Gapless Modes Research Articles

Spontaneous Quantum Hall States and Novel Luttinger Liquids in Chiral Graphene

Chirally stacked neutral N-layer graphene systems with N ≥ 2 exhibit spontaneous inversion symmetry breaking and interaction induced charge gaps due to interlayer exchange interactions. A variety of distinct broken symmetry states are distinguished by their charge, spin, and valley Hall conductivities, by their orbital magnetizations, and by their edge state properties. In the spinless case, quantum anomalous and valley Hall states are favored by a weak magnetic field and by an electric field between the graphene layers, respectively. At interfaces between different phases one dimensional gapless modes exhibit novel Luttinger liquid behaviors.

Majorana spin liquids on a two-leg ladder

We realize a gapless Majorana Orbital Liquid (MOL) using orbital degrees of freedom and also an SU(2)-invariant Majorana Spin Liquid (MSL) using both spin and orbital degrees of freedom in Kitaev-type models on a 2-leg ladder. The models are exactly solvable by Kitaev's parton approach, and we obtain long-wavelength descriptions for both Majorana liquids. The MOL has one gapless mode and power law correlations in energy at incommensuare wavevectors, while the SU(2) MSL has three gapless modes and power law correlations in spin, spin-nematic, and local energy observables. We study the stability of such states to perturbations away from the exactly solvable points. We find that both MOL and MSL can be stable against allowed short-range parton interactions. We also argue that both states persist upon allowing $Z_2$ gauge field fluctuations, in that the number of gapless modes is retained, although with an expanded set of contributions to observables compared to the free parton mean field.

Bose metals and insulators on multileg ladders with ring exchange

We establish compelling evidence for the existence of new quasi-one-dimensional descendants of the d-wave Bose liquid (DBL), an exotic two-dimensional quantum phase of uncondensed itinerant bosons characterized by surfaces of gapless excitations in momentum space [O. I. Motrunich and M. P. A. Fisher, Phys. Rev. B {\bf 75}, 235116 (2007)]. In particular, motivated by a strong-coupling analysis of the gauge theory for the DBL, we study a model of hard-core bosons moving on the $N$-leg square ladder with frustrating four-site ring exchange. Here, we focus on four- and three-leg systems where we have identified two novel phases: a compressible gapless Bose metal on the four-leg ladder and an incompressible gapless Mott insulator on the three-leg ladder. The former is conducting along the ladder and has five gapless modes, one more than the number of legs. This represents a significant step forward in establishing the potential stability of the DBL in two dimensions. The latter, on the other hand, is a fundamentally quasi-one-dimensional phase that is insulating along the ladder but has two gapless modes and incommensurate power law transverse density-density correlations. In both cases, we can understand the nature of the phase using slave-particle-inspired variational wave functions consisting of a product of two distinct Slater determinants, the properties of which compare impressively well to a density matrix renormalization group solution of the model Hamiltonian. Stability arguments are made in favor of both quantum phases by accessing the universal low-energy physics with a bosonization analysis of the appropriate quasi-1D gauge theory. We will briefly discuss the potential relevance of these findings to high-temperature superconductors, cold atomic gases, and frustrated quantum magnets.

Magnetic structure ofEr2Ti2O7

We employ the previously published neutron-scattering experiments to suggest that ${{\mathrm{Er}}_{2}{\mathrm{Ti}}_{2}\mathrm{O}}_{7}$ has a broken-symmetry multiple-q state with tetragonal magnetic symmetry. The ordered moments do not appear to lie close to the crystal-field anticipated directions, and we suggest that the low-energy gapless mode, visible in specific-heat measurements, is not of the usual transverse isotropic Goldstone-mode type but is longitudinal and is associated with the internal transfer of magnetism between distinct magnetic Bragg spots.

Some Hamiltonian models of friction

Mathematical results on a model describing the motion of a tracer particle through a non-interacting Bose-Einstein condensate are described. In the limit of a very dense gas and for a very large particle mass, the dynamics of the coupled system is determined by classical nonlinear Hamiltonian equations of motion. The particle's motion exhibits deceleration corresponding to friction with memory caused by the emission of Cerenkov radiation of gapless modes into the gas. A more general class of models involving interacting Bose gases will be studied in forthcoming papers.

Anomalous tunneling of collective excitations and effects of superflow in the polar phase of a spin-1 spinor Bose-Einstein condensate

We investigate tunneling properties of collective modes in the polar phase of a spin-1 spinor Bose-Einstein condensate. This spinor BEC state has two kinds of gapless modes, i.e., Bogoliubov mode and spin-wave. Within the framework of the mean-field theory at T=0, we show that these Goldstone modes exhibit the perfect transmission in the low-energy limit. Their anomalous tunneling behaviors still hold in the presence of superflow, except in the critical current state. In the critical current state, while the tunneling of Bogoliubov mode is accompanied by finite reflection, the spin-wave still exhibit the perfect transmission, unless the strengths of a spin-dependent and spin-independent interactions take the same value.

Model Characterization of Gapless Edge Modes of Topological Insulators Using Intermediate Brillouin-Zone Functions

We characterize gapless edge modes in translation invariant topological insulators. We show that the edge mode spectrum is a continuous deformation of the spectrum of a certain gluing function defining the occupied state bundle over the Brillouin zone. Topologically nontrivial gluing functions, corresponding to nontrivial bundles, then yield edge modes exhibiting spectral flow. We illustrate our results for the case of chiral edge states in two-dimensional Chern insulators, as well as helical edges in quantum spin Hall states.

Effective action approach to thep-band Mott insulator and superfluid transition

Motivated by the recent experiment on $p$-orbital-band bosons in optical lattices, we study theoretically the quantum phases of Mott insulator and superfluidity in two dimensions. The system features a superfluid phase with transversely staggered orbital current at weak interaction and a Mott insulator phase with antiferro-orbital order at strong coupling and commensurate filling. We go beyond mean-field theory and derive from a microscopic model an effective action that is capable of describing both the $p$-band Mott insulating and superfluid phases in strong coupling. We further calculate the excitation spectra near the quantum critical point and find two gapless modes away from the tip of the Mott lobe but four gapless modes at the tip. Our effective theory reveals how the phase coherence peak builds up in the Mott regime when approaching the critical point. We also discuss the finite-temperature phase transition of $p$-band superfluidity.

Polar phase of one-dimensional bosons with large spin

Spinor ultracold gases in one dimension (1D) represent an interesting example of strongly correlated quantum fluids. They have a rich phase diagram and exhibit a variety of quantum phase transitions. We consider a 1D spinor gas of bosons with a large spin S. A particular example is the gas of chromium atoms (S=3), where the dipolar collisions efficiently change the magnetization and make the system sensitive to the linear Zeeman effect. We argue that in 1D the most interesting effects come from the pairing interaction. If this interaction is negative, it gives rise to a (quasi)condensate of singlet bosonic pairs with an algebraic order at zero temperature, and for (2S+1)≫1 the saddle point approximation leads to physically transparent results. Since in 1D one needs a finite energy to destroy a pair, the spectrum of spin excitations has a gap. Hence, in the absence of a magnetic field, there is only one gapless mode corresponding to phase fluctuations of the pair quasicondensate. Once the magnetic field exceeds the gap, another condensate emerges, namely the quasicondensate of unpaired bosons with spins aligned along the magnetic field. The spectrum then contains two gapless modes corresponding to the singlet-paired and spin-aligned unpaired Bose condensed particles, respectively. At T=0, the corresponding phase transition is of the commensurate–incommensurate type.

Phase diagram of hard-core bosons on a zigzag ladder

We study hard-core bosons with unfrustrated nearest-neighbor hopping $t$ and repulsive interaction $V$ on a zigzag ladder. As a function of the boson density $\ensuremath{\rho}$ and $V/t$, the ground state displays different quantum phases. A standard one-component Tomonaga-Luttinger liquid is stable for $\ensuremath{\rho}<1/3$ (and $\ensuremath{\rho}>2/3$) at any value of $V/t$. At commensurate densities $\ensuremath{\rho}=1/3$, $1/2$, and $2/3$ insulating (crystalline) phases are stabilized for a sufficiently large interaction $V$. For intermediate densities $1/3<\ensuremath{\rho}<2/3$ and large $V/t$, the ground state shows a clear evidence of a bound state of two bosons, implying gapped single-particle excitations but gapless excitations of boson pairs. These properties can be understood by the fact that the antisymmetric sector acquires a gap and a single gapless mode survives. Finally, for the same range of boson densities and weak interactions, the system is again a one-component Tomonaga-Luttinger liquid with no evidence of any breaking of discrete symmetries, in contrast to the frustrated case, where a ${\mathcal{Z}}_{2}$ symmetry breaking has been predicted.

Exotic Gapless Mott Insulators of Bosons on Multileg Ladders

We present evidence for an exotic gapless insulating phase of hard-core bosons on multileg ladders with a density commensurate with the number of legs. In particular, we study in detail a model of bosons moving with direct hopping and frustrating ring exchange on a 3-leg ladder at ν=1/3 filling. For sufficiently large ring exchange, the system is insulating along the ladder but has two gapless modes and power law transverse density correlations at incommensurate wave vectors. We propose a determinantal wave function for this phase and find excellent comparison between variational Monte Carlo and density matrix renormalization group calculations on the model Hamiltonian, thus providing strong evidence for the existence of this exotic phase. Finally, we discuss extensions of our results to other N-leg systems and to N-layer two-dimensional structures.

Marginality of bulk-edge correspondence for single-valley Hamiltonians

We study the correspondence between the non-trivial topological properties associated with the individual valleys of gapped bilayer graphene (BLG), as a prototypical multi-valley system, and the gapless modes at its edges and other interfaces. We find that the exact connection between the valley-specific Hall conductivity and the number of gapless edge modes does not hold in general, but is dependent on the boundary conditions, even in the absence of intervalley coupling. This non-universality is attributed to the absence of a well-defined topological invariant within a given valley of BLG; yet, a more general topological invariant may be defined in certain cases, which explains the distinction between the BLG-vacuum and BLG-BLG interfaces.

Topological defects and gapless modes in insulators and superconductors

We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider Hamiltonians $\mathcal{H}(\mathbf{k},\mathbf{r})$ that vary slowly with adiabatic parameters $\mathbf{r}$ surrounding the defect and belong to any of the ten symmetry classes defined by time-reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three-dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions, and helical Majorana fermions, as well as zero-dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes.

Spectral Properties near the Mott Transition in the One-Dimensional Hubbard Model

The single-particle spectral properties near the Mott transition in the one-dimensional Hubbard model are investigated by using the dynamical density-matrix renormalization group method and the Bethe ansatz. The pseudogap, hole-pocket behavior, spectral-weight transfer, and upper Hubbard band are explained in terms of spinons, holons, antiholons, and doublons. The Mott transition is characterized by the emergence of a gapless mode whose dispersion relation extends up to the order of hopping t (spin exchange J) in the weak (strong) interaction regime caused by infinitesimal doping.

Entanglement Entropy and the Fermi Surface

Free fermions with a finite Fermi surface are known to exhibit an anomalously large entanglement entropy. The leading contribution to the entanglement entropy of a region of linear size L in d spatial dimensions is S∼L(d-1)logL, a result that should be contrasted with the usual boundary law S∼L(d-1). This term depends only on the geometry of the Fermi surface and on the boundary of the region in question. I give an intuitive account of this anomalous scaling based on a low energy description of the Fermi surface as a collection of one-dimensional gapless modes. Using this picture, I predict a violation of the boundary law in a number of other strongly correlated systems.

Properties of CFTs dual to charged BTZ black hole

We study properties of strongly coupled CFT's with non-zero background electric charge in 1 + 1 dimensions by studying the dual gravity theory—which is a charged BTZ black hole. Correlators of operators dual to scalars, gauge fields and fermions are studied at both T = 0 and T ≠ 0 . In the T = 0 case we are also able to compare with analytical results based on AdS 2 and find reasonable agreement. In particular the correlation between log periodicity and the presence of finite spectral density of gapless modes is seen. The real part of the conductivity (given by the current–current correlator) also vanishes as ω → 0 as expected. The fermion Green's function shows quasiparticle peaks with approximately linear dispersion but the detailed structure is neither Fermi liquid nor Luttinger liquid and bears some similarity to a “Fermi–Luttinger” liquid. This is expected since there is a background charge and the theory is not Lorentz or scale invariant. A boundary action that produces the observed non-Luttinger liquid like behavior ( k-independent non-analyticity at ω = 0 ) in the Green's function is discussed.

Crossover of the three-dimensional topological insulator Bi2Se3 to the two-dimensional limit

The gapless surface states of topological insulators could enable quantitatively different types of electronic device. A study of the topological insulating Bi2Se3 thin films finds that a gap in these states opens up in films below a certain thickness. This in turn suggests that in thicker films, gapless states exist on both upper and lower surfaces. A topological insulator1,2,3,4,5,6,7,8,9 is a new state of quantum matter that is characterized by a finite energy gap in the bulk and gapless modes flowing along the boundaries that are robust against disorder scattering. The topological protection of the surface state could be useful for both low-power electronics10 and error-tolerant quantum computing11,12. For a thin slab of three-dimensional topological insulator, the boundary modes from the opposite surfaces may be coupled by quantum tunnelling, so that a small, thickness-dependent gap is opened up13,14,15. Here we report such results from angle-resolved photoemission spectroscopy on Bi2Se3 films of various thicknesses grown by molecular beam epitaxy. The energy gap opening is clearly seen when the thickness is below six quintuple layers. The gapped surface states also exhibit sizeable Rashba-type spin–orbit splitting because of the substrate-induced potential difference between the two surfaces. The tunable gap and the spin–orbit coupling make these topological thin films ideal for electronic and spintronic device applications.

Metal-insulator phase transition in an exactly solvable model of nondegenerate interacting fermion chains

A two parametric family of the models of two coupled nondegenerate fermion chains is proposed and solved by the means of the Bethe ansatz. We give a detailed analysis of the exact zero-temperature phase diagram at half filling for different parameters of the interactions. It is shown that insulator and ``interchain ferromagnetic'' phase states are separated by ``intermediate metallic'' state, these phase states coexist in a tricritical point. The critical fluctuations are described by a conformal field theory with the central charge $c=1$. In the tricritical point the fermion state is described as a Luttinger liquid state with one gapless mode that is characterized by abnormal large density-density correlations and correlations of the momentum distribution function.

Topological insulators on the decorated honeycomb lattice

We show that the decorated honeycomb (star) lattice supports a number of topological insulating phases with a nontrivial ${Z}_{2}$ invariant and time-reversal symmetry protected gapless edge modes. We investigate the stability of these phases with respect to various symmetry-breaking perturbations and demonstrate the connection to the recent discovery of an exactly solvable $S=1/2$ chiral spin-liquid model [Yao and Kivelson, Phys. Rev. Lett. 99, 247203 (2007)] with non-Abelian and Abelian excitations on the same lattice at strong interaction strength. Our work highlights the relationship between noninteracting topological band insulators and strongly interacting topologically ordered spin systems, and points to promising avenues for enlarging the number of known examples of both.

Compactz=2Electrodynamics in2+1Dimensions: Confinement with Gapless Modes

We consider 2+1-dimensional compact U(1) gauge theory at the Lifshitz point with a dynamical critical exponent z=2. As in the usual z=1 theory, monopoles proliferate the vacuum for any value of the coupling, generating a mass scale. The theory of the dilute monopole gas is written in terms of a nonrelativistic sine-Gordon model with two real fields. While monopoles remove some of the massless poles of the perturbative field strength propagator, a gapless mode representing the incomplete screening of monopoles remains, and is protected by a shift invariance of the original theory. Timelike Wilson loops still obey area laws, implying that minimal charges are confined, but the action of spacelike Wilson loops of linear size L goes instead as L(3).