A Berry curvature is an imaginary component of the quantum geometric tensor (QGT) and well studied in many branches modern physics; however, metric as a real QGT less explored. Here, by using tunable superconducting circuits, we experimentally demonstrate two methods to directly measure for characterizing geometry topology underlying states parameter space. The first method probe transition probability after sudden quench, second one detect excitation rate under weak periodic driving....

This is an introductory review of the physics topological quantum matter with cold atoms. Topological phases, originally discovered and investigated in condensed physics, have recently been explored a range different systems, which produced both fascinating findings exciting opportunities for applications. Among physical systems that considered to realize probe these intriguing ultracold atoms become promising platforms due their high flexibility controllability. Quantum simulation phases...

We experimentally explore the topological Maxwell metal bands by mapping momentum space of condensed-matter models to tunable parameter superconducting quantum circuits. An exotic band structure that is effectively described spin-1 equations imaged. Threefold degenerate points dubbed are observed in bands. Moreover, we engineer and observe phase transition from a trivial insulator, report first experiment measure Chern numbers higher than one.Received 15 September 2017Revised 1 January...

Magnetic monopoles play a central role in areas of physics that range from electromagnetism to topological matter. String theory promotes conventional vector gauge fields electrodynamics tensor and predicts the existence more exotic monopoles. Here, we report synthesis monopole four-dimensional parameter space defined by spin degrees freedom single solid-state defect diamond. Using two complementary methods, characterized measuring its quantized charge emanating Kalb-Ramond field. By...

The discovery of relativistic spin-1/2 fermions such as Dirac and Weyl in condensed-matter or artificial systems opens a new era modern physics. An interesting but rarely explored question is whether other spinal excitations could be realized with systems. Here, we construct twoand three-dimensional tight-binding models realizable cold fermionic atoms optical lattices, where the low energy are effectively described by spin-1 Maxwell equations Hamiltonian form. These (linear dispersion)...

We investigate the non-Abelian Thouless pumping in a disorder tunable Lieb chain with degenerate flat bands. The results reveal that quasiperiodic will cause topological phase transition from trivial (without pumping) to nontrivial (with phase. underlying mechanism is monopole originally outside region can be driven into due introduction of disorder. Moreover, since corresponding turn nodal line spread beyond boundaries region, system large strength result disappearance pumping. Furthermore,...

Among non-Hermitian systems, pseudo-Hermitian phases represent a special class of physical models characterized by real energy spectra and the absence skin effects. Here we show that several in two three dimensions can be built employing $q$-deformed matrices, which are related to representation deformed algebras. Through this algebraic approach, present study version well-known Hermitian topological phases, ranging from two-dimensional Chern insulators time-reversal-invariant...

Quantum anomalies offer a useful guide for the exploration of transport phenomena in topological semimetals. In this work, we introduce model describing semimetal four spatial dimensions, whose nodal points act like tensor monopoles momentum space. This system is shown to exhibit monopole-to-monopole phase transitions, as signaled by change value Dixmier-Douady invariant well associated surface states on its boundary. We use reveal an intriguing "4D parity magnetic effect", which stems from...

The complete geometry of quantum states in parameter space is characterized by the geometric tensor, which contains metric and Berry curvature as real imaginary parts, respectively. When are degenerate, take non-Abelian forms. (Abelian) Abelian have been experimentally measured. However, an feasible scheme to extract all components tensor still lacking. Here we propose a generic protocol directly arbitrary degenerate any dimensional based on measuring transition probabilities after quenches....

We present a successful realization of four-dimensional semimetal bands featuring tensor monopoles, achieved using superconducting quantum circuits. Our experiment involves the creation highly tunable diamond energy diagram with four coupled transmons, and parametric modulation their couplers, effectively mapping momentum space to parameter space. This approach enables us establish Dirac-like Hamiltonian fourfold degenerate points. Moreover, we manipulate monopoles by introducing an...

Topological phases associated with non-Abelian charges can exhibit a distinguished bulk-edge correspondence compared Abelian phases, although elucidating this relationship remains challenging in traditional solid-state systems. In paper, we propose theoretical framework for synthesizing quaternion ultracold atomic gases. By designing artificial spin-orbit coupling patterns, the topological edge modes demonstrate clear band topology determined by various charges. This paves way observing...

Higher-dimensional topological phases play a key role in understanding the lower-dimensional and related responses through dimensional reduction procedure. In this work, we present Dirac-type model of four-dimensional Z_{2} insulator (TI) protected by CP symmetry, whose 3D boundary supports an odd number Dirac cones. A specific perturbation splits each bulk massive cone into two valleys separated energy-momentum space with opposite second Chern numbers, which modes become nodal sphere or...

Topological quantum state described by the global invariant has been extensively studied in theory and experiment. In this letter, we investigate relationship between \emph{Zitterbewegung} topology of systems that reflect properties local whole energy bands, respectively. We generalize usual two-band effective Hamiltonian to characterize topological phase transition spin-$J$ insulator. By studying dynamics before after transition, find direction quasiparticles' oscillation can well...

Novel fermionic quasiparticles with integer pseudospins in some energy bands, such as pseudospin-1 triple-point fermions, are attracting increasing interest since they beyond the conventional spin-$1/2$ Dirac and Weyl counterparts. In this paper, we propose a class of excitations emerging topological metal dubbed double-triple-point (DTP) fermions. We first present general three-band continuum model ${C}_{4}$ symmetry three dimensions, which has types threefold degenerate points bands...

We propose to uncover the topology of a pseudo-Hermitian Chern insulator by quantum quench dynamics. The Bloch Hamiltonian is defined in basis $q$-deformed Pauli matrices, which are related representation deformed algebras. show bulk-surface duality phases, then further build concrete relation between static band and dynamics, terms time-averaged spin textures. results also generalized into fully nonequilibrium case where postquench evolution governed Floquet Hamiltonian. Furthermore, we...

Two-dimensional Euler insulators are novel kinds of systems that host multigap topological phases, quantified by a quantized first number in their bulk. Recently, these phases have been experimentally realized suitable two-dimensional synthetic matter setups. Here, we introduce the second invariant, familiar invariant both differential topology (Chern-Gauss-Bonnet theorem) and four-dimensional Euclidean gravity, whose existence still needs to be explored condensed systems. Specifically,...

本文计算了简单立方铁磁体中由于磁性杂质的存在而形成的局域自旋波。其空间对称有s型模、p型模和dr型模三种。求出了各类局域模的波函数和自旋偏离的相角分布,其相应的能级与参量J′S′/JS的关系均由曲线表出。其次分析了在体心立方和面心立方铁磁体中可能出现的局域模的对称类型。我们指出,s型局域自旋波是最有可能实际观测到的。作为观测局域自旋波的样品的铁磁体,其居里点应随渗杂浓度的增加而上升。按照这一判据,我们举出一些可以试用的实验样品。最后,通过总自旋量子数的分析,一般地断定了辐射场不可能激

We propose that a kind of four-dimensional (4D) Hamiltonians, which host tensor monopoles related to the quantum metric in even dimensions, can be simulated by ultracold atoms optical lattices. The topological properties and bulk-boundary correspondence are investigated detail. By fixing momentum along one it reduced an effective three-dimensional model manifesting with nontrivial chiral insulator phase. Using semiclassical Boltzmann equation, we calculate longitudinal resistance against...

We report the discovery of several classes novel topological insulators (TIs) with hybrid-order boundary states generated from first-order TIs additional crystalline symmetries. Unlike current studies on where different-order topology arises merging in various energy, these exhibit a remarkable coexsitence gapless modes and higher-order Fermi-arc states, behaving as hybrid between semimetals within single bulk gap. Our findings establish profound connection $d$-dimensional ($d$D) ($d-1$)D...

We investigate the non-Abelian Thouless pumping in a disorder tunable Lieb chain with degenerate flat bands. The results reveal that quasiperiodic will cause topological phase transition from trivial (without pumping) to non-trivial (with phase. mechanism behind is monopole originally outside region can be driven into due introduction of disorder. Moreover, since corresponding turn nodal line spread beyond boundaries region, system large strength result disappearance pumping. Furthermore, we...

Topological phases associated with non-Abelian charges can exhibit a distinguished bulk-edge correspondence compared Abelian phases, although elucidating this relationship remains challenging in traditional solid-state systems. In paper, we propose theoretical framework for synthesizing quaternion ultracold atomic gases. By designing artificial spin-orbit coupling patterns, the topological edge modes demonstrate clear band topology determined by various charges. This paves way observing...

The essential role of synthetic spin-orbit coupling in discovering new topological matter phases with cold atoms is widely acknowledged. However, the engineering remains unclear for arbitrary-spin models due to complexity spin matrices. In this work, we develop a more general but relatively straightforward method achieve multi-spin models. Our approach hinges on controlling between distinct pseudo-spins through two intermediary states, resulting tunneling flips that have direction-dependent...