Three-dimensional topological Weyl semimetals can generally support a zero-dimensional Weyl point characterized by a quantized Chern number or a one-dimensional Weyl nodal ring (or line) characterized by a quantized Berry phase in the momentum space. Here, in a dissipative system with particle gain and loss, we discover a new type of topological ring, dubbed Weyl exceptional ring consisting of exceptional points at which two eigenstates coalesce. Such a Weyl exceptional ring is characterized by both a quantized Chern number and a quantized Berry phase, which are defined via the Riemann surface. We propose an experimental scheme to realize and measure the Weyl exceptional ring in a dissipative cold atomic gas trapped in an optical lattice.Recently, condensed matter systems have proven to be a powerful platform to study low energy gapless particles by using momentum space band structures to mimic the energy-momentum relation of relativistic particles [1,2] and beyond [3][4][5][6]. One celebrated example in three dimensions is the zero-dimensional Weyl point [7][8][9][10][11][12][13][14][15][16][17][18] described by the Weyl Hamiltonian, which has been long sought-after in particle physics but only experimentally observed in condensed matter materials [19][20][21]. Such a Weyl point can be viewed as a magnetic monopole [22] in the momentum space and possesses a quantized Chern number on a surface enclosing the point. Another example is the one-dimensional Weyl nodal ring [3,[23][24][25], which has no counterpart in particle physics. It can be regarded as the generalization of zero-dimensional Dirac cones in two-dimensional systems, such as in graphene, to three-dimensional systems. Such a nodal ring has a quantized Berry phase over a closed path encircling it but does not possess a nonzero quantized Chern number. This leads to a natural question of whether there exists a topological ring exhibiting both a quantized Chern number and a quantized Berry phase in the momentum space.So far, studies on those gapless states focus on closed and lossless systems. However, particle gain and loss are generally present in natural systems. Such systems can often be described by non-Hermitian Hamiltonians [26][27][28][29], which are widely applied to many different systems [30][31][32][33][34][35][36][37][38][39][40]. Due to the non-Hermiticity, eigenvalues of the Hamiltonian are generically complex unless the PT symmetry [41] is conserved and the imaginary part of energy is associated with either decay or growth. Another intriguing feature of a non-Hermitian system is the existence of exceptional points (EPs) [26][27][28][29] at which two eigenstates coalesce and the Hamiltonian becomes defective, leading to many novel phenomena, such as lossinduced transparency [30], single-mode lasers [36,37], and reversed pump dependence of lasers [33].In this paper, we investigate a system of Weyl points in the presence of a spin-dependent non-Hermitian term and find a Weyl exceptional ring composed of EPs. In stark contrast to a Weyl nodal ...

Cold-atom experiments in optical lattices offer a versatile platform to realize various topological quantum phases. A key challenge in those experiments is to unambiguously probe the topological order. We propose a method to directly measure the characteristic topological invariants (order) based on the time-of-flight imaging of cold atoms. The method is generally applicable to detection of topological band insulators in one, two, or three dimensions characterized by integer topological invariants. Using detection of the Chern number for the 2D anomalous quantum Hall states and the Chern-Simons term for the 3D chiral topological insulators as examples, we show that the proposed detection method is practical, robust to typical experimental imperfections such as limited imaging resolution, inhomogeneous trapping potential, and disorder in the system. The study of topological phases of matter, such as topological band insulators and superconductors, has attracted a lot of interest in recent years [1][2][3]. Various topological phases have been found associated with the free-fermion band theory and classified into a periodic table according to the system symmetry and dimensionality [4][5][6]. The topology of the band structure is characterized by a topological invariant taking only integer values, which gives the most direct and unambiguous signal of the corresponding topological order. To experimentally probe the topological order, it is desirable to have a way to measure the underlying topological invariant. For some phase, the topological invariant may manifest itself through certain quantized transport property or characteristic edge state behavior [7]. For instance, the quantized Hall conductivity is proportional to the underlying topological Chern number that characterizes the integer quantum Hall states [7][8][9]. For many other topological phases in the periodic table, it is not clear yet how to experimentally extract information of the underlying topological invariants.Cold atoms in optical lattices provide a powerful experimental platform to simulate various quantum states of matter. In particular, recent experimental advance in engineering of spin-orbit coupling and artificial gauge field for cold atoms [10][11][12][13][14][15] has pushed this system to the forefront for realization of various topological quantum phases [16][17][18][19][20][21][22]. The detection method for cold-atom experiments is usually quite different from those for conventional solid-state materials. A number of intriguing proposals have been made for detection of certain topological order in cold-atom experiments, such as those based on the dynamic response [23][24][25][26], the Bragg spectroscopy [27,28], imaging of the edge states [29], counting peaks in the momentum distribution [30] or detection of the Berry phase or curvature [1,26,[32][33][34][35][36][37][38]. Most of these proposals are targeted to detection of the quantum Hall * dldeng@umich.edu phase. Similar to solid-state systems, it is not clear yet how to probe the topo...

Besides the conventional bosons and fermions, in synthetic two-dimensional (2D) materials there could exist more exotic quasi-particles with non-abelian statistics, meaning that the quantum states in the system will be transformed by non-commuting unitary operators when we adiabatically braid the particles one around another. Here, we propose an experimental scheme to observe non-abelian statistics with cold atoms in a 2D optical lattice. We show that the Majorana-Schockley modes associated with line defects can be braided with non-abelian statistics through adiabatic shift of the local potentials. We also demonstrate that the braiding operations are robust against typical experimental imperfections and the readout of topological qubits can be accomplished by local measurement of the atom number.