One of the most important outstanding problems in particle physics is to describe the finite density and finite temperature behavior of strongly interacting systems involving fermions. The primary motivation is to understand quantum chromodynamics (QCD), the theory of the strong force that binds quarks and gluons, as a function of temperature and density. I have worked recently on the role of chiral symmetry breaking for the phase diagrams of the 1+1 dimensional field theories known as Gross-Neveu models. These models provide unique analytic insight into asymptotic freedom, dimensional transmutation, mass generation and chiral symmetry breaking. Their phase diagram analysis at finite temperature and density has only been solved recently. Surprisingly, we were able to find a complete analytic solutions to this problem using the gap equation technique, showing that at low temperatures and sufficiently high density the system spontaneously forms a crystalline state. This work has been done largely in collaboration with my graduate student, Gokce Basar, and with Michael Thies of the University of Erlangen.

The following paper was featured with a Synopsis by the APS:

* Gerald V. Dunne, Christian Fitzner, Michael Thies, “Baryon-baryon scattering in the Gross-Neveu model: the large N solution”: arXiv:1108.5888 [hep-th] , Phys. Rev. D 84, 105014 (2011)

The phase diagram of the Gross-Neveu models is derived in:

* Gokce Basar, Gerald V. Dunne, Michael Thies, “Inhomogeneous Condensates in the Thermodynamics of the Chiral NJL_2 model”, arXiv:0903.1868 , published in PhysRevD.79.105012.

This solution was made possible by the discovery of a new exact crystalline solution to the gap equation:

* Gokce Basar, Gerald V. Dunne, “Self-consistent crystalline condensate in chiral Gross-Neveu and Bogoliubov-de Gennes systems”, [arXiv:0803.1501 [hep-th]], published in Phys.Rev.Lett. 100 (2008) 200404.

* Gokce Basar, Gerald V. Dunne, “A Twisted Kink Crystal in the Chiral Gross-Neveu model”, [arXiv:0806.2659 [hep-th]], published in Phys.Rev. D78 (2008) 065022.

A novel geometric formulation of the gap equation problem, and its connection to string theory is explained in:

* Gokce Basar, Gerald V. Dunne, “Gross-Neveu Models, Nonlinear Dirac Equations, Surfaces and Strings”, arXiv:1011.3835 , published in JHEP 1101:127,2011

A formulation of the problem of finding inhomogeneous condensates in the language of world line path integrals:

* Gerald Dunne, Holger Gies, Klaus Klingmuller, Kurt Langfeld, “Worldline Monte Carlo for fermion models at large N(f)”, arXiv:0903.4421 [hep-th]]arXiv:0903.4421 [hep-th]], published in JHEP 0908 (2009) 010

The explanation of why an analytic solution is possible for this system, and its relation to quantum mechanical supersymmetry:

* Francisco Correa, Gerald V. Dunne, Mikhail S. Plyushchay, “The Bogoliubov-de Gennes system, the AKNS hierarchy, and nonlinear quantum mechanical supersymmetry”, arXiv:0904.2768, published in Annals of Physics Volume 324, Issue 12, December 2009, Pages 2522-2547

* See a conference summary of our work.