Complex Wannier centers and drifting Wannier functions in non-Hermitian Hamiltonians

Abstract

The extension of topological band theory to non-Hermitian Hamiltonians with line energy gaps remains largely unexplored, despite early indications of rich underlying physics. In this setting, Wilson loops-the quantities underlying polarization-generally become nonunitary, yet the physical consequences of this nonunitarity have remained unclear. Within the framework of biorthonormal quantum mechanics, we introduce the concept of complex Wannier centers, defined from the gauge-invariant eigenvalues of nonunitary Wilson loops. Complex Wannier centers acquire physical meaning through reciprocity breaking in their associated Wannier functions: when the centroid of a Wannier function shifts into the complex plane, it acquires an effective momentum offset that produces directional drift over time. We analyze how symmetries constrain complex Wannier centers and identify symmetry-protected Wannier configurations in pseudo-Hermitian Hamiltonians, where the centers are either real or form complex-conjugate pairs, as determined by conserved "Krein signatures" of the projected metric operator of pseudo-Hermiticity. We further show that the Krein structure of the Wilson loop can establish a bulk-boundary correspondence: in a system with anticommuting pseudo-Hermitian metric and (pseudo) inversion symmetries, the behavior of complex Wannier centers predicts the existence of a filling anomaly in the occupied bands and whether the resulting edge modes experience gain or loss. Finally, we propose a photonic waveguide implementation of this system that enables experimental tests of our predictions.

Figures (9)

• Figure 1: Flowchart summarizing key properties of Wannier functions (understood here as eigenstates of the projected position operator) in one dimension. In Hermitian Hamiltonians, the momentum-space distribution $\|w_k\|^2$ is uniform and consistent with the Bloch--Fourier construction, leading to reciprocal wave-packet dynamics. In non-Hermitian Hamiltonians, $\|w_k\|^2$ is generically non-uniform and Wannier centers can become complex, $z=\nu+\mathrm{i}\kappa$. The non-uniform momentum distribution may be symmetric or asymmetric about $k=0$, leading respectively to reciprocal or drifting (nonreciprocal) dynamics characterized by a nonzero average momentum $k_{\mathrm{eff}}$. In many cases of interest, and for weak non-Hermiticity, $k_{\mathrm{eff}}$ scales linearly with $\kappa$.
• Figure 2: Complex Wannier centers and nonreciprocal Wannier function dynamics of the lowest band of Hamiltonian \ref{['ToyModel1']}. (a) Energy spectrum \ref{['DispersionToyModel']} highlighting the lower band $E_{k,-}$, from which panels (b-f) are extracted. (b) imaginary part $\kappa(m)$ of the Wannier center as a function of the mass parameter $m$ for fixed non-Hermiticity $\varepsilon=0.2$. (c) Absolute value of the components of a Wannier function in the topological phase $(m=0.5)$. Vertical gray lines delineate unit cells. (d) Center of mass of time-evolved Wannier functions for several $\varepsilon$ at fixed $m=0.5$; the finite $\kappa$ produces nonreciprocal drift. (e) Wannier function in the trivial phase $(m=10)$. (f) Corresponding center-of-mass evolution for several $\varepsilon$, showing the absence of drift in the trivial phase as $m\to \infty$, where $\kappa \to 0$. Initial Wannier functions are taken at different positions to avoid overlap. System size $\ell=400$ and PBC are used in dynamical simulations.
• Figure 3: Dynamics of Wannier functions of the model \ref{['ToyModel1']} in the Hermitian case $(\varepsilon=0)$. Position-space profiles (real parts) of the Wannier wave packets at $t=0$ for the lower (a) and upper (b) bands. There are two sites per unit cell, whose boundaries are indicated by the vertical gray lines. Momentum-weight distribution $\|w_{k}\|^2$ of the Wannier functions in the lower (c) and upper (d) band. Evolution of the Wannier wave packets for the lower (e) and upper (f) bands. The gray color scale indicates the amplitude $|\psi(x,t)|$ of the wave packet at site $x$, and the blue (e) and red (f) curves track the wave packets' center of mass. We employed $\ell=100$ unit cells and periodic boundary conditions (PBC) in the simulations.
• Figure 4: Dynamics of Wannier functions of the model \ref{['ToyModel1']} in the non-Hermitian case $(\varepsilon=0.532)$. Position-space profiles (real parts) of the Wannier wave packets at $t=0$ for the lower (a) and upper (b) bands. There are two sites per unit cell, whose boundaries are indicated by the vertical gray lines. Momentum-weight distribution $\|w_{k}\|^2$ of the Wannier functions in the lower (c) and upper (d) band. Evolution of the Wannier wave packets for the lower (e) and upper (f) bands, showing asymmetric spreading and a net drift of the center of mass. The gray color scale indicates the amplitude $|\psi(x,t)|$ of the wave packet at site $x$, and the blue (e) and red (f) curves track the wave packets' center of mass. We employed $\ell=100$ unit cells and periodic boundary conditions (PBC) in the simulations.
• Figure 5: (a) Momentum distribution $\|w_{k}\|^2$ of Wannier functions from direct diagonalization of $PXP$ (circles) and from equation \ref{['Eq:WeightDistributionGamma']} (red line) for system size $\ell=800$. (b) Maximum absolute deviation $\max_k |\|w_k^{PXP}\|^2 - \|w_k^{\Gamma}\|^2|$ as a function of system size $\ell$.