We analyze the sharpness of crossing (“isosbestic”) points of a family of curves which are observed in many quantities described by a function $f(x,p)$, where $x$ is a variable (e.g., the frequency) and $p$ is a parameter (e.g., the temperature). We show that if a narrow crossing region is observed near $x_{*}$ for a range of parameters $p$, then $f(x,p)$ can be approximated by a perturbative expression in $p$ for a wide range of $x$. This allows us, e.g., to extract the temperature dependence of several experimentally obtained quantities, such as the Raman response of HgBa${}_2$CuO${}_{4+\delta }$, photoemission spectra of thin VO${}_2$ films, and the reflectivity of CaCu${}_3$Ti${}_4$O${}_{12}$, all of which exhibit narrow crossing regions near certain frequencies. We also explain the sharpness of isosbestic points in the optical conductivity of the Falicov-Kimball model and the spectral function of the Hubbard model.

• Received 21 December 2012

DOI:https://doi.org/10.1103/PhysRevB.87.195140