QuantumPhysics.dvi

and thus,

f=−

1

β

lnλ+ when J > 0

f=−

1

β

ln|λ−| when J < 0 (8.85)

In both cases, these functions are analytic functions ofβ, so there are no phase transitions

for any finite value ofβ. In other words, the system is in the same thermodynamic phase

for all temperatures.

8.14 Ordered versus disordered phases

An important qualitative characteristic of the dynamics of statistical magnetic systems is

order versus disorder. For magnetic systems, this property maybe understood systematically

in terms of the magnetization, defined as the thermodynamic average of the spin,

Sz= lim

N→∞

1

N

∑

i

Siz (8.86)

The magnetizationM(B) per unit volume is then defined by

M(B) =

Tr

(

Sze−βH

)

Tr (e−βH)

=− lim

N→∞

1

N

∂lnZ

∂(βB)

(8.87)

In the case of the 1-dimensional Ising model, and forB= 0, this quantity is easily computed

for bothJ >0 orJ <0. The eigenvaluesλ±both depend onβBthrough an even function

ofβB, and thus, the magnetizationM(B) at zero external magnetic field always vanishes.

This result is interpreted as the fact that the spins in the system, on average, point in all

directions randomly, so that their total contribution to magnetization vanishes in the bulk.

When can a system be ordered then? We have seen previously that forJ > 0, the

minimum energy states are

|+〉 σi= +1 i∈Λ

|−〉 σi=− 1 i∈Λ (8.88)

Note that, these states are mapped into one another R|±〉 = |∓〉 by the spin reversal

symmetryRof the Hamiltonian forB = 0. At high temperatures, the spins are allowed

to fluctuate away wildly from these minimum energy states, but one would expect that, as

temperature is lowered, that fluctuations away from these ground states are suppressed.