Critical phenomenon

A critical phenomenon is scale-invariant behavior near a continuous phase transition, described by universal critical exponents.

Universality of critical phenomena is the agreement of critical exponents or normalized scaling forms between microscopically different systems. Dimension, symmetry and interaction range help determine the universality class. A renormalization-group flow can explain this agreement by showing that different microscopic models approach the same fixed-point description, while irrelevant differences fade at large scales.

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.

A multicritical point where disordered, spatially uniform ordered and spatially modulated phases meet. The quadratic stiffness in one or more directions vanishes, so a positive higher-derivative term stabilises the Landau-Ginzburg theory. For a uniaxial case the kernel at the point has $p_{\perp }^2+p_{\parallel }^4$, with the coefficient of $p_{\parallel }^2$ tuned to zero.

For kernel ${\kappa }^{-1}{p}_{\perp }^2+{\mu }^{-1}{p}_{\parallel }^4+m^2$ with positive stiffnesses, preserve both derivative coefficients by $q_{\perp }=b{p}_{\perp }$ and $q_{\parallel }=b^{1/2}{p}_{\parallel }$. The Fourier field multiplier is $\tilde{Z}=b^{(2D+3)/4}$, the mass becomes $bm$, and a uniform source becomes $b^{(2D+3)/4}h$. The anisotropic effective dimension is $D-1/2$ and the Gaussian exponents are $\alpha =9/4-D/2$, $\Delta =(2D+3)/8$.

The endpoint of a gas-liquid phase coexistence curve, where the distinct liquid and gas states merge. For a smooth fluid equation of state, the critical isotherm has $p_v=p_{vv}=0$ at this point; this thermodynamic meaning differs from a stationary critical point of an arbitrary function.

A quantum critical point is a continuous zero-temperature phase transition driven by a nonthermal control parameter. Quantum and thermal fluctuations near it are organized by a dynamical critical exponent.

The quantum-critical regime is the finite-temperature region in which temperature exceeds the energy scale generated by detuning from a quantum critical point. Temperature then controls leading masses, relaxation rates, and correlation lengths.

Quantum critical scaling expresses observables as powers of temperature or detuning times universal functions of dimensionless ratios such as $\omega /T$, $vk/T$, and $\Delta /T$.

The dynamical critical exponent relates critical frequency and wavenumber scales by $\omega \sim k^z$. Relativistic critical theories have $z=1$.

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.

Melting is a solid-to-liquid phase transition that absorbs latent heat. It can be driven by heating, changing phase pressures, or adding solute that causes freezing-point depression. The ice layer between freshwater and cold brine illustrates melting into a cold liquid through dilution and a composition-dependent liquidus.

Freezing is a liquid-to-solid phase transition. A moving phase boundary releases latent heat; its rate is constrained by heat transport and the Stefan condition. In a solution the liquidus depends on composition, so being below the pure solvent's melting temperature does not imply that the solution must freeze.

A morphological instability amplifies shape perturbations of a solidification interface. Protrusions may receive a larger diffusive supply or encounter larger undercooling, while curvature-induced melting-temperature depression suppresses fine corrugations. Solute rejection can produce constitutional supercooling, destabilizing a planar front and leading to cells or crystal dendrites.

A direct transition from a solid phase to a gaseous phase. Heated interplanetary dust or a comet can lose material through this process, changing its size and radiation-pressure coefficient or destroying it before gravitational or drag-driven evolution is complete.

For a locally linear ice temperature with positive gradient $G$ into the solid, a corrugated sublimation front receives a perturbation in heat flux proportional to its horizontal wavenumber $\alpha$. The Gibbs-Thomson effect suppresses short waves, giving the displayed quasi-static growth rate. It requires $V/(\kappa \alpha )\ll 1$ and $|\sigma |/(\kappa {\alpha }^2)\ll 1$, as well as a perturbation depth short enough for the linear background approximation. Unstable modes have $0<\alpha <\sqrt{G/\Gamma }$.

For a sublimating surface with exponentially absorbed volumetric heating, use a frame moving into the solid at speed $V$. Its steady heat equation is $k{T}^{\prime \prime }+\rho c_{p}V{T}^{\prime }+q_0{e}^{-\lambda z}=0$. The surface Stefan condition is $k{T}^{\prime }(0)=\rho LV$. Integrating over the half-space gives the displayed energy-limited speed. The surface gradient is positive, but the far field is colder, so the temperature has a subsurface maximum where melting can begin.

An irradiated grain loses solid material by sublimation. Its lifetime depends strongly on temperature and composition. In a planetary dust tail, a short lifetime can truncate the distribution well before a relative orbital wrap.

Phase coexistence occurs when two or more thermodynamic phases have equal free energy under the same external conditions. A first-order transition crosses a coexistence locus, where the competing phases are simultaneously stable.

A system can split into macroscopic regions with distinct local order parameters, densities or compositions. In a short-range system at phase coexistence, regions of two phases with means $M_1,M_2$ and volume fractions $q,1-q$ have spatial mean $M=q{M}_1+(1-q)M_2$. Their interface contribution is subextensive, so their thermodynamic free-energy density is the corresponding convex combination of the phase densities. This can lie below the homogeneous-branch Landau free energy at the same $M$. The equilibrium constrained order-parameter free energy is consequently convexified, with a straight coexistence segment described by the Maxwell construction. A nonconvex local potential remains useful for homogeneous branches, metastability and interfaces, but must not be identified with the full equilibrium constrained density throughout a coexistence interval.

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.

For a bulk free-energy density $f(\varphi )$, two compositions coexist across a flat interface whenThe first condition equates the chemical potentials, and the second equates the pressure. Equivalently the same straight line of slope $\mu$ is tangent to the graph of $f$ at both compositions. A stable coexistence pair uses a supporting tangent, so the resulting mixture lowers the bulk free energy.

Adding $d\varphi$ to a free-energy density adds a constant at fixed total compositional order parameter. In the common-tangent construction for phase coexistence, it shifts the tangent slope by $d$ while preserving its points of contact. Thus coexistence compositions are unchanged, whereas their chemical potential shifts. This does not assert unchanged equilibrium composition at a fixed externally imposed chemical potential.

A continuous phase transition has no latent heat and develops fluctuations on a diverging correlation length. Its singular observables are characterized by critical exponents.

A critical exponent describes a power-law singularity near a continuous phase transition, such as $\xi \sim |t{|}^{-\nu }$ or $\chi \sim |t{|}^{-\gamma }$.

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.

The mean-field predictions model large clusters by branching processes. Rigorous results apply in specified high-dimensional or sufficiently spread-out regimes, using tools such as lace expansion and the percolation triangle condition. The full list involves different observables; the triangle condition alone should not be presented as proving every spatial exponent. With these exponents $2\beta +\gamma =3$, whereas $d\nu =d/2$, explaining the failure of naive hyperscaling relation above six.

The critical cluster mass-radius scaling exponent $D_f$ describes clusters of mass roughly $r^{D_f}$ at radius $r$. The displayed relation is the below-upper-critical-dimension scaling prediction, not an identity from the definition. In the planar scaling description $D_f=91/48$.

The exponent $\delta$ describes the tail of the critical cluster seen from a specified vertex of a graph. If the expected number of size-$s$ clusters per site scales as $s^{-\tau }$, weighting by cluster size gives $1/\delta =\tau -2$ when these scaling laws hold.

The leading power $\beta$ of the percolation probability approached from above the percolation critical probability. It quantifies how the density of the infinite percolation cluster first becomes positive.

The reduced temperature is a dimensionless distance from a critical temperature. Multiplying it by a nonzero constant does not change any critical exponent.

The heat-capacity critical exponent is defined by the singular part $c_{\mathrm{s}}\sim |t{|}^{-\alpha }$ as the reduced temperature tends to zero.

The order-parameter critical exponent is defined in the ordered phase by $m\sim (-t{)}^{\beta }$ at zero conjugate field.

The magnetic-susceptibility critical exponent is defined by the zero-field magnetic susceptibility $\chi \sim |t{|}^{-\gamma }$. Separate amplitudes, and occasionally separate exponents, may occur on the two sides of the transition.

The critical-isotherm exponent is defined at the critical temperature by $m\sim |B{|}^{1/\delta }$ as the conjugate field $B$ tends to zero.

The correlation-length critical exponent is defined by $\xi \sim |t{|}^{-\nu }$. At a renormalization-group fixed point, its reciprocal is the positive eigenvalue of the temperature-like scaling direction.

In the analysis of a phase transition, a critical dimension is a spatial dimension at which the character of fluctuations and critical behavior changes. Below a lower critical dimension the relevant ordered phase cannot persist; above an upper critical dimension mean-field critical exponents apply. This is distinct from the critical dimension of string theory, where cancellation of quantum anomalies selects a target-spacetime dimension.

The lower critical dimension is the spatial dimension at or below which fluctuations prevent the ordered phase associated with a proposed finite-temperature transition. A short-range model with discrete symmetry normally has $d_{\mathrm{l}}=1$, whereas continuous $O(2)$ symmetry has $d_{\mathrm{l}}=2$.

The Berezinskii–Kosterlitz–Thouless transition in the two-dimensional XY model separates a low-temperature phase with algebraically decaying correlations from a high-temperature phase with unbound vortices and exponentially decaying correlations. It does not produce conventional long-range order.

In the dilute-vortex description of the two-dimensional XY model, $K={\rho }_s/(k_{B}T)$ is dimensionless stiffness and $y$ is phase vortex fugacity. A phase vortex has logarithmic energy $\pi {\rho }_s\log (L/a)$, while its placement entropy is $2k_B\log (L/a)$. Small phase vortex pairs screen the logarithmic interaction, lowering the stiffness. Near $K=2/\pi$, the rescaled variables $X=2-\pi K$, $Y=4\pi y$ obey $X^{\prime }=Y^2$, $Y^{\prime }=XY$ to leading order. The invariant $X^2-Y^2$ leads to an essential correlation-length singularity $\xi \sim a\exp (C/\sqrt{T-T_c})$, not a finite power-law exponent.

The renormalized phase stiffness has this discontinuity at the Berezinskii–Kosterlitz–Thouless transition. The algebraic correlation function has exponent $1/4$ at the transition from the ordered side.

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.

Quasi-long-range order means that correlations decay algebraically rather than approaching a nonzero constant. It occurs in the low-temperature phase of the two-dimensional XY model, where spin waves destroy true long-range order but bound vortex pairs preserve power-law correlations.

A correlation function measures the joint fluctuation of observables at separated points. For a scalar field, the connected two-point function is $G(x-y)=⟨\varphi (x)\varphi (y)⟩-⟨\varphi (x)⟩⟨\varphi (y)⟩$.

The four-point correlation function measures a time-ordered product of four field insertions. For a centered scalar field its disconnected terms are the three products of two-point functions, and subtracting those products leaves the connected correlation function. In a free Gaussian scalar vacuum the connected four-point function vanishes, by Wick's theorem, so only the three paired products remain. The connected four-point function of phi-fourth theory starts with one quartic vertex; amputation removes its external propagators. The full correlator, its connected part and the four-point one-particle-irreducible correlation function should therefore be distinguished.

The correlation function with a single field insertion is its expectation value. In a translation-invariant vacuum it is independent of position. A tadpole subtraction chooses a linear counterterm, or shifts the expansion point of the field, so that this expectation is zero. This removes attached tadpole diagrams from higher connected correlation functions without discarding the physical need to renormalize the chosen vacuum.

The connected correlation of two observables is $⟨AB{⟩}_c=⟨AB⟩-⟨A⟩⟨B⟩$. It vanishes in a product state when $A$ and $B$ act on disjoint factors.

When an energy Hamiltonian contains $-h{\sum }_n{\sigma }_n$, differentiating its partition function gives $\chi =({\beta }_{\mathrm{th}}/N)\mathrm{Var}({\sum }_n{\sigma }_n)$. Translation invariance converts this to the connected-correlation sum. A dimensionless source ${\beta }_{\mathrm{th}}h$ removes the explicit inverse-temperature factor. In continuum physical coordinates the lattice sum includes the site-density factor $a^{-D}$. A selected pure thermodynamic phase is needed below an ordered transition to avoid a macroscopic mixture contribution.

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.

In the scalar-field source Legendre transform, differentiating $h=\delta \Gamma /\delta m$ with respect to the source and using $G=\delta m/\delta h$ yields this inverse-kernel relation. For a scalar quartic Landau-Ginzburg theory, ${\Gamma }_{\mathrm{L}}^{(2)}=-{\nabla }^2+r_0+u_0{m}^2/2$, so the connected response is the Green kernel of that operator. A one-momentum representation requires a translationally invariant background.

An amputated connected correlation function is obtained by removing external propagators. Amputation using full propagators eliminates external self-energy decorations; internal exchange diagrams remain. It is distinct from a one-particle-irreducible correlation function. In a scalar theory with vanishing one-point function, at zero momentum the four-point quantity satisfies ${\mathcal{A}}_4=-{\Gamma }^{(4)}+3[{\Gamma }^{(3)}{]}^2/{\Gamma }^{(2)}$ with a convention in which connected diagram vertices are minus effective-action derivatives. For a constant scalar source, the Legendre relation $J={\Gamma }^{\prime }(\varphi )$ gives $G^{\prime \prime }=1/{\Gamma }^{\prime \prime }$ and $G^{\prime \prime \prime \prime }=-{\Gamma }^{\prime \prime \prime \prime }/({\Gamma }^{\prime \prime }{)}^4+3({\Gamma }^{\prime \prime \prime }{)}^2/({\Gamma }^{\prime \prime }{)}^5$. Dividing by the four full external propagators proves the formula.

A two-point correlation function is the expectation of a product of fields or observables at two points. Its connected version subtracts the product of the one-point expectations.

The dynamic structure factor is the space-time Fourier transform of an equilibrium two-point correlation function. It resolves fluctuations by angular frequency and wavevector and is related to the dissipative part of a retarded Green function by the Fluctuation-dissipation theorem.

The static structure factor is the equal-time spatial Fourier transform of a two-point correlation function. Depending on convention it is obtained either by integrating the dynamic structure factor over frequency or by taking an appropriate zero-frequency limit.

For $N$ equal scattering sites on one polymer, define $g(\mathbf{k})=N^{-1}{\sum }_{m,n}⟨e^{i\mathbf{k}\cdot ({\mathbf{R}}_n-{\mathbf{R}}_m)}⟩$. It has $g(0)=N$; its normalized form factor is $g/N$. Under a rotationally invariant joint distribution, each separation vector has a uniform direction at fixed magnitude $r$. The angular average is $\frac{1}{2}{\int }_{-1}^1{e}^{ikr\mu }d\mu =\sin (kr)/(kr)$, with value one at $r=0$. Thus $g$ depends only on $k=|\mathbf{k}|$, and $g(k)=N^{-1}{\sum }_{m,n}⟨\sin (k|{\mathbf{R}}_n-{\mathbf{R}}_m|)/(k|{\mathbf{R}}_n-{\mathbf{R}}_m|)⟩$. The diagonal self terms are essential in a finite chain and give the large-$k$ limit one for a discrete Gaussian chain.

Expand the isotropic angular factor as $\sin (kr)/(kr)=1-k^2{r}^2/6+O(k^4{r}^4)$. For a polymer scattering function and finite fourth pair moments, the constant double sum gives $g(0)=N$, and the quadratic sum is determined by the radius of gyration:In the range $k{R}_g\ll 1$, this is also approximated by $e^{-k^2{R}_g^2/3}$. The coefficient follows from isotropy and the definition of $R_g$ and does not require a Gaussian chain.

The correlation length is the characteristic distance over which a connected correlation function decays. It diverges at an ordinary continuous critical point as $\xi \sim |t{|}^{-\nu }$.

A volume over which order-parameter fluctuations are correlated. For one isotropic correlation length $\xi$, its scale is ${\xi }^D$; with anisotropic lengths it is the product of the appropriate directional lengths. A Ginzburg criterion compares fluctuations averaged over this volume with the squared mean order parameter.

For a stable homogeneous scalar quartic saddle with gradient coefficient one, the Ornstein--Zernike correlation function has denominator $q^2+r_0+u_0{m}_0^2/2$. At zero source and $u_0>0$, the correlation length is $r_0^{-1/2}$ in the disordered phase and $(-2r_0{)}^{-1/2}$ on either selected ordered branch. Linear thermal tuning of $r_0$ gives correlation-length critical exponent $\nu =1/2$ with different amplitudes on the two sides.

An order parameter vanishes in a symmetric phase and becomes nonzero when that symmetry is spontaneously broken.

A field conjugate to a scalar order parameter $m$ contributes $-hm$ to its free-energy density. Differentiating the equilibrium free energy with respect to $h$ gives $-m$, with density or extensive normalization stated consistently. In a ferromagnet the conjugate field is magnetic; for a fluid-density order parameter it is related to a chemical-potential difference. It can explicitly break the symmetry whose spontaneous breaking defines the ordered phase.

A vector order parameter has several components transforming together as a vector under spatial rotations. A polar order parameter is an example. Its functional derivative $\delta F/\delta \mathbf{p}$ has one component for each component of the order parameter.

A conserved order parameter is a field whose spatial integral remains fixed under the dynamics. A local composition difference in a closed mixture is a standard example.

A topological defect is a localized obstruction to extending an ordered field smoothly and nonvanishingly through space. Its charge is determined by the homotopy class of the order parameter around the defect.

A spontaneous symmetry breaking chooses order-parameter directions independently in regions that have not been in causal contact. Mismatches around disconnected components, loops or enclosing spheres can trap domain walls, cosmic strings or magnetic monopoles. The correlation length cannot exceed the causal horizon, so one monopole per horizon volume estimates the smallest causal initial abundance up to geometrical and efficiency factors.

// Target: classical-field-theory-soliton.bigb

A domain wall is a codimension-one configuration interpolating between disconnected components of a vacuum manifold. In a one-dimensional system with broken discrete symmetry it is a point defect of finite energy, so positional entropy produces a nonzero density at every positive temperature.

For a Landau-Ginzburg theory with positive gradient stiffness $\kappa$, $r<0$ and quartic coupling $u>0$, the degenerate minima are $\pm M_0$, $M_0^2=|r|/u$. A planar domain wall solves $\kappa {\varphi }^{\prime \prime }=r\varphi +u{\varphi }^3$. Multiplying by ${\varphi }^{\prime }$ and using the limiting minima gives $\kappa ({\varphi }^{\prime }{)}^2/2=u({\varphi }^2-M_0^2{)}^2/4$. Separating this equation yields the displayed profile, and integrating its excess energy gives the positive surface tension $2\sqrt{2\kappa }|r{|}^{3/2}/(3u)$. Its area cost is subextensive compared with bulk volume, permitting phase-separated common-tangent construction states when the average order parameter is constrained.

A vortex is a topological defect around which the phase of a complex order parameter has nonzero integer winding. In two dimensions an isolated vortex in a phase-only model has energy proportional to the logarithm of the system size.

A quantum vortex has integer winding number of a condensate's complex argument. For atomic mass $m$, its superfluid velocity gives circulation $2\pi \hslash q/m$ for integer $q$. A straight vortex has a density-depleted core and, outside it, tangential velocity $q\hslash /(mr)$. Its long-range flow energy is proportional to $q^2\log (R/a)$, while its angular momentum has the sign of $q$.

In healing-length units a straight quantum vortex obeys $f^{\prime \prime }+f^{\prime }/r-{\mathcal{N}}^{2}f/r^2+(1-f^2)f=0$ and $f\to 1$. Inserting $f=1-a/r^2+\cdots$ makes its order-$r^{-2}$ balance $-{\mathcal{N}}^2+2a=0$, proving the tail. The physical healing length convention must be stated; the deficit of density $1-f^2$ has twice the leading coefficient of the amplitude deficit $1-f$.

For a centered quantum vortex in a two-dimensional Thomas–Fermi condensate, use an intermediate radius ${\xi }_{\mathrm{core}}\ll L\ll R$. The approximately uniform core contributes $\pi n_c{\hslash }^2{\mathcal{N}}^2[\log (L/\xi )+L_{0|\mathcal{N}|}]/m$. The outer superfluid velocity is $\hslash \mathcal{N}/(mr)$, so its kinetic contribution is $(\pi {\hslash }^2{\mathcal{N}}^2/m){\int }_L^R{n}_c(1-r^2/R^2)dr/r$. Integration cancels $L$ and produces the constant $-1/2$. The displayed result is the matched small-core approximation at fixed particle number, not a formula for arbitrarily large winding at fixed cloud size. The angular momentum is exactly $L_z=N\hslash \mathcal{N}$ for a normalized axisymmetric single-winding state.

For a cubic-quintic Gross–Pitaevskii equation with normalized bulk amplitude one, a straight quantum vortex of integer winding number $\mathcal{N}$ has radial equation $R^{\prime \prime }+R^{\prime }/r-{\mathcal{N}}^{2}R/r^2+(1-\alpha R^2-\beta R^4)R=0$. With $\alpha +\beta =1$, the derivative of the nonlinear term at $R=1$ is $-2(1+\beta )$. Balancing the order-$r^{-2}$ centrifugal term with the bulk amplitude restoring term proves the displayed tail for a stable nondegenerate bulk state, $1+\beta >0$. A vortex's amplitude tends to one; its full complex field tends to $e^{i\mathcal{N}\theta }$, not a direction-independent constant. At $\beta =-1$ the stated inverse-square expansion cannot cancel the centrifugal term.

If a continuous nonzero complex field tends uniformly to one on large circles, its values on a sufficiently large circle lie in a disk about one that excludes zero. That disk is contractible, so the field's winding number on that circle is zero. A lone unit-charge quantum vortex therefore cannot satisfy the literal boundary $\psi \to 1$. For a vortex it is the wave amplitude, rather than the entire complex field, that can approach one; $f(r)e^{i\theta }$ gives a simple illustration of direction-dependent far-field complex argument.

A quantum vortex becomes energetically favourable in a rotating reference frame when $\Delta E-\Omega \Delta L_z<0$. For a singly quantized vortex in a long radially Thomas–Fermi condensate, the cutoff-shell estimate isIntegrating the trapped number density against $u_{\theta }^2$ produces the logarithm and constant; integrating it against angular momentum per particle gives $\Delta L_z\simeq \pi H{n}_0\hslash R_r^2/2$. Resolved-core corrections can alter the nonlogarithmic constant. This criterion compares equilibrium rotating-frame energies and does not remove a dynamical vortex-entry barrier.

Single-valuedness of the condensate field gives total complex argument change $2\pi q$ on a closed loop avoiding zeros, where $q$ is an integer winding number. Integrating the superfluid velocity around that loop gives the displayed circulation. It is unchanged by smooth deformations of the loop that do not cross a quantum vortex.

A quench is a change of control parameters faster than the system can equilibrate. The subsequent relaxation can trap domains and topological defects.

A quench abruptly changes temperature or another control parameter and then studies the subsequent nonequilibrium relaxation of the order parameter.

A liquid crystal has orientational or partial translational order while retaining some fluid mobility.

An isotropic phase has no preferred spatial direction. In a liquid crystal, its orientational nematic order parameter vanishes, although the molecules themselves may be anisotropic.

A polar liquid crystal distinguishes head from tail and is described at leading order by a vector polar order parameter $\mathbf{p}$.

A nematic liquid crystal identifies the director $\mathbf{n}$ with $-\mathbf{n}$. Its rotational order is described by a symmetric traceless tensor such as $Q_{ij}=S(n_i{n}_j-{\delta }_{ij}/3)$.

The twisted nematic field effect switches a nematic liquid crystal between a twisted optically rotating state and a largely field-aligned state. With adiabatic optical following in a twisted nematic and crossed polarizers aligned to the two planar surface directors, a quarter-turn cell passes light at low voltage and suppresses it at high voltage. A rear reflector permits reflective operation with little power expended on light generation.

When the birefringence separation of optical propagation constants $\Delta k=2\pi \Delta n/\lambda$ is large compared with the spatial rate of rotation of the optical axes, light with linear polarization in a local eigenpolarization follows the twisting nematic director. Thickness alone is insufficient if birefringence vanishes. Strong field alignment along the propagation direction suppresses the transverse optical anisotropy and hence the quarter-turn polarization rotation.

The Fréedericksz transition is a field-driven orientational instability of a confined nematic liquid crystal. Elasticity and surface anchoring resist the electric field or magnetic torque; a lowest allowed distortion mode becomes unstable at a finite threshold. Positive dielectric anisotropy of a nematic permits a normal electric field to tilt a planar anchored director.

For planar strong anchoring and a quarter-turn base profile, ${\varphi }^{\prime }=\pi /(2L)$ and $\theta =0$. Expanding the one-dimensional twisted nematic energy through quadratic order gives $\Delta F=\frac{1}{2}\int [K_1{\theta }^{\prime 2}+\{(K_3-2K_2)(\pi /(2L){)}^2-\Delta ϵE^2/(4\pi )\}{\theta }^2]dz$. The lowest zero-endpoint mode is $\sin (\pi z/L)$. Its coefficient vanishes at the displayed threshold, with $V=EL$, provided the bracket and anisotropy are positive. Halving the electric coupling doubles $V_c^2$.

The dielectric response differs along and perpendicular to the nematic director. In Gaussian units the orientational electric contribution at prescribed field is $-\Delta ϵ(\mathbf{E}\cdot \mathbf{n}{)}^2/(8\pi )$; positive anisotropy favors alignment with the electric field. The electric term does not receive the additional one-half convention used in writing the Frank elastic terms.

A nematic director is a unit vector specifying the local unoriented axis of a nematic liquid crystal. Head-tail symmetry identifies $\mathbf{n}$ with $-\mathbf{n}$, so the order-parameter space is a Real projective space rather than a sphere.

For a unit nematic director, the bulk elastic density is $[K_1(\nabla \cdot \mathbf{n}{)}^2+K_2(\mathbf{n}\cdot \nabla \times \mathbf{n}{)}^2+K_3|\mathbf{n}\times \nabla \times \mathbf{n}{|}^2]/2$. The three positive moduli penalize nematic splay, nematic twist, and nematic bend. Boundary elastic terms, chirality, or varying nematic order require additional contributions.