Motivated by quantum quenches in spin chains, a one-dimensional toy-model of fermionic particles evolving in imaginary-time from a domain-wall initial state is solved.
The main interest of this toy-model is that it exhibits the arctic circle phenomenon, namely a spatial phase separation between a critically fluctuating region and a frozen region. Large-scale correlations inside the critical region are expressed in terms of correlators in a euclidean two-dimensional massless Dirac field theory. It is observed that this theory is inhomogenous: technique used to solve the toy-model is then extended to deal with the transfer matrices of other models: In all cases, explicit expressions are given for the long-range correlations in the critical region, as well as for the underlying Dirac action.
Although the setup developed here is heavily based on fermionic observables, the results can be translated into the language of height configurations and of the gaussian free field, via bosonization. Correlations close to the phase boundary and the generic appearance of Airy processes in all these models are also briefly revisited in the appendix.
The phenomenon consists in the appearance, in the thermodynamic limit, of a curve that separates two macroscopic domains, one in which the dimers appear to be frozen, meaning that one knows their exact position with a probability that goes quickly to one in the thermodynamic limit, and another domain where the dimers fluctuate. The same phenomenon was observed in other models [ 4 ], in particular in the six-vertex model with domain-wall boundary conditions [ 5 — 8 ], a system that already possessed a long history connected to integrability and to the evaluation of the norm of Bethe states [ 9 — 11 ].
In fact, the phenomenon itself appears to have been known as early asin a different context though: It was also well studied in the early eighties in the context of the physics of crystal growth, see for instance [ 14 ]. Since the late nineties, there has been an intense activity in this area, which lies at the frontier between physics, mathematics and computer science.
Among the many results that were collected are the limiting distribution of dimers around the origin in the thermodynamic limit [ 1617 ], the fluctuations of the boundary described by the Airy process [ 17 ], a general theory for dimer models [ 18 ] and the calculation of "Guillaume vasseur ijl dating" corresponding arctic curves in connection with algebraic geometry [ 19 ], calculations of various correlation functions [ 20 — 22 ], steps towards extension to interacting models [ 5Guillaume vasseur ijl dating23 ] i.
There are also connections with the physics of glassy systems [ 28 ]. Our motivation for revisiting this ancient problem comes from the physics of quantum Guillaume vasseur ijl dating in spin chains, and in particular their description by two-dimensional conformal field theory CFT. The basic idea is the following: In that setup, the initial state of the quantum system becomes a boundary condition in the classical model. Then the game is to chose the quantum system and the initial state such that i the bulk is critical, which is achieved when the Hamiltonian of the spin chain is gapless and ii the boundary condition flows in the RG sense towards a conformal boundary condition see appendix 8.