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The Blume–Capel model is a spin model with ${\sigma }_i\in \{-1,0,1\}$ and HamiltonianCompetition between the Ising interaction and the single-ion anisotropy $D$ produces a tricritical point in its phase diagram.

For a classical spin model with Hamiltonian $H_h=H_0-hM$, $M={\sum }_n{\sigma }_n$, define the static magnetic susceptibility per site by $\chi =N^{-1}{\partial }_h⟨M⟩$ at fixed temperature. Differentiating its partition function givesFor a translation-invariant array of spins, write $G(r)=⟨{\sigma }_0{\sigma }_r⟩-⟨{\sigma }_0⟩⟨{\sigma }_r⟩$. Expanding the variance of the sum then yields $\chi ={\beta }_{\mathrm{th}}{\sum }_{r}G(r)$. The derivative with respect to the dimensionless source ${\beta }_{\mathrm{th}}h$ removes the explicit inverse-temperature factor. Replacing the sum by an integral over physical coordinates requires the site-density factor $a^{-D}$.

The equal-time variance formula also holds in a quantum system when $[H_0,M]=0$. Without that commutation assumption, differentiating a matrix exponential instead givesThis is an integrated imaginary-time-ordered correlation function, not generally ${\beta }_{\mathrm{th}}$ times the equal-time variance. It follows by insertinginto the derivative of the normalized expectation value and using the cyclic matrix trace.

For example, a single qubit with $H_h=\Delta {\sigma }_z-h{\sigma }_x$, $\Delta >0$, has $\chi =\tanh ({\beta }_{\mathrm{th}}\Delta )/\Delta$ at $h=0$, whereas ${\beta }_{\mathrm{th}}\mathrm{Var}({\sigma }_x)={\beta }_{\mathrm{th}}$. Their difference directly exhibits the quantum noncommutation correction. Below an ordered transition, selecting a pure thermodynamic phase also avoids a macroscopic phase-mixture contribution to the classical sum.

The Ising model is a spin model with an Ising spin ${\sigma }_v\in \{-1,1\}$ at every graph vertex $v$. For a finite graph with vertex set $V$ and edge set $E$, a uniform coupling $J$ and field parameter $h$ give the HamiltonianEach edge is counted once. Here $T>0$ is temperature, $k_B$ is the Boltzmann constant, and $\beta$ is inverse temperature; $J$ and $h$ have units of energy. For magnetic applications, $h$ includes the magnetic moment multiplying the applied magnetic field. The canonical ensemble assigns the Boltzmann distributionThe partition function $Z$ makes the probabilities sum to one. Positive $J$ rewards equal neighboring Ising spins, modeling ferromagnetism; negative $J$ rewards opposite neighbors, giving an antiferromagnetic Ising model. Positive $h$ rewards $+1$, while negative $h$ rewards $-1$.^[ref]

A single local rule can organize a whole graph: rewarding agreement on each edge makes all-equal Ising spin patterns favorable even though no edge directly couples distant vertices. The four-spin Ising ring makes this visible with only sixteen microstates. It also separates three different questions: whether neighbors agree, whether the mean spin magnetization is nonzero, and how large its thermal fluctuations are.

A motivating experiment is the University of Colorado Boulder's Curie-temperature demonstration. An iron rod suspended from a twisting wire on a stand is initially held near a permanent magnet by ferromagnetism. A gas burner supplied from a propane cylinder heats the rod; the readout is its motion away from the magnet, followed by return as it cools. Repeated cooling supplies a reversible comparison on the same sample. The source explicitly calls for heating the rod while keeping the magnet cool, since heating the magnet can reduce its strength. The inference is that warming the sample weakens its magnetic attraction across the Curie temperature. This is a qualitative force readout, with no calibrated temperature or magnetization trace supplied: it motivates competition between alignment and thermal fluctuations, but does not identify $J$, measure individual Ising spins, or realize a four-spin Ising ring.^[ref]

Ernst Ising's 1925 paper examined a linear chain of two-orientation magnetic elements with neighboring interactions. In its calculation for a large chain, the zero-field mean magnetization vanishes and the model has no spontaneous magnetization. The inspected English translation states this immediately after equation (8) in §2. This historical result concerns the chain; it does not establish the behavior of a two-dimensional Ising model.^[ref]

Source paper: www.whatdotheyknow.com/request/oxford_physics_past_papers_2/response/2340696/attach/5/Physics%20Papers%202016.pdf?cookie_passthrough=1. The cover identifies Trinity Term 2017, 13 June; the bundle filename does not determine the examination year. We use $\hslash =c=1$ in the classical field theory and quantum field theory and lattice spacing one for the spin models.

A spin configuration specifies the spin variable at every site of a spin model. For $N$ Ising spins, it is a member of $\{-1,1{\}}^N$.

Spin inversion symmetry means that simultaneous reversal of every spin leaves the probability distribution or energy unchanged. For a finite spin model with this symmetry, the weights of opposite configurations pair up, so every observable that changes sign under the reversal has zero expected value. An applied magnetic field generally breaks this symmetry.

A spin model assigns a discrete or continuous spin variable to each site of a spin lattice or vertex of a graph and gives interacting configurations a Boltzmann factor. Spin models describe collective order and phase transitions in magnetic and other many-body systems.

The Transverse-field Ising model is a quantum spin model with competing Ising exchange and a perpendicular field, for example $H=-J{\sum }_i{X}_i{X}_{i+1}-h{\sum }_i{Z}_i$. In one dimension its ordered and paramagnetic phases meet at $|h/J|=1$.