We explore the non-equilibrium response of Chern insulators. Focusing on the Haldane model, we study the dynamics induced by quantum quenches between topological and non-topological phases. A notable feature is that the Chern number, calculated for an infinite system, is unchanged under the dynamics following such a quench. However, in finite geometries, the initial and final Hamiltonians are distinguished by the presence or absence of edge modes. We study the edge excitations and describe their impact on the experimentally-observable edge currents and magnetization. We show that, following a quantum quench, the edge currents relax towards new equilibrium values, and that there is light-cone spreading of the currents into the interior of the sample.PACS numbers: 03.65. Vf, 73.43.Nq, 71.10.Fd Topological phases of matter display many striking features, ranging from the precise quantization of macroscopic properties, to the emergence of fractional excitations and gapless edge states. An important class of topological systems is provided by the so-called Chern insulators realized in two-dimensional settings [1]. A famous example is the Haldane model [2], which describes spinless fermions hopping on a honeycomb lattice. The Haldane model exhibits both topological and non-topological phases, and its behavior is closely related to the integer quantum Hall effect. Recent advances using ultra cold atoms [3][4][5][6][7] have led to the experimental realization of the Haldane model [8]. Proposals also exist for realizing other states of topological matter using cold atoms [9].A fundamental characteristic of topological systems is their robustness to local perturbations, making them ideal candidates for applications in metrology and quantum computation. However, much less is known about their dynamical response to global perturbations and time-dependent driving. This issue is of relevance in a variety of contexts, ranging from the time-evolution and controlled manipulation of prepared topological states, to the dynamics of topological systems coupled to their environment. Understanding the impact of topology on the out of equilibrium response is crucial for further developments, and is the motivation for this present work. For recent progress in this direction see Refs. [10][11][12][13][14][15][16][17].In this manuscript we investigate the non-equilibrium dynamics of the paradigmatic Haldane model. In particular, we consider quantum quenches and sweeps between topological and non-topological phases. Key questions that we will address include: What happens to the topological properties on transiting between different phases? What happens to the edge excitations following a quantum quench? How do the topological characteristics influence the non-equilibrium dynamics?Model.-The Haldane model describes spinless fermions hopping on a honeycomb lattice with both nearest and next nearest neighbor hopping parameters. The Hamiltonian is given by [2]where the fermionic operators obey the anticommutation relations {ĉ j ,ĉ † j } = ...

We investigate the transport properties of Chern insulators following a quantum quench between topological and non-topological phases. Recent works have shown that this yields an excited state for which the Chern number is preserved under unitary evolution. However, this does not imply the preservation of other physical observables, as we stressed in our previous work. Here we provide an analysis of the Hall response following a quantum quench in an isolated system, with explicit results for the Haldane model. We show that the Hall conductance is no longer related to the Chern number in the post-quench state, in agreement with previous work. We also examine the dynamics of the edge currents in finite-size systems with open boundary conditions along one direction. We show that the late-time behavior is captured by a Generalized Gibbs Ensemble, after multiple traversals of the sample. We discuss the effects of generic open boundary conditions and confinement potentials.

Topological states of matter exhibit many novel properties due to the presence of robust topological invariants such as the Chern index. These global characteristics pertain to the system as a whole and are not locally defined. However, local topological markers can distinguish between topological phases, and they can vary in space. In equilibrium, we show that the topological marker can be used to extract the critical behavior of topological phase transitions. Out of equilibrium, we show that the topological marker spreads via a flow of currents, with a bounded maximum propagation speed. We discuss the possibilities for measuring the topological marker and its flow in experiment.Topological quantum systems exhibit many striking phenomena due to the inherent topological properties of their ground state wavefunctions. Experimental signatures in two-dimensions include the robust quantization of the transverse charge and spin transport, with direct links to topological invariants [1]. The observation of the Quantum Hall Effect in graphene highlights that topology can be relevant at room temperature [2], widening the scope for practical applications. The recent discovery of topological insulators [3-5] extends the reach of topology to a wider class of materials and dimensionalities, giving rise to exotic phases such as topological superconductors [6]. Discoveries of topological phases in photonic systems [7][8][9][10][11] and cold atomic gases [12][13][14][15][16][17][18][19][20][21] have expanded the range of experimental probes and measurement techniques, providing access to a much broader range of physical observables. These diverse systems could also play an important role in fault tolerant quantum information processing [22]. arXiv:1808.10463v1 [cond-mat.str-el]

A quantum computer needs the assistance of a classical algorithm to detect and identify errors that affect encoded quantum information. At this interface of classical and quantum computing the technique of machine learning has appeared as a way to tailor such an algorithm to the specific error processes of an experiment-without the need for a priori knowledge of the error model. Here, we apply this technique to topological color codes. We demonstrate that a recurrent neural network with long short-term memory cells can be trained to reduce the error rate ò L of the encoded logical qubit to values much below the error rate ò phys of the physical qubits-fitting the expected power law scaling   µ + ( ) d L phys 1 2 , with d the code distance. The neural network incorporates the information from 'flag qubits' to avoid reduction in the effective code distance caused by the circuit. As a test, we apply the neural network decoder to a density-matrix based simulation of a superconducting quantum computer, demonstrating that the logical qubit has a longer life-time than the constituting physical qubits with near-term experimental parameters.

The continuous effort towards topological quantum devices calls for an efficient and non-invasive method to assess the conformity of components in different topological phases. Here, we show that machine learning paves the way towards noninvasive topological quality control. To do so, we use a local topological marker, able to discriminate between topological phases of one-dimensional wires. The direct observation of this marker in solid state systems is challenging, but we show that an artificial neural network can learn to approximate it from the experimentally accessible local density of states. Our method distinguishes different non-trivial phases, even for systems where direct transport measurements are not available and for composite systems. This new approach could find significant use in experiments, ranging from the study of novel topological materials to high-throughput automated material design.