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An isotropic, noninteracting network of domain walls with energy density proportional to $a^{-1}$ has effective equation-of-state parameter $w=-2/3$. This follows from constant-equation-of-state density scaling $\rho \propto a^{-3(1+w)}$.

A relativistic membrane sweeps out a three-dimensional worldvolume with induced worldvolume metric $h_{ij}={\partial }_i{X}^m{\partial }_j{X}^n{\eta }_{mn}$. Its leading geometric action is tension times its invariant volume. In static gauge in four-dimensional spacetime, one transverse field $Y$ gives $h_{ij}={\eta }_{ij}+{\partial }_{i}Y{\partial }_{j}Y$, and the matrix determinant lemma gives $S=-T\int \sqrt{1+{\partial }_{i}Y{\partial }^{i}Y}$. Expanding supplies the translation-zero-mode kinetic term. A domain wall of finite thickness also has higher-curvature corrections outside the long-wavelength limit.

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]

At fixed finite $L$, the Helmholtz free energy of a periodic Potts chain receives its first thermal correction from a pair of domain walls. The two-domain-wall degeneracy of a Potts ring divided by the ground state degeneracy factor supplies the coefficient. Expanding the exact canonical partition function proves the formula. The expansion is not uniform in $L$: taking the thermodynamic limit first gives $F/L=-J-{\beta }^{-1}\log [1+(Q-1)e^{-\beta J}]$, whose leading correction is of order $e^{-\beta J}$.

Take a cycle graph with four labeled graph vertices and four distinct edges $\{1,2\},\{2,3\},\{3,4\},\{4,1\}$. Set $h=0$, $J>0$, and impose periodic boundary conditions by writing ${\sigma }_5={\sigma }_1$. The square drawing represents a one-dimensional ring: it has no diagonal interactions and is not a two-dimensional lattice.

Call an edge between opposite Ising spins a domain wall. If $d$ is their number, the four bond products sum to $4-2d$, so the energy isEach domain wall costs $2J$ relative to an aligned bond. Returning around the ring must return to the starting Ising spin, so $d$ is even: equivalently, ${\prod }_{i=1}^4({\sigma }_i{\sigma }_{i+1})=1$. Choosing an even set of $d$ wall edges and then choosing ${\sigma }_1$ determines the entire pattern. The state degeneracy is therefore $g_d=2\left(\genfrac{}{}{0ex}{}{4}{d}\right)$, using a binomial coefficient.

The twelve middle microstates consist of eight patterns with one exceptional Ising spin and four patterns with two adjacent $+1$ Ising spins and two adjacent $-1$ Ising spins. The two alternating patterns have four domain walls. All $2+12+2=16$ microstates are shown below.

Writing $K=\beta J$, the partition function and the probability of each energy class areFor an original exact example, choose $K=(\ln 2)/4$. Then $e^{4K}=2$, $Z=17$, and $P(d=0)=4/17$, $P(d=2)=12/17$, $P(d=4)=1/17$. Each aligned microstate has probability $2/17$, each middle microstate $1/17$, and each alternating microstate $1/34$: multiply by state degeneracy before comparing whole classes. As $K\to 0$, all sixteen microstates become equally likely, but the class probabilities approach $1/8,3/4,1/8$. As $K\to \infty$, the two all-aligned microstates each have limiting probability $1/2$.

A physicist associated with the Barkhausen effect: the magnetization of a ferromagnet can change in discrete jumps even when the applied magnetic field varies smoothly. A pickup coil detects the accompanying changes in magnetic flux; amplification makes the irregular signal audible. The experiment connects a macroscopic noise measurement with sudden rearrangements of magnetic domains and the motion of domain walls.^[ref] His work at Dresden also developed the experimental teaching of electrical engineering.^[ref]

For nonnegative field $g$, the bulk quasiparticle gap is $2J|g-1|$. At $g=0$ the dispersion is flat, $2J$; at large $g$ it is $2Jg-2J\cos k+O(J/g)$. At $g=1$ it becomes $4J|\sin (k/2)|$, linear near the gapless point. The finite periodic spin chain has additional global excitation constraints; a single isolated domain wall is not an allowed closed-chain spin configuration.

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]

For a one-dimensional periodic spin chain, a Kramers--Wannier intertwiner maps vertex spins in the $X$ basis to edge domain walls. It obeys $D(X_i{X}_{i+1})={\tilde{Z}}_{i+1/2}D$ and $D{Z}_i=({\tilde{X}}_{i-1/2}{\tilde{X}}_{i+1/2})D$ and identifies spin configurations related by the global ${\mathbb{Z}}_2$ symmetry.

In two dimensions, vertex COPY tensors and edge XOR tensors form a PEPO that maps vertex spins to edge domain walls. Its image obeys a zero-flux constraint around every plaquette.

Landau-Ginzburg theory promotes an order parameter to a spatial field and includes an energy cost for its variation. For a scalar field,The local terms select preferred phases, while the gradient term makes abrupt spatial changes costly. Minimizing this free-energy functional can describe a smooth domain wall between two ordered regions.

Expanding around a uniform stable minimum gives a quadratic fluctuation kernel and an Ornstein--Zernike correlation function. The framework therefore connects an equilibrium phase diagram with measurable spatial correlations and interfaces. Minimization alone is a mean-field treatment; integrating fluctuations can change the predictions and is central to the renormalization group. The derivative expansion assumes fields varying slowly compared with microscopic lengths.

A magnetic domain wall separates differently oriented magnetic domains. Its motion changes their volumes and average magnetization. It is a magnetic domain wall.

An original two-magnetic domain model makes the collective change explicit. Suppose opposite regions have magnetizations $+M_s$ and $-M_s$ along the measuring axis, where $M_s$ is the local saturation magnetization. If a magnetic domain wall sweeps volume $\Delta V$ from the negative region into the positive one, in a specimen of fixed volume $V$, thenThe factor two comes from replacing $-M_s$ by $+M_s$. The local magnitude $M_s$ stays fixed while the average magnetization changes; the event size depends on the swept volume, rather than a universal single-spin increment.

Let $w$ count the unequal neighbouring pairs, or domain walls. The energy is $E=-JL+Jw$. A single domain wall is impossible with periodic boundary conditions: following all the other, equal-state bonds around the ring would force its endpoints to be equal too. Two domain walls are possible, splitting the ring into two nonempty uniform arcs carrying different states. Thus the first excited state hasChoose the two wall bonds in $\left(\genfrac{}{}{0ex}{}{L}{2}\right)$ ways. For that choice, identify one of the resulting arcs by its smallest site label. Its state has $Q$ choices and the other arc has $Q-1$. This counts each configuration exactly once, giving the two-domain-wall degeneracy of a Potts ringFor example, $Q=2,L=4$ gives $d_0=2$ and $d_1=12$; the remaining two configurations are alternating and have four domain walls. The formula also covers $L=2$, where the specified ring energy counts the two directed neighbouring bonds, and an unequal pair breaks both. If $Q=1$ or $L=1$, every allowed configuration is a ground state and there is no first excited state. For $Q\ge 3$, the number of domain walls need not be even: three distinct states on a three-site ring give three, although one remains impossible.

Put $x=e^{-\beta J}$. Factoring out the ground state Boltzmann factor from the exact canonical partition function givesAt fixed finite $L$, the binomial theorem givesTaking the logarithm yields the finite-ring low-temperature expansion of the Helmholtz free energy,The constant term is the ground state energy. The term $-{\beta }^{-1}\log Q$ is the contribution of the ground state degeneracy factor. The first thermal correction follows directly by retaining the first excited state manifold:Each pair of domain walls costs $2J$, explaining the absence of an $e^{-\beta J}$ term. For $Q=2$, every configuration has an even number of domain walls, so the displayed remainder improves to $O_L(e^{-4\beta J}/\beta )$.

This is an expansion for fixed $L$, not a uniform thermodynamic limit. In particular, the small correction requires $\left(\genfrac{}{}{0ex}{}{L}{2}\right)(Q-1)x^2\ll 1$, with higher terms also controlled as $x\to 0$ at fixed $L,Q$. Taking $L\to \infty$ first instead selects ${\lambda }_{+}$ and givesIts leading correction is of order $x$. The finite-ring pair constraint and the extensive population of domain walls in an arbitrarily long chain explain the different ordering of these two limits.

Finite-energy configurations approach the vacuum manifold outside defect cores. The surrounding boundary in the transverse directions is a sphere, and noncontractible boundary data prevent a smooth deformation to the uniform vacuum.

Nontrivial ${\pi }_0(M)$ means disconnected vacuum components. Choosing different components on the two sides of an interface produces domain walls. Here ${\pi }_0$ labels components and need not itself carry a group structure. Nontrivial ${\pi }_1(M)$ means noncontractible loops: a circle surrounding a line can wind around the vacuum manifold, producing cosmic strings. Nontrivial ${\pi }_2(M)$ means noncontractible maps from an enclosing two-sphere, producing pointlike magnetic monopoles in three spatial dimensions.During the transition, initially uncorrelated vacuum choices in causally separated regions can generate these boundary classes through the Kibble mechanism. Nontrivial topology allows stable defects; it does not specify their exact formation efficiency or abundance.

With lattice spacing set to one and a smoothly varying auxiliary field, the discrete action becomesThis is precisely the Euclidean path integral of a quantum particle with $\hslash =1$, mass $M$ and potential $U$. With fixed endpoints it gives the transition amplitude $⟨m_f|e^{-\mathcal{L}{\hat{H}}_Q}|m_i⟩$, while periodic classical boundary conditions give its trace. The corresponding operator isThis is classical Ising chain as a quantum particle. It is the continuum approximation to the exact positive transfer kernel, so its numerical spectrum is accurate when the relevant auxiliary configurations vary slowly. The transfer-kernel reasoning and the domain-wall argument below also support the qualitative conclusion without treating a continuum approximation as an exact finite-lattice identity.

At zero field write $a=\tanh (\kappa /2)/J>0$. Expansion givesA double well exists when $a<2$; the stationary points satisfy $am=\tanh (2m)$. For $a\ge 2$ there is a single minimum at zero, with the quadratic curvature marginal at equality. Thus a double well is the low-effective-temperature regime of the model, not an unconditional property for every $J$. The exact potential remains confining because $U(m)=a{m}^2-2|m|+O(1)$ at large $|m|$.

Even when two wells exist, a quantum particle with finite mass and a finite barrier tunnels between them. Its one-dimensional confining Schrödinger operator has a unique, strictly positive ground-state wavefunction. At $h=0$ reflection symmetry makes that state even, so every odd observable has zero expectation. An antisymmetric first excited state is split from it by a positive gap. A spectral expansion of an odd-observable correlation has factors $e^{-x(E_n-E_0)}$; the smallest contributing gap therefore gives a finite classical correlation length. The original spin observable, after decoupling, is $\tanh (2m+h)$, which is odd at zero field. HenceA deep well can give an exponentially small quantum tunnelling gap and an extremely long correlation length, which can mimic order in a finite sample, but it does not produce true long-range order.

The classical counterpart is the finite cost of a domain wall. Because the pair sum is doubled, a sharp wall has dimensionless energyAt any finite temperature such walls have nonzero fugacity, divide a long chain into domains and destroy spontaneous magnetization. The range is large for $\kappa \ll 1$ but finite. The zero-temperature limit and an infinite-range limit are singular and must not be interchanged casually with the thermodynamic limit; an infinite-range interaction also requires an extensive normalization before comparing models.

A nonzero $h$ tilts the potential: to first order $U_h(m)=U_0(m)-h\tanh (2m)+O(h^2)$. A positive field favors the positive well and produces positive magnetization. For fixed finite parameters the ground state and classical free energy vary smoothly, so there is no zero-field discontinuity due merely to a double-well bare potential. A discrete broken-symmetry state appears only in an appropriate singular limit, not by ignoring quantum tunnelling.

More generally a local $d$-dimensional classical statistical model can be sliced along one coordinate and represented by a transfer operator. That coordinate becomes imaginary time; the other $d-1$ coordinates become the spatial coordinates of a quantum system. Infinite extent in the transfer direction corresponds to its ground-state, or zero-quantum-temperature, limit. The classical Ising model is thus related to a quantum Ising system in $d-1$ spatial dimensions, with transverse dynamics generated by the transfer operator. For $d=1$ this is quantum mechanics of finitely many local variables, where quantum tunnelling restores symmetry. For $d\ge 2$ the quantum system has infinitely many spatial degrees of freedom and can have distinct symmetry-broken ground states, allowing a classical finite-temperature ordering transition. Locality and anisotropy of the mapping determine the quantum couplings; the dimension correspondence alone does not fix them.

A canonical Wess–Zumino model with a cubic superpotential already supplies a microscopic membrane-like object. Choose positive real $\mu ,g$ andThere are two isolated supersymmetric vacua, ${\varphi }_{\pm }=\pm \mu /g$. A planar domain wall has a static scalar profile depending on the transverse coordinate $z=X^3$, with one vacuum at either end. Its energy per unit area isChoosing $\alpha =\arg \Delta W$ gives $T\ge 2|\Delta W|$. The real increasing solution saturates the bound:The tension follows either from integrating $2({\varphi }^{\prime }{)}^2$ or from $W(\mu /g)-W(-\mu /g)=4{\mu }^3/(3g^2)$. This is a supersymmetric domain wall, and the first-order equation also implies the second-order scalar equation. Its thickness is of order $1/\mu$.

The wall breaks transverse translations: changing $z_0$ gives a normalizable scalar zero mode proportional to ${\varphi }^{\prime }(z)$. It also preserves only half the four real supercharges. In the chiral-fermion transformation, $F=-\overline{W^{\prime }}=-{\varphi }^{\prime }$ and the derivative term combine into a rank-half condition on the constant supersymmetry parameter, for example $ϵ=i{\sigma }^3\overline{ϵ}$ in a compatible Weyl convention. The orthogonal two real parameters are broken and generate localized fermion zero modes. One such profile satisfies $({\partial }_z+2g\varphi )f=0$, giving $f\propto {\mathrm{sech}}^2(\mu (z-z_0))$. The scalar and a three-dimensional Majorana fermion therefore form the low-energy wall multiplet. The domain-wall charge in N=1 supersymmetry is a boundary/tensor charge contributing to the extended algebra; the preserved combinations saturate $T=2|\Delta W|$. This avoids falsely applying the unextended particle-algebra argument of Question 3 to a wall with nonzero tension.

Promote the translation collective coordinate to a slowly varying worldvolume field $Y({\xi }^0,{\xi }^1,{\xi }^2)$. At wavelengths long compared with the wall thickness, its leading relativistic geometric action is the Dirac membrane actionIt is invariant under worldvolume reparametrizations and the target Poincaré group. In static gauge $X^i={\xi }^i$, $X^3=Y$, the matrix determinant lemma givesDirectly substituting the leading collective-coordinate profile into the scalar kinetic term gives the same coefficient, since $\int dz({\varphi }^{\prime }{)}^2=T/2$. Target Lorentz symmetry fixes the leading first-derivative nonlinear completion to the invariant area; finite-thickness physics can add higher-curvature terms. The area action is therefore a long-wavelength effective field theory, not an exact replacement for the microscopic wall at arbitrary momenta. Its embedding equation is ${\partial }_i(\sqrt{-h}{h}^{ij}{\partial }_j{X}^m)=0$.

For the four-dimensional supermembrane, extend the embedding to $(X^m(\xi ),{\theta }^{\alpha }(\xi ))$ with a four-real-component Majorana spinor. Pull back the invariant coframe of Question 7:Replacing $\partial X$ by $\Pi$ makes the area term invariant under rigid target supersymmetry, but does not yet give the appropriate fermionic gauge symmetry. Add a Wess-Zumino brane coupling to a superspace three-form potential $C_3$. In one orientation and matching normalization,The supermembrane closed four-form uses the distinct four-dimensional Fierz identity $(C{\gamma }_{mn}{)}_{(\alpha \beta }(C{\gamma }^n{)}_{\gamma \delta )}=0$. Differentiating $G_4$ reduces its coefficient to this symmetric tensor, so $d{G}_4=0$. A local potential therefore exists. Since the curvature is supertranslation invariant, the potential can vary by an exact form; its pullback integral is invariant modulo boundary terms. This is why the three-form $H_3$ of Question 7 is not itself the membrane field strength: integrating a membrane potential requires three-form degree and hence four-form curvature.

The required additional local fermionic gauge invariance is kappa symmetry. Define the worldvolume gamma matrices ${\gamma }_i={\Pi }_i^m{\gamma }_m$ andWith mostly-plus target signature, a local orthonormal worldvolume frame gives ${\Gamma }_{\kappa }={\gamma }_0{\gamma }_1{\gamma }_2$, so ${\Gamma }_{\kappa }^2=1$ and $\mathrm{tr}{\Gamma }_{\kappa }=0$. Thus $(1\pm {\Gamma }_{\kappa })/2$ are rank-two kappa symmetry projectors. The embedding variation gives $\delta {\Pi }_i^m=2i\delta \bar{\theta }{\gamma }^m{\partial }_i\theta$. The area variation is consequently proportional to $-2iT\sqrt{-h}\delta \bar{\theta }{\gamma }^i{\partial }_i\theta$. Contracting the displayed $G_4$ with the fermionic variation and using ${ϵ}^{ijk}{\gamma }_{ij}=2\sqrt{-h}{\Gamma }_{\kappa }{\gamma }^k$ gives the opposite Wess-Zumino contribution with an extra ${\Gamma }_{\kappa }$. Their sum is proportional toFor the boxed kappa variation, $\delta \bar{\theta }=\bar{\kappa }(1+{\Gamma }_{\kappa })$, so it vanishes by $(1+{\Gamma }_{\kappa })(1-{\Gamma }_{\kappa })=0$. Reversing membrane orientation reverses the Wess-Zumino sign and the matching projector. The cancellation fixes the relative charge magnitude to the tension; rigid supersymmetry alone would not fix it.

Worldvolume reparametrizations remove three of the four embedding scalars, leaving one transverse mode. Kappa symmetry removes half the four real embedding spinor components, leaving a two-component three-dimensional Majorana field, whose first-order equation has one on-shell fermionic polarization. The physical worldvolume modes are one scalar and one Majorana fermion, matching the wall zero modes. Without kappa symmetry the fermionic degrees would be doubled. A planar configuration preserves the target supersymmetries that can be compensated by the kappa projector, reproducing the wall's half-supersymmetry condition. This is the four-dimensional effective domain-wall interpretation of a supermembrane, rather than the eleven-dimensional M2-brane particle-content problem.

The Maxwell construction replaces a nonconvex homogeneous free-energy density by its coexistence envelope. If $U_0(M)$ excludes the source, the coexisting values $M_1,M_2$ satisfyThus one straight line is tangent at both endpoints. Equivalently,the equal-area form of the common-tangent construction. At fixed average $\overline{M}$ between the endpoints, the lever rule gives $\overline{M}=p{M}_1+(1-p)M_2$. Such a mixture has the tangent-line bulk energy, below the energy of an unstable homogeneous state.

The two phases occupy spatial domains separated by domain walls. The gradient term gives a positive wall energy, but a macroscopic interface has area of order $L^{D-1}$ rather than bulk volume $L^D$, so its cost per volume vanishes in the thermodynamic limit. For the symmetric quartic example a scalar quartic domain wall obeys $\kappa {\varphi }^{\prime \prime }=r\varphi +u{\varphi }^3$ and interpolates between $\pm M_0$ as $\varphi =M_0\tanh (x/\ell )$, with $M_0^2=|r|/u$ and $\ell =\sqrt{2\kappa /|r|}$. Multiplication of the differential equation by ${\varphi }^{\prime }$ gives the first integral $\kappa ({\varphi }^{\prime }{)}^2/2=u({\varphi }^2-M_0^2{)}^2/4$, explaining the finite positive interface cost. A uniform unconstrained system need not form both domains: phase separation is particularly relevant when the average order parameter is fixed or boundaries enforce competing phases.

Hysteresis occurs when the system follows a metastable local minimum after another minimum has become the equilibrium one. For $r<0,u>0$, the equation of state is $h=rM+u{M}^3$. The hysteresis spinodals of a scalar quartic potential follow from $r+3u{M}^2=0$:Following local minima until these limits gives different switching fields on increasing and decreasing $h$. Equilibrium instead switches at $h=0$. Actual switching can occur before a mean-field spinodal through nucleation, and depends on time scales; hysteresis is not a property of a globally minimized equilibrium potential. The second panel below distinguishes the local branches and their limiting switches.

Assume short-range ferromagnetic exchange $J>0$ and zero external field. In the one-dimensional Ising model, a domain wall costs only $2J$, independent of the lengths of the domains it separates. There are order-$N$ positions for one wall, so its positional entropy offsets any finite energy cost at nonzero temperature as the system grows. Domain walls consequently have a positive density at every $T>0$, destroying a uniform spontaneous spin magnetization. Periodic boundaries require an even number of walls, but that constraint does not remove their nonzero thermodynamic-limit density.

The One-dimensional Ising correlation function makes this argument explicit. A bond is broken with probability $p_w=e^{-2{\beta }_{\mathrm{th}}J}/(1+e^{-2{\beta }_{\mathrm{th}}J})$, and the infinite-chain correlation isThe dominant eigenvalue of the transfer matrix for the one-dimensional Ising model is $2\cosh {\beta }_{\mathrm{th}}J$, so the free energy per spin $-k_{B}T\log [2\cosh {\beta }_{\mathrm{th}}J]$ is analytic for $T>0$. There is no finite-temperature transition in one dimension; only the zero-temperature limit orders.

In two dimensions a reversed cluster is enclosed by a Peierls contour. A contour of length $\ell$ costs $2J\ell$, while the number of such contours surrounding a fixed site grows at most as a polynomial in $\ell$ times $3^{\ell }$: after the initial edge, a nonbacktracking square-lattice walk has at most three choices per step. With plus boundary conditions, flipping the interior of a specified contour yields the Boltzmann suppression $e^{-2{\beta }_{\mathrm{th}}J\ell }$. ThereforeFor sufficiently low temperature this bound is below $1/2$, giving a positive spontaneous magnetization. This is the energy-entropy mechanism of the Peierls argument.

At sufficiently high temperature, the graphical expansion bounds correlations by sums of connecting paths weighted by $(\tanh {\beta }_{\mathrm{th}}J{)}^{\ell }$. The same nonbacktracking count converges geometrically when $3\tanh {\beta }_{\mathrm{th}}J<1$, giving exponential decay and a disordered phase. Since an ordered low-temperature phase and a disordered high-temperature phase both occur, the two-dimensional model has a finite-temperature phase transition. The argument establishes its existence and the dimensional distinction, not the exact value of its critical temperature.

A fixed comoving volume has physical volume proportional to $a^3$. The stretched domain walls within it have area proportional to $a^2$, and their energy per physical area is constant. The cosmological domain wall network therefore has $\rho \propto a^{-1}$, equivalently a barotropic equation of state with $w=-2/3$.

The flat Friedmann equation becomes ${\dot{a}}^2=Ca$ for a positive constant $C$. On the expanding branch, integration gives $a=(C/4)(t-t_B{)}^2$. Measuring age from the initial time $t_B=0$, $H=\dot{a}/a=2/t$. Thus