John Cardy’s scale-invariant journey in low dimensions: a special issue for his 70th birthday

By Phil Brown on November 3, 2017

Journal of Physics A: Mathematical and Theoretical celebrates the 70th birthday of Professor John Cardy with a special issue collection spearheaded by guest editors Pasquale Calabrese, Paul Fendley and Uwe C Täuber. The issue is nearing final completion and JPhys+ would like to highlight some of the papers submitted to this high quality issue.

John Cardy has affected several aspects of theoretical physics, making seminal contributions across a variety of fields. His work particularly concerned the field theoretical description of statistical physics and condensed matter; focusing on conformal field theory, percolation and integrate field theories, but also stretching beyond. John Cardy’s work has been recognized with the award of the Boltzmann medal, the Dirac medal by IOP, the Dirac medal by ICTP, the Onsager prize by APS and many more honors. This special issue forms a collection of articles which reflect the continuing impact of John Cardy’s work.

Critical Phenomena and Conformal Field Theory

In their contribution to this special issue Olalla A Castro-Alvaredo at City, University of London, Benjamin Doyon at King’s College London and Francesco Ravanini at Università di Bologna investigate the Irreversibility of the renormalization group flow in non-unitary quantum field theory.

We show irreversibility of the renormalization group flow in non-unitary but ${{\mathcal P}T}$ -invariant quantum field theory in two space-time dimensions. In addition to unbroken $\mathcal{PT}$ -symmetry and a positive energy spectrum, we assume standard properties of quantum field theory including a local energy-momentum tensor and relativistic invariance. This generalizes Zamolodchikov’s c-theorem to ${{\mathcal P}T}$ -symmetric Hamiltonians. Our proof follows closely Zamolodchikov’s arguments. Irreversibility of the renormalization group flow in non-unitary quantum field theory.

The key role played by the central charge c in the description of critical phenomena was discussed by John Cardy in his Boltzmann Medal lecture.

Percolation

Part of a planar or spherical percolation front propagating from left to right. Red: an activated cell (the agent), black: inactive agents, green: susceptible sites, yellow: weakened site. Both, the susceptible sites and the weakened site, are attackable. In the next time step the agent is deactivated, and zero, one, two or three of the attackable places become agents. Note the asymmetry of the number of the attackable sites in the direction of the propagating front. From Hans-Karl Janssen and Olaf Stenull 2017 J. Phys. A: Math. Theor. 50 324002 Copyright IOP Publishing

Cardy and Grassberger used a field theoretic approach to study epidemic models and percolation in their 1985 paper. Hans-Karl Janssen at Heinrich-Heine-Universität and Olaf Stenull at University of Pennsylvania contribute a study to the issue in the same field looking at Attacks and infections in percolation processes.

We discuss attacks and infections at propagating fronts of percolation processes based on the extended general epidemic process. The scaling behavior of the number of the attacked and infected sites in the long time limit at the ordinary and tricritical percolation transitions is governed by specific composite operators of the field-theoretic representation of this process. We calculate corresponding critical exponents for tricritical percolation in mean-field theory and for ordinary percolation to 1-loop order. Our results agree well with the available numerical data.

Entanglement entropy and Quantum Quench

The usefulness of quantum entanglement in understanding condensed matter systems is mitigated by the difficulty of computing it. John Cardy and Pasquele Calabrese pioneered the development of a ‘playground’ for entanglement using 1 + 1 dimensional conformal field theories. The ‘playground’ used a field theory replica technique to provide a practical computational framework. In their paper, Entanglement entropy in excited states of the quantum Lifshitz model, Daniel E Parker at Berkeley, Romain Vasseur at University of Massachusetts, Amherst and Joel E Moore from Berkeley investigate the entanglement properties of an infinite class of excited states in the quantum Lifshitz model (QLM).

The presence of a conformal quantum critical point in the QLM makes it unusually tractable for a model above one spatial dimension, enabling the ground state entanglement entropy for an arbitrary domain to be expressed in terms of geometrical and topological quantities. Here we extend this result to excited states and find that the entanglement can be naturally written in terms of quantities which we dub ‘entanglement propagator amplitudes’ (EPAs). EPAs are geometrical probabilities that we explicitly calculate and interpret. A comparison of lattice and continuum results demonstrates that EPAs are universal.

The Special Issue is full of high quality articles around John Cardy’s work and Journal of Physics A would like to thank everyone who contributed to the issue’s success; authors, reviewers and the guest editors. We finally wish John Cardy a very happy 70th birthday.

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Figure and Featured Image From Hans-Karl Janssen and Olaf Stenull 2017 J. Phys. A: Math. Theor. 50 324002. Quotes taken from John Cardy special issue papers mentioned immediately above. Copyright IOP Publishing All right reserved.