Order parameter

An order parameter is a quantity chosen to distinguish phases and measure their degree of order. For a conventional spontaneous symmetry breaking transition, it vanishes in the symmetric phase and acquires a nonzero value in an ordered phase once a state or symmetry-breaking limit is selected.

The magnetization of a ferromagnet is a vector example. In a nematic liquid crystal, an ordinary average molecular axis can vanish because opposite directions are equivalent, so the nematic order parameter is a tensor instead. Choosing the correct object is therefore part of identifying the order. In a finite symmetric ensemble, averaging over differently oriented ordered states can also give zero; correlations or an infinitesimal conjugate field reveal the underlying order. Some phases require nonlocal diagnostics beyond this conventional local-order-parameter description.

An order-parameter space is the set of allowed equilibrium values of an order parameter after accounting for the relevant symmetry breaking. When a symmetry group $G$ leaves a selected state invariant only under a subgroup $H$, the distinct symmetry-related states are often represented by the quotient $G/H$.

For a planar magnet with a fixed nonzero magnetization magnitude and free direction, the space is a circle. For a nematic orientation, opposite directions represent the same state, giving a different space. Its topology constrains possible topological defects; the physical identification of states matters as much as the number of coordinates.

For a two-component order parameter of nonzero magnitude, write $m_x+i{m}_y=\bar{m}{e}^{i\theta }$. Its phase is the orientation angle modulo $2\pi$, not a thermodynamic phase such as an ordered or disordered state. A global continuous symmetry rotates this angle uniformly; spin waves are smooth spatial changes, while a phase vortex has nonzero winding around an excluded core. At a zero of the magnitude the angle is undefined, allowing the topology of an orientation field to change.

For a homogeneous bosonic condensate, the one-body density matrix retains a nonzero long-distance component. More generally, macroscopic occupation is identified by an eigenvalue of that density matrix proportional to particle number. This gives a number-conserving description even when the one-point annihilation-field expectation vanishes. In a paired fermionic system the corresponding macroscopic order belongs to the two-particle density matrix, rather than an electron single-particle condensate.

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 spatial configuration whose surrounding ordered field cannot be smoothly extended through a core while staying everywhere in its order-parameter space. The field may lose its order or leave that space inside the core. Winding around a phase vortex is a simple example.

The topology of a surrounding loop can protect a line defect in three dimensions; a surrounding sphere can characterize a point defect. A nematic disclination uses an unoriented director, whereas a phase vortex uses an angle defined modulo $2\pi$, so their possible charges differ. Protection prevents a local smooth removal under the stated conditions, but a defect can still annihilate with another defect, escape through a boundary or disappear when order is destroyed. Its core and long-range distortion also have an energy cost, which affects formation and motion.

Disconnected vacuum components support walls, noncontractible vacuum loops support line defects, and nontrivial vacuum two-spheres support monopoles in three spatial dimensions. Allowed topology does not alone predict the number formed or their subsequent survival.

An extended order-parameter winding associated with a nontrivial third homotopy group of a vacuum manifold in three spatial dimensions. A texture can collapse and unwind when its field leaves the vacuum manifold; it differs from a permanent monopole or string core.

A three-dimensional point defect characterized by how the surrounding nematic director covers orientation space. Its hedgehog charge is not the same quantity as a planar disclination winding strength.

A point-defect texture that can compensate the radial boundary condition of a sphere with homeotropic anchoring, giving a dipolar particle–defect configuration in a uniform far field. It carries the opposite hedgehog charge to the radial particle boundary under a consistent convention.

A defect localized at a boundary of an ordered medium. A spherical inclusion with degenerate planar anchoring in a nematic liquid crystal can have a bipolar texture with two surface boojums, unlike a bulk point-defect pair.

The Kibble mechanism explains how a spontaneous symmetry breaking transition can form topological defects. Regions that order without coordinating their choices can settle at different points of a vacuum manifold. Smooth relaxation removes many mismatches, but some cannot be removed while the field remains in that vacuum manifold everywhere.^[ref]

The possible defects depend on the homotopy groups of the vacuum manifold: disconnected components permit domain walls, noncontractible loops permit cosmic strings, and nontrivial surrounding two-spheres permit magnetic monopoles in three spatial dimensions. This is the homotopy classification of cosmological defects, rather than a prediction that every permitted defect must occur.^[ref]

For a direct winding example, an order parameter represented by a complex scalar field with fixed nonzero magnitude has a circular vacuum manifold. If its order-parameter phase changes by $2\pi$ around a closed spatial loop, it cannot extend continuously across a spanning disk while keeping its magnitude nonzero. A zero of the order parameter supplies the core; in three dimensions the corresponding defect can be a string.

A formation estimate needs a domain correlation length $\xi$ and a model for the chance of trapping a defect. For point defects, a dimensional estimate is $n_{\mathrm{defect}}\sim p_{\mathrm{defect}}/{\xi }^3$, with dimensionless formation efficiency $p_{\mathrm{defect}}$. Causality restricts how ordering can spread; taking a horizon-sized $\xi$ supplies a scale estimate, not by itself a guaranteed number of defects in every horizon volume. Subsequent annihilation or expansion can further change the abundance. Kibble's original discussion separately estimates the initial domain scale from fluctuations and follows later evolution.^[ref]

A domain wall is a topological defect separating regions whose order parameter approaches different disconnected components of a vacuum manifold. In $d$ spatial dimensions, a flat domain wall extends in $d-1$ directions: a surface in three dimensions, a line in two, and a localized defect in one. A scalar quartic domain wall provides an explicit smooth interpolation between two degenerate minima.

For the one-dimensional, zero-field Ising model with nearest-neighbor coupling $J>0$, a reversed bond costs $2J$. A single domain wall can occupy order $N$ positions on a chain of $N$ sites, giving a free-energy balance $\Delta F\simeq 2J-k_{B}T\log N$. At every positive temperature, positional entropy defeats a fixed wall cost as $N\to \infty$. Thermally excited domain walls then destroy long-range order, despite two ordered zero-temperature ground states. This argument assumes short-range interactions and finite wall cost; it does not apply unchanged to long-range interacting chains. In higher dimensions the wall cost grows with its area, allowing different ordering behavior.^[ref]

If the typical wall separation grows as cosmic time, surface energy per volume is of order wall tension divided by time. This decreases more slowly than the usual radiation-era total density and can lead to a wall-domination problem.

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.

The integer $q$ is a winding number: a counterclockwise circuit returns the same complex order parameter after its angle changes by $2\pi q$. Smooth changes that keep the order parameter nonzero on that circuit cannot change $q$. In a continuum superfluid, the core suppresses the magnitude, allowing the winding obstruction to end there; in an XY model the microscopic core instead lies beyond the smooth-field approximation. The vortex-antivortex binding figure illustrates a locally nonzero charge with zero charge on a surrounding large circuit.

A Rankine vortex models a finite rotating core joined continuously to an irrotational exterior. In cylindrical polar coordinates, its radial and vertical velocities vanish and its azimuthal velocity has the displayed form, where $a$ is the core radius and $\Omega$ is a signed angular velocity. The axial vorticity is $r^{-1}{\partial }_r(r{u}_{\theta })$, equal to $2\Omega$ inside and zero outside. The velocity is continuous at $r=a$, although its first radial derivative has different one-sided values. The circulation outside the core is $2\pi \Omega a^2$.

For a steady Rankine vortex in a fluid of constant density $\rho$ with gravity $-gk$, the Euler equations give $p_r=\rho u_{\theta }^2/r$ and $p_z=-\rho g$. Integrating on the two sides and matching pressure at $r=a$ givesSetting a constant surface pressure yieldsThus the free surface and its radial slope match at the core edge. The central depression is ${\Omega }^2{a}^2/g$, twice the depression at $r=a$. Bernoulli's theorem applies throughout the exterior; the rotational core instead uses the radial momentum balance above.

An XY model orientation takes values on a circle, so a circuit around an excluded defect core can carry integer winding number. Smooth single-valued spin waves cannot change that integer without a core event or boundary crossing. The locally quadratic order-parameter phase action therefore describes a sector, rather than automatically summing every compact-field configuration. Vortex-antivortex unbinding changes the large-distance correlation function and renormalized phase stiffness, even though ordinary long-range order is already absent in two dimensions at positive temperature.

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 $q$.

Rotating a trap anisotropy can nucleate vortices in an atomic condensate. Expansion followed by absorption imaging resolves depleted cores, while varying rotation and waiting after removing the stirrer tests their preparation and persistence. The interpretation is quantized rotational response, rather than only the appearance of a condensed density profile.

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.

If a condensate order parameter stays nonzero along a closed loop, its phase changes by an integer multiple of $2\pi$. Broken rotational symmetry does not destroy this single-valuedness constraint. In an inhomogeneous ring the winding integer need not label an angular-momentum eigenstate.

An event changing the phase winding of a condensate around a loop. A continuous nonzero order parameter cannot change its winding integer; a phase-slip path must pass through a locally vanishing amplitude or an equivalent vortex-crossing process. This use concerns a superfluid field, rather than only the relative phase of two classical oscillators.

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 is a phase with orientational or partial positional order while retaining some fluid-like motion. Its structure lies between a fully disordered isotropic fluid and a crystal with positional order in all spatial directions.

In a nematic liquid crystal, molecular axes prefer a common unoriented nematic director without forming a three-dimensional lattice of fixed positions. The nematic order parameter measures that alignment and can vary across the sample. Orientational distortions cost elastic energy and can form topological defects. Applied fields and boundary anchoring can change the director pattern, thereby changing optical propagation. The phase should be specified, since different liquid-crystal phases break different symmetries and need different order parameters.

A strongly chiral liquid-crystal phase containing local double-twist order and frustrated regions. Ordered blue phases I and II have body-centered cubic and simple-cubic director/defect structures respectively, while blue phase III is disordered. The periodicity is orientational/defect order, not a molecular positional crystal lattice.

A columnar liquid crystal has columns arranged with two-dimensional positional order transverse to the column axes and relatively fluid positional order along them. Orientational order alone, as in a nematic liquid crystal, does not specify this structure.

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 has orientational order without a periodic three-dimensional lattice of molecular positions. Its local preferred axis is the nematic director $\mathbf{n}$, with $\mathbf{n}$ and $-\mathbf{n}$ representing the same physical alignment.

Because the two directions are equivalent, an ordinary vector average can vanish even in an aligned sample. A symmetric traceless nematic order parameter instead measures the alignment; a common convention is $Q_{ij}=S(n_i{n}_j-{\delta }_{ij}/3)$. Spatial variations produce nematic splay, nematic twist and nematic bend, while boundary anchoring constrains the texture. Optical transmission through polarizers can reveal director changes because aligned molecules generally respond differently along and across the axis. Such a texture measurement must be distinguished from direct imaging of fixed molecular positions.

A surface preference for the nematic director to lie tangent to the local boundary. Degenerate planar anchoring leaves the in-plane direction unselected; a directionally treated surface can instead select an azimuth. Uniform axial alignment satisfies degenerate planar anchoring on every side point of a cylindrical channel.

A surface boundary preference for the nematic director to align normal to the local surface. Around a sphere it favors radial orientation near the surface, which must match the far-field texture.

A cholesteric liquid crystal is a chiral nematic liquid crystal whose local nematic director rotates helically in space. The pitch is the distance over which the director completes a full turn. It retains locally nematic orientational order without smectic density layering; the helical modulation should not be mistaken for a layer lattice.

For slender hard rods of length $L$ and diameter $D\ll L$, the leading pair excluded volume is the displayed function of their mutual angle $\gamma$. Alignment lowers mutual obstruction and increases positional entropy, while narrowing the orientation distribution lowers orientational entropy. Their competition can drive an isotropic-to-nematic transition without attractive forces. The natural concentration scale is $c{L}^{2}D$ of order one. Finite rod diameter, flexibility and interactions beyond the hard-core second-virial approximation change quantitative predictions.

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 optic 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.

An apolar axis model with pair energy $-({\mathbf{s}}_i\cdot {\mathbf{s}}_j{)}^2$ is invariant when each molecular representative is independently reversed. The vector average changes under these replacements, but the nematic order parameter made from second moments does not. In an unbiased statistical ensemble the vector average vanishes; an individual finite representative need not have zero average. For independently chosen signs of a perfectly aligned axis, $⟨|N^{-1}{\sum }_i{\mathbf{s}}_i{|}^2⟩=1/N$. This distinguishes exact head-tail invariance of an observable from vanishing only after an average.^[ref]

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.

For a laterally uniform nematic director $(\cos \theta \cos \varphi ,\cos \theta \sin \varphi ,\sin \theta )$, its divergence is $\cos \theta {\theta }^{\prime }$ and its twist invariant is $-{\cos }^2\theta {\varphi }^{\prime }$. Thus $f=K_1{\cos }^2\theta +K_3{\sin }^2\theta$ and $g=K_2{\cos }^4\theta +K_3{\sin }^2\theta {\cos }^2\theta$. A normal electric field contributes $-c{\sin }^2\theta$. The Euler-Lagrange equations for two fields are $(g{\varphi }^{\prime }{)}^{\prime }=0$ and $f{\theta }^{\prime \prime }+f_{\theta }{\theta }^{\prime 2}/2-g_{\theta }{\varphi }^{\prime 2}/2+2c\sin \theta \cos \theta =0$.

Nematic bend curves the integral lines of the nematic director. For a unit field, $(\mathbf{n}\cdot \nabla )\mathbf{n}=-\mathbf{n}\times (\nabla \times \mathbf{n})$, so the bend energy measures curvature along those lines.

Nematic twist rotates the nematic director around an axis perpendicular to it. The helix $\mathbf{n}=(\cos qz,\sin qz,0)$ has zero nematic splay and nematic bend, and twist density $K_2{q}^2/2$.

Nematic splay describes local spreading or convergence of the nematic director field. Its leading elastic cost is quadratic in the divergence and invariant under the head-tail identification $\mathbf{n}\sim -\mathbf{n}$.

A nematic disclination is a topological line defect of the unoriented director field. In a two-dimensional sample its point cross-section can have half-integer winding because $\mathbf{n}$ and $-\mathbf{n}$ represent the same state.

An equatorial disclination loop surrounding a roughly spherical inclusion with homeotropic anchoring. A meridional section cuts the loop twice. It can match a radial near-surface director to a uniform far field.

A neutral pair of opposite nematic disclinations can annihilate under dissipative relaxation, because the total topological charge of a two-dimensional nematic disclination is zero. Its far-field elastic energy increases as $2\pi K{\lambda }_0^2{q}^2\log (R/r_0)$ with separation $R$, so the pair has an attractive elastic force proportional to $-1/R$. As opposite cores meet, the nematic order parameter may vanish locally and the two windings disappear. Integer defects may first undergo energetic fission of an integer nematic disclination, leading to encounters of elementary half-charge defects. The actual paths and annihilation time require a dynamical law; the free energy alone does not specify core mobility. Pinning, imposed boundary winding or thermal defect production can obstruct a defect-free final state.

With constant amplitude outside a core, the planar nematic divergence-elasticity identity reduces the elastic density to $K{\lambda }_0^2|\nabla \chi {|}^2/2$. A topological charge of a two-dimensional nematic disclination $q$ has $\chi =q\theta +{\chi }_0$ and $|\nabla \chi {|}^2=q^2/r^2$. Integrating over an annulus gives the displayed energy, plus finite core and boundary terms. The outer cutoff is the sample size, compensating-defect separation, or another screening length. An isolated defect in an infinite ordered plane has divergent energy. A neutral pair $q,-q$ separated by $R$ instead has $E_{\mathrm{pair}}=2\pi K{\lambda }_0^2{q}^2\log (R/r_0)$ plus core terms, since its far-field gradients cancel.

In a strictly planar nematic liquid crystal, a charge-one nematic disclination is a winding-two map into the circle ${\mathbb{RP}}^1$. Its nonzero fundamental group class prevents complete unwinding while the surrounding loop stays fixed, but permits splitting into two charge-$1/2$ defects. For separation $r_0\ll d\ll L$, the two near fields contribute $\pi K{\lambda }_0^2(1/4+1/4)\log (d/r_0)$ while their common charge-one far field contributes $\pi K{\lambda }_0^2\log (L/d)$. Compared with the unsplit defect, the change isIt is negative at sufficiently large scale separation. Thus integer-defect fission is energetically favorable despite conservation of total topological charge. This leading-log argument proves failure of global energetic stability, not absence of kinetic barriers or pinning. If escape into the third dimension is allowed, an integer winding can also become topologically trivial in ${\mathbb{RP}}^2$.

For a planar tangential charge-one nematic disclination, $\nabla \cdot Q=-({\lambda }^{\prime }/2+\lambda /r){\mathbf{e}}_r$. This follows by differentiating the tensor dyad, using $\nabla \cdot {\mathbf{e}}_{\theta }=0$ and $({\mathbf{e}}_{\theta }\cdot \nabla ){\mathbf{e}}_{\theta }=-{\mathbf{e}}_r/r$. With bulk density $a{\lambda }^2/2+b{\lambda }^4/4$ and divergence elastic density $K({\lambda }^{\prime }/2+\lambda /r{)}^2/2$, radial variation of $2\pi \int rfdr$ givesFor $a<0$, a regular melted core has $\lambda =O(r^2)$ and an ordered far field has ${\lambda }_0=\sqrt{-a/b}$. Linearizing the bulk derivative gives $\lambda /{\lambda }_0=1-K/(2|a|r^2)+O(r^{-4})$. The core scale is of order $\sqrt{K/|a|}$; the amplitude disturbance has an integrable algebraic tail, rather than exactly compact support.

For a continuous director angle around a closed counterclockwise circuit, the enclosed topological charge is $q=\Delta \theta /(2\pi )$. Since $\theta$ is defined modulo $\pi$, elementary nematic disclinations can have $q=\pm 1/2$.

A two-dimensional director winding can sometimes unwind when the director may tilt out of the plane. For a three-dimensional nematic the fundamental group of the order-parameter space is ${\pi }_1({\mathbb{RP}}^2)=\mathbb{Z}/2\mathbb{Z}$, so two disclination lines are topologically equivalent to none.

In the one-elastic-constant approximation, slow director distortions have a quadratic gradient energy with one modulus. For a planar angle field it reduces to $F=(K/2)\int |\nabla \theta {|}^2$ up to the convention-dependent effective modulus.

For a planar nematic director $\mathbf{n}=(\cos \chi ,\sin \chi )$ and constant amplitude ${\lambda }_0$, the nematic order parameter isDifferentiation gives $\nabla \cdot Q={\lambda }_0(-\sin 2\chi {\chi }_x+\cos 2\chi {\chi }_y,\cos 2\chi {\chi }_x+\sin 2\chi {\chi }_y)$. The coefficient matrix is orthogonal, proving the identity. Hence $K|\nabla \cdot Q{|}^2/2$ is the angle-gradient energy with effective modulus $K{\lambda }_0^2$. The full tensor-gradient norm instead obeys ${\sum }_{ijk}({\partial }_k{Q}_{ij}{)}^2=2{\lambda }_0^2|\nabla \chi {|}^2$ in this normalization; the two local densities must not be confused.

A nematic order parameter measures orientational alignment without distinguishing the two ends of a molecular axis. For unit molecular axes $\mathbf{u}$ in three dimensions, a common tensor convention isIt is symmetric and traceless, and is unchanged by $\mathbf{u}\mapsto -\mathbf{u}$. An isotropic distribution gives $Q=0$ even though individual molecules retain definite axes.

For a uniaxial nematic liquid crystal, this convention gives $Q=S(\mathbf{nn}-I/3)$, where $\mathbf{n}$ is the unoriented nematic director and $S=⟨(3{\cos }^2\theta -1)/2⟩$. Perfect alignment has $S=1$ and eigenvalues $2/3,-1/3,-1/3$. Other normalizations multiply $Q$ by a constant, so coefficients in a Landau-Ginzburg theory must use the same convention. The tensor representation handles head–tail symmetry and permits alignment to vanish at defect cores.

For a narrow nematic liquid crystal orientation distribution about its unoriented nematic director, expand $S=⟨(3{\cos }^2\theta -1)/2⟩=1-3⟨{\theta }^2⟩/2+O(⟨{\theta }^4⟩)$. This proves the displayed root-mean-square width. It does not determine the full distribution or a full width at half maximum. If the two small transverse tilt components are independent equal-variance Gaussians with variance ${\sigma }^2$, then $⟨{\theta }^2⟩\simeq 2{\sigma }^2$, so $\sigma \simeq \sqrt{(1-S)/3}$.

For scalar nematic order parameter $S=⟨(3{\cos }^2\theta -1)/2⟩$, the displayed angle has ${\cos }^2{\theta }_{\mathrm{eff}}=⟨{\cos }^2\theta ⟩$. It summarizes a second moment, not the literal mean angle. For example an isotropic orientation distribution has $S=0$, giving ${\theta }_{\mathrm{eff}}=\arccos (1/\sqrt{3})$; the average acute angle to its unoriented axis is instead one radian, obtained by integrating $\theta \sin \theta$ over $[0,\pi /2]$.

A uniaxial nematic order parameter has one distinguished nematic director and equal eigenvalues in its perpendicular plane. In the displayed normalization the eigenvalues are $\lambda ,-\lambda /2,-\lambda /2$, giving $\mathrm{Tr}{Q}^2=3{\lambda }^2/2$ and $\mathrm{Tr}{Q}^3=3{\lambda }^3/4$. Positive and negative $\lambda$ describe prolate and oblate orientational anisotropy, respectively; reversing the nematic director leaves the tensor unchanged.

For $S=⟨3\mathbf{s}{\mathbf{s}}^{\mathsf{T}}-I⟩=Q(3\mathbf{n}{\mathbf{n}}^{\mathsf{T}}-I)$ and unit molecular orientations $\mathbf{s}$, contraction with $\mathbf{n}$ gives $2Q=3⟨(\mathbf{s}\cdot \mathbf{n}{)}^2⟩-1$. Since the squared projection is between zero and one, the displayed bounds follow. The eigenvalues in this normalization are $2Q,-Q,-Q$. A differently normalized nematic order parameter has correspondingly rescaled bounds; its numerical coefficients cannot be transferred unchanged.

Under rotation by a rotation matrix, a symmetric second-rank tensor transforms by conjugation, so its traces of powers are invariant. For a three-dimensional traceless second-rank tensor, the Cayley-Hamilton theorem implies $\mathrm{Tr}{Q}^4=(\mathrm{Tr}{Q}^2{)}^2/2$. In two dimensions its eigenvalues are $s,-s$, so $\mathrm{Tr}{Q}^3=0$. These identities organize the Landau-de Gennes free energy.

The Landau-de Gennes free energy is a rotational symmetry preserving Landau free energy of the nematic order parameter. One three-dimensional normalization through fourth order isThe cubic invariant generically produces a first-order phase transition. Its absence in two dimensions is an algebraic identity, not a guarantee that a mean-field approximation captures all fluctuations.

The leading scalar coupling of an electric field to the nematic order parameter is quadratic in the field and linear in the tensor. For uniaxial nematic order, it is $-3\chi \lambda E^2({\cos }^2\vartheta -1/3)/2$. When $\chi \lambda >0$, minimization aligns the nematic director with the field axis and adds $-\chi E^2\lambda$ to the scalar Landau free energy.

For a scalar Landau-de Gennes free energy with $\bar{b}>0$, $\bar{c}<0$, and coupling $-\chi E^2\lambda$ with $\chi >0$, the critical endpoint of a quartic Landau free energy is reached at the displayed field strength and ${\bar{a}}_c=3{\bar{c}}^2/(8\bar{b})$, ${\lambda }_c=-\bar{c}/(4\bar{b})$. It terminates the field-biased first-order phase transition when both field and quadratic coefficient can be tuned. This is a quartic small-field mean-field approximation; higher terms may alter its location.

An unoriented local axis: $\mathbf{n}$ and $-\mathbf{n}$ describe the same orientation.

Nonconserved relaxational order-parameter dynamics has ${\partial }_t\psi =-\Gamma \delta F/\delta \psi$ plus thermal noise. Its zero-wavenumber mode may decay.

For quadratic free energy with positive $a,\kappa$ and kinetic coefficient $\Gamma$, a deterministic initial Dirac delta distribution evolves under nonconserved order-parameter dynamics to the meanTaking the expectation removes the centered Gaussian white noise; the resulting diffusion-reaction equation is solved by the heat kernel. The integrated mean is $A{e}^{-\Gamma at}$, so spreading does not conserve the initial excess. The connected Fourier mode covariance grows from zero to $k_{B}T/(a+\kappa q^2)$ as $1-e^{-2\Gamma (a+\kappa q^2)t}$ by the Itô isometry. A realization retains equilibrium fluctuations after its mean has decayed. Pointwise continuum white-noise quantities require a coarse-graining cutoff.

For nonconserved order-parameter dynamics, the positive kinetic coefficient $\Gamma$ converts the functional derivative of the free energy into a relaxation rate through $\dot{p}=-\Gamma \delta F/\delta p$. Microscopic reversibility fixes the accompanying Gaussian white noise variance by the Model A fluctuation-dissipation relation.

For additive Gaussian white noise of variance ${\sigma }^2$, a Model A trajectory has path weight proportional toComparing a path with its time reverse yields the free-energy change and hence the thermal fluctuation-dissipation relation.

For additive Gaussian noise, a stochastic trajectory has probability proportional to the exponential of minus the squared noise history required to generate it. Rewriting that history in terms of the trajectory gives an Onsager--Machlup action; comparing it with the time-reversed action yields a path-probability ratio.

The Onsager–Machlup functional is the action in the exponential of an additive Gaussian stochastic path probability. For $\dot{x}=b(x)+\sigma \Lambda$, its elementary discretization contains $\int |\dot{x}-b(x){|}^2/(2{\sigma }^2)dt$, with convention-dependent Jacobian terms for multiplicative or state-dependent drift.

For additive Gaussian white noise and a time-even order parameter, a midpoint discretization of the Onsager–Machlup functional evaluates drift derivatives at the midpoint configurations. Reversing a trajectory visits the same midpoints in reverse order, so the noise-to-path Jacobian determinant is invariant under reversal. It may depend on the trajectory, and must be distinguished from the constant normalization of the noise measure.

For nonconserved order-parameter dynamics with $\dot{p}=-\Gamma \mu [p]+f$, the factor at one time step, apart from a path-independent power of the step size, isHere the fields have first been restricted to a finite spatial grid, and ${\mathcal{H}}_n$ is the Hessian matrix of the free energy.

For mixed conserved and nonconserved order-parameter dynamics, the required noises for a joint configuration and flux trajectory arewhere $\mathsf{D},\mathsf{G}$ represent the divergence and gradient, and $\Delta =\mathsf{DG}$ represents the Laplacian. The Jacobian matrix isTaking its Schur complement gives the joint factorFor periodic boundary conditions, discretizing the two spatial operators compatibly gives $\mathsf{D}=-{\mathsf{G}}^{\mathsf{T}}$, so $\Gamma I-M\Delta$ is the positive relaxation operator when $\Gamma ,M>0$. Reversal leaves ${\mathcal{H}}_n$ unchanged and reverses the sign of the flux. The product of the joint factors is consequently the same for both histories, justifying its cancellation from the joint path probability of an order parameter and its flux. The functional chain rule used in the action ratio holds in the continuum limit of this midpoint convention.

For nonconserved order-parameter dynamics, microscopic reversibility with equilibrium weight proportional to $e^{-F/(k_{B}T)}$ requiresIndeed the Onsager--Machlup path probability gives $\log (P_F/P_B)=-2\Gamma \int \dot{p}\delta F/\delta pdt/{\sigma }^2=-2\Gamma (F_2-F_1)/{\sigma }^2$. Equating this to the detailed balance value proves the relation. The functional chain rule is understood with the same midpoint convention as the path action.

After a sudden change from quadratic kernel $K_I(q)$ to $K_F(q)$, each Fourier mode is an Ornstein-Uhlenbeck process with decay rate $r(q)=\Gamma K_F(q)$. Its equal-time covariance interpolates exponentially between $k_{B}T/K_I(q)$ and $k_{B}T/K_F(q)$.

A conserved scalar order parameter evolves by transporting it rather than creating or destroying its total amount. A simple relaxational model has chemical potential $\mu =\delta F/\delta \varphi$, current $\mathbf{J}=-M\nabla \mu$ andwith compatible boundary conditions and optional conserved thermal noise. Integrating gives constant total $\int \varphi$ when boundary flux vanishes.

Linearizing a quadratic Landau-Ginzburg theory gives a decay rate $\Gamma (q)=M{q}^2(a+\kappa q^2)$. Away from criticality, positive $a$ yields ordinary diffusive $q^2$ scaling at small wavevector. At the Gaussian critical point $a=0$, the leading rate is instead proportional to $q^4$. Conservation forces the zero-wavevector mode to remain unchanged but does not by itself fix every critical decay exponent.

The positive order-parameter mobility $M$ relates the diffusive transport flux to the chemical potential gradient: $\mathbf{J}=-M\nabla \mu$. Combining this law with local conservation gives the Cahn--Hilliard equation when $\mu$ is the functional derivative of a composition free energy.

For the flux $W_{ij}=-M{\partial }_i{\mu }_j+N_{ij}$ of a vector order parameter, microscopic reversibility requires independent Gaussian white noise components with covarianceThe current changes sign under time reversal. The resulting path-probability ratio contributes $-(k_{B}T{)}^{-1}\int W_{ij}{\partial }_i{\mu }_j$ to its logarithm; integration by parts converts this to the free-energy loss from the conserved dynamics.

A hydrodynamic mode is a long-wavelength collective mode whose decay rate tends to zero because of a conservation law. Coupling to fast nonconserved variables renormalizes its diffusivity and eigenvector.