Antiferromagnetism and spin excitations in a two-dimensional non-Hermitian Hatano-Nelson flux model

The one-dimensional Hatano-Nelson model with nonreciprocal hoppings is a prominent example of a relatively simple non-Hermitian quantum-mechanical system, which allows to study various phenomena in open quantum systems without adding extra gain and loss terms. Here we propose to use it as a building block to construct a correlated non-Hermitian Hamiltonian in two dimensions. It has the characteristic form of a flux model with clock-anticlockwise nonreciprocal hopping on each plaquette. Adding the on-site Hubbard type interaction, we analyze the formation of the longe-range antiferromagnetic order and its spin excitations. Such a model is non-Hermitian, but <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi mathvariant="script">PT</a:mi></a:math>-symmetric, which leads to the existence of two regions: a region of unbroken <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi mathvariant="script">PT</c:mi></c:math> symmetry (real-valued spectrum) and a region of broken <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi mathvariant="script">PT</e:mi></e:math> symmetry with exceptional lines and complex-valued energy spectrum. The transition from one region to another is controlled by the value of the on-site interaction parameter and coincides with the metal-insulator transition. We also analyze the spin wave spectrum, which is characterized by two diffusive <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>d</g:mi></g:math>-wave type of modes corresponding to gain and loss.