Hofstadter butterfly as quantum phase diagram

Abstract

The Hofstadter butterfly is viewed as a quantum phase diagram with infinitely many phases, labeled by their (integer) Hall conductance, and a fractal structure. We describe various properties of this phase diagram: We establish Gibbs phase rules; count the number of components of each phase, and characterize the set of multiple phase coexistence.Introduction.-Azbel [1] recognized that the spectral properties of two-dimensional, periodic, quantum systems have sensitive dependence on the magnetic flux through a unit cell. A simple model conceived by Peierls and put to the eponymous Harper as a thesis problem, gained popularity with D. Hofstadter Ph.D. thesis [2], where a wonderful diagram, reminiscent of a fractal butterfly, provided a source of inspiration and a tool for spectral analysis [3][4][5][6][7][8][9].The Hofstadter butterfly can also be viewed as the quantum (zero temperature) phase diagram for the integer quantum Hall effect. It is a fractal phase diagram with infinitely many phases [11,12]. The diagram leads to certain natural questions: Count the number of components of a given phase; Classify which phases coexist and where. It also leads to the general question: What form does the Gibbs phase rule [14,13] take for quantum phase transitions.Fractal phase diagrams and/or infinitely many phases appear in dynamical systems [17,18]. In classical lattice systems fractal phase diagrams [14,19,20,13] are commonly viewed as a pathology due to either long range interactions, or, as is the case for spin glasses, loss of translation invariance. The Hofstadter model, when viewed as a statistical mechanical model, is both short range and translation invariant in a natural way. But, it is quantum and the translation group is non-commutative. It suggests that fractal phase diagram may be more common in quantum phase transitions than in classical phase transitions.The Hofstadter Model.-The model conceived by Peierls has two versions. For the sake of concreteness we shall focus here on the tight binding version. On the lattice Z 2 , define magnetic shifts

Applications of Magnetic Ψdo Techniques to Sapt

“…The Hofstadter model is a paradigm in 4 G. De Nittis and M. Lein the study of fractal spectra (Hofstadter butterfly) and was used by Thouless et al in the seminal paper [28] to give the first theoretical explanation of the topological quantization of the QHE. More recently Avron [22] interpreted the results by Thouless et al from the viewpoint of thermodynamics and connected the QHE to anomalous phase transition diagrams (colored quantum butterflies). Even though this list of publications is very much incomplete, it shows the significance of the Peierls substitution in the study of the QHE.…”

Equality of the Bulk and Edge Hall Conductances in a Mobility Gap

“…We refer the reader to ref. [26] and references therein for further discussion of the Harper Hamiltonian.…”

Fractional Chern Insulators in Harper-Hofstadter Bands with Higher Chern Number

“…(1) is seen to be the net Chern number of the bands contributing to the n s states below the energy gap. Solutions to (1) exist for any C in which n s arises from a single band (see below), establishing the presence of bands of any Chern number in the Hofstadter spectrum (albeit with rapidly decreasing gaps for large C) [36,37].To realize the nontrivial Chern bands of the HarperHofstadter model, it is sufficient to create a tight-binding arXiv:1504.06623v2 [cond-mat.str-el] …”