Hofstadter Model Research Articles

Hofstadter’s Cocoon

Hofstadter showed that the energy levels of electrons on a lattice plotted as a function of magnetic field form an beautiful structure now referred to as "Hofstadter's butterfly". We study a non-Hermitian continuation of Hofstadter's model; as the non-Hermiticity parameter $g$ increases past a sequence of critical values the eigenvalues successively go complex in a sequence of "double-pitchfork bifurcations" wherein pairs of real eigenvalues degenerate and then become complex conjugate pairs. The associated wavefunctions undergo a spontaneous symmetry breaking transition that we elucidate. Beyond the transition a plot of the real parts of the eigenvalues against magnetic field resembles the Hofstadter butterfly; a plot of the imaginary parts plotted against magnetic fields forms an intricate structure that we call the Hofstadter cocoon. The symmetries of the cocoon are described. Hatano and Nelson have studied a non-Hermitian continuation of the Anderson model of localization that has close parallels to the model studied here. The relationship of our work to that of Hatano and Nelson and to PT transitions studied in PT quantum mechanics is discussed.

The Colored Hofstadter Butterfly for the Honeycomb Lattice

We rely on a recent method for determining edge spectra and we use it to compute the Chern numbers for Hofstadter models on the honeycomb lattice having rational magnetic flux per unit cell. Based on the bulk-edge correspondence, the Chern number $\sigma_H$ is given as the winding number of an eigenvector of a $2 \times 2$ transfer matrix, as a function of the quasi-momentum $k \in (0,2 \pi)$. This method is computationally efficient (of order $O(n^4)$ in the resolution of the desired image). It also shows that for the honeycomb lattice the solution for $\sigma_H $ for flux $p/q$ in the $r$-th gap conforms with the Diophantine equation $r=\sigma_H\cdot p+ s\cdot q$, which determines $\sigma_H \mod q$. A window such as $\sigma_H \in(-q/2,q/2)$, or possibly shifted, provides a natural further condition for $\sigma_H$, which however turns out not to be met. Based on extensive numerical calculations, we conjecture that the solution conforms with the relaxed condition $\sigma_H\in(-q,q)$.

A study of the ambiguity in the solutions to the Diophantine equation for Chern numbers

The Chern numbers for Hofstadter models with rational flux 2πp/q are partially determined by a Diophantine equation. A mod q ambiguity remains. The resolution of this ambiguity is only known for the rectangular lattice with nearest neighbors hopping where it has the form of a ‘window condition’. We study a Hofstadter butterfly on the triangular lattice for which the resolution of ambiguity is open. In the model many pairs (p, q) satisfy a window condition which is shifted relative to the window of the square model. However, we also find pairs (p, q) where the Chern numbers do not belong to any contiguous window. This shows that the rectangular model and the one we study on the triangular lattice are not adiabatically connected: many gaps must close. Our results suggest the conjecture that the mod q ambiguity in the Diophantine equation generically reduces to a sign ambiguity.

Characterization of the quantized Hall insulator phase in the quantum critical regime

The conductivity σ and resistivity ρ tensors of the disordered Hofstadter model are mapped as functions of Fermi energy EF and temperature T in the quantum critical regime of the plateau-insulator transition (PIT). The finite-size errors are eliminated by using the non-commutative Kubo formula. The results reproduce all the key experimental characteristics of this transition in integer quantum Hall effect (IQHE) systems. In particular, the quantized Hall insulator (QHI) phase is detected and analyzed. The presently accepted characterization of the QHI phase in the quantum critical regime, based entirely on experimental data, is fully supported by our theoretical investigation.

Seeing Hofstadter's butterfly in atomic Fermi gases

We propose a novel way to detect the fractal energy spectrum of the Hofstadter model from the density distributions of ultracold fermions in an external trap. At low temperature, the local compressibility is proportional to the density of states of the system which reveals the fractal energy spectrum. However, thermal broadening and noises in the real experimental situation inevitably smear out fine features in the density distribution. To overcome this difficulty, we use the maximum entropy method to extract the density of states directly from the noisy thermal density distributions. Simulations show that one is able to restore the core feature of the Hofstadter's butterfly spectrum with current experimental techniques. By further reducing the noise or the temperature, one can refine the resolution and observe fine structures of the butterfly spectrum.

Coupling-matrix approach to the Chern number calculation in disordered systems

The Chern number is often used to distinguish different topological phases of matter in two-dimensional electron systems. A fast and efficient coupling-matrix method is designed to calculate the Chern number in finite crystalline and disordered systems. To show its effectiveness, we apply the approach to the Haldane model and the lattice Hofstadter model, and obtain the correct quantized Chern numbers. The disorder-induced topological phase transition is well reproduced, when the disorder strength is increased beyond the critical value. We expect the method to be widely applicable to the study of topological quantum numbers.

Tunable Band Topology Reflected by Fractional Quantum Hall States in Two-Dimensional Lattices

Two-dimensional lattice models subjected to an external effective magnetic field can form nontrivial band topologies characterized by nonzero integer band Chern numbers. In this Letter, we investigate such a lattice model originating from the Hofstadter model and demonstrate that the band topology transitions can be realized by simply introducing tunable longer-range hopping. The rich phase diagram of band Chern numbers is obtained for the simple rational flux density and a classification of phases is presented. In the presence of interactions, the existence of fractional quantum Hall states in both |C| = 1 and |C| > 1 bands is confirmed, which can reflect the band topologies in different phases. In contrast, when our model reduces to a one-dimensional lattice, the ground states are crucially different from fractional quantum Hall states. Our results may provide insights into the study of new fractional quantum Hall states and experimental realizations of various topological phases in optical lattices.

Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices

We demonstrate the experimental implementation of an optical lattice that allows for the generation of large homogeneous and tunable artificial magnetic fields with ultracold atoms. Using laser-assisted tunneling in a tilted optical potential, we engineer spatially dependent complex tunneling amplitudes. Thereby, atoms hopping in the lattice accumulate a phase shift equivalent to the Aharonov-Bohm phase of charged particles in a magnetic field. We determine the local distribution of fluxes through the observation of cyclotron orbits of the atoms on lattice plaquettes, showing that the system is described by the Hofstadter model. Furthermore, we show that for two atomic spin states with opposite magnetic moments, our system naturally realizes the time-reversal-symmetric Hamiltonian underlying the quantum spin Hall effect; i.e., two different spin components experience opposite directions of the magnetic field.

Semiclassical Approximations for Hamiltonians with Operator-Valued Symbols

We consider the semiclassical limit of quantum systems with a Hamiltonian given by the Weyl quantization of an operator valued symbol. Systems composed of slow and fast degrees of freedom are of this form. Typically a small dimensionless parameter $\varepsilon\ll 1$ controls the separation of time scales and the limit $\varepsilon\to 0$ corresponds to an adiabatic limit, in which the slow and fast degrees of freedom decouple. At the same time $\varepsilon\to 0$ is the semiclassical limit for the slow degrees of freedom. In this paper we show that the $\varepsilon$-dependent classical flow for the slow degrees of freedom first discovered by Littlejohn and Flynn, coming from an $\epsi$-dependent classical Hamilton function and an $\varepsilon$-dependent symplectic form, has a concrete mathematical and physical meaning: Based on this flow we prove a formula for equilibrium expectations, an Egorov theorem and transport of Wigner functions, thereby approximating properties of the quantum system up to errors of order $\varepsilon^2$. In the context of Bloch electrons formal use of this classical system has triggered considerable progress in solid state physics. Hence we discuss in some detail the application of the general results to the Hofstadter model, which describes a two-dimensional gas of non-interacting electrons in a constant magnetic field in the tight-binding approximation.

Entanglement spectrum and Wannier center flow of the Hofstadter problem

We examine the quantum entanglement spectra and Wannier functions of the square lattice Hofstadter model. Consistent with previous work on entanglement spectra of topological band structures, we find that the entanglement levels exhibit a spectral flow similar to that of the full system's energy spectrum. While the energy spectra are continuous, with open boundary conditions the entanglement spectra exhibit discontinuities associated with the passage of an energy edge state through the Fermi level. We show how the entanglement spectrum can be understood by examining the band projectors of the full system and their behavior under adiabatic pumping. In so doing we make connections with the original TKNN work on topological two-dimensional band structures and their Chern numbers. Finally we consider Wannier states and their adiabatic flows, and draw connections to the entanglement properties.

SU(3) Spin-Orbit Coupling in Systems of Ultracold Atoms

Motivated by the recent experimental success in realizing synthetic spin-orbit coupling in ultracold atomic systems, we consider N-component atoms coupled to a non-Abelian SU(N) gauge field. More specifically, we focus on the case, referred to here as "SU(3) spin-orbit-coupling," where the internal states of three-component atoms are coupled to their momenta via a matrix structure that involves the Gell-Mann matrices (in contrast to the Pauli matrices in conventional SU(2) spin-orbit-coupled systems). It is shown that the SU(3) spin-orbit-coupling gives rise to qualitatively different phenomena and in particular we find that even a homogeneous SU(3) field on a simple square lattice enables a topologically nontrivial state to exist, while such SU(2) systems always have trivial topology. In deriving this result, we first establish an equivalence between the Hofstadter model with a 1/N Abelian flux per plaquette and a homogeneous SU(N) non-Abelian model. The former is known to have a topological spectrum for N>2, which is thus inherited by the latter. It is explicitly verified by an exact calculation for N=3, where we develop and use a new algebraic method to calculate topological indices in the SU(3) case. Finally, we consider a strip geometry and establish the existence of three gapless edge states-the hallmark feature of such an SU(3) topological insulator.

Wave functions for fractional Chern insulators

We provide a parton construction of wavefunctions and effective field theories for fractional Chern insulators. We also analyze a strong coupling expansion in lattice gauge theory that enables us to reliably map the parton gauge theory onto the microsopic Hamiltonian. We show that this strong coupling expansion is useful because of a special hierarchy of energy scales in fractional quantum Hall physics. Our procedure is illustrated using the Hofstadter model and then applied to bosons at 1/2 filling and fermions at 1/3 filling in a checkerboard lattice model recently studied numerically. Because our construction provides a more or less unique mapping from microscopic model to effective parton description, we obtain wavefunctions in the same phase as the observed fractional Chern insulators without tuning any continuous parameters.

Fractality of Hofstadter Butterfly in Specific Heat Oscillation

We calculate thermodynamical properties of the Hofstadter model using a recently developed quantum transfer matrix method. We find intrinsic oscillation features in specific heat that manifest the fractal structure of the Hofstadter butterfly. We also propose experimental approaches which use specific heat as an access to detect the Hofstadter butterfly.

STRING THEORY AND DUALITIES IN THE QUANTUM DISSIPATIVE HOFSTADTER SYSTEM

We study the dualities of the quantum dissipative Hofstadter system which describes particles moving in two dimensions, subject to a uniform magnetic field, a periodic potential and a dissipative force. Using the string theory formulation, we show that the system has two kinds of dualities. The duality, previously known as the exact duality in the literature is shown to correspond to a subgroup of the T-dual symmetry group unbroken by the periodic boundary potential in string theory. The other duality is a particle–kink duality in the noncommutative open string theory which is a generalized Schmid duality in the presence of the uniform magnetic field. The kinks of the dissipative Hofstadter model are found to be noncommutative objects. The particle–kink duality, which is called previously the approximate duality, is shown to be also exact. In contrast to the previous derivation, which is based on the Coulomb gas expansion of the partition function and asserts that the duality holds only approximately in the regime of strong magnetic field, the string theory formulation proves that duality holds always exactly in the off-critical regions where the periodic potential becomes strong, regardless of the strength of the magnetic field. The dualities of the DHM may also be useful for studying the rolling tachyon in string theory in the presence of the Neveu–Schwarz (NS) B-field, since both DHM and rolling tachyon in the presence of NS B-field are described by the same action.

Probing the Hofstadter butterfly with the quantum oscillation of magnetization

We have developed a different quantum transfer-matrix method to accurately determine thermodynamic properties of the Hofstadter model. This method resolves a technical problem which is intractable by other methods and makes the calculation of physical quantities of the Hofstadter model in the thermodynamic limit at finite temperatures feasible. It is shown that the quantum correction to the de Haas--van Alphen oscillation of magnetization bears the energy structure of the Hofstadter butterfly. The measurement of this quantum correction, which can be materialized on the superlattice or cold atom systems, can reveal unambiguously the Hofstadter fractal energy spectrum.

Dissipative Hofstadter Model at the Magic Points and Critical Boundary Sine-Gordon Model

The dissipative Hofstadter model describes quantum particles moving in two dimensions subject to a uniform magnetic field, a periodic potential and a dissipative force. We discuss the dissipative Hofstadter model in the framework of the boundary state formulation in string theory and construct exact boundary states for the model at the magic points by using the fermion representation. The dissipative Hofstadter model at magic points is shown to be equivalent to the critical boundary sine-Gordon model.

Junctions of three quantum wires

We study a junction of three quantum wires enclosing a magnetic flux. The wires aremodelled as single-channel spinless Tomonaga–Luttinger liquids. This is the simplestproblem of a quantum junction between Tomonaga–Luttinger liquids in which Fermistatistics enter in a non-trivial way. We study the problem using a mapping onto thedissipative Hofstadter model, describing a single particle moving on a plane in a magneticfield and a periodic potential coupled to a harmonic oscillator bath. Alternatively we studythe problem by identifying boundary conditions corresponding to the low energy fixedpoints. We obtain a rich phase diagram including a chiral fixed point in which theasymmetric current flow is highly sensitive to the sign of the flux and a phase in whichelectron pair tunnelling dominates. We also study the effects on the conductance tensor ofthe junction of contacting the three quantum wires to Fermi liquid reservoirs.

Tunneling through two resonant levels: Fixed points and conductances

We study point contact tunneling between two leads of a Tomonaga-Luttinger liquid through two degenerate resonant levels in parallel. This is one of the simplest cases of a quantum junction problem where the Fermi statistics of the electrons plays a non-trivial role through the Klein factors appearing in bosonization. Using a mapping to a `generalized Coulomb model' studied in the context of the dissipative Hofstadter model, we find that any asymmetry in the tunneling amplitudes from the two leads grows at low temperatures, so that ultimately there is no conductance across the system. For the symmetric case, we identify a non-trivial fixed point of this model; the conductance at that point is generally different from the conductance through a single resonant level.

An Observation on Hofstadter’s Butterfly

Joseph Avron, Daniel Osadchy, and Ruedi Seiler have nicely highlighted the relevance of topological invariants, or Chern numbers, to the integer quantum Hall effect and to the conductance of a Hofstadter model when the Fermi energy lies in a gap (Physics Today, Physics Today 0031-9228 56 8 2003 38 https://doi.org/10.1063/1.1611351. August 2003, page 38 ).As an addition to their remarks, it is intriguing and suggestive that one can go beyond these facts and construct, for the quantum Hall effect, a mean-field theory that has the unique property of covering both the integer and fractional regimes in a single framework. In such a model, Hofstadter’s butterfly (figure 5 in the article) maps out the essence of what is observed in the lab: the odd-denominator selection rule and hierarchy in the plateau widths and resolution. 1 1. F. Claro, Phys. Rev. B 35, 7980 (1987). https://doi.org/10.1103/PhysRevB.35.7980 The surprising unity of the effect results simply from the unity of the butterfly spectrum, with its self similar features reflecting the hierarchies seen at very low temperatures.As Douglas Hofstadter mentioned in his PhD thesis, his friend David Jennings, on seeing the spectrum, described it as “a picture of God.” 2 2. D. D. Hofstadter, “The Energy Levels of Bloch Electrons in a Magnetic Field,” PhD thesis, U. of Oregon (1975), p. vii. Perhaps that is going too far, but one can certainly see this particular butterfly flying very high.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. F. Claro, Phys. Rev. B 35, 7980 (1987). https://doi.org/10.1103/PhysRevB.35.7980 , Google ScholarCrossref, ISI2. D. D. Hofstadter, “The Energy Levels of Bloch Electrons in a Magnetic Field,” PhD thesis, U. of Oregon (1975), p. vii. Google Scholar© 2004 American Institute of Physics.

Junctions of three quantum wires and the dissipative Hofstadter model.

We study a junction of three quantum wires enclosing a magnetic flux. This is the simplest problem of a quantum junction between Tomonaga-Luttinger liquids in which Fermi statistics enter in a nontrivial way. We present a direct connection between this problem and the dissipative Hofstadter problem, or quantum Brownian motion in two dimensions in a periodic potential and an external magnetic field, which in turn is connected to open string theory in a background electromagnetic field. We find nontrivial fixed points corresponding to a chiral conductance tensor leading to an asymmetric flow of the current.