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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]

An experimental physicist who coauthored the self-assembly of ordered colloidal crystals in a thin nematic liquid crystal. Particles distort the surrounding orientational order and create topological defects; sharing those distortions can bind neighboring particles. The observed structures depended on the type of local ordering, showing how a host material's defects can organize suspended particles into larger patterns.^[ref]

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 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.

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.

The Berezinskii–Kosterlitz–Thouless transition is a defect-unbinding transition of a two-dimensional system with a compact continuous phase, such as the short-range XY model. At long wavelengths, smooth spin waves have energywhere ${\rho }_s$ is the phase stiffness. Gaussian averaging givesThus smooth fluctuations already destroy true nonzero magnetization in an infinite two-dimensional system, in accordance with the Mermin-Wagner theorem. The low-temperature phase instead has quasi-long-range order, with algebraically decaying correlations. The stiffness in its asymptotic exponent is the renormalized long-distance stiffness, not merely the microscopic coupling.

The phase is periodic modulo $2\pi$, so it also permits topological defects with integer winding $n$. A phase vortex has $|\nabla \theta |=|n|/r$, and integration givesA singly charged phase vortex can be placed in roughly $(L/a{)}^2$ locations, giving entropy $2k_B\log (L/a)$. This energy-entropy balance suggests that isolated vortices become favorable when the effective stiffness falls to $2k_{B}T/\pi$. Below the transition, vortex-antivortex binding predominates; above it their unbinding screens the logarithmic interaction and produces a finite correlation length. A single-vortex estimate identifies the mechanism but does not include screening by smaller pairs.

The dilute-vortex vortex-pair renormalization flow incorporates this screening. With $K={\rho }_s/(k_{B}T)$ and dimensionless core fugacity $y$, one conventional leading-order normalization givesThe first equation says polarizable phase vortex pairs reduce stiffness. The second compares the phase vortex's position scaling dimension two with its logarithmic energy cost $\pi K$. Numerical coefficients in $y$ depend on its definition, but the zero-fugacity threshold $K=2/\pi$ is invariant. For $K>2/\pi$, sufficiently small fugacity flows to zero, leaving a line of Gaussian spin-wave renormalization-group fixed points with varying algebraic exponent. On the other side of the separatrix, fugacity grows and the dilute approximation eventually ceases to apply.

Near the endpoint set $X=2-\pi K$ and $Y=4\pi y$. To leading order $X^{\prime }=Y^2$ and $Y^{\prime }=XY$, so $X^2-Y^2$ is invariant. The critical separatrix approaches $(0,0)$. Just on the disordered side this invariant is $-{\epsilon }^2$, proportional to the reduced temperature deviation for a generic path through the transition. Hence $X^{\prime }=X^2+{\epsilon }^2$, whose integration gives a time in logarithmic scale of order $1/\epsilon$. Stopping when the fugacity becomes order one yieldswith nonuniversal $B>0$. This derives the essential singularity and explains why assigning a finite ordinary correlation-length exponent is inappropriate.

At the transition the limiting stiffness from below has the universal stiffness jumpAbove $T_c$, the thermodynamic long-distance stiffness vanishes and correlations decay exponentially. The singular free-energy contribution is of order ${\xi }^{-2}$ and is exponentially small in $1/\sqrt{\tau }$, so this infinite-order transition is unlike a Landau transition with a discontinuous order parameter or an ordinary power-law heat-capacity singularity. The renormalization mechanism is developed in Kosterlitz's original vortex analysis. The transition separates bound vortices and algebraic order from unbound vortices and short-range order, while neither finite-temperature phase has conventional spontaneous XY magnetization.

For a uniform field the gradient term vanishes and minimization over $p=|\mathbf{p}|$ givesWhen $a<0$, the nonzero minimum haswhile its direction is arbitrary by rotational symmetry. A sudden quench creates many independently oriented domains. Their mismatches contain gradients and topological defects, whose motion and pair annihilation are slow, so the uniform state is not reached promptly.

A topological defect occurs where the local order cannot be continued smoothly while retaining its surrounding texture. At a nematic disclination core, the director model fails and the orientational order may decrease.

For a planar texture $\mathbf{n}=(\cos \varphi ,\sin \varphi )$,Because $\mathbf{n}\equiv -\mathbf{n}$, half-integer winding strengths are allowed. The original sketches use $\varphi =s\alpha +{\varphi }_0$ to show $s=+1/2,-1/2,+1$, corresponding to director turns $+\pi ,-\pi ,+2\pi$. A radial $+1$ pattern and an azimuthally rotated version have the same winding.

These are planar labels. In unrestricted three dimensions an integer line texture may escape by tilting out of the plane; half-integer lines have a different classification. A planar disclination strength must not be identified automatically with a three-dimensional hedgehog point charge.

A topological defect can remain after spontaneous symmetry breaking when the field cannot be continuously put into the vacuum everywhere. If the vacuum manifold is $\mathcal{M}=G/H$, its topology classifies possible obstructions. Causally separated regions can settle into different local order-parameter configurations; joining them during a phase transition can trap defects. This is a formation mechanism, not an assertion that every transition creates every possible defect.

In three spatial dimensions the homotopy classification of cosmological defects gives areDisconnected vacua can meet across a wall; winding around a line can obstruct filling its core; a sphere around a point can carry monopole charge. Textures differ from permanent localized cores: they can collapse and unwind when the field leaves the vacuum manifold. Local gauge and global symmetry breaking also differ in their long-range fields and radiation. The original cosmological analysis connects vacuum topology with domain and string formation and their subsequent evolution.^[ref]

The energy budget determines cosmological importance. Nonrelativistic monopoles dilute approximately as $a^{-3}$, so a large primordial abundance can become unacceptable as radiation falls faster. Inflation can dilute an earlier population, provided the same defect-producing transition is not repeated afterwards. A slowly evolving wall network dilutes especially slowly and can dominate the energy budget or create excessive anisotropy.

Strings can lose energy by intercommutation and loop formation, with loops radiating gravitational waves or field quanta. A horizon-scaling cosmic string network whose characteristic separation is of order the horizon can have density ${\rho }_{\mathrm{string}}\sim \mu /t^2$ in natural units, keeping a limited fraction of the total rather than accumulating indefinitely. This differs from an ideal frozen noninteracting network's simple $a^{-2}$ dilution law. Similarly, a horizon-scaling domain wall network has ${\rho }_{\mathrm{wall}}\sim \sigma /t$, while a frozen wall network follows $a^{-1}$. The evolutionary assumptions must accompany any quoted scaling.

Defects can seed density wakes and other inhomogeneities, perturb the microwave background and generate non-Gaussian signatures. A sufficiently straight ideal cosmic string has conical spacetime and can create doubled background images; a cosmic-string network can generate gravitational-wave bursts and a stochastic background. Textures can produce localized microwave-background temperature features while collapsing. These signatures constrain defect abundance and symmetry-breaking scale; an isolated suggestive feature is not a unique defect detection.

The predictions depend on network evolution, reconnection, loop sizes, radiation channels and the relation between symmetry breaking and inflation. topology identifies which configurations are possible; formation and evolution determine whether they survive and whether their gravitational or radiative effects are observable. A defect's unseen energy does not automatically make it the dominant cold dark matter.

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.

Tom Kibble related spontaneous symmetry breaking in the early universe to the formation of topological defects. Different regions can choose different states on a vacuum manifold. Its homotopy groups determine whether boundaries, loops or surrounding surfaces can be continuously unwound, giving possible domain walls, cosmic strings or monopoles. His 1976 paper also examines their subsequent cosmological effects. The connection makes the global shape of a space of field states relevant to large-scale physical structures.^[ref]

A vacuum manifold is the space of field values minimizing the potential in the specified homogeneous classical description. For a global symmetry group $G$ acting transitively on one orbit of minima, a chosen vacuum with stabilizer $H$ gives that orbit as $G/H$.

The orbit need not be connected: a real scalar field with two symmetry-related minima provides a disconnected example. Restricting to the identity component requires the corresponding connected subgroup and stabilizer, rather than automatically identifying every $G/H$ with one connected component. Continuous vacuum directions motivate Goldstone modes, while nontrivial surrounding maps can classify topological defects. In a gauge theory, some minimum-field values are gauge-equivalent, so the distinction between field-space minima and physical vacuum labels must also be stated.

A theoretical physicist who developed early ideas about phase transitions in two-dimensional systems with continuous symmetry. His work helped establish how strong fluctuations can remove ordinary long-range order while topological defects still control a sharp transition. John Michael Kosterlitz explicitly credits Berezinskii's earlier related ideas in the history of the Berezinskii–Kosterlitz–Thouless transition.^[ref]

A fluid vortex is a region in which velocity field circulates about an axis or center. Vorticity measures local rotation, while circulation measures the integral of velocity around a closed path. A phase vortex is instead a topological defect defined by winding of an order parameter; superfluid examples connect the two notions.