Gauge fields are central in our modern understanding of physics at all scales. At the highest energy scales known, the microscopic universe is governed by particles interacting with each other through the exchange of gauge bosons. At the largest length scales, our Universe is ruled by gravity, whose gauge structure suggests the existence of a particle—the graviton—that mediates the gravitational force. At the mesoscopic scale, solid-state systems are subjected to gauge fields of different nature: materials can be immersed in external electromagnetic fields, but they can also feature emerging gauge fields in their low-energy description. In this review, we focus on another kind of gauge field: those engineered in systems of ultracold neutral atoms. In these setups, atoms are suitably coupled to laser fields that generate effective gauge potentials in their description. Neutral atoms ‘feeling’ laser-induced gauge potentials can potentially mimic the behavior of an electron gas subjected to a magnetic field, but also, the interaction of elementary particles with non-Abelian gauge fields. Here, we review different realized and proposed techniques for creating gauge potentials—both Abelian and non-Abelian—in atomic systems and discuss their implication in the context of quantum simulation. While most of these setups concern the realization of background and classical gauge potentials, we conclude with more exotic proposals where these synthetic fields might be made dynamical, in view of simulating interacting gauge theories with cold atoms.

https://iopscience.iop.org/article/10.1088/0034-4885/77/12/126401/pdf

Light-induced gauge fields for ultracold atoms

Non-Hermitian systems can exhibit a counterintuitive phenomenon where a single local boundary or disorder modifies the entire spectrum, no matter how large the system is. In such cases, all bulk modes become localized ``skin'' modes, and usual bulk topological invariants no longer correctly predict topological boundary modes. Generalizing Laughlin's gauge argument to complex fluxes, the authors derive a geometrical approach for the exact determination of the skin-mode spectrum. They also devise a new topological criterion for non-Hermitian particle-hole symmetric Hamiltonians based on complex analysis.

Anatomy of skin modes and topology in non-Hermitian systems

A review is given on the foundations and applications of non-Hermitian classical and quantum physics. First, key theorems and central concepts in non-Hermitian linear algebra, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity and parity-time symmetry, are delineated in a pedagogical and mathematically coherent manner. Building on these, we provide an overview of how diverse classical systems, ranging from photonics, mechanics, electrical circuits, acoustics to active matter, can be used to simulate non-Hermitian wave physics. In particular, we discuss rich and unique phenomena found therein, such as unidirectional invisibility, enhanced sensitivity, topological energy transfer, coherent perfect absorption, single-mode lasing, and robust biological transport. We then explain in detail how non-Hermitian operators emerge as an effective description of open quantum systems on the basis of the Feshbach projection approach and the quantum trajectory approach. We discuss their applications to physical systems relevant to a variety of fields, including atomic, molecular and optical physics, mesoscopic physics, and nuclear physics with emphasis on prominent phenomena/subjects in quantum regimes, such as quantum resonances, superradiance, continuous quantum Zeno effect, quantum critical phenomena, Dirac spectra in quantum chromodynamics, and nonunitary conformal field theories. Finally, we introduce the notion of band topology in complex spectra of non-Hermitian systems and present their classifications by providing the proof, firstly given by this review in a complete manner, as well as a number of instructive examples. Other topics related to non-Hermitian physics, including nonreciprocal transport, speed limits, nonunitary quantum walk, are also reviewed.

Non-Hermitian Physics

We develop the topological band theory for systems described by non-Hermitian Hamiltonians, whose energy spectra are generally complex. After generalizing the notion of gapped band structures to the non-Hermitian case, we classify ``gapped'' bands in one and two dimensions by explicitly finding their topological invariants. We find nontrivial generalizations of the Chern number in two dimensions, and a new classification in one dimension, whose topology is determined by the energy dispersion rather than the energy eigenstates. We then study the bulk-edge correspondence and the topological phase transition in two dimensions. Different from the Hermitian case, the transition generically involves an extended intermediate phase with complex-energy band degeneracies at isolated ``exceptional points'' in momentum space. We also systematically classify all types of band degeneracies.

Topological Band Theory for Non-Hermitian Hamiltonians

We show that the bulk-boundary correspondence for topological insulators can be modified in the presence of non-Hermiticity. We consider a one-dimensional tight-binding model with gain and loss as well as long-range hopping. The system is described by a non-Hermitian Hamiltonian that encircles an exceptional point in momentum space. The winding number has a fractional value of 1/2. There is only one dynamically stable zero-energy edge state due to the defectiveness of the Hamiltonian. This edge state is robust to disorder due to protection by a chiral symmetry. We also discuss experimental realization with arrays of coupled resonator optical waveguides.

Anomalous Edge State in a Non-Hermitian Lattice

We show that one-dimensional quasiperiodic optical lattice systems can exhibit edge states and topological phases which are generally believed to appear in two-dimensional systems. When the Fermi energy lies in gaps, the Fermi system on the optical superlattice is a topological insulator characterized by a nonzero topological invariant. The topological nature can be revealed by observing the density profile of a trapped fermion system, which displays plateaus with their positions uniquely determined by the ration of wavelengths of the bichromatic optical lattice. The butterflylike spectrum of the superlattice system can be also determined from the finite-temperature density profiles of the trapped fermion system. This finding opens an alternative avenue to study the topological phases and Hofstadter-like spectrum in one-dimensional optical lattices.

http://www.iop.cas.cn/xwzx/kydt/201206/P020120612419783630245.pdf

Edge states and topological phases in one-dimensional optical superlattices.

Non-Hermiticity from nonreciprocal hoppings has been shown recently to demonstrate the non-Hermitian skin effect (NHSE) under open boundary conditions (OBCs). Here we study the interplay of this effect and the Anderson localization (AL) in a nonreciprocal quasiperiodic lattice, dubbed nonreciprocal Aubry-Andr\'e model, and a rescaled transition point is exactly proved. The nonreciprocity can induce not only NHSEs but also the asymmetry in localized states, characterized by two Lyapunov exponents. Meanwhile, this transition is also topological, in the sense of a winding number associated with complex eigenenergies under periodic boundary conditions (PBCs), establishing a bulk-bulk correspondence. This interplay can be realized straightforwardly by an electrical circuit with only linear passive RLC components instead of elusive nonreciprocal ones, showing the transport of a continuous wave undergoes a transition between insulating and amplifying. This paradigmatic scheme can be immediately accessed in experiments even for more nonreciprocal models and will definitely inspire the study of interplay of NHSEs and ALs as well as more other quantum/topological phenomena in various systems.

Interplay of non-Hermitian skin effects and Anderson localization in nonreciprocal quasiperiodic lattices

We study the competition of disorder and superconductivity for a one-dimensional p-wave superconductor in incommensurate potentials. With the increase in the strength of the incommensurate potential, the system undergoes a transition from a topological superconducting phase to a topologically trivial localized phase. The phase boundary is determined both numerically and analytically from various aspects and the topological superconducting phase is characterized by the presence of Majorana edge fermions in the system with open boundary conditions. We also calculate the topological Z(2) invariant of the bulk system and find it can be used to distinguish the different topological phases even for a disordered system. DOI: 10.1103/PhysRevLett.110.176403

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Topological superconductor to Anderson localization transition in one-dimensional incommensurate lattices.

Non-Hermitian systems can exhibit exotic topological and localization properties. Here we elucidate the non-Hermitian effects on disordered topological systems using a nonreciprocal disordered Su-Schrieffer-Heeger model. We show that the non-Hermiticity can enhance the topological phase against disorders by increasing bulk gaps. Moreover, we uncovera topological phase which emerges under both moderate non-Hermiticity and disorders, and is characterized by localized insulating bulk states with a disorder-averaged winding number and zero-energy edge modes. Such topological phases induced by the combination of non-Hermiticity and disorders are dubbed non-Hermitian topological Anderson insulators . We reveal that the system has unique non-monotonous localization behavior and the topological transition is accompanied by an Anderson transition. These properties are general in other non-Hermitian models.

Non-Hermitian topological Anderson insulators

Nonlinear transmission lines (NLTLs) are nonlinear electronic circuits used for parametric amplification and pulse generation, and it is known that left-handed NLTLs support enhanced harmonic generation while suppressing shock wave formation. We show experimentally that in a left-handed NLTL analogue of the Su-Schrieffer-Heeger (SSH) lattice, harmonic generation is greatly increased by the presence of a topological edge state. Previous studies of nonlinear SSH circuits focused on solitonic behaviours at the fundamental harmonic. Here, we show that a topological edge mode at the first harmonic can produce strong propagating higher-harmonic signals, acting as a nonlocal cross-phase nonlinearity. We find maximum third-harmonic signal intensities five times that of a comparable conventional left-handed NLTL, and a 250-fold intensity contrast between topologically nontrivial and trivial configurations. This work advances the fundamental understanding of nonlinear topological states, and may have applications for compact electronic frequency generators.

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Li-Jun Lang

Papers

Edge states and topological phases in one-dimensional optical superlattices.

Interplay of non-Hermitian skin effects and Anderson localization in nonreciprocal quasiperiodic lattices

Topological superconductor to Anderson localization transition in one-dimensional incommensurate lattices.

Non-Hermitian topological Anderson insulators

Topologically enhanced harmonic generation in a nonlinear transmission line metamaterial.