Statistical mechanics approach to the electric polarization and dielectric constant of band insulators

We develop a theory for the analytic computation of the free energy of band insulators in the presence of a uniform and constant electric field. The two key ingredients are a perturbation-like expression of the Wannier-Stark energy spectrum of electrons and a modified statistical mechanics approach involving a local chemical potential in order to deal with the unbounded spectrum and impose the physically relevant electronic filling. At first order in the field, we recover the result of King-Smith, Vanderbilt, and Resta for the electric polarization in terms of a Zak phase—albeit at finite temperature—and, at second order, deduce a general formula for the electric susceptibility, or equivalently for the dielectric constant. Advantages of our method are the validity of the formalism both at zero and finite temperature and the easy computation of higher order derivatives of the free energy. We verify our findings on two different one-dimensional tight-binding models.

Figure 1

Inset: Spectrum of the Rice-Mele chain for $\theta =\frac{3}{4}\pi$, in the presence of an electric field. For readability, the spectrum corresponds to a small chain of 20 unit cells. At zero field, the chain is a two-band insulator, while at finite field, the bands evolve into a WSL with energies $E_n^{\pm }\left(\mathcal{E}\right)=\overline{E^{\pm }}+\mathcal{E}n+\mathcal{E}\overline{{\mathcal{A}}^{\pm \pm }}+\cdots$; see Eq. (31). Main panel: Standard deviation of the numerical energy levels with respect to the analytical levels of an infinite chain [up to second order; see Eq. (31)], i.e., $\sqrt{\frac{1}{N}{\sum }_n{(E_{n,\text{numeric}}^{-}-E_{n,\text{analytic}}^{-})}^2}$, for $N=20$ (highest deviation), 40, 80, and 160 (lowest deviation) unit cells. The initial decrease of the deviation is exponential $e^{-\mathcal{E}/{\mathcal{E}}_c}$ and characterized by the field ${\mathcal{E}}_c^{}\sim \frac{W}{N}$ where $W$ is the bandwidth. High-field decrease of the deviation with respect to the Wannier-Stark ladder of the infinite system is governed by a second characteristic value of the field. Units are such that $e=1, \hslash =1, t=1$ and $a=1$.

Figure 2

Energy center $E_n^{\alpha }-\mathcal{E}{X}_n^{\alpha }$ as a function of the average position $X_n^{\alpha }$ for eigenstates of a finite chain with $N=320$ unit cells in an electric field. $n$ is the unit cell index and $\alpha$ is the index labeling the two bands. At weak electric field, $\mathcal{E}\ll W/N$, the energy center spans the zero-field bandwidth and the average position is the chain center for all eigenstates (the two bands are visible in red in the figure). When increasing the electric field and once the WSL regime is reached (green), bulk eigenstates form a plateau and all have the same energy center. Edge effects are seen on the two ends of the chain and tend to disappear with increasing field (blue and magenta). The curves in different colors are shifted vertically for clarity; the typical (vertical) distance between curves of the same color is of the order of the band gap. Units are such that $e=1, \hslash =1, t=1$ and $a=1$.

Figure 3

Zero-temperature spontaneous polarization $\mathcal{P}$ (in units such that $e=1$) as a function of the Rice-Mele parameter $\theta$: $\theta =0$ and $\pi$ correspond to a CDW chain and $\theta =\pi /2$ and $3\pi /2$ to an SSH chain. The numerically computed first order of the WSL $-{\partial }_{\mathcal{E}}{E}^{-}{|}_{\mathcal{E}=0}$ emerging from the valence band of a finite Rice-Mele chain with 80 unit cells is shown with red crosses. The analytical prediction of Eq. (31) for the infinite system $\mathcal{P}=-X^{-}$ is shown as a blue line, where $X^{-}$ is the Wannier center of the lower band (in units such that $a=1$). For the infinite system (blue line), the polarization is defined modulo the quantum of polarization, which is 1 here, such that $-1/2\le \mathcal{P}<1/2$. For the finite chain, the polarization has an absolute meaning and is not defined modulo a quantum of polarization. When $\pi <\theta <2\pi$, the finite chain has two edge states with opposite energies inside the bulk gap. Units are such that $e=1, \hslash =1, t=1$ and $a=1$.

Figure 4

Numerically computed second order of the energy of the WSL $-2{\partial }_{\mathcal{E}}^2{E}^{-}{|}_{\mathcal{E}=0}$ emerging from the lowest band of the Rice-Mele model as a function of the angular parameter $\theta$ (red crosses) compared with the analytical value of Eq. (31) (blue line). This is essentially the susceptibility $\chi$ at zero temperature. The calculation was done using 80 unit cells; finite-size effects explain the small difference between the numerical and analytical curves around $\theta =0,\pi$. Units are such that $e=1, \hslash =1, t=1$ and $a=1$.