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At fixed average order parameter, a nonconvex local potential need not describe the equilibrium free energy. A mixture of two phases can lower it to the common-tangent convex envelope. If their magnetizations are $m_1,m_2$, the coexistence field satisfiesEquivalently, ${\int }_{m_1}^{m_2}[f^{\prime }(m)-h_{\mathrm{coex}}]dm=0$, the equal-area Maxwell construction. An average $\bar{m}$ between $m_1$ and $m_2$ is obtained by choosing their volume fractions according to the lever rule. The gradient term in Landau-Ginzburg theory gives a finite interfacial energy and smooth domain walls; in the thermodynamic limit the bulk gain dominates the interface cost. Domains are therefore especially natural under a constrained mean magnetization or suitable boundary conditions. Without such a constraint, one pure phase is also an equilibrium state at coexistence.

Write $m_i={\mu }_i^2$ and $r^2={\varphi }_1^2+{\varphi }_2^2$. The uniform Landau free energy isso stationarity requiresFor $m_1,m_2>0$, the unique ground state is $(0,0)$ and the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ discrete symmetry is unbroken. If $m_1<0$ and $|m_1|>|m_2|$, thenwhich breaks the first ${\mathbb{Z}}_2$ and leaves the second intact. This includes the case in which both masses are negative, because $m_1$ is then the more negative one. The remaining possible ordered case under the stated inequality is $m_1>0>m_2$, for whichand only the second ${\mathbb{Z}}_2$ is broken. The Hessian matrix in each ordered state is positive because the uncondensed direction has squared mass $m_k-m_j>0$, where $m_j$ is the condensed, more negative mass.

Near either continuous transition the nonzero order parameter is proportional to $(-m_i{)}^{1/2}$. Hence the mean-field critical exponents are

The lower critical dimension is the dimension at or below which fluctuations destroy the proposed finite-temperature ordered phase. Here the broken symmetry is discrete, so $d_{\mathrm{l}}=1$. In one dimension a domain wall interpolating between the two signs has finite energy, whereas its possible position gives an entropy growing as $\log L$. Domain walls therefore occur with nonzero density at every positive temperature and split the system into domains of finite typical length. Thus there is no finite-temperature ordered phase in one dimension.

When a cosmological phase transition breaks a symmetry $G$ to a subgroup $H$, the degenerate vacuum configurations form a vacuum manifold $\mathcal{M}=G/H$. Causally separated regions need not choose the same vacuum orientation. The Kibble mechanism explains how joining those choices can leave a field configuration that cannot be continuously deformed to a uniform vacuum without leaving $\mathcal{M}$. The field then stores energy in a defect core, gradients or gauge fields.^[ref]

The relevant homotopy classification of cosmological defects uses the vacuum map on a surface surrounding a proposed defect:A domain wall separates different disconnected vacua. A cosmic string has nonzero winding number on a loop around its core. A magnetic monopole has a nontrivial vacuum map on an enclosing sphere. Nontrivial ${\pi }_3(\mathcal{M})$ can instead support cosmological texture configurations extending over a volume; cosmological textures can unwind dynamically by leaving the vacuum manifold and need not possess a permanently localized stable core. Nontrivial topology permits a defect; formation efficiency, interactions and the cosmological history still determine its abundance.

For example, a complex scalar with a circular set of minima can wind by $2\pi n$ around a cosmic string. Continuity forces the field away from its vacuum value near the core. A local gauge symmetry allows gauge fields to screen much of the exterior gradient and trap flux, whereas a global symmetry leaves long-range Goldstone-field gradients. Thus local and global cosmic strings have different energy-loss channels; they should not be treated as identical networks.

Relic defects can be tightly constrained because their energy dilutes differently from radiation. Nonrelativistic magnetic monopoles approximately dilute as $a^{-3}$, so excessive production causes the cosmological monopole problem. A roughly frozen wall network has energy density proportional to $a^{-1}$, even slower dilution. An evolving cosmic string network can approach a scaling regime with characteristic separation proportional to cosmic time and density of order $\mu /t^2$ in $c=1$ units, losing energy into loops and radiation. That is a dynamical network statement, not the dilution law of a fixed collection of cosmic strings.

cosmic inflation can dilute defects made before it; reheating and later symmetry breaking must not regenerate an unacceptable abundance. Observational tests include microwave-background anisotropy, gravitational lensing by cosmic strings, gravitational radiation from string loops, and searches for magnetic monopoles. Each is sensitive to model-dependent tensions, couplings and evolution. Defects are possible relics of symmetry breaking, not an inevitable dominant dark-matter population. Their absence constrains a transition and its subsequent history rather than every theory possessing a nontrivial vacuum topology.

A coarse-grained scalar order parameter $\varphi (\mathbf{x})$ describes a short-range Ising-like transition with sign symmetry. Write the dimensionless Landau-Ginzburg functional asHere $r=a_{t}t$, with the relevant critical-mass shift understood when fluctuations are included. Higher local powers and gradients are negligible only within a justified long-distance expansion. The partition function is $Z=\int \mathcal{D}\varphi e^{-\mathcal{H}}$ with a microscopic cutoff. The local functional is not already the fluctuation-integrated thermodynamic free energy.

Mean-field theory replaces this integral by its stable uniform saddle, $r{\varphi }_0+u{\varphi }_0^3=h$. At $h=0$,The branch curvature is $r_L=r+3u{\varphi }_0^2$: it is $r$ above and $2|r|$ below. HenceTogether with the finite heat-capacity jump from $f_{\min }$, this yieldsThe equal-$|t|$ susceptibility amplitude ratio is two. Uniform saddles also miss domain walls and spatial texture. For an $n$-component order parameter, symmetry allows a vector generalization; ordered transverse directions additionally carry Goldstone fluctuations.

To include small fluctuations set $\varphi ={\varphi }_0+\delta \varphi$. The Gaussian action is $\frac{1}{2}\int [c(\nabla \delta \varphi {)}^2+r_L(\delta \varphi {)}^2]$, givingThe displayed $G(q)$ is a spectral covariance per Fourier delta, avoiding an extra volume factor. As $r_L\to 0$, a growing correlation length lets increasingly large correlated regions fluctuate. The singular Gaussian heat correction contains $\int d^{d}q/(c{q}^2+r_L{)}^2$, which is infrared singular for $d<4$ and logarithmic at four. This signals failure of a simple saddle expansion; the Gaussian heat exponent obtained by extrapolation is not the exact interacting critical exponent.

The Ginzburg criterion compares the fluctuation of a correlation-volume average with ${\varphi }_0^2$, rather than using all ultraviolet fluctuations at a point. Keeping $q\lesssim {\xi }^{-1}$ below the transition gives, up to finite convention-dependent constants,Dividing by ${\varphi }_0^2=|r|/u$ yieldsFor $d<4$, writing $r=a_{t}t$ gives a nonuniversal Ginzburg reduced-temperature windowInside $|t|\lesssim \mathrm{Gi}$, interacting fluctuations must be treated. For example, in three dimensions $\mathrm{Gi}\sim u^2/(c^3{a}_t)$ in this normalization. Above four dimensions the long-wavelength ratio vanishes at criticality; at four it is marginal and interaction-generated logarithms require further analysis.

The spatial gradient term supplies correlations absent from uniform Landau theory. The Ginzburg test bounds where its mean-field saddle remains self-consistent; it does not compute nonclassical exponents or a universal material-independent window.^[ref]

For a ferromagnetic nearest-neighbour Ising model on a chain let $s_i=\pm 1$, $K=J/(k_{B}T)>0$ and $h=H/(k_{B}T)$. The dimensionless Hamiltonian is $-K{\sum }_i{s}_i{s}_{i+1}-h{\sum }_i{s}_i$. In one dimension, decimating every other spin closes exactly within the nearest-neighbour family, including its additive constant. This is the exact chain building block of the Migdal-Kadanoff approach; bond moving becomes an additional approximation in higher-dimensional lattices.

For exact zero-field Ising-chain decimation, sum over an intermediate spin $\sigma$ between two retained spins:Parallel and antiparallel end spins give $A{e}^{K^{\prime }}=2\cosh 2K$ and $A{e}^{-K^{\prime }}=2$. HenceFor an even periodic chain, $Z_N(K)=A^{N/2}{Z}_{N/2}(K^{\prime })$. Rescaling the lattice spacing $2a$ back to $a$ completes the renormalization step. Decimation of $b$ successive links gives $\tanh K^{\prime }=(\tanh K{)}^b$, as follows by multiplying the transfer matrix $b$ times.

Let $v=\tanh K$. Then $v^{\prime }=v^2$ on $0\le v\le 1$. The infinite-temperature renormalization-group fixed point $v=0$ attracts every $0\le v<1$; the zero-temperature renormalization-group fixed point $v=1$ is unstable to heating, with slope two. There is no intermediate ferromagnetic renormalization-group fixed point. Equivalently $K^{\prime }\simeq K^2$ for small $K$, while $K^{\prime }\simeq K-\frac{1}{2}\log 2$ for large finite $K$. Thus there is no finite-temperature ordered thermodynamic phase in the short-range chain.

The domain-wall scaling coordinate of an Ising chain near $T=0$ is the domain-wall Boltzmann factor $w=e^{-2K}$. Its recursion and the zero-field correlation length areSince $\xi (K^{\prime })/a=[\xi (K)/a]/2$, the recursion has the correct length scaling. Its exponent is one with respect to $w$; it is not a finite power of physical temperature about $T_c=0$. In the thermodynamic limit, a bond is a domain wall with probability $(1-\tanh K)/2\sim e^{-2K}$, giving the same exponentially large length scale. Their finite energy allows entropy to proliferate walls at any positive temperature.

The Ising-chain decimation constant matters for thermodynamics. With dimensionless free energy per spin $f=-\lim N^{-1}\log Z_N$,The second expression follows independently from transfer eigenvalues $2\cosh K$ and $2\sinh K$ and satisfies the recursion. Multiplying it by $k_{B}T$ restores the physical free energy. Dropping $A$ would preserve normalized retained-spin probabilities but lose the eliminated-spin entropy.

Finite-field Ising-chain decimation is also exact. Use the symmetric transfer matrixand read the new couplings fromHere $W_{++}=e^{2K+2h}+e^{-2K}$, $W_{--}=e^{2K-2h}+e^{-2K}$ and $W_{+-}=2\cosh h$. The latter off-diagonal entry, the field convention split between adjacent bonds and the constant must all be tracked consistently. Near the zero-temperature zero-field renormalization-group fixed point $h^{\prime }\simeq 2h$, while at $K=0$ the independent-spin field is unchanged.

In the higher-dimensional Migdal-Kadanoff approximation, moving bonds creates chains that can be decimated. With bond moving first on a $d$-dimensional hypercubic lattice and length factor $b$, the strengthened links have $b^{d-1}K$, giving the conventionFor $d=1$ no transverse bonds move, so this reduces to the exact chain result. In higher dimensions it discards geometric correlations and is generally approximate on the original lattice; moving and decimating in another order yields another approximation, not a contradictory exact chain formula.^[ref]

Let $S=\mathrm{diag}(1,-1)$ be the base-spin insertion matrix. In the eigenbasis from solution, its diagonal matrix elements vanish: $⟨+|S|+⟩=⟨-|S|-⟩=0$. Hencenot only at large $L$ but for every finite ring at zero field.

This vanishing is not merely an averaging artifact hiding a finite-temperature ordered phase. The two-point correlation function is $⟨{\sigma }_1{\sigma }_{1+r}⟩=({\lambda }_{-}^r{\lambda }_{+}^{L-r}+{\lambda }_{+}^r{\lambda }_{-}^{L-r})/Z_L$. In the thermodynamic limit it becomes $({\lambda }_{-}/{\lambda }_{+}{)}^r$, with finite correlation length ${\xi }^{-1}=\log ({\lambda }_{+}/{\lambda }_{-})$ for every $T>0$ and $J>0$. Thermal domain walls destroy long-range alignment in this one-dimensional short-range system. At zero temperature, the two fully aligned ground states are degenerate; that limiting case must be distinguished from any fixed positive temperature.

Choose a coarse-grained magnetization $m(\mathbf{x})$ as the order parameter: it vanishes in the paramagnetism phase and develops a spontaneous nonzero value in the ferromagnetism phase. In zero applied magnetic field, spin reversal sends $m\mapsto -m$, so the local Landau free energy contains only even powers. A minimal spatially inhomogeneous free-energy functional isThe positive gradient coefficient penalizes rapid spatial variations. Near an ordinary transition take $a(T)=a_0(T-T_0)$, $a_0>0$, and positive $b$; retain the sextic term when $b$ can approach or cross zero. The source $h$ is conjugate to the magnetization and explicitly breaks spin reversal.

Determine equilibrium by minimizing this free-energy functional, imposing the appropriate boundary conditions. Its Euler-Lagrange equation isFor a uniform system at $h=0$, compare the values of all stable stationary solutions, rather than accepting every solution of this equation. The selected global minimum gives the equilibrium magnetization; derivatives of its free energy give entropy, heat capacity and magnetic susceptibility. At an ordinary transition the two ordered minima emerge continuously from zero. Spatial dependence also permits domain walls and an estimate of the correlation length.

This is a mean-field approximation: it selects a minimizing profile and neglects statistical fluctuations around it. The symmetry constrains the expansion, but its numerical coefficients are phenomenological. Close enough to a transition, thermal fluctuations near a critical point can invalidate the resulting mean-field critical exponents.^[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 ferromagnetic particle with no internal domain wall. Magnetization reversal can proceed by rotation or nonuniform modes; thermal activation and anisotropy determine blocking and coercivity. Small size alone does not guarantee a permanent hard magnet.

Stefano Zapperi and Gianfranco Durin studied the statistics of Barkhausen effect avalanches in soft ferromagnets. Slowly changing the applied magnetic field produces bursts of magnetization as pinned domain walls move. Their 2000 work measured avalanche-size distributions in polycrystalline and amorphous materials and found distinct scaling classes in the slow-drive limit. Interpreting the data through the release of pinned domain walls connects irregular magnetic noise with a collective phase transition, including a quantitative relation between the distribution cutoff and the demagnetizing factor.^[ref]

A physicist and professor of theoretical physics of matter at Milan.^[ref]

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]

For $Q,L\ge 2$, a periodic Potts chain has $Q$ uniform ground states with energy $-JL$. A single domain wall cannot close around the ring. Its first excited states have two domain walls and energy $-JL+2J$. Choose the wall bonds in $\left(\genfrac{}{}{0ex}{}{L}{2}\right)$ ways and assign distinct states to the two resulting arcs in $Q(Q-1)$ ways. This proves the displayed degeneracy factor without identifying globally relabelled configurations.