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A measure of spatial nonuniformity in the order parameter, with averages taken over one lattice cell. Set $X=|\psi {|}^2$. ThenEquality requires uniform $|\psi {|}^2$, so a nonconstant core pattern costs more in the quartic term than a uniform amplitude with the same mean squared amplitude. This identity alone does not compare two different vortex arrays: their admissible profiles must also satisfy the field equations.

An Abrikosov vortex has a core where the superconducting order parameter is suppressed, surrounding circulating electric currents, and a winding of its quantum phase. A singly wound vortex carries one superconducting flux quantum in the appropriate bulk flux-quantization limit. The exact contour statement is fluxoid and flux quantization in a superconductor; the enclosed magnetic flux alone need not equal a flux quantum on a contour with appreciable electric current.

Alexei Abrikosov showed how a type-II superconductor can admit magnetic flux while remaining superconducting over most of its volume. In the mixed state, circulating supercurrents surround vortex cores where the superconducting order parameter is suppressed. An array of these vortices can minimize the free energy between the lower and upper critical magnetic fields. The resulting vortex lattice explains a regime that lies between complete magnetic-field expulsion and destruction of superconductivity.^[ref]

A physicist who studied superconductivity at Argonne.^[ref]

For an isotropic vector order parameter, an analytic scalar Landau free energy near the origin contains powers of $\mathbf{m}\cdot \mathbf{m}$. Odd powers of its magnitude are rotationally invariant but nonanalytic as functions of the Cartesian components at zero; excluding them requires analyticity, not merely writing the magnitude as a nonnegative scalar. With $d>0$, changing $b$ from positive to negative continuously creates minima at $|\mathbf{m}|=\sqrt{-b/(2d)}$, with an arbitrary direction.^[ref]

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.

How can something resist a twist when its orientation never settles down? The distinction is between long-range order, a nonzero limiting correlation function, and phase stiffness, a free energy cost for a slow spatial twist. In a uniform two-dimensional XY model at positive temperature, smooth spin waves can erase the first while leaving the second finite. Vortex-antivortex binding then decides whether that resistance survives on arbitrarily large scales.

Vadim Lvovich Berezinskii developed an earlier description of two-dimensional order. John Michael Kosterlitz and D. J. Thouless independently developed the defect mechanism; Kosterlitz's 2016 lecture explicitly acknowledges their initial ignorance of Berezinskii's work. The central step was to include rare vortices alongside smooth spin waves.^[ref]

The Mermin-Wagner theorem concerns continuous symmetry and sufficiently short-range interactions in the thermodynamic limit. It does not say that every two-dimensional system is disordered in every sense, nor that every continuous symmetry gives a Berezinskii–Kosterlitz–Thouless transition. Here the order parameter has a circular phase, permitting integer winding numbers. Finite-scale vortex renormalization explains why a finite sample can look coherent even when its infinite counterpart lacks long-range order.

An experimental condensed matter physicist who coauthored a phase-sensitive test of d-wave superconductivity using a tricrystal ring. A ring containing three grain-boundary Josephson junctions exhibited spontaneous half-quantum magnetic flux, while two-junction control rings did not. The geometry converts a sign change of the pairing order parameter into a measurable magnetic signal.^[ref]

A coarse-grained variable retains selected collective information while averaging over microscopic details. Dividing a magnetic sample into small cells and assigning each cell its mean magnetization gives a field that ignores individual spins but still describes variations over many lattice sites. The cell size should be specified relative to both microscopic spacing and the spatial changes of interest.

Different microstates can give the same coarse-grained variable. At thermal equilibrium, their statistical weights contribute an entropic term to the effective free energy; the effective cost is therefore not simply the microscopic energy evaluated at a smoothed configuration. In statistical field theory, a local order parameter is a common retained variable. Changing the averaging scale can change its effective parameters while leaving long-distance predictions consistent. Coarse graining is useful precisely because the retained field answers a chosen macroscopic question without reconstructing every microscopic coordinate.

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.

Fix the spatial mean of a scalar order parameter to $M$ and sum the Boltzmann weights of all configurations with that mean. The resulting constrained partition function $Z_M$ defines this free-energy density, with an additive normalization fixed consistently. For an applied conjugate field $h$, $Z(h)=\int dM{e}^{-{\beta }_{T}V[F_c(M)-hM]}$, up to the chosen measure normalization. The thermodynamic limit gives $f(h)={\inf }_M[F_c(M)-hM]$. A homogeneous Landau free energy is an approximation to this constrained potential; phase-separated configurations can convexify it in the full thermodynamic limit.

A correlation function quantifies how measurements at different points or times vary together. For a scalar order parameter, the connected equal-time correlation isThe subtraction removes a contribution due solely to the mean values. Positive or negative values indicate enhanced alignment or anticorrelation relative to that independent baseline.

Its spatial decay defines a correlation length, while its Fourier transform gives the corresponding static structure factor. Scattering measurements therefore reveal collective structure through correlations even when individual microscopic configurations cannot be imaged. Equal-time, time-dependent and quantum time-ordered correlations are different objects; the measurement and operator ordering must identify which one is intended.

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.

The critical region is the range of control parameters near a thermodynamic critical point where correlated order parameter fluctuations invalidate a simple mean-field approximation. For short-range scalar quartic theory below four dimensions, the dimensionless Ginzburg criterion ratio grows as $g{\xi }^{4-D}$ in fixed microscopic units. The range where this ratio is of order one or larger is fluctuation dominated. Its width depends on nonuniversal microscopic coefficients even though the limiting critical exponents can be universal.

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]

A superconductor whose pairing order parameter has the angular or crystal-symmetry structure of d-wave superconductivity. The sign of the gap depends on momentum direction and nodes can permit low-energy fermionic Bogoliubov quasiparticles.

A fermionic excitation obtained by a canonical particle–hole Bogoliubov transformation of a paired mean-field Hamiltonian. The uniform spin-singlet problem has the displayed positive energy and weights $|u{|}^2=(1+\xi /E)/2$, $|v{|}^2=(1-\xi /E)/2$. It interpolates between particle-like and hole-like excitations across the Fermi surface. These single-particle excitations differ from collective phase and amplitude modes of the order parameter.

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.

The field crossover exponent describes the field scale $h\sim |t{|}^{\Delta }$ at which the zero-field critical regime changes. In the two-variable scaling hypothesis for critical phenomena, its value is the ratio $y_h/y_t$ of magnetic and thermal scaling dimensions. It is distinct from either $y_h$ or the scaling dimension of the order parameter. The Widom scaling relation and differentiation of the homogeneous free-energy density give the displayed identities when ordinary scaling applies.

Freeze the magnitude of an order parameter and retain just one common orientation $\theta$ for $N$ rigidly aligned cells. Give this collective coordinate energy $E(\theta )=-Nh\cos \theta$ at positive temperature $T$. The Boltzmann distribution is the displayed normalized probability density function, where $k_B$ is the Boltzmann constant. Here $h$ is energy per cell, a different normalization from the source density in continuous vacuum selection by an external source. This is a deliberately restricted model of selection among already ordered orientations: it does not derive alignment or an ordering critical temperature from a spatial interaction.

Its mean orientation is an elementary ratio of integrals:At $h=0$ every angle has equal probability, and the numerator vanishes exactly. For small $x$, expansion of the exponential gives $m_N(h)=x/2+O(x^3)$ because the angular mean of ${\cos }^2\theta$ is $1/2$. For $x=1$ and $5$ the means are approximately $0.446390$ and $0.893383$. At fixed $h>0$, increasing $N$ concentrates the probability density function near $\theta =0$; expanding $\cos \theta =1-{\theta }^2/2+O({\theta }^4)$ gives angular variance approximately $1/x$. ConsequentlyThe uniform angular average remains a symmetric mixture if one sets $h=0$ first. The other order selects a definite direction. This is the continuous-orientation version of the symmetry-breaking order of limits; it illustrates selection without replacing the separate problem of stability against spatial fluctuations.

For a real unrestricted scalar order parameter and $A=A_2{m}^2/2+A_3{m}^3/3+A_4{m}^4/4$ with $A_3\ne 0$ and $A_4>0$, the disordered global minimum is replaced discontinuously by a nonzero minimum at the displayed phase coexistence condition. At coexistence $A=A_4{m}^2[m+2A_3/(3A_4){]}^2/4$. Ordered stationary points first appear at the spinodal point $A_2=A_3^2/(4A_4)$, while the disordered point loses local stability at $A_2=0$; neither local limit is the equilibrium transition condition.

For uniform superconducting carrier density, the displayed combination of enclosed magnetic flux and the circulating supercurrent is quantized on a closed contour where the order parameter does not vanish. It follows by integrating the gauge-invariant phase gradient. Magnetic flux alone is quantized only when the current contribution on that contour is negligible.