Fig. 2: Self-normal and biorthogonal DQPTs during a quench between distinct topological phases in the PT-symmetry-unbroken region.

Loschmidt rate of LR(t) as a function of time in non-unitary QWs with self-normal bases (a) and biorthogonal bases (e), respectively. DTOPs of ν_m(t) as a function of time in non-unitary QWs for b with self-normal bases and f biorthogonal bases, respectively. The green and dark red dot-dashed lines represent the critical times for self-normal DQPTs with \({t}_{c}^{S}\approx 0.869,1.180,2.607\), and 3.54 in (a, b), corresponding to the Fisher zeros (n = 0, 1) crossing the axis in (c). The magenta and goldenrod dot-dashed lines represent the critical times for biorthogonal DQPTs, with \({t}_{c}^{B}\approx 1.025\) and 3.075 in (e, f), corresponding to the Fisher zeros (n = 0, 1) crossing the axis in (g). c Real and imaginary parts of lines of self-normal Fisher zeros \({z}_{n,k}^{S}\) with n = 0 (green), n = 1 (dark red), n = 2 (cyan), n = 3 (purple). g Real and imaginary parts of lines of biorthogonal Fisher zeros \({z}_{n,k}^{B}\) with n = 0 (magenta), n = 1 (golden rod), n = 2 (dark green), n = 3 (deep skyblue). d \({A}_{k}^{S}={a}_{k,+}^{S}-{a}_{k,-}^{S}\) as a function of k, where self-normal fixed points are at \({k}_{m}^{S}=\{0.189,\pi ,3.331,2\pi \}\) with \({a}_{k+}^{S}=0\) (cyan) and \({k}_{m}^{S}=\{\pi /2,1.761,3\pi /2,4.491\}\) with \({a}_{k,-}^{S}=0\) (purple). h \({A}_{k}^{B}={a}_{k,+}^{B}-{a}_{k,-}^{B}\) as a function of k, where biorthogonal fixed points are at \({k}_{m}^{S}=\{0,\pi ,2\pi \}\) with \({a}_{k,+}^{B}=0\) (deepskyblue) and \({k}_{m}^{S}=\{1.371,1.766,4.513,4.908\}\) with \({a}_{k,-}^{B}=0\) (darkgreen). In all panels, the loss parameter is l = 0.5. The initial state of the QWs is \(\left\vert x=0\right\rangle \otimes \left\vert {\psi }_{-}^{i}\right\rangle\) of \({\hat{U}}^{i}\) with the coin parameters \(({\theta }_{1}^{i}=\pi /4,{\theta }_{2}^{i}=-\pi /2)\) and the QWs are governed by the final Floquet operator \({\hat{U}}^{f}\) with \(({\theta }_{1}^{f}=-\pi /4,{\theta }_{2}^{f}=0)\). All quantities are unitless