Theory of electron spin resonance in one-dimensional topological insulators with spin-orbit couplings: Detection of edge states

Edge/surface states often appear in a topologically nontrivial phase when the system has a boundary. The edge state of a one-dimensional topological insulator is one of the simplest examples. Electron spin resonance (ESR) is an ideal probe to detect and analyze the edge state for its high sensitivity and precision. We consider ESR of the edge state of a generalized Su-Schrieffer-Heeger model with a next-nearest-neighbor (NNN) hopping and a staggered spin-orbit coupling. The spin-orbit coupling is generally expected to bring about nontrivial changes on the ESR spectrum. Nevertheless, in the absence of the NNN hoppings, we find that the ESR spectrum is unaffected by the spin-orbit coupling thanks to the chiral symmetry. In the presence of both the NNN hopping and the spin-orbit coupling, on the other hand, the edge ESR spectrum exhibits a nontrivial frequency shift. We derive an explicit analytical formula for the ESR shift in the second-order perturbation theory, which agrees very well with a nonperturbative numerical calculation.

Figure 1

Band structure of the SSH model in Eq. (2) with a periodic boundary condition [1]. The solid and dashed lines represent band structures of the insulating case with ${\delta }_0\ne 0$ and the gapless point at ${\delta }_0=0$, respectively. The low-energy physics around $k=\pi /2$ can be described by the one-dimensional Dirac fermion model.

Figure 2

Amplitude $\left|\psi \right|$ of the edge-state wave function of the SSH model in Eq. (2) under an open boundary condition [8]. Blue and red represent the spatial distribution of the existing probability for left and right localized edge states, respectively. The total site number is set to be even, the dimerization parameter ${\delta }_0>0$, and symbols $v$ and $w$ denote hopping amplitudes $t(1-{\delta }_0)$ and $t(1+{\delta }_0)$, respectively. The wave-function amplitude decays exponentially into the bulk [4].

Figure 3

${\delta }_0$ dependence of the energy spectrum of the SSH model with $t=1$ and $N=40$ under an open boundary condition. As ${\delta }_0\to 0^{+}$, two localized edge states at the ends merge into the bulk. For the limit $N\to +\infty$, the edge states are strictly at zero energy as $0<\delta \le t$, which are protected by the chiral symmetry [8].

Figure 4

Angle $\theta$ dependence of the ESR frequency shift $\mathrm{\Delta }\omega$. We set the parameters $t=1.0$, $\mathrm{\Delta }t={\delta }_0=0.2$, and $H=0.05$. Red and blue curves are obtained by the second-order perturbation calculation in Sec. 4b. Square and circle points are the results of direct numerical diagonalization for a system (of 100 sites) with an open boundary condition. The definition of angle $\theta$ is indicated in Fig. 5. The zeros of $\mathrm{\Delta }\omega$ occur at $\varphi \vec{n}\times {\vec{n}}_H=\gamma {\vec{n}}_{\gamma }\times {\vec{n}}_H$.

Figure 5

Geometric relation among some vectors in Fig. 4, where the direction of the magnetic field is in the plane spanned by $\vec{n}=\widehat{y}$ and ${\vec{n}}_{\gamma }=-\widehat{z}$. The parameter $\theta$ is defined as the angle between ${\vec{n}}_H$ and $\widehat{y}$ in the $\widehat{y}\text{−}\widehat{z}$ plane.