arXiv subjects

B. Andrei Bernevig

Publications and source records attributed to B. Andrei Bernevig.

At least 181 records · Page 10Linked to original sources

arXiv2019

Higher-Order Topology, Monopole Nodal Lines, and the Origin of Large Fermi Arcs in Transition Metal Dichalcogenides XTe$_2$ (X=Mo,W)

In recent years, transition metal dichalcogenides (TMDs) have garnered great interest as topological materials -- monolayers of centrosymmetric $β$-phase TMDs have been identified as 2D topological insulators (TIs), and bulk crystals of noncentrosymmetric $γ$-phase MoTe$_2$ and WTe$_2$ have been identified as type-II Weyl semimetals. However, ARPES and STM probes of these TMDs have revealed huge, "arc-like" surface states that overwhelm, and are sometimes mistaken for, the much smaller topological surface Fermi arcs of bulk type-II Weyl points. In this letter, we use first-principles calculations and (nested) Wilson loops to analyze the bulk and surface electronic structure of both $β$- and $γ$-MoTe$_2$, finding that $β$-MoTe$_2$ ($γ$-MoTe$_2$ gapped with symmetry-preserving distortion) is an inversion-symmetry-indicated $\mathbb{Z}_{4}$-nontrivial ($noncentrosymmetric, non$-$symmetry$-$indicated$) higher-order TI (HOTI) driven by double band inversion. Both structural phases of MoTe$_2$ exhibit the same surface features as WTe$_2$, revealing that the large Fermi arcs are in fact not topologically trivial, but are rather the characteristic split and gapped fourfold surface states of a HOTI. We also show that, when the effects of SOC are neglected, $β$-MoTe$_2$ is a nodal-line semimetal with $\mathbb{Z}_{2}$-nontrivial monopole nodal lines (MNLSM). This finding confirms that MNLSMs driven by double band inversion are the weak-SOC limit of HOTIs, implying that MNLSMs are higher-order topological $semimetals$ with flat-band-like hinge states, which we find to originate from the corner modes of 2D "fragile" TIs.

cond-mat.mtrl-sci↗

arXiv2019

Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian

We study the quantum dynamics of a simple translation invariant, center-of-mass (CoM) preserving model of interacting fermions in one dimension (1D), which arises in multiple experimentally realizable contexts. We show that this model naturally displays the phenomenology associated with fractonic systems, wherein single charges can only move by emitting dipoles. This allows us to demonstrate the rich Krylov fractured structure of this model, whose Hilbert space shatters into exponentially many dynamically disconnected subspaces. Focusing on exponentially large Krylov subspaces, we show that these can be either be integrable or non-integrable, thereby establishing the notion of Krylov-restricted thermalization. We analytically find a tower of integrable Krylov subspaces of this Hamiltonian, all of which map onto spin-1/2 XX models of various system sizes. We also discuss the physics of the non-integrable subspaces, where we show evidence for weak Eigenstate Thermalization Hypothesis (ETH) restricted to each non-integrable Krylov subspace. Further, we show that constraints in some of the thermal Krylov subspaces cause the long-time expectation values of local operators to deviate from behavior typically expected from translation-invariant systems. Finally, we show using a Schrieffer-Wolff transformation that such models naturally appear as effective Hamiltonians in the large electric field limit of the interacting Wannier-Stark problem, and comment on connections of our work with the phenomenon of Bloch many-body localization.

cond-mat.str-el↗

arXiv2019

Multi-Layer Restricted Boltzmann Machine Representation of 1D Quantum Many-Body Wave Functions

We consider representing two classes of 1D quantum wave functions of spin systems, including the AKLT and CFT correlator wave functions, in terms of multi-layer restricted Boltzmann machines. In our prescription, the AKLT wave function can be exactly represented by a 2-layer restricted Boltzmann machine with five hidden spins per visible spin. The construction can be generalized to prove that any MPS wave function on $N$ unit cells with finite bond dimension can be approximated by a 2-layer restricted Boltzmann machine with $\mathcal{O}(N)$ hidden spins within an error which scales linearly with $N$. The Haldane-Shastry wave function or a chiral boson CFT correlator wave function, as any Jastrow type of wave functions, can be exactly written as a 1-layer Boltzmann machine with $\mathcal{O}(N^2)$ hidden spins and $N$ visible spins. Applying the cumulant expansion, we further find that the chiral boson CFT correlator wave function (with small vertex operator conformal dimension $α$, i.e., $α<0.1$) can be approximated, within 99.9\% accuracy up to 22 visible spins, by a 1-layer RBM with $\mathcal{O}(N)$ hidden spins. The cumulant expansion also leads us to a physically inspiring result in which the hidden spins of the restricted Boltzmann machine can be interpreted as the conformal tower of the chiral boson CFT on the cylinder.

cond-mat.str-el↗

arXiv2019

Real Space Invariants: Twisted Bulk-Boundary Correspondence of Fragile Topology

In this paper, we propose a new type of bulk-boundary correspondence as a generic approach to theoretically and experimentally detect fragile topological states. When the fragile phase can be written as a difference of a trivial atomic insulator and the so-called obstructed atomic insulator, the gap between the fragile phase and other bands must close under a specific novel twist of the boundary condition of the system. We explicitly work out all the twisted boundary conditions (TBC) that can detect all the 2D fragile phases implied by symmetry eigenvalues in all wallpaper groups. We develop the concept of real space invariants - local good quantum numbers in real space - which fully characterize the eigenvalue fragile phases. We show that the number of unavoidable level crossings under the twisted boundary condition is completely determined by the real space invariants. Possible realizations of the TBC of the fragile band in metamaterial systems are discussed.

cond-mat.mes-hall↗

arXiv2019

Observation of a Majorana zero mode in a topologically protected edge channel

Superconducting proximity pairing in helical edge modes, such as those of topological insulators (TI), is predicted to provide a unique platform for realizing Majorana zero modes (MZMs). We use scanning tunneling microscopy measurements to probe the influence of proximity induced superconductivity and magnetism on the helical hinge states of Bi(111) films, grown on a superconducting Nb substrate and decorated with magnetic Fe clusters. Consistent with model calculations, our measurements reveal the emergence of a localized MZM at the interface between the superconducting helical edge channel and the Fe clusters with strong magnetization component along the edge. Our experiments also resolve the MZM spin signature that distinguishes it from trivial in-gap states that may accidently occur at zero energy in a superconductor.

cond-mat.mes-hall↗

arXiv2019

Spectroscopic signatures of many-body correlations in magic angle twisted bilayer graphene

The discovery of superconducting and insulating states in magic angle twisted bilayer graphene (MATBG) has ignited considerable interest in understanding the nature of electronic interactions in this chemically pristine material system. The phenomenological similarity of the MATBG transport properties as a function of doping with those of the high-Tc cuprates and other unconventional superconductors suggests the possibility that MATBG may be a highly interacting system. However, there have not been any direct experimental evidence for strong many-body correlations in MATBG. Here we provide such evidence from using high-resolution spectroscopic measurements, as a function of carrier density, with a scanning tunneling microscopy (STM). We find MATBG to display unusual spectroscopic characteristics that can be attributed to electron-electron interactions over a wide range of doping, including when superconductivity emerges in this system. We show that our measurements cannot be explained with a mean-field approach for modeling electron-electron interaction in MATBG. The breakdown of a mean-field approach for understanding the properties of other correlated superconductors, such as cuprates, has long inspired the study of highly correlated Hubbard model. We show that a phenomenological extended Hubbard model cluster calculation, motivated by the nearly localized nature of the relevant electronic states of MATBG produces spectroscopic features similar to those we observe experimentally. Our findings demonstrate the critical role of many-body correlations in understanding the properties of MATBG.

cond-mat.mes-hall↗

arXiv2019

Topology-Bounded Superfluid Weight In Twisted Bilayer Graphene

While regular flat bands are good for enhancing the density of states and hence the gap, they are detrimental to the superfluid weight. We show that the predicted nontrivial topology of the two lowest flat bands of twisted bilayer graphene plays an important role in the enhancement of the superfluid weight and hence of superconductivity. We derive the superfluid weight (phase stiffness) of the TBLG superconducting flat bands with a uniform pairing, and show that it can be expressed as an integral of the Fubini-Study metric of the flat bands. This mirrors results already obtained for nonzero Chern number bands even though the TBLG flat bands have zero Chern number. We further show the metric integral is lower bounded by the topological $C_{2z}T$ Wilson loop winding number of the TBLG flat bands, which renders the superfluid weight has a topological lower bound proportional to the pairing gap. In contrast, trivial flat bands have a zero superfluid weight. The superfluid weight is crucial in determining the BKT transition temperature of the superconductor. Based on the transition temperature measured in TBLG experiments, we estimate the topological contribution of the superfluid weight in TBLG.

cond-mat.supr-con↗

arXiv2019

Restricted Boltzmann Machines and Matrix Product States of 1D Translational Invariant Stabilizer Codes

We discuss the relations between restricted Boltzmann machine (RBM) states and the matrix product states (MPS) for the ground states of 1D translational invariant stabilizer codes. A generic translational invariant and finitely connected RBM state can be expressed as an MPS, and the matrices of the resulting MPS are of rank 1. We dub such an MPS as an RBM-MPS. This provides a necessary condition for exactly realizing a quantum state as an RBM state, if the quantum state can be written as an MPS. For generic 1D stabilizer codes having a non-degenerate ground state with periodic boundary condition, we obtain an expression for the lower bound of their MPS bond dimension, and an upper bound for the rank of their MPS matrices. In terms of RBM, we provide an algorithm to derive the RBM for the cocycle Hamiltonians whose MPS matrices are proved to be of rank 1. Moreover, the RBM-MPS produced by our algorithm has the minimal bond dimension. A family of examples is provided to explain the algorithm. We finally conjecture that these features hold true for all the 1D stabilizer codes having a non-degenerate ground state with periodic boundary condition, as long as their MPS matrices are of rank 1.

cond-mat.str-el↗

arXiv2019

Glide-resolved photoemission spectroscopy: Measuring topological invariants in nonsymmorphic space groups

The two classes of 3D, time-reversal-invariant insulators are known to subdivide into four classes in the presence of glide symmetry. Here, we extend this classification of insulators to include glide-symmetric Weyl metals, and find a finer $Z_4\oplus Z$ classification. We further elucidate the smoking-gun experimental signature of each class in the photoemission spectroscopy of surface states. Measuring the $Z_4$ topological invariant by photoemission relies on identifying the glide representation of the initial Bloch state before photo-excitation - we show how this is accomplished with relativistic selection rules, combined with standard spectroscopic techniques to resolve both momentum and spin. Our method relies on a novel spin-momentum locking that is characteristic of all glide-symmetric solids (inclusive of insulators and metals in trivial and topological categories). As an orthogonal application, given a glide-symmetric solid with an ideally symmetric surface, we may utilize this spin-momentum locking to generate a source of FULLY spin-polarized photoelectrons, which have diverse applications in solid-state spectroscopy. Our ab-initio calculations predict Ba$_2$Pb, stressed Na$_3$Bi, and KHgSb to realize all three, nontrivial insulating phases in the $Z_4$ classification.

cond-mat.mtrl-sci↗

arXiv2019

Three-dimensional superconductors with hybrid higher order topology

We consider three dimensional superconductors in class DIII with a four-fold rotation axis and inversion symmetry. It is shown that such systems can exhibit higher order topology with helical Majorana hinge modes. In the case of even-parity superconductors we show that higher order topological superconductors can be obtained by adding a small pairing with the appropriate $C_4$ symmetry implementation to a topological insulator. We also show that a hybrid case is possible, where Majorana surface cones resulting from non-trivial strong topology coexist with helical hinge modes. We propose a bulk invariant detecting this hybrid scenario, and numerically analyse a tight binding model exhibiting both Majorana cones and hinge modes.

cond-mat.supr-con↗

arXiv2019

Disconnected Elementary Band Representations, Fragile Topology, and Wilson Loops as Topological Indices: An Example on the Triangular Lattice

In this work, we examine the topological phases that can arise in triangular lattices with disconnected elementary band representations. We show that, although these phases may be "fragile" with respect to the addition of extra bands, their topological properties are manifest in certain nontrivial holonomies (Wilson loops) in the space of nontrivial bands. We introduce an eigenvalue index for fragile topology, and we show how a nontrivial value of this index manifests as the winding of a hexagonal Wilson loop; this remains true even in the absence of time-reversal or sixfold rotational symmetry. Additionally, when time-reversal and twofold rotational symmetry are present, we show directly that there is a protected nontrivial winding in more conventional Wilson loops. Crucially, we emphasize that these Wilson loops cannot change without closing a gap to the nontrivial bands. By studying the entanglement spectrum for the fragile bands, we comment on the relationship between fragile topology and the "obstructed atomic limit" of B. Bradlyn et al., Nature 547, 298--305 (2017). We conclude with some perspectives on topological matter beyond the K-theory classification.

cond-mat.mes-hall↗

arXiv2019

Flat bands and perfect metal in trilayer moiré graphene

We investigate the electronic structure of a twisted multilayer graphene system forming a moiré pattern. We consider small twist angles separating the graphene sheets and develop a low-energy theory to describe the coupling of Dirac Bloch states close to the K point in each individual plane. Extending beyond the bilayer case, we show that, when the ratio of the consecutive twist angles is rational, a periodicity emerges in quasimomentum space with moiré Bloch bands even when the system does not exhibit a crystalline lattice structure in real space. For a trilayer geometry, we find flat bands in the spectrum at certain rotation angles. Performing a symmetry analysis of the band model for the trilayer, we prove that the system is a perfect metal in the sense that it is gapless at all energies. This striking result originates from the three Dirac cones which can only gap in pairs and produce bands with an infinite connectivity. The full gapless property is protected by an emergent particle-hole symmetry valid at sufficiently small angles.

cond-mat.mes-hall↗

arXiv2019

Twisted Bilayer Graphene: A Phonon Driven Superconductor

We study the electron-phonon coupling in twisted bilayer graphene (TBG), which was recently experimentally observed to exhibit superconductivity around the magic twist angle $θ\approx 1.05^\circ$. We show that phonon-mediated electron electron attraction at the magic angle is strong enough to induce a conventional intervalley pairing between graphene valleys $K$ and $K'$ with a superconducting critical temperature $T_c\sim1K$, in agreement with the experiment. We predict that superconductivity can also be observed in TBG at many other angles $θ$ and higher electron densities in higher Moiré bands, which may also explain the possible granular superconductivity of highly oriented pyrolytic graphite. We support our conclusions by \emph{ab initio} calculations.

cond-mat.mes-hall↗

arXiv2018

The Axion Insulator as a Pump of Fragile Topology

The axion insulator (AXI) has long been recognized as the simplest example of a 3D magnetic topological insulator (TI). The most familiar AXI results from magnetically gapping the surface states of a 3D $\mathbb{Z}_{2}$ TI while preserving the bulk gap. Like the 3D TI, it exhibits a quantized magnetoelectric polarizability of $θ=π$, and can be diagnosed from bulk symmetry eigenvalues when inversion symmetric. However, whereas a 3D TI is characterized by bulk Wilson loop winding, 2D surface states, and the pumping of the 2D $\mathbb{Z}_{2}$ TI index, we show that an AXI with a large number of bulk bands displays no Wilson loop winding, exhibits chiral hinge states, and does not pump any previously identified quantity. Crucially, as the AXI exhibits the topological angle $θ=π$, its occupied bands cannot be formed into maximally localized symmetric Wannier functions, despite its absence of Wilson loop winding. In this letter, we revisit the AXI from the perspective of the recently introduced notion of "fragile" topology, and discover that it in fact can be generically expressed as the cyclic pumping of a "trivialized" fragile phase: a 2D inversion-symmetric insulator with no Wilson loop winding which nevertheless carries a nontrivial topological index, the nested Berry phase $γ_{2}$. We numerically show that the nontrivial value $γ_{2}=π$ indicates the presence of anomalous 0D corner charges in a 2D insulator, and therefore, that the chiral pumping of $γ_{2}$ in a 3D AXI corresponds to the presence of chiral hinge states. We also briefly generalize our results to time-reversal-symmetric higher-order TIs, and discuss the related appearance of nontrivial $γ_{2}$ protected by $C_{2}\times\mathcal{T}$ symmetry in twisted bilayer graphene, and its implications for the presence of 0D corner states.

cond-mat.mes-hall↗

arXiv2018

All "Magic Angles" Are "Stable" Topological

We show that the electronic structure of the low-energy bands in the small angle-twisted bilayer graphene consists of a series of semi-metallic and topological phases. In particular we are able to prove, using an approximate low-energy particle-hole symmetry, that the gapped set of bands that exist around all magic angles has what we conjecture to be a stable topological index stabilized by a magnetic symmetry and reflected in the odd winding of the Wilson loop in the Moiré BZ. The approximate, emergent particle-hole symmetry is essential to the topology of graphene: when strongly broken, non-topological phases can appear. Our paper underpins topology as the crucial ingredient to the description of low-energy graphene. We provide a $4$-band short range tight-binding model whose $2$ lower bands have the same topology, symmetry, and flatness as those of the twisted graphene, and which can be used as an effective low-energy model. We then perform large-scale ($11000$ atoms per unit cell, 40 days per $\bf k$-point computing time) ab-initio calculations of a series of small angles, from $3^\circ$ to $1^\circ$, which show a more complex and somewhat qualitatively different evolution of the symmetry of the low-energy bands than that of the theoretical Moiré model, but which confirms the topological nature of the system. At certain angles, we find no insulating filling in graphene at $-4$ electrons per Moiré unit cell. The ab-initio evolution of gaps tends to differ from that of the continuum Moiré model.

cond-mat.mes-hall↗

arXiv2018

Wallpaper Fermions and the Nonsymmorphic Dirac Insulator

Recent developments in the relationship between bulk topology and surface crystal symmetry have led to the discovery of materials whose gapless surface states are protected by crystal symmetries. In fact, there exists only a very limited set of possible surface crystal symmetries, captured by the 17 "wallpaper groups." We show that a consideration of symmetry-allowed band degeneracies in the wallpaper groups can be used to understand previous topological crystalline insulators, as well as to predict new examples. In particular, the two wallpaper groups with multiple glide lines, $pgg$ and $p4g$, allow for a new topological insulating phase, whose surface spectrum consists of only a single, fourfold-degenerate, true Dirac fermion. Like the surface state of a conventional topological insulator, the surface Dirac fermion in this "nonsymmorphic Dirac insulator" provides a theoretical exception to a fermion doubling theorem. Unlike the surface state of a conventional topological insulator, it can be gapped into topologically distinct surface regions while keeping time-reversal symmetry, allowing for networks of topological surface quantum spin Hall domain walls. We report the theoretical discovery of new topological crystalline phases in the A$_2$B$_3$ family of materials in SG 127, finding that Sr$_2$Pb$_3$ hosts this new topological surface Dirac fermion. Furthermore, (100)-strained Au$_2$Y$_3$ and Hg$_2$Sr$_3$ host related topological surface hourglass fermions. We also report the presence of this new topological hourglass phase in Ba$_5$In$_2$Sb$_6$ in SG 55. For orthorhombic space groups with two glides, we catalog all possible bulk topological phases by a consideration of the allowed non-abelian Wilson loop connectivities, and we develop topological invariants for these systems. Finally, we show how in a particular limit, these crystalline phases reduce to copies of the SSH model.

cond-mat.mes-hall↗

arXiv2018

Entanglement of Exact Excited States of AKLT Models: Exact Results, Many-Body Scars and the Violation of Strong ETH

We obtain multiple exact results on the entanglement of the exact excited states of non-integrable models we introduced in arXiv:1708.05021. We first discuss a general formalism to analytically compute the entanglement spectra of exact excited states using Matrix Product States and Matrix Product Operators and illustrate the method by reproducing a general result on single-mode excitations. We then apply this technique to analytically obtain the entanglement spectra of the infinite tower of states of the spin-$S$ AKLT models in the zero and finite energy density limits. We show that in the zero density limit, the entanglement spectra of the tower of states are multiple shifted copies of the ground state entanglement spectrum in the thermodynamic limit. We show that such a resemblance is destroyed at any non-zero energy density. Furthermore, the entanglement entropy $\mathcal{S}$ of the states of the tower that are in the bulk of the spectrum is sub-thermal $\mathcal{S} \propto \log L$, as opposed to a volume-law $\mathcal{S} \propto L$, thus indicating a violation of the strong Eigenstate Thermalization Hypothesis (ETH). These states are examples of what are now called many-body scars. Finally, we analytically study the finite-size effects and symmetry-protected degeneracies in the entanglement spectra of the excited states, extending the existing theory.

cond-mat.str-el↗

arXiv2018

Exact Excited States of Non-Integrable Models

We discuss a method of numerically identifying exact energy eigenstates for a finite system, whose form can then be obtained analytically. We demonstrate our method by identifying and deriving exact analytic expressions for several excited states, including an infinite tower, of the one dimensional spin-1 AKLT model, a celebrated non-integrable model. The states thus obtained for the AKLT model can be interpreted as one-to-an extensive number of quasiparticles on the ground state or on the highest excited state when written in terms of dimers. Included in these exact states is a tower of states spanning energies from the ground state to the highest excited state. To our knowledge, this is the first time that exact analytic expressions for a tower of excited states have been found in non-integrable models. Some of the states of the tower appear to be in the bulk of the energy spectrum, allowing us to make conjectures on the strong Eigenstate Thermalization Hypothesis (ETH). We also generalize these exact states including the tower of states to the generalized integer spin AKLT models. Furthermore, we establish a correspondence between some of our states and those of the Majumdar-Ghosh model, yet another non-integrable model, and extend our construction to the generalized integer spin AKLT models.

cond-mat.str-el↗