The Peierls argument

Starting around 1933, Peierls published scaling arguments why a phase transition should occur in 2d as opposed to 1d. Here we report a few of these kinds of arguments to demonstrate their spirit. Note that their validity also comes from the fact that thanks to Onsager, we have an exact solution and thus can check back if they describe the core of

Simple argument for 1d

We consider an Ising chain with all spins up and then select a few neighboring spins and flip the whole island over. This creates twodomain walls(also calledgrain boundaries ordefects) in the chain. The change in energy is

∆E=2·2J

because there are two defects, each with an energy penalty 2J. The change in entropy corresponds to the number of ways to choose the positions of the two defects:

∆S= kBlnN(N−1)

2 ≈2kBlnN

where we assume the number of lattice sitesN1 in the thermodynamic limit. Thus the change in free energy reads

∆F=4J−2k_BTlnN <0

for any temperatureTin the thermodynamic limit. This means that it is always favor- able to create grain boundaries due to entropic reasons and a phase transition to order cannot occur at finite temperature.

More complex argument for 1d

We now look at an arbitrary number of domain walls, not only at one island with two of them. We introduce the number of such domain wallsM and write the free energy in the domain wall picture:

F=2J M−k_BTln N

M

In the thermodynamic limit and with the Stirling formula we get F

N =2Jx+k_BT(xlnx+ (1−x)ln(1−x))

wherex= M/Nis the domain wall density. If we minimizeFforxwe get x_eq = ¹

e^2J/kBT+1

thus at finiteTthere is always a finite domain wall density and correlations decay over a finite distance. Moreover the system will not feel the effect of the boundary conditions.

Only atT =0 we havexeq =0, because then entropy does not matter.

Simple argument for 2d

We now want to make the simple argument for 2d rather than for 1d. We immediately encounter the problem that now there are two processes we have to account for: where to place the domain walls, and which shape to assign to them. With some intuition, we anticipate that shape fluctuations are now more important than where the islands are located. Thus we consider one island of down spins in a sea of up spins. The change in energy is

∆E= L·2J

whereLis the contour length of the domain. A crude estimate for the number of pos- sible shapes is 3^L, assuming a random walk on a 2d cubic lattice and neglecting inter- sections and the fact that it has to close onto itself (at each lattice site, there are three possibilities to proceed). Thus for entropy we have

∆S=kBln 3^L. Together we get

∆F= L(2J−k_BTln 3)

and thus ∆F < 0 only for T > T_c = 2J/(ln 3k_B) even in the thermodynamic limit L→_∞. Thus this simple argument predicts that in 2d a phase transition can take place at finiteT, and the reason is a feature that is only present in two and higher dimensions, namely shape fluctuations of the domain walls.

More complex argument for 2d

Another way to identify a phase transition is to investigate the effects of boundaries.

We consider a quadratic field of spins and fix all the ones at the boundary to point up.

We then consider the spin in the middle and ask if it keeps the up-preference of the boundary in the TD-limit (p+ > 1/2 ?). One can show that for sufficiently low but finite T, indeed this happens. This means that correlations do not decay completely and that spontaneous magnetisation can emerge, indicating a phase transition.

We consider the quantitym = p+−p− =2p+−1, which will be finite if spontaneous magnetization exists and vanish otherwise. We can write

m= ¹ Z

∑

Σ+

e^−βH− ¹ Z

∑

Σ−

e^−βH= ¹ Z

∑

Σ+

e^−βH(1−_Σ)

The first and second terms are sums over all configurations with a positive and nega- tive central spin, respectively. The basic idea of the newly defined quantity Σis that each configuration with a positive central spin can be turned into one with a negative central spin by flipping all spins in the surrounding positive domain. Importantly, the difference in energy is simply 2Jl, wherelis the length of the domain wall surrounding this domain. Therefore one can write

Σ=

∑

^{e−2Jβl =}

∑

^{∞ g(l)e−2Jβl}

where the sum is now over all configurations which have been obtained by flipping. In the second step we have rewritten the sum in terms of the length of the boundary. Here g(l)is the number of domains with lengthl. We note that the minimumlis 4 (one spin flipped) and that one only will have even values (l = 4, 6, . . . ), because adding spins one by one to the domain increaseslby 2.

In order to prove the polarization, we have to show thatΣcan be smaller than 1. We do this by establishing an upper bound forg(l):

g(l)<(^l

4)²·4·3^l−1· ¹ 2l = ^l

243^l

The first term is the maximal area corresponding to the contour lengthl. The second term is the number of possible paths starting from each point within this area: 4 for the first step and 3 for each additional step (on a 2d simple cubic lattice). The last term corrects for the fact that a path can go in two directions and can start at any point along the contour of a boundary. We now transfer this into an upper bound forΣ:

Σ<

∑

∞ l=₄

l

24w^l = ¹ 24

∑

∞ n=2

(2n)w⁽²ⁿ⁾ = ^w

4(2−w²) 12(1−w²)²

wherew=3e^−2βJ. We thus obtainΣ<1 forw<wc=0.87. This in turn translates into a critical temperature

Tc = ^2J

k_Bln(3/wc) =_1.6J/kB

The exact result for the 2d Ising model isT_c =2.269J/k_B (see below). Thus the Peierls argument does not only prove the transition, but even gives a reasonable first estimate for its value. Note that here we have established only an upper bond forΣ. This does not mean thatΣwill be different from 1 above the critical temperature, we only showed that it will certainly become smaller than this value at sufficiently low temperature. Our argument is obviously very crude because we neglect interactions between boundary loops, which will strongly bring down the number of possible paths.