arXiv subjects

Emil Prodan

Publications and source records attributed to Emil Prodan.

At least 55 records · Page 3Linked to original sources

arXiv2020

Fermionic Topological Order on Generic Triangulations

Consider a finite triangulation of a surface $M$ of genus $g$ and assume that spin-less fermions populate the edges of the triangulation. The quantum dynamics of such particles takes place inside the algebra of canonical anti-commutation relations (CAR). Following Kitaev's work on toric models, we identify a sub-algebra of CAR generated by elements associated to the triangles and vertices of the triangulation. We show that any Hamiltonian drawn from this sub-algebra displays topological spectral degeneracy. More precisely, if $\mathcal P$ is any of its spectral projections, the Booleanization of the fundmental group $π_1(M)$ can be embedded inside the group of invertible elements of the corner algebra $\mathcal P \, {\rm CAR} \, \mathcal P$. As a consequence, $\mathcal P$ decomposes in $4^g$ lower projections. Furthermore, a projective representation of $\mathbb Z_2^{4g}$ is also explicitly constructed inside this corner algebra. Key to all these is a presentation of CAR as a crossed product with the Boolean group $(2^X,Δ)$, where $X$ is the set of fermion sites and $Δ$ is the symmetric difference.

cond-mat.str-el↗

arXiv2020

Bulk-Boundary Correspondence for Topological Insulators with Quantized Magneto-Electric Effect

We study bulk-boundary correspondences and related surface phenomena stabilized by the second Chern number in three-dimensional insulators driven in adiabatic cycles. Magnetic fields and disorder effects are incorporated in our analysis using operator algebraic methods. We use the connecting maps between the $K$-theories of bulk and boundary algebras as engines for the bulk-boundary correspondences. We discovered that both the exponential and the index connecting maps are relevant for the context considered here as they lead to distinct experimentally observable surface phenomena, such as pumping and transfer of quantum surface Hall states or proximity induced Hall effect. The surface Hall physics of time-reversal symmetric topological insulators is also investigated using the new tools, which can model irrational magnetic fluxes and arbitrary large surface disorder.

cond-mat.str-el↗

arXiv2020

Topological braiding of non-Abelian mid-gap defects in classical meta-materials

Non-trivial braid-group representations appear as non-Abelian quantum statistics of emergent Majorana zero modes in one and two-dimensional topological superconductors. Here, we generate such representations with topologically protected domain-wall modes in a classical analogue of the Kitaev superconducting chain, with a particle-hole like symmetry and a Z2 topological invariant. The mid-gap modes are found to exhibit distinct fusion channels and rich non-Abelian braiding properties, which are investigated using a T-junction setup. We employ the adiabatic theorem to explicitly calculate the braiding matrices for one and two pairs of these mid-gap topological defects.

cond-mat.mes-hall↗

arXiv2020

A Computer Code for Topological Quantum Spin Systems over Triangulated Surfaces

We derive explicit closed-form matrix representations of Hamiltonians drawn from tensored algebras, such as quantum spin Hamiltonians. These formulas enable us to soft-code generic Hamiltonian systems and to systematize the input data for uniformly structured as well as for un-structured Hamiltonians. The result is an optimal computer code that can be used as a black box that takes in certain input files and returns spectral information about the Hamiltonian. The code is tested on Kitaev's toric code deployed on triangulated surfaces of genus 0 and 1. The input file corresponding to the minimal triangulation of genus 2 is also supplied.

cond-mat.str-el↗

arXiv2019

Braiding Flux-Tubes in Topological Quantum and Classical Lattice Models from Class-D

We use magnetic flux-tubes to stabilize zero-energy modes in a lattice realization of a 2-dimensional superconductor from class D of classification table of topological condensed matter systems. The zero modes are exchanged by slowly displacing the flux-tubes and an application of the adiabatic theorem demonstrates the geometric nature of the resulting unitary time-evolution operators. Furthermore, an explicit numerical evaluation reveals that the evolutions are in fact topological, hence supplying a representation of the braid group, which turns out to be non-abelian. This physical representation is further formalized using single-strand planar diagrams. Lastly, we discuss how these predictions can be implemented with and observed in classical meta-materials and how the standard Majorana representation of the braid group can be generated by measuring derived physical observables.

cond-mat.str-el↗

arXiv2019

Valley-Chern Effect with LC-Resonators: A Modular Platform

The valley Chern-effect is theoretically demonstrated with a novel alternating current circuitry, where closed-loop LC-resonators sitting at the nodes of a honeycomb lattice are inductively coupled along the bonds. This enables us to generate a dynamical matrix which copies identically the Hamiltonian driving the electrons in graphene. The valley-Chern effect is generated by splitting the inversion symmetry of the lattice. After a detailed study of the Berry curvature landscape and of the localization of the interface modes, we derive an optimal configuration of the circuit. Furthermore, we show that Q-factors as high as $10^4$ can be achieved with reasonable materials and configurations.

cond-mat.mes-hall↗

arXiv2019

Observation of topological edge modes in a quasi-periodic acoustic waveguide

Topological boundary and interface modes are generated in an acoustic waveguide by simple quasi-periodic patternings of the walls. The procedure opens many topological gaps in the resonant spectrum and qualitative as well as quantitative assessments of their topological character are supplied. In particular, computations of the bulk invariant for the continuum wave equation are performed. The experimental measurements reproduce the theoretical predictions with high fidelity. In particular, acoustic modes with high Q-factors localized in the middle of a breathable waveguide are engineered by a simple patterning of the walls.

cond-mat.mes-hall↗

arXiv2019

Flat Energy Bands within Antiphase and Twin Boundaries and at Open Edges in Topological Materials

A model for two-dimensional electronic, photonic, and mechanical metamaterial systems is presented, which has flat one-dimensional zero-mode energy bands and stable localized states of a topological origin confined within twin boundaries, antiphase boundaries, and at open edges. Topological origins of these flat bands are analyzed for an electronic system as a specific example, using a two-dimensional extension of the Su-Schrieffer-Heeger Hamiltonian with alternating shift of the chains. It is demonstrated that the slow group velocities of the localized flat band states are sensitively controlled by the distance between the boundaries and the propagation can be guided through designed paths of these boundaries. We also discuss how to realize this model in metamaterials.

cond-mat.mes-hall↗

arXiv2018

Topological Classification Table Implemented with Classical Passive Meta-Materials

Topological condensed matter systems from class A and class AII of the classification table have received classical electromagnetic and mechanical analogs and protected wave-guiding with such systems has been demonstrated experimentally. Here we introduce a map which generates classical analogs for all entries of the classification table, using only passive elements. Physical mechanical models are provided for all strong topological phases in dimension 2, as well as for three classes in dimension 3. This includes topological super-conducting phases, which have never been attempted with classical systems.

cond-mat.mes-hall↗

arXiv2018

Topological Edge Modes by Smart Patterning

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.

physics.app-ph↗

arXiv2018

The K-Theoretic Bulk-Boundary Principle for Dynamically Patterned Resonators

Starting from a dynamical system $(Ω,G)$, with $G$ a generic topological group, we devise algorithms that generate families of patterns in the Euclidean space, which densely embed $G$ and on which $G$ acts continuously by rigid shifts. We refer to such patterns as being dynamically generated. For $G=\mathbb Z^d$, we adopt Bellissard's $C^\ast$-algebraic formalism to analyze the dynamics of coupled resonators arranged in dynamically generated point patterns. We then use the standard connecting maps of $K$-theory to derive precise conditions that assure the existence of topological boundary modes when a sample is halved. We supply four examples for which the calculations can be carried explicitly. The predictions are supported by many numerical experiments.

math-ph↗

arXiv2018

Non-Commutative Chern Numbers for Generic Aperiodic Discrete Systems

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.

math-ph↗

arXiv2018

Theory and Experimental Investigation of the Quantum Valley Hall Effect

The 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.

cond-mat.mes-hall↗

arXiv2017

Disordered Crystals from First Principles I: Quantifying the Configuration Space

This work represents the first chapter of a project on the foundations of first-principle calculations of the electron transport in crystals at finite temperatures. We are interested in the range of temperatures, where most electronic components operate, that is, room temperature and above. The aim is a predictive first-principle formalism that combines ab-initio molecular dynamics and a finite-temperature Kubo-formula for homogeneous thermodynamic phases. The input for this formula is the ergodic dynamical system $(Ω,\mathbb G,{\rm d}\mathbb P)$ defining the crystalline phase, where $Ω$ is the configuration space for the atomic degrees of freedom, $\mathbb G$ is the space group acting on $Ω$ and ${\rm d}\mathbb P$ is the ergodic Gibbs measure relative to the $\mathbb G$-action. The present work develops an algorithmic method for quantifying $(Ω,\mathbb G,{\rm d}\mathbb P)$ from first principles. Using the silicon crystal as a working example, we find the Gibbs measure to be extremely well characterized by a multivariate normal distribution, which can be quantified using a small number of parameters. The latter are computed at various temperatures and communicated in the form of a table. Using this table, one can generate large and accurate thermally-disordered atomic configurations to serve, for example, as input for subsequent simulations of the electronic degrees of freedom.

physics.comp-ph↗

arXiv2017

Bulk-boundary correspondance for Sturmian Kohmoto like models

We consider one dimensional tight binding models on $\ell^2(\mathbb Z)$ whose spatial structure is encoded by a Sturmian sequence $(ξ_n)_n\in \{a,b\}^\mathbb Z$. An example is the Kohmoto Hamiltonian, which is given by the discrete Laplacian plus an onsite potential $v_n$ taking value $0$ or $1$ according to whether $ξ_n$ is $a$ or $b$. The only non-trivial topological invariants of such a model are its gap-labels. The bulk-boundary correspondence we establish here states that there is a correspondence between the gap label and a winding number associated to the edge states, which arises if the system is augmented and compressed onto half space $\ell^2(\mathbb N)$. This has been experimentally observed with polaritonic waveguides. A correct theoretical explanation requires, however, first a smoothing out of the atomic motion via phason flips. With such an interpretation at hand, the winding number corresponds to the mechanical work through a cycle which the atomic motion exhibits on the edge states.

math-ph↗

arXiv2017

Dynamical Majorana Edge Modes in a Broad Class of Topological Mechanical Systems

Mechanical 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.

cond-mat.soft↗

arXiv2017

A Computational Non-Commutative Geometry Program for Disordered Topological Insulators

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.

math-ph↗

arXiv2016

Generalized Connes-Chern characters in KK-theory with an application to weak invariants of topological insulators

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 $\mathbb Z^k$, $\mathcal A = \mathcal B \rtimes_ξ^θ\mathbb Z^k$, we derive a local formula for the character corresponding to the fundamental class of a properly defined Dirac cycle. Furthermore, when $\mathcal B = C(Ω) \rtimes_{ξ'}^ϕ \mathbb Z^j$, with $C(Ω)$ 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.

math.OA↗