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Physical Review Letters

Non-Abelian Holonomy in Degenerate Non-Hermitian Systems

Zhong‐Lei Shan, Yi‐Ke Sun, Ran Tao, Qi‐Dai Chen, Zhen‐Nan Tian, Xu‐Lin Zhang

Published Aug 2, 2024• 19 citations
Publisher

Abstract

Non-Abelian holonomy, a noncommutative process that measures the parallel transport of non-Abelian gauge fields, has so far been realized in degenerate Hermitian systems with degenerate eigenstates or nondegenerate non-Hermitian systems with exceptional points. Here, we introduce non-Abelian holonomy into degenerate non-Hermitian systems possessing degenerate exceptional points and degenerate energy topologies. The interplay between energy degeneracy and energy topology around exceptional points leads to a non-Abelian holonomy with multiple energy levels and multiple degenerate levels simultaneously, going beyond that in degenerate Hermitian systems with a single energy level, or in nondegenerate non-Hermitian systems with a single degenerate level. We exploit an on-chip photonic platform to experimentally demonstrate the holonomy induced non-Abelian phenomenon, including the switching of eigenstates associated with different degenerate exceptional points and sequence-dependent holonomic outcomes. Our work shifts the paradigm of non-Abelian holonomy and adds new degrees of freedom for non-Abelian applications.

DOI
10.1103/physrevlett.133.053802

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Non-Abelian holonomy, a noncommutative process that measures the parallel transport of non-Abelian gauge fields, has so far been realized in degenerate Hermitian systems with degenerate eigenstates or nondegenerate non-Hermitian systems with exceptional points. Here, we introduce non-Abelian holonomy into degenerate non-Hermitian systems possessing degenerate exceptional points and degenerate energy topologies. The interplay between energy degeneracy and energy topology around exceptional points leads to a non-Abelian holonomy with multiple energy levels and multiple degenerate levels simultaneously, going beyond that in degenerate Hermitian systems with a single energy level, or in nondegenerate non-Hermitian systems with a single degenerate level. We exploit an on-chip photonic platform to experimentally demonstrate the holonomy induced non-Abelian phenomenon, including the switching of eigenstates associated with different degenerate exceptional points and sequence-dependent holonomic outcomes. Our work shifts the paradigm of non-Abelian holonomy and adds new degrees of freedom for non-Abelian applications.
An interplay of an exotic quantum holonomy and exceptional points is examined in one-dimensional Bose systems. The eigenenergy anholonomy, in which Hermitian adiabatic cycle induces nontrivial change in eigenenergies, can be interpreted as a manifestation of eigenenergy's Riemann surface structure, where the branch points are identified as the exceptional points which are degeneracy points in the complexified parameter space. It is also shown that the exceptional points are the divergent points of the non-Abelian gauge connection for the gauge theoretical formulation of the eigenspace anholonomy. This helps us to evaluate anti-path-ordered exponentials of the gauge connection to obtain gauge covariant quantities.
Non-Hermitian physics endows the non-Abelian systems with exceptional topology characterized by non-commutative braid patterns. Interplay of distinct competing sources of non-Hermiticity may induce novel topological effects. Here, we provide a generalized Hatano-Nelson model with higher-order nonreciprocal hoppings, non-Abelian gauge fields, and staggered gain-loss processes showing the exceptional topological structure of the Hopf-link braids that undergoes a EP-mediated topological phase transition. We demonstrate that mixing multiple nonreciprocal channels drives the system into highly intricate, nested complex energy Hopf-link braids and expands the topological landscape up to higher-order braiding sectors. Furthermore, utilizing biorthogonal eigenvector tracking, we map the structural evolution of the exceptional phase boundaries via the maximum Petermann factor in the phase angle parameter planes. We show that the gain-loss non-Hermiticity drives a topological crossover where the extended exceptional contours constrict into isolated regimes. The demonstration of EP-mediated topological phase transition of Hopf-link braids and the associated rich exceptional phase portraits may offer new physical insights with promising applications in robust, fault-tolerant communication channels and quantum computing platforms.
Abstract When a quantum system undergoes slow changes, the evolution of its state depends only on the corresponding trajectory in Hilbert space. This phenomenon, known as quantum holonomy, brings to light the geometric aspects of quantum theory. Depending on the number of degrees of freedom involved, these purely geometric entities can be scalar or belong to a matrix-valued symmetry group. In their various forms, holonomies are vital elements in the description of the fundamental forces in particle physics as well as theories beyond the standard model such as loop quantum gravity or topological quantum field theory. Yet, implementing matrix-valued holonomies thus far has proven challenging, being further complicated by the difficulties involved in identifying suitable dark states for their construction in bosonic systems. Here we develop a representation of holonomic theory founded on the Heisenberg picture and leverage these insights for the experimental realization of a three-dimensional quantum holonomy. Its non-Abelian geometric phase is implemented via the judicious manipulation of bosonic modes constructed from indistinguishable photons and obeys the U(3) symmetry relevant to the strong interaction. Our findings could enable the experimental study of higher-dimensional non-Abelian gauge symmetries and the exploration of exotic physics on a photonic chip.
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.
Eigenstates of a non-Hermitian system exist on complex Riemannian manifolds, with multiple sheets connecting at branch cuts and exceptional points (EPs). These eigenstates can evolve across different sheets-a process that naturally corresponds to state permutation. Here, we report the first experimental realization of non-Abelian permutations in a three-state non-Hermitian system. Our approach relies on the stroboscopic encircling of two different exceptional arcs (EAs), which are smooth trajectories of order-2 EPs appearing from the coalescence of two adjacent states. The non-Abelian characteristics are confirmed by encircling the EAs in opposite sequences. A total of five non-trivial permutations are experimentally realized, which together comprise a non-Abelian group. Our approach provides a reliable way of investigating non-Abelian state permutations and the related exotic winding effects in non-Hermitian systems.