Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 69 / 4

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.