Hofstadter Model Research Articles

Critical magnetic flux for Weyl points in the three-dimensional Hofstadter model

Critical magnetic flux for Weyl points in the three-dimensional Hofstadter model

Hofstadter-Toda spectral duality and quantum groups

The Hofstadter model allows to describe and understand several phenomena in condensed matter such as the quantum Hall effect, Anderson localization, charge pumping, and flat-bands in quasiperiodic structures, and is a rare example of fractality in the quantum world. An apparently unrelated system, the relativistic Toda lattice, has been extensively studied in the context of complex nonlinear dynamics, and more recently for its connection to supersymmetric Yang-Mills theories and topological string theories on Calabi-Yau manifolds in high-energy physics. Here we discuss a recently discovered spectral relationship between the Hofstadter model and the relativistic Toda lattice which has been later conjectured to be related to the Langlands duality of quantum groups. Moreover, by employing similarity transformations compatible with the quantum group structure, we establish a formula parametrizing the energy spectrum of the Hofstadter model in terms of elementary symmetric polynomials and Chebyshev polynomials. The main tools used are the spectral duality of tridiagonal matrices and the representation theory of the elementary quantum group.

Revealing the spatial nature of sublattice symmetry

The sublattice symmetry on a bipartite lattice is commonly regarded as the chiral symmetry in the AIII class of the tenfold Altland–Zirnbauer classification. Here, we reveal the spatial nature of sublattice symmetry and show that this assertion holds only if the periodicity of primitive unit cells agrees with that of the sublattice labeling. In cases where the periodicity does not agree, sublattice symmetry is represented as a glide reflection in energy–momentum space, which inverts energy and simultaneously translates some k by π, leading to substantially different physics. Particularly, it introduces novel constraints on zero modes in semimetals and completely alters the classification table of topological insulators compared to class AIII. Notably, the dimensions corresponding to trivial and nontrivial classifications are switched, and the nontrivial classification becomes Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{Z}}}_{2}$$\\end{document} instead of Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb{Z}}$$\\end{document}. We have applied these results to several models, including the Hofstadter model both with and without dimerization.

Dimerized Hofstadter model in two-leg ladder quasi-crystals

We theoretically study topological features, band structure, and localization properties of a dimerized two-leg ladder with an oscillating on-site potential. The periodicity of the on-site potential can take either rational or irrational values. We consider two types of dimerized configurations; symmetric and asymmetric models. For rational values of the periodicity as long as inversion symmetry is preserved both symmetric and asymmetric ladders can host topological phases. Additionally, the energy spectrum of the models exhibits a fractal structure known as the Hofstadter butterfly spectrum, dependent on the dimerization of the hopping and the strength of the on-site potential. In the case of irrational values for the periodicity, a metal-insulator phase transition occurs with small values of the critical strength of the on-site potential in the dimerized cases. Our models incorporate the effects of lattice configuration and quasi-periodicity, paving the way for establishing platforms that host both topological and non-topological phase transitions.

Injection spectroscopy of momentum state lattices

The energy spectrum of quantum systems contain a wealth of information about their underlying properties. Spectroscopic techniques, especially those with access to spatially resolved measurements, can be challenging to implement in real-space systems of cold atoms in optical lattices. Here we explore a technique for probing energy spectra in synthetic lattices that is analogous to scanning tunneling microscopy. Using one-dimensional synthetic lattices of coupled atomic momentum states, we explore this spectroscopic technique and observe qualitative agreement between the measured and simulated energy spectra for small two- and three-site lattices as well as a uniform many-site lattice. Finally, through simulations, we show that this technique should allow for the exploration of the topological bands and the fractal energy spectrum of the Hofstadter model as realized in synthetic lattices.

Unusual magnetotransport in twisted bilayer graphene from strain-induced open Fermi surfaces

Anisotropic hopping in a toy Hofstadter model was recently invoked to explain a rich and surprising Landau spectrum measured in twisted bilayer graphene away from the magic angle. Suspecting that such anisotropy could arise from unintended uniaxial strain, we extend the Bistritzer-MacDonald model to include uniaxial heterostrain and present a detailed analysis of its impact on band structure and magnetotransport. We find that such strain strongly influences band structure, shifting the three otherwise-degenerate van Hove points to different energies. Coupled to a Boltzmann magnetotransport calculation, this reproduces previously unexplained nonsaturating [Formula: see text] magnetoresistance over broad ranges of density near filling [Formula: see text] and predicts subtler features that had not been noticed in the experimental data. In contrast to these distinctive signatures in longitudinal resistivity, the Hall coefficient is barely influenced by strain, to the extent that it still shows a single sign change on each side of the charge neutrality point-surprisingly, this sign change no longer occurs at a van Hove point. The theory also predicts a marked rotation of the electrical transport principal axes as a function of filling even for fixed strain and for rigid bands. More careful examination of interaction-induced nematic order versus strain effects in twisted bilayer graphene could thus be in order.

Thermal Hall effect and the Wiedemann–Franz law in Chern insulator

Thermal Hall effect, where a transverse temperature difference is generated by implementing a longitudinal temperature gradient and an external magnetic field in the perpendicular direction to systems, is a useful tool to reveal transport properties of quantum materials. A systematic study of the thermal Hall effect in a Chern insulator is still lacking. Here, using the Landauer–Büttiker formula, we investigated the thermal Hall transport of the Harper–Hofstadter model with flux φ = 1/2 and its generalizations. We demonstrated that the Wiedemann–Franz law, which states that the thermal Hall conductivity is linearly proportional to the quantum Hall conductivity in the low temperature limit, is still valid in this Chern insulator, and that the thermal Hall conductivity can be used to characterize the topological properties of quantum materials.

Dirac fermion spectrum of the fractional quantum Hall states

Applying a unified approach, we study the integer quantum Hall effect (IQHE) and fractional quantum Hall effect (FQHE) in the Hofstadter model with short range interactions between fermions. An effective field, that takes into account the interaction between fermions, is determined by both amplitude and phase. Its amplitude is proportional to the interaction strength, the phase corresponds to the minimum energy. In fact, the problem is reduced to the Harper equation with two different scales: the first is a magnetic scale with the cell size corresponding to a unit quantum magnetic flux, the second scale determines the inhomogeneity of the effective field, forms the steady fine structure of the Hofstadter spectrum and leads to the realization of fractional quantum Hall states. In a sample of finite size with open boundary conditions, the fine structure of the Hofstadter spectrum consists of the Dirac branches of the fermion excitations and includes the fine structure of the edge chiral modes. The Chern numbers of the topological Hofstadter bands are conserved during the formation of their fine structure. The edge modes are formed into the Hofstadter bands. They connect the nearest-neighbor subbands and determine the conductance for the fractional filling.

Fractional Disclination Charge and Discrete Shift in the Hofstadter Butterfly.

In the presence of crystalline symmetries, topological phases of matter acquire a host of invariants leading to nontrivial quantized responses. Here we study a particular invariant, the discrete shift 𝒮, for the square lattice Hofstadter model of free fermions. 𝒮 is associated with a Z_{M} classification in the presence of M-fold rotational symmetry and charge conservation. 𝒮 gives quantized contributions to (i)the fractional charge bound to a lattice disclination and (ii)the angular momentum of the ground state with an additional, symmetrically inserted magnetic flux. 𝒮 forms its own "Hofstadter butterfly," which we numerically compute, refining the usual phase diagram of the Hofstadter model. We propose an empirical formula for 𝒮 in terms of density and flux per plaquette for the Hofstadter bands, and we derive a number of general constraints. We show that bands with the same Chern number may have different values of 𝒮, although odd and even Chern number bands always have half-integer and integer values of 𝒮, respectively.

Finding spectral gaps in quasicrystals

We present an algorithm for reliably and systematically proving the existence of spectral gaps in Hamiltonians with quasicrystalline order, based on numerical calculations on finite domains. We apply this algorithm to prove that the Hofstadter model on the Ammann-Beenker tiling of the plane has spectral gaps at certain energies, and we are able to prove the existence of a spectral gap where previous numerical results were inconclusive. Our algorithm is applicable to more general systems with finite local complexity and eventually finds all gaps, circumventing an earlier no-go theorem regarding the computability of spectral gaps for general Hamiltonians.

Non-Abelian nonsymmorphic chiral symmetries

The Hofstadter model exemplifies a large class of physical systems characterized by particles hopping on a lattice immersed in a gauge field. Recent advancements on various synthetic platforms have enabled highly controllable simulations of such systems with tailored gauge fields featuring complex spatial textures. These synthetic gauge fields could introduce synthetic symmetries that do not appear in electronic materials. Here, in an $\mathrm{SU}(2)$ non-Abelian Hofstadter model, we theoretically show the emergence of multiple nonsymmorphic chiral symmetries, which combine an internal unitary antisymmetry with fractional spatial translation. Depending on the values of the gauge fields, the nonsymmorphic chiral symmetries can exhibit non-Abelian algebra and protect Kramers quartet states in the bulk band structure, creating general fourfold degeneracy at all momenta. These nonsymmorphic chiral symmetries protect double Dirac semimetals at zero energy, which become gapped into quantum confined insulating phases upon introducing a boundary. Moreover, the parity of the system size can determine whether the resulting insulating phase is trivial or topological. Our work indicates a pathway for creating topology via synthetic symmetries emergent from synthetic gauge fields.

Robust nonequilibrium surface currents in the three-dimensional Hofstadter model

Genuinely two-dimensional robust crosscurrents -- which flow against the natural direction of heat flux -- have been missing since the discovery of their one-dimensional counterpart. We provide a setup to realize them on a cubic three-dimensional (3D) lattice hosting a Hofstadter model coupled to two heat baths with different temperatures. We show that these currents exhibit dissipative robustness: they are stable against the presence of impurities and tilting of the gauge field in certain nonequilibrium configurations. Moreover, we find protected boundary currents with genuinely 3D robustness, i.e. they are only stable if tunnelling can occur in all three spatial directions. The model also presents generic surface currents, which are robust for both bosonic and fermionic systems. We identify the underlying qualitative mechanism responsible for the robustness of the surface currents and the crucial role played by certain discrete symmetries.

Winding vectors of topological defects: multiband Chern numbers

Chern numbers can be calculated within a frame of vortex fields related to phase conventions of a wave function. In a band protected by gaps the Chern number is equivalent to the total number of flux carrying vortices. In the presence of topological defects like Dirac cones this method becomes problematic, in particular if they lack a well-defined winding number. We develop a scheme to include topological defects into the vortex field frame. A winding number is determined by the behavior of the phase in reciprocal space when encircling the defect’s contact point. To address the possible lack of a winding number we utilize a more general concept of winding vectors. We demonstrate the usefulness of this ansatz on Dirac cones generated from bands of the Hofstadter model.

Thermodynamics of correlated electrons in a magnetic field

The Hofstadter–Hubbard model captures the physics of strongly correlated electrons in an applied magnetic field, which is relevant to many recent experiments on Moiré materials. Few large-scale, numerically exact simulations exists for this model. In this work, we simulate the Hubbard–Hofstadter model using the determinant quantum Monte Carlo (DQMC) algorithm. We report the field and Hubbard interaction strength dependence of charge compressibility, fermion sign, local moment, magnetic structure factor, and specific heat. The gross structure of magnetic Bloch bands and band gaps determined by the non-interacting Hofstadter spectrum is preserved in the presence of U. Incompressible regions of the phase diagram have improved fermion sign. At half filling and intermediate and larger couplings, a strong orbital magnetic field delocalizes electrons and reduces the effect of Hubbard U on thermodynamic properties of the system.

Linking topological features of the Hofstadter model to optical diffraction figures

In two, three and even four spatial dimensions, the transverse responses experienced by a charged particle on a lattice in a uniform magnetic field are fully controlled by topological invariants called Chern numbers, which characterize the energy bands of the underlying Hofstadter Hamiltonian. These remarkable features, solely arising from the magnetic translational symmetry, are captured by Diophantine equations which relate the fraction of occupied states, the magnetic flux and the Chern numbers of the system bands. Here we investigate the close analogy between the topological properties of Hofstadter Hamiltonians and the diffraction figures resulting from optical gratings. In particular, we show that there is a one-to-one relation between the above mentioned Diophantine equation and the Bragg condition determining the far-field positions of the optical diffraction peaks. As an interesting consequence of this mapping, we discuss how the robustness of diffraction figures to structural disorder in the grating is a direct analogue of the robustness of transverse conductance in the quantum Hall effect.

Reformulation of gauge theories in terms of gauge invariant fields

We present a reformulation of gauge theories in terms of gauge invariant fields. Focusing on abelian theories, we show that the gauge and matter covariant fields can be recombined to introduce new gauge invariant degrees of freedom. Starting from the (1+1) dimensional case on the lattice, with both periodic and open boundary conditions, we then generalize to higher dimensions and to the continuum limit. To show explicit and physically relevant examples of the reformulation, we apply it to the Hamiltonian of a single particle in a (static) magnetic field, to pure abelian lattice gauge theories, to the Lagrangian of quantum electrodynamics in (3+1) dimensions and to the Hamiltonian of the 2d and the 3d Hofstadter model. In the latter, we show that the particular construction used to eliminate the gauge covariant fields enters the definition of the magnetic Brillouin zone. Finally, we briefly comment on relevance of the presented reformulation to the study of interacting gauge theories.

Topological van Hove singularities at phase transitions in Weyl metals

We show that in three-dimensional (3D) topological metals, a subset of the van Hove singularities of the density of states sits exactly at the transitions between topological and trivial gapless phases. We may refer to these as topological van Hove singularities. By investigating two minimal models, we show that they originate from energy saddle points located between Weyl points with opposite chiralities, and we illustrate their topological nature through their magnetotransport properties in the ballistic regime. We exemplify the relation between van Hove singularities and topological phase transitions in Weyl systems by analyzing the 3D Hofstadter model, which offers a simple and interesting playground to consider different kinds of Weyl metals and to understand the features of their density of states. In this model, as a function of the magnetic flux, the occurrence of topological van Hove singularities can be explicitly checked.

Dissipative preparation of fractional Chern insulators

We 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 single-particle spectrum. While pure (up to topological degeneracy) steady states are only reached in the low-flux 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 well-controlled flux densities. This may be seen as an encouraging step towards addressing the long-standing challenge of preparing strongly correlated topological phases in quantum simulators.

Three-dimensional non-Abelian generalizations of the Hofstadter model: Spin-orbit-coupled butterfly trios

We theoretically introduce and study a three-dimensional Hofstadter model with linearly varying non-Abelian gauge potentials along all three dimensions. The model can be interpreted as spin-orbit coupling among a trio of Hofstadter butterfly pairs since each Cartesian surface ($xy$, $yz$, or $zx$) of the model reduces to a two-dimensional non-Abelian Hofstadter problem. By evaluating the commutativity among arbitrary loop operators around all axes, we derive its genuine (necessary and sufficient) non-Abelian condition, namely, at least two out of the three hopping phases should be neither 0 nor $\pi$. Under different choices of gauge fields in either the Abelian or the non-Abelian regime, both weak and strong topological insulating phases are identified in the model.

Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model

The Hofstadter model is a popular choice for theorists investigating the fractional quantum Hall effect on lattices, due to its simplicity, infinite selection of topological flat bands, and increasing applicability to real materials. In particular, fractional Chern insulators in bands with Chern number $|C|>1$ can demonstrate richer physical properties than continuum Landau level states and have recently been detected in experiments. Motivated by this, we examine the stability of fractional Chern insulators with higher Chern number in the Hofstadter model, using large-scale infinite density matrix renormalization group (iDMRG) simulations on a thin cylinder. We confirm the existence of fractional states in bands with Chern numbers $C=1,2,3,4,5$ at the filling fractions predicted by the generalized Jain series [Phys. Rev. Lett. 115, 126401 (2015)]. Moreover, we discuss their metal-to-insulator phase transitions, as well as the subtleties in distinguishing between physical and numerical stability. Finally, we comment on the relative suitability of fractional Chern insulators in higher Chern number bands for proposed modern applications.