arXiv subjects

Emil J. Bergholtz

Publications and source records attributed to Emil J. Bergholtz.

At least 91 records · Page 5Linked to original sources

arXiv2017

Symmetry Enforced Stability of Interacting Weyl and Dirac Semimetals

The nodal and effectively relativistic dispersion featuring in a range of novel materials including two- dimensional graphene and three-dimensional Dirac and Weyl semimetals has attracted enormous interest during the past decade. Here, by studying the structure and symmetry of the diagrammatic expansion, we show that these nodal touching points are in fact perturbatively stable to all orders with respect to generic two-body interactions. For effective low-energy theories relevant for single and multilayer graphene, type-I and type-II Weyl and Dirac semimetals as well as Weyl points with higher topological charge, this stability is shown to be a direct consequence of a spatial symmetry that anti-commutes with the effective Hamiltonian while leaving the interaction invariant. A more refined argument is applied to the honeycomb lattice model of graphene showing that its Dirac points are also perturbatively stable to all orders. We also give examples of nodal Hamiltonians that acquire a gap from interactions as a consequence of symmetries different from those of Weyl and Dirac materials.

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arXiv2017

Charge density wave instabilities of type-II Weyl semimetals in a strong magnetic field

Shortly after the discovery of Weyl semimetals properties related to the topology of their bulk band structure have been observed, e.g. signatures of the chiral anomaly and Fermi arc surface states. These essentially single particle phenomena are well understood but whether interesting many-body effects due to interactions arise in Weyl systems remains much less explored. Here, we investigate the effect of interactions in a microscopic model of a type-II Weyl semimetal in a strong magnetic field. We identify a charge density wave (CDW) instability even for weak interactions stemming from the emergent nesting properties of the type-II Weyl Landau level dispersion. We map out the dependence of this CDW on magnetic field strength. Remarkably, as a function of decreasing temperature a cascade of CDW transitions emerges and we predict characteristic signatures for experiments.

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arXiv2017

Anatomy of Topological Surface States: Exact Solutions from Destructive Interference on Frustrated Lattices

The hallmark of topological phases is their robust boundary signature whose intriguing properties---such as the one-way transport on the chiral edge of a Chern insulator and the sudden disappearance of surface states forming open Fermi arcs on the surfaces of Weyl semimetals---are impossible to realize on the surface alone. Yet, despite the glaring simplicity of non-interacting topological bulk Hamiltonians and their concomitant energy spectrum, the detailed study of the corresponding surface states has essentially been restricted to numerical simulation. In this work, however, we show that exact analytical solutions of both topological and trivial surface states can be obtained for generic tight-binding models on a large class of geometrically frustrated lattices in any dimension without the need for fine-tuning of hopping amplitudes. Our solutions derive from local constraints tantamount to destructive interference between neighboring layer lattices perpendicular to the surface and provide microscopic insights into the structure of the surface states that enable analytical calculation of many desired properties. We illustrate our general findings on a large number of examples in two and three spatial dimensions. Notably, we derive exact chiral Chern insulator edge states on the spin orbit-coupled kagome lattice, and Fermi arcs relevant for various recently synthesized pyrochlore iridate slabs. Remarkably, each of the pyrochlore slabs exhibit Fermi arcs although only the ones with a magnetic one-in-three-out configuration feature bulk Weyl nodes in realistic parameter regimes. Our approach furthermore signal the absence of topological surface states, which we illustrate for a class of models akin to the trivial surface of Hourglass materials KHg$X$.

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arXiv2017

Exotic Non-Abelian Topological Defects in Lattice Fractional Quantum Hall States

We investigate extrinsic wormholelike twist defects that effectively increase the genus of space in lattice versions of multicomponent fractional quantum Hall systems. Although the original band structure is distorted by these defects, leading to localized midgap states, we find that a new lowest flat band representing a higher genus system can be engineered by tuning local single-particle potentials. Remarkably, once local many-body interactions in this new band are switched on, we identify various Abelian and non-Abelian fractional quantum Hall states, whose ground-state degeneracy increases with the number of defects, i.e, with the genus of space. This sensitivity of topological degeneracy to defects provides a "proof of concept" demonstration that genons, predicted by topological field theory as exotic non-Abelian defects tied to a varying topology of space, do exist in realistic microscopic models. Specifically, our results indicate that genons could be created in the laboratory by combining the physics of artificial gauge fields in cold atom systems with already existing holographic beam shaping methods for creating twist defects.

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arXiv2017

Transport properties of an interacting Majorana chain

We study a one-dimensional (1D) chain of $2N$ Majorana bound states, which interact through a local quartic interaction. This model describes for example the edge physics of a quasi 1D stack of $2N$ Kitaev chains with modified time-reversal symmetry $Tγ_iT^{-1}=γ_i$, which precludes the presence of quadratic coupling. The ground state of our 1D Majorana chain displays a four-fold periodicity in $N$, corresponding to the four distinct topological classes of the stacked Kitaev chains. We analyze the transport properties of the 1D Majorana chain, when probed by local conductors located at its ends. We find that for finite but large $N$, the scattering matrix partially reflects the four-fold periodicity, and the chain exhibits strikingly different transport properties for different chain lengths. In the thermodynamic limit, the 1D Majorana chain hosts a robust many-body zero mode, which indicates that the corresponding stacked two-dimensional bulk system realizes a weak topological phase.

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arXiv2017

Josephson effect in a Weyl SNS junction

We calculate the Josephson current density j(phi) for a Weyl superconductor--normal-metal--superconductor junction for which the outer terminals are superconducting Weyl metals and the normal layer is a Weyl (semi)metal. We describe the Weyl (semi)metal using a simple model with two Weyl points. The model has broken time-reveresal symmetry, but inversion symmetry is present. We calculate the Josephson current for both zero and finite temperature for the two pairing mechanisms inside the superconductors that have been proposed in the literature, zero-momentum BCS-like pairing and finite-momentum FFLO-like pairing, and assuming the short-junction limit. For both pairing types we find that the current is proportional to the normal-state junction conductivity, with a proportionality coefficient that shows quantitative differences between the two pairing mechanisms. The current for the BCS-like pairing is found to be independent of the chemical potential, whereas the current for the FFLO-like pairing is not.

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arXiv2017

Tilted disordered Weyl semimetals

Although Lorentz invariance forbids the presence of a term that tilts the energy-momentum relation in the Weyl Hamiltonian, a tilted dispersion is not forbidden and, in fact, generic for condensed matter realizations of Weyl semimetals. We here investigate the combined effect of such a tilted Weyl dispersion and the presence of potential disorder. In particular, we address the influence of a tilt on the disorder-induced phase transition between a quasi-ballistic phase at weak disorder, in which the disorder is an irrelevant perturbation, and a diffusive phase at strong disorder. Our main result is that the presence of a tilt leads to a reduction of the critical disorder strength for this transition or, equivalently, that increasing the tilt at fixed disorder strength drives the system through the phase transition to the diffusive strong-disorder phase. Notably this obscures the tilt induced Lifshitz transition to an over tilted type-II Weyl phase at any finite disorder strength. Our results are supported by analytical calculations using the self-consistent Born approximation and numerical calculations of the density of states and of transport properties.

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arXiv2016

Model Fractional Chern Insulators

We devise local lattice models whose ground states are model fractional Chern insulators---Abelian and non-Abelian topologically ordered states characterized by exact ground state degeneracies at any finite size and infinite entanglement gaps. Most saliently, we construct exact parent Hamiltonians for two distinct families of bosonic lattice generalizations of the $\mathcal{Z}_k$ parafermion quantum Hall states: (i) color-entangled fractional Chern insulators at band filling fractions $ν=k/(\mathcal{C}+1)$ and (ii) nematic states at $ν=k/2$, where $\mathcal{C}$ is the Chern number of the lowest band in our models. In spite of a fluctuating Berry curvature, our construction is partially frustration free: the ground states reside entirely within the lowest band and exactly minimize a local $(k+1)$-body repulsion term by term. In addition to providing the first known models hosting intriguing states such as higher Chern number generalizations of the Fibonacci anyon quantum Hall states, the remarkable stability and finite-size properties make our models particularly well-suited for the study of novel phenomena involving e.g. twist defects and proximity induced superconductivity, as well as being a guide for designing experiments.

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arXiv2016

Disordered double Weyl node

Double Weyl nodes are topologically protected band crossing points which carry chiral charge $\pm2$. They are stabilized by $C_4$ point group symmetry and are predicted to occur in $\mathrm{SrSi_2}$ or $\mathrm{HgCr_{2}Se_{4}}$. We study their fate in the presence of quenched disorder by numerical transport calculations and scaling arguments. We find that a double Weyl node is unstable in a disordered environment and splits into a pair of emergent simple Weyl nodes which carry equal chiral charge of $\pm1$. This pair of simple Weyl nodes is robust to disorder up to a certain critical disorder strength, in complete analogy to their `elementary' counterparts.

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arXiv2015

Quantum Critical Exponents for a Disordered Three-Dimensional Weyl Node

Three-dimensional Dirac and Weyl semimetals exhibit a disorder-induced quantum phase transition between a semimetallic phase at weak disorder and a diffusive-metallic phase at strong disorder. Despite considerable effort, both numerically and analytically, the critical exponents $ν$ and $z$ of this phase transition are not known precisely. Here we report a numerical calculation of the critical exponent $ν=1.47\pm0.03$ using a minimal single-Weyl node model and a finite-size scaling analysis of conductance. Our high-precision numerical value for $ν$ is incompatible with previous numerical studies on tight-binding models and with one- and two-loop calculations in an $ε$-expansion scheme. We further obtain $z=1.49\pm0.02$ from the scaling of the conductivity with chemical potential.

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arXiv2015

Quantum transport in Dirac materials: signatures of tilted and anisotropic Dirac and Weyl cones

We calculate conductance and noise for quantum transport at the nodal point for arbitrarily tilted and anisotropic Dirac or Weyl cones. Tilted and anisotropic dispersions are generic in absence of certain discrete symmetries, such as particle-hole and lattice point group symmetries. Whereas anisotropy affects the conductance g, but leaves the Fano factor F (the ratio of shot noise power and current) unchanged, a tilt affects both g and F. Since F is a universal number in many other situations, this finding is remarkable. We apply our general considerations to specific lattice models of strained graphene and a pyrochlore Weyl semi-metal.

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arXiv2014

Quantum transport of disordered Weyl semimetals at the nodal point

Weyl semimetals are paradigmatic topological gapless phases in three dimensions. We here address the effect of disorder on charge transport in Weyl semimetals. For a single Weyl node with energy at the degeneracy point and without interactions, theory predicts the existence of a critical disorder strength beyond which the density of states takes on a nonzero value. Predictions for the conductivity are divergent, however. In this work, we present a numerical study of transport properties for a disordered Weyl cone at zero energy. For weak disorder our results are consistent with a renormalization group flow towards an attractive pseudoballistic fixed point with zero conductivity and a scale-independent conductance; for stronger disorder diffusive behavior is reached. We identify the Fano factor as a signature that discriminates between these two regimes.

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arXiv2014

Correlations and entanglement in flat band models with variable Chern numbers

We discuss a number of illuminating results for tight binding models supporting a band with variable Chern number, and illustrate them explicitly for a simple class of two-banded models. First, for models with a fixed number of bands, we show that the minimal hopping range needed to achieve a given Chern number $C$ is increasing with $C$, and that the band flattening requires an exponential tail of long-range processes. We further verify that the entanglement spectrum corresponding to a real-space partitioning contains $C$ chiral modes and thereby complies with the archetypal correspondence between the bulk entanglement and the edge energetics. Finally, we address the issue of interactions and study the problem of two interacting particles projected to the flattened band as a function of the Chern number. Our results provide valuable insights for the full interacting problem of a partially filled Chern band at variable filling fractions and Chern numbers.

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arXiv2013

Non-Abelian Fractional Chern Insulators from Long-Range Interactions

The recent theoretical discovery of fractional Chern insulators (FCIs) has provided an important new way to realize topologically ordered states in lattice models. In earlier works, on-site and nearest neighbor Hubbard-like interactions have been used extensively to stabilize Abelian FCIs in systems with nearly flat, topologically nontrivial bands. However, attempts to use two-body interactions to stabilize non-Abelian FCIs, where the ground state in the presence of impurities can be massively degenerate and manipulated through anyon braiding, have proven very difficult in uniform lattice systems. Here, we study the remarkable effect of long-range interactions in a lattice model that possesses an exactly flat lowest band with a unit Chern number. When spinless bosons with two-body long-range interactions partially fill the lowest Chern band, we find convincing evidence of gapped, bosonic Read-Rezayi (RR) phases with non-Abelian anyon statistics. We characterize these states through studying topological degeneracies, the overlap between the ground states of two-body interactions and the exact RR ground states of three- and four-body interactions, and state counting in the particle-cut entanglement spectrum. Moreover, we demonstrate how an approximate lattice form of Haldane's pseudopotentials, analogous to that in the continuum, can be used as an efficient guiding principle in the search for lattice models with stable non-Abelian phases.

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arXiv2013

Topological equivalence of crystal and quasicrystal band structures

A number of recent articles have reported the existence of topologically non-trivial states and associated end states in one-dimensional incommensurate lattice models that would usually only be expected in higher dimensions. Using an explicit construction, we here argue that the end states have precisely the same origin as their counterparts in commensurate models and that incommensurability does not in fact provide a meaningful connection to the topological classification of systems in higher dimensions. In particular, we show that it is possible to smoothly interpolate between states with commensurate and incommensurate modulation parameters without closing the band gap and without states crossing the band gap.

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arXiv2013

Topological Flat Band Models and Fractional Chern Insulators

Topological insulators and their intriguing edge states can be understood in a single-particle picture and can as such be exhaustively classified. Interactions significantly complicate this picture and can lead to entirely new insulating phases, with an altogether much richer and less explored phenomenology. Most saliently, lattice generalizations of fractional quantum Hall states, dubbed fractional Chern insulators, have recently been predicted to be stabilized by interactions within nearly dispersionless bands with non-zero Chern number, $C$. Contrary to their continuum analogues, these states do not require an external magnetic field and may potentially persist even at room temperature, which make these systems very attractive for possible applications such as topological quantum computation. This review recapitulates the basics of tight-binding models hosting nearly flat bands with non-trivial topology, $C\neq 0$, and summarizes the present understanding of interactions and strongly correlated phases within these bands. Emphasis is made on microscopic models, highlighting the analogy with continuum Landau level physics, as well as qualitatively new, lattice specific, aspects including Berry curvature fluctuations, competing instabilities as well as novel collective states of matter emerging in bands with $|C|>1$. Possible experimental realizations, including oxide interfaces and cold atom implementations as well as generalizations to flat bands characterized by other topological invariants are also discussed.

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arXiv2013

Bulk-edge correspondence in fractional Chern insulators

It has been recently realized that strong interactions in topological Bloch bands give rise to the appearance of novel states of matter. Here we study connections between these systems -- fractional Chern insulators and the fractional quantum Hall states -- via generalization of a gauge-fixed Wannier-Qi construction in the cylinder geometry. Our setup offers a number of important advantages compared to the earlier exact diagonalization studies on a torus. Most notably, it gives access to edge states and to a single-cut orbital entanglement spectrum, hence to the physics of bulk-edge correspondence. It is also readily implemented in the state-of-the-art density matrix renormalisation group method that allows for numerical simulations of significantly larger systems. We demonstrate our general approach on examples of flat-band models on ruby and kagome lattices at bosonic filling fractions $ν=1/2$ and $ν=1$, which show the signatures of (non)-Abelian phases, and establish the correspondence between the physics of edge states and the entanglement in the bulk. Notably, we find that the non-Abelian $ν=1$ phase can be stabilized by purely on-site interactions in the presence of a confining potential.

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arXiv2013

From fractional Chern insulators to Abelian and non-Abelian fractional quantum Hall states: adiabatic continuity and orbital entanglement spectrum

The possibility of realizing lattice analogs of fractional quantum Hall (FQH) states, so-called fractional Chern insulators (FCIs), in nearly flat topological (Chern) bands has attracted a lot of recent interest. Here, we make the connection between Abelian as well as non-Abelian FQH states and FCIs more precise. Using a gauge-fixed version of Qi's Wannier basis representation of a Chern band, we demonstrate that the interpolation between several FCI states, obtained by short-range lattice interactions in a spin-orbit-coupled kagome lattice model, and the corresponding continuum FQH states is smooth: the gap remains approximately constant and extrapolates to a finite value in the thermodynamic limit, while the low-lying part of the orbital entanglement spectrum remains qualitatively unaltered. The orbital entanglement spectra also provide a first glimpse of the edge physics of FCIs via the bulk-boundary correspondence. Corroborating these results, we find that the squared overlaps between the FCI and FQH ground states are as large as 98.7% for the 8-electron Laughlin state at $ν=1/3$ (consistent with an earlier study) and 97.8% for the 10-electron Moore-Read state at $ν=1/2$. For the bosonic analogs of these states, the adiabatic continuity is also shown to hold, albeit with somewhat smaller associated overlaps, etc. Although going between the Chern bands to the Landau level problem is often smooth, we show that this is not always the case by considering fermions at filling fraction $ν=4/5$, where the interpolation between Hamiltonians describing the two systems results in a phase transition.

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