Professor Rodney Baxter, from the Mathematical Sciences Institute at ANU, received both the 2006 Onsager Prize of the American Physical Society (APS) and the separate Onsager Lectureship and Medal for 2006 from the Norwegian University of Science and Technology.

Both prizes are named for widely respected theoretical physicist and Nobel Laureate Lars Onsager, who exactly calculated the order parameter of the Ising model in 1949. This was the first such calculation for a statistical mechanical model of magnetism. He received the 1968 Nobel Prize in Chemistry for his earlier work on irreversible thermodynamics.

“These awards are particularly pleasing for me as it is recognition of work on the order parameters of the chiral Potts model, which is research in the Lars Onsager tradition,” Professor Baxter said.

In his research, Professor Baxter showed by careful mathematical analysis that numerical predictions about the order parameters of the chiral Potts model were exactly right, something which had been elusive for mathematicians in the 15 years before his proof.

Simply, the complicated chiral Potts model is a prototype of theoretical descriptions of the interaction and behaviour of materials at the molecular level. It includes the Ising model as a special case.

The “exact solution” of the chiral Potts model achieved by Professor Baxter has important implications in the physical sciences. It greatly increases confidence in theoretical models, particularly in materials science, where physicists around the world, and at ANU, are building next generation electronic devices using two-dimensional layers in ‘chips’. These specialised ‘chips’ may eventually be used in computing, audiovisual technologies and advanced telecommunications.

The American Physical Society prize is awarded to recognise outstanding research in theoretical statistical physics. It awarded Professor Baxter the Prize for “his original and groundbreaking contributions to the field of exactly solved models in statistical mechanics, which continue to inspire profound developments in statistical physics and related fields”.

Full Story from ANU

The supersymmetric t-J model is a mathematical model of strongly correlated electrons. Analytical solutions to integrable models provide insight into the dynamics of systems of strongly correlated electrons. Recently, the search for a theory explaining high Tc superconductivity has resulted in increased interest in integrable models describing correlated electrons.

A University of Queensland research team led by senior mathematics lecturer Dr Yao-Zhong Zhang has successfully solved a major long-standing problem in mathematical physics.

Dr Zhang and his postdoctoral fellows Wen-Li Yang and Shao-You Zhao, from the North West University in China, have discovered the determinant representation of correlation functions of the supersymmetric t-J model.

Dr Zhang said the theoretical problem had been around for many years.

“The analytic computation of correlation functions was arguably one of the most challenging and notoriously difficult problems in mathematical physics and its solution will have important implications.”

“It opens doors to further research in the theory of exactly soluble models as well as in pure mathematics, statistical mechanics and condensed matter physics.”

Dr Zhang said the work had attracted a great deal of academic interest and had been described by world-leading authorities in the field as a “major breakthrough” and a “great discovery”.

A paper on the solution will appear in the January 2006 issue of the Journal of Mathematical Physics, a leading international scientific journal published by the American Institute of Physics.

A second paper has also been submitted to the prestigious mathematical physics journal Communications in Mathematical Physics.

Dr Zhang and his research team have also been approached by the International Journal of Modern Physics B to write a review article on their research.

Source: University of Queensland

Electrons are elementary particles. Experiment indicates that they have no sub-structure.

They have a mass of [tex]9.11 \times 10^{-31}[/tex] kg and an electric charge of [tex]-1[/tex]. They have a spin of [tex]\frac{1}{2}[/tex].

The most interesting thing is the (intrinsic) spin [tex]\frac{1}{2}[/tex]. This spin is not like the spin we usually associate to a spinning object.

For a classical object, the spin can take any (reasonable) value we wish. For an electron, the spin is quantised. The experiment that demonstrates this is the Stern-Gerlach experiment, where a beam of silver atoms is passed through an inhomogeneous magnetic field. The experiment reveals that rather than fanning out as would be expected classically, the beam is split into two beams, one going up in the direction of the z-component of the magnetic field and the other down.

These two spins are called up [tex]\frac{1}{2}[/tex] and down [tex]-\frac{1}{2}[/tex]. The funny number [tex]\frac{1}{2}[/tex] comes from the Lie algebra [tex]su(2)[/tex].

In an effort to understand the nature of electron spin we turn to the study of the Lie algebra [tex]su(2)[/tex].

We begin with the defining relations of [tex]su(2)[/tex]. The algebra has three generators, [tex]S_x, S_y[/tex] and [tex]S_z[/tex] which obey the following commutation relations

[tex] [S_x, S_y] = i S_z [/tex] (and cyclic permutations).

We are working in natural units here, so that [tex]\hbar[/tex] is set to unity.

It is convenient to introduce the ladder operators [tex]S_+[/tex] and [tex]S_-[/tex] defined by
[tex] S_+ = S_x + i S_y [/tex]
[tex] S_- = S_x - i S_y . [/tex]

We also set

[tex]S_0 = S_z [/tex].

In terms of these new operators, the [tex]su(2)[/tex] commutation relations become

[tex] [S_0 ,S_\pm]= \pm S_\pm [/tex]
and
[tex] [S_+ ,S_-]= 2 S_0 . [/tex]

Let [tex]V[/tex] be a vector space over a field [tex]F[/tex]. Since we are doing quantum mechanics, we take the field to be [tex]C[/tex], the field of complex numbers.

Let [tex]L[/tex] be a Lie algebra, in our case [tex]L = su(2)[/tex]. What we want to do is make the elements of our Lie algebra act on the elements of the vector space [tex]V[/tex] in a way that is consistent with the commutation relations of the algebra. In other words we want a homomorphism from [tex]L[/tex] into [tex]V[/tex].

Now since we are working over the complex field [tex]C[/tex], we can be sure that there is at least one solution to the eigenvector equation

[tex] S_0 v_k = k v_k. [/tex]

Now, [tex] [ S_0 , S_+ ] = S_+ [/tex], so

[tex] S_0 S_+ = S_+ S_0 + S_+ [/tex]

and the action of [tex]S_0[/tex] on the vector [tex]S_+ v_k[/tex] is

[tex] S_0 (S_+ v_k ) = k S_+ v_k + S_+ v_k = (k+1) S_+ v_k .[/tex]

So, [tex]S_+ v_k[/tex] is an eigenvector of [tex]S_0[/tex], with eigenvalue [tex]k+1[/tex].

We define

[tex] S_+ v_k = v_{k+1} , [/tex]

giving

[tex] S_0 v_{k+1} = (k+1) v_{k+1} . [/tex]

Now we see why [tex]S_+[/tex] is called a ladder operator (raising operator), since it raises the eigenvalue of [tex]v_k[/tex].

We can do a similar thing with [tex]S_-[/tex], going throught the calculations, we find

[tex] S_0 (S_- v_k ) = (k-1) S_- v_k [/tex]

and we set

[tex] S_- v_k = v_{k-1} . [/tex]

The ladder operator [tex]S_-[/tex] is called a lowering operator, since it acts on [tex]v_k[/tex] , to reduce its eigenvalue by 1.

To summarise, we have three equations describing how our Lie algebra operators, [tex]S[/tex] act on vectors from [tex]v \in V[/tex],

[tex] S_0 v_k = k v_k [/tex]

[tex] S_+ v_k = v_{k+1} [/tex]

[tex] S_- v_k = v_{k-1} .[/tex]

We introduce the following operator, called the Casimir operator for [tex]su(2)[/tex]

[tex] S^2=S_x ^2+S_y ^2+S_z ^2. [/tex]

This operator isn’t really part of [tex]su(2)[/tex], however. The algebra [tex]su(2)[/tex] doesn’t contain elements like [tex]S_x ^2[/tex]. We need a “bigger” algebra, the Universal Eneveloping algebra [tex]U(su(2))[/tex] of [tex]su(2)[/tex], which consists of all products of the form [tex]S_x ^p S_y ^q S_z ^r[/tex], where [tex]p,q,r[/tex] are integers. For now, we’ll just ignore this and pretend [tex]S^2[/tex] is an element of [tex]su(2)[/tex].

Consider now the commutator,

[tex] [S^2, S_x ] = [S_x^2+S_y^2+S_z^2, S_x ] [/tex]
[tex] = [S_x^2, S_x ] + [S_y^2, S_x ] + [S_z ^2, S_x ][/tex]
[tex] = [S_x S_x ,S_x ]+[S_y S_y ,S_x ]+[S_z S_z , S_x ]. [/tex]

To expand the last expression we note that

[tex] [ab,c] = abc - cab = abc - acb + acb -cab [/tex]

[tex] = a(bc-cb) + (ac -ca)b [/tex]

[tex] = a[b,c] + [a,c]b. [/tex]

Hence,

[tex] [S^2, S_x ] = S_x [S_x ,S_x ] + [S_x ,S_x ]S_x + S_y [S_y ,S_x ] + [S_y ,S_x ]S_y [/tex][tex] + S_z [S_z , S_x] + [S_z ,S_x ]S_z[/tex]

[tex]= -i S_y S_z - i S_z S_y + i S_z S_y + i S_y S_z [/tex]

[tex] = -i S_y S_z + i S_y S_z + i S_z S_y - i S_z S_y = 0. [/tex]

We have shown that [tex]S^2[/tex] commutes with [tex]S_x[/tex] . A similar calculation shows that [tex]S^2[/tex] commutes with [tex]S_y[/tex] and [tex]S_z[/tex] .

We’ve seen that the [tex]su(2)[/tex] Casimir, namely [tex]S^2[/tex] commutes with the generators of [tex]su(2)[/tex]. In order to proceed we now express [tex]S^2[/tex] in a form which will make our calculations easier.

Recall that,

[tex] S^2 = S_x^2 + S_y^2 + S_z^2 [/tex]

and that our raising and lowering operators were defined as

[tex] S_+ = S_x + i S_y [/tex]

[tex] S_- = S_x - i S_y . [/tex]

(We also set [tex]S_0 = S_z[/tex] .)

Consider the following,

[tex] S_+ S_- = (S_x + i S_y )(S_x - i S_y ) [/tex]
[tex] = S_x ^2 -i S_x S_y + iS_y S_x + S_y ^2 [/tex]
[tex] = S_x ^2 -i [S_x ,S_y ] + S_y ^2 [/tex]
[tex] = S_x ^2 -i^2 S_z + S_y ^2 [/tex]
[tex] = S_x ^2 + S_y ^2 + S_z [/tex]

similarly

[tex] S_- S_+ = S_x ^2 + S_y ^2 - S_z . [/tex]

This gives us two expressions for [tex]S^2[/tex],

[tex] S^2 = S_+ S_- + S_0 ^2 - S_0 = S_+ S_- + S_0 (S_0 - 1) [/tex]

and

[tex] S^2 = S_- S_+ + S_0 ^2 + S_0 = S_- S_+ + S_0 (S_0 + 1). [/tex]