Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 69 (Phase Transitions and Collective Phenomena)

Spontaneous symmetry breaking occurs when the statistical Hamiltonian is invariant under a symmetry but a chosen equilibrium pure state is not. A small selecting field or a boundary condition chooses one of several states related by the symmetry. For a finite unconstrained system the symmetric ensemble generally has zero order parameter; spontaneous order is defined by taking the thermodynamic limit before removing the selecting field. A ferromagnet with a selected magnetization direction is the standard example.

If the broken symmetry is continuous, nearby symmetry-related states cost no energy for a uniform displacement along the manifold of states. Locality and a nonzero stiffness then make a slowly varying displacement cost gradient energy rather than a mass term. These gapless long-wavelength fluctuations are Goldstone modes. In the conventional relativistic internal-symmetry setting their count is the number of broken generators. Nonrelativistic dynamics can pair broken generators, as in a ferromagnet; gauge symmetries and broken space-time symmetries require additional qualifications. A discrete broken symmetry alone does not produce a Goldstone mode.

For an interface with a selected mean height, the continuous displacement is a rigid translation normal to the mean plane. The tension gives a stiffness for spatially varying translations. The following calculation concerns this static Goldstone sector. It determines the excitation energy and equilibrium fluctuations, but a frequency dispersion requires a separate inertial or dissipative dynamics. In low dimensions thermal roughness also limits the existence of a strictly localized flat interface in an infinite system.

Put $D=d-1$ for the membrane's internal dimension and assume $\sigma >0$. The small-slope expansion isThe zeroth-order term is the projected area and is independent of the height field. Discarding it changes only an overall normalization of the partition function. Thus the leading fluctuation energy and its statistical weight areOne obtains this Gaussian field theory by discretizing the height field on a microscopic mesh, integrating every height with its Boltzmann weight, and then writing the continuum measure. A microscopic cutoff is retained for quantities dominated by short wavelengths. Quadratic order here really means small slopes: a large rigid height shift is still free because the energy has no dependence on $h$ itself.

The uniform height mode has zero energy. Fix its mean, impose a boundary condition, or divide out the integral over global translations to normalize the partition function. Height differences are independent of that choice. The approximation is the tension-only part of small-slope elastic membrane energy; there is no bending rigidity or confining potential in this part.

Use the Fourier transform conventionParseval's identity and $\nabla \mapsto i\mathbf{q}$ give the diagonal quadratic formIn a finite periodic box one can equivalently use orthonormal discrete Fourier modes, for which each independent real amplitude has a positive stiffness $\sigma q^2$. Fourier orthogonality eliminates all cross terms between different modes. The zero mode is removed as explained in part (a).

The stiffness vanishes as $q\to 0$, so capillary waves at long wavelength have arbitrarily low energy. The statistical Hamiltonian is invariant under $h(\mathbf{x})\mapsto h(\mathbf{x})+a$ for any constant $a$. Selecting a particular mean membrane height breaks this continuous normal-translation symmetry. A varying local translation therefore has a gradient cost but no restoring mass: these are Goldstone modes. The diagonal static kernel is $q^2$; it should not be identified with a dynamical frequency without specifying the surrounding fluid or membrane inertia.

Each nonzero mode is Gaussian, giving the covarianceConsequently the capillary height-difference correlation isThe difference removes the zero mode and makes the integral infrared-finite at each fixed separation. Where required, cut off $q$ at $\Lambda \sim 1/a$ with microscopic length $a$. The question's $d$ is the embedding dimension, so the momentum integral has dimension $D=d-1$, not $d$.

For $d=2$, $D=1$, use ${\int }_{-\infty }^{\infty }dq(1-\cos qr)/(2\pi q^2)=|r|/2$. HenceThe root-mean-square relative height grows as $r^{1/2}$: the interface wanders increasingly far from a fixed height.

For $d=3$, $D=2$, angular integration givesThe Bessel function expression separates the universal logarithmic growth from a cutoff-dependent constant. The membrane is logarithmically rough, so its absolute height fluctuations are unbounded in the thermodynamic limit even though long-wavelength slopes need not be large.

For $d=4$, $D=3$, the massless Green function at nonzero separation is $1/(4\pi r)$. Thuswhere $B_{\infty }$ is finite once the microscopic cutoff is supplied. For a spherical sharp cutoff, $B_{\infty }=k_{B}T\Lambda /({\pi }^2\sigma )$. Relative height fluctuations saturate at large distance, allowing a localized interface height selected by boundaries. The roughness distinction is linear growth in $d=2$, logarithmic growth in $d=3$, and bounded long-distance fluctuations in $d=4$. The short-distance cutoff and the small-slope assumption remain necessary; bounded height fluctuations do not imply a perfectly flat microscopic membrane.

Assume the pinning coefficient $t>0$. Here $t$ belongs to an actual energy, so its Boltzmann coefficient is $\beta t$. The quadratic kernel and covariance becomeTherefore the pinned capillary-wave correlation isThe normal-translation symmetry is now explicitly broken, and the formerly massless Goldstone mode has a restoring term. For $r\ll \xi$, but beyond the microscopic cutoff, the unpinned behavior is recovered. For $r\gg \xi$, the connected height covariance decays and $B_t$ approaches $2⟨h^2⟩$, so roughness no longer grows without bound even for $d=2,3$.

In internal dimension one the exact continuum answer isIn internal dimension two the connected covariance at $r>0$ is $k_{B}T{K}_0(r/\xi )/(2\pi \sigma )$, where $K_0$ is a modified Bessel function; its zero-separation value needs the cutoff and the saturation value grows as $(k_{B}T/\pi \sigma )\log (\xi /a)$ for $\xi \gg a$. In internal dimension three the connected covariance is approximately $k_{B}T{e}^{-r/\xi }/(4\pi \sigma r)$ beyond the cutoff. Thus pinning replaces the long-distance massless tails by exponential decay. A negative $t$ would destabilize the quadratic statistical Hamiltonian rather than bind the membrane.

The renormalization group compares descriptions at successively longer length scales. First eliminate short-wavelength degrees of freedom, then rescale coordinates and fields to restore the reference cutoff. This changes the couplings, including interactions and additive free-energy terms generated during elimination. A renormalization-group fixed point is a scale-invariant description. Linearized directions growing under coarse graining are relevant, shrinking directions are irrelevant, and zero-eigenvalue directions are marginal and require higher-order analysis.

A diverging correlation length means that many coarse-graining steps can approach a renormalization-group fixed point before a relevant perturbation becomes large. Systems with different microscopic details but the same renormalization-group fixed point and relevant directions share universal critical behavior. For the quadratic model below, Fourier modes decouple, so eliminating a high-momentum shell produces a Gaussian determinant but does not couple the remaining modes. The anisotropic gradient kernel requires different length rescalings in the parallel and transverse directions. This is an anisotropic renormalization group, not an isotropic rescaling of all $d$ coordinates.

ChooseFor a real scalar field $\tilde{m}(-q)=\tilde{m}(q{)}^{\ast }$. The parallel derivative becomes $i{q}_{\parallel }$, and the transverse Laplacian becomes $-q_{\perp }^2$. The latter is squared in the energy, producing $q_{\perp }^4$, not $q_{\perp }^2$. ThusThe zero-momentum source is understood in a finite box or as a uniform-field distribution before taking the thermodynamic limit. With an orthonormal box transform the source is $-h\sqrt{V}{m}_0$. The positive coefficients and $t>0$ give a normalizable Gaussian field theory, with mean uniform field $\overline{m}=h/t$ and connected Fourier covariance $[K{q}_{\parallel }^2+L{q}_{\perp }^4+t{]}^{-1}$.

Let $b,c>1$, rescale $q_{\parallel }^{\prime }=b{q}_{\parallel }$, ${\mathbf{q}}_{\perp }^{\prime }=c{\mathbf{q}}_{\perp }$, and set $B=b{c}^{d-1}$. Use the Fourier-amplitude conventionThe momentum measure contributes $B^{-1}$, and substitution into the quadratic statistical Hamiltonian givesIn the uniform source the momentum delta function cancels the Jacobian determinant of the measure, explaining why $h^{\prime }$ does not have an additional $B^{-1}$. In a full momentum-shell renormalization group transformation integrate out the fast modes first. Their uncoupled Gaussian integrals contribute an additive free-energy density; the displayed transformations describe the slow-mode statistical Hamiltonian after coordinate and field rescaling.

The field convention must be stated. In real space, $x_{\parallel }^{\prime }=x_{\parallel }/b$ and ${\mathbf{x}}_{\perp }^{\prime }={\mathbf{x}}_{\perp }/c$, and Fourier inversion gives $m^{\prime }(x^{\prime })=Bm(x)/z$. Thus if instead the symbol $m^{\prime }=m/z_x$ refers to real-space amplitudes, $z=B{z}_x$. Then the same transformation reads $K^{\prime }=B{z}_x^{2}K/b^2$, $L^{\prime }=B{z}_x^{2}L/c^4$, $t^{\prime }=B{z}_x^{2}t$, $h^{\prime }=B{z}_{x}h$. These are equivalent normalizations, not different critical exponents.

Keeping both gradient coefficients fixed requires $z^2=B{b}^2=B{c}^4$. HenceEquivalently the real-space field normalization is $z_x=z/B=b^{(3-d)/4}$. The transformed thermal and source couplings areThis is the Gaussian fixed point at $t=h=0$, with anisotropic effective volume exponent $D_{\mathrm{eff}}=(d+1)/2$. The stiffnesses remain positive parameters labeling the normalization of coordinates and fields. A strictly positive $t$ is a relevant perturbation away from the renormalization-group fixed point, not itself a critical renormalization-group fixed point.

The two correlation lengths also follow directly by balancing the kernel: $K{q}_{\parallel }^2\sim t$ gives ${\xi }_{\parallel }\sim (K/t{)}^{1/2}$, while $L{q}_{\perp }^4\sim t$ gives ${\xi }_{\perp }\sim (L/t{)}^{1/4}$. Their different exponents are consistent with the required anisotropic scaling. An omitted scalar quartic interaction has scaling eigenvalue $4-D_{\mathrm{eff}}=(7-d)/2$, so the quadratic renormalization-group fixed point alone does not establish the interacting smectic's critical behavior below physical dimension seven.

Let $f=-V^{-1}\log \mathcal{Z}$ be the dimensionless free-energy density. The transformed volume is $V^{\prime }=V/B$. Including the normalization from eliminated modes giveswhere $g_b$ is an additive shell contribution. After subtraction of regular backgrounds, the leading singular part obeys the scaling hypothesis for critical phenomenaChoose $b=t^{-1/2}$ for the high-temperature side $t>0$. Then the nominal homogeneous form isThe volume exponent is $D_{\mathrm{eff}}$, so inserting $d$ in the isotropic hyperscaling formula would give the wrong answer.

There are two qualifications to interpreting this as the total free energy. Completing the Gaussian square gives the source contribution exactly as $-h^2/(2t)$, and the zero-field determinant contributesRescaling $q_{\parallel }\sim t^{1/2}$ and $q_{\perp }\sim t^{1/4}$ gives the power $t^{(d+1)/4}$, while $2\Delta -1=(d+1)/4$ makes the source term consistent with it. Cutoff-dependent analytic terms need not obey the homogeneous law. Furthermore Gaussian free-energy logarithms at integer singular powers replace the pure zero-field power by $t^n\log t$ when $(d+1)/4=n$ is a positive integer, including $d=3,7$. This is the usual logarithmic qualification of the nominal exponents. The quadratic model also has no stable negative-$t$ phase unless a stabilizing interaction is added.

The scaling hypothesis for critical phenomena describes the singular part of the equilibrium free-energy density, rather than its analytic background. For an isotropic system in spatial dimension $D$, with thermal scaling field $t$ and conjugate field $h$, writeThis follows from coarse graining near a renormalization-group fixed point: a correlation volume is enlarged by $b^D$, while the relevant thermal and field perturbations grow with their eigenvalues. Choosing $b=|t{|}^{-1/y_t}$ givesThe two functions describe the two sides of the transition. Nonuniversal metric factors multiplying $t,h,f_s$ must be allowed when comparing microscopic models. Exponents and suitably normalized scaling functions or amplitude ratios characterize a universality class.

Differentiating the free-energy density gives the order parameter $M=-{\partial }_h{f}_s$, magnetic susceptibility $\chi ={\partial }_{h}M$ and singular heat capacity proportional to $-{\partial }_t^2{f}_s$. ThusThe subscript distinguishes the order-parameter exponent from inverse temperature. At $t=0$, choose $b=|h{|}^{-1/y_h}$ to obtain $M\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$, with $\delta =y_h/(D-y_h)$. These derivatives imply the Rushbrooke scaling relation $\alpha +2{\beta }_{\mathrm{mag}}+\gamma =2$ and the Widom scaling relation $\gamma ={\beta }_{\mathrm{mag}}(\delta -1)$. Equivalently, the equation of state has $h=|t{|}^{\Delta }{\mathcal{H}}_{\pm }(M/|t{|}^{{\beta }_{\mathrm{mag}}})$, with $\Delta ={\beta }_{\mathrm{mag}}\delta ={\beta }_{\mathrm{mag}}+\gamma$.

The correlation length scales as $\xi \sim |t{|}^{-\nu }$, $\nu =1/y_t$. At criticality the connected correlation function behaves as $G(r)\sim r^{-(D-2+\eta )}$, giving $y_h=(D+2-\eta )/2$. Integrating $G$ over a correlation volume gives $\chi \sim {\xi }^{2-\eta }$ and hence the Fisher scaling relation $\gamma =(2-\eta )\nu$. The hyperscaling relation $2-\alpha =D\nu$ expresses one singular free-energy contribution per correlation volume.

These laws organize measurements: plotting $M/|t{|}^{{\beta }_{\mathrm{mag}}}$ against $h/|t{|}^{\Delta }$ collapses near-critical curves onto the appropriate scaling function. A finite system replaces the divergent length by its size $L$; the corresponding prediction is $f_s=L^{-D}\mathcal{F}(t{L}^{y_t},h{L}^{y_h})$ under the same scaling assumptions. Finite size rounds a sharp thermodynamic singularity.

The hypothesis has a domain of validity. At a renormalization-group fixed point one generally needs $f_s=b^{-D}{f}_s(b^{y_t}t,b^{y_h}h,b^{y_u}u,\dots )$, including other scaling fields. Ordinary irrelevant couplings can be set to zero in a nonsingular limiting scaling function. A dangerously irrelevant coupling cannot: above the upper critical dimension of scalar quartic theory, the quartic term is irrelevant at the Gaussian point but still stabilizes the ordered phase. Consequently naive hyperscaling fails there even though other exponent relations can remain valid. Marginal variables can produce logarithms, anisotropic systems require the appropriate volume exponent, and the Berezinskii–Kosterlitz–Thouless transition has an essential correlation-length singularity instead of a finite power-law $\nu$. Scaling theory relates observables through a small number of fixed-point dimensions; it does not make analytic backgrounds, marginal logarithms or dangerously irrelevant variables disappear.

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.

The Landau-Ginzburg theory for a scalar order parameter supplements a local symmetry-allowed potential with a gradient penalty. For a ${\mathbb{Z}}_2$ symmetry at zero field, a stable quartic model iswith $r=a(T-T_c)$ near a mean-field transition. Equilibrium fluctuations have weight $e^{-F/(k_{B}T)}$. In the Landau approximation, minimize the functional. A uniform stationary state obeys $r\varphi +u{\varphi }^3=h$. At zero field the minimum is $\varphi =0$ for $r>0$ and $\varphi =\pm \sqrt{-r/u}$ for $r<0$. The latter pair is spontaneous symmetry breaking of the discrete symmetry.

The uniform free-energy density at its minimum is zero above the transition and $-r^2/(4u)$ below, up to regular backgrounds. Differentiating the equation of state gives ${\chi }_{+}=1/r$ and ${\chi }_{-}=1/(2|r|)$. At $r=0$, $\varphi =(h/u{)}^{1/3}$. Gaussian fluctuations around the minimum have inverse covariance $(K{q}^2+r)/(k_{B}T)$ above and $(K{q}^2+2|r|)/(k_{B}T)$ below. Their correlation lengths are ${\xi }_{+}=(K/r{)}^{1/2}$ and ${\xi }_{-}=(K/(2|r|){)}^{1/2}$. These results give the usual mean-field critical exponentsThe heat capacity has a finite jump rather than a power-law divergence. Negative quartic coefficient requires higher stabilizing terms and can produce a first-order transition; a vanishing quartic coefficient requires a tricritical analysis. Thus the assumed positivity of $u$ matters.

The Ginzburg criterion tests whether this saddle description is self-consistent. Below the transition compare fluctuations of the order parameter averaged over a correlation volume with its squared mean-field value ${\varphi }_0^2=|r|/u$. Modes with $q\lesssim {\xi }^{-1}$ dominate this coarse-grained fluctuation estimate:where the positive dimensionless $A_D$ depends on the averaging window and the factor of two used in $\xi$. Dividing by ${\varphi }_0^2$ givesIt is the fluctuation of a correlation-volume average, not the full cutoff-dominated microscopic variance, that should be compared with the mean-field order parameter.

For $D<4$, $\mathcal{G}$ diverges as $r\to 0$, so the asymptotic critical region is fluctuation dominated. Away from it, mean-field theory can still work whenFor $D>4$, the long-wavelength ratio tends to zero near criticality, consistently with Gaussian control and mean-field thermodynamic exponents. At $D=4$, quartic coupling is marginal; scale-dependent logarithms require a renormalization-group analysis rather than interpreting a constant bare ratio as a proof of no corrections. This identifies four as the upper critical dimension of the short-range scalar quartic model.

The Landau-Ginzburg theory is valuable because symmetries restrict a manageable list of local operators, the gradient term describes correlations and interfaces, and the Ginzburg criterion identifies when neglected fluctuations become decisive. Below the upper critical dimension an interacting renormalization-group fixed point, rather than the bare saddle, determines universal exponents. For continuous order parameters the manifold of ordered minima also has Goldstone modes, and lower-critical-dimension fluctuations require separate attention. Mean-field minimization supplies an approximation; the Ginzburg ratio decides whether its neglect of fluctuations is justified.

Write the interaction matrix as $\mathsf{J}$ to distinguish it from its scalar strength. The printed pair sum contains both orientations and has no factor $1/2$. For a positive-definite matrix $\mathsf{J}$, completion of the square gives the Hubbard–Stratonovich transformationsince $m^T{\mathsf{J}}^{-1}m-2m^T\sigma =(m-\mathsf{J}\sigma {)}^T{\mathsf{J}}^{-1}(m-\mathsf{J}\sigma )-{\sigma }^T\mathsf{J}\sigma$. For the exponential kernel with $J>0$ and $\kappa >0$, positivity follows from the positive Fourier symbol derived in part (b), or from its finite open-chain covariance matrix.

Insert this identity into the Ising model partition sum and interchange the finite spin sum with the convergent Gaussian integral. The spin variables now factor independently:It follows thatwith $C=({\pi }^N\det \mathsf{J}{)}^{-1/2}$ for the measure used here. The factor two in the spin-field coupling is required by the double pair sum; replacing it by one would decouple a statistical Hamiltonian with a different normalization. The diagonal $J_{ii}{\sigma }_i^2=J$ is a spin-independent energy. Including it as printed makes the positive Gaussian identity directly applicable; if diagonal self-couplings are removed they should first be restored as an irrelevant constant before this real auxiliary-field decoupling.

Work first on the infinite chain, or with a periodized exponential kernel. Put $\rho =e^{-\kappa }$ and use lattice Fourier momentum $q\in [-\pi ,\pi ]$. Summing the geometric series gives the exponentially coupled Ising inverse kernel:ThusAlthough the original kernel is long-ranged, its inverse contains only a constant and a cosine and therefore is nearest-neighbor. Fourier orthogonality givesSubstitution into part (a) yieldsThis exact bulk identity does not require replacing $\sinh \kappa$ by $\kappa$; small $\kappa$ is used later to motivate a smooth continuum field.

For a finite open chain the inverse has off-diagonal entries $-\rho /[J(1-{\rho }^2)]$, interior diagonal entries $(1+{\rho }^2)/[J(1-{\rho }^2)]$ and endpoint diagonal entries $1/[J(1-{\rho }^2)]$. Accordingly the displayed homogeneous potential acquires the endpoint correction $\rho (m_1^2+m_N^2)/[J(1+\rho )]$. These boundary terms do not change the bulk thermodynamic conclusion. Stating infinite-chain or periodized boundary conditions is necessary if the translation-invariant expression is asserted exactly.

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.