1549380323-Statistical Mechanics Theory and Molecular Simulation

614 Critical phenomena

a scalar variableσzithat can take values of±1 only. At this point, since there is only
one relevant spatial axis, we can henceforth drop the “z” label on the spin variables.
Finally, we will make the assumption that the spin-spin couplings, whichare now
simple numbersJij, couple only nearest neighboring spins. Thus, the Hamiltonian for
this idealized situation becomes

H=−

1

2

∑

<i,j>

Jijσiσj−h

∑

i

σi, (16.3.3)

where the notation

∑

<i,j>signifies that only nearest-neighbor interactions are in-
cluded in the first term. Theh= 0 limit of eqn. (16.3.3) gives the unperturbed Hamil-
tonian

H 0 =−

1

2

∑

<i,j>

Jijσiσj. (16.3.4)

The model described by eqn. (16.3.3) is known as theIsing model, named after the
German physicist Ernst Ising (1900–1998). We have dropped the hat fromHin eqns.
(16.3.3) and (16.3.4), signifying that the Ising model is a type of generic discrete
model obtainable as idealizations of actual classical or quantum Hamiltonians (see
Problems 10.9 and 16.5 for examples of the former).
As for the gas–liquid phase transition, where the densityρwas used to distin-
guish one phase from another, we seek a variable that can distinguish a disordered
phase from an ordered, magnetized phase. To this end, we introduce the average total
magnetization

M=

〈N

∑

i=1

σi

〉

, (16.3.5)

which is an extensive thermodynamic observable. The equivalent intensive observable
is the average magnetization per spinm=M/N. In a perfectly ordered state,m=± 1
andM =±N, depending on whether the spins are aligned along the positive or
negativez-direction. The applied magnetic fieldhplays the same role as the applied
external pressureP in the gas–liquid case and therefore the phase diagram is a plot
of the phases in theh–T plane, as shown in Fig. 16.4. As the figure suggests, the
uniaxial nature of the Ising model leads to a single line,h= 0, along which a first-
order phase transition between spin-up and spin-down ordered states can occur. All
thermodynamic functions are smooth functions ofhandT everywhere else in the
phase diagram. Theh= 0 coexistence line terminates at a critical point, the only
point at which the phase transition becomes second order. ForT > Tc, the system is
in a disordered state forh= 0. It could still order in the presence of a finite applied
field and, therefore, is paramagnetic. ForT < Tc, ash→0, a finite magnetization
persists down toh= 0. Ifh→ 0 +, then the magnetization will be positive, and if
h→ 0 −, it will be negative. As this analysis implies, ath= 0, the magnetization
can be either positive or negative, and indeed, the Ising model exhibits a two-phase
coexistence ath= 0. Since a plot ofmvs.hforT < Tcis an isotherm of the equation
of state, such an isotherm shows a discontinuous change inm(see Fig. 16.5). For
T > Tc, the magnetization vanishes ash→0. The critical isotherm shown in Fig. 16.5