In this Letter, we provide a general methodology to directly measure topological order in cold atom systems. As an application, we propose the realization of a characteristic topological model, introduced by Haldane, using optical lattices loaded with fermionic atoms in two internal states. We demonstrate that time-of-flight measurements directly reveal the topological order of the system in the form of momentumspace Skyrmions. DOI: 10.1103/PhysRevLett.107.235301 PACS numbers: 67.85.Àd, 03.65.Vf Different phases of matter can be distinguished by their symmetries. This information is usually captured by locally measurable order parameters that summarize the essential properties of the phase. Topological insulators are materials with symmetries that depend on the topology of the energy eigenstates of the system [1]. These materials are of interest because they give rise to robust spin transport effects with potential applications ranging from sensitive detectors to quantum computation [2,3]. However, direct observation and measurement of topological order has been up to now impossible due to its nonlocal character. Instead, experiments have relied so far on indirect manifestations of this order, such as edge states and the quantization of conductivity.Ultracold atoms facilitate the implementation of artificial gauge fields [4]. Here, we distinguish proposals that generate continuous fields [5], such as the recent experiment by Lin et al. [6], from those that rely on optical lattices and engineering of hopping [7]. We will concentrate on the latter, introducing a method based on standard time-of-flight (TOF) measurements that can identify a topological character in the quantum state of the system. Our starting point is a possible implementation of Haldane's model using fermionic atoms in two internal states. The topological nature of its ground state is witnessed by the Chern number. This number counts the times the ground state, written as a spinor, wraps around the sphere, as a function of momentum. We demonstrate that TOF measurements reconstruct the Chern number in a way which is robust against the presence of external perturbations or state preparation. Our method can be adapted to other quantum simulations of topological order in optical lattices [8][9][10][11][12][13][14][15][16][17], as many already use internal degrees of freedom of the atoms to encode the order.One common mechanism for the appearance of topological order is based on the topology of the eigenstate manifolds. Consider a real-space lattice whose unit cell has d quantum degrees of freedom-position of the particle, spin, etc. In the case of the quantum Hall effect, the energy bands are separated from each other and the material becomes an insulator for appropriate Fermi energies, E F . In a real setup, with finite boundaries, the sample can have a quantized nonzero conductivity given by the topological invariant m xy ¼ e 2 =h P E m

The Chern number, ν, as a topological invariant that identifies the winding of the ground state in the particle-hole space, is a definitive theoretical signature that determines whether a given superconducting system can support Majorana zero modes. Here we show that such a winding can be faithfully identified for any superconducting system (p-wave or s-wave with spin-orbit coupling) through a set of time-of-flight measurements, making it a diagnostic tool also in actual cold atom experiments. As an application, we specialize the measurement scheme for a chiral topological model of spinless fermions. The proposed model only requires the experimentally accessible s-wave pairing and staggered tunnelling that mimics spin-orbit coupling. By adiabatically connecting this model to Kitaev's honeycomb lattice model, we show that it gives rise to ν = ±1 phases, where vortices bind Majorana fermions, and ν = ±2 phases that emerge as the unique collective state of such vortices. Hence, the preparation of these phases and the detection of their Chern numbers provide an unambiguous signature for the presence of Majorana modes. Finally, we demonstrate that our detection procedure is resilient against most inaccuracies in experimental control parameters as well as finite temperature.

We analyze a tight-binding model of ultracold fermions loaded in an optical square lattice and subjected to a synthetic non-Abelian gauge potential featuring both a magnetic field and a translationally invariant SU(2) term. We consider in particular the effect of broken time-reversal symmetry and its role in driving non-trivial topological phase transitions. By varying the spin-orbit coupling parameters, we find both a semimetal/insulator phase transition and a topological phase transition between insulating phases with different numbers of edge states. The spin is not a conserved quantity of the system and the topological phase transitions can be detected by analyzing its polarization in time of flight images, providing a clear diagnostic for the characterization of the topological phases through the partial entanglement between spin and lattice degrees of freedom.

Topological invariants, such as the Chern number, characterize topological phases of matter. Here we provide a method to detect Chern numbers in systems with two distinct species of fermion, such as spins, orbitals or several atomic states. We analytically show that the Chern number can be decomposed as a sum of component specific winding numbers, which are themselves physically observable. We apply this method to two systems, the quantum spin Hall insulator and a staggered topological superconductor, and show that (spin) Chern numbers are accurately reproduced. The measurements required for constructing the component winding numbers also enable one to probe the entanglement spectrum with respect to component partitions. Our method is particularly suited to experiments with cold atoms in optical lattices where timeof-flight images can give direct access to the relevant observables.

In this work we present an optical lattice setup to realize a full Dirac Hamiltonian in 2+1 dimensions. We show how all possible external potentials coupled to the Dirac field can arise from perturbations of the existing couplings of the honeycomb lattice pattern. This greatly simplifies the proposed implementations, requiring only spatial modulations of the intensity of the laser beams to induce complex non-Abelian potentials. We finally suggest several experiments to observe the properties of the quantum field theory in the setup.