Can STM detect nematic ordering in underdoped Bi$_2$Sr$_2$CaCu$_2$O$_{\mathrm{8+x}}$ or other correlated systems?

Electronic nematic phases, where, for example, the electronic states undergo a spontaneous four-fold (C$_{4})$ to two-fold (C$_{2})$ symmetry breaking, have recently gained vast interest as a possible candidate for various hidden order states in several correlated electron systems such as cuprates, pnictides, and heavy fermions. Such states are difficult to detect using non-local probes because of possible twin domain structures in macroscopic samples. STM spectroscopy has been proposed as a possible approach to detect such nematic orders, with several recent experiments reporting signals in the cuprates and iron-based superconductors. We specifically investigate the situation in which STM topographic data shows C$_{4}$ symmetry while energy-resolved spectroscopic maps signal C$_{2}$ symmetry. We find that such behavior can in fact occur for asymmetric tip geometries and discuss both model calculations and experimental results that provide evidence for this false nematic signature. We discuss possible future STM experiments that could unambiguously detect electronic nematic order.