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In the paper [X. Tan et al., Phys. Rev. Lett. 122, 210401 (2019)], we have studied the Euler characteristic number $\chi$ for a two-band model; However, the calculated $\chi$ there is not an integer when the parameter $|h|>1$ and then may not be considered as a suitable topological number for the system. In this note, we find that the Bloch vectors of the ground state for $|h|>1$ do not cover the whole Bloch sphere and thus the Euler characteristic number should be effectively associated with the manifold of a disk, rather than a sphere. After taking into account the boundary contribution, we derive the correct Euler characteristic number. Unfortunately, the Euler characteristic number does not change when crossing the critical points $|h|=1$ and thus can not be used to characterize the topological phase transition of the present model., Comment: 4 pages, 3 figures

Non-Hermitian systems can exhibit unique topological and localization properties. Here we elucidate the non-Hermitian effects on disordered topological systems by studying a non-Hermitian disordered Su-Schrieffer-Heeger model with nonreciprocal hoppings. We show that the non-Hermiticity can enhance the topological phase against disorders by increasing energy gaps. Moreover, we uncover a topological phase which emerges only 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 also find that the system has non-monotonous localization behaviour and the topological transition is accompanied by an Anderson transition. These properties are general in other non-Hermitian models., Comment: 6+4 pages, 4+6 figures; close to the published version

Berry curvature is an imaginary component of the quantum geometric tensor (QGT) and is well studied in many branches of modern physics; however, the quantum metric as a real component of the QGT is less explored. Here, by using tunable superconducting circuits, we experimentally demonstrate two methods to directly measure the quantum metric tensor for characterizing the geometry and topology of underlying quantum states in parameter space. The first method is to probe the transition probability after a sudden quench, and the second one is to detect the excitation rate under weak periodic driving. Furthermore, based on quantum-metric and Berry-curvature measurements, we explore a topological phase transition in a simulated time-reversal-symmetric system, which is characterized by the Euler characteristic number instead of the Chern number. The work opens up a unique approach to explore the topology of quantum states with the QGT., Comment: 6 pages, 4 figures; newly cited two experimental references

We propose a simple method to simulate and detect topological insulators with cold atoms trapped in a one-dimensional bichromatic optical lattice subjected to a time-periodic modulation. The tight-binding form of this shaken system is equivalent to the periodically driven Aubry-Andre model. We demonstrate that this model can be mapped into a two-dimensional Chern insulator model, whose energy spectrum hosts a topological phase within an experimentally accessible parameter regime. By tuning the laser phase adiabatically, such one-dimensional system constitutes a natural platform to realize topological particle pumping. We show that the Chern number characterizing the topological features of this system can be measured by detecting the density shift after one cycle of pumping., Comment: 8 pages, 2 figures

We propose an experimental scheme to simulate and observe the Klein tunneling of relativistic Maxwell particles with trapped ions. We explore the scattering dynamics of the pseudospin-one Maxwell particles and demonstrate that the scattered state should be a superposition of a reflection state, a localization state, and a transmission state. The probabilities of these states can be analytically obtained by the approach of Landau-Zener transition. We further show that the Maxwell Hamiltonian and the associated scattering dynamics can be mimicked with two trapped ions. The Maxwell spinors are encoded by three internal states of the first ion, the position and momentum are described by those of the motional modes, and the desired linear potential barrier is built by the second ion., Comment: 6 pages, 4 figures

Novel fermionic quasiparticles with integer pseudospins in some energy bands, such as pseudospin-1 triple-point fermions, recently attract increasing interest since they are beyond the conventional spin-$1/2$ Dirac and Weyl counterparts. In this paper, we propose a class of pseudospin-1 fermioic excitations emerging in topological metal bands, dubbed double-triple-point (DTP) fermions. We first present a general three-band continuum model with $C_4$ symmetry in three dimensions, which has three types of threefold degenerate points in the bands classified by their topological charges $C=\pm4,\pm2,0$, respectively. They are dubbed DTPs as spin-1 generalization of double-Weyl points. We then construct two-dimensional and three-dimensional tight-binding lattice models of topological metal bands with exotic DTP fermions near the DTPs. In two dimensions, the band gaps close at a trivial DTP with zero Berry phase, which occurs at the transition between the normal and topological insulator phases. In three dimensions, the topological properties of three different DTP fermions in lattice systems are further investigated, and the effects of breaking $C_4$ symmetry are also studied, which generally leads to splitting each quadratic DTP into two linear triple points and gives topological phase diagrams. Using ultracold fermionic atoms in optical lattices, the proposed models can be realized and the topological properties of the DTP fermions can be detected., Comment: 11 pages, 9 figures; accepted in PRA

This is an introductory review of the physics of topological quantum matter with cold atoms. Topological quantum phases, originally discovered and investigated in condensed matter physics, have recently been explored in a range of different systems, which produced both fascinating physics findings and exciting opportunities for applications. Among the physical systems that have been considered to realize and probe these intriguing phases, ultracold atoms become promising platforms due to their high flexibility and controllability. Quantum simulation of topological phases with cold atomic gases is a rapidly evolving field, and recent theoretical and experimental developments reveal that some toy models originally proposed in condensed matter physics have been realized with this artificial quantum system. The purpose of this article is to introduce these developments. The article begins with a tutorial review of topological invariants and the methods to control parameters in the Hamiltonians of neutral atoms. Next, topological quantum phases in optical lattices are introduced in some detail, especially several celebrated models, such as the Su-Schrieffer-Heeger model, the Hofstadter-Harper model, the Haldane model and the Kane-Mele model. The theoretical proposals and experimental implementations of these models are discussed. Notably, many of these models cannot be directly realized in conventional solid-state experiments. The newly developed methods for probing the intrinsic properties of the topological phases in cold atom systems are also reviewed. Finally, some topological phases with cold atoms in the continuum and in the presence of interactions are discussed, and an outlook on future work is given., Comment: Close to the published version. arXiv admin note: text overlap with arXiv:cond-mat/0210564, arXiv:1308.6533, arXiv:1703.07222 by other authors

Dipolar parity anomaly can be induced by spatiotemporally weak-dependent energy-momentum separation of paired Dirac points in two-dimensional Dirac semimetals. Here we reveal topological currents arising from this kind of anomaly. A corresponding lattice model is proposed to emulate the topological currents by using two-component ultracold atoms in a two-dimensional optical Raman lattice. In our scheme, the topological currents can be generated by varying on-site coupling between the two atomic components in time and tuned via the laser fields. Moreover, we show that the topological particle currents can directly be detected from measuring the drift of the center of mass of the atomic gases., Comment: 10 pages, 8 figures; published version

The interaction between a photon and a qubit in the Janeys-Cummings (JC) model generates a kind of quasiparticle called polariton. While they are widely used in quantum optics, difficulties in engineering controllable coupling of them severely limit their applications to simulate spinful quantum systems. Here we show that, in the superconducting quantum circuit context, polariton states in the single-excitation manifold of a JC lattice can be used to simulate a spin-1/2 system, based on which tunable synthetic spin-orbit coupling and novel topological polaritons can be generated and explored. The lattice is formed by a sequence of coupled transmission line resonators, each of which is connected to a transmon qubit. Synthetic spin-orbit coupling and effective Zeeman field of the polariton can both be tuned by modulating the coupling strength between neighbouring resonators, allowing for the realization of a large variety of polaritonic topological semimetal bands. Methods for detecting the polaritonic topological edge states and topological invariants are also proposed. Therefore, our work suggests that the JC lattice is a versatile platform for exploring spinful topological states of matter, which may inspire developments of topologically protected quantum optical and information processing devices., Comment: V2: Extended rewritten version; V3: Accepted version; V4 published version with corrections

The discovery of relativistic spin-1/2 fermions such as Dirac and Weyl fermions in condensed-matter or artificial systems opens a new era in modern physics. An interesting but rarely explored question is whether other relativistic spinal excitations could be realized with artificial systems. Here, we construct twoand three-dimensional tight-binding models realizable with cold fermionic atoms in optical lattices, where the low energy excitations are effectively described by the spin-1 Maxwell equations in the Hamiltonian form. These relativistic (linear dispersion) excitations with unconventional integer pseudospin, beyond the Dirac-Weyl-Majorana fermions, are an exotic kind of fermions named asMaxwell fermions.We demonstrate that the systems have rich topological features. For instance, the threefold degenerate points called Maxwell points may have quantized Berry phases and anomalous quantum Hall effects with spin-momentum locking may appear in topological Maxwell insulators in the two-dimensional lattices. In three dimensions,Maxwell points may have nontrivial monopole charges of \pm2 with two Fermi arcs connecting them, and the merging of the Maxwell points leads to topological phase transitions. Finally, we propose realistic schemes for realizing the model Hamiltonians and detecting the topological properties of the emergent Maxwell quasiparticles in optical lattices., Comment: 16 pages, 10 figures; arXiv admin note: substantial text of the 2D results overlap with arXiv:1610.05993

We experimentally explore the topological Maxwell metal bands by mapping the momentum space of condensed-matter models to the tunable parameter space of superconducting quantum circuits. An exotic band structure that is effectively described by the spin-1 Maxwell equations is imaged. Three-fold degenerate points dubbed Maxwell points are observed in the Maxwell metal bands. Moreover, we engineer and observe the topological phase transition from the topological Maxwell metal to a trivial insulator, and report the first experiment to measure the Chern numbers that are higher than one., Comment: 6 pages, 4 figures; accepted version

We explore the topological properties of double-Weyl semimetals with cold atoms in optical lattices. We first propose to realize a tight-binding model of simulating the double-Weyl semimetal with a pair of double-Weyl points by engineering the atomic hopping in a three-dimensional optical lattice. We show that the double-Weyl points with topological charges of \$pm2$ behave as sink and source of Berry flux in momentum space connecting by two Fermi arcs and they are stabilized by the \$C_{4h}$ point-group symmetry. By applying a realizable \$C_4$ breaking term, we find that each double-Weyl point splits into two single-Weyl points and obtain rich phase diagrams in the parameter space spanned by the strengths of an effective Zeeman term and the \$C_4$ breaking term, which contains a topological and a normal insulating phases and two topological Weyl semimetal phases with eight and four single-Weyl points, apart from the double-Weyl semimetal phase. Furthermore, we demonstrate with numerical simulations that (i) the mimicked double- and single-Weyl points can be detected by measuring the atomic transfer fractions after a Bloch oscillation; (ii) the Chern number of different quantum phases in the phase diagram can be extracted from the center shift of the hybrid Wannier functions, which can be directly measured with the time-of-flight imaging; (iii) the band topology of the \$C_4$ symmetric Bloch Hamiltonian can be detected simply from measuring the spin polarization at the high symmetry momentum points with a condensate in the optical lattice. The proposed system would provide a promising platform for elaborating the intrinsic exotic physics of double-Weyl semimetals and the related topological phase transitions., Comment: 12 pages, 7 figures; accepted by PRA

We propose that a tunable generalized three-dimensional Hofstadter Hamiltonian can be realized by engineering the Raman-assisted hopping of ultracold atoms in a cubic optical lattice. The Hamiltonian describes a periodic lattice system under artificial magnetic fluxes in three dimensions. For certain hopping configurations, the bulk bands can have Weyl points and nodal loops, respectively, allowing the study of both the two nodal semimetal states within this system. Furthermore, we illustrate that with proper rational fluxes and hopping parameters, the system can exhibit the three-dimensional quantum Hall effect when the Fermi level lies in the band gaps, which is topologically characterized by one or two nonzero Chern numbers. Our proposed optical-lattice system provides a promising platform for exploring various exotic topological phases in three dimensions., Comment: 10 pages, 5 figures

We study the dynamics of a transverse-field XY chain driven across quantum critical points by noisy control fields. We characterize the defect density as a function of the quench time and the noise strength, and demonstrate that the defect productions for three quench protocols with different scaling exponents exhibit the anti-Kibble-Zurek behavior, whereby slower driving results in more defects. The protocols are quenching through the boundary line between paramagnetic and ferromagnetic phases, quenching across the isolated multicritical point and along the gapless line, respectively. We also show that the optimal quench time to minimize defects scales as a universal power law of the noise strength in all the three cases. Furthermore, by using quantum simulation of the quench dynamics in the spin system with well-designed Landau-Zener crossings in pseudo-momentum space, we propose an experimentally feasible scheme to test the predicted anti-Kibble-Zurek behavior of this noisy transverse-field XY chain with two-level systems under controllable fluctuations., Comment: 9 pages, 4 figures; Revised version, accepted by PRB

Implementing holonomic quantum computation is a challenging task as it requires complicated interaction among multilevel systems. Here we propose to implement nonadiabatic holonomic quantum computation based on dressed-state qubits in circuit QED. An arbitrary holonomic single-qubit gate can be conveniently achieved using external microwave fields and tuning their amplitudes and phases. Meanwhile, nontrivial two-qubit gates can be implemented in a coupled-cavities scenario assisted by a grounding SQUID with tunable interaction, where the tuning is achieved by modulating the ac flux threaded through the SQUID. In addition, our proposal is directly scalable, up to a two-dimensional lattice configuration. In the present scheme, the dressed states involve only the lowest two levels of each transmon qubit and the effective interactions exploited are all of resonant nature. Therefore, we release the main difficulties for physical implementation of holonomic quantum computation on superconducting circuits., Comment: v3: to be published at Physical Review Applied

We reveal a novel topological property of the exceptional points in a two-level parity-time symmetric system and then propose a scheme to detect the topological exceptional points in the system, which is embedded in a larger Hilbert space constructed by a four-level cold atomic system. We show that a tunable parameter in the presented system for simulating the non-Hermitian Hamiltonian can be tuned to swept the eigenstates through the exceptional points in parameter space. The non-trivial Berry phases of the eigenstates obtained in this loop from the exceptional points can be measured by the atomic interferometry. Since the proposed operations and detection are experimentally feasible, our scheme may pave a promising way to explore the novel properties of non-Hermitian systems.

We construct a two-dimensional tight-binding model of an optical lattice, where the low energy excitations should be described by the spin-1 Maxwell equations in the Hamiltonian form, and such linear dispersion excitations with pesudospin-1 are so called as the Maxwell quasiparticles. The system has rich topological features, for examples, the threefold degeneracy points called Maxwell points may have nontrivial $\pm 2\pi$ Berry phases and the anomalous quantum Hall effect with spin-momentum locking may appear in topological Maxwell insulators. We propose realistic schemes for realizing the Maxwell metals/insulators and detecting the intrinsic properties of the topological Maxwell quasiparticles with ultracold atoms in optical lattices., Comment: 5 pages, 4 figures. Supplement: 10 pages, 5 figures