Ising model

A model [a1] defined by the following Hamiltonian (cf. Hamilton function) $\mathcal{H}$ (i.e. energy functional of variables; in this case the "spins" $S _ { i } = \pm 1$ on the $N$ sites of a regular lattice in a space of dimension $d$)

\begin{equation} \tag{a1} \mathcal{H} = - \sum _ { i < j = 1 } ^ { N } J _ { i j } S _ { i } S _ { j } - H \sum _ { i = 1 } ^ { N } S _ { i }. \end{equation}

Here, $J _ {i j }$ are "exchange constants" , $H$ is a (normalized) magnetic field, involving an interpretation of the model to describe magnetic ordering in solids ($M = \sum _ { i = 1 } ^ { N } S _ { i }$ is "magnetization" , the Zeeman energy $- H M$ in (a1) is the energy gained due to application of the field).

Since its solution for $d = 1$ in 1925 [a1], the model became a "fruitfly" for the development of both concepts and techniques in statistical thermodynamics. It appears also in other interpretations in lattice statistics: defining occupation variables $\rho _ { i } = ( 1 - S _ { i } ) / 2$, where lattice site $i$ is empty ($\rho _ { i } = 0$) if $S _ { i } = 1$ or occupied ($\rho _ { i } = 1$) if $S _ { i } = - 1$. This is the lattice gas model of a fluid. One can also interpret the cases $S _ { i } = \pm 1$ as two chemical species $A$, $B$ for describing ordering or unmixing of binary alloys $( A B )$, etc.

Statistical thermodynamics [a2] aims to compute average properties of systems with a large number of degrees of freedom (i.e., in the thermodynamic limit $N = \infty$). These averages at a temperature $T$ are obtained from the free energy $F ( T , H )$ (per spin) or the partition function $Z$,

\begin{equation} \tag{a2} F = - \frac { k _ { B } T \operatorname { ln } Z } { N } , \quad Z = \operatorname { Tr } \operatorname { exp } \left( - \frac { \mathcal{H} } { k _ { B } T } \right). \end{equation}

Here, $k _ { B }$ is the Boltzmann constant [a2], and the trace operation $\operatorname { Tr}$ stands for a sum over all the states in the phase space of the system (which here is the set of $2 ^ { N }$ states $S _ { 1 } = \pm 1 , \dots , S _ { N } = \pm 1$). Magnetization per spin $m \equiv \langle M \rangle _ { T } / N$, susceptibility $\chi$, entropy $S$, etc. are then found as partial derivatives of $F$ [a2]:

\begin{equation} \tag{a3} \left\{ \begin{array}{l}{ m = - \left( \frac { \partial F } { \partial H } \right) _ { T }, }\\{ \chi = \left( \frac { \partial m } { \partial H } \right) _ { T }, }\\{ S = - \left( \frac { \partial F } { \partial T } \right) _ { H }, }\end{array} \right. \end{equation}

where $\langle A \rangle _ { T }$ stands for a canonical average of a quantity $A$:

\begin{equation*} \langle A \rangle _ { T } = Z ^ { - 1 } \operatorname { Tr } \left[ \operatorname { exp } ( - \frac { \mathcal{H} } { k _ { B } T } ) A \right]. \end{equation*}

The Ising model is important since for $d \geq 2$ it exhibits phase transitions. In the simplest case, $J _ { i j } = J$ if sites $i$, $j$ are nearest neighbours on the lattice and zero elsewhere, a transition occurs for $J > 0$ from a paramagnet ($T > T _ { c }$) to a ferromagnet ($T < T _ { c }$) at a critical temperature $T _ { c }$. In the disordered paramagnet $\operatorname { lim } _ { H \rightarrow 0 } m ( T , H ) = 0$, while in the ordered ferromagnet the spontaneous magnetization $m_S$ occurs:

\begin{equation} \tag{a4} m _ { s } = \operatorname { lim } _ { H \rightarrow 0 } m ( T , H ) > 0. \end{equation}

This is an example of spontaneous symmetry breaking: $\mathcal{H}$ for $H = 0$ does not single out a sign of $m$ (replacing all $\{ S _ { i } \}$ by $\{ - S _ { i } \}$ leaves $\mathcal{H}$ invariant). However, for $T < T _ { c }$ and $H = 0$ the equilibrium state of the system is two-fold degenerate ($\pm m _ { s }$). This degeneracy is already obvious from the groundstate of (a1), for $T \rightarrow 0$, found from the absolute minimum of $H$ as a functional of the $\{ S _ { i } \}$: for $H = 0$ this minimum occurs for either all $S _ { i } = + 1$ or all $S _ { i } = - 1$.

Interestingly, for $d = 1$ no such phase transition at $T_{c} > 0$ occurs; rather $T _ { c } = 0$ [a1]. The problem (a1)–(a3) is solved exactly by transfer matrix methods [a3]. Rewriting (a1) as $H = - J \sum _ { i = 1 } ^ { N } S _ { i } S _ { i+ 1 } - {\cal H} \sum _ { i = 1 } ^ { N } S _ { i }$ with the periodic boundary condition $S _ { N + 1 } = S _ { 1 }$, one finds

\begin{equation} \tag{a5} Z = \sum _ { S _ { 1 } = \pm 1 } \ldots \sum _ { S _ { N } = \pm 1 } \end{equation}

\begin{equation*} \operatorname { exp } \left\{ \frac { 1 } { k _ { B } T } \sum _ { i = 1 } ^ { N } [ J S _ { i } S _ { i+ 1 } + \frac { H } { 2 } ( S _ { i } + S _ { i+ 1 } ) ] \right\} = \end{equation*}

\begin{equation*} = \sum _ { S _ { 1 } = \pm 1 } \cdots \sum _ { S _ { N } = \pm 1 } \prod _ { i = 1 } ^ { N } \end{equation*}

\begin{equation*} \operatorname { exp } \left\{ \frac { 1 } { k _ { B } T } \left[ J S _ { i } S _ { i + 1 } + \frac { H } { 2 } ( S _ { i } + S _ { i + 1 } ) \right] \right\} = \end{equation*}

\begin{equation*} = \sum _ { S _ { 1 } = \pm 1 } \ldots \sum _ { S _ { N } = \pm 1 } \prod _ { i = 1 } ^ { N } \langle S _ { i } | \mathcal{P} | S _ { i+ 1 } \rangle \end{equation*}

The $( 2 \times 2 )$-matrix ${\cal P} = ( P _ { s s ^ { \prime } } ) = ( \langle S | {\cal P} | S ^ { \prime } \rangle )$ is defined as

\begin{equation} \tag{a6} \mathcal{P} \equiv \left( \begin{array} { c c } { \operatorname { exp } \left( \frac { J + H } { k _ { B } T } \right) } & { \operatorname { exp } \left( \frac { - J } { k _ { B } T } \right) } \\ { \operatorname { exp } \left( \frac { - J } { k _ { B } T } \right) } & { \operatorname { exp } \left( \frac { J - H } { k _ { B } T } \right) } \end{array} \right). \end{equation}

Now $Z$ is simply the trace of an $N$-fold matrix product,

\begin{equation} \tag{a7} Z = \sum _ { S _ { 1 } = \pm 1 } \left( S _ { 1 } | \mathcal{P} ^ { N } | S _ { 1 } \right) = \lambda _ { + } ^ { N } + \lambda ^ { N }_{-}, \end{equation}

where the property was used that the trace of a symmetric matrix is independent of the representation, and so one can evaluate the trace by first diagonalizing $\mathcal{P}$,

\begin{equation} \tag{a8} \mathcal{P} = \left( \begin{array} { c c } { \lambda _ { + } } & { 0 } \\ { 0 } & { \lambda _ { - } } \end{array} \right) , \quad \mathcal{P} ^ { N } = \left( \begin{array} { c c } { \lambda _ { + } ^ { N } } & { 0 } \\ { 0 } & { \lambda ^ { N } } \end{array} \right), \end{equation}

where the eigenvalues $\lambda _ { + }$, $\lambda_{-}$ are found from the vanishing of the determinant, $\operatorname { det } ( \mathcal{P} - \lambda \mathcal{I} ) = 0$, $\cal I$ being the unit $( 2 \times 2 )$-matrix:

\begin{equation} \tag{a9} \lambda _ { \pm } = \operatorname { exp } \left( \frac { J } { k _ { B } T } \right) \operatorname { cosh } \left( \frac { H } { k _ { B } T } \right) \pm \end{equation}

\begin{equation*} \pm \left[ \operatorname { exp } ( \frac { 2 J } { k _ { B } T } ) \operatorname { cosh } ^ { 2 } ( \frac { H } { k _ { B } T } ) - 2 \operatorname { sinh } ( \frac { 2 J } { k _ { B } T } ) \right] ^ { 1 / 2 }. \end{equation*}

In the limit $N \rightarrow \infty$ the largest eigenvalue dominates, $Z \rightarrow \lambda _ { + } ^ { N } $, and hence

\begin{equation} \tag{a10} F = - k _ { B } T \operatorname { ln } \lambda _ { + } = \end{equation}

\begin{equation*} = - J - k _ { B }T \operatorname { ln } \left\{ \operatorname { cosh } \left( \frac { H } { k _ { B } T } \right) + + \left[ \operatorname { sinh } ^ { 2 } \left( \frac { H } { k _ { B } T } \right) + \operatorname { exp } \left( - \frac { 4 J } { k _ { B } T } \right) \right] ^ { 1 / 2 } \right\}, \end{equation*}

\begin{equation} \tag{a11} m = \frac { \operatorname { sinh } \left( \frac { H } { k _ { B } T } \right) } { [ \operatorname { sinh } ^ { 2 } \left( \frac { H } { k _ { B } T } \right) + \operatorname { exp } \left( - \frac { 4 J } { k _ { B } T } \right) ] ^ { 1 / 2 } }. \end{equation}

Indeed, for $T > 0$ there is no spontaneous magnetization, and for $H \rightarrow 0$ the susceptibility becomes $\chi = ( k _ { B } T ) ^ { - 1 } \operatorname { exp } ( 2 J / k _ { B } T )$.

It is remarkable that (a11) strongly contradicts the popular molecular field approximation (MFA). In the molecular field approximation one replaces in the interaction of every spin $S _ { i }$ with its neighbours, $[ S _ { i } ( S _ { i - 1 } + S _ { i + 1 } ) ]$, the spins by their averages, $S _ { i - 1 } \rightarrow \langle m \rangle$; $S_{i + 1 }\rightarrow \langle m \rangle$, the problem becomes a single-site Hamiltonian where $S _ { i }$ is exposed to an effective field $H _ { \text{eff} } = H + 2 m J$, which needs to be calculated self-consistently; carrying out the average over the two states $S _ { i } = \pm 1$ one finds

\begin{equation} \tag{a12} m = \frac { \operatorname { exp } \Bigl( \frac { H _ { \text{eff} } } { k _ { B } T }\Bigr ) - \operatorname { exp } \Bigl( - \frac { H _ {\text{eff} } } { k _ { B } T }\Bigr ) } { \operatorname { exp }\Bigl ( \frac { H _ { \text{eff} } } { k _ { B } T }\Bigr ) + \operatorname { exp } \Bigl( - \frac { H _ { \text{eff} } } { k _ { B } T } \Bigr) } = \end{equation}

\begin{equation*} = \operatorname { tanh } [ \frac { H + 2 m J } { k _ { B } T } ], \end{equation*}

which yields $T _ { c } = 2 J / k _ { B }$ and $m _ { s } \propto ( 1 - T / T _ { c } ) ^ { \beta }$ with a critical exponent $\beta = 1 / 2$, and a Curie–Weiss law for $\chi ( \chi \propto ( T / T _ { c } - 1 ) ^ { - \gamma } \text { with } \gamma = 1 )$. Thus, the Ising model shows that the molecular field approximation in this case yields unreliable and misleading results!

For the Ising-model in $d = 2$, exact transfer matrix methods are applicable for $H = 0$; they show that a phase transition at $T _ { c } > 0$ does exist [a3], [a4], [a5]. But the critical exponents $\beta$, $\gamma$ differ very much from their molecular field approximation values; namely, $\beta = 1 / 8$ and $\gamma = 7 / 4$. This is important, since the exponents $\beta = 1 / 2$ and $\gamma = 1$ also follow from the Landau theory of phase transitions [a2], which only requires that $F$ can be expanded in a power series in $m$, with the coefficient at the $m ^ { 2 }$ term changing sign at $T _ { c }$ as $T / T _ { c } \rightarrow 1$, which are plausible assumptions on many grounds. The $d = 2$ Ising model testifies that neither molecular field approximation nor Landau theory are correct. The Ising model then prompted the development of entirely new theoretical concepts, namely renormalization group analysis [a6], by which one can understand how non-mean-field critical behaviour arises. The Ising model also became a very useful testing ground for many numerical methods: e.g. systematic expansions of $F$ at low $T$ (in the variable $u = \operatorname { exp } ( - 4 J / k _ { B } T )$) or at high $T$ in the variable $v = \operatorname { tanh } ( J / k _ { B } T )$ [a7], or Monte-Carlo methods [a8]. It also played a pivotal role for the concepts on surface effects on phase transitions, and for phase coexistence (domains of oppositely oriented magnetization, separated by walls). Such problems were described with a mathematical rigor that is seldomly found in the statistical thermodynamics of many-body systems. Rigorous work includes the existence of a spontaneous magnetization for $d \geq 2$ ( "Peierls proof" ), inequalities between spin correlations, theorems on the zeros of the partition function, etc.; see [a9]. The Ising model is the yardstick against which each new approach is measured.

Finally, there are extensions of the Ising model. One direction is to make the $J _ {i j }$ more complicated rather than uniformly ferromagnetic ($J _ { i j } > 0$). E.g., if in one lattice direction $J _ { 1 } > 0$ between nearest neighbours but $J _ { 2 } < 0$ between next nearest neighbours, the resulting anisotropic next nearest neighbour Ising model (ANNNI model) is famous [a10] for its phase diagram with infinitely many phases and transitions; choosing the $J _ { i j } = \pm J$ at random from a prescribed distribution, the resulting Ising spin glass [a11] is a prototype model of glasses and other disordered solids.

Another extension adds "time t" as a variable: by a transition probability $w ( \{ S _ { i } \} \rightarrow \{ S _ { i } ^ { \prime } \} )$ per unit time one is led to a master equation for the probability that a state $\{ S _ { 1 } , \ldots , S _ { N } \}$ occurs at time $t$. Such kinetic Ising models [a12] are most valuable to test concepts of non-equilibrium statistical mechanics, and provide the basis for simulations of unmixing in alloys ( "spinodal decomposition" ), etc. Finally, one can generalize the Ising model by replacing the spin $S _ { i } = \pm 1$ by a more complex variable, e.g. in the Potts model [a13] each site may be in one of $p$ states where $p$ is integer (also, the limit $p \rightarrow 1$ is of interest; the so-called "percolation problem" [a14]). The techniques for the Ising model (transfer matrix, series expansions, renormalization, Monte Carlo, etc.) are valuable for all these related problems, too.

References

[a1]	E. Ising, "Beitrag zur Theorie des Ferromagnetismus" Z. Phys. , 31 (1925) pp. 253–258
[a2]	L.D. Landau, E.M. Lifshitz, "Statistical physics" , Pergamon (1958)
[a3]	R.J. Baxter, "Exactly solved models in statistical mechanics" , Acad. Press (1982)
[a4]	L. Onsager, "Crystal statistics I. A two-dimensional model with an order-disorder transition" Phys. Rev. , 65 (1944) pp. 117–149
[a5]	B.M. McCoy, T.T. Wu, "The two-dimensional Ising model" , Harvard Univ. Press (1973)
[a6]	M.E. Fisher, "The renormalization group in the theory of critical behavior" Rev. Mod. Phys. , 46 (1974) pp. 597–616
[a7]	"Phase Transitions and Critical Phenomena" C. Domb (ed.) M.S. Green (ed.) , 3 , Acad. Press (1974)
[a8]	"Monte Carlo methods in statistical physics" K. Binder (ed.) , Springer (1979)
[a9]	R.B. Griffiths, "Rigorous results and theorems" C. Domb (ed.) M.S. Green (ed.) , Phase Transitions and Critical Phenomena , 1 , Acad. Press (1972) pp. 7–109
[a10]	W. Selke, "The Annni model-theoretical analysis and experimental application" Phys. Rep. , 170 (1988) pp. 213–264
[a11]	K. Binder, A.P. Young, "Spin glasses: experimental facts, theoretical concepts, and open questions" Rev. Mod. Phys. , 58 (1986) pp. 801–976
[a12]	K. Kawasaki, "Kinetics of Ising models" C. Domb (ed.) M.S. Green (ed.) , Phase Transitions and Critical Phenomena , 2 , Acad. Press (1972) pp. 443–501
[a13]	F.Y. Wu, "The Potts model" Rev. Mod. Phys. , 54 (1982) pp. 235–268
[a14]	D. Stauffer, A. Aharony, "Introduction to percolation theory" , Taylor&Francis (1992)