Adiabatic charge transport through non-Bloch bands

TL;DR

The paper develops a non-Bloch-momentum framework for an extended NH SSH model with second-nearest-neighbor hopping to restore bulk-boundary correspondence under open boundaries. By constructing the aGBZ/GBZ and employing open-bulk invariants like the real-space Bott index and its spatio-temporal extension, it unifies static and driven NH topology, linking gap structure of non-Bloch bands to adiabatic charge pumping through non-Bloch Chern numbers. Analytic roots of the characteristic equation provide insight into winding and phase structure deep inside phases, while numerical GBZ constructions capture boundary-sensitive features and extended critical regions. The work suggests that quantized pumping in NH systems is governed by the gapped/non-gapped nature of non-Bloch bands and that BBC can be consistently described via bulk non-Bloch invariants in both static and driven settings, with potential experimental realizations in photonic or topolectrical platforms.

Abstract

We explore the non-reciprocal intracell hopping mediated non-Hermitian topological phases of an extended Su-Schrieffer-Heeger model hosting second-nearest-neighbour hopping. We microscopically analyze the phase boundaries using the non-Bloch momentum while the off-critical (critical) phases are directly associated with the gapped (gapless) nature of the non-Bloch bands that we derive from the characteristic equation using the gauge freedom. The non-Bloch momentum accurately reflects the bulk boundary correspondence (BBC) explaining the winding number profile under open boundary conditions. We examine the adiabatic dynamics to promote the concept of adiabatic charge transport in a non-Hermitian scenario justifying the BBC in spatio-temporal Bott index and non-Bloch Chern number. Once the non-Bloch bands experience no (a) gap-closing during the evolution of time, quantized flow of is preserved (broken). Our study systematically unifies the concept of non-Bloch bands for both static and driven situations.

Figures (13)

• Figure 1: Schematic diagram of the SSHLR model with two sub-lattices $A$ and $B$ per unit cell. The inter(intra)-cell hoppings $b,c$, and $d$ are designated by dashed and dotted (solid) lines where $\gamma$ introduces non-reciprocity in the intra-cell hopping.
• Figure 2: We show the phase diagram of open-bulk winding number ${\rm Re}[W]$ (color bar), using Eq. (\ref{['eq:rwind']}), in (a) as a function of $c$ and $d$. The solid (dashed) lines represent the Hermitian (NH) phase boundaries for $k=0$, $\pi$ and $2\pi/3$. We consider $a=b=1$, $\gamma=0.5$, $L=70, l=15$. (b) shows the GBZs corresponding to each parameter point as marked by red symbols $\star$, $\square$, $\triangle$, $\circ$, $\diamond$, $\triangledown$, $+$, and $\times$ in (a). We depict ${\rm Im}[W]$ in (c) where the extended critical regions acquire finite ${\rm Im}[W]$. The insets in (c) show the parametric plot in $d^x_R(\beta)$-$d^y_R(\beta)$ plane for different regions of the phase diagram as marked by red symbols in (a). The continuous non-Bloch energy spectra ${\tilde{E}}$ with polar angle $\phi=\tan^{-1}({\rm Im[\beta]/{\rm Re[\beta]}})$ are shown in (d) for the above marked points, see SM supp for more details.
• Figure 3: We show the evolution of energy levels (left axis ${\rm Re}[E]$), computed from real space NH SSHLR model under OBC, and winding number, obtained from bulk non-Bloch model $H(\beta)$ (right axis $W_{\beta}$), with $d$ and $c$ in (a,b) and (c,d), respectively. We consider $c=-1$, $c=1$, $d=-1$ and $d=1$ in (a,b,c,d), respectively. For $W_{\beta}$, we integrate over 200 values of $\phi \in[0, 2 \pi]$ values. We color-code (a,b,d) with the average position of the individual energy level and (c) with the ${\rm Im}[E]$.
• Figure 4: We show the topological phase diagram, obtained using open-bulk Bott index ${\rm Re}[B]$ given by Eq.(\ref{['eq:bott_obc']}) (color bar), over $d_0$-$t_0$ plane in (a). We consider $a = b = h_0 = T = 1$, and $L_x = L_y = 50$. We depict ${\rm Im}[B]$ to highlight the extended critical phase in (b). We show the time evolution of the effective dispersion $\tilde{E}$-$\phi$ associated with non-Bloch bands in (c1,c2) for $(t_0,d_0)=(0.5,1)$ and $(1.5,1.5)$, respectively. We demonstrate the variation of non-Bloch Chern number $C_\beta$, associated with the ground state, in $(d_1,d_2,d_3,d_4)$, corresponding to the lines $d_0=-1.5$ (black), $d_0=0.5$ (green), $d_0=-t_0$ (purple), $t_0=0$ (yellow) in (a), respectively.
• Figure S1: The main figure [inset] represents the imaginary part of the open-bulk winding number ${\rm Im}[W]$ in $c$-$d$ plane [${\rm Re}(E)$ vs ${\rm Im}(E)$ corresponding to their symbol taken from Fig. 2(a) of the main text.