model, whose low-temperature phase has no long-range order but algebraically decaying correlations; the transition as the unbinding of vortex–antivortex pairs when the entropy of a free vortex overcomes its logarithmic energy [Berezinskii:1971] [Kosterlitz:1973]; an essential rather than power-law singularity, so the exponent language of Section 123.5 does not apply; the predicted universal jump in superfluid density, \(\rho_{s}/T_{c}=2mk_{B}^{2}/\pi\hbar^{2}\) [Nelson:1977].]
Observation in helium films
[Reserved: the torsional-oscillator measurement of superfluid density in adsorbed helium-4 films of varying thickness, which found the onset temperature proportional to the areal superfluid density with the predicted universal coefficient [Bishop:1978]; why this is one of the very few parameter-free confirmations in the whole of critical phenomena; later realizations in two-dimensional atomic gases (Experiment: Bose–Einstein Condensation) and in Josephson junction arrays.]
Liquid crystals
Discovery and the mesomorphic states
[Reserved: Reinitzer's observation that cholesteryl benzoate melts twice, at \(418.6\,\mathrm{K}\) to a turbid fluid and at \(451.6\,\mathrm{K}\) to a clear one [Reinitzer:1888]; Lehmann's polarizing-microscope identification of the intermediate phase as a flowing crystal [Lehmann:1889]; Friedel's classification into nematic, smectic and cholesteric mesophases and his demonstration that they are true thermodynamic states, not emulsions [Friedel:1922].]
Cholesteryl benzoate has two melting points. Heated, the crystal becomes a turbid but freely flowing fluid at \(418.6\,\mathrm{K}\), and that fluid becomes an ordinary clear liquid at \(451.6\,\mathrm{K}\) [Reinitzer:1888]. The intermediate state is birefringent under crossed polarizers, so it is optically anisotropic like a crystal while flowing like a liquid [Lehmann:1889]. It is a genuine thermodynamic phase and not an emulsion or a mixture: its transitions are reversible and occur at reproducible temperatures, and its several varieties — nematic, smectic, cholesteric — are distinguished by their own textures and transitions [Friedel:1922]. Positional and orientational order are therefore separately destroyable, and a phase may possess one without the other.
The orientational order parameter as a traceless symmetric second-rank tensor, forced by the equivalence of the director and its negative; the cubic invariant that this symmetry permits and that therefore makes the isotropic–nematic transition weakly first order, with a small latent heat and a jump in the order parameter; and the purely entropic hard-rod mechanism that produces the same transition with no attraction at all
The nematic transition
[Reserved: the nematic order parameter as a traceless symmetric tensor, since the director \(\vect{n}\) and \(-\vect{n}\) are equivalent, and the cubic invariant that symmetry therefore permits — forcing the isotropic–nematic transition to be weakly first order [deGennes:1971]; Onsager's purely entropic transition for hard rods, driven by excluded volume alone [Onsager:1949b]; the Maier–Saupe mean-field theory with attractive coupling [Maier:1958]; the measured latent heat and order-parameter jump, both small.]
Defects, elasticity and textures
[Reserved: the Frank free energy with its splay, twist and bend elastic constants [Frank:1958]; disclinations of half-integer strength, whose existence is a direct consequence of the director's headless symmetry and whose classification is the homotopy theory of Topological and Metric Spaces; the schlieren textures by which they are counted under crossed polarizers; the monograph [deGennes:1993]; electro-optic switching as the application that made the phase ubiquitous.]