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Eigenfunction avoiding for the PLBRM with b=0.42(stars), 3D Anderson insulator (red triangles) and metal (blue triangles). The dotted line corresponds to the limit of uncorrelated eigenfunctions; the solid line corresponds to the power law 1/ω 2. Points below the dotted line correspond to eigenfunction avoiding.
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We consider the correlation of two single-particle probability densities $|\Psi_{E}({\bf r})|^{2}$ at coinciding points ${\bf r}$ as a function of the energy separation $\omega=|E-E'|$ for disordered tight-binding lattice models (the Anderson models) and certain random matrix ensembles. We focus on the models in the parameter range where they are c...
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... of both numerical and analytical calculation presented in Fig.3 reveal another unexpected feature of eigenfunction correlation which appears to be common to all ADRM. Surprisingly it is also present for the 3D Anderson model both in the metal and in the insula- tor phase (see Fig.6). This is the negative eigenfunc- ton correlations for ω = |E − E ′ | > E 0 ∼ b. ...
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... is the negative eigenfunc- ton correlations for ω = |E − E ′ | > E 0 ∼ b. In- deed, one can see from Fig.3 and Fig.6 that for large enough ω the correlation function C(ω) goes below the uncorrelated limit C(ω) = 1/N which corresponds to ...
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... the picture with a stratified coordinate space is possible for PLBRM Eq.(3) with small enough b < 1 when the single shell bandwidth E 0 ∼ b is small com- pared to the total bandwidth ∼ 1. Amazingly, the 3D Anderson model which low-frequency critical features are well described by the critical PLBRM with b ≈ 0.42, also follows the predictions of the critical PLBRM for high frequencies ω > E 0 . This is a consequence of a rela- tively large value W c = 16.5 of the critical disorder which results in E 0 considerably smaller than the conduction bandwidth D. In particular its coordinate space must be stratified to explain the observed (see Fig.6) mutual avoiding of eigenstates. ...
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Citations
... This implies a close relationship with the critical regime of the Anderson metal-insulator localization transition [9]. Indeed, vacancies in graphene, irrespective of the disorder concentration, can be expected to behave as a multifractal insulator [48], with enhanced correlations between states with small energy separation. This is consistent with the enhancement of Ω α at small energies in Fig. 2(b). ...
Random vacancies in a graphene monolayer induce defect states that are known to form a narrow impurity band centered around zero energy at half-filling. We use a space-resolved formulation of the quantum metric and establish a strong enhancement of the electronic correlations in this impurity band. The enhancement is primarily due to strong correlations between pairs of vacancies situated on different sublattices at anomalously large spatial distances. We trace the strong enhancement to both the multifractal vacancy wave functions, which ties the system exactly at the Anderson insulator transition for all defect concentrations, and preserving the chiral symmetry.
... This universal scaling is limited to systems with a shortrange hopping, while the hopping decreasing with the distance as V (r) ∝ r −d or slower leads to delocalization of all states [1,[10][11][12][13][14] except for the marginal case of diverging Fourier transform of a hopping amplitude [15][16][17][18][19][20][21][22][23][24][25][26]. If * aburin@tulane.edu ...
Although it is recognized that Anderson localization takes place for all states at a dimension d less than or equal to 2, while delocalization is expected for hopping V (r) decreasing with the distance slower or as r −d , the localization problem in the crossover regime for the dimension d = 2 and hopping V (r) ∝ r −2 is not resolved yet. Following earlier suggestions we show that for the hopping determined by two-dimensional anisotropic dipole-dipole interactions in the presence of time-reversal symmetry there exist two distinguishable phases at weak and strong disorder. The first phase is characterized by ergodic dynamics and superdiffusive transport, while the second phase is characterized by diffusive transport and delocalized eigenstates with fractal dimension less than 2. The transition between phases is resolved analytically using the extension of scaling theory of localization and verified numerically using an exact numerical diagonalization.
... In this example, we take the half-filled Aubry-André model with τ = 1/ √ 2, that has z = 1.575, and set w = 3, finding that DV (w = 3) = 2z − 1. The relation between the energy, α , and the interaction strength,V 0α0α , follows a generalized Chalker scaling [70][71][72][73][74]. However, a significant difference to previous Chalker scaling analyses is the full antisymmetrization of the interaction term that follows from fermionic statistics. ...
... We show that generic short-range (and some long-range) interactions are irrelevant at the critical point of the Aubry-André model in the U → 0 limit, by unveiling the existence of a generalized Chalker scaling [70][71][72][73][74] at this point. All the results that we present in this section are for the parameters studied in the main text, namely τ = 1/ √ 2 and at half-filling, with N p = N/2 particles. ...
... This shows that there is a generalized Chalker scaling [70][71][72][73][74] at the critical point of the Aubry-André model, manifested by power-law correlations (on average) between the single-particle eigenfunctions with respect to their energy difference. ...
We show that short-range interactions are irrelevant around gapless ground-state delocalization-localization transitions driven by quasiperiodicity in interacting fermionic chains. In the presence of interactions, these transitions separate Luttinger liquid and Anderson glass phases. Remarkably, close to criticality, we find that excitations become effectively noninteracting. By formulating a many-body generalization of a recently developed method to obtain single-particle localization phase diagrams, we carry out precise calculations of critical points between Luttinger liquid and Anderson glass phases and find that the correlation length critical exponent takes the value ν=1.001±0.007, compatible with ν=1 known exactly at the noninteracting critical point. We also show that other critical exponents, such as the dynamical exponent z and a many-body analog of the fractal dimension are compatible with the exponents obtained at the noninteracting critical point. Noteworthy, we find that the transitions are accompanied by the emergence of a many-body generalization of previously found single-particle hidden dualities. Finally, we show that in the limit of vanishing interaction strength, all finite-range interactions are irrelevant at the noninteracting critical point.
... On the other hand, the field theoretical approach with the nonlinear sigma model has been developed, and interactions in a system have been proven to provide vital roles for the MIT [1][2][3][12][13][14]. To understand the criticality of the MIT, two-point correlations and multifractal analysis of the local density of states (LDOS) have been proposed to characterize the quantum critical phenomena [1][2][3][15][16][17]. The distribution of LDOS or the amplitude of wave functions at an energy level is closely related to the order parameter function of the Anderson transitions [3,18]. ...
Our previous study observed the localization-delocalization transition and critical quantum fluctuations of the local density of states (LDOS) on the structurally disordered two-dimensional (2D) semiconductor MoS2. This transition corresponds to the metal-insulator transition (MIT) reported in transport measurements. The structural disorder in MoS2 caused curvature-induced band gap fluctuations, leading to charge localization and unusual band edge flattening through doping. The critical behavior for the MIT was analyzed using autocorrelation and multifractality of LDOS mapping results. However, the effect of structural disorder on critical points has not been fully explored. Here, we systematically investigated the impact of structural disorder on band tail formation and critical doping concentration by examining the radial-averaged autocorrelation and multifractality of LDOS in 2D semiconductors. Our finding indicates that the radial-averaged autocorrelation and multifractality of LDOS characterize the band tail ranges and band edge flattening in disordered 2D semiconductors. Decaying regions in the radial-averaged autocorrelation profile and first-order derivative of singularity peak positions determine band tail ranges. Increased structural disorder led to larger band tail widths near valence and conduction band edges, while the doping-induced band edge flattening altered band tail widths for each valence and conduction band. As the band edge is flattened due to doping, the LDOS map near the critical energy becomes uniform, exhibiting a divergence in the localization length. The average value of conduction and valence band tail widths remained almost constant regardless of doping control, serving as a representative value for the degree of structural disorder. For the MIT, we found that the critical doping concentration depends on the degree of structural disorder in 2D semiconductors. Our findings provide valuable insights into the fundamental physics of structurally disordered 2D semiconductors in relevance to quantum phase transitions, which could have important implications for designing and optimizing electronic/optoelectronic devices based on 2D materials.
... where L d is the system volume, ψ n (r) and ϵ n are the eigenstate and eigenenergy of the Hamiltonian (1), respectively. This function demonstrates power-law decay at the Anderson transition [14,32], ...
We study the interplay between quasi-periodic disorder and superconductivity in a 1D tight-binding model with the quasi-periodic modulation of on-site energies that follow the Fibonacci rule and all the eigenstates are multifractal. As a signature of multifractality, we observe the power-law dependence of the correlation between different single-particle eigenstates as a function of their energy difference. By computing numerically the superconducting transition temperature, we find the distribution of critical temperatures, analyze their statistics and estimate the mean value and variance of critical temperatures for various regimes of the attractive coupling strength and quasi-periodic disorder. We find an enhancement of the critical temperature as compared to the analytical results that are based on strong assumptions of absence of correlations and self-averaging of multiple characteristics of the system, which are not justified for the Fibonacci chain. For the very weak coupling regime, we observe a crossover where the self-averaging of the critical temperature breaks down completely and a strong sample-to-sample fluctuations emerge.
... often called the Thouless energy [49,50], within which GOE-like spectral correlations (and in particular level repulsion) have been established. All the moments of the wave-functions' coefficients (the so-called generalized inverse participation ratios, IPR) behave as ...
The generalized Rosenzweig-Porter model with real (GOE) off-diagonal entries arguably constitutes the simplest random matrix ensemble displaying a phase with fractal eigenstates, which we characterize here by using replica methods. We first derive analytical expressions for the average spectral density in the limit in which the size N N of the matrix is large but finite. We then focus on the number of eigenvalues in a finite interval and compute its cumulant generating function as well as the level compressibility, i.e., the ratio of the first two cumulants: these are useful tools to describe the local level statistics. In particular, the level compressibility is shown to be described by a universal scaling function, which we compute explicitly, when the system is probed over scales of the order of the Thouless energy. Interestingly, the same scaling function is found to describe the level compressibility of the complex (GUE) Rosenzweig-Porter model in this regime. We confirm our results with numerical tests.
... Following the pioneering works of the "Anderson transition", the localization-delocalization transition (or MIT) in disordered systems has emerged as a continuous quantum phase transition accompanied by global symmetry and dimensionality 1,5,6,16 . Since the discovery of MIT in the two-dimensional (2D) electron gas in semiconductor devices, the scaling theory and renormalization group approaches based on the non-linear sigma model have been established to explain MIT in disordered 2D systems with interacting pictures [1][2][3][4][5]7,8,[12][13][14]17,[22][23][24] . This is in contrast to the prior prediction of the scaling theory, which anticipates that all states of 2D non-interacting systems are weakly localized and no true metallic states at 0 K exist 17 . ...
... In the randomly corrugated MoS 2 on SiO 2 , the localization of the doping charge was confirmed by a spatial flattening of band edge and a modulation of the local tunneling barrier height (TBH). The direct evidence of the localization-delocalization transition in the structurally disordered monolayer was provided by the autocorrelation (localization length), distribution, and multifractality of wavefunctions near or at the critical point, showing the power law, the change from log-normal (insulating) to normal (metallic) distributions, and the singularity spectra towards the metallic limit of normalized STS mapping results, respectively [1][2][3][4][33][34][35] . In addition, STS and theoretical calculation results confirm that the structural disorder in TMdCs is the origin of the band tail. ...
... In Fig. 4c, the criticality is observed at the critical energy which is determined by autocorrelation as shown in Fig. 4e (see below). The strong fluctuations of LDOS at the critical energy were exhibited over the entire area, which is predicted by theoretical results [1][2][3][4]33 . The extended states above the critical energy (Fig. 4d) show a uniform intensity regardless of the structural disorder, as a result of the band edge flattening by the doping charge. ...
Quantum fluctuations of wavefunctions in disorder-driven quantum phase transitions (QPT) exhibit criticality, as evidenced by their multifractality and power law behavior. However, understanding the metal-insulator transition (MIT) as a continuous QPT in a disordered system has been challenging due to fundamental issues such as the lack of an apparent order parameter and its dynamical nature. Here, we elucidate the universal mechanism underlying the structural-disorder-driven MIT in 2D semiconductors through autocorrelation and multifractality of quantum fluctuations. The structural disorder causes curvature-induced band gap fluctuations, leading to charge localization and formation of band tails near band edges. As doping level increases, the localization-delocalization transition occurs when states above a critical energy become uniform due to unusual band bending by localized charge. Furthermore, curvature induces local variations in spin-orbit interactions, resulting in non-uniform ferromagnetic domains. Our findings demonstrate that the structural disorder in 2D materials is essential to understanding the intricate phenomena associated with localization-delocalization transition, charge percolation, and spin glass with both topological and magnetic disorders.
... In this example, we take the half-filled Aubry-André model with τ = 1/ √ 2, that has z = 1.575, and set w = 3, finding that DV (w = 3) = 2z−1. The relation between the energy, α , and the interaction strength,V 0α0α , follows a generalized Chalker scaling [75][76][77][78][79]. However, a significant difference to previous Chalker scaling analyses is the full antisymmetrization of the interaction term that follows from fermionic statistics. ...
... We show that generic short-range (and some long-range) interactions are irrelevant at the critical point of the Aubry-André model in the U → 0 limit, by unveiling the existence of a generalized Chalker scaling [75][76][77][78][79] at this point. All the results that we present in this section are for the parameters studied in the main text, namely τ = 1/ √ 2 and at half-filling, with N p = N/2 particles. ...
... We will discuss this dependence below in more detail below. At this point we also note that when averaged over minibands,V 0α0α ∼ (| (m) α |) µ , where µ > 0. This shows that there is a generalized Chalker scaling [75][76][77][78][79] at the critical point of the Aubry-André model, manifested by power-law correlations (on average) between the single-particle eigenfunctions with respect to their energy difference. ...
We show that short-range interactions are irrelevant around gapless ground-state delocalization-localization transitions driven by quasiperiodicity in interacting fermionic chains. In the presence of interactions, these transitions separate Luttinger Liquid and Anderson glass phases. Remarkably, close to criticality, we find that excitations become effectively non-interacting. By formulating a many-body generalization of a recently developed method to obtain single-particle localization phase diagrams, we carry out precise calculations of critical points between Luttinger Liquid and Anderson glass phases and find that the correlation length critical exponent takes the value $\nu = 1.001 \pm 0.007$, compatible with $\nu=1$ known exactly at the non-interacting critical point. We also show that other critical exponents, such as the dynamical exponent $z$ and a many-body analog of the fractal dimension are compatible with the exponents obtained at the non-interacting critical point. Noteworthy, we find that the transitions are accompanied by the emergence of a many-body generalization of previously found single-particle hidden dualities. Finally, we show that in the limit of vanishing interaction strength, all finite range interactions are irrelevant at the non-interacting critical point.
... On the other hand, the states of diffusive metals present Wigner-Dyson level statistics which correspond to the fluctuations of critical quantum spectra with irregular behavior and reveal a Gaussian distribution [63]. Near a mobility edge, where the localization-delocalization transition occurs, extended states record quantum critical scaling in the overlap of wave function probabilities at various energies [64,65]. ...
... A valuable probe of the level statistics and of the statistics of wave-functions' amplitudes is provided by the spectral correlation function K 2 (ω; E ) between eigenstates at different energy, but on the same site, which allows one to distinguish between ergodic, localized, and multifractal states [40][41][42][43][44]: ...
We consider critical eigenstates in a two-dimensional quasicrystal and their evolution as a function of disorder. By exact diagonalization of finite-size systems we show that the evolution of properties of a typical wave function is nonmonotonic. That is, disorder leads to states delocalizing, until a certain crossover disorder strength is attained, after which they start to localize. Although this nonmonotonic behavior is only present in finite-size systems and vanishes in the thermodynamic limit, the crossover disorder strength decreases logarithmically slowly with system size, and is quite large even for very large approximants. The nonmonotonic evolution of spatial properties of eigenstates can be observed in the anomalous dimensions of the wave-function amplitudes, in their multifractal spectra, and in their dynamical properties. We compute the two-point correlation functions of wave-function amplitudes and show that these follow power laws in distance and energy, consistent with the idea that wave functions retain their multifractal structure on a scale which depends on disorder strength. Dynamical properties are studied as a function of disorder. We find that the diffusion exponents do not reflect the nonmonotonic wave-function evolution. Instead, they are essentially independent of disorder until disorder increases beyond the crossover value, after which they decrease rapidly, until the strong localization regime is reached. The differences between our results and earlier studies on geometrically disordered “phason-flip” models lead us to propose that the two models are in different universality classes. We conclude by discussing some implications of our results for transport and a proposal for a Mott hopping mechanism between power-law localized wave functions, in moderately disordered quasicrystals.
