at half maximum equal to $0.5t$, simulating the experimental resolution of ARPES on Ca$_2$RuO$_4$ [@Sutter2017]. The yellow rectangles mark the high binding energy parts of the spectra that are incoherent and almost momentum-independent, are identified in ARPES as having a dominant $xz$/$yz$ orbital character [@Sutter2017], and are theoretically modelled by panel (b) \[The dispersive branch visible in (a), both inside and outside of the yellow rectangle and not discussed here, is associated with the $xy$ orbital [@Sutter2017]. See main text of the paper\]. Theoretical spectra (b-c) are normalised in the same manner as the ARPES spectrum \[(a)\] of Ref. . []{data-label="result"}](fig1.png){width="50.00000%"}

${\bf t}$-${\bf J}$ model. {#sec:tjmodel}
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In order to model (the high-binding energy part of) the photoemission spectrum of Ca$_2$RuO$_4$ we follow the scheme that was already developed ca. 30 years ago and, as mentioned in the introduction, was successfully used to describe, [*inter alia*]{}, the photoemission spectra of several undoped copper oxides [@Wells1995; @LaRosa1997; @Kim1998; @Damascelli2003; @Shen2007]. Thus, we consider an appropriate $t$–$J$-like Hamiltonian constructed as a sum of two parts ${\mathcal H} = {\mathcal H}_J+ {\mathcal H}_t$.

The first part, ${\mathcal H}_J$, describes the low energy physics of the Mott insulating Ca$_2$RuO$_4$ in terms of the interaction 