Non-Hermitian singularities are ubiquitous in non-conservative open systems. These singularities are often points of measure zero in the eigenspectrum of the system which make them difficult to access without careful engineering. Despite that, they can remotely induce observable effects when some of the system's parameters are varied along closed trajectories in the parameter space. To date, a general formalism for describing this process beyond simple cases is still lacking. Here, we bridge this gap and develop a general approach for treating this problem by utilizing the power of permutation operators and representation theory. This in turn allows us to reveal the following surprising result which contradicts the common belief in the field: loops that enclose the same singularities starting from the same initial point and traveling in the same direction, do not necessarily share the same end outcome. Interestingly, we find that this equivalence can be formally established only by invoking the topological notion of homotopy. Our findings are general with far reaching implications in various fields ranging from photonics and atomic physics to microwaves and acoustics.
In a recent paper entitled "Winding around non-Hermitian singularities" by Zhong et al., published in Nat. Commun. 9, 4808 (2018), a formalism is proposed for calculating the permutations of eigenstates that arise upon encircling (multiple) exceptional points (EPs) in the complex parameter plane of an analytic non-Hermitian Hamiltonian. The authors suggest that upon encircling EPs one should track the eigenvalue branch cuts that are traversed, and multiply the associated permutation matrices accordingly. In this comment we point out a serious shortcoming of this approach, illustrated by an explicit example that yields the wrong result for a specific loop. A more general method that has been published earlier by us and that does not suffer from this problem, is based on using fundamental loops. We briefly explain the method and list its various advantages. In addition, we argue that this method can be verified in a three wave-guide system, which then also unambiguously establishes the noncommutativity associated with encircling multiple EPs.
Chirality is more than a geometric curiosity; it governs measurable asymmetries across nature, from enantiomer-selective drugs and left-handed fermions in particle physics to handed charge transport in Weyl semimetals. We extend this universal concept to non-Hermitian systems by defining topological chirality, an invariant that emerges whenever an exceptional-points (EP) pair is present. Built from the non-commutative fundamental group and its braid representation, topological chirality acts as a singularity selector: clockwise EP loops occupy a homotopy class that avoids EPs, whereas counter-clockwise mirrors are equivalent only if they cross the EPs themselves. We confirm this binary rule in an optical microcavity and a non-Hermitian topological band. The same two-sheeted topology governs EP pairs in spin systems, photonic crystals and hybrid light-matter structures, where EP encirclements have already been demonstrated, so the framework transfers without alteration and confirms its experimental viability. Our findings lay the cornerstone for interpreting loop-sensitive observables such as spectral vorticity, the complex Berry phase and the non-Abelian holonomy. Finally, a gluing-of-planes construction extends the invariant to an n-sheeted surface hosting 2m EPs, unifying higher-order EP pairs.
Abstract Describing systems with non-Hermitian (NH) operators remains a challenge in quantum theory due to singularities (e.g. exceptional points and decoherence) arising from interactions with the environment. Here, we introduce a well-defined reference or computational basis for representing the NH Hamiltonian eigenstates, where singularities are shifted from the basis states to the expansion coefficients. This approach simplifies the mathematical treatment of open quantum systems. Furthermore, we propose a local ‘space-time’ transformation on the computational basis that defines a generic dual space mapping. Interestingly, this transformation reveals a static/global symmetry for real/imaginary energy values, unveiling inherent conserved quantities in open quantum systems. Our formalism provides new insights into key features such as exceptional points, dual space maps, and the origin of symmetry-enforced real eigenvalues. The framework is broadly applicable across various areas of physics where NH operators appear as ladder operators, order parameters, self-energies, projectors, and other entities.
Building on the infinite-component Chern--Simons theory of three-dimensional fracton phases by Ma et al. [Phys. Rev. B 105, 195124 (2022)] and the Toeplitz braiding of anyons by Li et al.~[Phys. Rev B 110, 205108 (2024)], we show that stacking
(3+1)D
BF topological field theories, which serve as low-energy effective descriptions of a class of three-dimensional topological orders, along a fourth spatial direction gives rise to an exotic class of four-dimensional fracton phases. Their low-energy physics is governed by a new field-theoretic framework, namely \textit{infinite-component
BF} (i
BF) \textit{theories}, characterized by asymmetric integer Toeplitz
K matrices. Under open boundary conditions along the stacking direction, i
BF theories with properly chosen
K matrices exhibit a striking phenomenon termed \textit{Toeplitz particle--loop braiding}, where a particle and a loop placed on opposite three-dimensional boundaries acquire a strongly oscillating yet robustly nonvanishing braiding phase even at infinite separation. This nonlocal braiding admits a geometric interpretation: adiabatically transporting the particle induces a winding boundary trajectory on the opposite boundary that encircles the loop. We show that this robustness originates from boundary zero singular modes (ZSMs) of Toeplitz
K matrices revealed by singular value decomposition, rather than from boundary zero eigenmodes responsible for previously known Toeplitz braiding of anyons, and that the same ZSM mechanism also underlies directional amplification in the rapidly developing field of non-Hermitian physics. We analytically and numerically study representative i
BF theories with Hatano--Nelson--type and non-Hermitian Su--Schrieffer--Heeger--type
K matrices, establishing a universal correspondence between ZSMs and Toeplitz particle--loop braiding.
Non-Hermitian systems are widespread in both classical and quantum physics. The dynamics of such systems has recently become a focal point of research, showcasing surprising behaviors that include apparent violation of the adiabatic theorem and chiral topological conversion related to encircling exceptional points (EPs). These have both fundamental interest and potential practical applications. Yet the current literature features a number of apparently irreconcilable results. Here we develop a general theory for slow evolution of non-Hermitian systems and resolve these contradictions. We prove an analog of the adiabatic theorem for non-Hermitian systems and generalize it in the presence of uncontrolled environmental fluctuations (noise). The effect of noise turns out to be crucial due to inherent exponential instabilities present in non-Hermitian systems. Disproving common wisdom, the end state of the system is determined by the final Hamiltonian only, and is insensitive to other details of the evolution trajectory in parameter space. Our quantitative theory, leading to transparent physical intuition, is amenable to experimental tests. It provides efficient tools to predict the outcome of the system's evolution, avoiding the need to follow costly time-evolution simulations. Our approach may be useful for designing devices based on non-Hermitian physics and may stimulate analyses of classical and quantum non-Hermitian-Hamiltonian dynamics, as well as that of quantum Lindbladian and hybrid-Liouvillian systems.