Nonreciprocal non-Hermitian Su-Schrieffer-Heeger chain modulated by the influence of $\mathcal{PT}$-symmetric imaginary potentials

We concentrate on the skin effects and topological edge states in the nonreciprocal non-Hermitian Su-Schrieffer-Heeger (SSH) chain, by taking into account the presence of $\mathcal{PT}$-symmetric imaginary potentials. Following a detailed demonstration of the theoretical method, we find that, regardless of the application manner of $\mathcal{PT}$-symmetric imaginary potentials, they take weak impacts from the skin effect of bulk states, but efficiently modulate the topological edge states. On the other hand, if such potentials are applied according to two domains, the skin effect and new edge states can be induced along the domain wall, which are also dependent on the structural parameters. Therefore, the findings in this paper promote understanding of the independent modulations of the skin effects and topological edge states via the interplay between the imaginary potentials and nonreciprocal couplings in the SSH lattices.

Figure 1

Three models of our considered nonreciprocal non-Hermitian SSH chain. In case I, $\mathcal{PT}$-symmetric imaginary potentials are introduced according to the cells. In case II, the $\mathcal{PT}$-symmetric imaginary potentials are introduced to the end sites, whereas in case III such imaginary potentials are applied according to the left and right domains of the SSH chain.

Figure 2

The direction of the skin effect in the nonreciprocal SSH chain is highly related to the ratio of $t_l$ and $t_r$. Relevant parameters are taken as $\epsilon =0, t_2=1.0$. (a) $t_l=1.5$ and $t_r=0.5$ for $t_l>t_r$. (b) $t_l=0.5$ and $t_r=1.5$ for $t_l<t_r$.

Figure 3

(a)–(c) Spectra of the real and imaginary parts of the energy-band structures. Parameters are taken as $t_2=1.0, \gamma =2/3$, and (a) $\epsilon =0$, (b) $\epsilon =0.15$, and (c) $\epsilon =0.85$. (d) Real and imaginary parts of energy as functions of $\epsilon$. Relevant parameters are set to be $t_1=0.2, t_2=1.0$, and $\gamma =2/3$.

Figure 4

(a), (b) The curves of ${\beta }_j$ as a function of $t_1$ in the cases of $\epsilon =0.15$ and 0.85, respectively. (c) The distribution of edge state number with the change of $\epsilon$ and $t_1$. Two pink dots are ($t_1=0.5, \epsilon =0.9$) and ($t_1=1.0, \epsilon =0.15$). (d) Phase diagram according to panel (c) with the change of $\epsilon$ and $t_1$.

Figure 5

The index of eigenstates and density-of-state distribution of the bulk states and topological edge states (a) in the central zero-gap phase ($t_1=0.5, \epsilon =0.90$) and (b) in the topological phase ($t_1=1.0, \epsilon =0.15$). In panels (a) and (b), the colorscale corresponds to the value of the imaginary part of energy. Eigenenergies of edge states for panel (a) are taken as (i) $E=0.9000i$, (ii) $E=-0.9000i$, (iii) $E=0.1133i$, and (iv) $E=-0.1133i$; for panel (b) they are taken as (i) $E=0.1500i$ and (ii) $E=-0.1500i$. The red lines describe the density-of-state distribution of the edge states, and cyan lines denote the skin effect of the bulk states.

Figure 6

(a) Phase diagram with the change of $\epsilon$ and $t_1$. The spectra of energy bands are (b) $\epsilon =0.4$ and (c) $\epsilon =3.6$.

Figure 7

Real and imaginary parts of energy with (a) $t_1=0.2$ and (b) $t_1=1.5$. We can clearly find that two bulk states become edge states induced by imaginary potential.

Figure 8

Index of eigenstates and DOS selected from (a) phase region II ($t_1=1.9, \epsilon =3.6$) and (b) phase region I ($t_1=0.5, \epsilon =3.6$). The eigenvalues of edge states are [(a)(i)–(a)(iv)] $E=2.4317i, -2.4317i, 0.8906i$, and $-0.8906i$ and [(b)(i), (b)(ii)] $E=3.6495i$ and $-3.6495i$. Because there is a negative imaginary potential on the right edge which creates a LP for negative energy, negative-energy edge states localize at the right edge.

Figure 9

(a) Phase diagram with the change of $\epsilon$ and $t_1$. Two red dots are ($t_1=1.0, \epsilon =0.2$) and ($t_1=1.0, \epsilon =1.8$). The energy-band spectra are (b) $\epsilon =0.2$ and (c) $\epsilon =1.8$.

Figure 10

Index of eigenstates and DOS selected from (a) phase region I ($t_1=1.0, \epsilon =1.8$) and (b) phase region II ($t_1=1.0, \epsilon =0.2$). The eigenvalues of edge states are [(a)(i)–(a)(iv)] $E=1.8000i, -1.8000i, 1.6720i$, and $-1.6720i$ and [(b)(i), (b)(ii)] $E=0.2000i$ and $-0.2000i$.

Figure 11

(a) The DOS of edge states of $\epsilon =0.9$ at $t_1=0.2$ in $U_1^{\prime }$ where (i) $E=0.9000i$, (ii) $E=-0.900i$, (iii) $E=0.4836i$, and (iv) $E=-0.4836i$. (b) The DOS of edge states of $\epsilon =1.8$ at $t_1=1.0$ in $U_2^{\prime }$ where (i) $E=1.7999i$, (ii) $E=-1.7999i$, (iii) $E=1.6719i$, and (iv) $E=-1.6719i$.