From PT-symmetric quantum mechanics

to conformal field theory

PATRICK DOREY¹, CLARE DUNNING² and ROBERTO TATEO^3,*

¹Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK

²IMSAS, University of Kent, Canterbury CT2 7NF, UK

3Dip.\ di Fisica Teorica

and INFN, Universit\`a di Torino, Via P. Giuria 1, 10125 Torino, Italy

^*Corresponding author

E-mail: p.e.dorey@durham.ac.uk; t.c.dunning@kent.ac.uk; tateo@to.infn.it

Abstract. One of the simplest examples of a PT-symmetric quantum system is

the scaling Yang--Lee model, a quantum field theory with cubic

interaction and purely imaginary coupling.

We give a historical review of some

facts about this model in $d \le 2$ dimensions,

from its original definition in connection with phase transitions

in the Ising model and its relevance to polymer physics, to

the role it has played in studies of integrable quantum field

theory and PT-symmetric quantum mechanics. We also

discuss some more general results on PT-symmetric quantum

mechanics and the ODE/IM correspondence, and mention applications

to magnetic systems and cold atom physics.

Keywords. Parity; time-reversal; PT symmetry; integrable models;

quantum mechanics.

PACS Nos 11.30.Er; 02.30.Hq