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The phase stiffness is the coefficient of $\frac{1}{2}\int d^{d}x(\nabla \theta {)}^2$ in the long-distance free energy. Its renormalized value determines the thermal phase fluctuations and algebraic correlation function of a two-dimensional O(N) model with $N=2$.

Operationally impose a small phase difference $\varphi$ between opposite sides of a square sample of side $L$. The smooth minimum has $\theta =\varphi x/L$, so the two-dimensional excess free energy isThe derivative concerns a prescribed twist at the stated observation scale; in $d$ dimensions its conventional size factor is $L^{d-2}$. This response definition includes thermal fluctuations and vortices after equilibration, whereas a microscopic gradient coefficient need not. A finite phase stiffness requires no nonzero spatially uniform order parameter; the Berezinskii–Kosterlitz–Thouless transition is the instructive example.

A phase transition is a change between equilibrium phases associated with nonanalytic thermodynamic behavior in an appropriate infinite-system limit. Melting, liquid–vapor coexistence and magnetic ordering illustrate different kinds. A phase diagram specifies the control parameters and the boundaries between phases.

At a first-order phase transition, a first derivative of the relevant thermodynamic potential changes discontinuously; thermal examples can have latent heat and phase coexistence. At a continuous phase transition, an order parameter can develop continuously while response functions and a correlation length become singular. Real finite systems show rounded signatures, so measurements usually compare heat capacity, density, structural scattering or an order parameter across a controlled scan. Hysteresis is evidence of delayed conversion or metastability, rather than the definition of an equilibrium transition.

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 pseudogap is a partial suppression of low-energy spectral weight without all the properties of a fully developed energy gap. In high-temperature superconductors it is identified through several probes and can occur outside the superconducting phase.

A depletion in a spectrum does not by itself establish a particular order parameter or pairing mechanism. Different measurements weight different momenta and excitations, so temperature and momentum dependence help distinguish competing explanations. The term names an observed class of behavior while leaving its microscopic interpretation to further evidence.^[ref]

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.

Giorgio Parisi's 1979 work replaced a single replica order parameter by an infinite hierarchy in the mean-field theory of spin glasses. The claim concerns a thermodynamic model, rather than inequivalence of the labels attached to two finite laboratory copies.^[ref]

For equilibrium states with weights $w_{\alpha }$ and local means $m_i^{\alpha }$, define $Q_{\alpha \gamma }=N^{-1}{\sum }_i{m}_i^{\alpha }{m}_i^{\gamma }$. The distribution $P_J(q)={\sum }_{\alpha ,\gamma }{w}_{\alpha }{w}_{\gamma }\delta (q-Q_{\alpha \gamma })$ records their similarities. This state-based overlap distribution is distinct from the finite instantaneous spin overlap used in the triangle example.^[ref]

A theoretical physicist who, with Philip Warren Anderson, developed a description of spin glasses using an order parameter based on persistent local-spin correlations. A material can lack a net magnetization while individual spins retain memory of their orientation. Edwards's own account connects this problem with ideas from disordered polymer networks and explains the use of replicas in calculating thermodynamic behavior.^[ref]

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 scalar order parameter with symmetry $q\mapsto -q$ has only even powers in its zero-field free energy. A conjugate field contributes $-hq$. Without this symmetry, odd terms can be allowed and change the transition structure.

Spin-triplet p-wave pairing has pair spin $S=1$ and orbital angular momentum $L=1$, allowing multiple order parameters beyond conventional BCS theory with zero orbital angular momentum. ^[ref]

Specifying this symmetry does not determine its microscopic attraction mechanism. A short exchange-symmetry check explains why these quantum numbers fit fermions. The spin-one-half triplet states $|\uparrow \uparrow ⟩$, $(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩)/\sqrt{2}$ and $|\downarrow \downarrow ⟩$ are symmetric under exchange. A stipulated orbital factor proportional to relative momentum $k_x$ changes sign under $\mathbf{k}\mapsto -\mathbf{k}$. Multiplying any symmetric spin function by that odd orbital factor gives an antisymmetric pair wave function, as required by Fermi-Dirac statistics. By comparison, an even s-wave orbital factor needs the antisymmetric spin singlet state. This checks compatibility with exchange statistics; it neither selects a stable phase nor derives its transition temperature.

Spontaneous symmetry breaking occurs when the action and equations have a symmetry but a chosen ground state is invariant only under a proper subgroup. Acting with the broken symmetry generates a degenerate vacuum manifold. An order parameter distinguishes the selected states: the equations do not prefer its direction, yet a particular state does. This lets one use the symmetry of the laws to predict excitations around a state that has less symmetry, notably the Goldstone bosons of a broken continuous symmetry.

A finite system with symmetry-preserving boundary conditions can have a symmetric ground state; a symmetric statistical mixture of broken states also has vanishing mean order parameter. A nonzero mean therefore requires a stated selection procedure. The symmetry-breaking order of limits takes the thermodynamic limit in a small applied source before removing that source. The finite-volume angular source selection example makes the distinction explicit for a continuous symmetry.

Statistical field theory assigns probability distributions to spatial field configurations, often obtained by coarse graining an interacting many-particle system. A local order parameter $\varphi (\mathbf{x})$ can encode a slowly varying magnetization or a density difference. In thermal equilibrium, a typical effective description useswhere $\mathcal{F}$ is an effective free energy and the functional measure specifies the configurations being counted.

Minimizing $\mathcal{F}$ gives a mean-field theory, while integrating over neighbouring configurations includes thermal fluctuations. Landau-Ginzburg theory adds a gradient cost, expressing the energetic penalty for rapidly varying the order parameter.^[ref] The field description makes correlation length and critical phenomenon accessible without tracking each microscopic particle. It still requires a microscopic cutoff or other definition of its short-distance degrees of freedom; a continuum integral is not automatically a complete microscopic model.

A superconducting fluctuation is a variation of the superconducting order parameter about its mean value. Near a transition, transient pairing correlations can influence transport or magnetic response even when long-range phase-coherent superconductivity is absent.

This makes it necessary to distinguish a local pairing tendency from a fully established zero-resistance state. The strength and extent of fluctuation effects depend on dimensionality, disorder and the distance from the transition. A measured excess conductivity needs comparison with the normal-state background and alternative mechanisms.^[ref]

The magnitude of one magnetic flux unit for an electron-pair order parameter, whose electric charge has magnitude $2e$. The enclosing contour must satisfy the flux condition in fluxoid and flux quantization in a superconductor; it is not a statement that every finite area inside the material contains an integer number of these units.

The superconducting mixed state is the equilibrium vortex regime of a type-II superconductor, between its lower critical magnetic field and upper critical magnetic field in the ideal bulk description. Magnetic flux enters through Abrikosov vortices, while the surrounding material retains a nonzero superconducting order parameter.

Each vortex has a suppressed-order core and circulating electric currents; in the appropriate bulk limit, a singly wound vortex carries one superconducting flux quantum. Their mutual interactions can produce an Abrikosov vortex lattice. Increasing the field increases the vortex density until superconductivity is destroyed at the upper critical field. Defects can immobilize vortices through flux pinning, so a prepared sample's magnetization need not coincide with the equilibrium curve. Spatial coexistence here means vortex cores within superconducting matter, rather than arbitrary macroscopic patches of two phases.^[ref]

A thermal phase fluctuation is a finite-temperature fluctuation of the phase of a complex order parameter. For a linearly dispersing phase mode, its infrared variance is proportional to $T\int d^{d}k/k^2$, which prevents true finite-temperature long-range order when $d\le 2$.

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 thermodynamic phase is a form of macroscopic equilibrium characterized by its structure, symmetries and thermodynamic response. Away from phase transitions, its free-energy density varies smoothly with the control parameters. Different phases can share a phase coexistence boundary. An ordered magnetic phase is distinguished from a disordered phase by its spontaneous order parameter; positive and negative ordered states are different pure thermodynamic phases related by spin inversion symmetry.

Thermodynamic stability means that allowed small perturbations do not lower the appropriate thermodynamic potential from an equilibrium state. The constraints select that potential: an isolated system maximizes entropy, a system at fixed temperature and volume minimizes Helmholtz free energy, and one at fixed temperature and pressure minimizes Gibbs free energy.

For an unconstrained scalar order parameter $\varphi$, a stationary state has ${\partial }_{\varphi }F=0$, and positive curvature ${\partial }_{\varphi }^{2}F>0$ gives local stability. A distant lower minimum can still make it metastable rather than globally stable. In an ordinary additive fluid, positive isothermal compressibility expresses resistance to density perturbations: increasing pressure should reduce volume. Loss of the corresponding curvature can mark a spinodal. These statements depend on the ensemble and permitted perturbations; mechanical equilibrium alone does not establish thermal, chemical or global stability.

A phenomenological near-transition relaxation equation for a charged complex order parameter, with a positive real relaxation coefficient in its simplest dissipative form. Under $\Psi \mapsto e^{iq\chi /\hslash }\Psi$, $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$ and $\varphi \mapsto \varphi -{\partial }_t\chi$, the displayed time derivative transforms covariantly, as does the spatial derivative in the Ginzburg-Landau theory of superconductivity. Microscopic applicability depends on regime and scattering processes; this equation is distinct from a universal conservative condensate wave equation.