Hofstadter Model Research Articles

Measuring the spin Chern number in time-reversal-invariant Hofstadter optical lattices

We propose an experimental scheme to directly measure the spin Chern number of the time-reversal-invariant Hofstadter model in optical lattices. We first show that this model can be realized by using ultracold Fermi atoms with two pseudo-spin states encoded by the internal Zeeman states in a square optical lattice and the corresponding topological Bloch bands are characterized by the spin Chern number. We then propose and numerically demonstrate that this topological invariant can be extracted from the shift of the hybrid Wannier center in the optical lattice. By spin-resolved in situ detection of the atomic densities along the transverse direction combined with time-of-flight measurement along another spatial direction, the spin Chern number in this system is directly measured.

Dynamical Quantum Phase Transitions: Role of Topological Nodes in Wave Function Overlaps.

A sudden quantum quench of a Bloch band from one topological phase toward another has been shown to exhibit an intimate connection with the notion of a dynamical quantum phase transition (DQPT), where the returning probability of the quenched state to the initial state-i.e., the Loschmidt echo-vanishes at critical times {t^{*}}. Analytical results to date are limited to two-band models, leaving the exact relation between topology and DQPT unclear. In this Letter, we show that, for a general multiband system, a robust DQPT relies on the existence of nodes (i.e., zeros) in the wave function overlap between the initial band and the postquench energy eigenstates. These nodes are topologically protected if the two participating wave functions have distinctive topological indices. We demonstrate these ideas in detail for both one and two spatial dimensions using a three-band generalized Hofstadter model. We also discuss possible experimental observations.

Anatomy of a Periodically Driven p-Wave Superconductor

Abstract The topological properties of periodically driven many-body systems often have no static analogs and defy a simple description based on the effective Hamiltonian. To explore the emergent edge modes in driven p-wave superconductors in two dimensions, we analysed a toy model of Kitaev chains (one-dimensional spinless p-wave superconductors with Majorana edge states) coupled by time-periodic hopping. We showed that with proper driving, the coupled Kitaev chains can turn into a fully gapped superconductor, which is analogous to the p x +ip y state but has two, rather than one, chiral edge modes. A different driving protocol turns it into a gapless superconductor with isolated point nodes and completely flat edge states at quasienergy ω=0 or π/T, with T as the driving period. The time evolution operator U(k x , k y , t) of the toy model is computed exactly to yield the phase bands. And the “topological singularities” of the phase bands are exhausted and compared to those of a periodically driven Hofstadter model, which features counter-propagating chiral edge modes. These examples demonstrate the unique edge states in driven superconducting systems and suggest driving as a potentially fruitful route to engineer new topological superconductors.

Peierls substitution for magnetic Bloch bands

We consider the Schr\"odinger operator in two dimensions with a periodic potential and a strong constant magnetic field perturbed by slowly varying non-periodic scalar and vector potentials, $\phi(\epsilon x)$ and $A(\epsilon x)$, for $\epsilon\ll 1$. For each isolated family of magnetic Bloch bands we derive an effective Hamiltonian that is unitarily equivalent to the restriction of the Schr\"odinger operator to a corresponding almost invariant subspace. At leading order, our effective Hamiltonian can be interpreted as the Peierls substitution Hamiltonian widely used in physics for non-magnetic Bloch bands. However, while for non-magnetic Bloch bands the corresponding result is well understood, for magnetic Bloch bands it is not clear how to even define a Peierls substitution Hamiltonian beyond a formal expression. The source of the difficulty is a topological obstruction: magnetic Bloch bundles are generically not trivializable. As a consequence, Peierls substitution Hamiltonians for magnetic Bloch bands turn out to be pseudodifferential operators acting on sections of non-trivial vector bundles over a two-torus, the reduced Brillouin zone. Part of our contribution is the construction of a suitable Weyl calculus for such pseudos. As an application of our results we construct a new family of canonical one-band Hamiltonians $H^B_{\theta,q}$ for magnetic Bloch bands with Chern number $\theta\in \mathbb{Z}$ that generalizes the Hofstadter model $H^B_{\rm Hof} = H^B_{0,1}$ for a single non-magnetic Bloch band. It turns out that $H^B_{\theta,q}$ is isospectral to $H^{q^2B}_{\rm Hof}$ for any $\theta$ and all spectra agree with the Hofstadter spectrum depicted in his famous black and white butterfly. However, the resulting Chern numbers of subbands, corresponding to Hall conductivities, depend on $\theta$ and $q$, and thus the models lead to different colored butterflies.

Quantum geometry and stability of the fractional quantum Hall effect in the Hofstadter model

We study how the stability of the fractional quantum Hall effect (FQHE) is influenced by the geometry of band structure in lattice Chern insulators. We consider the Hofstadter model, which converges to continuum Landau levels in the limit of small flux per plaquette. This gives us a degree of analytic control not possible in generic lattice models, and we are able to obtain analytic expressions for the relevant geometric criteria. These may be differentiated by whether they converge exponentially or polynomially to the continuum limit. We demonstrate that the latter criteria have a dominant effect on the physics of interacting particles in Hofstadter bands in this low flux density regime. In particular, we show that the many-body gap depends monotonically on a band-geometric criterion related to the trace of the Fubini-Study metric.

Density matrix renormalization group on a cylinder in mixed real and momentum space

We develop a variant of the density matrix renormalization group (DMRG) algorithm for two-dimensional cylinders that uses a real space representation in the direction along the axis of the cylinder and a momentum space representation in the direction around the circumference. The mixed representation allows us to use the momentum around the cylinder as a conserved quantity in the DMRG algorithm. Compared with the traditional purely real-space approach, we find a significant speedup in computation time and a considerable reduction in memory usage. Applying the method to the interacting fermionic Hofstadter model, we demonstrate a reduction in computation time by over 20-fold, in addition to a sixfold memory reduction.

Quons in a quantum dissipative system

String theory proves to be an imperative tool to explore the critical behavior of the quantum dissipative system. We discuss the quantum particles moving in two dimensions, in the presence of a uniform magnetic field, subject to a periodic potential and a dissipative force, which are described by the dissipative Wannier–Azbel–Hofstadter (DWAH) model. Using string theory formulation of the model, we find that the elementary excitations of the system at the generic points of the off-critical regions, in the zero temperature limit are quons, which satisfy [Formula: see text]-deformed statistics.

Mapping the current–current correlation function near a quantum critical point

The current–current correlation function is a useful concept in the theory of electron transport in homogeneous solids. The finite-temperature conductivity tensor as well as Anderson’s localization length can be computed entirely from this correlation function. Based on the critical behavior of these two physical quantities near the plateau–insulator or plateau–plateau transitions in the integer quantum Hall effect, we derive an asymptotic formula for the current–current correlation function, which enables us to make several theoretical predictions about its generic behavior. For the disordered Hofstadter model, we employ numerical simulations to map the current–current correlation function, obtain its asymptotic form near a critical point and confirm the theoretical predictions.

Canonical Quantization of the Dissipative Hofstadter Model

We perform canonical quantization of the dissipative Hofstadter model, which has a wide range of applications in condensed matter physics and string theory. The target space duality and the non-commutative algebra developed in string theory are discussed for the model. We show that the target space duality transformation of closed string theory, $O(2,2;R)$, removes the interaction with a uniform magnetic field. The $O(2,2;R)$ dual transformation changes the basis of oscillator operators so that the algebra of the string coordinate operators at the boundary become non-commutative. In the zero temperature limit, the non-commutative algebra of open string theory emerges. We also developed the boundary state formulation for the dissipative Hofstadter model.

Baryon squishing in synthetic dimensions by effectiveSU(M)gauge fields

We investigate few body physics in a cold atomic system with synthetic dimensions (Celi et al., PRL 112, 043001 (2014)) which realizes a Hofstadter model with long-ranged interactions along the synthetic dimension. We show that the problem can be mapped to a system of particles (with $SU(M)$ symmetric interactions) which experience an $SU(M)$ Zeeman field at each lattice site {\em and} a non-Abelian $SU(M)$ gauge potential that affects their hopping from one site to another. This mapping brings out the possibility of generating {\em non-local} interactions (interaction between particles at different physical sites). It also shows that the non-Abelian gauge field, which induces a flavor-orbital coupling, mitigates the "baryon breaking" effects of the Zeeman field. For $M$ particles, the $SU(M)$ singlet baryon which is site localized, is "deformed" to be a nonlocal object ("squished" baryon) by the combination of the Zeeman and the non-Abelian gauge potential, an effect that we conclusively demonstrate by analytical arguments and exact (numerical) diagonalization studies. These results not only promise a rich phase diagram in the many body setting, but also suggests possibility of using cold atom systems to address problems that are inconceivable in traditional condensed matter systems. As an example, we show that the system can be adapted to realize Hamiltonians akin to the $SU(M)$ random flux model.

Topological Edge States with Zero Hall Conductivity in a Dimerized Hofstadter Model.

The Hofstadter model is a simple yet powerful Hamiltonian to study quantum Hall physics in a lattice system, manifesting its essential topological states. Lattice dimerization in the Hofstadter model opens an energy gap at half filling. Here we show that even if the ensuing insulator has a Chern number equal to zero, concomitantly a doublet of edge states appear that are pinned at specific momenta. We demonstrate that these states are topologically protected by inversion symmetry in specific one-dimensional cuts in momentum space, define and calculate the corresponding invariants, and identify a platform for the experimental detection of these novel topological states.

Viscoelastic response of topological tight-binding models in two and three dimensions

The topological response to external perturbations is an effective probe to characterize different topological phases of matter. Besides the Hall conductance, the Hall viscosity is another example of such a response that measures how electronic wave functions respond to changes in the underlying geometry. Topological (Chern) insulators are known to have a quantized Hall conductance. A natural question is how the Hall viscosity behaves for these materials. So far, most of studies on the Hall viscosity of Chern insulators have focused on the continuum limit. The presence of lattice breaks the continuous translational symmetry to a discrete group and this causes two complications: it introduces a new length scale associated with the lattice constant, and makes the momentum periodic. We develop two different methods of how to implement a lattice deformation: (1) a lattice distortion is encoded as a shift in the lattice momentum, and (2) a lattice deformation is treated microscopically in the gradient expansion of the hopping matrix elements. After establishing the method of deformation we can compute the Hall viscosity through a linear response (Kubo) formula. We examine these methods for three models: the Hofstadter model, the Chern insulator, and the surface of a 3D topological insulator. Our results in certain regimes of parameters, where the continuum limit is relevant, are in agreement with previous calculations. We also provide possible experimental signatures of the Hall viscosity by studying the phononic properties of a single crystal 3D topological insulator.

Statistics of two-dimensional random walks, the cyclic sieving phenomenon and the Hofstadter model

We focus on the algebraic area probability distribution of planar random walks on a square lattice with and l2 steps right, left, up and down. We aim, in particular, at the algebraic area generating function evaluated at a root of unity, when both and multiples of q. In the simple case of staircase walks, a geometrical interpretation of in terms of the cyclic sieving phenomenon is illustrated. Then, an expression for which is relevant to the Stembridge case, is proposed. Finally, the related problem of evaluating the nth moments of the Hofstadter Hamiltonian in the commensurate case is addressed.

Fermionic non-Abelian fractional Chern insulators from dipolar interactions

We study fermions on a triangular lattice model that exhibits topological flatbands characterized by nonzero Chern numbers. Our scheme stems from the well-known Hofstadter model but the next-nearest-neighbor hopping is introduced, which is crucial for tuning the lowest band to be nearly flat. Differing from previous proposals with the necessity of multiparticle interactions, we consider the more realistic long-range dipolar interaction combined with two-body short-range attractions between fermions. We show the realization of the non-Abelian $\nu=1/2$ Moore-Read fractional Chern insulators, and strong evidence for the existence of the more exotic $\nu=3/5$ Read-Rezayi fractional Chern insulators. Our results provide insights for the experimental realization of these exotic states by realistic two-body interactions and thus facilitates the implementation of the universal topological quantum computation.

Fractional topological phases in generalized Hofstadter bands with arbitrary Chern numbers

We construct generalized Hofstadter models that possess "color-entangled" flat bands and study interacting many-body states in such bands. For a system with periodic boundary conditions and appropriate interactions, there exist gapped states at certain filling factors for which the ground-state degeneracy depends on the number of unit cells along one particular direction. This puzzling observation can be understood intuitively by mapping our model to a single-layer or a multilayer system for a given lattice configuration. We discuss the relation between these results and the previously proposed "topological nematic states," in which lattice dislocations have non-Abelian braiding statistics. Our study also provides a systematic way of stabilizing various fractional topological states in $C>1$ flat bands and provides some hints on how to realize such states in experiments.

Floquet edge states in a harmonically driven integer quantum Hall system

Recent theoretical work on time-periodically kicked Hofstadter model found robust counterpropagating edge modes. It remains unclear how ubiquitously such anomalous modes can appear, and what dictates their robustness against disorder. Here we shed further light on the nature of these modes by analyzing a simple type of periodic driving where the hopping along one spatial direction is modulated sinusoidally with time while the hopping along the other spatial direction is kept constant. We obtain the phase diagram for the quasienergy spectrum at flux 1/3 as the driving frequency $\ensuremath{\omega}$ and the hopping anisotropy are varied. A series of topologically distinct phases with counterpropagating edge modes appear due to the harmonic driving, similar to the case of a periodically kicked system studied earlier. We analyze the time dependence of the pair of Floquet edge states localized at the same edge and compare their Fourier components in the frequency domain. In the limit of small modulation, one of the Floquet edge mode within the pair can be viewed as the edge mode originally living in the other energy gap shifted in quasienergy by $\ensuremath{\hbar}\ensuremath{\omega}$, i.e., by absorption or emission of a ``photon'' of frequency $\ensuremath{\omega}$. Our result suggests that counterpropagating Floquet edge modes are generic features of periodically driven integer quantum Hall systems, and not tied to any particular driving protocol. It also suggests that the Floquet edge modes would remain robust to any static perturbations that do not destroy the chiral edge modes of static quantum Hall states.

Exact solutions of fractional Chern insulators: Interacting particles in the Hofstadter model at finite size

We show that all the bands of the Hofstadter model on the torus have an exactly flat dispersion and Berry curvature when a special system size is chosen. This result holds for any hopping and Chern number. Our analysis therefore provides a simple rule for choosing a particularly advantageous system size when designing a Hofstadter system whose size is controllable, like a qubit lattice or an optical cavity array. The density operators projected onto the flat bands obey exactly the Girvin-MacDonald-Platzman algebra, like for Landau levels in the continuum in the case of $C=1$, or obey its straightforward generalization in the case of $C>1$. This allows a mapping between density-density interaction Hamiltonians for particles in the Hofstatder model and in a continuum Landau level. By using the well-known pseudopotential construction in the latter case, we obtain fractional Chern insulator phases, the lattice counterpart of fractional quantum Hall phases, that are exact zero-energy ground states of the Hofstadter model with certain interactions. Finally, the addition of a harmonic trapping potential is shown to lead to an appealingly symmetric description in which a new Hofstadter model appears in momentum space.

Perturbative approach to flat Chern bands in the Hofstadter model

We present a perturbative approach to the study of the Hofstadter model for when the amount of flux per plaquette is close to a rational fraction. Within this approximation certain eigenstates of the system are shown to be multi-component wavefunctions that connect smoothly to the Landau levels of the continuum. The perturbative corrections to these are higher Landau level contributions that break rotational invariance and allow the perturbed states to adopt the symmetry of the lattice. In the presence of interactions, this approach allows for the calculation of generalised Haldane pseudopotentials, and in turn, the many-body properties of the system. The method is sufficiently general that it can apply to a wide variety of lattices, interactions and magnetic field strengths.

Critical integer quantum Hall topology and the integrable Maryland model as a topological quantum critical point

One-dimensional tight binding models such as the Aubry-Andr\'e-Harper (AAH) model (with an on-site cosine potential) and the integrable Maryland model (with an on-site tangent potential) have been the subject of extensive theoretical research in localization studies. AAH can be directly mapped onto the two-dimensional (2D) Hofstadter model which manifests the integer quantum Hall topology on a lattice. However, such a connection needs to be made for the Maryland model (MM). Here we describe a generalized model that contains AAH and MM as the limiting cases with the MM lying precisely at a topological quantum phase transition (TQPT) point. A remarkable feature of this critical point is that the one-dimensional MM retains well defined energy gaps whereas the equivalent 2D model becomes gapless, signifying the 2D nature of the TQPT.

Efficient algorithm to compute the Berry conductivity

We propose and construct a numerical algorithm to calculate the Berry conductivity in topological band insulators. The method is applicable to cold atom systems as well as solid state setups, both for the insulating case where the Fermi energy lies in the gap between two bulk bands as well as in the metallic regime. This method interpolates smoothly between both regimes. The algorithm is gauge-invariant by construction, efficient, and yields the Berry conductivity with known and controllable statistical error bars. We apply the algorithm to several paradigmatic models in the field of topological insulators, including Haldaneʼs model on the honeycomb lattice, the multi-band Hofstadter model, and the BHZ model, which describes the 2D spin Hall effect observed in CdTe/HgTe/CdTe quantum well heterostructures.