Preprints

BibTeX: @unpublished{nowforever,, title = { 
Zhao Liu, Gunnar Möller and Emil J. Bergholtz
Lattice Genons arXiv:1702.05115v1 [condmat.strel]. 
Abstract: Topological degeneracy is, together with fractionalization, the hallmark of topological order. While fractionalization has been demonstrated both theoretically and in laboratories leading to deep insights into our basic understanding of matter, there has been no corresponding success in tracking the evolution of groundstate degeneracy due to the lack of tractable microscopic models with arbitrarily tunable topology of space. In this work however, by computer simulations, we show that lattice versions of multicomponent fractional quantum Hall systems are novel playgrounds to fill this void. With wormholelike defects that connect separated "universes" of different components, thus changing the global topology of space, we observe a nontrivial dependence of groundstate degeneracy on the number of defects, equivalent to adding exotic nonAbelian quasiparticles called "genons" proposed recently in terms of effective topological field theory. We identify a number of different lattice genons in both Abelian and nonAbelian phases, with quantum dimensions exceeding those of fundamental quasiparticles in the host states, and provide a promising platform to study their intriguing behavior. 
BibTeX: @unpublished{1702.05115v1, author = Zhao Liu, Gunnar Möller and Emil J. Bergholtz, title = Lattice Genons, year = 2017, note = arXiv:1702.05115v1 [condmat.strel], url = http://arxiv.org/abs/1702.05115v1 } 
Björn Sbierski, Maximilian Trescher, Emil J. Bergholtz and Piet W. Brouwer
Disordered double Weyl node: Comparison of transport and densityofstates calculations arXiv:1606.06941 [condmat.meshall]. 
Abstract: 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 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 predictions of Goswami and Nevidomskyy [Phys. Rev. B 92, 214504 (2015)] and Bera, Sau, and Roy [Phys. Rev. B 93, 201302(R) (2016)]. However, for finite system sizes a crossover between pseudodiffusive and diffusive quantum transport can be observed. 
BibTeX: @unpublished{Sbierski2016a, author = Björn Sbierski, Maximilian Trescher, Emil J. Bergholtz and Piet W. Brouwer, title = Disordered double Weyl node: Comparison of transport and densityofstates calculations, year = 2016, note = arXiv:1606.06941 [condmat.meshall], url = https://arxiv.org/abs/1606.06941 } 
2017

BibTeX: @unpublished{20172018,, title = { 
Kevin A. Madsen, Emil J. Bergholtz and Piet W. Brouwer
Josephson effect in a Weyl SNS junction Phys. Rev. B 95(6), 064511 (2017). 
Abstract: We calculate the Josephson current density j(ϕ) for a Weyl superconductor–normalmetal–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 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. 
BibTeX: @article{PhysRevB.95.064511, author = Kevin A. Madsen, Emil J. Bergholtz and Piet W. Brouwer, title = Josephson effect in a Weyl SNS junction, journal = Phys. Rev. B, year = 2017, volume = 95, number = 6, pages = 064511, url = http://link.aps.org/doi/10.1103/PhysRevB.95.064511, doi = http://dx.doi.org/10.1103/PhysRevB.95.064511 } 
Maximilian Trescher, Björn Sbierski, Piet W. Brouwer and Emil J. Bergholtz
Tilted disordered Weyl semimetals Phys. Rev. B 95(4), 045139 (2017). 
Abstract: Although 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. 
BibTeX: @article{PhysRevB.95.045139, author = Maximilian Trescher, Björn Sbierski, Piet W. Brouwer and Emil J. Bergholtz, title = Tilted disordered Weyl semimetals, journal = Phys. Rev. B, year = 2017, volume = 95, number = 4, pages = 045139, url = http://link.aps.org/doi/10.1103/PhysRevB.95.045139, doi = http://dx.doi.org/10.1103/PhysRevB.95.045139 } 
2016

BibTeX: @unpublished{20162017,, title = { 
W. Zhu, Zhao Liu, F. D. M. Haldane and D. N. Sheng
Fractional quantum Hall bilayers at half filling: Tunnelingdriven nonAbelian phase Phys. Rev. B 94(24), 245147 (2016). 
Abstract: Multicomponent quantum Hall systems with internal degrees of freedom provide a fertile ground for the emergence of exotic quantum liquids. Here, we investigate the possibility of nonAbelian topological order in the halffilled fractional quantum Hall (FQH) bilayer system driven by the tunneling effect between two layers. By means of the stateoftheart densitymatrix renormalization group, we unveil “fingerprint” evidence of the nonAbelian MooreRead Pfaffian state emerging in the intermediatetunneling regime, including the groundstate degeneracy on the torus geometry and the topological entanglement spectroscopy (entanglement spectrum and topological entanglement entropy) on the spherical geometry, respectively. Remarkably, the phase transition from the previously identified Abelian (331) Halperin state to the nonAbelian MooreRead Pfaffian state is determined to be continuous, which is signaled by the continuous evolution of the universal part of the entanglement spectrum, and discontinuities in the excitation gap and the derivative of the groundstate energy. Our results not only provide a “proofofprinciple” demonstration of realizing a nonAbelian state through coupling different degrees of freedom, but also open up a possibility in FQH bilayer systems for detecting different chiral p−wave pairing states. 
BibTeX: @article{PhysRevB.94.245147, author = W. Zhu, Zhao Liu, F. D. M. Haldane and D. N. Sheng, title = Fractional quantum Hall bilayers at half filling: Tunnelingdriven nonAbelian phase, journal = Phys. Rev. B, year = 2016, volume = 94, number = 24, pages = 245147, url = http://link.aps.org/doi/10.1103/PhysRevB.94.245147, doi = http://dx.doi.org/10.1103/PhysRevB.94.245147 } 
Zhao Liu and R. N. Bhatt
Quantum Entanglement as a Diagnostic of Phase Transitions in Disordered Fractional Quantum Hall Liquids Physical Review Letters 117(20), (2016). 
Abstract: We investigate the disorderdriven phase transition from a fractional quantum Hall state to an Anderson insulator using quantum entanglement methods. We find that the transition is signaled by a sharp increase in the sensitivity of a suitably averaged entanglement entropy with respect to disorder—the magnitude of its disorder derivative appears to diverge in the thermodynamic limit. We also study the level statistics of the entanglement spectrum as a function of disorder. However, unlike the dramatic phasetransition signal in the entanglement entropy derivative, we find a gradual reduction of level repulsion only deep in the Anderson insulating phase. 
BibTeX: @article{Liu2016, author = Zhao Liu and R. N. Bhatt, title = Quantum Entanglement as a Diagnostic of Phase Transitions in Disordered Fractional Quantum Hall Liquids, journal = Physical Review Letters, year = 2016, volume = 117, number = 20, pages = , url = http://dx.doi.org/10.1103/PhysRevLett.117.206801, doi = http://dx.doi.org/10.1103/physrevlett.117.206801 } 
Frédéric Combes, Maximilian Trescher, Frédéric Piéchon and JeanNoël Fuchs
Statistical mechanics approach to the electric polarization and dielectric constant of band insulators Phys. Rev. B 94(15), 155109 (2016). 
Abstract: We develop a theory for the analytic computation of the free energy of band insulators in the presence of a uniform and constant electric field. The two key ingredients are a perturbationlike expression of the WannierStark energy spectrum of electrons and a modified statistical mechanics approach involving a local chemical potential in order to deal with the unbounded spectrum and impose the physically relevant electronic filling. At first order in the field, we recover the result of KingSmith, Vanderbilt, and Resta for the electric polarization in terms of a Zak phase—albeit at finite temperature—and, at second order, deduce a general formula for the electric susceptibility, or equivalently for the dielectric constant. Advantages of our method are the validity of the formalism both at zero and finite temperature and the easy computation of higher order derivatives of the free energy. We verify our findings on two different onedimensional tightbinding models. 
BibTeX: @article{PhysRevB.94.155109, author = Frédéric Combes, Maximilian Trescher, Frédéric Piéchon and JeanNoël Fuchs, title = Statistical mechanics approach to the electric polarization and dielectric constant of band insulators, journal = Phys. Rev. B, year = 2016, volume = 94, number = 15, pages = 155109, url = http://link.aps.org/doi/10.1103/PhysRevB.94.155109, doi = http://dx.doi.org/10.1103/PhysRevB.94.155109 } 
M. Udagawa and E. J. Bergholtz
FieldSelective Anomaly and Chiral Mode Reversal in TypeII Weyl Materials Phys. Rev. Lett. 117(8), 086401 (2016). 
Abstract: Threedimensional condensed matter incarnations of Weyl fermions generically have a tilted dispersion—in 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 typeII Weyl 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. 
BibTeX: @article{PhysRevLett.117.086401, author = M. Udagawa and E. J. Bergholtz, title = FieldSelective Anomaly and Chiral Mode Reversal in TypeII Weyl Materials, journal = Phys. Rev. Lett., year = 2016, volume = 117, number = 8, pages = 086401, url = http://link.aps.org/doi/10.1103/PhysRevLett.117.086401, doi = http://dx.doi.org/10.1103/PhysRevLett.117.086401 } 
RobertJan Slager, Vladimir Juričić, Ville Lahtinen and Jan Zaanen
Selforganized pseudographene on grain boundaries in topological band insulators Phys. Rev. B 93(24), 245406 (2016). 
Abstract: Semimetals are characterized by nodal band structures that give rise to exotic electronic properties. The stability of Dirac semimetals, such as graphene in two spatial dimensions, requires the presence of lattice symmetries, while akin to the surface states of topological band insulators, Weyl semimetals in three spatial dimensions are protected by band topology. Here we show that in the bulk of topological band insulators, selforganized topologically protected semimetals can emerge along a grain boundary, a ubiquitous extended lattice defect in any crystalline material. In addition to experimentally accessible electronic transport measurements, these states exhibit a valley anomaly in two dimensions influencing edge spin transport, whereas in three dimensions they appear as graphenelike states that may exhibit an oddinteger quantum Hall effect. The general mechanism underlying these semimetals—the hybridization of spinon modes bound to the grain boundary—suggests that topological semimetals can emerge in any topological material where lattice dislocations bind localized topological modes. 
BibTeX: @article{PhysRevB.93.245406, author = RobertJan Slager, Vladimir Juričić, Ville Lahtinen and Jan Zaanen, title = Selforganized pseudographene on grain boundaries in topological band insulators, journal = Phys. Rev. B, year = 2016, volume = 93, number = 24, pages = 245406, url = http://link.aps.org/doi/10.1103/PhysRevB.93.245406, doi = http://dx.doi.org/10.1103/PhysRevB.93.245406 } 
Jörg Behrmann, Zhao Liu and Emil J. Bergholtz
Model Fractional Chern Insulators Phys. Rev. Lett. 116(21), 216802 (2016). 
Abstract: We devise local lattice models whose ground states are model fractional Chern insulators—Abelian and nonAbelian 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) colorentangled fractional Chern insulators at band filling fractions $\nu =k/(\mathcal{C}+1)$ and (ii) nematic states at $\nu =k/2$, where $\mathcal{C}$ is the Chern number of the lowest band. 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 finitesize 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. 
BibTeX: @article{PhysRevLett.116.216802, author = Jörg Behrmann, Zhao Liu and Emil J. Bergholtz, title = Model Fractional Chern Insulators, journal = Phys. Rev. Lett., year = 2016, volume = 116, number = 21, pages = 216802, url = http://link.aps.org/doi/10.1103/PhysRevLett.116.216802, doi = http://dx.doi.org/10.1103/PhysRevLett.116.216802 } 
Zhao Liu, Abolhassan Vaezi, Cécile Repellin and Nicolas Regnault
Phase diagram of ν=1/2+1/2 bilayer bosons with interlayer couplings Phys. Rev. B 93(8), 085115 (2016). 
Abstract: We present the quantitative phase diagram of the bilayer bosonic fractional quantum Hall system on the torus geometry at total filling factor ν=1 in the lowest Landau level. We consider shortrange interactions within and between the two layers, as well as the interlayer tunneling. In the fully polarized regime, we provide an updated detailed numerical analysis to establish the presence of the MooreRead phase of both even and odd numbers of particles. In the actual bilayer situation, we find that both interlayer interactions and tunneling can provide the physical mechanism necessary for the lowenergy physics to be driven by the fully polarized regime, thus leading to the emergence of the MooreRead phase. Interlayer interactions favor a ferromagnetic phase when the system is SU(2) symmetric, while the interlayer tunneling acts as a Zeeman field polarizing the system. Besides the MooreRead phase, the (220) Halperin state and the coupled MooreRead state are also realized in this model. We study their stability against each other. 
BibTeX: @article{PhysRevB.93.085115, author = Zhao Liu, Abolhassan Vaezi, Cécile Repellin and Nicolas Regnault, title = Phase diagram of ν=1/2+1/2 bilayer bosons with interlayer couplings, journal = Phys. Rev. B, year = 2016, volume = 93, number = 8, pages = 085115, url = http://link.aps.org/doi/10.1103/PhysRevB.93.085115, doi = http://dx.doi.org/10.1103/PhysRevB.93.085115 } 
Diana Prychynenko and Sebastian D. Huber
Z_2 slavespin theory of a strongly correlated Chern insulator Physica B: Condensed Matter 481(53), 53–58 (2016). 
Abstract: We calculate the phase diagram of the topological honeycomb model in the presence of strong interactions. We concentrate on half filling and employ a Z2 slavespin method to find a band insulator with staggered density, a spindensitywave and a Mott insulating phase. Both the band insulator and the spindensity wave come in various topological varieties. Finally, we calculate the response function relevant for lattice modulation spectroscopy with cold atomic gases in optical lattices. 
BibTeX: @article{Prychynenko2016, author = Diana Prychynenko and Sebastian D. Huber, title = Z_2 slavespin theory of a strongly correlated Chern insulator, journal = Physica B: Condensed Matter, year = 2016, volume = 481, number = 53, pages = 53–58, url = http://dx.doi.org/10.1016/j.physb.2015.10.027, doi = http://dx.doi.org/10.1016/j.physb.2015.10.027 } 
2015

BibTeX: @unpublished{20152016,, title = { 
Ville Lahtinen and Eddy Ardonne
Realizing All so(N)_1 Quantum Criticalities in Symmetry Protected Cluster Models Physical Review Letters 115(23), 237203 (2015). 
Abstract: We show that all so(N)1 universality class quantum criticalities emerge when onedimensional generalized cluster models are perturbed with Ising or Zeeman terms. Each critical point is described by a lowenergy theory of N linearly dispersing fermions, whose spectrum we show to precisely match the prediction by so(N)1 conformal field theory. Furthermore, by an explicit construction we show that all the cluster models are dual to nonlocally coupled transverse field Ising chains, with the universality of the so(N)1 criticality manifesting itself as N of these chains becoming critical. This duality also reveals that the symmetry protection of cluster models arises from the underlying Ising symmetries and it enables the identification of local representations for the primary fields of the so(N)1 conformal field theories. For the simplest and experimentally most realistic case that corresponds to the original onedimensional cluster model with local threespin interactions, our results show that the su(2)2≃so(3)1 WessZuminoWitten model can emerge in a local, translationally invariant, and JordanWigner solvable spin1/2 model. 
BibTeX: @article{Lahtinen2015, author = Ville Lahtinen and Eddy Ardonne, title = Realizing All so(N)_1 Quantum Criticalities in Symmetry Protected Cluster Models, journal = Physical Review Letters, year = 2015, volume = 115, number = 23, pages = 237203, url = http://dx.doi.org/10.1103/PhysRevLett.115.237203, doi = http://dx.doi.org/10.1103/physrevlett.115.237203 } 
Flore K. Kunst, Christophe Delerue, Cristiane Morais Smith and Vladimir Juričić
Kekule versus hidden superconducting order in graphenelike systems: Competition and coexistence Phys. Rev. B 92(16), 165423 (2015). 
Abstract: We theoretically study the competition between two possible exotic superconducting orders that may occur in graphenelike systems, assuming dominant nearestneighbor attraction: the gapless hidden superconducting order, which renormalizes the Fermi velocity, and the Kekule order, which opens a superconducting gap. We perform an analysis within the meanfield theory for Dirac electrons, at finite temperature and finite chemical potential, as well as at half filling and zero temperature, first excluding the possibility of the coexistence of the two orders. In that case, we find the dependence of the critical (more precisely, crossover) temperature and the critical interaction on the chemical potential. As a result of this analysis, we find that the Kekule order is preferred over the hidden order at both finite temperature and finite chemical potential. However, when the coexistence of the two superconducting orders is allowed, by solving the coupled meanfield gap equations, we find that above a critical value of the attractive interaction a mixed phase sets in, in which these orders coexist. We show that the critical value of the interaction for this transition is greater than the critical coupling for the hidden superconducting state in the absence of the Kekule order, implying that there is a region in the phase diagram where the Kekule order is favored as a result of the competition with the hidden superconducting order. The latter, however, eventually sets in and coexists with the Kekule state. According to our meanfield calculations, the transition from the Kekule to the mixed phase is of the second order, but it may become first order when fluctuations are considered. Finally, we investigate whether these phases could be possible in honeycomb superlattices of selfassembled semiconducting nanocrystals, which have been recently experimentally realized with CdSe and PbSe. 
BibTeX: @article{PhysRevB.92.165423, author = Flore K. Kunst, Christophe Delerue, Cristiane Morais Smith and Vladimir Juričić, title = Kekule versus hidden superconducting order in graphenelike systems: Competition and coexistence, journal = Phys. Rev. B, year = 2015, volume = 92, number = 16, pages = 165423, url = http://link.aps.org/doi/10.1103/PhysRevB.92.165423, doi = http://dx.doi.org/10.1103/PhysRevB.92.165423 } 
Björn Sbierski, Emil J. Bergholtz and Piet W. Brouwer
Quantum critical exponents for a disordered threedimensional Weyl node Phys. Rev. B 92(11), 115145 (2015). 
Abstract: Threedimensional Dirac and Weyl semimetals exhibit a disorderinduced quantum phase transition between a semimetallic phase at weak disorder and a diffusivemetallic 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±0.03 using a minimal singleWeyl node model and a finitesize scaling analysis of conductance. Our highprecision numerical value for ν is incompatible with previous numerical studies on tightbinding models and with one and twoloop calculations in an εexpansion scheme. We further obtain z=1.49±0.02 from the scaling of the conductivity with chemical potential. 
BibTeX: @article{PhysRevB.92.115145, author = Björn Sbierski, Emil J. Bergholtz and Piet W. Brouwer, title = Quantum critical exponents for a disordered threedimensional Weyl node, journal = Phys. Rev. B, year = 2015, volume = 92, number = 11, pages = 115145, url = http://link.aps.org/doi/10.1103/PhysRevB.92.115145, doi = http://dx.doi.org/10.1103/PhysRevB.92.115145 } 
Maximilian Trescher, Björn Sbierski, Piet W. Brouwer and Emil J. Bergholtz
Quantum transport in Dirac materials: Signatures of tilted and anisotropic Dirac and Weyl cones Phys. Rev. B 91(11), (2015). 
Abstract: 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 the absence of certain discrete symmetries, such as particlehole 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 semimetal. 
BibTeX: @article{Trescher2015, author = Maximilian Trescher, Björn Sbierski, Piet W. Brouwer and Emil J. Bergholtz, title = Quantum transport in Dirac materials: Signatures of tilted and anisotropic Dirac and Weyl cones, journal = Phys. Rev. B, year = 2015, volume = 91, number = 11, pages = , url = http://dx.doi.org/10.1103/PhysRevB.91.115135, doi = http://dx.doi.org/10.1103/physrevb.91.115135 } 
E. J. Bergholtz, Zhao Liu, M. Trescher, R. Moessner and M. Udagawa
Topology and Interactions in a Frustrated Slab: Tuning from Weyl Semimetals to $\mathcal{C}\>1$ Fractional Chern Insulators Phys. Rev. Lett. 114(1), 016806 (2015). 
Abstract: We show that, quite generically, a [111] slab of spinorbit coupled pyrochlore lattice exhibits surface states whose constant energy curves take the shape of Fermi arcs, localized to different surfaces depending on their quasimomentum. Remarkably, these persist independently of the existence of Weyl points in the bulk. Considering interacting electrons in slabs of finite thickness, we find a plethora of known fractional Chern insulating phases, to which we add the discovery of a new higher Chern number state which is likely a generalization of the MooreRead fermionic fractional quantum Hall state. By contrast, in the threedimensional limit, we argue for the absence of gapped states of the flat surface band due to a topologically protected coupling of the surface to gapless states in the bulk. We comment on generalizations as well as experimental perspectives in thin slabs of pyrochlore iridates. 
BibTeX: @article{PhysRevLett.114.016806, author = E. J. Bergholtz, Zhao Liu, M. Trescher, R. Moessner and M. Udagawa, title = Topology and Interactions in a Frustrated Slab: Tuning from Weyl Semimetals to $\mathcal{C}\>1$ Fractional Chern Insulators, journal = Phys. Rev. Lett., year = 2015, volume = 114, number = 1, pages = 016806, url = http://link.aps.org/doi/10.1103/PhysRevLett.114.016806, doi = http://dx.doi.org/10.1103/PhysRevLett.114.016806 } 
2014

BibTeX: @unpublished{20142015,, title = { 
Masafumi Udagawa and Emil J. Bergholtz
Correlations and entanglement in flat band models with variable Chern numbers Journal of Statistical Mechanics: Theory and Experiment 2014(10), P10012 (2014). 
Abstract: We discuss a number of illuminating results for tightbinding models supporting a band with variable Chern numbers and illustrate them explicitly for a simple class of twobanded 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 increases with C and that the band flattening requires an exponential tail of longrange 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. 
BibTeX: @article{17425468201410P10012, author = Masafumi Udagawa and Emil J. Bergholtz, title = Correlations and entanglement in flat band models with variable Chern numbers, journal = Journal of Statistical Mechanics: Theory and Experiment, year = 2014, volume = 2014, number = 10, pages = P10012, url = http://stacks.iop.org/17425468/2014/i=10/a=P10012, doi = http://dx.doi.org/10.1088/17425468/2014/10/P10012 } 
J. C. Budich, J. Eisert, E. J. Bergholtz, S. Diehl and P. Zoller
Search for localized Wannier functions of topological band structures via compressed sensing Phys. Rev. B 90(11), 115110 (2014). 
Abstract: We investigate the interplay of band structure topology and localization properties of Wannier functions. To this end, we extend a recently proposed compressed sensing based paradigm for the search for maximally localized Wannier functions [Ozolins et al., Proc. Natl. Acad. Sci. USA 110, 18368 (2013)]. We develop a practical toolbox that enables the search for maximally localized Wannier functions which exactly obey the underlying physical symmetries of a translationally invariant quantum lattice system under investigation. Most saliently, this allows us to systematically identify the most localized representative of a topological equivalence class of band structures, i.e., the most localized set of Wannier functions that is adiabatically connected to a generic initial representative. We also elaborate on the compressed sensing scheme and find a particularly simple and efficient implementation in which each step of the iteration is an O(NlogN) algorithm in the number of lattice sites N. We present benchmark results on onedimensional topological superconductors demonstrating the power of these tools. Furthermore, we employ our method to address the open question of whether compact Wannier functions can exist for symmetryprotected topological states such as topological insulators in two dimensions. The existence of such functions would imply exact flatband models with finite range hopping. Here, we find numerical evidence for the absence of such functions. We briefly discuss applications in dissipativestate preparation and in devising variational sets of states for tensor network methods. 
BibTeX: @article{PhysRevB.90.115110, author = J. C. Budich, J. Eisert, E. J. Bergholtz, S. Diehl and P. Zoller, title = Search for localized Wannier functions of topological band structures via compressed sensing, journal = Phys. Rev. B, year = 2014, volume = 90, number = 11, pages = 115110, url = http://link.aps.org/doi/10.1103/PhysRevB.90.115110, doi = http://dx.doi.org/10.1103/PhysRevB.90.115110 } 
Björn Sbierski, Gregor Pohl, Emil J. Bergholtz and Piet W. Brouwer
Quantum Transport of Disordered Weyl Semimetals at the Nodal Point Phys. Rev. Lett. 113(2), 026602 (2014). 
Abstract: 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 scaleindependent conductance; for stronger disorder, diffusive behavior is reached. We identify the Fano factor as a signature that discriminates between these two regimes. 
BibTeX: @article{PhysRevLett.113.026602, author = Björn Sbierski, Gregor Pohl, Emil J. Bergholtz and Piet W. Brouwer, title = Quantum Transport of Disordered Weyl Semimetals at the Nodal Point, journal = Phys. Rev. Lett., year = 2014, volume = 113, number = 2, pages = 026602, url = http://link.aps.org/doi/10.1103/PhysRevLett.113.026602, doi = http://dx.doi.org/10.1103/PhysRevLett.113.026602 } 
J. C. Budich, J. Eisert and E. J. Bergholtz
Topological insulators with arbitrarily tunable entanglement Phys. Rev. B 89(19), 195120 (2014). 
Abstract: We elucidate how Chern and topological insulators fulfill an area law for the entanglement entropy. By explicit construction of a family of lattice Hamiltonians, we are able to demonstrate that the area law contribution can be tuned to an arbitrarily small value but is topologically protected from vanishing exactly. We prove this by introducing novel methods to bound entanglement entropies from correlations using perturbation bounds, drawing intuition from ideas of quantum information theory. This rigorous approach is complemented by an intuitive understanding in terms of entanglement edge states. These insights have a number of important consequences: The area law has no universal component, no matter how small, and the entanglement scaling cannot be used as a faithful diagnostic of topological insulators. This holds for all Renyi entropies which uniquely determine the entanglement spectrum, which is hence also nonuniversal. The existence of arbitrarily weakly entangled topological insulators furthermore opens up possibilities of devising correlated topological phases in which the entanglement entropy is small and which are thereby numerically tractable, specifically in tensor network approaches. 
BibTeX: @article{PhysRevB.89.195120, author = J. C. Budich, J. Eisert and E. J. Bergholtz, title = Topological insulators with arbitrarily tunable entanglement, journal = Phys. Rev. B, year = 2014, volume = 89, number = 19, pages = 195120, url = http://link.aps.org/doi/10.1103/PhysRevB.89.195120, doi = http://dx.doi.org/10.1103/PhysRevB.89.195120 } 
2013

BibTeX: @unpublished{20132014,, title = { 
Zhao Liu, Emil J. Bergholtz and Eliot Kapit
NonAbelian fractional Chern insulators from longrange interactions Phys. Rev. B 88(20), 205101 (2013). 
Abstract: 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, onsite and nearestneighbor Hubbardlike interactions have been used extensively to stabilize Abelian FCIs in systems with nearly flat, topologically nontrivial bands. However, attempts to use twobody interactions to stabilize nonAbelian 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 longrange interactions in a lattice model that possesses an exactly flat lowest band with a unit Chern number. When spinless bosons with twobody longrange interactions partially fill the lowest Chern band, we find convincing evidence of gapped, bosonic ReadRezayi (RR) phases with nonAbelian anyon statistics. We characterize these states through studying topological degeneracies, the overlap between the ground states of twobody interactions and the exact RR ground states of three and fourbody interactions, and state counting in the particlecut 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 nonAbelian phases. 
BibTeX: @article{PhysRevB.88.205101, author = Zhao Liu, Emil J. Bergholtz and Eliot Kapit, title = NonAbelian fractional Chern insulators from longrange interactions, journal = Phys. Rev. B, year = 2013, volume = 88, number = 20, pages = 205101, url = http://link.aps.org/doi/10.1103/PhysRevB.88.205101, doi = http://dx.doi.org/10.1103/PhysRevB.88.205101 } 
Emil J. Bergholtz and Zhao Liu
Topological Flat Band Models and Fractional Chern Insulators International Journal of Modern Physics B 27(24), 1330017 (2013). 
Abstract: Topological insulators and their intriguing edge states can be understood in a singleparticle 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 nonzero 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 tightbinding models hosting nearly flat bands with nontrivial topology, C≠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. 
BibTeX: @article{emilzhao2013, author = Emil J. Bergholtz and Zhao Liu, title = Topological Flat Band Models and Fractional Chern Insulators, journal = International Journal of Modern Physics B, year = 2013, volume = 27, number = 24, pages = 1330017, url = http://www.worldscientific.com/doi/abs/10.1142/S021797921330017X, doi = http://dx.doi.org/10.1142/S021797921330017X } 
A. M. Läuchli, Zhao Liu, E. J. Bergholtz and R. Moessner
Hierarchy of Fractional Chern Insulators and Competing Compressible States Phys. Rev. Lett. 111(12), 126802 (2013). 
Abstract: We study the phase diagram of interacting electrons in a dispersionless Chern band as a function of their filling. We find hierarchy multiplets of incompressible states at fillings ν=1/3, 2/5, 3/7, 4/9, 5/9, 4/7, 3/5 as well as ν=1/5, 2/7. These are accounted for by an analogy to Haldane pseudopotentials extracted from an analysis of the twoparticle problem. Important distinctions to standard fractional quantum Hall physics are striking: in the absence of particlehole symmetry in a single band, an interactioninduced singlehole dispersion appears, which perturbs and eventually destabilizes incompressible states as ν increases. For this reason, the nature of the state at ν=2/3 is hard to pin down, while ν=5/7, 4/5 do not seem to be incompressible in our system. 
BibTeX: @article{PhysRevLett.111.126802, author = A. M. Läuchli, Zhao Liu, E. J. Bergholtz and R. Moessner, title = Hierarchy of Fractional Chern Insulators and Competing Compressible States, journal = Phys. Rev. Lett., year = 2013, volume = 111, number = 12, pages = 126802, url = http://link.aps.org/doi/10.1103/PhysRevLett.111.126802, doi = http://dx.doi.org/10.1103/PhysRevLett.111.126802 } 
Kevin A. Madsen, Emil J. Bergholtz and Piet W. Brouwer
Topological equivalence of crystal and quasicrystal band structures Phys. Rev. B 88(12), 125118 (2013). 
Abstract: A number of recent articles have reported the existence of topologically nontrivial states and associated end states in onedimensional 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. 
BibTeX: @article{PhysRevB.88.125118, author = Kevin A. Madsen, Emil J. Bergholtz and Piet W. Brouwer, title = Topological equivalence of crystal and quasicrystal band structures, journal = Phys. Rev. B, year = 2013, volume = 88, number = 12, pages = 125118, url = http://link.aps.org/doi/10.1103/PhysRevB.88.125118, doi = http://dx.doi.org/10.1103/PhysRevB.88.125118 } 
Zhao Liu, D. L. Kovrizhin and Emil J. Bergholtz
Bulkedge correspondence in fractional Chern insulators Phys. Rev. B 88(8), 081106 (2013). 
Abstract: 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 gaugefixed WannierQi 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 singlecut orbital entanglement spectrum, hence to the physics of bulkedge correspondence. It is also readily implemented in the stateoftheart density matrix renormalization group method that allows for numerical simulations of significantly larger systems. We demonstrate our general approach on examples of flatband 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 nonAbelian ν=1 phase can be stabilized by purely onsite interactions in the presence of a confining potential. 
BibTeX: @article{PhysRevB.88.081106, author = Zhao Liu, D. L. Kovrizhin and Emil J. Bergholtz, title = Bulkedge correspondence in fractional Chern insulators, journal = Phys. Rev. B, year = 2013, volume = 88, number = 8, pages = 081106, url = http://link.aps.org/doi/10.1103/PhysRevB.88.081106, doi = http://dx.doi.org/10.1103/PhysRevB.88.081106 } 
Zhao Liu and Emil J. Bergholtz
From fractional Chern insulators to Abelian and nonAbelian fractional quantum Hall states: Adiabatic continuity and orbital entanglement spectrum Phys. Rev. B 87(3), 035306 (2013). 
Abstract: The possibility of realizing lattice analogs of fractional quantum Hall (FQH) states, socalled 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 nonAbelian FQH states and FCIs more precise. Using a gaugefixed version of Qi's Wannier basis representation of a Chern band, we demonstrate that the interpolation between several FCI states, obtained by shortrange lattice interactions in a spinorbitcoupled 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 lowlying 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 bulkboundary 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 8electron Laughlin state at ν=1/3 (consistent with an earlier study) and 97.8\% for the 10electron MooreRead 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 Landaulevel 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. 
BibTeX: @article{Liu2013, author = Zhao Liu and Emil J. Bergholtz, title = From fractional Chern insulators to Abelian and nonAbelian fractional quantum Hall states: Adiabatic continuity and orbital entanglement spectrum, journal = Phys. Rev. B, year = 2013, volume = 87, number = 3, pages = 035306, url = http://link.aps.org/doi/10.1103/PhysRevB.87.035306, doi = http://dx.doi.org/10.1103/PhysRevB.87.035306 } 
2012

BibTeX: @unpublished{20122013,, title = { 
Maximilian Trescher and Emil J. Bergholtz
Flat bands with higher Chern number in pyrochlore slabs Phys. Rev. B 86(24), 241111 (2012). 
Abstract: A large number of recent works point to the emergence of intriguing analogs of fractional quantum Hall states in lattice models due to effective interactions in nearly flat bands with Chern number C=1. Here, we provide an intuitive and efficient construction of almost dispersionless bands with higher Chern numbers. Inspired by the physics of quantum Hall multilayers and pyrochlorebased transitionmetal oxides, we study a tightbinding model describing spinorbit coupled electrons in N parallel kagome layers connected by apical sites forming Nâ1 intermediate triangular layers (as in the pyrochlore lattice). For each N, we find finite regions in parameter space giving a virtually flat band with C=N. We analytically express the states within these topological bands in terms of singlelayer states and thereby explicitly demonstrate that the C=N wave functions have an appealing structure in which layer index and translations in reciprocal space are intricately coupled. This provides a promising arena for new collective states of matter. 
BibTeX: @article{Trescher2012, author = Maximilian Trescher and Emil J. Bergholtz, title = Flat bands with higher Chern number in pyrochlore slabs, journal = Phys. Rev. B, year = 2012, volume = 86, number = 24, pages = 241111, url = http://link.aps.org/doi/10.1103/PhysRevB.86.241111, doi = http://dx.doi.org/10.1103/PhysRevB.86.241111 } 
Zhao Liu, Emil J. Bergholtz, Heng Fan and Andreas M. Läuchli
Fractional Chern Insulators in Topological Flat Bands with Higher Chern Number Phys. Rev. Lett. 109(18), 186805 (2012). 
Abstract: Lattice models forming bands with higher Chern number offer an intriguing possibility for new phases of matter with no analogue in continuum Landau levels. Here, we establish the existence of a number of new bulk insulating states at fractional filling in flat bands with a Chern number C=N>1, forming in a recently proposed pyrochlore model with strong spinorbit coupling. In particular, we find compelling evidence for a series of stable states at ν=1/(2N+1) for fermions as well as bosonic states at ν=1/(N+1). By examining the topological ground state degeneracies and the excitation structure as well as the entanglement spectrum, we conclude that these states are Abelian. We also explicitly demonstrate that these states are nevertheless qualitatively different from conventional quantum Hall (multilayer) states due to the novel properties of the underlying band structure. 
BibTeX: @article{Liu2012, author = Zhao Liu, Emil J. Bergholtz, Heng Fan and Andreas M. Läuchli, title = Fractional Chern Insulators in Topological Flat Bands with Higher Chern Number, journal = Phys. Rev. Lett., year = 2012, volume = 109, number = 18, pages = 186805, url = http://link.aps.org/doi/10.1103/PhysRevLett.109.186805, doi = http://dx.doi.org/10.1103/PhysRevLett.109.186805 } 
Masaaki Nakamura, ZhengYuan Wang and Emil J. Bergholtz
Exactly Solvable Fermion Chain Describing a $$\nu${}=1/3$ Fractional Quantum Hall State Phys. Rev. Lett. 109(1), 016401 (2012). 
Abstract: We introduce an exactly solvable fermion chain that describes a ν=1/3 fractional quantum Hall (FQH) state beyond the thintorus limit. The ground state of our model is shown to be unique for each centerofmass sector, and it has a matrix product representation that enables us to exactly calculate order parameters, correlation functions, and entanglement spectra. The ground state of our model shows striking similarities with the BCS wave functions and quantum spin1 chains. Using the variational method with matrix product ansatz, we analytically calculate excitation gaps and vanishing of the compressibility expected in the FQH state. We also show that the above results can be related to a ν=1/2 bosonic FQH state. 
BibTeX: @article{Nakamura2012, author = Masaaki Nakamura, ZhengYuan Wang and Emil J. Bergholtz, title = Exactly Solvable Fermion Chain Describing a $$\nu${}=1/3$ Fractional Quantum Hall State, journal = Phys. Rev. Lett., year = 2012, volume = 109, number = 1, pages = 016401, url = http://link.aps.org/doi/10.1103/PhysRevLett.109.016401, doi = http://dx.doi.org/10.1103/PhysRevLett.109.016401 } 
Emma Wikberg, Jonas Larson, Emil J. Bergholtz and Anders Karlhede
Fractional domain walls from onsite softening in dipolar bosons Phys. Rev. A 85(3), 033607 (2012). 
Abstract: We study dipolar bosons in a 1D optical lattice and identify a region in parameter space—strong coupling but relatively weak onsite repulsion—hosting a series of stable chargedensitywave (CDW) states whose lowenergy excitations, built from “fractional domain walls,” have remarkable similarities to those of nonAbelian fractional quantum Hall states. Here, a conventional domain wall between translated CDW's may be split by inserting strings of degenerate, but inequivalent, CDW states. Outside these insulating regions, we find numerous supersolids as well as a superfluid regime. The mentioned phases should be accessible experimentally and, in particular, the fractional domain walls can be created in the ground state using singlesite addressing, i.e., by locally changing the chemical potential. 
BibTeX: @article{Wikberg2012, author = Emma Wikberg, Jonas Larson, Emil J. Bergholtz and Anders Karlhede, title = Fractional domain walls from onsite softening in dipolar bosons, journal = Phys. Rev. A, year = 2012, volume = 85, number = 3, pages = 033607, url = http://link.aps.org/doi/10.1103/PhysRevA.85.033607, doi = http://dx.doi.org/10.1103/PhysRevA.85.033607 } 
Zhao Liu, Emil J. Bergholtz, Heng Fan and Andreas M. Läuchli
Edgemode combinations in the entanglement spectra of nonAbelian fractional quantum Hall states on the torus Phys. Rev. B 85(4), 045119 (2012). 
Abstract: We present a detailed analysis of bipartite entanglement in the nonAbelian MooreRead fractional quantum Hall state of bosons and fermions on the torus. In particular, we show that the entanglement spectra can be decomposed into intricate combinations of different sectors of the conformal field theory describing the edge physics, and that the edge level counting and tower structure can be microscopically understood by considering the vicinity of the thin torus limit. We also find that the boundary entropy density of the MooreRead state is markedly higher than in the Laughlin states investigated so far. Despite the torus geometry being somewhat more involved than in the sphere geometry, our analysis and insights may prove useful when adopting entanglement probes to other systems that are more easily studied with periodic boundary conditions, such as fractional Chern insulators and lattice problems in general. 
BibTeX: @article{PhysRevB.85.045119, author = Zhao Liu, Emil J. Bergholtz, Heng Fan and Andreas M. Läuchli, title = Edgemode combinations in the entanglement spectra of nonAbelian fractional quantum Hall states on the torus, journal = Phys. Rev. B, year = 2012, volume = 85, number = 4, pages = 045119, url = http://link.aps.org/doi/10.1103/PhysRevB.85.045119, doi = http://dx.doi.org/10.1103/PhysRevB.85.045119 } 
2011

BibTeX: @unpublished{20112012,, title = { 
Emil J. Bergholtz, Masaaki Nakamura and Juha Suorsa
Effective spin chains for fractional quantum Hall states Physica E: Lowdimensional Systems and Nanostructures 43(3), 755  760 (2011). 
Abstract: Fractional quantum Hall (FQH) states are topologically ordered which indicates that their essential properties are insensitive to smooth deformations of the manifold on which they are studied. Their microscopic Hamiltonian description, however, strongly depends on geometrical details. Recent work has shown how this dependence can be exploited to generate effective models that are both interesting in their own right and also provide further insight into the quantum Hall system. We review and expand on recent efforts to understand the \{FQH\} system close to the solvable thintorus limit in terms of effective spin chains. In particular, we clarify how the difference between the bosonic and fermionic \{FQH\} states, which is not apparent in the thintorus limit, can be seen at this level. Additionally, we discuss the relation of the Haldane–Shastry chain to the socalled \{QH\} circle limit and comment on its significance to recent entanglement studies. 
BibTeX: @article{Bergholtz2011755, author = Emil J. Bergholtz, Masaaki Nakamura and Juha Suorsa, title = Effective spin chains for fractional quantum Hall states, journal = Physica E: Lowdimensional Systems and Nanostructures, year = 2011, volume = 43, number = 3, pages = 755  760, url = http://www.sciencedirect.com/science/article/pii/S1386947710004340, doi = http://dx.doi.org/http://dx.doi.org/10.1016/j.physe.2010.07.044 } 
Masaaki Nakamura, ZhengYuan Wang and Emil J. Bergholtz
Beyond the TaoThouless limit of the fractional quantum Hall effect: spin chains and Fermi surface deformation Journal of Physics: Conference Series 302(1), 012020 (2011). 
Abstract: We discuss the relationship between the fractional quantum Hall effect in the vicinity of the thintorus, a.k.a. TaoThouless (TT), limit and quantum spin chains. We argue that the energetics of fractional quantum Hall states in Jain sequence at filling fraction ν = p /(2 p + 1) (and ν = 1 − p /(2 p + 1)) in the lowest Landau level is captured by S = 1 spin chains with p spins in the unit cell. These spin chains naturally arise at subleading order in e −2π 2 / L 2 1 which serves as an expansion parameter away from the TT limit ( L 1 → 0). We also corroborate earlier results on the smooth Fermi surface deformation of the gapless state at ν = 1/2, interpolating between a state described by a critical S = 1/2 chain and the bulk. 
BibTeX: @article{174265963021012020, author = Masaaki Nakamura, ZhengYuan Wang and Emil J. Bergholtz, title = Beyond the TaoThouless limit of the fractional quantum Hall effect: spin chains and Fermi surface deformation, journal = Journal of Physics: Conference Series, year = 2011, volume = 302, number = 1, pages = 012020, url = http://stacks.iop.org/17426596/302/i=1/a=012020, doi = http://dx.doi.org/10.1088/17426596/302/1/012020 } 
2010

BibTeX: @unpublished{20102011,, title = { 
E. J. Bergholtz, A. M. Läuchli and R. Moessner
Symmetry Breaking on the ThreeDimensional Hyperkagome Lattice of ${\mathrm{Na}}_{4}{\mathrm{Ir}}_{3}{\mathrm{O}}_{8}$ Phys. Rev. Lett. 105(23), 237202 (2010). 
Abstract: We study the antiferromagnetic spin1/2 Heisenberg model on the highly frustrated, threedimensional, hyperkagome lattice of Na4Ir3O8 using a series expansion method. We propose a valence bond crystal with a 72 site unit cell as a ground state that supports many, very low lying, singlet excitations. Low energy spinons and triplons are confined to emergent lowerdimensional motifs. Here, and for analogous kagome and pyrochlore states, we suggest finite temperature signatures, including an Ising transition, in the magnetic specific heat due to a multistep breaking of discrete symmetries. 
BibTeX: @article{PhysRevLett.105.237202, author = E. J. Bergholtz, A. M. Läuchli and R. Moessner, title = Symmetry Breaking on the ThreeDimensional Hyperkagome Lattice of ${\mathrm{Na}}_{4}{\mathrm{Ir}}_{3}{\mathrm{O}}_{8}$, journal = Phys. Rev. Lett., year = 2010, volume = 105, number = 23, pages = 237202, url = http://link.aps.org/doi/10.1103/PhysRevLett.105.237202, doi = http://dx.doi.org/10.1103/PhysRevLett.105.237202 } 
Andreas M. Läuchli, Emil J. Bergholtz and Masudul Haque
Entanglement scaling of fractional quantum Hall states through geometric deformations New Journal of Physics 12(7), 075004 (2010). 
Abstract: We present a new approach for obtaining the scaling behavior of the entanglement entropy in fractional quantum Hall (FQH) states from finitesize wavefunctions. By employing the torus geometry and the fact that the torus aspect ratio can be readily varied, we can extract the entanglement entropy of a spatial block as a continuous function of the block boundary length. This approach allows us to extract the topological entanglement entropy with an accuracy superior to that possible for the spherical or disc geometry, where no natural continuously variable parameter is available. Other than the topological information, the study of entanglement scaling is also useful as an indicator of the difficulty posed by FQH states for various numerical techniques. 
BibTeX: @article{13672630127075004, author = Andreas M. Läuchli, Emil J. Bergholtz and Masudul Haque, title = Entanglement scaling of fractional quantum Hall states through geometric deformations, journal = New Journal of Physics, year = 2010, volume = 12, number = 7, pages = 075004, url = http://stacks.iop.org/13672630/12/i=7/a=075004, doi = http://dx.doi.org/10.1088/13672630/12/7/075004 } 
Andreas M. Läuchli, Emil J. Bergholtz, Juha Suorsa and Masudul Haque
Disentangling Entanglement Spectra of Fractional Quantum Hall States on Torus Geometries Phys. Rev. Lett. 104(15), 156404 (2010). 
Abstract: We analyze the entanglement spectrum of Laughlin states on the torus and show that it is arranged in towers, each of which is generated by modes of two spatially separated chiral edges. This structure is present for all torus circumferences, which allows for a microscopic identification of the prominent features of the spectrum by perturbing around the thintorus limit. 
BibTeX: @article{PhysRevLett.104.156404, author = Andreas M. Läuchli, Emil J. Bergholtz, Juha Suorsa and Masudul Haque, title = Disentangling Entanglement Spectra of Fractional Quantum Hall States on Torus Geometries, journal = Phys. Rev. Lett., year = 2010, volume = 104, number = 15, pages = 156404, url = http://link.aps.org/doi/10.1103/PhysRevLett.104.156404, doi = http://dx.doi.org/10.1103/PhysRevLett.104.156404 } 
Masaaki Nakamura, Emil J. Bergholtz and Juha Suorsa
Link between the hierarchy of fractional quantum Hall states and Haldane's conjecture for quantum spin chains Phys. Rev. B 81(16), 165102 (2010). 
Abstract: We study a strong coupling expansion of the ν=1/3 fractional quantum Hall state away from the TaoThouless limit and show that the leading quantum fluctuations lead to an effective spin1 Hamiltonian that lacks parity symmetry. By analyzing the energetics, discrete symmetries of lowlying excitations, and string order parameters, we demonstrate that the ν=1/3 fractional quantum Hall state is adiabatically connected to both Haldane and largeD phases and is characterized by a string order parameter which is dual to the ordinary one. This result indicates a close relation between (a generalized form of) the Haldane conjecture for spin chains and the fractional quantum Hall effect. 
BibTeX: @article{PhysRevB.81.165102, author = Masaaki Nakamura, Emil J. Bergholtz and Juha Suorsa, title = Link between the hierarchy of fractional quantum Hall states and Haldane's conjecture for quantum spin chains, journal = Phys. Rev. B, year = 2010, volume = 81, number = 16, pages = 165102, url = http://link.aps.org/doi/10.1103/PhysRevB.81.165102, doi = http://dx.doi.org/10.1103/PhysRevB.81.165102 } 
2009

BibTeX: @unpublished{20092010,, title = { 
Emma Wikberg, Emil J. Bergholtz and Anders Karlhede
Spin chain description of rotating bosons at ν = 1 Journal of Statistical Mechanics: Theory and Experiment 2009(07), 19 (2009). 
Abstract: We consider bosons at Landau level filling ν = 1 on a thin torus. In analogy with previous work on fermions at filling ν = 1/2, we map the lowenergy sector onto a spin1/2 chain. While the fermionic system may realize the gapless XY phase, we show that typically this does not happen for the bosonic system. Instead, both delta function and Coulomb interaction lead to gapped phases in the bosonic system, and in particular we identify a phase corresponding to the nonAbelian Moore–Read state. In the spin language, the Hamiltonian is dominated by a ferromagnetic nextnearestneighbor interaction, which leads to a description consistent with the nontrivial degeneracies of the ground and excited states of this phase of matter. In addition we comment on the similarities and differences of the two systems mentioned above and fermions at ν = 5/2. 
BibTeX: @article{emilP07038, author = Emma Wikberg, Emil J. Bergholtz and Anders Karlhede, title = Spin chain description of rotating bosons at ν = 1, journal = Journal of Statistical Mechanics: Theory and Experiment, year = 2009, volume = 2009, number = 07, pages = 19, url = http://iopscience.iop.org/17425468/2009/07/P07038/fulltext/, doi = http://dx.doi.org/10.1088/17425468/2009/07/P07038 } 
E. J. Bergholtz and A. Karlhede
Quantum Hall Circle Journal of Statistical Mechanics: Theory and Experiment 2009(04), P04015 (2009). 
Abstract: We consider spinpolarized electrons in a single Landau level on a cylinder as the circumference of the cylinder goes to infinity. This gives a model of interacting electrons on a circle where the momenta of the particles are restricted and there is no kinetic energy. Quantum Hall states are exact ground states for appropriate short range interactions, and there is a gap to excitations. These states develop adiabatically from this onedimensional quantum Hall circle to the bulk quantum Hall states and further on into the Tao–Thouless states as the circumference goes to zero. For low filling fractions a gapless state is formed which we suggest is connected to the Wigner crystal expected in the bulk. 
BibTeX: @article{17425468200904P04015, author = E. J. Bergholtz and A. Karlhede, title = Quantum Hall Circle, journal = Journal of Statistical Mechanics: Theory and Experiment, year = 2009, volume = 2009, number = 04, pages = P04015, url = http://stacks.iop.org/17425468/2009/i=04/a=P04015, doi = http://dx.doi.org/10.1088/17425468/2009/04/P04015 } 
2008

BibTeX: @unpublished{20082009,, title = { 
Eddy Ardonne, Emil J. Bergholtz, Janik Kailasvuori and Emma Wikberg
Degeneracy of nonAbelian quantum Hall states on the torus: domain walls and conformal field theory Journal of Statistical Mechanics: Theory and Experiment 2008(04), P04016 (2008). 
Abstract: We analyze the nonAbelian Read–Rezayi quantum Hall states on the torus, where it is natural to employ a mapping of the manybody problem onto a onedimensional lattice model. On the thin torus—the Tao–Thouless (TT) limit—the interacting manybody problem is exactly solvable. The Read–Rezayi states at filling ν = k /( kM +2) are known to be exact ground states of a local repulsive k +1body interaction, and in the TT limit this is manifested in that all states in the ground state manifold have exactly k particles on any kM +2 consecutive sites. For #"IMG"# [http://ej.iop.org/images/17425468/2008/04/P04016/jstat275930ieqn1.gif] {M\neq 0} the twobody correlations of these states also imply that there is no more than one particle on M adjacent sites. The fractionally charged quasiparticles and quasiholes appear as domain walls between the ground states, and we show that the number of distinct domain wall patterns gives rise to the nontrivial degeneracies, required by the nonAbelian statistics of these states. In the second part of the paper we consider the quasihole degeneracies from a conformal field theory (CFT) perspective, and show that the counting of the domain wall patterns maps one to one on the CFT counting via the fusion rules. Moreover we extend the CFT analysis to topologies of higher genus. 
BibTeX: @article{17425468200804P04016, author = Eddy Ardonne, Emil J. Bergholtz, Janik Kailasvuori and Emma Wikberg, title = Degeneracy of nonAbelian quantum Hall states on the torus: domain walls and conformal field theory, journal = Journal of Statistical Mechanics: Theory and Experiment, year = 2008, volume = 2008, number = 04, pages = P04016, url = http://stacks.iop.org/17425468/2008/i=04/a=P04016, doi = http://dx.doi.org/10.1088/17425468/2008/04/P04016 } 
E. J. Bergholtz, T. H. Hansson, M. Hermanns, A. Karlhede and S. Viefers
Quantum Hall hierarchy wave functions: From conformal correlators to TaoThouless states Phys. Rev. B 77(16), 165325 (2008). 
Abstract: Laughlin’s wave functions, which describe the fractional quantum Hall effect at filling factors ν=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. 
BibTeX: @article{PhysRevB.77.165325, author = E. J. Bergholtz, T. H. Hansson, M. Hermanns, A. Karlhede and S. Viefers, title = Quantum Hall hierarchy wave functions: From conformal correlators to TaoThouless states, journal = Phys. Rev. B, year = 2008, volume = 77, number = 16, pages = 165325, url = http://link.aps.org/doi/10.1103/PhysRevB.77.165325, doi = http://dx.doi.org/10.1103/PhysRevB.77.165325 } 
E. J. Bergholtz and A. Karlhede
Quantum Hall system in TaoThouless limit Phys. Rev. B 77(15), 155308 (2008). 
Abstract: We consider spinpolarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a onedimensional lattice model with lattice constant 2π/L1, where L1 is a circumference of the torus (in units of the magnetic length). In the TaoThouless limit L1→0, the interacting manyelectron problem is exactly diagonalized at any rational filling factor ν=p/q≤1. For odd q, the ground state has the same qualitative properties as a bulk (L1→∞) quantum Hall hierarchy state and the lowestenergy quasiparticle excitations have the same fractional charges as in the bulk. These states are the L1→0 limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd q, are continuously connected to the twodimensional bulk quantum Hall hierarchy states—i.e., that there is no phase transition as L1→∞ for filling factors where such states can be observed. For evendenominator fractions, a phase transition occurs as L1 increases. For ν=1/2 this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite L1; this then develops continuously into the composite fermion wave function that is believed to describe the bulk ν=1/2 system. The analysis generalizes to nonAbelian quantum Hall states. 
BibTeX: @article{PhysRevB.77.155308, author = E. J. Bergholtz and A. Karlhede, title = Quantum Hall system in TaoThouless limit, journal = Phys. Rev. B, year = 2008, volume = 77, number = 15, pages = 155308, url = http://link.aps.org/doi/10.1103/PhysRevB.77.155308, doi = http://dx.doi.org/10.1103/PhysRevB.77.155308 } 
M. Hermanns, J. Suorsa, E. J. Bergholtz, T. H. Hansson and A. Karlhede
Quantum Hall wave functions on the torus Phys. Rev. B 77(12), 125321 (2008). 
Abstract: We 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. 
BibTeX: @article{PhysRevB.77.125321, author = M. Hermanns, J. Suorsa, E. J. Bergholtz, T. H. Hansson and A. Karlhede, title = Quantum Hall wave functions on the torus, journal = Phys. Rev. B, year = 2008, volume = 77, number = 12, pages = 125321, url = http://link.aps.org/doi/10.1103/PhysRevB.77.125321, doi = http://dx.doi.org/10.1103/PhysRevB.77.125321 } 
2007

BibTeX: @unpublished{20072008,, title = { 
E. J. Bergholtz, T. H. Hansson, M. Hermanns and A. Karlhede
Microscopic Theory of the Quantum Hall Hierarchy Phys. Rev. Lett. 99(25), 256803 (2007). 
Abstract: We solve the quantum Hall problem exactly in a limit and show that the ground states can be organized in a fractal pattern consistent with the HaldaneHalperin hierarchy, and with the global phase diagram. We present wave functions for a large family of states, including those of Laughlin and Jain and also for states recently observed by Pan et al., and show that they coincide with the exact ones in the solvable limit. We submit that they establish an adiabatic continuation of our exact results to the experimentally accessible regime, thus providing a unified approach to the hierarchy states. 
BibTeX: @article{PhysRevLett.99.256803, author = E. J. Bergholtz, T. H. Hansson, M. Hermanns and A. Karlhede, title = Microscopic Theory of the Quantum Hall Hierarchy, journal = Phys. Rev. Lett., year = 2007, volume = 99, number = 25, pages = 256803, url = http://link.aps.org/doi/10.1103/PhysRevLett.99.256803, doi = http://dx.doi.org/10.1103/PhysRevLett.99.256803 } 
2006

BibTeX: @unpublished{20062007,, title = { 
E. J. Bergholtz, J. Kailasvuori, E. Wikberg, T. H. Hansson and A. Karlhede
Pfaffian quantum Hall state made simple: Multiple vacua and domain walls on a thin torus Phys. Rev. B 74(8), 081308 (2006). 
Abstract: We analyze the MooreRead Pfaffian state on a thin torus. The known sixfold degeneracy is realized by two inequivalent crystalline states with a four and twofold degeneracy, respectively. The fundamental quasihole and quasiparticle excitations are domain walls between these vacua, and simple counting arguments give a Hilbert space of dimension 2n−1 for 2n−k holes and k particles at fixed positions and assign each a charge ±e∕4. This generalizes the known properties of the hole excitations in the Pfaffian state as deduced using conformal field theory techniques. Numerical calculations using a model Hamiltonian and a small number of particles support the presence of a stable phase with degenerate vacua and quartercharged domain walls also away from the thintorus limit. A spinchain Hamiltonian encodes the degenerate vacua and the various domain walls. 
BibTeX: @article{PhysRevB.74.081308, author = E. J. Bergholtz, J. Kailasvuori, E. Wikberg, T. H. Hansson and A. Karlhede, title = Pfaffian quantum Hall state made simple: Multiple vacua and domain walls on a thin torus, journal = Phys. Rev. B, year = 2006, volume = 74, number = 8, pages = 081308, url = http://link.aps.org/doi/10.1103/PhysRevB.74.081308, doi = http://dx.doi.org/10.1103/PhysRevB.74.081308 } 
Emil J. Bergholtz and Anders Karlhede
`Onedimensional' theory of the quantum Hall system Journal of Statistical Mechanics: Theory and Experiment 2006(04), 4 (2006). 
Abstract: We consider the lowest Landau level on a torus as a function of its circumference L1. When L1 → 0, the ground state at general rational filling fraction is a crystal with a gap  a TaoThouless state. For filling fractions ν = p/(2pm + 1), these states are the limits of Laughlin's or Jain's wavefunctions describing the gapped quantum Hall states when L1 → ∞. For the halffilled Landau level, there is a transition to a Fermi sea of noninteracting neutral fermions (dipoles), or rather to a Luttinger liquid modification thereof, at L1 ∼ 5 magnetic lengths. Using exact diagonalization we identify this state as a version of the RezayiRead state, and find that it develops continuously into the state that is believed to describe the observed metallic phase as L1 → ∞. Furthermore, the effective Landau level structure that emerges within the lowest Landau level is found to be a consequence of the magnetic symmetries. 
BibTeX: @article{emilL04001, author = Emil J. Bergholtz and Anders Karlhede, title = `Onedimensional' theory of the quantum Hall system, journal = Journal of Statistical Mechanics: Theory and Experiment, year = 2006, volume = 2006, number = 04, pages = 4, url = http://iopscience.iop.org/17425468/2006/04/L04001/fulltext/, doi = http://dx.doi.org/10.1088/17425468/2006/04/L04001 } 
2005

BibTeX: @unpublished{20052006,, title = { 
Emil J. Bergholtz and Anders Karlhede
HalfFilled Lowest Landau Level on a Thin Torus Phys. Rev. Lett. 94(2), 026802 (2005). 
Abstract: We solve a model that describes an interacting electron gas in the halffilled lowest Landau level on a thin torus, with radius of the order of the magnetic length. The lowenergy sector consists of noninteracting, onedimensional, neutral fermions. The ground state, which is homogeneous, is the Fermi sea obtained by filling the negative energy states, and the excited states are gapless neutral excitations out of this onedimensional sea. Although the limit considered is extreme, the solution has a striking resemblance to the composite fermion description of the bulk ν=1/2 state—the ground state is homogeneous and the excitations are neutral and gapless. This suggests a onedimensional Luttinger liquid description, with possible observable effects in transport experiments, of the bulk state where it develops continuously from the state on a thin torus as the radius increases. 
BibTeX: @article{PhysRevLett.94.026802, author = Emil J. Bergholtz and Anders Karlhede, title = HalfFilled Lowest Landau Level on a Thin Torus, journal = Phys. Rev. Lett., year = 2005, volume = 94, number = 2, pages = 026802, url = http://link.aps.org/doi/10.1103/PhysRevLett.94.026802, doi = http://dx.doi.org/10.1103/PhysRevLett.94.026802 } 