Helical Liquid Research Articles

Synthetic helical liquid in a quantum wire

We show that the combination of a Dresselhaus interaction and a spatially periodic Rashba interaction leads to the formation of a helical liquid in a quantum wire when the electron-electron interaction is weakly screened. The effect is sustained by a helicity-dependent effective band gap which depends on the size of the Dresselhaus and Rashba spin-orbit couplings. We propose a design for a semiconductor device in which the helical liquid can be realized and probed experimentally.

Signatures of Majorana bound states in transport properties of hybrid structures based on helical liquids

Majorana bound states can emerge as zero-energy modes at the edge of a two-dimensional topological insulator in proximity to an ordinary s-wave superconductor. The presence of an additional ferromagnetic domain close to the superconductor can lead to their localization. We consider both N-S and S-N-S junctions based on helical liquids and study their spectral properties for arbitrary ferromagnetic scatterers in the normal region. Thereby, we explicitly compute Andreev wave-functions at zero energy. We show under which conditions these states form localized Majorana bound states in N-S and S-N-S junctions. Interestingly, we can identify Majorana-specific signatures in the transport properties of N-S junctions and the Andreev bound levels of S-N-S junctions that are robust against external perturbations. We illustrate these findings with the example of a ferromagnetic double barrier (i.e. a quantum dot) close to the N-S boundaries.

Nonequilibrium spectroscopy of topological edge liquids

We develop a theory for energy and spatially resolved tunneling spectroscopy of topological quantum spin Hall helical states driven out of equilibrium. When a helical liquid is constrained between two superconducting reservoirs transport at the edge is governed by multiple Andreev reflections. The resulting quasiparticle distribution functions of the edge channels exhibit multiple discontinuities at subgap energies with the periodicity of an applied voltage. The combined effect of interactions, disorder, and normal scattering off the superconducting interface leads to the inelastic processes mixing different helicity modes, thus causing smearing of these singularities. If equilibration is strong, then the distribution functions of the edge channels tend to collapse into a Fermi-like function with an effective temperature determined by the superconducting gap, applied voltage, and intraedge interaction parameter. We conclude that mapping out nonequilibrium distribution functions may help to quantify the relative importance of various relevant perturbations that spoil ideally ballistic edge transport.

Edge physics of the quantum spin Hall insulator from a quantum dot excited by optical absorption.

The gapless edge modes of the quantum spin Hall insulator form a helical liquid in which the direction of motion along the edge is determined by the spin orientation of the electrons. In order to probe the Luttinger liquid physics of these edge states and their interaction with a magnetic (Kondo) impurity, we consider a setup where the helical liquid is tunnel coupled to a semiconductor quantum dot that is excited by optical absorption, thereby inducing an effective quantum quench of the tunneling. At low energy, the absorption spectrum is dominated by a power-law singularity. The corresponding exponent is directly related to the interaction strength (Luttinger parameter) and can be computed exactly using boundary conformal field theory thanks to the unique nature of the quantum spin Hall edge.

Controlled Helicity of the Rigid-Flexible Molecular Assembly Triggered by Water Addition: From Nanocrystal to Liquid Crystal Gel and Aqueous Nanofibers

Despite recent advances in synthetic nanometer-scale helical assembly, control of supramolecular chirality remains a challenge. Here, we describe the fine-tuning of the shape and morphology transitions of twisted and helical assembly from dumbbell-shaped rigid-flexible amphiphile triggered by concentration. The amphiphile 2 self-assembles into nonchiral 3D columnar crystals with alternative packing of aromatic segment in solid state. Remarkably, with the addition of water into the solid, the achiral crystal transforms into 2D hexagonally ordered liquid crystal gel with supramolecular chirality due to increased entropy of flexible coil in aqueous environment. Notably, the helical liquid crystal gel was observed to dissolve into optically active aqueous nanofibers caused by a conformational change of hydrophobic aromatic rods and enhanced hydro-volume of the ethylene oxide chains.

Charge Generation Measured for Fullerene–Helical Nanofilament Liquid Crystal Heterojunctions

The helical nanofilament (HNF) liquid crystal phase is an ordered architecture exhibiting interesting properties for charge transport. It is a small molecule self-assembly of stacked and twisted crystalline layers, which form alignable organic nanorods with half the surface area of the filaments consisting of aromatic sublayer edges. HNFs mixed with an electron acceptor generate an intriguing network for photoinduced electron transfer (PET). In this work, we characterize the structure of the HNF phase as processed into thin films with transmission electron microscopy (TEM) and X-ray diffraction (XRD). Additionally, we measure the flash-photolysis time-resolved microwave conductivity (TRMC) in samples where the HNF phase is fabricated into heterojunctions with the fullerenes C60 and PC60BM, prototypical electron acceptors for organic photovoltaics. Two distinct microstructures of the thin films were identified and compared for PET. A near-unity charge generation yield is observed in a bilayer of HNFs with C60. Moreover, the HNF phase is shown to be 10× better at charge generation than a lamellar structuring of the same components. Thus, the HNF phase is shown to be a good charge-generation interface.

Magnetic ac control of the spin textures in a helical Luttinger liquid

We demonstrate the possibility to induce and control peculiar spin textures in a helical Luttinger liquid, by means of a time-dependent magnetic scatterer. The presence of a perturbation that breaks the time-reversal symmetry opens a gap in the spectrum, inducing single-particle backscattering and a peculiar spin response. We show that in the weak backscattering regime asymmetric spin textures emerge at the left and right sides of the scatterer, whose spatial oscillations are controlled by the ratio between the magnetization frequency and the Fermi energy and by the electron interaction. This peculiar spin response marks a strong difference between helical and nonhelical liquids, which are expected to produce symmetric spin textures even in the ac regime.

Tunneling between a topological superconductor and a Luttinger liquid

We study the quantum point contact between the topological superconductor and the helical Luttinger liquid. The effects of the electron-electron interactions in the helical Luttinger liquid on the low-energy physics of this system are analyzed by the renormalization group. Among the various couplings at the point contact which arise from the tunneling via the Majorana edge channel, the induced backscattering in the helical Luttinger liquid is the most relevant for repulsive interactions. Hence, at low temperatures, the helical Luttinger liquid is effectively cut into two separated half wires. As a result, the low-temperature physics is described by a fixed point consisting of two leads coupled to the topological superconductor, and the electrical transport properties through the point contact at low temperature and low bias are dominated by the tunneling via the Majorana edge channel. We compute the temperature dependence of the zero-bias tunneling conductance and study the full counting statistics for the tunneling current at zero temperature.

Complex superstructures in chiral liquid crystals: surface-induced helix destruction.

The influence of surface anchoring in thin chiral ferroelectric liquid crystals on helical superstructures imposed onto smectic layers is studied by means of a simple model allowing surface anchoring through the self-depolarization effect. It is shown that, for narrow temperature ranges close to temperatures of tilting phase transitions, these interactions can destroy helices, leading to the emergence of complex, intertwined undulatory, and helical substructures with different, mainly short, pitches. Regions with both undulated and helical orderings are demonstrated to reveal different sizes and random distributions along the smectic layer normal. Assuming values of material model parameters, typical for liquid crystals being close to tilting phase transitions, the free energy of thin helical smectic liquid crystal systems is determined as a function of two variables. One of them specifies the helix period of the appropriate bulk system (unperturbed by surfaces), while the second describes the interplay between the surface anchoring and the liquid crystal elasticity. It is shown that the free energy exhibits a very complicated behavior within variable regions for which the regular helices or complex superstructures occur. The resulting sensitivity of the free energy on changes of these variables corresponds to the appearance of a large variety of complex superstructures with coexisting helices of different periods. This gives an explanation of the occurrence of intricate superstructures and provides new insight into the interplay between molecular and surface interactions near the tilting phase transitions.

Fractional helical liquids in quantum wires

We study one-dimensional wires with spin-orbit coupling. We show that in the presence of Zeeman field and strong electron-electron interaction a clean wire may form fractional helical liquid states with phenomenology similar to fractional quantum Hall liquids. Most notably, the wire's two-terminal conductance is predicted to show fractional quantized conductance plateaus at low electron density. When the system is proximity coupled to a superconductor, fractional Majorana bound states may be stabilized, giving rise to a one-dimensional analog of non-Abelian anyons. We discuss how disorder destabilizes these fractional phases. Possible experimental realizations of similar states in double wire systems are discussed.

Direct proportionality between the Kondo cloud and current cross correlations in helical liquids

The Kondo cloud, represented by the correlations between the magnetic moment and the spin density in the leads of a Kondo setup, is until now eluding its observation. We exploit the unique coupling of spin and direction of motion of the recently discovered helical liquids in a setup with two leads to establish a proportionality between the Kondo cloud and the time resolved current cross correlations. This relation holds around a specific choice of model parameters. Thereby, we provide a new and direct way to detect the Kondo cloud.

Hanbury Brown and Twiss correlations of Cooper pairs in helical liquids

We propose a Hanbury Brown and Twiss (HBT) experiment of Cooper pairs on the edge channels of quantum spin Hall insulators. The helical edge channels provide a well-defined beam of Cooper pairs and perfect Andreev reflections from superconductors. This allows our setup to be identical in spirit to the original HBT experiment. Interestingly, the cross correlation is always negative and provides no hint of the bosonic nature of Cooper pairs. This counterintuitive result is attributed to the perfect Andreev reflection and the true beam splitter in the setup.

Structure factor of interacting one-dimensional helical systems

We calculate the dynamical structure factor $S(q,\ensuremath{\omega})$ of a weakly interacting helical edge state in the presence of a magnetic field $B$. The latter opens a gap of width $2B$ in the single-particle spectrum, which becomes strongly nonlinear near the Dirac point. For chemical potentials $|\ensuremath{\mu}|>B$, the system then behaves as a nonlinear helical Luttinger liquid, and a mobile-impurity analysis reveals power-law singularities in $S(q,\ensuremath{\omega})$ which depend on the interaction strength as well as on the spin texture of the edge states. For $|\ensuremath{\mu}|<B$, the low-energy excitations are gapped, and we determine $S(q,\ensuremath{\omega})$ by using an analogy to exciton physics.

Low-energy properties of fractional helical Luttinger liquids

We investigate the low-energy properties of (quasi) helical and fractional helical Luttinger liquids. In particular, we calculate the Drude peak of the optical conductivity, the density of states, as well as charge transport properties of the interacting system with and without attached Fermi liquid leads at small and large (compared to the gap) frequencies. For fractional wires, we find that the low-energy tunneling density of states vanishes. The conductance of a fractional helical Luttinger liquid is noninteger. It is independent of the Luttinger parameters in the wire, despite the intricate mixing of charge and spin degrees of freedom, and only depends on the relative locking of charge and spin degrees of freedom.

Transport properties of the Coulomb–Majorana junction

We provide a comprehensive theoretical description of low-energy quantum transport for a Coulomb–Majorana junction, where several helical Luttinger liquid nanowires are coupled to a joint mesoscopic superconductor with finite charging energy. Including the Majorana bound states formed near the ends of superconducting wire parts, we derive and analyze the Keldysh phase action describing non-equilibrium charge transport properties of the junction. The low-energy physics corresponds to a two-channel Kondo model with symmetry group SO(M), where M is the number of leads connected to the superconductor. Transport observables, such as the conductance tensor or current noise correlations, display non-trivial temperature or voltage dependences reflecting non-Fermi liquid behavior.

Back Cover: Spin textures of strongly correlated spin Hall quantum dots (Phys. Status Solidi RRL 12/2013)

When quantum confinement is applied to topological phases of matter, very interesting physics can emerge. Majorana fermions at the boundaries of topological superconductors are the most striking example. Another interesting system is a quantum dot defined into the edges of two-dimensional topological insulators. These edges represent a new state of matter, the helical Luttinger liquid, characterized by spin-momentum locking. In their work, Dolcetto et al. inspect the spin resolved charge density and the spin resolved density–density correlation functions of such quantum dots (see the Letter on pp. 1059–1063). The interplay between helicity and quantum confinement induced by magnetic barriers leads to strong oscillations in the spin resolved density even when Coulomb interaction is weak. The main finding is that when Coulomb interaction is strong, a peculiar spin texture emerges as signalled by the spin-resolved correlation functions. This crystal of spins is the counterpart of the charge Wigner crystal occurring in ordinary one-dimensional quantum dots.

Helical liquid of snake states

We derive an exact solution to the problem of spin snake states induced in a nonhomogeneous magnetic field by a combined action of the Rashba spin-orbit and Zeeman fields. In an antisymmetric magnetic field the spin snake states are nonlocal composite particles, originating from spatially separated entangled spins. Adding an external homogeneous magnetic field breaks the spin-parity symmetry gapping out the spectral branches, which results in a regular beating pattern of the spin current. These new phenomena in a helical liquid of snake states are proposed for an experimental realization.

Point contacts and localization in generic helical liquids

We consider two helical liquids on opposite edges of a narrow two-dimensional topological insulator, which are connected by one or several local tunnel junctions. In the presence of spatially inhomogeneous Rashba spin-orbit coupling, the spin textures of the helical states on opposite edges are different. We demonstrate that this has a strong impact on the electron transport between the edges. In particular, in the case of many random tunnel contacts, the localization length depends strongly on the spin textures of the edge states.

Nonequilibrium transport of helical Luttinger liquids through a quantum dot

We study a steady state non-equilibrium transport between two interacting helical edge states of a two dimensional topological insulator, described by helical Luttinger liquids, through a quantum dot. For non-interacting dot the current is obtained analytically by including the self-energy correction to the dot Green's function. For interacting dot we use equation of motion method to study the influence of weak on-site Coulomb interaction on the transport. We find the metal-to-insulator quantum phase transition for attractive or repulsive interactions in the leads when the magnitude of the interaction strength characterized by a charge sector Luttinger parameter $K$ goes beyond a critical value. The critical Luttinger parameter $K_{cr}$ depends on the hoping strength between dot and the leads as well as the energy level of the dot with respect to the Fermi levels of the leads, ranging from weak interaction regime for dot level off resonance to strong interaction regime for dot in resonance with the equilibrium Fermi level. Nearby the transition various singular behaviors of current noise, dot density of state, and the decoherence rate (inverse of lifetime) of the dot are briefly discussed.

Magnetic impurities in the Kane-Mele model

The realization of the spin-Hall effect in quantum wells has led to a plethora of studies regarding the properties of the edge states of a 2D topological insulator. These edge states constitute a class of one-dimensional liquids, called the helical liquid, where an electron's spin quantization axis is tied to its momentum. In contrast to one dimensional conductors, magnetic impurities - below the Kondo temperature - cannot block transport and one expects the current to circumvent the impurity. To study this phenomenon, we consider the single impurity Anderson model embedded into an edge of a Kane-Mele ribbon with up to 512x80 sites and use the numerically exact continuous time QMC method to study the Kondo effect. We present results on the temperature dependence of the spectral properties of the impurity and the bulk system that show the behaviour of the system in the various regimes of the Anderson model. A view complementary to the single particle spectral functions can be obtained using the spatial behaviour of the spin-spin correlation functions. Here we show the characteristic, algebraic decay in the edge channel near the impurity.