Guest blog: Murray Batchelor revisits Rodney Baxter’s publications

By Murray Batchelor

murray headshotI am Topical Reviews Editor for Journal of Physics A: Mathematical and Theoretical. I have been working primarily at the Centre for Modern Physics, Chongqing University, China since 2013. My work involves various aspects of mathematical and theoretical physics, chiefly on integrability.

With the special issue celebrating Rodney Baxter’s 75th birthday currently underway I have listed here the publications of Baxter in JPhysA over the past four decades. These papers illustrate the remarkable depth and significance of Baxter’s work. Some of my specific highlights include:

1. Hard hexagons: exact solution

R J Baxter 1980 J. Phys. A: Math. Gen. 13 L61 (free until 13/11/15)

This letter outlined the exact solution of the hard hexagon model using the generalized star-triangle or Yang-Baxter relation and uncovered remarkable unexpected connections with the Rogers-Ramanujan identities. Read more about the historical background and comments on this work by Rodney Baxter here.

Hard hexagon particles on a triangular lattice

Hard hexagon particles on a triangular lattice

 

2. Equivalence of the Potts model or Whitney polynomial with an ice-type model

R J Baxter et al 1976 J. Phys. A: Math. Gen. 9 397 (free until 13/11/15)

In this work Baxter, Kelland and Wu demonstrated the graphical equivalence between the Potts model and the six-vertex model by introducing a surrounding lattice as in the figure and its polygon decomposition. Temperley and Lieb had earlier demonstrated the algebraic equivalence. The relationship with the Whitney or dichromatic polynomial was also explained. At the time Stuart Kelland was Baxter’s PhD student and Fred Wu was on sabbatical leave in Canberra.

An irregular lattice and its surrounding graph

An irregular lattice and its surrounding graph

 

3. q colourings and chromatic polynomials of large triangular lattices

R J Baxter 1986 J. Phys. A: Math. Gen. 19 2821 (free until 13/11/15)

R J Baxter 1987 J. Phys. A: Math. Gen. 20 5241 (free until 13/11/15)

The first of these papers solved the critical O(n) model on the honeycomb lattice by the Bethe ansatz method, thereby giving the large-lattice limit of the chromatic polynomial of the triangular lattice. In the second paper the results were extended to the full complex q plane, giving the limiting distribution of the zeros of the chromatic polynomial shown in the figure below. These exact results continue to inspire work on other loop models and lattice chromatic polynomials.

Chromatic zeros in the large lattice limit

Chromatic zeros in the large lattice limit

 

4. Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models

F C Alcaraz et al 1987 J. Phys. A: Math. Gen. 20 6397 (free until 13/11/15)

This paper with Alcaraz, Barber, Batchelor and Quispel solved the XXZ Heisenberg chain with diagonal open boundary conditions by the Bethe ansatz method and used the equivalence with the Potts and Ashkin-Teller hamiltonians to obtain various surface critical exponents. Soon after, E K Sklyanin gave a seminal treatment of this and more general problems using the boundary Yang-Baxter (reflection) equation. Considerable progress has been made recently on solving this and related problems for non-diagonal open boundary conditions. On a personal note, this paper formed part of my PhD thesis.

 

In selecting the above highlights I have simply chosen the most well-known and highly cited Baxter publications in JPhysA. There are many other gems, as can be seen by glancing through the full list of publications below. They illustrate the original and inspirational nature of Baxter’s work and the power of exact results. The full ramifications of some of them, like Baxter’s Perimeter Bethe ansatz, remain to be fully appreciated and further explored.

See also:

First articles for R J Baxter’s 75th birthday issue are in!

Some academic and personal reminiscences of Rodney James Baxter

 

The full list of publications by R J Baxter in JPhysA:

Some academic and personal reminiscences of Rodney James Baxter

R J Baxter 2015 J. Phys. A: Math. Theor. 48 254001

The τ2 model and parafermions

R J Baxter 2014 J. Phys. A: Math. Theor. 47 315001

Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant DPQ

R J Baxter 2010 J. Phys. A: Math. Theor. 43 145002

Corner transfer matrices in statistical mechanics

R J Baxter 2007 J. Phys. A: Math. Theor. 40 12577

Some hyperelliptic function identities that occur in the chiral Potts model

R J Baxter 1998 J. Phys. A: Math. Gen. 31 6807

A direct proof of Kim’s identities

R J Baxter 1998 J. Phys. A: Math. Gen. 31 1105

Exact solution and interfacial tension of the six-vertex model with anti-periodic boundary conditions

M T Batchelor et al 1995 J. Phys. A: Math. Gen. 28 2759

Interfacial tension of the chiral Potts model

R J Baxter 1994 J. Phys. A: Math. Gen. 27 1837

Surface free energy of the critical six-vertex model with free boundaries

A L Owczarek and R J Baxter 1989 J. Phys. A: Math. Gen. 22 1141

Series expansion of the percolation probability for the directed square lattice

R J Baxter and A J Guttmann 1988 J. Phys. A: Math. Gen. 21 3193

Spontaneous magnetisations of the Ising model on the bathroom tile lattice

R J Baxter and T C Choy 1988 J. Phys. A: Math. Gen. 21 2143

Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models

F C Alcaraz et al 1987 J. Phys. A: Math. Gen. 20 6397

Generalised percolation probabilities for the self-dual Potts model

A L Owczarek and R J Baxter 1987 J. Phys. A: Math. Gen. 20 5263

Chromatic polynomials of large triangular lattices

R J Baxter 1987 J. Phys. A: Math. Gen. 20 5241

Perimeter Bethe ansatz

R J Baxter 1987 J. Phys. A: Math. Gen. 20 2557

q colourings of the triangular lattice

R J Baxter 1986 J. Phys. A: Math. Gen. 19 2821

Is the Zamolodchikov model critical?

R J Baxter and P J Forrester 1985 J. Phys. A: Math. Gen. 18 1483

Disorder points of the IRF and checkerboard Potts models

R J Baxter 1984 J. Phys. A: Math. Gen. 17 L911

A variational approximation for cubic lattice models in statistical mechanics

R J Baxter and P J Forrester 1984 J. Phys. A: Math. Gen. 17 2675

Deviations from critical density in the generalised hard hexagon model

P A Pearce and R J Baxter 1984 J. Phys. A: Math. Gen. 17 2095

Hard squares with diagonal attractions

R J Baxter and P A Pearce 1983 J. Phys. A: Math. Gen. 16 2239

Magnetisation discontinuity of the two-dimensional Potts model

R J Baxter 1982 J. Phys. A: Math. Gen. 15 3329

Hard hexagons: interfacial tension and correlation length

R J Baxter and P A Pearce 1982 J. Phys. A: Math. Gen. 15 897

An investigation of the high-field series expansions for the square lattice Ising model

I G Enting and R J Baxter 1980 J. Phys. A: Math. Gen. 13 3723

Entropy of hard hexagons

R J Baxter and S K Tsang 1980 J. Phys. A: Math. Gen. 13 1023

Hard hexagons: exact solution

R J Baxter 1980 J. Phys. A: Math. Gen. 13 L61

399th solution of the Ising model

R J Baxter and I G Enting 1978 J. Phys. A: Math. Gen. 11 2463

A special series expansion technique for the square lattice

I G Enting and R J Baxter 1977 J. Phys. A: Math. Gen. 10 L117

The three-spin Ising model as an eight-vertex model

R J Baxter and I G Enting 1976 J. Phys. A: Math. Gen. 9 L149

Equivalence of the Potts model or Whitney polynomial with an ice-type model

R J Baxter et al 1976 J. Phys. A: Math. Gen. 9 397

Magnetization of the three-spin triangular Ising model

R J Baxter et al 1975 J. Phys. A: Math. Gen. 8 245

 


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This work is licensed under a Creative Commons Attribution 3.0 Unported License.  Image 1 Courtesy of Murray Batchelor, Image 2 R J Baxter 1980 J. Phys. A: Math. Gen. 13 L61, Image 3 R J Baxter et al 1976 J. Phys. A: Math. Gen. 9 397, Image 4 R J Baxter 1987 J. Phys. A: Math. Gen. 20 5241. Copyright IOP publishing.



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