results for au:Prodan_E in:cond-mat

- May 16 2018 cond-mat.mes-hall arXiv:1805.05828v1For any model belonging to the classification table of strong topological condensed matter systems, we define a mapping which transforms the hermitean Hamiltonian into a real symmetric positive dynamical matrix, at the expense of doubling the degrees of freedom per site. This dynamical matrix has a certain built-in symmetry and, when projected on one of the invariant subspaces, the dynamical matrix inherits all the symmetries of the original Hamiltonian. Using an ultra-local property of the mapping, we also show that the bulk-boundary correspondence principle also transfers, entirely. Concrete procedures for exciting and driving modes from one of the invariant subspaces is given. We package these results into an algorithm for translating any topological condensed matter system into a classical topological meta-material. Several concrete topological mechanical systems are worked out in detail using the algorithm.
- Mar 26 2018 cond-mat.mes-hall arXiv:1803.08781v2The quantum valley Hall effect (QVHE) has been observed in a variety of experimental setups, both quantum and classical. While extremely promising for applications, one should be reminded that QVHE is not an exact topological phenomenon and that, so far, it has been fully understood only qualitatively in certain extreme limits. Here we present a technique to relate QVHE systems with exact quantum spin-Hall insulators that accept real-space representations, without taking any extreme limit. Since the bulk-boundary correspondence is well understood for the latter, we are able to formulate precise quantitative statements about the QVHE regime and its robustness against disorder. We further investigate the effect using a novel experimental platform based on magnetically coupled spinners. Visual renderings, quantitative data and various tests of the domain-wall modes are supplied, hence giving an unprecedented insight into the effect.
- The research in topological materials and meta-materials reached maturity and is now gradually entering the phase of practical applications and devices. However, scaling down the experimental demonstrations definitely presents a challenge. In this work, we study coupled identical resonators whose collective dynamics is fully determined by the pattern in which the resonators are arranged. We call a pattern topological if boundary resonant modes fully fill all existing spectral gaps whenever the pattern is halved. This is a characteristic of the pattern and is entirely independent of the structure of the resonators and the details of the couplings. Existence of such patterns is proven using $K$-theory and exemplified using a novel experimental platform based on magnetically coupled spinners. Topological meta-materials built on these principles can be easily engineered at any scale, providing a practical platform for applications and devices.
- The search for strong topological phases in generic aperiodic materials and meta-materials is now vigorously pursued by the condensed matter physics community. In this work, we first introduce the concept of patterned resonators as a unifying theoretical framework for topological electronic, photonic, phononic etc. (aperiodic) systems. We then discuss, in physical terms, the philosophy behind an operator theoretic analysis used to systematize such systems. A model calculation of the Hall conductance of a 2-dimensional amorphous lattice is given, where we present numerical evidence of its quantization in the mobility gap regime. Motivated by such facts, we then present the main result of our work, which is the extension of the Chern number formulas to Hamiltonians associated to lattices without a canonical labeling of the sites, together with index theorems that assure the quantization and stability of these Chern numbers in the mobility gap regime. Our results cover a broad range of applications, in particular, those involving quasi-crystalline, amorphous as well as synthetic (i.e. algorithmically generated) lattices.
- Apr 24 2017 cond-mat.soft arXiv:1704.06542v1Mechanical systems can display topological characteristics similar to that of topological insulators. Here we report a large class of topological mechanical systems related to the BDI symmetry class. These are self-assembled chains of rigid bodies with an inversion center and no reflection planes. The particle-hole symmetry characteristic to the BDI symmetry class stems from the distinct behavior of the translational and rotational degrees of freedom under inversion. This and other generic properties led us to the remarkable conclusion that, by adjusting the gyration radius of the bodies, one can always simultaneously open a gap in the phonon spectrum, lock-in all the characteristic symmetries and generate a non-trivial topological invariant. The particle-hole symmetry occurs around a finite frequency, hence we can witness a dynamical topological Majorana edge mode. Contrasting a floppy mode occurring at zero frequency, a dynamical edge mode can absorb and store mechanical energy, potentially opening new applications of topological mechanics.
- It has been some time since non-commutative geometry was proposed by Jean Bellissard as a theoretical framework for the investigation of homogeneous condensed matter systems. Recently, Bellissard's approach has been enthusiastically adopted in the relatively young field of topological insulators, where it facilitated many rigorous results concerning the stability of the topological invariants against disorder. In this work we present a computational program based on the principles of non-commutative geometry and showcase several applications to topological insulators. In the first part we introduce the notion of a homogeneous material and define the class of disordered crystals together with the classification table which conjectures all topological phases from this class. We continue with a discussion of electron dynamics in disordered crystals and we briefly review the theory of topological invariants in the presence of strong disorder. We show how all these can be captured in the language of non-commutative geometry using the concept of non-commutative Brillouin torus, and present a list of known formulas for various physical response functions. In the second part, we introduce auxiliary algebras and develop a canonical finite-volume approximation of the non-commutative Brillouin torus. Explicit numerical algorithms for computing generic correlation functions are discussed. In the third part we derive upper bounds on the numerical errors and demonstrate that the canonical-finite volume approximation converges extremely fast to the thermodynamic limit. Convergence tests and various applications concludes our presentation.
- Aug 25 2016 cond-mat.str-el arXiv:1608.06634v1Using exact diagonalization and quantum Monte Carlo calculations we investigate the effects of disorder on the phase diagram of both non-interacting and interacting models of two-dimensional topological insulators. In the fermion sign problem-free interacting models we study, electron-electron interactions are described by an on-site repulsive Hubbard interaction and disorder is included via the one-body hopping operators. In both the non-interacting and interacting models we make use of recent advances in highly accurate real-space numerical evaluation of topological invariants to compute phase boundaries, and in the non-interacting models determine critical exponents of the transitions. We find different models exhibit distinct stability conditions of the topological phase with respect to interactions and disorder. We provide a general analytical theory that accurately predicts these trends.
- We use constructive bounded Kasparov K-theory to investigate the numerical invariants stemming from the internal Kasparov products $K_i(\mathcal A) \times KK^i(\mathcal A, \mathcal B) \rightarrow K_0(\mathcal B) \rightarrow \mathbb R$, $i=0,1$, where the last morphism is provided by a tracial state. For the class of properly defined finitely-summable Kasparov $(\mathcal A,\mathcal B)$-cycles, the invariants are given by the pairing of K-theory of $\mathcal B$ with an element of the periodic cyclic cohomology of $\mathcal B$, which we call the generalized Connes-Chern character. When $\mathcal A$ is a twisted crossed product of $\mathcal B$ by $\mathbb Z^k$, $\mathcal A = \mathcal B \rtimes_\xi^\theta \mathbb Z^k$, we derive a local formula for the character corresponding to the fundamental class of a properly defined Dirac cycle. Furthermore, when $\mathcal B = C(\Omega) \rtimes_{\xi'}^{\phi} \mathbb Z^j$, with $C(\Omega)$ the algebra of continuous functions over a disorder configuration space, we show that the numerical invariants are connected to the weak topological invariants of the complex classes of topological insulators, defined in the physics literature. The end products are generalized index theorems for these weak invariants, which enable us to predict the range of the invariants and to identify regimes of strong disorder in which the invariants remain stable. The latter will be reported in a subsequent publication.
- Dec 09 2015 cond-mat.dis-nn arXiv:1512.02476v1The current-current correlation function is a useful concept in the theory of electron transport in homogeneous solids. The finite-temperature conductivity tensor as well as Anderson's localization length can be computed entirely from this correlation function. Based on the critical behavior of these two physical quantities near the plateau-insulator or plateau-plateau transitions in the integer quantum Hall effect, we derive an asymptotic formula for the current-current correlation function, which enables us to make several theoretical predictions about its generic behavior. For the disordered Hofstadter model, we employ numerical simulations to map the current-current correlation function, obtain its asymptotic form near a critical point and confirm the theoretical predictions.
- This monograph offers an overview on the topological invariants in fermionic topological insulators from the complex classes. Tools from K-theory and non-commutative geometry are used to define bulk and boundary invariants, to establish the bulk-boundary correspondence and to link the invariants to physical observables.
- Jul 10 2015 cond-mat.dis-nn arXiv:1507.02605v3In the strictly periodic setting, the electric polarization of inversion-symmetric solids with and without time-reversal symmetry and the isotropic magneto-electric response function of time-reversal symmetric insulators are known to be topological invariants displaying an exact $\mathbb Z_2$ quantization. This quantization is stabilized by the symmetries. In the present work, we investigate the fate of such symmetry-stabilized topological invariants in the presence of a disorder which breaks the symmetries but restores them on average. Using a rigorous analysis, we conclude that the strict quantization still holds in these conditions. Numerical calculations confirm this prediction.
- Mar 17 2015 cond-mat.str-el arXiv:1503.04757v3A concrete strategy is presented for generating strong topological insulators in $d+d'$ dimensions which have quantized physics in $d$ dimensions. Here, $d$ counts the physical and $d'$ the virtual dimensions. It consists of seeking $d$-dimensional representations of operator algebras which are usually defined in $d+d'$ dimensions where topological elements display strong topological invariants. The invariants are shown, however, to be fully determined by the physical dimensions, in the sense that their measurement can be done at fixed virtual coordinates. We solve the bulk-boundary correspondence and show that the boundary invariants are also fully determined by the physical coordinates. We analyze the virtual Chern insulator in $(1+1)$-dimensions realized in Ref.~\citeKrausPRL2012hh and predict quantized forces at the edges. We generate a novel topological system in $(3+1)$-dimensions, which is predicted to have quantized magneto-electric response.
- We present a natural imbedding of the crossed product $\mathcal A \rtimes_\xi \mathbb Z^d$ into the $C^\ast$-algebra of adjointable operators over the standard Hilbert $\mathcal A$-module $\mathcal H_{\mathcal A}$. By replacing the representations on Hilbert spaces with this canonical imbedding, we define Fredholm modules and corresponding Chern-Connes characters that are intrinsic to the $C^\ast$-dynamical system $(\mathcal A,\xi,\mathbb Z^d)$. The compression of the Dirac operator against projectors from $\mathcal A \rtimes_\xi \mathbb Z^d$ produces generalized Fredholm operators over $\mathcal H_{\mathcal A}$ and Mingo's index defines a $KK$-map from $K_0(\mathcal A \rtimes_\xi \mathbb Z^d)$ to $K(\mathcal A)$. Using a generalized Fedosov principle and a generalized Fedosov formula, we prove an index formula for the pairing of the intrinsic Chern-Connes characters and $K_0(\mathcal A \rtimes_\xi \mathbb Z^d)$. This pairing takes values in the image of $K_0(\mathcal A)$ in $\mathbb R$ under a canonical trace. A local index formula enables new applications in condensed matter physics to the so called weak topological invariants.
- Aug 12 2014 cond-mat.dis-nn arXiv:1408.2446v1The effect of strong disorder on chiral-symmetric 3-dimensional lattice models is investigated via analytical and numerical methods. The phase diagrams of the models are computed using the non-commutative winding number, as functions of disorder strength and model's parameters. The localized/delocalized characteristic of the quantum states is probed with level statistics analysis. Our study re-confirms the accurate quantization of the non-commutative winding number in the presence of strong disorder, and its effectiveness as a numerical tool. Extended bulk states are detected above and below the Fermi level, which are observed to undergo the so called "levitation and pair annihilation" process when the system is driven through a topological transition. This suggests that the bulk invariant is carried by these extended states, in stark contrast with the 1-dimensional case where the extended states are completely absent and the bulk invariant is carried by the localized states.
- Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, $C$-selfadjoint extensions of $C$-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of $C$-orthonormal vectors, and conjugate-linear symmetric operators. The main results are complemented by a variety of natural examples arising in field theory, quantum physics, and complex variables.
- Alain Connes' Non-Commutative Geometry program [Connes 1994] has been recently carried out [Prodan, Leung, Bellissard 2013, Prodan, Schulz-Baldes 2014] for the entire A- and AIII-symmetry classes of topological insulators, in the regime of strong disorder where the insulating gap is completely filled with dense localized spectrum. This is a short overview of these results, whose goal is to highlight the methods of Non-Commutative Geometry involved in these studies. The exposition proceeds gradually through the cyclic cohomology, quantized calculus with Fredholm-modules, local formulas for the odd and even Chern characters and index theorems for the odd and even Chern numbers. The characterization of the A- and AIII-symmetry classes in the presence of strong disorder and magnetic fields emerges as a natural application of these tools.
- Mar 03 2014 cond-mat.dis-nn arXiv:1402.7116v3Using an explicit 1-dimensional model, we provide direct evidence that the one-dimensional topological phases from the AIII and BDI symmetry classes follow a $\mathbb Z$-classification, even in the strong disorder regime when the Fermi level is embedded in a dense localized spectrum. The main tool for our analysis is the winding number $\nu$, in the non-commutative formulation introduced in I. Mondragon-Shem, J. Song, T. L. Hughes, and E. Prodan, arXiv:1311.5233. For both classes, by varying the parameters of the model and/or the disorder strength, a cascade of sharp topological transitions $\nu=0 \rightarrow \nu=1 \rightarrow \nu=2$ is generated, in the regime where the insulating gap is completely filled with the localized spectrum. We demonstrate that each topological transition is accompanied by an Anderson localization-delocalization transition. Furthermore, to explicitly rule out a $\mathbb Z_2$ classification, a topological transition between $\nu=0$ and $\nu=2$ is generated. These two phases are also found to be separated by an Anderson localization-delocalization transition, hence proving their distinct identity.
- An odd index theorem for higher odd Chern characters of crossed product algebras is proved. It generalizes the Noether-Gohberg-Krein index theorem. Furthermore, a local formula for the associated cyclic cocycle is provided. When applied to the non-commutative Brillouin zone, this allows to define topological invariants for condensed matter phases from the chiral unitary (or AIII-symmetry) class in the presence of strong disorder and magnetic fields whenever the Fermi level lies in region of Anderson localization.
- Nov 22 2013 cond-mat.dis-nn cond-mat.str-el arXiv:1311.5233v2The chiral AIII symmetry class in the periodic table of topological insulators contains topological phases classified by a winding number $\nu$ for each odd space-dimension. An open problem for this class is the characterization of the phases and phase-boundaries in the presence of strong disorder. In this work, we derive a covariant real-space formula for $\nu$ and, using an explicit 1-dimensional disordered topological model, we show that $\nu$ remains quantized and non-fluctuating when disorder is turned on, even though the bulk energy-spectrum is completely localized. Furthermore, $\nu$ remains robust even after the insulating gap is filled with localized states, but when the disorder is increased even further, an abrupt change of $\nu$ to a trivial value is observed. Using exact analytic calculations, we show that this marks a critical point where the localization length diverges. As such, in the presence of disorder, the AIII class displays a markedly different physics from everything known to date, with robust invariants being carried entirely by localized states and bulk extended states emerging from an absolutely localized spectrum. Detailed maps and a clear physical description of the phases and phase boundaries are presented based on numerical and exact analytic calculations.
- The theory of the higher Chern numbers in the presence of strong disorder is developed. Sharp quantization and homotopy invariance conditions are provided. The relevance of the result to the field of strongly disordered topological insulators is discussed.
- Feb 07 2013 cond-mat.mes-hall cond-mat.dis-nn arXiv:1302.1470v3Using the non-commutative Kubo formula for aperiodic solids [1-3] and a recently developed numerical implementation [4], we study the conductivity $\sigma$ and resistivity $\rho$ tensors as functions of Fermi level and temperature, for models of strongly disordered Chern insulators. The formalism enabled us to converge the transport coefficients at temperatures low enough to enter the quantum critical regime at the Chern-to-trivial insulator transition. We find that the $\rho_{xx}$-curves at different temperatures intersect each other at one single critical point, and that they obey a single-parameter scaling law with an exponent close to the universally accepted value for the unitary symmetry class. However, when compared with the established experimental facts on the plateau-insulator transition in the Integer Quantum Hall Effect, we find a universal critical conductance $\sigma_{xx}^c$ twice as large, an ellipse rather than a semi-circle law, and absence of the quantized Hall insulator phase.
- Jan 23 2013 cond-mat.mes-hall cond-mat.dis-nn arXiv:1301.5305v4The conductivity $\sigma$ and resistivity $\rho$ tensors of the disordered Hofstadter model are mapped as functions of Fermi energy $E_F$ and temperature $T$ in the quantum critical regime of the plateau-insulator transition (PIT). The finite-size errors are eliminated by using the non-commutative Kubo-formula. The results reproduce all the key experimental characteristics of this transition in Integer Quantum Hall (IQHE) systems. In particular, the Quantized Hall Insulator (QHI) phase is detected and analyzed. The presently accepted characterization of the QHI phase in the quantum critical regime, based entirely on experimental data, is fully supported by our theoretical investigation.
- A non-commutative formula for the isotropic magneto-electric response of disordered insulators under magnetic fields is derived using the methods of non-commutative geometry. Our result follows from an explicit evaluation of the Ito derivative with respect to the magnetic field of the non-commutative formula for the electric polarization reported in Ref. 1. The quantization, topological invariance and connection to a second Chern number of the magneto-electric response are discussed in the context of 3-dimensional, disordered, time-reversal or inversion symmetric topological insulators.
- The non-commutative theory of charge transport in mesoscopic aperiodic systems under magnetic fields, developed by Bellissard, Shulz-Baldes and collaborators in the 90's, is complemented with a practical numerical implementation. The scheme, which is developed within a $C^*$-algebraic framework, enable efficient evaluations of the non-commutative Kubo formula, with errors that vanish exponentially fast in the thermodynamic limit. Applications to a model of a 2-dimensional Quantum spin-Hall insulator are given. The conductivity tensor is mapped as function of Fermi level, disorder strength and temperature and the phase diagram in the plane of Fermi level and disorder strength is quantitatively derived from the transport simulations. Simulations at finite magnetic field strength are also presented.
- The linear conductivity tensor for generic homogeneous, microscopic quantum models was formulated as a noncommutative Kubo formula in Refs. \citeBELLISSARD:1994xj,Schulz-Baldes:1998vm,Schulz-Baldes:1998oq. This formula was derived directly in the thermodynamic limit, within the framework of $C^*$-algebras and noncommutative calculi defined over infinite spaces. As such, the numerical implementation of the formalism encountered fundamental obstacles. The present work defines a $C^*$-algebra and an approximate noncommutative calculus over a finite real-space torus, which naturally leads to an approximate finite-volume noncommutative Kubo formula, amenable on a computer. For finite temperatures and dissipation, it is shown that this approximate formula converges exponentially fast to its thermodynamic limit, which is the exact noncommutative Kubo formula. The approximate noncommutative Kubo formula is then deconstructed to a form that is implementable on a computer and simulations of the quantum transport in a 2-dimensional disordered lattice gas in a magnetic field are presented.
- Feb 10 2012 cond-mat.dis-nn cond-mat.str-el arXiv:1202.2108v3We study the effect of strong disorder in a 3-dimensional topological insulators with time-reversal symmetry and broken inversion symmetry. Firstly, using level statistics analysis, we demonstrate the persistence of delocalized bulk states even at large disorder. The delocalized spectrum is seen to display the levitation and pair annihilation effect, indicating that the delocalized states continue to carry the Z2 invariant after the onset of disorder. Secondly, the Z2 invariant is computed via twisted boundary conditions using an efficient numerical algorithm. We demonstrate that the Z2 invariant remains quantized and non-fluctuating even after the spectral gap becomes filled with dense localized states. In fact, our results indicate that the Z2 invariant remains quantized until the mobility gap closes or until the Fermi level touches the mobility edges. Based on such data, we compute the phase diagram of the Bi2Se3 topological material as function of disorder strength and position of the Fermi level.
- Aug 16 2011 cond-mat.mes-hall arXiv:1108.2929v1We demonstrate the existence of robust bulk extended states in the disordered Kane-Mele model with vertical and horizontal Zeeman fields, in the presence of a large Rashba coupling. The phase diagrams are mapped out by using level statistics analysis and computations of the localization length and spin-Chern numbers $C_\pm$. $C_\pm$ are protected by the finite energy and spin mobility gaps. The latter is shown to stay open for arbitrarily large vertical Zeeman fields, or for horizontal Zeeman fields below a critical strength or at moderate disorder. In such cases, a change of $C_\pm$ is necessarily accompanied by the closing of the mobility gap at the Fermi level. The numerical simulations reveal sharp changes in the quantized values of $C_\pm$ when crossing the regions of bulk extended states, indicating that the topological nature of the extended states is indeed linked to the spin-Chern numbers. For large horizontal Zeeman fields, the spin-gap closes at strong disorder prompting a change in the quantized spin-Chern numbers without a closing of the energy mobility gap.
- Mar 24 2011 cond-mat.mtrl-sci arXiv:1103.4603v5We use a "monodromy" argument to derive new expressions for the ${\bm Z}_2$ invariants of topological insulators with time-reversal symmetry in 2 and 3 dimensions. The derivations and the final expressions do not require any gauge choice and the calculation of the invariants is based entirely on the projectors onto the occupied states. Explicit numerical tests for tight-binding models with strongly broken inversion symmetry are presented in 2 and 3-dimensions.
- Feb 23 2011 cond-mat.dis-nn cond-mat.mes-hall arXiv:1102.4535v2We compute the phase diagram of the HgTe/CdTe quantum wells in the 3 dimensional (3D) parameter space of Dirac mass, Fermi level and disorder strength. The phase diagram reveals the Quantum spin-Hall, the metallic and the normal insulating phases. The phase boundary of the Quantum spin-Hall state is shown to be strongly deformed by the disorder. Taking specific cuts into this 3D phase diagram, we recover the so called topological Anderson insulator (TAI) phase, but now we can demonstrate explicitly that TAI is not a distinct phase and instead it is part of the Quantum spin-Hall phase. The calculations are performed with $S_z$-conserving and $S_z$-nonconserving Hamiltonians.
- This paper demonstrates the existence of topological models with gapped edge states but protected extended bulk states against disorder. Such systems will be labeled as trivial by the current classification of topological insulators. Our finding calls for a re-examination of the definition of a topological insulator. The analysis is supported by extensive numerical data for a model of non-interacting electrons in the presence of strong disorder. In the clean limit, the model displays a topological insulating phase with spin-Chern number $C_s$=2 and gapped edge states. In the presence of disorder, level statistics on energy spectrum reveals regions of extended states displaying levitation and pair annihilation. Therefore, the extended states carry a topological invariant robust against disorder. By driving the Fermi level over the mobility edges, it is shown that this invariant is precisely the spin-Chern number. The protection mechanism for the extended state is explained.
- Oct 27 2010 cond-mat.soft q-bio.QM arXiv:1010.5407v2Topological phonon modes are robust vibrations localized at the edges of special structures. Their existence is determined by the bulk properties of the structures and, as such, the topological phonon modes are stable to changes occurring at the edges. The first class of topological phonons was recently found in 2-dimensional structures similar to that of Microtubules. The present work introduces another class of topological phonons, this time occurring in quasi one-dimensional filamentous structures with inversion symmetry. The phenomenon is exemplified using a structure inspired from that of actin Microfilaments, present in most live cells. The system discussed here is probably the simplest structure that supports topological phonon modes, a fact that allows detailed analysis in both time and frequency domains. We advance the hypothesis that the topological phonon modes are ubiquitous in the biological world and that living organisms make use of them during various processes.
- Oct 22 2010 cond-mat.mes-hall cond-mat.other arXiv:1010.4508v1We study translationally-invariant insulators with inversion symmetry that fall outside the established classification of topological insulators. These insulators are not required to have gapless boundary modes in the energy spectrum. However, they do exhibit protected modes in the entanglement spectrum localized on the cut between two entangled regions. Their entanglement entropy cannot be made to vanish adiabatically, and hence the insulators can be called topological. There is a direct connection between the inversion eigenvalues of the band structure and the mid-gap states in the entanglement spectrum. The classification of protected entanglement levels is given by an integer $n\in Z$, which is the difference between the negative inversion eigenvalues at inversion symmetric points in the Brillouin zone, taken in sets of two. When the Hamiltonian describes a Chern insulator or a non-trivial T-invariant topological insulator, the entanglement spectrum exhibits spectral flow. If the Chern number is zero for the former, or T is broken in the latter, the entanglement spectrum does \emphnot have spectral flow, but, depending on the inversion eigenvalues, can still have protected midgap bands. Although spectral flow is broken, the mid-gap entanglement bands cannot be adiabatically removed, and the insulator is `topological.' In 1D, we establish a link between the product of the inversion eigenvalues of all occupied bands at all inversion momenta and charge polarization. In 2D, we prove a link between the product of the inversion eigenvalues and the parity of the Chern number. In 3D, we find a topological constraint on the product of the inversion eigenvalues indicating that some 3D materials are topological metals, and we show the link between the inversion eigenvalues and the 3D Quantum Hall Effect and the magnetoelectric polarization in the absence of T-symmetry.
- This review deals with strongly disordered topological insulators and covers some recent applications of a well established analytic theory based on the methods of Non-Commutative Geometry (NCG) and developed for the Integer Quantum Hall-Effect. Our main goal is to exemplify how this theory can be used to define topological invariants in the presence of strong disorder, other than the Chern number, and to discuss the physical properties protected by these invariants. Working with two explicit 2-dimensional models, one for a Chern insulator and one for a Quantum spin-Hall insulator, we first give an in-depth account of the key bulk properties of these topological insulators in the clean and disordered regimes. Extensive numerical simulations are employed here. A brisk but self-contained presentation of the non-commutative theory of the Chern number is given and a novel numerical technique to evaluate the non-commutative Chern number is presented. The non-commutative spin-Chern number is defined and the analytic theory together with the explicit calculation of the topological invariants in the presence of strong disorder are used to explain the key bulk properties seen in the numerical experiments presented in the first part of the review.
- May 28 2010 cond-mat.mes-hall quant-ph arXiv:1005.5148v2How much information is stored in the ground-state of a system without \emphany symmetry and how can we extract it? This question is investigated by analyzing the behavior of a topological Chern Insulator (CI) in the presence of disorder, with a focus on its entanglement spectrum (EtS) constructed from the ground state. For systems with symmetries, the EtS was shown to contain explicit information revealed by sorting the EtS against the conserved quantum numbers. In the absence of any symmetry, we demonstrate that statistical methods such as the level statistics of the EtS can be equally insightful, allowing us to distinguish when an insulator is in a topological or trivial phase and to map the boundary between the two phases, where EtS becomes entirely delocalized. The phase diagram of a CI is explicitly computed as function of Fermi level ($E_F$) and disorder strength using the level statistics of the EtS and energy spectrum (EnS), together with a computation of the Chern number via an efficient real-space formula.
- Jan 13 2010 cond-mat.str-el cond-mat.stat-mech arXiv:1001.1930v1In this paper we explore the braiding properties of the Moore-Read fractional Hall sequence, which amounts to computing the adiabatic evolution of the Hall liquid when the anyons are moved along various trajectories. In this work, the anyons are pinned to precise spatial configurations by using specific external potentials. Such external potentials break the translational symmetry and it appears that one will be forced to simulate the braidings on the entire many-body Hilbert space, an absolutely prohibitive scenario. We demonstrate how to overcome this difficulty and obtain the exact braidings for fairly large Hall systems. For this, we show that the incompressible state of a general $(k,m)$ fractional Hall sequence can be viewed as the unique zero mode of a specific Hamiltonian $H^{(k,m)}$, whose form is explicitly derived by using k-particles creation operators. The compressible Hall states corresponding to $n$$\times$$k$ anyons fixed at $w_1$,...,$w_{nk}$ are shown to be the zero modes of a pinning Hamiltonian $H^{(k,m)}_{w_1,...,w_{nk}}$, which is also explicitly derived. The zero modes of $H^{(k,m)}_{w_1,...,w_{nk}}$ are shown to be contained in the space of the zero modes of $H^{(k,m)}$. Therefore, the computation of the braidings can be done entirely within this space, which we map out for a number of Hall systems. Using this efficient computational method, we study various properties of the Moore-Read states. In particular, we give direct confirmation of their topological and non-abelian properties that were previously implied from the underlying Conformal Field Theory (CFT) structure of the Moore-Read state.
- Nov 17 2009 cond-mat.stat-mech cond-mat.dis-nn arXiv:0911.2816v4This paper reviews several analytic tools for the field of topological insulators, developed with the aid of non-commutative calculus and geometry. The set of tools includes bulk topological invariants defined directly in the thermodynamic limit and in the presence of disorder, whose robustness is shown to have non-trivial physical consequences for the bulk states. The set of tools also includes a general relation between the current of an observable and its edge index, relation that can be used to investigate the robustness of the edge states against disorder. The paper focuses on the motivations behind creating such tools and on how to use them.
- Nov 12 2009 cond-mat.quant-gas cond-mat.str-el arXiv:0911.2184v1This work presents a many-fermion Hamiltonian with the following properties: 1) is exactly solvable, 2) has a second order insulator-metal quantum phase transition, 3) has a well defined mean field approximation and 4) its mean-field ground state displays a liquid-solid transition. The phenomenon of symmetry breaking in fermionic self-consistent models is discussed in the light of these remarkable properties of the many-body model.
- Sep 21 2009 cond-mat.soft cond-mat.stat-mech arXiv:0909.3492v3Microtubules (MTs) are self-assembled hollow protein tubes playing important functions in live cells. Their building block is a protein called tubulin, which self-assembles in a particulate 2 dimensional lattice. We study the vibrational modes of this lattice and find Dirac points in the phonon spectrum. We discuss a splitting of the Dirac points that leads to phonon bands with nonzero Chern numbers, signaling the existence of topological vibrational modes localized at MTs edges, which we indeed observe after explicit calculations. Since these modes are robust against the large changes occurring at the edges during the dynamic cycle of the MTs, we can build a simple mechanical model to illustrate how they would participate in this phenomenon.
- Jul 28 2009 cond-mat.mes-hall arXiv:0907.4636v2This paper extends the modern theory of tunneling transport to finite temperatures. The extension enables applications to molecular electronic devices connected to semiconducting leads. The paper presents an application of the theory to molecular devices made of alkyl chains connected to silicon nano-wires, mapping their transport characteristics as functions of temperature and alkyl chain's length. Based on these calculations and on the analytic theory, it is found that the tunneling decay constant is determined not by the Fermi level, but by the edge of the valence or conductance band, whichever is closer to the Fermi level. Further insight is provided by mapping the evanescent transport channels of the alkyl chains and few other physical quantities appearing in the analytic formula for conductance. A good qualitative agreement with the experimental data is obtained.
- Apr 21 2009 cond-mat.mes-hall cond-mat.stat-mech arXiv:0904.3007v1We present an extended discussion of a recently proposed theoretical approach for off-resonance tunneling transport. The proofs and the arguments are explained at length and simple analogies and illustrations are used where possible. The result is an analytic formula for the asymptotic tunneling conductance which involves the overlap of three well defined physical quantities. We argue that the formula can be used to gain fresh insight into the tunneling transport characteristics of various systems. The formalism is applied here to molecular devices consisting of planar phenyl chains connected to gold electrodes via amine linkers.
- The Spin-Chern ($C_s$) was originally introduced on finite samples by imposing spin boundary conditions at the edges. This definition lead to confusing and contradictory statements. On one hand the original paper by Sheng and collaborators revealed robust properties of $C_s$ against disorder and certain deformations of the model and, on the other hand, several people pointed out that $C_s$ can change sign under special deformations that keep the bulk Hamiltonian gap open. Because of the later findings, the Spin-Chern number was dismissed as a true bulk topological invariant and now is viewed as something that describes the edge where the spin boundary conditions are imposed. In this paper, we define the Spin-Chern number directly in the thermodynamic limit, without using any boundary conditions. We demonstrate its quantization in the presence of strong disorder and we argue that $C_s$ is a true bulk topological invariant whose robustness against disorder and smooth deformations of the Hamiltonian have important physical consequences. The properties of the Spin-Chern number remain valid even when the time reversal invariance is broken.
- Quantum Spin-Hall systems are topological insulators displaying dissipationless spin currents flowing at the edges of the samples. In contradistinction to the Quantum Hall systems where the charge conductance of the edge modes is quantized, the spin conductance is not and it remained an open problem to find the observable whose edge current is quantized. In this paper, we define a particular observable and the edge current corresponding to this observable. We show that this current is quantized and that the quantization is given by the index of a certain Fredholm operator. This provides a new topological invariant that is shown to take same values as the Spin-Chern number previously introduced in the literature. The result gives an effective tool for the investigation of the edge channels' structure in Quantum Spin-Hall systems. Based on a reasonable assumption, we also show that the edge conducting channels are not destroyed by a random edge.
- Chern insulators are periodic band insulators with the property that their projector onto the occupied bands have non-zero Chern number. Chern insulator with a homogeneous boundary display continuum spectrum that fills the entire insulating gap. The local density of states corresponding to this part of the spectrum is localized near the boundary, hence the terminology edge spectrum. An interesting question arises, namely, if a rough boundary, which can be seen as a strong random potential acting on these quasi 1-dimensional states, would destroy the continuum edge spectrum. This paper shows how such question can be answered via a newly formulated abstract framework in which the expectation value of the current of a general observable is connected to the index of a specific Fredholm operator. For the present application, we will connect the expectation value of the charge edge current with the index of a Fredholm operator that remains invariant under arbitrary deformations of the boundary.
- May 14 2008 cond-mat.soft cond-mat.other arXiv:0805.1745v1We develop a theoretical framework to describe the dielectric response of live cells in suspensions when placed in low external electric fields. The treatment takes into account the presence of the cell's membrane and of the charge movement at the membrane's surfaces. For spherical cells suspended in aqueous solutions, we give an analytic solution for the dielectric function, which is shown to account for the alpha and beta plateaus seen in many experimental data. The effect of different physical parameters on the dielectric curves is methodically analyzed.
- Apr 10 2008 cond-mat.mes-hall cond-mat.mtrl-sci arXiv:0804.1351v1We present periodic Density Functional Theory calculations of the electronic properties of molecular junctions formed by amine-, and thiol-terminated alkane chains attached to two metal (Au, Ag) electrodes. Based on extensive analysis that includes molecular monolayers of varying densities, we establish a relationship between the alignment of the molecular energy levels and the interface dipoles, which shows that the band alignment (BA) in the limit of long, isolated chains is independent of the link group and can be computed from a reference system of non interacting molecule + metal electrodes. The main difference between the amine and thiol linkers is the effective dipole moment at the contact. This is very large, about 4.5 D, for amine linkers, leading to a strong dependence of the BA on the monolayer density and a slow convergence to the isolated molecule limit. Instead, this convergence is fast for S anchors due to the very small, ~ 0.2 D, effective dipoles at the contacts.
- We define the current of a quantum observable and, under well defined conditions, we connect its ensemble average to the index of a Fredholm operator. The present work builds on a formalism developed by Kellendonk and Schulz-Baldes \citeKellendonk:2004p597 to study the quantization of edge currents for continuous magnetic Schroedinger operators. The generalization given here may be a useful tool to scientists looking for novel manifestations of the topological quantization. As a new application, we show that the differential conductance of atomic wires is given by the index of a certain operator. We also comment on how the formalism can be used to probe the existence of edge states.
- Mar 06 2008 cond-mat.mes-hall arXiv:0803.0710v2The tunneling transport theory developed in Phys. Rev. B \bf 76, 115102 (2007) is applied to molecular devices made of alkyl chains linked to gold electrodes via amine groups. Using the analytic expression of the tunneling conductance derived in our previous work, we identify the key physical quantities that characterize the conductance of these devices. By investigating the transport characteristics of three devices, containing 4, 6, and 8 methyl groups, we extract the dependence of the tunneling conductance on the chain's length, which is an exponential decay law in close agreement with recent experimental data.
- Nov 12 2007 cond-mat.mtrl-sci cond-mat.stat-mech arXiv:0711.1447v2We consider single particle Schrodinger operators with a gap in the en ergy spectrum. We construct a complete, orthonormal basis function set for the inv ariant space corresponding to the spectrum below the spectral gap, which are exponentially localized a round a set of closed surfaces of monotonically increasing sizes. Estimates on the exponential dec ay rate and a discussion of the geometry of these surfaces is included.
- Feb 09 2007 cond-mat.mtrl-sci arXiv:cond-mat/0702192v1Inspired by the work of Kamenev and Kohn, we present a general discussion of the two-terminal dc conductance of molecular devices within the framework of Time Dependent Current-Density Functional Theory. We derive a formally exact expression for the adiabatic conductance and we discuss the dynamical corrections. For junctions made of long molecular chains that can be either metallic or insulating, we derive the exact asymptotic behavior of the adiabatic conductance as a function of the chain's length. Our results follow from the analytic structure of the bands of a periodic molecular chain and a compact expression for the Green's functions. In the case of an insulating chain, not only do we obtain the exponentially decaying factors, but also the corresponding amplitudes, which depend very sensitively on the electronic properties of the contacts. We illustrate the theory by a numerical study of a simple insulating structure connected to two metallic jellium leads.
- This paper deals with Hamiltonians of the form $H=-{\bf \nabla}^2+v(\rr)$, with $v(\rr)$ periodic along the $z$ direction, $v(x,y,z+b)=v(x,y,z)$. The wavefunctions of $H$ are the well known Bloch functions $\psi_{n,\lambda}(\rr)$, with the fundamental property $\psi_{n,\lambda}(x,y,z+b)=\lambda \psi_{n,\lambda}(x,y,z)$ and $\partial_z\psi_{n,\lambda}(x,y,z+b)=\lambda \partial_z\psi_{n,\lambda}(x,y,z)$. We give the generic analytic structure (i.e. the Riemann surface) of $\psi_{n,\lambda}(\rr)$ and their corresponding energy, $E_n(\lambda)$, as functions of $\lambda$. We show that $E_n(\lambda)$ and $\psi_{n,\lambda}(x,y,z)$ are different branches of two multi-valued analytic functions, $E(\lambda)$ and $\psi_\lambda(x,y,z)$, with an essential singularity at $\lambda=0$ and additional branch points, which are generically of order 1 and 3, respectively. We show where these branch points come from, how they move when we change the potential and how to estimate their location. Based on these results, we give two applications: a compact expression of the Green's function and a discussion of the asymptotic behavior of the density matrix for insulating molecular chains.
- Jun 28 2005 cond-mat.stat-mech cond-mat.mtrl-sci arXiv:cond-mat/0506687v2The concept of nearsightedeness of electronic matter (NEM) was introduced by W. Kohn in 1996 as the physical principal underlining Yang's electronic structure alghoritm of divide and conquer. It describes the fact that, for fixed chemical potential, local electronic properties at a point $r$, like the density $n(r)$, depend significantly on the external potential $v$ only at nearby points. Changes $\Delta v$ of that potential, \it no matter how large, beyond a distance $\textsf{R}$, have \it limited effects on local electronic properties, which tend to zero as function of $\textsf{R}$. This remains true even if the changes in the external potential completely surrounds the point $r$. NEM can be quantitatively characterized by the nearsightedness range, $\textsf{\textsf{R}}(r,\Delta n)$, defined as the smallest distance from $r$, beyond which \it any change of the external potential produces a density change, at $r$, smaller than a given $\Delta n$. The present paper gives a detailed analysis of NEM for periodic metals and insulators in 1D and includes sharp, explicit estimates of the nearsightedness range. Since NEM involves arbitrary changes of the external potential, strong, even qualitative changes can occur in the system, such as the discretization of energy bands or the complete filling of the insulating gap of an insulator with continuum spectrum. In spite of such drastic changes, we show that $\Delta v$ has only a limited effect on the density, which can be quantified in terms of simple parameters of the unperturbed system.
- In an earlier paper, W. Kohn had qualitatively introduced the concept of "nearsightedness" of electrons in many-atom systems. It can be viewed as underlying such important ideas as Pauling's "chemical bond," "transferability" and Yang's computational principle of "divide and conquer." It describes the fact that, for fixed chemical potential, local electronic properties, like the density $n(r)$, depend significantly on the effective external potential only at nearby points. Changes of that potential, \it no matter how large, beyond a distance $\textsf{R}$ have \it limited effects on local electronic properties, which rapidly tend to zero as function of $\textsf{R}$. In the present paper, the concept is first sharpened for representative models of uncharged fermions moving in external potentials, followed by a discussion of the effects of electron-electron interactions and of perturbing external charges.
- Feb 03 2005 cond-mat.stat-mech math-ph math.MP physics.chem-ph physics.comp-ph arXiv:cond-mat/0502065v3The Kohn-Sham (KS) equations determine, in a self-consistent way, the particle density of an interacting fermion system at thermal equilibrium. We consider a situation when the KS equations are known to have a unique solution at high temperatures and this solution is a uniform particle density. We show that, at zero temperature, there are stable solutions that are not uniform. We provide the general principles behind this phenomenon, namely the conditions when it can be observed and how to construct these non-uniform solutions. Two concrete examples are provided, including fermions on the sphere which are shown to crystallize in a structure that resembles the C$_{60}$ molecule.
- Jul 14 2003 cond-mat.mtrl-sci arXiv:cond-mat/0307289v1The electronic structure and optical properties of metallic nanoshells are investigated using a jellium model and the Time Dependent Local Density Approximation (TDLDA). An efficient numerical implementation enables applications to nanoshells of realistic size with up to a million electrons. We demonstrate how a frequency dependent background polarizability of the jellium shell can be included in the TDLDA formalism. The energies of the plasmon resonances are calculated for nanoshells of different sizes and with different dielectric cores, dielectric embedding media, and dielectric shell backgrounds. The plasmon energies are found to be in good agreement with the results from classical Mie scattering theory using a Drude dielectric function. A comparison with experimental data shows excellent agreement between theory and the measured frequency dependent absorption spectra.
- Mar 26 2003 cond-mat.mtrl-sci cond-mat.stat-mech arXiv:cond-mat/0303518v1We study the question of existence and uniqueness for the finite temperature Kohn-Sham equations. For finite volumes, a unique soluion is shown to exists if the effective potential satisfies a set of general conditions and the coupling constant is smaller than a certain value. For periodic background potentials, this value is proven to be volume independent. In this case, the finite volume solutions are shown to converge as the thermodynamic limit is considered. The local density approximation is shown to satisfy the general conditions mentioned above.
- A model is discussed in which an electric field induces quantum nucleation of soliton-antisoliton pairs in a pinned charge or spin density wave. Coulomb blockade prevents pair creation until the electric field exceeds a sharp threshold value, which can be much smaller than the classical depinning field. We calculate the vacuum state energy and expectation value of the phase, which is treated as a quantum scalar field. We find that the phase can also be much smaller, below threshold, than predicted by classical ``sliding'' density wave models.