Zero energy modes with Gaussian, exponential, or polynomial decay: Exact solutions in hermitian and nonhermitian regimes

Abstract

Topological zero modes in topological insulators or superconductors are exponentially localized at the phase transition between a topologically trivial and nontrivial phase. These modes are solutions of a Jackiw-Rebbi equation modified with an additional term which is quadratic in the momentum. Moreover, localized fermionic modes can also be induced by harmonic potentials in superfluids and superconductors or in atomic nuclei. Here, by using inverse methods, we consider in the same framework exponentially-localized zero modes, as well as Gaussian modes induced by harmonic potentials (with superexponential decay) and polynomially decaying modes (with subexponential decay), and derive the explicit and analytical form of the modified Jackiw-Rebbi equation (and of the Schrödinger equation) which admits these modes as solutions. We find that the asymptotic behavior of the mass term is crucial in determining the decay properties of the modes. Furthermore, these considerations naturally extend to the nonhermitian regime. These findings allow us to classify and understand topological and nontopological boundary modes in topological insulators and superconductors.

Figures (5)

• Figure 1: Modes with real wavefunction and Gaussian, exponential, or polynomial decay on the infinite interval $-\infty< x<\infty$ and associated fields for some choice of the parameters. (i) wavefunction $\varphi_1^s(x)=\varphi(x)$ as in \ref{['eq:gauss', 'eq:sech', 'eq:witch']}, (ii) associated field $V(x)$ satisfying \ref{['eq:SchrodingerGeneral']}, (iii) the wavefunction $\varphi_1^{-s}(x)$ of the associated mode in \ref{['eq:twin']} with opposite pseudospin (if bounded) and (iv) the associated field $m(x)$ for $s=\pm1$ satisfying \ref{['eq:diffeq']} for a constant field $v(x)=v=\mathop{\mathrm{const}}\nolimits$ (where $v=1$), (v) the wavefunction $\varphi_2^{s}(x)$ of the associated mode in \ref{['eq:2ndmode']} with same pseudospin and (vi) associated fields $v(x)$ for $s=\pm1$ satisfying \ref{['eq:diffeq']} for a constant field $m(x)=m=\mathop{\mathrm{const}}\nolimits$ where $m =m^*$, such that the field $v(x)$ is continuous and bounded ($m^*=-\beta$, $(\alpha^2-\beta^2\gamma^2)/\gamma$, and $-2\beta^2$ respectively for Gaussian, exponentially decaying, and polynomially decaying modes). For Gaussian modes, the fields $V(x)$ and $m(x)$ (for $v(x)=v=\mathop{\mathrm{const}}\nolimits$) diverge quadratically, with the field $m(x)$ changing sign twice on the real axis, while the field $v(x)$ (for $m(x)=m=\mathop{\mathrm{const}}\nolimits$) diverges linearly, changing sign only once. In this case, both the associated mode with opposite pseudospin for $v(x)=v=\mathop{\mathrm{const}}\nolimits$ and the associated mode with same pseudospin for $m(x)=m=\mathop{\mathrm{const}}\nolimits$ with $m=m^*$ are bounded and thus normalizable. For exponentially decaying modes, the fields $V(x)$, $m(x)$, and $v(x)$ converge to finite values for $x\to\pm\infty$. In this case, the associated mode with the opposite pseudospin is bounded only for $\mathrm{Re}(\beta\gamma+\alpha)>2s \mathrm{Re}(v)>-\mathrm{Re}(\beta\gamma-\alpha)$ while the associated mode with the same pseudospin is bounded only if $2s\mathrm{Re}(v_{L})<-\mathrm{Re}(\beta\gamma+\alpha)$ and $2s\mathrm{Re}(v_{R})>\mathrm{Re}(\beta\gamma-\alpha)$. For polynomially decaying modes, the fields $V(x)$ and $m(x)$ (for $v(x)=v=\mathop{\mathrm{const}}\nolimits$) converge to zero, with the field $m(x)$ changing sign twice on the real axis, while the field $v(x)$ (for $m(x)=m=\mathop{\mathrm{const}}\nolimits$) diverges linearly, changing sign only once. In this case, the associated mode with opposite pseudospin is bounded only in the limiting case where $\mathrm{Re}(v)=0$, while the associated mode with the same pseudospin is bounded as long as $s\mathrm{Re}(v_R)>0>s\mathrm{Re}(v_L)$.
• Figure 2: Same as in \ref{['fig:real']} but for modes with nonuniform complex phases and nonhermitian fields.
• Figure 3: Couples of modes with real wavefunction and Gaussian, exponential, or polynomial decay and associated fields for some choice of the parameters. (i) wavefunction $\varphi_1^{s_1}(x)=\varphi(x+a)$ of the first mode with pseudospin $s_1$, (ii) wavefunction $\varphi_2^{s_2}(x)=\varphi(x-a)$ of the second mode with pseudospin $s_1$ with $\varphi(x)$ as in \ref{['eq:gauss', 'eq:sech', 'eq:witch']}, (iii) and (iv) associated field $m(x)$ and the corresponding associated field $v(x)$ for $s_1s_2=\pm1$ satisfying \ref{['eq:diffeq']}. In general, for couples of modes with real wavefunction, the associated fields are not continuous (and thus not physical).
• Figure 4: Same as in \ref{['fig:doublereal']} but for modes with nonuniform complex phases and nonhermitian fields.
• Figure 5: Periodic modes with real wavefunction and Gaussian, exponential, or polynomial decay on the infinite interval $-\infty< x<\infty$ and associated fields: (i) wavefunction $\varphi(x)$ as in \ref{['eq:periodicgauss', 'eq:periodicsech', 'eq:periodicwitch']}, (ii) associated field $V(x)$ satisfying \ref{['eq:SchrodingerGeneral']}, (iii) associated field $m(x)$ for $s=\pm1$ satisfying \ref{['eq:diffeq']} for constant field $v(x)=v=\mathop{\mathrm{const}}\nolimits$ (where $v=1$).