Exciton Berryology

Abstract

In translationally invariant semiconductors that host exciton bound states, one can define an infinite number of possible exciton Berry connections. These correspond to the different ways in which a many-body exciton state, at fixed total momentum, can be decomposed into free electron and hole Bloch states that are entangled by an exciton envelope wave function. Inspired by the modern theory of polarization, we define an exciton projected position operator whose eigenvalues single out two unique choices of exciton Berry phase and associated Berry connection - one for electrons, and one for holes. We clarify the physical meaning of these exciton Berry phases and provide a discrete Wilson loop formulation that allows for their numerical calculation without a smooth gauge. As a corollary, we obtain a gauge-invariant expression for the exciton polarisation at a given total momentum, i.e. the mean separation of the electron and hole within the exciton wave function. In the presence of crystalline inversion symmetry, the electron and hole exciton Berry phases are quantized to the same value and we derive how this value can be expressed in terms of inversion eigenvalues of the many-body exciton state. We then consider $C_2 \mathcal{T}$ symmetry, for which no symmetry eigenvalues are available as it is anti-unitary, and confirm that the exciton Berry phase remains quantized and still diagnoses topologically distinct exciton bands. The notion of shift excitons, whose exciton Wannier states are displaced from those of the non-interacting bands by a quantized amount, can therefore be generalised beyond symmetry indicators.

Figures (3)

• Figure 1: Exciton Wannier states. $(a)$ An illustrative semiconductor electronic band structure. Energy $E$ is plotted against momentum $k$. At half filling the valence band ($v$) is occupied and the conduction band ($c$) is empty. An electron $e^-$ can be excited from the conduction band leaving a hole in the valence band $h^+$. $(b)$ The $e^-$ and $h^+$ can become bound due to the Coulomb interaction. The electron-hole excitation energy ($E$) plotted against total momentum ($p$) shows an exciton band (red) gapped from the continuum of unbound electron-hole states (purple). $(c)$ Exciton Wannier functions can maximally localise the hole (red) or the electron (purple). These are not always equal and can have different centres of mass. The different exciton Wannier centres are associated with the different exciton Berry phases we introduce.
• Figure 2: Excitons in $C_2\mathcal{T}$ symmetric systems. $(a)$ Exciton dispersion relation for a simple $C_2\mathcal{T}$ symmetric system (see Hamiltonian in Eq. \ref{['eq:C2THamiltonian']}). The shift exciton band is highlighted in red. The hopping parameters are $v = 0.6 + 0.4\,\mathrm{i}$, $w = 0.03$, and $t = 0.01$. The interaction parameters are $U = 0.1, U' = 0.15$. $(c)$ The components of the corresponding exciton Wannier function in the basis of the electronic Wannier functions i.e.$\ket{\mathcal{W}^{R'=0}_{\mathrm{exc}, c}} =\ket{\mathcal{W}^{R'=0}_{\mathrm{exc}, v}} = \sum_{R, r} \mathcal{W}^{R' = 0}_{c/v}({R, r})\: c^\dagger_{R+r, c} c_{R, v}\ket{\mathrm{GS}}$. The red line marks the exciton Wannier centre ($s_{\mathrm{exc}, c/v} = 1/2$), corresponding to an exciton Berry phase of $\gamma_{\mathrm{exc}, c/v} = \pi$.
• Figure 3: Excitons in a system without $\mathcal{I}$ and $C_2 \mathcal{T}$ symmetry. $(a)$ is the tight-binding model which consists of two stacked SSH models, both with hoppings $v = 1.0, w=0.2$. In addition the SSH model constructed from the sites $C, D$ has a uniform on-site potential $V=4.5$. At the single-particle level, the two SSH models are uncoupled. The Hubbard interaction $U(R)$ between the unit cells separated by distance $R$ couples them. We consider excitons at half filling. See Appendix \ref{['sec:ApxNoSymmetries']} for the full model specification. $(b)$ The exciton band structure. The red band highlights the exciton band studied in the remaining panels. $(c)$ The electronic polarisation of the exciton [$\mathcal{F}(p)$]. $(d), (e)$ are the spatial profiles of the maximally localised exciton Wannier functions in the basis of the electronic Wannier functions i.e.$\ket{\mathcal{W}^{R'=0}_{\mathrm{exc}, c/v}} = \sum_{R, r} \mathcal{W}^{R' = 0}_{c/v}({R, r})\: c^\dagger_{R+r, c} c_{R, v}\ket{\mathrm{GS}}$. $(d)$ is the hole maximally localised Wannier function $\ket{\mathcal{W}_{\mathrm{exc}, v}^{R'=0}}$, $(e)$ is the electron maximally localised Wannier function $\ket{\mathcal{W}_{\mathrm{exc}, c}^{R'=0}}$ . The Wannier centres extracted from the Wilson loops $W_{\mathrm{exc}, c/v}$ are indicated by a red line.