Gil Refael

Helical Liquids and Majorana Bound States in Quantum Wires

We show that the combination of spin-orbit coupling with a Zeeman field or strong interactions may lead to the formation of a helical electron liquid in single-channel quantum wires, with spin and velocity perfectly correlated. We argue that zero-energy Majorana bound states are formed in various situations when such wires are situated in proximity to a conventional s-wave superconductor. This occurs when the external magnetic field, the superconducting gap, or, most simply, the chemical potential vary along the wire. These Majorana states do not require the presence of a vortex in the system. Experimental consequences of the helical liquid and the Majorana states are also discussed.

Non-Abelian statistics and topological quantum information processing in 1D wire networks

The synthesis of a quantum computer remains an ongoing challenge in modern physics. Whereas decoherence stymies most approaches, topological quantum computation schemes evade decoherence at the hardware level by storing quantum information non-locally. Here we establish that a key operation—braiding of non-Abelian anyons—can be implemented using one-dimensional semiconducting wires. Such wires can be driven into a topological phase supporting long-sought particles known as Majorana fermions that can encode topological qubits. We show that in wire networks, Majorana fermions can be meaningfully braided by simply adjusting gate voltages, and that they exhibit non-Abelian statistics like vortices in a p+ip superconductor. We propose experimental set-ups that enable probing of the Majorana fusion rules and the efficient exchange of arbitrary numbers of Majorana fermions. This work should open a new direction in topological quantum computation that benefits from physical transparency and experimental feasibility.

Floquet topological insulator in semiconductor quantum wells

Topological phases of matter have captured our imagination over the past few years, with tantalizing properties such as robust edge modes and exotic non-Abelian excitations, and potential applications ranging from semiconductor spintronics to topological quantum computation. Despite recent advancements in the field, our ability to control topological transitions remains limited, and usually requires changing material or structural properties. We show, using Floquet theory, that a topological state can be induced in a semiconductor quantum well, initially in the trivial phase. This can be achieved by irradiation with microwave frequencies, without changing the well structure, closing the gap and crossing the phase transition. We show that the quasi-energy spectrum exhibits a single pair of helical edge states. We discuss the necessary experimental parameters for our proposal. This proposal provides an example and a proof of principle of a new non-equilibrium topological state, the Floquet topological insulator, introduced in this paper.

Entanglement Entropy of Random Quantum Critical Points in One Dimension

For quantum critical spin chains without disorder, it is known that the entanglement of a segment of N>>1 spins with the remainder is logarithmic in N with a prefactor fixed by the central charge of the associated conformal field theory. We show that for a class of strongly random quantum spin chains, the same logarithmic scaling holds for mean entanglement at criticality and defines a critical entropy equivalent to central charge in the pure case. This effective central charge is obtained for Heisenberg, XX, and quantum Ising chains using an analytic real-space renormalization-group approach believed to be asymptotically exact. For these random chains, the effective universal central charge is characteristic of a universality class and is consistent with a c-theorem.

Fractionalizing Majorana fermions: non-Abelian Statistics on the Edges of Abelian Quantum Hall States

We study the non-abelian statistics characterizing systems where counter-propagating gapless modes on the edges of fractional quantum Hall states are gapped by proximity-coupling to superconductors and ferromagnets. The most transparent example is that of a fractional quantum spin Hall state, in which electrons of one spin direction occupy a fractional quantum Hall state of

Adiabatic manipulations of Majorana fermions in a three-dimensional network of quantum wires

\nu= 1/m

Topological polaritons and excitons in garden-variety systems

, while electrons of the opposite spin occupy a similar state with

Anomalous Floquet-Anderson Insulator as a Nonadiabatic Quantized Charge Pump

\nu = -1/m

Hilbert-Glass Transition: New Universality of Temperature-Tuned Many-Body Dynamical Quantum Criticality

. However, we also propose other examples of such systems, which are easier to realize experimentally. We find that each interface between a region on the edge coupled to a superconductor and a region coupled to a ferromagnet corresponds to a non-abelian anyon of quantum dimension

Disorder-induced Floquet topological insulators.

\sqrt{2m}