Density of States and Lifshitz Tails for Discrete 1D Random Dirac Operators

IF 0.9 3区 数学 Q3 MATHEMATICS, APPLIED Mathematical Physics, Analysis and Geometry Pub Date : 2021-09-14 DOI:10.1007/s11040-021-09403-4
Roberto A. Prado, César R. de Oliveira, Edmundo C. de Oliveira
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引用次数: 4

Abstract

We study the density of states and Lifshitz tails for a family of random Dirac operators on the one-dimensional lattice \(\mathbb {Z}\). These operators consist of the sum of a discrete free Dirac operator with a random potential. The potential is a diagonal matrix formed by two different scalar potentials, which are sequences of independent and identically distributed random variables according to a Borel probability measure of compact support in \(\mathbb {R}\). The existence of the density of state measure for these Dirac operators is obtained through two approaches by finite-volume quantities. By using one of these approaches, we show that the distribution function of the density of states decays exponentially for energies near the spectral band edges, i.e., we establish Lifshitz tails for these operators. Lifshitz tails are established first for Dirac operators restricted to appropriate subspaces of energies and, using this, extended to the full operators, including the occurrence of internal tails in the case of spectral gap.

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离散1D随机Dirac算子的态密度和Lifshitz尾
我们研究了一维晶格\(\mathbb {Z}\)上随机狄拉克算子族的态密度和Lifshitz尾。这些算子由具有随机势的离散自由狄拉克算子的和组成。势是由两个不同的标量势组成的对角矩阵,这两个标量势是根据\(\mathbb {R}\)中紧支持的Borel概率度量的独立的、同分布的随机变量的序列。通过两种有限体积量的方法,得到了这些狄拉克算子的态密度测度的存在性。通过使用其中一种方法,我们证明了态密度分布函数在谱带边缘附近的能量呈指数衰减,即我们建立了这些算符的Lifshitz尾。首先为限制在适当能量子空间的狄拉克算符建立Lifshitz尾,并以此推广到全算符,包括谱隙情况下的内尾的出现。
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来源期刊
Mathematical Physics, Analysis and Geometry
Mathematical Physics, Analysis and Geometry 数学-物理:数学物理
CiteScore
2.10
自引率
0.00%
发文量
26
审稿时长
>12 weeks
期刊介绍: MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.
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