Direct probe of topological order for cold atoms

Abstract

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...

Topological metal bands with double-triple-point fermions in optical lattices

“…The Berry phase of a 2D DTP can be directly measured using an interferometric approach in momentum space [38]. Moreover, it has been demonstrated that the full tomography of Bloch states (vectors) can be achieved with cold atoms in optical lattices to reveal the band topology [72][73][74][75][76][77], which would be applicable in our proposed system.…”

Probe Knots and Hopf Insulators with Ultracold Atoms

“…In experiment, one discretizes the BZ and a pixelized version ofŜ(k) can be obtained through time-of-flight imaging. Refs [61,62] describe in detail how to measureŜ(k) in experiment. One astute observation [62] was that the spin component is related to the density distributions aŝ S z (k) = [n ↑ (k) − n ↓ (k)]/[n ↑ (k) + n ↓ (k)].…”

“…For the simplest case of p = q = 1, we numerically diagonalize the realistic real-space Hamiltonian and compute the corresponding Hopf index for different h based on the method introduced in Ref. [61]. Our results are summarized in Table I.…”

Topological states with broken translational and time-reversal symmetries in a honeycomb-triangular lattice

“…Due to the periodicity of the momentum space Hamiltonian, the Brillouin zone can be regarded as a two-dimensional torus. Remarkably, the field strengths F 12 (k l ) can also be directly measured by using time-of-flight imaging [57]. We found that the Chern number for the lowest band is c 3 = 2.…”