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arXiv:1402.3222 · New Journal of Physics

Detection of Chern numbers and entanglement in topological two-species systems through subsystem winding numbers

James de Lisle, Suvabrata De, Emilio Alba, Alex Bullivant, Juan J Garcia-Ripoll, Ville Lahtinen, Jiannis K. Pachos

Published Aug 8, 2014• 22 citations
arXivHTMLPDF

Abstract

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 time-of-flight images can give direct access to the relevant observables.

DOI
10.1088/1367-2630/16/8/083022

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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 time-of-flight images can give direct access to the relevant observables.
In part I, we describe a protected qubit which is realized in a two-dimensional array of Josephson junctions. Our construction is the magnetic analogue of (‘dual’ to) a suggestion of a superconducting current mirror qubit (Kitaev,2006b). Our proposal therefore inherits the intrinsic fault-tolerance of the current mirror qubit, but may perform better than it in the laboratory, since magnetic \nnoise is generally less of a problem than electric noise. We adapt the scheme for universal fault-tolerant quantum computation proposed by Kitaev to our construction. \n \nIn part II, we describe a method of detecting the Chern number and entanglement properties of topological four-component free-fermion systems in cold atom experiments. We show that the Chern number of these systems decomposes into a sum of subsystem winding numbers which can be measured from time-of-flight images. Such images also enable the degree of subsystem entanglement in, and the component entanglement spectra of, these systems to be measured. The method is applied to the quantum spin-Hall insulator and \na staggered topological superconductor. We find that the phase diagrams are accurately reproduced, except when the subsystems are highly entangled.
We present a novel technique for detecting topological invariants ‐ Chern numbers ‐ from time-of-flight images of ultra-cold atoms. We show that the Chern numbers of integer quantum Hall states of lattice fermions leave their fingerprints as oscillations in the atoms’ momentum distribution. The number of oscillations within the lowest Brillouin zone is simply equal to the Chern number. The ubiquitous signature is strikingly enhanced for anisotropic hopping, where the momentum distribution assumes a sinusoidal form with period determined by the Chern number. Thus, an everyday observable in cold atom experiments can serve as a useful tool to characterize and visualize quantum states with non-trivial topology. Topological insulators (TI) and superfluids are many-body quantum systems with energy gaps in the bulk but topologically protected, gapless excitations on the boundary [1, 2]. Each class of topological phases is characterized by a topological invariant which can be either an integer or a binary quantity. For example, the integer quantum Hall insulators [3] are distinguished from other two-dimensional insulators with the same symmetry by the Chern number, an integer that capture the global properties of all occupied bands [4]. The Chern number coincides with the number of chiral edge modes, and the Hall conductance in units of the conductance quantum. A central problem in exploring new forms of topological matter is how to extract the topological invariants from experiments, and unambiguously determine the topological phase. Experiments with ultracold atoms has brought us opportunities to creating topological phases of matter in parameter regimes unreachable in solids. In addition to band-structure engineering using optical lattices and tuning the atom-atom interaction by Feshbach resonance, artificial magnetic fields for neutral atoms have been produced by either rotation [5] or atom-laser coupling [6, 7], making way to realizing quantum Hall states of cold atoms [8]. Moreover, non-Abelian synthetic gauge fields [7, 9] and spin-orbit coupling [10] have been demonstrated. These developments motivated theories on properties of ultracold topological matter, e.g., the fractional quantum Hall effect of interacting bosons [5, 11, 12], the quantum spin Hall effect of fermions [9], and the quantum anomalous Hall states ofp-orbital fermions [13]. While cold atom systems offer new detection techniques and great tunability, it is challenging to measure the topological invariants directly. Measurements of local density of states (of the edge states) or mass transport are marred with complexity in contrast to solid state systems. Thus, creative methods have to be invented. For example, Ref. [8] proposed to extract the quantized Hall conductance from the derivatives
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 two-dimensional anomalous quantum Hall states and the Chern-Simons term for the three-dimensional chiral topological insulators as examples, we show that the proposed detection method is practical, and robust to typical experimental imperfections such as limited imaging resolution, inhomogeneous trapping potential, and disorder in the system.
We present a technique for detecting topological invariants---Chern numbers---from time-of-flight images of ultracold atoms. We show that the Chern numbers of integer quantum Hall states of lattice fermions leave their fingerprints in the atoms' momentum distribution. We analytically demonstrate that the number of local maxima in the momentum distribution is equal to the Chern number in two limiting cases, for large hopping anisotropy and in the continuum limit. In addition, our numerical simulations beyond these two limits show that these local maxima persist for a range of parameters. Thus, an everyday observable in cold atom experiments can serve as a useful tool to characterize and visualize quantum states with nontrivial topology.
We propose an experimental technique for classifying the topology of band structures realized in optical lattices, based on a generalization of topological charge pumping in quantum Hall systems to cold atoms in optical lattices. Time-of-flight measurement along one spatial direction combined with in situ detection along the transverse direction provides a direct measure of the system's Chern number, as we illustrate by calculations for the Hofstadter lattice. Based on an analogy with Wannier function techniques of topological band theory, the method is very general and also allows the measurement of other topological invariants, such as the Z(2) topological invariant of time-reversal symmetric insulators.
Ultracold atoms in optical lattices form a clean quantum simulator platform which can be utilized to examine topological phenomena and test exotic topological materials. Here we propose an experimental scheme to measure the Chern numbers of two-dimensional multiband topological insulators with bosonic atoms. We show how to extract the topological invariants out of a sequence of time-of-flight images by applying a phase-retrieval algorithm to matter waves. We illustrate advantages of using bosonic atoms as well as efficiency and robustness of the method with two prominent examples: the Harper-Hofstadter model with an arbitrary commensurate magnetic flux and the Haldane model on a brick-wall lattice.