Quantum geometrical effects in non-Hermitian systems

TL;DR

This work develops a bi-orthogonal framework for quantum geometry in non-Hermitian systems, introducing complex-valued Berry connections and quantum metrics and applying them to three concrete problems. First, it derives non-Hermitian adiabatic potentials for fast-slow dynamics, showing how left-right Berry connections and metrics act as vector and scalar potentials and enabling tunable oscillatory or decaying slow dynamics. Second, it extends Wannier-state localization to non-Hermitian lattices, proving that the center remains gauge-invariant and that the spread is bounded below by the non-Hermitian quantum metric integrated over the Brillouin zone. Third, it presents a time-periodic-driving protocol to extract the right-right non-Hermitian quantum metric from the linear response of a two-band system, introducing a Peterman factor and showing that off-diagonal metric elements can be accessed with multi-parameter drives. The results offer a practical route to harness non-Hermitian geometry as a resource for designing gauge fields and probing topology in photonic and atomic systems, with explicit numerical validations on representative models.

Abstract

We explore the relation between quantum geometry in non-Hermitian systems and physically measurable phenomena. We highlight various situations in which the behavior of a non-Hermitian system is best understood in terms of quantum geometry, namely the notion of adiabatic potentials in non-Hermitian systems and the localization of Wannier states in periodic non-Hermitian systems. Further, we show that the non-Hermitian quantum metric appears in the response of the system upon time-periodic modulation, which one can use to experimentally measure the non-Hermitian quantum metric. We validate our results by providing numerical simulations of concrete exemplary systems.

Figures (3)

• Figure 1: Comparison between the numerical solution of the non-Hermitian two-level system and the solution obtained from the non-Hermitian Schrödinger equation with adiabatic potentials in Eq. (\ref{['eq:adia']}). In panels (a) and (d) the evolution of the wavefunction of the two-level systems are shown, where in each case the initial wavefunction (highlighted as black dashed line) is $\psi(x,0)\ket{\phi_0(x)}_\text{R}$, with $\psi(x,0)=\sqrt[4]{2/\pi} \exp(-(x+2)^2)$ and $\ket{\phi_0(x)}_\text{R}$ the respective non-decaying eigenstate of $H_\text{F}(x)$ and $\Tilde{H}_\text{F}(x)$. Shown in both panels are the projection to the non-decaying state defined by $\psi_0(x,t) = \leftindex_\text{L} {\ip{\phi_0}{\psi(t)}}$. Panels (b) and (e) show for both systems the solution of $\psi(x,t)$ obtained by considering the adiabatic potentials. Panel (c) compares the norm $n_{(0)}(t)=\int_x |\psi_{(0)}(x,t)|^2$ and the first three central moments of $|\psi_{(0)}(x,t)|^2$ during the evolution governed by $H_\text{F}(x)$. Here the expectation value is defined at $\langle f(x) \rangle = 1/n(t) \int_x f(x) |\psi_{(0)}(x,t)|^2$. The variance is given by Var$(x)=\langle (x-\langle x\rangle)^2\rangle$ and the skewness by Skew$(x)=\langle (x-\langle x\rangle)^3\rangle/(\text{Var}(x))^{3/2}$. The values obtained from the solution of the two-level system are shown in blue (dashed) and are compared to the values derived from the adiabatic potential solution in red. In panel (f) the same properties are compared for the evolution governed by $\Tilde{H}_\text{F}(x)$.
• Figure 2: Comparison of occupation of the excited state of the non-Hermitian system defined in Eq. (\ref{['eq:lin_resp_model']}) and the time-dependent perturbation given by Eq. (\ref{['eq:perturbation']}). The numerical solution $n^\text{num}_1(t)$ (red) is compared with the result obtained by non-Hermitian time-dependent perturbation theory $n^\text{per}_1(t)$ (blue), calculated from Eq. (\ref{['eq:firstorder']}). Below the difference $\Delta n_1(t)= n^\text{num}_1(t)-n^\text{per}_1(t)$ is shown.
• Figure 3: Numerically extracted non-Hermitian quantum metric $g_{ij,0}^{\text{R}\text{R}}$ via time-dependent perturbation. The driving frequency is $\omega=2.2$, the perturbation strength is $\epsilon=0.02$ and the time averaging is done over the time interval $t\in[24,24+2\pi/\omega]$. On the left, the measured values of $g_{xx,0}^{\text{R}\text{R}}$ (blue dots), $g_{yy,0}^{\text{R}\text{R}}$ (red dots), and $g_{xy,0}^{\text{R}\text{R}}$ (black dots) are compared with analytically determined exact values (solid lines) for two lines in parameter space. On the right the upper row shows the measured values as density plots and the comparison with the analytical values in the lower plots shows good agreement.