Quantum geometry has emerged as a central and ubiquitous concept in quantum sciences, with direct consequences on quantum metrology and many-body quantum physics. In this context, two fundamental geometric quantities play complementary roles: the Fub
ini-Study metric, which introduces a notion of distance between quantum states defined over a parameter space, and the Berry curvature associated with Berry-phase effects and topological band structures. In fact, recent studies have revealed direct relations between these two important quantities, suggesting that topological properties can, in special cases, be deduced from the quantum metric. In this work, we establish general and exact relations between the quantum metric and the topological invariants of generic Dirac Hamiltonians. In particular, we demonstrate that topological indices (Chern numbers or winding numbers) are bounded by the quantum volume determined by the quantum metric. Our theoretical framework, which builds on the Clifford algebra of Dirac matrices, is applicable to topological insulators and semimetals of arbitrary spatial dimensions, with or without chiral symmetry. This work clarifies the role of the Fubini-Study metric in topological states of matter, suggesting unexplored topological responses and metrological applications in a broad class of quantum-engineered systems.
Applying time-periodic modulations is routinely used to control and design synthetic matter in quantum-engineered settings. In lattice systems, this approach is explored to engineer band structures with non-trivial topological properties, but also to
generate exotic interaction processes. A prime example is density-assisted tunneling, by which the hopping amplitude of a particle between neighboring sites explicitly depends on their respective occupations. Here, we show how density-assisted tunneling can be tailored in view of simulating the effects of strain in synthetic graphene-type systems. Specifically, we consider a mixture of two atomic species on a honeycomb optical lattice: one species forms a Bose-Einstein condensate in an anisotropic harmonic trap, whose inhomogeneous density profile induces an effective uniaxial strain for the second species through density-assisted tunneling processes. In direct analogy with strained graphene, the second species experiences a pseudo magnetic field, hence exhibiting relativistic Landau levels and the valley Hall effect. Our proposed scheme introduces a unique platform for the investigation of strain-induced gauge fields and their possible interplay with quantum fluctuations and collective excitations.
Bloch oscillations (BOs) are a fundamental phenomenon by which a wave packet undergoes a periodic motion in a lattice when subjected to an external force. Observed in a wide range of synthetic lattice systems, BOs are intrinsically related to the geo
metric and topological properties of the underlying band structure. This has established BOs as a prominent tool for the detection of Berry phase effects, including those described by non-Abelian gauge fields. In this work, we unveil a unique topological effect that manifests in the BOs of higher-order topological insulators through the interplay of non-Abelian Berry curvature and quantized Wilson loops. It is characterized by an oscillating Hall drift that is synchronized with a topologically-protected inter-band beating and a multiplied Bloch period. We elucidate that the origin of this synchronization mechanism relies on the periodic quantum dynamics of Wannier centers. Our work paves the way to the experimental detection of non-Abelian topological properties in synthetic matter through the measurement of Berry phases and center-of-mass displacements.
Quantum anomalies offer a useful guide for the exploration of transport phenomena in topological semimetals. In this work, we introduce a model describing a semimetal in four spatial dimensions, whose nodal points act like tensor monopoles in momentu
m space. This system is shown to exhibit monopole-to-monopole phase transitions, as signaled by a change in the value of the topological Dixmier-Douady invariant as well as by the associated surface states on its boundary. We use this model to reveal an intriguing 4D parity magnetic effect, which stems from a parity-type anomaly. In this effect, topological currents are induced upon time-modulating the separation between the fictitious monopoles in the presence of a magnetic perturbation. Besides its theoretical implications in both condensed matter and quantum field theory, the peculiar 4D magnetic effect revealed by our model could be measured by simulating higher-dimensional semimetals in synthetic matter.
We introduce a scheme by which flat bands with higher Chern number $vert Cvert>1$ can be designed in ultracold gases through a coherent manipulation of Bloch bands. Inspired by quantum-optics methods, our approach consists in creating a dark Bloch ba
nd by coupling a set of source bands through resonant processes. Considering a $Lambda$ system of three bands, the Chern number of the dark band is found to follow a simple sum rule in terms of the Chern numbers of the source bands: $C_D!=!C_1+C_2-C_3$. Altogether, our dark-state scheme realizes a nearly flat Bloch band with predictable and tunable Chern number $C_D$. We illustrate our method based on a $Lambda$ system, formed of the bands of the Harper-Hofstadter model, which leads to a nearly flat Chern band with $C_D!=!2$. We explore a realistic sequence to load atoms into the dark Chern band, as well as a probing scheme based on Hall drift measurements. Dark Chern bands offer a practical platform where exotic fractional quantum Hall states could be realized in ultracold gases.
Bound states of two interacting particles moving on a lattice can exhibit remarkable features that are not captured by the underlying single-particle picture. Inspired by this phenomenon, we introduce a novel framework by which genuine interaction-in
duced geometric and topological effects can be realized in quantum-engineered systems. Our approach builds on the design of effective lattices for the center-of-mass motion of two-body bound states (emph{doublons}), which can be created through long-range interactions. This general scenario is illustrated on several examples, where flat-band localization, topological pumps and higher-order topological corner modes emerge from genuine interaction effects. Our results pave the way for the exploration of interaction-induced topological effects in a variety of platforms, ranging from ultracold gases to interacting photonic devices.
The interplay of $pi$-flux and lattice geometry can yield full localization of quantum dynamics in lattice systems, a striking interference phenomenon known as Aharonov-Bohm caging. At the level of the single-particle energy spectrum, this full-local
ization effect is attributed to the collapse of Bloch bands into a set of perfectly flat (dispersionless) bands. In such lattice models, the effects of inter-particle interactions generally lead to a breaking of the cages, and hence, to the spreading of the wavefunction over the lattice. Motivated by recent experimental realizations of analog Aharonov-Bohm cages for light, using coupled-waveguide arrays, we hereby demonstrate that caging always occurs in the presence of local nonlinearities. As a central result, we focus on special caged solutions, which are accompanied by a breathing motion of the field intensity, that we describe in terms of an effective two-mode model reminiscent of a bosonic Josephson junction. Moreover, we explore the quantum regime using small particle ensembles, and we observe quasi-caged collapse-revival dynamics with negligible leakage. The results stemming from this work open an interesting route towards the characterization of nonlinear dynamics in interacting flat band systems.
The present Chapter discusses methods by which topological Bloch bands can be prepared in cold-atom setups. Focusing on the case of Chern bands for two-dimensional systems, we describe how topological properties can be triggered by driving atomic gas
es, either by dressing internal levels with light or through time-periodic modulations. We illustrate these methods with concrete examples, and we discuss recent experiments where geometrical and topological band properties have been identified.
We investigate the properties of a two-dimensional quasicrystal in the presence of a uniform magnetic field. In this configuration, the density of states (DOS) displays a Hofstadter butterfly-like structure when it is represented as a function of the
magnetic flux per tile. We show that the low-DOS regions of the energy spectrum are associated with chiral edge states, in direct analogy with the Chern insulators realized with periodic lattices. We establish the topological nature of the edge states by computing the topological Chern number associated with the bulk of the quasicrystal. This topological characterization of the non-periodic lattice is achieved through a local (real-space) topological marker. This work opens a route for the exploration of topological insulating materials in a wide range of non-periodic lattice systems, including photonic crystals and cold atoms in optical lattices.
