Dynamical response theory of interacting Majorana fermions and its application to generic Kitaev quantum spin liquids in a field

Motivated by the appearance of Majorana fermions in a broad range of correlated and topological electronic systems, we develop a general method to compute the dynamical response of interacting Majorana fermions in the random-phase approximation (RPA). This can be applied self-consistently on top of Majorana mean-field theory backgrounds, thereby in particular providing a powerful tool to analyze behavior in the vicinity of (various heavily studied) exactly soluble models. Prime examples are quantum spin liquids (QSL) with emergent Majorana excitations, with the celebrated exact solution of Kitaev. We employ the RPA to study in considerable detail phase structure and dynamics of the extended Kitaev honeycomb <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>K</a:mi><a:mi>J</a:mi><a:mi mathvariant="normal">Γ</a:mi></a:mrow></a:math> model, with and without an applied field. First, we benchmark our method with Kitaev's exactly soluble model, finding a remarkable agreement. The interactions between Majorana fermions even turn out to mimic the effect of local <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi mathvariant="double-struck">Z</c:mi><c:mn>2</c:mn></c:msub></c:math> flux excitations, which we explain analytically. Second, we show how small non-Kitaev couplings <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>J</e:mi></e:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi mathvariant="normal">Γ</f:mi></f:math> induce Majorana bound states, resulting in sharp features in the dynamical structure factor in the presence of fractionalization: such “spinon excitons” naturally appear, and can coexist and interact with the broad Majorana continuum. Third, for increasing couplings or field, our theory predicts instabilities of the Kitaev QSL (KQSL) triggered by the condensation of the sharp modes. From the high-symmetry momenta of the condensation we can deduce which magnetically ordered phases surround the KQSL, in good agreement with previous finite-size numerics. We discuss implications for experiments and the broad range of applicability of our method to other QSL and Majorana systems.