Eigenfunctions in a two-particle Anderson tight binding model
Victor Chulaevsky, Yuri Suhov
我们建立了一个具有短程相互作用的d维晶格上的量子两粒子系统的Anderson局域化现象，并且存在一个具有足够规则的边缘分布的IID外部势。
We establish the phenomenon of Anderson localisation for a quantum two-particle system on a d-dimensional lattice with short-range interaction and in presence of an IID external potential with sufficiently regular marginal distribution.
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This short note is a complement to our recent paper [2] where we established strong dynamical localization for a class of multi-particle Anderson models in a Euclidean space with an alloy-type random potential and a sub-exponentially decaying interaction of infinite range. We show that the localized eigenfunctions at low energies actually decay exponentially fast. This improves the results by Fauser and Warzel who established sub-exponential decay of eigenfunctions in presence of a sub-exponentially decaying interaction.
We present an eigensystem multiscale analysis for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model in an energy interval. In particular, it yields localization for the Anderson model in a nonempty interval at the bottom of the spectrum. This eigensystem multiscale analysis in an energy interval treats all energies of the finite volume operator at the same time, establishing level spacing and localization of eigenfunctions with eigenvalues in the energy interval in a fixed box with high probability. In contrast …
We consider random Schrödinger operators on $\ell^2(\mathbb{Z}^d)$ when the distribution of single site potentials is $α$-Hölder continuous ($0<α\leq 1$). In localized regime we study the distribution of eigenfunctions simultaneously in space and energy. In a certain scaling limit we prove limits point are Poisson.
We introduce a new approach for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model at high disorder. In contrast to the usual strategy, we do not study finite volume Green's functions. Instead, we perform a multiscale analysis based on finite volume eigensystems (eigenvalues and eigenfunctions). Information about eigensystems at a given scale is used to derive information about eigensystems at larger scales. This eigensystem multiscale analysis treats all energies of the finite volume operator at the same time, e…
We use a bootstrap argument to enhance the eigensystem multiscale analysis, introduced by Elgart and Klein for proving localization for the Anderson model at high disorder. The eigensystem multiscale analysis studies finite volume eigensystems, not finite volume Green's functions. It yields pure point spectrum with exponentially decaying eigenfunctions and dynamical localization. The starting hypothesis for the eigensystem bootstrap multiscale analysis only requires the verification of polynomial decay of the finite volume eigenfunctions, at some sufficiently large scale, with some minimal pro…
We extend the bootstrap multi-scale analysis developed by Germinet and Klein to the multi-particle Anderson model, obtaining Anderson localization, dynamical localization, and decay of eigenfunction correlations.
It is known that the eigenfunctions of a random Schrödinger operator on a strip decay exponentially, and that the rate of decay is not slower than prescribed by the slowest Lyapunov exponent. A variery of heuristic arguments suggest that no eigenfunction can decay faster than at this rate. We make a step towards this conjecture (in the case when the distribution of the potential is regular enough) by showing that, for each eigenfunction, the rate of exponential decay along any subsequence is strictly slower than the fastest Lyapunov exponent, and that there exists a subsequence along which it …
Non-Lyapunov annealed decay for 1d Anderson eigenfunctions
针对一维 Anderson 模型期望动力学衰减是否等于传递矩阵余循环 Lyapunov 指数的问题，本文证明在强无序条件下该关系基本不成立。
We extend the bootstrap multiscale analysis developed by Germinet and Klein to the multi-particle continuous Anderson Hamiltonian, obtaining Anderson localization with finite multiplicity of eigenvalues, decay of eigenfunction correlations, and a strong form of dynamical localization.
We present a new approach to the eigensystem multiscale analysis (EMSA) for random Schrödinger operators that relies on the Wegner estimate. The EMSA treats all energies of the finite volume operator in an energy interval at the same time, simultaneously establishing localization of all eigenfunctions with eigenvalues in the energy interval with high probability. It implies all the usual manifestations of localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization). The new method removes the restrictive level spacing hypothesis used in the previous ver…
We give a proof of the exact asymptotic behaviour in Anderson's Orthogonality Catastrophe for Dirac-$δ$ perturbations. We prove the exact asymptotics of the scalar product of the ground states of two non-interacting Fermi gases confined to a $3$-dimensional ball $B_L$ of radius $L$ in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-$δ$ perturbation. More precisely, we show the algebraic decay of the correlation determinant $\big|\det\big(\langleφ_j^L, ψ_k^L\rangle\big)_{j,k=1,...,N}\big|^2= L^{-ζ(E)+ \text{o}(1)}$, as $N,L\to\infty$ and $N/|B_L|\to~ ρ>0$,…
We establish exponential localization for a multi-particle Anderson model in a Euclidean space of an arbitrary dimension, in presence of a non-trivial short-range interaction and an alloy-type random external potential. Specifically, we prove that all eigenfunctions with eigenvalues near the lower edge of the spectrum decay exponentially.
We study the bottom of the spectrum of the Anderson Hamiltonian $\mathcal{H}_L := -\partial_x^2 + ξ$ on $[0,L]$ driven by a white noise $ξ$ and endowed with either Dirichlet or Neumann boundary conditions. We show that, as $L\rightarrow\infty$, the point process of the (appropriately shifted and rescaled) eigenvalues converges to a Poisson point process on $\mathbb{R}$ with intensity $e^x dx$, and that the (appropriately rescaled) eigenfunctions converge to Dirac masses located at independent and uniformly distributed points. Furthermore, we show that the shape of each eigenfunction, recentere…
We prove a result of delocalization for the Anderson model on the regular tree (Bethe lattice). When the disorder is weak, it is known that large parts of the spectrum are a.s. purely absolutely continuous, and that the dynamical transport is ballistic. In this work, we prove that in such AC regime, the eigenfunctions are also delocalized in space, in the sense that if we consider a sequence of regular graphs converging to the regular tree, then the eigenfunctions become asymptotically uniformly distributed. The precise result is a quantum ergodicity theorem.
We establish exponential localization for a two-particle Anderson model in a Euclidean space ${\mathbb R}^{d}$, $d\ge 1$, in presence of a non-trivial short-range interaction and a random external potential of the alloy type. Specifically, we prove that all eigenfunctions with eigenvalues near the lower edge of the spectrum decay exponentially in $L^2$-norm.
Following [7,8], we analyze regularity properties of single-site probability distributions of the random potential and of the Integrated Density of States (IDS) in the Anderson models with infinite-range interactions and arbitrary nontrivial probability distributions of the site potentials. In the present work, we study $2$-particle Anderson Hamiltonians on a lattice and prove spectral and strong dynamical localization at low energies, with exponentially decaying eigenfunctions, for a class of site potentials featuring a power-law decay.
We establish strong dynamical localization for a class of multi-particle Anderson models in a Euclidean space with an alloy-type random potential and a sub-exponentially decaying interaction of infinite range. For the first time in the mathematical literature, the uniform decay bounds on the eigenfunction correlators at low energies are proved, in the multi-particle continuous configuration space, in the norm-distance and not in the Hausdorff pseudo-metric.
We consider a one-dimensional continuum Anderson model where the potential decays in average like $|x|^{-α}$, $α>0$. We show dynamical localization for $0<α<\frac12$ and provide control on the decay of the eigenfunctions.
We establish the complete spectral exponential, and the strong Hilbert-Schmidt dynamical localization for the one-dimensional multi-particle Anderson tight-binding model and for weakly interacting particles system. In other words, we show stability of the one-dimensional localization from the single-particle to multi-particle systems with an arbitrary large but finite number of particles and for sufficient weakly interacting models. The proof uses the multi-scale analysis estimates for multi-particle systems. The common probability distribution function of the random external potential in the …
We show that the spacing between eigenvalues of the discrete 1D Hamiltonian with arbitrary potentials which are bounded, and with Dirichlet or Neumann Boundary Conditions is bounded away from zero. We prove an explicit lower bound, given by $Ce^{-bN}$, where $N$ is the lattice size, and $C$ and $b$ are some finite constants. In particular, the spectra of such Hamiltonians have no degenerate eigenvalues. As applications we show that to leading order in the coupling, the solution of a nonlinearly perturbed Anderson model in one-dimension (on the lattice) remains exponentially localized, in proba…
Lyapunov exponents in continuum Bernoulli-Anderson models
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We propose a reformulation of the bootstrap version of the Multi-Scale Analysis (BMSA), developed by Germinet and Klein, to make explicit the fact that BMSA implies asymptotically exponential decay of eigenfunctions (EFs) and of EF correlators (EFCs), in the lattice Anderson models with diagonal disorder, viz. with an IID random potential. We also show that the exponential scaling limit of EFs and EFCs holds true for a class of marginal distributions of the random potential with regularity lower than Hölder continuity of any positive order.
We consider the Anderson model with Bernoulli potential on the 3D lattice, and prove localization of eigenfunctions corresponding to eigenvalues near zero, the lower boundary of the spectrum. We follow the framework by Bourgain-Kenig and Ding-Smart, and our main contribution is a 3D discrete unique continuation, which says that any eigenfunction of the harmonic operator with bounded potential cannot be too small on a significant fractional portion of all the points. Its proof relies on geometric arguments about the 3D lattice.
Localization Lengths and Boltzmann Limit for the Anderson Model at Small Disorders in Dimension 3
我们证明了三维Anderson模型在弱无序下本征函数的局部化长度的下界。我们的结果类似于Schlag，Shubin和Wolff对一维和二维的结果。我们证明，概率为1，大多数本征函数的本地化长度有界从下面的$O(\frac{λ^{-2}}{\log\frac1λ})$，其中$λ$是无序强度。这是通过依赖于时间的方法来实现的，该方法将Erdös和Yau开发的方法推广到晶格和非高斯情况。此外，我们表明，相应的晶格随机薛定谔动力学的宏观极限是由线性玻尔兹曼方程。
Recent work [G. David, M. Filoche, and S. Mayboroda, arXiv:1909.10558[Adv. Math. (to be published)]] has proved the existence of bounds from above and below for the Integrated Density of States (IDOS) of the Schrödinger operator throughout the spectrum, called the landscape law. These bounds involve dimensional constants whose optimal values are yet to be determined. Here, we investigate the accuracy of the landscape law in 1D and 2D tight-binding Anderson models, with binary or uniform random distributions. We show, in particular, that in 1D, the IDOS can be approximated with high accuracy th…
In this paper, we establish Anderson localization for the quantum kicked rotor model. More precisely, we proved that \begin{equation*} H=\tanπ\left(x_0+my_0+\frac{m(m-1)}{2}ω\right) δ_{mn}+εS_ϕ\end{equation*} has pure point spectrum with exponentially decaying eigenfunctions for almost all $ω\in DC$ (diophantine condition).
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We propose a simplified version of the Multi-Scale Analysis of tight-binding Anderson models with strongly mixing random potentials which leads directly to uniform exponential bounds on decay of eigenfunctions in arbitrarily large finite subsets of a lattice. Naturally, these bounds imply also dynamical localization and exponential decay of eigenfunctions on the entire lattice.
The conductivity measure for the Anderson model
本文研究了Anderson紧束缚模型下的线性响应理论中的交流电导。我们定义了交流电导率，并计算了任意费米能量下零温下的线性响应电流。特别地，费米能量可能位于被认为存在扩展态的光谱区域中。
We study multi-particle interactive quantum disordered systems on a polynomially-growing countable connected graph (Z,E). The novelty is to give localization bounds uniform in finite or infinite volumes (subgraphs) in Z^N as well as for the whole of Z^N. Such bounds are proved here by means of a comprehensive fixed-energy multi-particle multi-scale analysis. Another feature of the paper is that we consider -- for the first time in the literature -- an infinite-range (although fast-decaying) interaction between particles. For the models under consideration we establish (1) exponential spectral …
We consider the discrete Schrödinger operator $H=-Δ+V$ on a cube $M\subset \mathbb{Z}^d$, with periodic or Dirichlet (simple) boundary conditions. We use a hidden landscape function $u$, defined as the solution of an inhomogeneous boundary problem with uniform right-hand side for $H$, to predict the location of the localized eigenfunctions of $H$. Explicit bounds on the exponential decay of Agmon type for low energy modes are obtained. This extends the recent work of Agmon type of localization in [5] for $\mathbb{R}^d$ to a tight-binding Hamiltonian on $\mathbb{Z}^d$ lattice. Contrary to the c…
Mobility Edge for the Anderson Model on the Bethe Lattice
本文在足够大度数的Bethe晶格上研究带无界随机势的Anderson紧束缚模型，精确刻画其谱分解，证明存在有限条迁移率边分隔纯点谱与绝对连续谱，并借助转移算子主导特征值的单调性控制非对角预解式分式矩衰减，从而证实Abou-Chacra、Thouless和Anderson的预测。
This work establishes the Anderson localization in both the spectral exponential and the strong dynamical localization for the multi-particle Anderson tight-binding model with correlated but strongly mixing random external potential. The results are obtained near the lower edge of the spectrum of the multi-particle Hamiltonian. In particular, the exponential decay of the eigenfunctions is proved in the max-norm and the dynamical localization in the Hilbert-Schmidt norm. The proofs need the conditional probability distribution function of the random external stochastic processes to obey the uni…
Mobility Edge for the Anderson Model on Random Regular Graphs
研究确定了随机正则图上安德森紧束缚模型的相图，证明在固定度数和顶点数趋于无穷时，谱由有限非局域区间和两个无限局域成分组成。
Whereas much work in the mathematical literature on quantum chaos has focused on phenomena such as quantum ergodicity and scarring, relatively little is known at the rigorous level about the existence of eigenfunctions whose morphology is more complex. Quantum systems whose dynamics is intermediate between certain regimes - for example, at the transition between Anderson localized and delocalized eigenfunctions, or in systems whose classical dynamics is intermediate between integrability and chaos - have been conjectured in the physics literature to have eigenfunctions exhibiting multifractal,…
We consider a tight-binding Schroedinger equation with time dependent diagonal noise, given as a function of a Markov process. This model was considered previously by Kang and Schenker (J. Stat. Phys., 134(5-6):1005, arXiv:0808.2784), who proved that the wave propagates diffusively. We revisit the proof of diffusion so as to obtain a uniform bound on exponential moments of the wave amplitude and a central limit theorem that implies, in particular, diffusive scaling for all position moments of the mean wave amplitude.
We study the persistence of localization for a strongly disordered tight-binding Anderson model on the lattice $\mathbb{Z}^d$, periodically driven on each site. Under two different sets of conditions, we show that Anderson localization survives if the driving frequency is higher than some threshold value that we determine. We discuss the implication of our results for recent development in condensed matter physics, we compare them with the predictions issuing from adiabatic theory, and we comment on the connexion with Mott's law, derived within the linear response formalism.
Reflection Resonances in the One-Dimensional Anderson Localization: Finite-Length Statistics, Wigner Time Delay, and Boundary Eigenfunctions
该研究分析一维安德森局域化有限长度无序样品的反射共振，关联共振密度与反射系数，推导维格纳时间延迟统计等结果，通过计算验证相关规律。
Eigenfunctions and Quantum Transport with Applications to Trimmed Schrodinger Operators
本文利用预解恒等式和Combes-Thomas估计，在解满足增长条件时证明离散薛定谔算子的动力学离域，并将其应用于Γ-修剪薛定谔算子及Anderson模型，得到同时具有局域化与离域现象的随机遍历模型。
In 1990, Klein, Lacroix, and Speis proved (spectral) Anderson localisation for the Anderson model on the strip of width $W \geqslant 1$, allowing for singular distribution of the potential. Their proof employs multi-scale analysis, in addition to arguments from the theory of random matrix products (the case of regular distributions was handled earlier in the works of Goldsheid and Lacroix by other means). We give a proof of their result avoiding multi-scale analysis, and also extend it to the general quasi-one-dimensional model, allowing, in particular, random hopping. Furthermore, we prove a …
We consider the multi-particle Anderson tight-binding model and prove that its lower spectral edge is non-random under some mild assumptions on the inter-particle interaction and the random external potential. We also adapt to the low energy regime the multi-particle multi-scale analysis initially developed by Chulaevsky and Suhov in the high disorder limit, if the marginal probability distribution of the i.i.d. random variables is log-Hölder continuous and obtain the spectral exponential and strong dynamical localization near the bottom of the spectrum.
We prove a quantum-ergodicity theorem on large graphs, for eigenfunctions of Schrödinger operators in a very general setting. We consider a sequence of finite graphs endowed with discrete Schrödinger operators, assumed to have a local weak limit. We assume that our graphs have few short loops, in other words that the limit model is a random rooted tree endowed with a random discrete Schrödinger operator. We show that absolutely continuous spectrum for the infinite model, reinforced by a good control of the moments of the Green function, imply "quantum ergodicity", a form of spatial delocalizat…
Self-consistent equations and quantum diffusion for the Anderson model
本文针对低无序高斯噪声下的Anderson紧束缚模型，通过分析预解式的自洽方程并建立新的算子范数估计，在迄今最长时间尺度上证明了量子扩散，并改进了本征函数局域化长度的下界，首次在二维情形取得量子扩散结果。
We prove Anderson localization for all Diophantine frequencies and all non-resonant phases for a model that arises from 2D quantum walk model subject to an external magnetic field. This is the first localization result for all Diophantine frequencies in the magnetic quantum walk and the quasi-periodic CMV setting. We also obtain sharp asymptotics of the localized eigenfunctions.
We consider the distribution function $P(|ψ|^{2})$ of the eigenfunction amplitude at the center-of-band (E=0) anomaly in the one-dimensional tight-binding chain with weak uncorrelated on-site disorder (the one-dimensional Anderson model). The special emphasis is on the probability of the anomalously localized states (ALS) with $|ψ|^{2}$ much larger than the inverse typical localization length $\ell_{0}$. Using the solution to the generating function $Φ_{an}(u,ϕ)$ found recently in our works we find the ALS probability distribution $P(|ψ|^{2})$ at $|ψ|^{2}\ell_{0} >> 1$. As an auxiliary prelimi…
We study many-body properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective one-particle Hamiltonian. We show how state-of-the-art techniques for proving Anderson localization can be used to prove that these properties hold in a number of standard models. We also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one-particle eigenfunction c…
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