Emil Johansson BergholtzProfessor
Om mig
My interests are broad and varying with time but I am particularly fascinated by the world of quantum mechanics, in particular the by collective behaviour of many particles under extreme conditions, their exuberant phenomenology and possible impact on future technology. In search of new exciting phenomena our group brings together insights from different fields of science ranging from contemporary mathematics and high energy physics to materials science, photonics, cold atoms and complex systems.
With the help of the Wallenberg Academy Fellows program, I returned to my alma mater Stockholm University in 2016 after eight years abroad  as a Distinguished PKS fellow at the Max Planck Institute for the Physics of Complex Systems in Dresden and as an Emmy Noether Group Leader at the Free University of Berlin. I was promoted to full professor in 2019 and appointed Wallenberg Scholar in 2024.
I was awarded the 2022 Göran Gustafsson Prize in physics by the Royal Swedish Academy of Sciences with the motivation ”for his innovative research on topological phases and quantum materials. He has made significant and recognized contributions to the theory of lowdimensional materials, specifically to the description of open and dissipative topological phases in terms of nonHermitian topological models."
Undervisning
The past few years I have taught Advanced Quantum Mechanics, and presently I also lead more specialized research seminars.
Forskning
We work broadly in theoretical physics with an emphasis on collective quantum mechanical phenomena that occur in systems with a macroscopic number of particles. Most saliently, we study quantum manyparticle systems for which topology, entanglement and interactions play important roles. These include fractional quantum Hall states, geometrically frustrated magnets, nonequilibrium systems, Weyl semimetals and various instances of flat band models. A common feature in these systems is that their lowenergy quasiparticles bear little or no resemblance to their electronic constituents. Instead, the quasiparticles have intriguing properties such as fractional charge and statistics. To understand these notoriously complex systems we use a combinationof analytical and numerical methods, beyond standard manybody theory, including exact diagnalization, field theory, strong coupling expansions etc., we occasionally adopt new methods and concepts from quantum information theory, including entanglement quantifiers and tensor networks, and contemporary mathematics, such as compressed sensing.
Our research brings together several frontiers of basic science, while at the same time having the potential to provide the basis of future technological advances.
My research group presently (early 2022) consists of Elisabet Edvardsson (PhD student), Marcus Stålhammar (PhD student), Ahmed Abouelkomsan (PhD student), Lukas König (PhD student), Fan Yang (Postdoc), Kang Yang (Postdoc), Daniel Varjas (Postdoc/Researcher), Ipsita Mandal (Researcher).
Recent group alumi include Flore Kunst (PhD & winner of the Arrhenius prize in 2019, next postdoc at MPQHarvard), QingDong Jiang (Researcher, next Associate Prof. at the TD Lee Institute, Shanghai), Maximilian Trescher (PhD in 2018, next computer scientist in Berlin), Zhao Liu (next Thousand Talents Awardee & Asst. Prof. at Zhejiang University), Theresa Leistner (Master Student, next engineer in industry, Stockholm), Yaron Kedem (next postdoc with Frank Wilczek), and Johan Carlström (next independent PI at SU). Naemi Florin (next PhD with Bourennane) and Fanny Terrier (next UPMC Paris).
Earlier project students include Jörg Behrmann (Master 20122013), Samuel Sanchez (Master 20122013), Maximilian Trescher (Bachelor 2012), Diana Prychynenko (Bachelor 2012 & coadvised Master 20132014), Kevin Madsen (Bachelor 2013 & Master 20152016), Gregor Pohl (internship project 2013), Alexander Nietner (Master 20142015), Gunnar Riemenschneider (Bachelor 2014), Huaiyu Li (internship project 2015), Marlon Rueck (Bachelor 2015), Irina Gancheva (Master 20152016), Jann Launer (Master 20152016), David Schneider (Master 20152016) and Yann Salimi (Master 20152016).
Publications See Google Scholar for publications and links to frequent collaborators.
Forskningsprojekt
Publikationer
An updated list of publications is avaliable at Google Scholar.
I urval från Stockholms universitets publikationsdatabas

Biorthogonal BulkBoundary Correspondence in NonHermitian Systems
2018. Flore K. Kunst (et al.). Physical Review Letters 121 (2)
ArtikelNonHermitian systems exhibit striking exceptions from the paradigmatic bulkboundary correspondence, including the failure of bulk Bloch band invariants in predicting boundary states and the (dis) appearance of boundary states at parameter values far from those corresponding to gap closings in periodic systems without boundaries. Here, we provide a comprehensive framework to unravel this disparity based on the notion of biorthogonal quantum mechanics: While the properties of the left and right eigenstates corresponding to boundary modes are individually decoupled from the bulk physics in nonHermitian systems, their combined biorthogonal density penetrates the bulk precisely when phase transitions occur. This leads to generalized bulkboundary correspondence and a quantized biorthogonal polarization that is formulated directly in systems with open boundaries. We illustrate our general insights by deriving the phase diagram for several microscopic open boundary models, including exactly solvable nonHermitian extensions of the SuSchriefferHeeger model and Chern insulators.

Exceptional topology of nonHermitian systems
2021. Emil J. Bergholtz, Jan Carl Budich, Flore K. Kunst. Reviews of Modern Physics 93 (1)
ArtikelThe current understanding of the role of topology in nonHermitian (NH) systems and its farreaching physical consequences observable in a range of dissipative settings are reviewed. In particular, how the paramount and genuinely NH concept of exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, leads to phenomena drastically distinct from the familiar Hermitian realm is discussed. An immediate consequence is the ubiquitous occurrence of nodal NH topological phases with concomitant open FermiSeifert surfaces, where conventional bandtouching points are replaced by the aforementioned exceptional degeneracies. Furthermore, new notions of gapped phases including topological phases in singleband systems are detailed, and the manner in which a given physical context may affect the symmetrybased topological classification is clarified. A unique property of NH systems with relevance beyond the field of topological phases consists of the anomalous relation between bulk and boundary physics, stemming from the striking sensitivity of NH matrices to boundary conditions. Unifying several complementary insights recently reported in this context, a picture of intriguing phenomena such as the NH bulkboundary correspondence and the NH skin effect is put together. Finally, applications of NH topology in both classical systems including optical setups with gain and loss, electric circuits, and mechanical systems and genuine quantum systems such as electronic transport settings at material junctions and dissipative coldatom setups are reviewed.

NonHermitian Topological Sensors
2020. Jan Carl Budich, Emil J. Bergholtz. Physical Review Letters 125 (18)
ArtikelWe introduce and study a novel class of sensors whose sensitivity grows exponentially with the size of the device. Remarkably, this drastic enhancement does not rely on any finetuning, but is found to be a stable phenomenon immune to local perturbations. Specifically, the physical mechanism behind this striking phenomenon is intimately connected to the anomalous sensitivity to boundary conditions observed in nonHermitian topological systems. We outline concrete platforms for the practical implementation of these nonHermitian topological sensors ranging from classical metamaterials to synthetic quantum materials.

ParticleHole Duality, Emergent Fermi Liquids, and Fractional Chern Insulators in Moire Flatbands
2020. Ahmed Abouelkomsan, Zhao Liu, Emil J. Bergholtz. Physical Review Letters 124 (10)
ArtikelMoire flatbands, occurring, e.g., in twisted bilayer graphene at magic angles, have attracted ample interest due to their high degree of experimental tunability and the intriguing possibility of generating novel strongly interacting phases. Here we consider the core problem of Coulomb interactions within fractionally filled spin and valley polarized Moire flatbands and demonstrate that the dual description in terms of holes, which acquire a nontrivial hole dispersion, provides key physical intuition and enables the use of standard perturbative techniques for this strongly correlated problem. In experimentally relevant examples such as ABC stacked trilayer and twisted bilayer graphene aligned with boron nitride, it leads to emergent interactiondriven Fermi liquid states at electronic filling fractions down to around 1/3 and 2/3, respectively. At even lower filling fractions, the electron density still faithfully tracks the singlehole dispersion while exhibiting distinct nonFermi liquid behavior. Most saliently, we provide microscopic evidence that high temperature fractional Chern insulators can form in twisted bilayer graphene aligned with hexagonal boron nitride.

Black and white holes at material junctions
2020. Yaron Kedem, Emil J. Bergholtz, Frank Wilczek. Physical Review Research 2 (4)
ArtikelElectrons in type II Weyl semimetals display oneway propagation, which supports totally reflecting behavior at an endpoint, as one has for black hole horizons viewed from the inside. Junctions of type I and type II lead to equations identical to what one has near black hole horizons, but the physical implications, we suggest, are quite different from expectations which are conventional in that context. The timereversed, “white hole” configuration is also physically accessible.

Corner states of light in photonic waveguides
2019. Ashraf El Hassan (et al.). Nature Photonics 13 (10), 697700
ArtikelThe recently established paradigm of higherorder topological states of matter has shown that not only edge and surface states(1,2) but also states localized to corners, can have robust and exotic properties(39). Here we report on the experimental realization of novel corner states made out of visible light in threedimensional photonic structures inscribed in glass samples using femtosecond laser technology(10,11). By creating and analysing waveguide arrays, which form twodimensional breathing kagome lattices in various sample geometries, we establish this as a platform for corner states exhibiting a remarkable degree of flexibility and control. In each sample geometry we measure eigenmodes that are localized at the corners in a finite frequency range, in complete analogy with a theoretical model of the breathing kagome(79,1214). Here, measurements reveal that light can be 'fractionalized,' corresponding to simultaneous localization to each corner of a triangular sample, even in the presence of defects.

Phase Transitions and Generalized Biorthogonal Trace Polarization in nonHermitian Systems
Elisabet Edvardsson (et al.).

Classification of exceptional nodal topologies protected by PT symmetry
2021. Marcus Stålhammar, Emil J. Bergholtz. Physical Review B 104 (20)
ArtikelExceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, and paritytime (PT) symmetry, reflecting balanced gain and loss in photonic systems, are paramount concepts in nonHermitian systems. We here complete the topological classification of exceptional nodal degeneracies protected by PT symmetry in up to three dimensions and provide simple example models whose exceptional nodal topologies include previously overlooked possibilities such as secondorder knotted surfaces of arbitrary genus, thirdorder knots, and fourthorder points.

Dissipative preparation of fractional Chern insulators
2021. Zhao Liu, Emil J. Bergholtz, Jan Carl Budich. Physical Review Research 3 (4)
ArtikelWe report on the numerically exact simulation of the dissipative dynamics governed by quantum master equations that feature fractional quantum Hall states as unique steady states. In particular, for the paradigmatic Hofstadter model, we show how Laughlin states can be to good approximation prepared in a dissipative fashion from arbitrary initial states by simply pumping strongly interacting bosons into the lowest Chern band of the corresponding singleparticle spectrum. While pure (up to topological degeneracy) steady states are only reached in the lowflux limit or for extended hopping range, we observe a certain robustness regarding the overlap of the steady state with fractional quantum Hall states for experimentally wellcontrolled flux densities. This may be seen as an encouraging step towards addressing the longstanding challenge of preparing strongly correlated topological phases in quantum simulators.

Exceptional Spin Liquids from Couplings to the Environment
2021. Kang Yang, Siddhardh C. Morampudi, Emil J. Bergholtz. Physical Review Letters 126 (7)
ArtikelWe establish the appearance of a qualitatively new type of spin liquid with emergent exceptional points when coupling to the environment. We consider an open system of the Kitaev honeycomb model generically coupled to an external environment. In extended parameter regimes, the Dirac points of the emergent Majorana fermions from the original model are split into exceptional points with Fermi arcs connecting them. In glaring contrast to the original gapless phase of the honeycomb model that requires timereversal symmetry, this new phase is stable against all perturbations. The system also displays a large sensitivity to boundary conditions resulting from the nonHermitian skin effect with telltale experimental consequences. Our results point to the emergence of new classes of spin liquids in open systems that might be generically realized due to unavoidable couplings with the environment.

GateTunable Fractional Chern Insulators in Twisted Double Bilayer Graphene
2021. Zhao Liu, Ahmed Abouelkomsan, Emil J. Bergholtz. Physical Review Letters 126 (2)
ArtikelWe predict twisted double bilayer graphene to be a versatile platform for the realization of fractional Chern insulators readily targeted by tuning the gate potential and the twist angle. Remarkably, these topologically ordered states of matter, including spin singlet Halperin states and spin polarized states in Chern number C=1 and C=2 bands, occur at high temperatures and without the need for an external magnetic field.

Symmetry and HigherOrder Exceptional Points
2021. Ipsita Mandal, Emil J. Bergholtz. Physical Review Letters 127 (18)
ArtikelExceptional points (EPs), at which both eigenvalues and eigenvectors coalesce, are ubiquitous and unique features of nonHermitian systems. Secondorder EPs are by far the most studied due to their abundance, requiring only the tuning of two real parameters, which is less than the three parameters needed to generically find ordinary Hermitian eigenvalue degeneracies. Higherorder EPs generically require more finetuning, and are thus assumed to play a much less prominent role. Here, however, we illuminate how physically relevant symmetries make higherorder EPs dramatically more abundant and conceptually richer. More saliently, thirdorder EPs generically require only two real tuning parameters in the presence of either a paritytime (PT) symmetry or a generalized chiral symmetry. Remarkably, we find that these different symmetries yield topologically distinct types of EPs. We illustrate our findings in simple models, and show how thirdorder EPs with a generic ∼k1/3 dispersion are protected by PT symmetry, while thirdorder EPs with a ∼k1/2 dispersion are protected by the chiral symmetry emerging in nonHermitian Lieb lattice models. More generally, we identify stable, weak, and fragile aspects of symmetryprotected higherorder EPs, and tease out their concomitant phenomenology.

Synchronization in epidemic growth and the impossibility of selective containment
2021. Jan C. Budich, Emil J. Bergholtz. Mathematical Medicine and Biology 38 (4), 467473
ArtikelContainment, aiming to prevent the epidemic stage of communityspreading altogether, and mitigation, aiming to merely 'flatten the curve' of a wideranged outbreak, constitute two qualitatively different approaches to combating an epidemic through nonpharmaceutical interventions. Here, we study a simple model of epidemic dynamics separating the population into two groups, namely a lowrisk group and a highrisk group, for which different strategies are pursued. Due to synchronization effects, we find that maintaining a slower epidemic growth behaviour for the highrisk group is unstable against any finite coupling between the two groups. More precisely, the density of infected individuals in the two groups qualitatively evolves very similarly, apart from a small time delay and an overall scaling factor quantifying the coupling between the groups. Hence, selective containment of the epidemic in a targeted (highrisk) group is practically impossible whenever the surrounding society implements a mitigated communityspreading. We relate our general findings to the ongoing COVID19 pandemic.

MagnetoOptical Conductivity in Generic Weyl Semimetals
2020. Marcus Stålhammar (et al.). Physical Review B. Condensed Matter and Materials Physics 102 (23)
ArtikelMagnetooptical studies of Weyl semimetals have been proposed as a versatile tool for observing lowenergy Weyl fermions in candidate materials including the chiral Landau level. However, previous theoretical results have been restricted to the linearized regime around the Weyl node and are at odds with experimental findings. Here, we derive a closed form expression for the magnetooptical conductivity of generic Weyl semimetals in the presence of an external magnetic field aligned with the tilt of the spectrum. The systems are taken to have linear dispersion in two directions, while the tilting direction can consist of any arbitrary continuously differentiable function. This general calculation is then used to analytically evaluate the magnetooptical conductivity of Weyl semimetals expanded to cubic order in momentum. In particular, systems with arbitrary tilt, as well as systems hosting trivial Fermi pockets are investigated. The higherorder terms in momentum close the Fermi pockets in the typeII regime, removing the need for unphysical cutoffs when evaluating the magnetooptical conductivity. Crucially, the ability to take into account closed overtilted and additional trivial Fermi pockets allows us to treat model systems closer to actual materials and we propose a simple explanation why the presence of parasitic trivial Fermi pockets can mask the characteristic signature of Weyl fermions in magnetooptical conductivity measurements.

Phase transitions and generalized biorthogonal polarization in nonHermitian systems
2020. Elisabet Edvardsson (et al.). Physical Review Research 2 (4)
ArtikelNonHermitian (NH) Hamiltonians can be used to describe dissipative systems, notably including systems with gain and loss, and are currently intensively studied in the context of topology. A salient difference between Hermitian and NH models is the breakdown of the conventional bulkboundary correspondence, invalidating the use of topological invariants computed from the Bloch bands to characterize boundary modes in generic NH systems. One way to overcome this difficulty is to use the framework of biorthogonal quantum mechanics to define a biorthogonal polarization, which functions as a realspace invariant signaling the presence of boundary states. Here, we generalize the concept of the biorthogonal polarization beyond the previous results to systems with any number of boundary modes and show that it is invariant under basis transformations as well as local unitary transformations. Additionally, we focus on the anisotropic SuSchriefferHeeger chain and study gap closings analytically. We also propose a generalization of a previously developed method with which to find all the bulk states of the system with open boundaries to NH models. Using the exact solutions for the bulk and boundary states, we elucidate genuinely NH aspects of the interplay between the bulk and boundary at the phase transitions.

Boundaries of boundaries
2019. Flore K. Kunst, Guido van Miert, Emil J. Bergholtz. Physical Review B 99 (8)
ArtikelWe present a generic and systematic approach for constructing D−dimensional lattice models with exactly solvable d−dimensional boundary states localized to corners, edges, hinges, and surfaces. These solvable models represent a class of “sweet spots” in the space of possible tightbinding models—the exact solutions remain valid for any tightbinding parameters as long as they obey simple locality conditions that are manifest in the underlying lattice structure. Consequently, our models capture the physics of both (higher order) topological and nontopological phases as well as the transitions between them in a particularly illuminating and transparent manner.

Extended Bloch theorem for topological lattice models with open boundaries
2019. Flore K. Kunst, Guido van Miert, Emil J. Bergholtz. Physical Review B 99 (8)
ArtikelWhile the Bloch spectrum of translationally invariant noninteracting lattice models is trivially obtained by a Fourier transformation, diagonalizing the same problem in the presence of open boundary conditions is typically only possible numerically or in idealized limits. Here we present exact analytic solutions for the boundary states in a number of lattice models of current interest, including nodalline semimetals on a hyperhoneycomb lattice, spinorbit coupled graphene, and threedimensional topological insulators on a diamond lattice, for which no previous exact finitesize solutions are available in the literature. Furthermore, we identify spectral mirror symmetry as the key criterium for analytically obtaining the entire (bulk and boundary) spectrum as well as the concomitant eigenstates, and exemplify this for Chern and Z2 insulators with open boundaries of codimension one. In the case of the twodimensional Lieb lattice, we extend this further and show how to analytically obtain the entire spectrum in the presence of open boundaries in both directions, where it has a clear interpretation in terms of bulk, edge, and corner states.

Fractional quantum Hall states with gapped boundaries in an extreme lattice limit
2019. Zhao Liu, Emil J. Bergholtz. Physical Review B 99 (19)
ArtikelWe present a detailed microscopic investigation of fractional quantum Hall states with gapped boundaries in a coupled bilayer lattice model featuring holes whose counterpropagating chiral edge states are hybridized and gapped out. We focus on an lattice limit for coldatom experiments, in which each hole just consists of a single removed site. Although the holes distort the original band structure and lead to ingap remnants of the continuum edge modes, we find that the lowest nearly flat band representing a highergenus system may naturally form by controlling the local hopping terms that gap out the boundaries. Remarkably, local interactions in this new flat band lead to various Abelian and nonAbelian fractional quantum Hall states with gapped boundaries residing on emergent highergenus surfaces, which we identify by extracting the nontrivial topological groundstate degeneracies and the fractional statistics of quasiparticles. These results demonstrate the feasibility of realizing novel fractional quantum Hall states with gapped boundaries even in the extreme lattice limit, thus enabling a possible new route towards universal topological quantum computation.

Hyperbolic nodal band structures and knot invariants
2019. Marcus Stålhammar (et al.). SciPost Physics 7 (2)
ArtikelWe extend the list of known band structure topologies to include a large family of hyperbolic nodal links and knots, occurring both in conventional Hermitian systems where their stability relies on discrete symmetries, and in the dissipative nonHermitian realm where the knotted nodal lines are generic and thus stable towards any small perturbation. We show that these nodal structures, taking the forms of Turk's head knots, appear in both continuum and lattice models with relatively shortranged hopping that is within experimental reach. To determine the topology of the nodal structures, we devise an efficient algorithm for computing the Alexander polynomial, linking numbers and higher order Milnor invariants based on an approximate and well controlled parameterisation of the knot.

Knotted nonHermitian metals
2019. Johan Carlström (et al.). Physical Review B 99 (16)
ArtikelWe report on the occurrence of knotted metallic band structures as stable topological phases in nonHermitian (NH) systems. These knotted NH metals are characterized by open Fermi surfaces, known in mathematics as Seifert surfaces, that are bounded by knotted lines of exceptional points. Quite remarkably, and in contrast to the situation in Hermitian systems, no fine tuning or symmetries are required in order to stabilize these exotic phases of matter. By explicit construction, we derive microscopic tightbinding models hosting knotted NH metals with strictly shortranged hopping, and investigate the stability of their topological properties against perturbations. Building up on recently developed experimental techniques for the realization of NH band structures, we discuss how the proposed models may be experimentally implemented in photonic systems.

Mixed AxialTorsional Anomaly in Weyl Semimetals
2019. Yago Ferreiros (et al.). Physical Review Letters 122 (5)
ArtikelWe show that Weyl semimetals exhibit a mixed axialtorsional anomaly in the presence of axial torsion, a concept exclusive of these materials with no known natural fundamental interpretation in terms of the geometry of spacetime. This anomaly implies a nonconservation of the axial currentthe difference in the current of left and righthanded chiral fermionswhen the torsion of the spacetime in which the Weyl fermions move couples with opposite sign to different chiralities. The anomaly is activated by driving transverse sound waves through a Weyl semimetal with a spatially varying tilted dispersion, which can be engineered by applying strain. This leads to a sizable alternating current in the presence of a magnetic field that provides a clearcut experimental signature of our predictions.

NonHermitian extensions of higherorder topological phases and their biorthogonal bulkboundary correspondence
2019. Elisabet Edvardsson, Flore K. Kunst, Emil J. Bergholtz. Physical Review B 99 (8)
ArtikelNonHermitian Hamiltonians, which describe a wide range of dissipative systems, and higherorder topological phases, which exhibit novel boundary states on corners and hinges, comprise two areas of intense current research. Here we investigate systems where these frontiers merge and formulate a generalized biorthogonal bulkboundary correspondence, which dictates the appearance of boundary modes at parameter values that are, in general, radically different from those that mark phase transitions in periodic systems. By analyzing the interplay between corner/hinge, edge/surface and bulk degrees of freedom we establish that the nonHermitian extensions of higherorder topological phases exhibit an even richer phenomenology than their Hermitian counterparts and that this can be understood in a unifying way within our biorthogonal framework. Saliently this works in the presence of the nonHermitian skin effect, and also naturally encompasses genuinely nonHermitian phenomena in the absence thereof.

Symmetryprotected nodal phases in nonHermitian systems
2019. Jan Carl Budich (et al.). Physical Review B 99 (4)
ArtikelNonHermitian (NH) Hamiltonians have become an important asset for the effective description of various physical systems that are subject to dissipation. Motivated by recent experimental progress on realizing the NH counterparts of gapless phases such as Weyl semimetals, here we investigate how NH symmetries affect the occurrence of exceptional points (EPs), that generalize the notion of nodal points in the spectrum beyond the Hermitian realm. Remarkably, we find that the dimension of the manifold of EPs is generically increased by one as compared to the case without symmetry. This leads to nodal surfaces formed by EPs that are stable as long as a protecting symmetry is preserved, and that are connected by open Fermi volumes. We illustrate our findings with analytically solvable twoband lattice models in one and two spatial dimensions, and show how they are readily generalized to generic NH crystalline systems.

Corner state of light in photonic waveguides
2018. Ashraf Mohamed El Hassan (et al.).
Artikel 
Exceptional links and twisted Fermi ribbons in nonHermitian systems
2018. Johan Carlström, Emil J. Bergholtz. Physical Review A: covering atomic, molecular, and optical physics and quantum information 98 (4)
ArtikelThe generic nature of band touching points in threedimensional band structures is at the heart of the rich phenomenology, topological stability, and novel Fermi arc surface states associated with Weyl semimetals. Here we report on the corresponding scenario emerging in systems effectively described by nonHermitian Hamiltonians. Remarkably, threedimensional nonHermitian systems have generic band touching along onedimensional closed contours, forming exceptional rings and links in reciprocal space. The associated Seifert surfaces support open Fermi ribbons where the real part of the energy gap vanishes, providing a novel class of higherdimensional bulk generalizations of Fermi arcs which are characterized by an integer twist number. These results have possible applications to a plethora of physical settings, ranging from mechanical systems and optical metamaterials with loss and gain to heavy fermion materials with finitelifetime quasiparticles. In particular, photonic crystals provide fertile ground for simulating the exuberant phenomenology of exceptional links and their concomitant Fermi ribbons.

Lattice models with exactly solvable topological hinge and corner states
2018. Flore K. Kunst, Guido van Miert, Emil J. Bergholtz. Physical Review B 97 (24)
ArtikelWe devise a generic recipe for constructing Ddimensional lattice models whose ddimensional boundary states, located on surfaces, hinges, corners, and so forth, can be obtained exactly. The solvability is rooted in the underlying lattice structure and as such does not depend on fine tuning, allowing us to track their evolution throughout various phases and across phase transitions. Most saliently, our models provide boundary solvable examples of the recently introduced higherorder topological phases. We apply our general approach to breathing and anisotropic kagome and pyrochlore lattices for which we obtain exact corner eigenstates, and to periodically driven twodimensional models as well as to threedimensional lattices where we present exact solutions corresponding to onedimensional chiral states at the hinges of the lattice. We relate the higherorder topological nature of these models to reflection symmetries in combination with their provenance from lowerdimensional conventional topological phases.

Quantum oscillations and magnetoresistance in typeII Weyl semimetals
2018. Maximilian Trescher, Emil J. Bergholtz, Johannes Knolle. Physical Review B 98 (12)
ArtikelRecent experiments on typeII Weyl semimetals such as WTe2 , MoTe2 , MoxW1xTe2 , and WP2 reveal remarkable transport properties in the presence of a strong magnetic field, including an extremely large magnetoresistance and an unusual temperature dependence. Here, we investigate magnetotransport via the Kubo formula in a minimal model of a typeII Weyl semimetal taking into account the effect of a charge density wave (CDW) transition, which can arise even at weak coupling in the presence of a strong magnetic field because of the special Landau level dispersion of typeII Weyl systems. Consistent with experimental measurements we find an extremely large magnetoresistance with close to B2 scaling at particlehole compensation, while in the extreme quantum limit there is a transition to a qualitatively new scaling with approximately B0.75 . We also investigate the Shubnikovde Haas effect and find that the amplitude of the resistivity quantum oscillations are greatly enhanced below the CDW transition temperature which is accompanied by an unusual nonmonotonous (nonLifshitzKosevich) temperature dependence.

Strongly interacting Weyl semimetals
2018. Johan Carlström, Emil J. Bergholt. Physical Review B 98 (24)
ArtikelUsing a combination of analytical arguments and stateoftheart diagrammatic Monte Carlo simulations, we show that the corrections to the dispersion in interacting Weyl semimetals are determined by the ultraviolet cutoff and the inverse screening length. If both of these are finite, then the diagrammatic series is convergent even in the lowtemperature limit, which implies that the semimetallic phase remains stable. Meanwhile, the absence of a UV cutoff or screening results in logarithmic divergences at zero temperature. These results highlight the crucial impact of Coulomb interactions and screening, mediated, e.g., through the presence of parasitic bands, which are ubiquitous effects in realworld materials. Also, despite sizable corrections from Coulomb forces, the contribution from the frequencydependent part of the selfenergy remains extremely small, thus giving rise to a system of effectively almost free fermions with a strongly renormalized dispersion.

Symmetryenforced stability of interacting Weyl and Dirac semimetals
2018. Johan Carlström, Emil J. Bergholtz. Physical Review B 97 (16)
ArtikelThe nodal and effectively relativistic dispersion featuring in a range of novel materials including twodimensional graphene and threedimensional 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 twobody interactions. For effective lowenergy theories relevant for single and multilayer graphene, typeI and typeII 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 anticommutes 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.

Anatomy of topological surface states
2017. Flore K. Kunst, Maximilian Trescher, Emil J. Bergholtz. Physical Review B 96 (8)
ArtikelThe hallmark of topological phases is their robust boundary signature whose intriguing propertiessuch as the oneway 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 semimetalsare impossible to realize on the surface alone. Yet, despite the glaring simplicity of noninteracting 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 tightbinding models on a large class of geometrically frustrated lattices in any dimension without the need for finetuning 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 including correlation functions, surface dispersion, Berry curvature, and the system size dependent gap closing, which necessarily occurs when the spatial localization switches surface. This further provides a deepened understanding of the bulkboundary correspondence. 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 spinorbitcoupled kagome lattice, and Fermi arcs relevant for recently synthesized slabs of pyrochlorebased Eu2Ir2O7 and Nd2Ir2O7, which realize an allinallout spin configuration, as well as for spinicelike twointwoout and oneinthreeout configurations, which are both relevant for Pr2Ir2O7. Remarkably, each of the pyrochlore examples exhibit clearly resolved Fermi arcs although only the oneinthreeout configuration features bulk Weyl nodes in realistic parameter regimes. Our approach generalizes to symmetry protected phases, e.g., quantum spin Hall systems and Dirac semimetals with timereversal symmetry, and can furthermore signal the absence of topological surface states, which we illustrate for a class of models akin to the trivial surface of Hourglass materials KHgX where the exact solutions apply but, independently of Hamiltonian details, yield eigenstates delocalized over the entire sample.

Charge density wave instabilities of typeII Weyl semimetals in a strong magnetic field
2017. Maximilian Trescher (et al.). Physical Review B 96 (20)
ArtikelShortly 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 manybody 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 typeII 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 typeII 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.

Composite symmetryprotected topological order and effective models
2017. A. Nietner (et al.). Physical Review B 96 (23)
ArtikelStrongly correlated quantum manybody systems at low dimension exhibit a wealth of phenomena, ranging from features of geometric frustration to signatures of symmetryprotected topological order. In suitable descriptions of such systems, it can be helpful to resort to effective models, which focus on the essential degrees of freedom of the given model. In this work, we analyze how to determine the validity of an effective model by demanding it to be in the same phase as the original model. We focus our study on onedimensional spin1/2 systems and explain how nontrivial symmetryprotected topologically ordered (SPT) phases of an effective spin1 model can arise depending on the couplings in the original Hamiltonian. In this analysis, tensor network methods feature in two ways: on the one hand, we make use of recent techniques for the classification of SPT phases using matrix product states in order to identify the phases in the effective model with those in the underlying physical system, employing Kunneth's theorem for cohomology. As an intuitive paradigmatic model we exemplify the developed methodology by investigating the bilayered Delta chain. For strong ferromagnetic interlayer couplings, we find the system to transit into exactly the same phase as an effective spin1 model. However, for weak but finite coupling strength, we identify a symmetry broken phase differing from this effective spin1 description. On the other hand, we underpin our argument with a numerical analysis making use of matrix product states.

Disordered doubleWeyl node
2017. Bjoern Sbierski (et al.). Physical Review B 95 (11)
ArtikelDouble Weyl nodes are topologically protected band crossing points which carry chiral charge +/ 2. They are stabilized by C4 pointgroup symmetry and are predicted to occur in SrSi2 or HgCr2Se4. We study their stability and physical properties in the presence of a disorder potential. We investigate the density of states and the quantum transport properties at the nodal point. We find that, in contrast to their counterparts with unit chiral charge, double Weyl nodes are unstable to any finite amount of disorder and give rise to a diffusive phase, in agreement with the predictions of Goswami and Nevidomskyy [Phys. Rev. B 92, 214504 (2015)] and Bera, Sau, and Roy [Phys. Rev. B 93, 201302 (2016)]. However, for finite system sizes a crossover between pseudodiffusive and diffusive quantum transport can be observed.

Exotic NonAbelian Topological Defects in Lattice Fractional Quantum Hall States
2017. Zhao Liu, Gunnar Möller, Emil J. Bergholtz. Physical Review Letters 119 (10)
ArtikelWe 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 singleparticle potentials. Remarkably, once local manybody interactions in this new band are switched on, we identify various Abelian and nonAbelian fractional quantum Hall states, whose groundstate 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 nonAbelian 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.

Interacting Majorana chain
2017. Zhao Liu (et al.). Physical Review B 96 (20)
ArtikelWe study a onedimensional chain of 2N Majorana bound states, which interact through a local quartic interaction. This model describes for example the edge physics of a quasionedimensional (1D) stack of 2N Kitaev chains with modified timereversal symmetry T gamma iT1 = gamma(i), which precludes the presence of quadratic coupling. The ground state of our 1D Majorana chain displays a fourfold 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 fourfold periodicity, and the chain exhibits strikingly different transport properties for different chain lengths. In the thermodynamic limit, the 1D Majorana chain hosts a robust manybody zero mode, which indicates that the corresponding stacked twodimensional bulk system realizes a weak topological phase.

Josephson effect in a Weyl SNS junction
2017. Kevin A. Madsen, Emil J. Bergholtz, Piet W. Brouwer. Physical Review B 95 (6)
ArtikelWe calculate the Josephson current density j (phi) for a Weyl superconductornormalmetalsuperconductor junction for which the outer terminals are superconducting Weylmetals 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 timereversal 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, zeromomentum BCSlike pairing and finitemomentum FFLOlike pairing, and assuming the shortjunction limit. For both pairing types we find that the current is proportional to the normalstate junction conductivity, with a proportionality coefficient that shows quantitative differences between the two pairing mechanisms. The current for the BCSlike pairing is found to be independent of the chemical potential, whereas the current for the FFLOlike pairing is not.

Tilted disordered Weyl semimetals
2017. Maximilian Trescher (et al.). Physical Review B 95 (4), 18
ArtikelAlthough Lorentz invariance forbids the presence of a term that tilts the energymomentum 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 disorderinduced phase transition between a quasiballistic 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 strongdisorder phase. Notably this obscures the tiltinduced Lifshitz transition to an overtilted type II Weyl phase at any finite disorder strength. Our results are supported by analytical calculations using the selfconsistent Born approximation and numerical calculations of the density of states and of transport properties.

FieldSelective Anomaly and Chiral Mode Reversal in TypeII Weyl Materials
2016. M. Udagawa, Emil J. Bergholtz. Physical Review Letters 117 (8)
ArtikelThreedimensional condensed matter incarnations of Weyl fermions generically have a tilted dispersionin sharp contrast to their elusive highenergy relatives where a tilt is forbidden by Lorentz invariance, and with the lowenergy excitations of twodimensional graphene sheets where a tilt is forbidden by either crystalline or particlehole symmetry. Very recently, a number of materials (MoTe2, LaAlGe, and WTe2) have been identified as hosts of socalled typeIIWeyl fermions whose dispersion is so strongly tilted that a Fermi surface is formed, whereby the Weyl node becomes a singular point connecting electron and hole pockets. We here predict that these systems have remarkable properties in the presence of magnetic fields. Most saliently, we show that the nature of the chiral anomaly depends crucially on the relative angle between the applied field and the tilt, and that an inversionasymmetric overtilting creates an imbalance in the number of chiral modes with positive and negative slopes. The fieldselective anomaly gives a novel magnetooptical resonance, providing an experimental way to detect concealed Weyl nodes.

Quantum Hall hierarchy wave functions: From conformal correlators to TaoThouless states
2008. Emil Johansson Bergholtz (et al.). Physical Review B Condensed Matter 77 (16), 16532511653259
ArtikelLaughlin’s wave functions, which describe the fractional quantum Hall effect at filling factorsν=1/(2k+1), can be obtained as correlation functions in a conformal field theory, and recently, this construction was extended to Jain’s composite fermion wave functions at filling factors ν=n/(2kn+1). Here, we generalize this latter construction and present ground state wave functions for all quantum Hall hierarchy states that are obtained by successive condensation of quasielectrons (as opposed to quasiholes) in the original hierarchy construction. By considering these wave functions on a cylinder, we show that they approach the exact ground states, which are the TaoThouless states, when the cylinder becomes thin. We also present wave functions for the multihole states, make the connection to Wen’s general classification of Abelian quantum Hall fluids, and discuss whether the fractional statistics of the quasiparticles can be analytically determined. Finally, we discuss to what extent our wave functions can be described in the language of composite fermions.

Quantum Hall wave functions on the torus
2008. Maria Hermanns (et al.). Physical Review B Condensed Matter 77 (12), 125321112532116
ArtikelWe present explicit expressions for a large set of hierarchy wave functions on the torus. Included are the Laughlin states, the states in the positive Jain series, and recently observed states at, e.g., ν=4∕11. The techniques we use constitute a nontrivial extension of the conformal field theory methods developed earlier to construct the corresponding wave functions in disk geometry.
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