Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics

热霍夫斯塔德蝴蝶

Thermal Hofstadter Butterflies

研究通过热力学测量揭示了霍夫斯塔德蝴蝶的分形特性，发现电子熵和比热的自相似性及磁热振荡，为识别分形谱提供新方法。

Statistical mechanics of Bose gas in Sierpinski carpets

We carry out a mathematically rigorous investigation into the equilibrium thermodynamics of massless and massive bosons confined in generalized Sierpinski carpets (GSCs), a class of infinitely ramified fractals having non-integer Hausdorff dimensions $d_h$. Due to the anomalous walk dimension $d_w>2$ associated with Brownian motion on GSCs, all extensive thermodynamic quantities are shown to scale with the spectral volume with dimension $d_s = 2(d_h/d_w)$ rather than the Hausdorff volume. We prove that for a low-temperature, high-density ideal massive Bose gas in an unbounded GSC, Bose-Einstei…

The Ising model in planar lacunary and fractal lattices, a path counting approach

The method of counting loops for calculating the partition function of the Ising model on the two dimensional square lattice is extended to lacunary planar lattices, especially scale invariant fractal lattices, the Sierpińsky carpets with different values of the scale invariance ratio and of the number of deleted sites. The critical temperature of the Ising model on these lattices is exactly calculated for finite iteration steps, and for a range of the scale invariance ratio $n$ from 3 to 1000 and of the number of deleted sites from $(n-2)^2$ to $(n-10)^2$. The critical temperature at the limi…

Sierpiński 地毯上的磁化测量

Measurements of magnetization on the Sierpiński carpet

本文利用改进的高阶张量重正化群方法研究了 Sierpiński 地毯上经典 Ising 模型的相变，发现局部磁化临界指数随晶格位置变化显著，并通过自动微分计算得出全局临界指数。

Universality of Hofstadter butterflies on hyperbolic lattices

Motivated by recent experimental breakthroughs in realizing hyperbolic lattices in superconducting waveguides and electric circuits, we compute the Hofstadter butterfly on regular hyperbolic tilings. By utilizing large hyperbolic lattices with periodic boundary conditions, we obtain the true hyperbolic bulk spectrum that is unaffected by contributions from boundary states. Our results reveal that the butterfly spectrum with large extended gapped regions prevails and that its shape is universally determined by the number of edges of the fundamental tile, while the fractal structure is lost in s…

Hofstadter-Toda谱对偶性与量子群

Hofstadter-Toda spectral duality and quantum groups

本文探讨了Hofstadter模型与相对论性Toda格子之间的谱对偶性及其与量子群Langlands对偶性的联系，并利用三对角矩阵谱对偶性和初等量子群表示理论，建立了用初等对称多项式和Chebyshev多项式参数化Hofstadter模型能谱的公式。

Entanglement Spectra and Entanglement Thermodynamics of Hofstadter Bilayers

We study Hofstadter bilayers, i.e. coupled hopping models on two-dimensional square lattices in a perpendicular magnetic field. Upon tracing out one of the layers, we find an explicit expression for the resulting entanglement spectrum in terms of the energy eigenvalues of the underlying monolayer system. For strongly coupled layers the entanglement Hamiltonian is proportional to the energetic Hamiltonian of the monolayer system. The proportionality factor, however, cannot be interpreted as the inverse thermodynamic temperature, but represents a phenomenological temperature scale. We derive an …

Entanglement spectrum of fermionic bilayer honeycomb lattice: Hofstadter butterfly

We perform an analytical study of the energy and entanglement spectrum of non-interacting fermionic bilayer honeycomb lattices in the presence of trigonal warping in the energy spectrum, on-site energy difference and uniform magnetic field. Employing single particle correlation functions, we present an explicit form for layer-layer entanglement Hamiltonian whose spectrum is entanglement spectrum. We demonstrate that in the absence of trigonal warping, at zero on-site energy difference exact correspondence is established between entanglement spectrum and energy spectrum of monolayer which means…