Quantum Geometric Bounds in Non-Hermitian Systems

TL;DR

The work addresses whether quantum geometric bounds derived for Hermitian, closed systems survive in non-Hermitian and open quantum systems and how they constrain response functions. It introduces non-Hermitian quantum geometric tensors (QGTs) and derives local and global bounds on NH Berry curvature, QGTs, and optical weights, with explicit demonstration in a Rice-Mele model carrying non-Hermitian Chern numbers $C_{\text{NH}}$. Through Keldysh-based open-system response diagrams and Lindbladian dynamics, it shows that positivity of bath self-energies enforces these geometric bounds and that NH topology, with $|C_{\text{NH}}|>0$, yields nontrivial lower bounds on time-dependent responses. The results connect geometric bounds to experimentally accessible observables in dissipative platforms such as quantum-optical and circuit-QED systems, and clarify how NH geometry constrains open-system steady-state responses beyond Hermitian limits.

Abstract

We identify quantum geometric bounds for observables in non-Hermitian systems. We find unique bounds on non-Hermitian quantum geometric tensors, generalized two-point response correlators, conductivity tensors, and optical weights. We showcase these findings in topological systems with non-Hermitian Chern numbers. We demonstrate that the non-Hermitian geometric constraints on response functions naturally arise in open quantum systems governed by out-of-equilibrium Lindbladian dynamics. Our findings are relevant to experimental observables and responses under the realistic setups that fall beyond the idealized closed-system descriptions.

Figures (2)

• Figure 1: Non-Hermitian quantum geometric bounds. Local Berry curvature bound from NH QGTs in NH RM model ($\gamma = 1$) (a)--(b). (a) NH Berry curvature over parameter space ($k_x, k_y$). (b) Difference of the NH QGTs and NH Berry curvature, demonstrating that the local bound, Eq. \ref{['eq:LocalBound']}, holds. (c) Bound on the optical weights due to NH Chern numbers in NH RM model.
• Figure 2: Non-Hermitian geometry from open quantum system response diagrammatics with Keldysh Green's functions ($G^K_n$). One-loop paramagnetic response diagrams. (a) Bubble diagram encoding the geometric bounds in the closed-system limit. (b)--(c) Geometric responses from Keldysh bubbles upon coupling to a bath inducing the non-Hermiticity. (b) Geometric polarization bubble from advanced and Keldysh Green's functions. (c) Geometric polarization bubble from retarded and Keldysh Green's functions. The positivity of the nonequilibrium Green's function sums ($h^\pm_{nm}$) provides for nontrivial geometric bounds in the non-Hermitian systems governed by the open system Lindbladians.