One-dimensional Insulators Research Articles

Universal properties of boundary and interface charges in continuum models of one-dimensional insulators

We study single-channel continuum models of one-dimensional insulators induced by periodic potential modulations which are either terminated by a hard wall (the boundary model) or feature a single region of dislocations and/or impurity potentials breaking translational invariance (the interface model). We investigate the universal properties of excess charges accumulated near the boundary and the interface, respectively. We find a rigorous analytic proof for the earlier observed linear dependence of the boundary charge on the phase of the periodic potential modulation as well as extend these results to the interface model. The linear dependence on the phase shows a universal value for the slope, and is intersected by discontinuous jumps by plus or minus one electron charge at the phase points where localized states enter or leave a band of extended states. Both contributions add up such that the periodicity of the excess charge in the phase over a $2\pi$-cycle is maintained. While in the boundary model this property is usually associated with the bulk-boundary correspondence, in the interface model a correspondence of scattering state and localized state contributions to the total interface charge is unveiled on the basis of the so-called nearsightedness principle.

Quantized transport induced by topology transfer between coupled one-dimensional lattice systems

We show that a topological pump in a one-dimensional (1D) insulator can induce a strictly quantized transport in an auxiliary chain of non-interacting fermions weakly coupled to the first. The transported charge is determined by an integer topological invariant of the ficticious Hamiltonian of the insulator, given by the covariance matrix of single-particle correlations. If the original system consists of non-interacting fermions, this number is identical to the TKNN (Thouless, Kohmoto, Nightinghale, den Nijs) invariant of the original system and thus the coupling induces a transfer of topology to the auxiliary chain. When extended to particles with interactions, for which the TKNN number does not exist, the transported charge in the auxiliary chain defines a topological invariant for the interacting system. In certain cases this invariant agrees with the many-body generalization of the TKNN number introduced by Niu, Thouless, and Wu (NTW). We illustrate the topology transfer to the auxiliary system for the Rice-Mele model of non-interacting fermions at half filling and the extended superlattice Bose-Hubbard model at quarter filling. In the latter case the induced charge pump is fractional.

Theory of bulk photovoltaic effect in Anderson insulator

The localization of wavefunction by disorder makes a conductive material an insulator with vanishing conductivity at zero temperature. A similar outcome is expected for the photocurrent in semiconductor p-n junctions because the photoexcited carriers cannot drift through the device. In contrast, we here show numerically that the bulk photovoltaic effect-the photovoltaic effect in noncentrosymmetric bulk materials-occurs in a noncentrosymmetric, disordered, one-dimensional insulator where all eigenstates are localized. We find this photocurrent remains, even when the energy scale of random potential is larger than the bandwidth. On the other hand, the photocurrent decays exponentially when the excitation is local, i.e., when only a part of the device is illuminated. The photocurrent also vanishes if the relaxation occurs only by contact with the electrodes. Our result implies that the ratio of the photovoltaic current and the direct current by the variable-range hopping increases with decreasing temperature. These results suggest a route to design high-efficiency solar cells and photodetectors.

Magnetic impurities along the edge of a quantum spin Hall insulator: Realizing a one-dimensional AIII insulator

In this paper we construct a one-dimensional insulator with an approximate chiral symmetry belonging to the AIII class and discuss its properties. The construction principle is the intentional pollution of the edge of a two-dimensional quantum spin Hall insulator with magnetic impurities. The resulting bound states hybridize and disperse along the edge. We discuss under which circumstances this chain possesses zero-dimensional boundary modes on the level of an effective low-energy theory. The main appeal of our construction is the independence on details of the impurity lattice: the zero modes are stable against disorder and random lattice configurations. We also show that in the presence of Rashba coupling, which changes the symmetry class to A, one can still expect localized half-integer boundary excess charges protected by mirror symmetry although there is no nontrivial topological index. All of the results are confirmed numerically in a microscopic model.

Quadrupole moments, edge polarizations, and corner charges in the Wannier representation

The modern theory of polarization allows for the determination of the macroscopic end charge of a truncated one-dimensional insulator, modulo the charge quantum $e$, from a knowledge of bulk properties alone. A more subtle problem is the determination of the corner charge of a two-dimensional insulator, modulo $e$, from a knowledge of bulk and edge properties alone. While previous works have tended to focus on the quantization of corner charge in the presence of symmetries, here we focus on the case that the only bulk symmetry is inversion, so that the corner charge can take arbitrary values. We develop a Wannier-based formalism that allows the corner charge to be predicted, modulo $e$, only from calculations on ribbon geometries of two different orientations. We elucidate the dependence of the interior quadrupole and edge dipole contributions upon the gauge used to construct the Wannier functions, finding that while these are individually gauge-dependent, their sum is gauge-independent. From this we conclude that the edge polarization is not by itself a physical observable, and that any Wannier-based method for computing the corner charge requires the use of a common gauge throughout the calculation. We satisfy this constraint using two Wannier construction procedures, one based on projection and another based on a gauge-consistent nested Wannier construction. We validate our theory by demonstrating the correct prediction of corner charge for several tight-binding models. We comment on the relations between our approach and previous ones that have appeared in the literature.

Rational boundary charge in one-dimensional systems with interaction and disorder

We study the boundary charge $Q_B$ of generic semi-infinite one-dimensional insulators with translational invariance and show that non-local symmetries (i.e., including translations) lead to rational quantizations $p/q$ of $Q_B$. In particular, we find that (up to an unknown integer) the quantization of $Q_B$ is given in integer units of $\frac{1}{2}\bar{\rho}$ and $\frac{1}{2}(\bar{\rho}-1)$, where $\bar{\rho}$ is the average charge per site (which is a rational number for an insulator). This is a direct generalization of the known half-integer quantization of $Q_B$ for systems with local inversion or local chiral symmetries to any rational value. Quite remarkably, this rational quantization remains valid even in the presence of short-ranged electron-electron interactions as well as static random disorder (breaking translational invariance). This striking stability can be traced back to the fact that local perturbations in insulators induce only local charge redistributions. We establish this result with complementary methods including density matrix renormalization group calculations, bosonization methods, and exact solutions for particular lattice models. Furthermore, for the special case of half-filling $\bar{\rho}=\frac{1}{2}$, we present explicit results in single-channel and nearest-neighbor hopping models and identify Weyl semimetal physics at gap closing points. Our general framework also allows us to shed new light on the well-known rational quantization of soliton charges at domain walls.

Topological invariants to characterize universality of boundary charge in one-dimensional insulators beyond symmetry constraints

In the absence of any symmetry constraints we address universal properties of the boundary charge $Q_B$ for a wide class of nearest-neighbor tight-binding models in one dimension with one orbital per site but generic modulations of on-site potentials and hoppings. We provide a precise formulation of the bulk-boundary correspondence relating the boundary charge of a single band uniquely to the Zak phase evaluated in a particular gauge. We reveal the topological nature of $Q_B$ by proving the quantization of a topological index $eI=\Delta Q_B - \bar{\rho}$, where $\Delta Q_B$ is the change of $Q_B$ when shifting the lattice by one site towards a boundary and $\bar{\rho}$ is the average charge per site. For a single band we find this index to be given by the winding number of the fundamental phase difference of the Bloch wave function between the two lattice sites defining the boundary of a half-infinite system. For a given chemical potential we establish a central topological constraint $I\in\{-1,0\}$ related only to charge conservation of particles and holes. Our results are shown to be stable against disorder and we propose generalizations to multi-channel and interacting systems.

Novel topological insulators from crystalline symmetries

We discuss recent advances in the study of topological insulators protected by spatial symmetries by reviewing three representative, theoretical examples. In three dimensions (3D), these states of matter are generally characterized by the presence of gapless boundary states at surfaces that respect the protecting spatial symmetry. We discuss the appearance of these topological states in both crystals with negligible spin–orbit coupling and a fourfold rotational symmetry, as well as in mirror-symmetric crystals with sizable spin–orbit interaction characterized by the so-called mirror Chern number. Finally, we also discuss similar topological crystalline states in one-dimensional (1D) insulators, such as nanowires or atomic chains, with mirror symmetry. There, the prime physical consequence of the non-trivial topology is the presence of quantized end charges.

The Bulk-Edge Correspondence for Disordered Chiral Chains

We study one-dimensional insulators obeying a chiral symmetry in the single-particle picture. The Fermi level is assumed to lie in a mobility gap. Topological indices are defined for infinite (bulk) or half-infinite (edge) systems, and it is shown that for a given Hamiltonian with nearest neighbor hopping the two indices are equal. We also give a new formulation of the index in terms of the Lyapunov exponents of the zero energy Schr\"odinger equation, which illustrates the conditions for a topological phase transition occurring in the mobility gap regime.

Machine Learning Topological Invariants with Neural Networks.

In this Letter we supervisedly train neural networks to distinguish different topological phases in the context of topological band insulators. After training with Hamiltonians of one-dimensional insulators with chiral symmetry, the neural network can predict their topological winding numbers with nearly 100% accuracy, even for Hamiltonians with larger winding numbers that are not included in the training data. These results show a remarkable success that the neural network can capture the global and nonlinear topological features of quantum phases from local inputs. By opening up the neural network, we confirm that the network does learn the discrete version of the winding number formula. We also make a couple of remarks regarding the role of the symmetry and the opposite effect of regularization techniques when applying machine learning to physical systems.

Excess charges as a probe of one-dimensional topological crystalline insulating phases

We show that in conventional one-dimensional insulators excess charges created close to the boundaries of the system can be expressed in terms of the Berry phases associated with the electronic Bloch wave functions. Using this correspondence, we uncover a link between excess charges and the topological invariants of the recently classified one-dimensional topological phases protected by spatial symmetries. Excess charges can be thus used as a probe of crystalline topologies.

Bott periodicity for the topological classification of gapped states of matter with reflection symmetry

Using a dimensional reduction scheme based on scattering theory, we show that the classification tables for topological insulators and superconductors with reflection symmetry can be organized in two period-2 and four period-8 cycles, similar to the Bott periodicity found for topological insulators and superconductors without spatial symmetries. With the help of the dimensional reduction scheme, the classification in arbitrary dimensions $d\ensuremath{\ge}1$ can be obtained from the classification in one dimension, for which we present a derivation based on relative homotopy groups and exact sequences to classify one-dimensional insulators and superconductors with reflection symmetry. The resulting classification is fully consistent with a comprehensive classification obtained recently by Shiozaki and Sato [Phys. Rev. B 90, 165114 (2014)]. The use of a scattering-matrix-inspired method allows us to address the second descendant ${\mathbb{Z}}_{2}$ phase, for which the topological nontrivial phase was previously reported to be vulnerable to perturbations that break translation symmetry.

Interband coherence induced correction to Thouless pumping: possible observation in cold-atom systems

In Thouless pump, the charge transport in a one-dimensional insulator over an adiabatic cycle is topologically quantized. For nonequilibrium initial states, however, interband coherence will induce a previously unknown contribution to Thouless pumping. Though not geometric in nature, this contribution is independent of the time scale of the pumping protocol. In this work, we perform a detailed analysis of our previous finding [Phys. Rev. B 91, 085420 (2015)] in an already available cold-atom setup. We show that initial states with interband coherence can be obtained via a quench of the system's Hamiltonian. Adiabatic pumping in the post-quench system are then examined both theoretically and numerically, in which the interband coherence is shown to play an important role and can hence be observed experimentally. By choosing adiabatic protocols with different switching-on speeds, we also show that the contribution of interband coherence to adiabatic pumping can be tuned. It is further proposed that the interband coherence induced correction to Thouless pumping may be useful in capturing a topological phase transition point. All our results have direct experimental interests.

Spectral correlations in finite-size Anderson insulators

We investigate spectral correlations in quasi one-dimensional Anderson insulators with broken time-reversal symmetry. While energy levels are uncorrelated in the thermodynamic limit of infinite wire-length, some correlations remain in finite-size Anderson insulators. Asymptotic behaviors of level-level correlations in these systems are known in the large- and small-frequency limits, corresponding to the regime of classical diffusive dynamics and the deep quantum regime of strong Anderson localization. Employing non-perturbative methods and a mapping to the Coulomb-scattering problem, recently introduced by {\it M.~A.~Skvortsov} and {\it P.~M.~Ostrovsky}, we derive a closed analytical expression for the spectral statistics in the classical-to-quantum region bridging the known asymptotic behaviors. We further discuss how Poisson statistics at large energies develop into Wigner-Dyson statistics as the wire-length decreases.

Dynamical correlation functions for the one-dimensional Bose-Hubbard insulator

We calculate the dynamical current and kinetic-energy correlation functions for the first Mott lobe of the one-dimensional Bose-Hubbard model. We employ the strong-coupling expansion up to sixth order in $x=t/U$ and the dynamical density-matrix renormalization group method on rings with 64 sites. The correlation functions are finite above the single-particle gap with a square-root onset, as also found from field theory close to the Mott transition. The correlation functions display a featureless superposition of the primary and tertiary Hubbard bands. We find very good agreement between all methods in the interaction and frequency regimes where they are applicable.

Electrons do the split

Interacting electrons that are confined to move in a one-dimensional structure do not simply jam together like cars in rush hour. Inelastic X-ray scattering shows that the electrons act as if they split into separate fractional entities. See Letter p.82 Electronic excitations in the solid state can exhibit properties quite distinct from those of free electrons, which carry both spin and charge. An extreme example is the case of one-dimensional systems, where the electronic excitations (quasiparticles) can each fragment into two independent particle-like entities, one carrying charge (the holon) and the other carrying spin (the spinon). Schlappa et al. describe a system, the one-dimensional insulator Sr2CuO3, in which this fragmentation proceeds differently, with the orbital degree of freedom separating from the spinon to yield a third distinct quasiparticle — the orbiton. This work confirms that all of the fundamental degrees of freedom of the electron can 'fractionalize', putting spin and orbital excitations on an equal footing.

TEMPERATURE DEPENDENCE OF SPIN CURRENTS CARRIED BY JORDAN–WIGNER FERMIONS AND MAGNONS IN INSULATORS

The temperature dependence of spin currents in insulators at the finite temperature near zero Kelvin is theoretically studied. The spin currents are carried by Jordan–Wigner fermions and magnons in one- and three-dimensional insulators. These spin currents are generated by the external magnetic field gradient along the quantization axis and also by the two-particle interaction gradient. In one-dimensional insulators, quantum fluctuations are strong and the spin current carried by Jordan–Wigner fermions shows the stronger dependence on temperatures than the one by magnons.

Static electric field in one-dimensional insulators without boundaries

In this brief report, we show that in a one-dimensional insulating system with periodic boundary conditions, the coefficient of the $\ensuremath{\theta}$ term in the effective theory is not only determined by the topological index $\ensuremath{\int}i{\ensuremath{\sum}}_{\ensuremath{\alpha}\ensuremath{\in}\mathrm{occ}}\ensuremath{\langle}{u}_{k\ensuremath{\alpha}}|\frac{\ensuremath{\partial}}{\ensuremath{\partial}k}\left|{u}_{k\ensuremath{\alpha}}\right\ensuremath{\rangle}\mathrm{d}k$. Specifically, the relative position between the electronic orbitals and the ions also alters the coefficient, as one would expect when one identifies $\ensuremath{-}e\ensuremath{\theta}/2\ensuremath{\pi}$ as the polarization. This resolves a paradox when we apply our previous result to the Su-Shreiffer-Heeger model, where the two ground states related by a lattice translation have $\ensuremath{\theta}$ differed by $\ensuremath{\pi}$. We also show that the static dielectric screening is the same with or without boundaries, in contrast to comments made in our previous paper.

Topological classification of interaction-driven spin pumps

When adiabatically varied in time, certain one-dimensional band insulators allow for the quantized noiseless pumping of spin even in the presence of strong spin orbit scattering. These spin pumps are closely related to the quantum spin Hall system, and their properties are protected by a time-reversal restriction on the pumping cycle. In this paper we study pumps formed of one-dimensional insulators with a time-reversal restriction on the pumping cycle and a bulk energy gap which arises due to interactions. We find that the correlated gapped phase can lead to novel pumping properties. In particular, systems with $d$ different ground states can give rise to $d+1$ different classes of spin pumps, including a trivial class which does not pump quantized spin and $d$ non-trivial classes allowing for the pumping of quantized spin $\hbar/n $ on average per cycle, where $1\leq n\leq d$. We discuss an example of a spin pump that transfers on average spin $ \hbar/2$ without transferring charge.

Spin-charge decoupling and the photoemission line-shape in one-dimensional insulators

The recent advances in angle resolved photoemission techniques allowed the unambiguous experimental confirmation of spin charge decoupling in quasi one dimensional (1D) Mott insulators. This opportunity stimulates a quantitative analysis of the spectral function $A(k,\omega)$ of prototypical one dimensional correlated models. Here we combine Bethe Ansatz results, Lanczos diagonalizations and field theoretical approaches to obtain $A(k,\omega)$ for the 1D Hubbard model as a function of the interaction strength. By introducing a {\it single spinon approximation}, an analytic expression is obtained, which shows the location of the singularities and allows, when supplemented by numerical calculations, to obtain an accurate estimate of the spectral weight distribution in the $(k,\omega)$ plane. Several experimental puzzles on the observed intensities and line-shapes in quasi 1D compounds, like ${\rm SrCuO_2}$, find a natural explanation in this theoretical framework.