Searcharxiv⌕ Search

arXiv subjects

Emil Prodan

Publications and source records attributed to Emil Prodan.

At least 73 records · Page 4Linked to original sources
arXiv2016

Disorder effects in correlated topological insulators

Using 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.

cond-mat.str-el↗
arXiv2016

Non-commutative odd Chern numbers and topological phases of disordered chiral systems

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.

math-ph↗
arXiv2016

Topological Insulators at Strong Disorder

Topological insulators are newly discovered materials with the defining property that any boundary cut into such crystal supports spectrum which is immune to the Anderson localization. The present paper summarizes our efforts on the rigorous characterization of these materials in the regime of weak and strong disorder. In particular, the defining property is rigorously proven under certain relevant conditions, for more than half of the classification table of topological insulators.

math-ph↗
arXiv2015

Mapping the Current-Current Correlation Function Near a Quantum Critical Point

The 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.

cond-mat.dis-nn↗
arXiv2015

Generalized Wannier Functions

We 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.

cond-mat.mtrl-sci↗
arXiv2015

Quantization of Topological Invariants under Symmetry-Breaking Disorder

In 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.

cond-mat.dis-nn↗
arXiv2015

Virtual Topological Insulators with Real Quantized Physics

A 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.~\cite{KrausPRL2012hh} 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.

cond-mat.str-el↗
arXiv2015

Intrinsic Chern-Connes Characters for Crossed Products by $\mathbb Z^d$

We present a natural imbedding of the crossed product $\mathcal A \rtimes_ξ\mathbb Z^d$ 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 $(\mathcal A,ξ,\mathbb Z^d)$. The compression of the Dirac operator against projectors from $\mathcal A \rtimes_ξ\mathbb Z^d$ produces generalized Fredholm operators over $\mathcal H_{\mathcal A}$ and Mingo's index defines a $KK$-map from $K_0(\mathcal A \rtimes_ξ\mathbb Z^d)$ 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 $K_0(\mathcal A \rtimes_ξ\mathbb Z^d)$. 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.

math.OA↗
arXiv2014

Effect of Strong Disorder on 3-Dimensional Chiral Topological Insulators: Phase Diagrams, Maps of the Bulk Invariant and Existence of Topological Extended Bulk States

The 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.

cond-mat.dis-nn↗
arXiv2014

AIII and BDI Topological Systems at Strong Disorder

Using 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 $ν$, 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 $ν=0 \rightarrow ν=1 \rightarrow ν=2$ 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 $ν=0$ and $ν=2$ is generated. These two phases are also found to be separated by an Anderson localization-delocalization transition, hence proving their distinct identity.

cond-mat.dis-nn↗
arXiv2014

Mathematical and physical aspects of complex symmetric operators

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.

math.FA↗
arXiv2014

The Non-Commutative Geometry of the Complex Classes of Topological Insulators

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.

math-ph↗
arXiv2014

Characterization of the Quantized Hall Insulator Phase in the Quantum Critical Regime

The conductivity $σ$ and resistivity $ρ$ 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.

cond-mat.mes-hall↗
arXiv2013

Topological Criticality in the Chiral-Symmetric AIII Class at Strong Disorder

The chiral AIII symmetry class in the periodic table of topological insulators contains topological phases classified by a winding number $ν$ 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 $ν$ and, using an explicit 1-dimensional disordered topological model, we show that $ν$ remains quantized and non-fluctuating when disorder is turned on, even though the bulk energy-spectrum is completely localized. Furthermore, $ν$ remains robust even after the insulating gap is filled with localized states, but when the disorder is increased even further, an abrupt change of $ν$ 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.

cond-mat.dis-nn↗
arXiv2013

The non-commutative n-th Chern number

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.

math-ph↗
arXiv2013

Quantum criticality at the Chern-to-normal insulator transition

Using the non-commutative Kubo formula for aperiodic solids [1-3] and a recently developed numerical implementation [4], we study the conductivity $σ$ and resistivity $ρ$ 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 $ρ_{xx}$-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 $σ_{xx}^c$ twice as large, an ellipse rather than a semi-circle law, and absence of the quantized Hall insulator phase.

cond-mat.mes-hall↗
arXiv2012

A Non-Commutative Formula for the Isotropic Magneto-Electric Response

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.

math-ph↗