Gapless Modes Research Articles

Classification of topological insulators and superconductors in the presence of reflection symmetry

We discuss a topological classification of insulators and superconductors in the presence of both (non-spatial) discrete symmetries in the Altland-Zirnbauer classification and spatial reflection symmetry in any spatial dimensions. By using the structure of bulk Dirac Hamiltonians of minimal matrix dimensions and explicit constructions of topological invariants, we provide the complete classification, which still has the same dimensional periodicities with the original Altland-Zirnbauer classification. The classification of reflection-symmetry-protected topological insulators and superconductors depends crucially on the way reflection symmetry operation is realized. When a boundary is introduced, which is reflected into itself, these non-trivial topological insulators and superconductors support gapless modes localized at the boundary.

Goldstone Modes in Bilayer Quantum Hall Systems at ν = 2

At the filling factor ν=2, the bilayer quantum Hall system has three phases, the spin-ferromagnet phase, the spin singlet phase and the canted antiferromagnet (CAF) phase, depending on the relative strength between the Zeeman energy and interlayer tunneling energy. We present a systematic method to derive the effective Hamiltonian for the Goldstone modes in these three phases. We then investigate the dispersion relations and the coherence lengths of the Goldstone modes. To explore a possible emergence of the interlayer phase coherence, we analyze the dispersion relations in the limit of zero tunneling energy. We find one gapless mode with the linear dispersion relation in the CAF phase.

Low lying modes of triplet-condensed neutron matter and their effective theory

The condensation of neutrons into a 3P2 superfluid phase occurs at densities relevant for the interior of neutron stars. The triplet pairing breaks rotational symmetry spontaneously and leads to the existence of gapless modes (angulons) that are relevant for many transport coefficients and to the star's cooling properties. We derive the leading terms of the low energy effective field theory, including the leading coupling to electroweak currents, valid for a variety of possible 3P2 phases.

Non-Abelian quasigapless modes localized on mass vortices in superfluidHe3-B

Kelvin waves or Kelvons have been known for a long time as gapless excitations propagating along superfluid vortices. These modes can be interpreted as the Nambu-Goldstone excitations arising from the spontaneous breaking of the translational symmetry. Recently a different type of gapless excitation localized on strings -- the so-called non-Abelian mode -- attracted much attention in high-energy physics. We discuss their relevance in condensed matter physics. Although we failed to find exactly gapless non-Abelian modes, non-Abelian rotational quasigapless excitations are argued to exist on the mass vortices in the B phase of the superfluid 3He, due to the fact that the order parameter in 3He-B is tensorial. While the U(1) rotational excitations are well established in vortices with asymmetric cores, the non-Abelian rotational excitations belonging to the same family were not considered. In the general case they are coupled with the translational modes.

SimplexZ2spin liquids on the kagome lattice with projected entangled pair states: Spinon and vison coherence lengths, topological entropy, and gapless edge modes

Gapped ${\mathbb{Z}}_{2}$ spin liquids have been proposed as candidates for the ground state of the $S=1/2$ quantum antiferromagnet on the kagome lattice. We extend the use of projected entangled pair states to construct (on the cylinder) resonating valence bond (RVB) states including both nearest-neighbor and next-nearest-neighbor singlet bonds. Our ansatz---dubbed ``simplex spin liquid''---allows for an asymmetry between the two types of triangles (of order $2%--3%$ in the energy density after optimization) leading to the breaking of inversion symmetry. We show that the topological ${\mathbb{Z}}_{2}$ structure is still preserved and, by considering the presence or the absence of spinon and vison lines along an infinite cylinder, we explicitly construct four orthogonal RVB minimally entangled states. The spinon and vison coherence lengths are extracted from a finite size scaling with regard to the cylinder perimeter of the energy splittings of the four sectors and are found to be of the order of the lattice spacing. The entanglement spectrum of a partitioned (infinite) cylinder is found to be gapless, suggesting the occurrence, on a cylinder with real open boundaries, of gapless edge modes formally similar to Luttinger liquid (nonchiral) spin and charge modes. When inversion symmetry is spontaneously broken, the RVB spin liquid exhibits an extra Ising degeneracy, which might have been observed in recent exact diagonalization studies.

Implications of Relativity on Nonrelativistic Goldstone Theorems: Gapped Excitations at Finite Charge Density

We adapt the Goldstone theorem to study spontaneous symmetry breaking in relativistic theories at finite charge density. It is customary to treat systems at finite density via nonrelativistic Hamiltonians. Here, we highlight the importance of the underlying relativistic dynamics. This leads to seemingly new results whenever the charge in question is spontaneously broken and does not commute with other broken charges. We find that that the latter interpolate gapped excitations. In contrast, all existing versions of the Goldstone theorem predict the existence of gapless modes. We derive exact nonperturbative expressions for their gaps, in terms of the chemical potential and of the symmetry algebra.

Anomalous Tunneling of Spin Wave in Heisenberg Ferromagnet

The ferromagnetic spin wave (FSW) in classical Heisenberg chain exhibits the perfect transmission in the long-wavelength limit in the transmission-reflection problem with an inhomogeneity of exchange integral. In the presence of local magnetic field, on the other hand, FSW undergoes the perfect reflection in the long-wavelength limit. This difference in the long-wavelength limit is attributed to the symmetry property of the scatterers; it is crucial whether the potential preserves or breaks the spin rotation symmetry. Our result implies that the anomalous tunneling (i.e., perfect transmission in the low-energy limit) found both in scalar and spinor BECs is not specific to gapless modes in superfluids but is a common property shared with generic Nambu-Goldstone modes in the presence of a symmetry-preserving potential scatterer.

Majorana edge modes at topological insulator-superconductor-ferromagnet junctions in three dimensions

Recently the condition for the existence of gapless modes in arbitrary topological defects is proposed [Jeffrey C. Y. Teo and C. L. Kane, Phys. Rev. B 82, 115120 (2010)]. It is suggested that the existence of gapless modes follows from topologically nontrivial Hamiltonian which varies with adiabatic material-parameters characterizing the defects. We show that such adiabatic approach fails in cases of the existence of Majorana edge modes at topological insulator-superconductor-ferromagnet junctions. It is pointed out that quantum corrections beyond adiabatic quasiclassical approximation are required for the evaluation of second Chern number associated with the line junction.

The study of Goldstone modes in ν = 2 bilayer quantum Hall systems

At the filling factor $\nu$=2, the bilayer quantum Hall system has three phases, the spin-ferromagnet phase, the spin singlet phase and the canted antiferromagnet (CAF) phase, depending on the relative strength between the Zeeman energy and interlayer tunneling energy. We present a systematic method to derive the effective Hamiltonian for the Goldstone modes in these three phases. We then investigate the dispersion relations and the coherence lengths of the Goldstone modes. To explore a possible emergence of the interlayer phase coherence, we analyze the dispersion relations in the zero tunneling energy limit. We find one gapless mode with the linear dispersion relation in the CAF phase.

Non-equilibrium field dynamics of an honest holographic superconductor

Most holographic models of superconducting systems neglect the effects of dynamical boundary gauge fields during the process of spontaneous symmetry-breaking. Usually a global symmetry gets broken. This yields a superfluid, which then is gauged "weakly" afterwards. In this work we build (and probe the dynamics of) a holographic model in which a local boundary symmetry is spontaneously broken instead. We compute two-point functions of dynamical non-Abelian gauge fields in the normal and in the broken phase, and find non-trivial gapless modes. Our AdS3 gravity dual realizes a p-wave superconductor in (1+1) dimensions. The ground state of this model also breaks (1+1)-dimensional parity spontaneously, while the Hamiltonian is parity-invariant. We discuss possible implications of our results for a wider class of holographic liquids.

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 $\nu= 1/m$, while electrons of the opposite spin occupy a similar state with $\nu = -1/m$. 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 $\sqrt{2m}$. We calculate the unitary transformations that are associated with braiding of these anyons, and show that they are able to realize a richer set of non-abelian representations of the braid group than the set realized by non-abelian anyons based on Majorana fermions. We carry out this calculation both explicitly and by applying general considerations. Finally, we show that topological manipulations with these anyons cannot realize universal quantum computation.

Superfluid-insulator transition of two-species bosons with spin-orbit coupling

Motivated by recent experiments [Y.J. Lin {\it et al.}, Nature {\bf 471}, 83 (2011)], we study Mott phases and superfluid-insulator (SI) transitions of two-species ultracold bosonic atoms in a two-dimensional square optical lattice with nearest neighbor hopping amplitude $t$ in the presence of a spin-orbit coupling characterized by a tunable strength $\gamma$. Using both strong-coupling expansion and Gutzwiller mean-field theory, we chart out the phase diagrams of the bosons in the presence of such spin-orbit interaction. We compute the momentum distribution of the bosons in the Mott phase near the SI transition point and show that it displays precursor peaks whose position in the Brillouin zone can be varied by tuning $\gamma$. Our analysis of the critical theory of the transition unravels the presence of unconventional quantum critical points at $t/\gamma=0$ which are accompanied by emergence of an additional gapless mode in the critical region. We also study the superfluid phases of the bosons near the SI transition using a Gutzwiller mean-field theory which reveals the existence of a twisted superfluid phase with an anisotropic twist angle which depends on $\gamma$. Finally, we compute the collective modes of the bosons and point out the presence of reentrant SI transitions as a function of $\gamma$ for non-zero $t$. We propose experiments to test our theory.

Some Hamiltonian models of friction II

In the present paper we consider the motion of a very heavy tracer particle in a medium of a very dense, non-interacting Bose gas. We prove that, in a certain mean-field limit, the tracer particle will be decelerated and come to rest somewhere in the medium. Friction is caused by emission of Cerenkov radiation of gapless modes into the gas. Mathematically, a system of semilinear integro-differential equations, introduced in Fröhlich et al. [“Some hamiltonian models of friction,” J. Math. Phys. 52(8), 083508 (2011)10.1063/1.3619799], describing a tracer particle in a dispersive medium is investigated, and decay properties of the solution are proven. This work is an extension of Fröhlich et al. [“Friction in a model of hamiltonian dynamics,” Commun. Math. Phys. 315(2), 401–444 (2012)10.1007/s00220-012-1564-2]; it is an extension because no weak coupling limit for the interaction between tracer particle and medium is assumed. The technical methods used are dispersive estimates and a contraction principle.

Effects of the edge states on conductance and thermopower for the bilayer graphene nanoribbons

By using the nonequilibrium Green’s function combined with the tight-binding Hamiltonian, we have studied the effects of the edge states on the conductance and the thermopower for zigzag bilayer graphene nanoribbons (ZBGNs). It is shows that the band structure, conductance, and thermopower can be modulated by the boundary potentials and the bias voltages to the layers of the ZBGNs. When the boundary potentials are adjusted to the nearest neighbor hopping energy, two gapless edge modes with opposite velocities appear in the vicinity of the two Dirac points, and the quantized conductance has a transition from 2(n+1)G0 to 2(n+1/2)G0 with G0=2e2/h being the conductance unit and n an integer. Particularly, under the strong bias voltage, compare to the gapless perfect ZBGN, the thermopower can be enhanced more than twice for the gapless edge modes with opposite velocities. Combining the reduced thermal conductivity in few-layer graphene, our results show that the modulated ZBGNs are more reliable in thermoelectric application.

Valley-Polarized Metals and Quantum Anomalous Hall Effect in Silicene

Silicene is a monolayer of silicon atoms forming a two-dimensional honeycomb lattice, which shares almost every remarkable property with graphene. The low-energy structure of silicene is described by Dirac electrons with relatively large spin-orbit interactions due to its buckled structure. The key observation is that the band structure is controllable by applying electric field to silicene. We explore the phase diagram of silicene together with exchange field M and by applying electric field E(z). A quantum anomalous Hall (QAH) insulator, valley polarized metal (VPM), marginal valley polarized metal (M-VPM), quantum spin Hall insulator, and band insulator appear. They are characterized by the Chern numbers and/or by the edge modes of a nanoribbon. It is intriguing that electrons have been moved from a conduction band at the K point to a valence band at the K' point for E(z) > 0 in the VPM. We find in the QAH phase that almost flat gapless edge modes emerge and that spins form a momentum-space Skyrmion to yield the Chern number. It is remarkable that a topological quantum phase transition can be induced simply by changing electric field in a single silicene sheet.

Propagation of sound on line vortices in superfluids: role of ergoregions

We (re)consider the propagation of small disturbances (sound waves) in the presence of an irrotational vortex in a superfluid pinned/wound on a stiff central wire, with the help of the formalism of acoustic spacetimes. We give closed formulas for the scattering angle for sound rays, formulate the sound-propagation problem in the Hamiltonian form and discuss the form of boundary conditions at the core of the vortex, where the Hamiltonian has a singular point. The wave equation is simplified to a single ordinary differential equation of Mathieu type. We give an extensive discussion of perturbations localized close to the core, which are similar to what is known as the Kelvin waves for vortices that are bendable (not pinned). The spectra of modes depend strongly on the type of boundary condition employed close to the vortex core. The gapless mode with the angular number −1, the Kelvin mode usually discussed in the context of unpinned vortices in superfluid helium or rotating Bose–Einstein condensates, turns out either not to exist or to have a completely different dispersion relation if the vortex is pinned. The question of whether or not the acoustic spacetime admits an ergoregion turns out to have a decisive (qualitative) influence on many aspects of sound-propagation phenomena.

Phases and phase transitions in aU(1)×U(1)system withθ=2π/3mutual statistics

We study a $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ system with short-range interactions and mutual $\ensuremath{\theta}=2\ensuremath{\pi}/3$ statistics in (2+1) dimensions. We are able to reformulate the model to eliminate the sign problem and perform a Monte Carlo study. We find a phase diagram containing a phase with only small loops and two phases with one species of proliferated loop. We also find a phase where both species of loop condense, but without any gapless modes. Lastly, when the energy cost of loops becomes small, we find a phase that is a condensate of bound states, each made up of three particles of one species and a vortex of the other. We define several exact reformulations of the model that allow us to precisely describe each phase in terms of gapped excitations. We propose field-theoretic descriptions of the phases and phase transitions, which are particularly interesting on the ``self-dual'' line where both species have identical interactions. We also define irreducible responses useful for describing the phases.

On the theory of slowing down gracefully

We discuss the transport of a tracer particle through the Bose Einstein condensate of a Bose gas. The particle interacts with the atoms in the Bose gas through two-body interactions. In the limiting regime where the particle is very heavy and the Bose gas is very dense, but very weakly interacting ("mean-field limit"), the dynamics of this system corresponds to classical Hamiltonian dynamics. We show that, in this limit, the particle is decelerated by emission of gapless modes into the condensate (Cerenkov radiation). For an ideal gas, the particle eventually comes to rest. In an interacting Bose gas, the particle is decelerated until its speed equals the propagation speed of the Goldstone modes of the condensate. This is a model of "Hamiltonian friction". It is also of interest in connection with the phenomenon of "decoherence" in quantum mechanics. It is based on work we have carried out in collaboration with D Egli, IM Sigal and A Soffer.

Phonon thermal conductivity in silicon nanowires: The effects of surface roughness at low temperatures

Using a Green’s function method based on an elastic wave equation, the effects of surface roughness and the nanowire-contact interface scattering on phonon thermal conductivity are studied at low temperatures. It is found that the interface geometry between a nanowire and its contacts affects the transmission function at small energies related to the gapless modes and it gives rise to deviated results from the universal conductance. It is also shown that the surface roughness is crucial in the suppression of phonon thermal conductivity with reducing the nanowire size by averaging the transmission function over the rough-surface configurations. Furthermore, the phonon mean free path is proportional to the ratio of the correlation length and roughness heights quadratically as well as the cross-section area of the nanowire.

Green's function method for line defects and gapless modes in topological insulators: Beyond the semiclassical approach

Defects which appear in heterostructure junctions involving topological insulators are sources of gapless modes governing the low energy properties of the systems, as recently elucidated by Teo and Kane [Physical Review B82, 115120 (2010)]. A standard approach for the calculation of topological invariants associated with defects is to deal with the spatial inhomogeneity raised by defects within a semiclassical approximation. In this paper, we propose a full quantum formulation for the topological invariants characterizing line defects in three-dimensional insulators with no symmetry by using the Green's function method. On the basis of the full quantum treatment, we demonstrate the existence of a nontrivial topological invariant in the topological insulator-ferromagnet tri-junction systems, for which a semiclassical approximation fails to describe the topological phase. Also, our approach enables us to study effects of electron-electron interactions and impurity scattering on topological insulators with spatial inhomogeneity which gives rise to the Axion electrodynamics responses.