Metal Binding Site Prediction.

  * •

Post-Translational Modification Site Prediction.

Graph-level Tasks

  * •

Fold Prediction.

  * •

Gene Ontology Prediction.

  * •

Reaction Class Prediction.

  * •

Antibody Developability Prediction.

###  D.9 SE(3) Equivariant Noise Predictor

Similar to Zhang et al. (2023c), for structure-based denoising tasks we use an
SE(3) equivariant noise predictor network to predict per-residue perturbations
from SE(3) invariant scalar embeddings and corrupted atomic coordinates
$\tilde{\bm{\vec{X}}}$. Each edge ${\bm{e}}_{ij}$ is featurised by
concatenating the two adjoining scalar node representations
$\tilde{{\bm{s}}_{i}},\tilde{{\bm{s}}_{j}}$ and the euclidean distance between
them $\|\tilde{{\bm{x}}_{i}}-\tilde{{\bm{x}}_{j}}\|_{2}$. We use a two-layer
MLP to produce a score ${\bm{m}}_{ij}$:

${\bm{m}}_{ij}=\mathrm{MLP}(\tilde{{\bm{s}}_{i}},\tilde{{\bm{s}}_{j}},\mathrm{MLP}(\|\tilde{{\bm{x}}_{i}}-\tilde{{\bm{x}}_{j}}\|_{2})).$
(13)

Subsequently, Equation 13 is used to aggregate normalised directional edge
vectors over the neighbourhood $\mathcal{N}_{i}$ of each node:

$\bm{\epsilon}_{\theta}(\tilde{\mathcal{G}})=\sum_{j\in\mathcal{N}_{i}}{\bm{m}}_{ij}\cdot\frac{\tilde{{\bm{x}}_{i}}-\tilde{{\bm{x}}_{j}}}{\|\tilde{{\bm{x}}_{i}}-\tilde{{\bm{x}}_{j}}\|_{2}}$
(14)

###  D.10 Hyperparameter Selection

Given the large number of models and featurisation schemes, we try our best to
do a consistent and fair hyperparameter search. We fix a consistently high
number of layers and large hidden dimension across models, as we wanted to
focus on scaling model size and dataset size via pre-training. We use the Fold
Classification task to select the best learning rate and dropout per model and
featurisation scheme for downstream tasks. While our best performing models
sometimes do not outperform the best reported results in the literature, we
have obtained these results in a consistent experimental setup. Our goal was
to demonstrate the utility of our benchmarking framework and uncover the
impact of architectural considerations such as featurisation schemes,
geometric GNN models, and pre-training/auxiliary tasks under fair and rigorous
experimental settings.

##### Protein Structure Encoders

  1. 1.

We use a consistent batch size of 32.

  2. 2.

For all models, we try to consistently use six layers, each with 512 hidden
channels. For tensor-based equivariant GNNs (TFN, MACE), we reduced the number
of layers and hidden channels to fit 80GB of GPU memory on one NVIDIA A100
GPU.

  3. 3.

For each encoder and featurisation baseline, we search over learning rates:
$0.00001,0.0001,0.0003,0.001$ and select the best based on the validation
performance on the fold classification task.

  4. 4.

For each finetuning configuration, we conduct a small sweep over learning
rates: $0.0001,0.0003,0.00001$ and perform model selection based on the
validation performance.

  5. 5.

We train all models to convergence or to a maximum of 24 hours on a single
A100 GPU.

##### Output Heads

  1. 1.

All primary output heads use a three-layer MLP with 512 as the hidden
dimension.

  2. 2.

For all auxiliary tasks we use a two-layer MLP with 128 as the hidden
dimension.

  3. 3.

For all structure denoising tasks we use a two-layer SE(3) equivariant noise
predictor network (Section D.9). The message and distance MLPs each consist of
two layers of 128 hidden units.

  4. 4.

For each encoder and featurisation, we search over decoder dropout:
$0.0,0.1,0.3,0.5$, and select the best based on the validation performance on
the Fold Classification task.

##  Appendix E Documentation for Datasets

Table 6: Overview of supervised tasks and datasets.

Task Dataset Origin Structures # Train # Validation # Test Node-level Inverse
Folding Ingraham et al. (Ingraham et al., 2019) Experimental 3.9 M 105 K 180 K
PPI Site Prediction Gainza et al. (2020) Experimental 478 K 53 K 117 K Metal
Binding Site Prediction Experimental 1.1 M 13.7 K 29.8 K Post-Trans.
Modification Site Prediction Yan et al. (2023) Predicted 44 K 2.4 K 2.5 K
Graph-level Fold Prediction Hou et al. (Hou et al., 2017) Experimental 12.3 K
0.7 K 1.3/0.7/1.3 K Gene Ontology Prediction Gligorijević et al. (2021)
Experimental 27.5 K 3.1 K 3.0 K Reaction Class Prediction Hermosilla et al.
(2020) Experimental 29.2 K 2.6 K 5.6 K Antibody Developability Prediction
Huang et al. (Huang et al., 2021) Experimental 1.7 K 0.24 K 0.48 K

Below, we provide detailed documentation for each dataset included in our
benchmark, summarised in Table 6. Each dataset will be made available for
download in processed and raw forms from Zenodo upon publication. Note that,
for all datasets, we authors bear all responsibility in case of any violation
of rights regarding the usage of such datasets, whether they were compiled
from existing sources or curated from scratch.

###  E.1 CATH - Inverse Folding

This is a common protein engineering task where the goal is to recover an
amino acid sequence given a structure up to backbone completeness. Formally,
this is a node-level classification task where the model learns a mapping for
each residue $r_{i}$ to an amino acid type $y\in\\{1,\dots,n\\}$, where $n$ is
the vocabulary size ($n=20$ for the canonical set of amino acids).

Note that inverse folding is a generic task that can be applied to any
dataset. In the literature, it is commonly evaluated on the CATH dataset
compiled by Ingraham et al. (2019). We additionally use inverse folding on
AlphaFoldDB as a pretraining task.

  * •

Motivation Several generative methods for protein design produce backbone
structures that require the design of an associated sequence. As a result,
inverse folding is an important part of _de novo_ design pipelines for
proteins.

  * •

Collection For this dataset, we adopt the commonly-used CATH dataset
originally compiled by Ingraham et al. (2019).

  * •

Composition The dataset consists of protein structures randomly split into
training, validation and test sets such that proteins in different sets do not
share the same CATH topology classification (i.e., CAT code).

  * •

Hosting A preprocessed version of the dataset can be downloaded from the
benchmark’s Zenodo data record at
https://zenodo.org/record/8282470/files/cath.tar.gz.

  * •

Licensing We have released a preprocessed version of the dataset under a
Creative Commons Attribution 4.0 International license. The original dataset
is available under a Creative Commons Attribution 4.0 International license at
http://cathdb.info.

  * •

Maintenance We will announce any errata discovered in or changes made to the
dataset using the benchmark’s GitHub repository at
https://github.com/a-r-j/ProteinWorkshop.

  * •

Uses This dataset can be used for multilabel node classification tasks where a
model learns a mapping for each residue $r_{i}$ to an amino acid type
$y\in\\{1,\dots,n\\}$, where $n$ is the vocabulary size (e.g., $n=20$ for the
canonical set of amino acids).

  * •

Metric Perplexity.

###  E.2 MaSIF-Site - PPI Site Prediction

This task is a node-level binary classification task where the goal is to
predict whether or not a residue is involved in a protein-protein interaction
interface.

  * •

Motivation Identifying protein-protein interaction (PPI) sites has important
applications in developing improved protein-protein interaction networks and
docking tools, providing biological context to guide protein engineering and
target identification in drug discovery campaigns (Jamasb et al., 2021).

  * •

Collection We adopt the dataset of experimental structures curated from the
PDB by Gainza et al. (2020) and retain the original splits, though we modify
the labelling scheme to be based on inter-atomic proximity (3.5 Å), which can
be user-defined, rather than solvent exclusion.

  * •

Composition The dataset is curated from the PDB by preprocessing such as the
presence of one of the seven specified ligands (e.g., ADP or FAD), clustering
based on 30% sequence identity and random subsampling. It contains 1,459
structures, which are randomly assigned to training (72%), validation (8%) and
test set (20%). 12 (Å) radius patches were extracted from the generated
structures and a patch labelled as part of a binding pocket if its centre
point was < 3 (Å) away from an atom of the corresponding ligand.

  * •

Hosting The original dataset is made available by the authors on Zenodo at
https://zenodo.org/record/2625420.

  * •

Licensing We have released a preprocessed version of the dataset under a
Creative Commons Attribution 4.0 International license. The original dataset
is available under an Apache 2.0 license at https://github.com/LPDI-
EPFL/masif/blob/master/LICENSE.

  * •

Maintenance We will announce any errata discovered in or changes made to the
dataset using the benchmark’s GitHub repository at
https://github.com/a-r-j/ProteinWorkshop.

  * •

Uses This dataset can be used for binary node classification tasks where the
goal is to predict whether or not a residue is involved in a protein-protein
interaction interface.

  * •

Metric AUPRC.

###  E.3 ccPDB - Metal Binding Site Prediction

This is a binary node classification task where each residue is mapped to a
label $y\in\\{0,1\\}$ indicating whether the residue (or its constituent
atoms) is within 3.5 (Å) of a user-defined metal ion or ligand heteroatom,
respectively.

  * •

Motivation Several proteins coordinate transition metal ions to carry out
their functions. As such, predicting the binding sites of metal ions can
elucidate the role of metal binding on protein function.

  * •

Collection The dataset is constructed from experimental structures curated
from the PDB, where binding site assignments for each residue are computed on-
the-fly. While the benchmark supports this task on arbitrary subsets of the
PDB and ligands, we provide the Zinc-binding dataset from Dürr et al. (2023)
specifically for this task.

  * •

Composition The dataset is constructed by sequence-based clustering of the PDB
at 30% sequence identity to remove sequence and structural redundancy.
Clusters with a member shorter than 3000 residues, containing at least one
zinc atom with resolution better than 2.5 (Å) determined by x-ray
crystallography and not containing nucleic acids are used to compose the
dataset. If multiple structures fulfill these criteria, the highest resolution
structure is used. The train (2,085) / validation (26) / test (59) splits are
constructed such that proteins in the validation and test sets have no partial
overlap with any protein in the training data.

  * •

Hosting A preprocessed version of the dataset can be downloaded from the
benchmark’s Zenodo data record at
https://zenodo.org/record/8282470/files/Metal3D.tar.gz.

  * •

Licensing We have released a preprocessed version of the dataset under a
Creative Commons Attribution 4.0 International license. The original dataset
is freely available without a license at
https://academic.oup.com/database/article/doi/10.1093/database/bay142/5298333#130010908.

  * •

Maintenance We will announce any errata discovered in or changes made to the
dataset using the benchmark’s GitHub repository at
https://github.com/a-r-j/ProteinWorkshop.

  * •

Uses This dataset can be used for binary node classification tasks where each
residue is mapped to a label $y\in\\{0,1\\}$ indicating whether the residue
(or its constituent atoms) is within 3.5 (Å) of a user-defined metal ion or
ligand heteroatom, respectively.

  * •

Metric Accuracy.

###  E.4 PTM - Post-Translational Modification Site Prediction

We frame prediction of post-translational modification (PTM) sites as a
multilabel classification task where each residue is mapped to a label
$y\in\\{1,\dots,13\\}$ distinguishing between modifications on different amino
acids (e.g. phosphorylation on S/T/Y and N-linked glycosylation on N).

  * •

Motivation Identifying the exact sites where post-translational modifications
(PTMs) occur is essential for understanding protein behaviour and designing
targeted therapeutic interventions.

  * •

Collection We adopt a dataset of 48,811 AlphaFold2-predicted structures
curated by Yan et al. (2023), where each structure contains the PTM metadata
necessary to construct residue-wise site prediction labels.

  * •

Composition The dataset is split into training (43,907), validation (2,393)
and test (2,511) sets based on 50% sequence identity and 80% coverage. In
total, there are 240,090 PTMs present in the dataset compared to 3,391,208
residues where PTMs could be possible but are not present. The most common
PTMs are phosphorylations on serine (93,734) and N-linked glycosylation at
asparagine (59,143) which together account for around 70% of the PTMs.

  * •

Hosting A preprocessed version of the dataset can be downloaded from the
benchmark’s Zenodo data record at
https://zenodo.org/record/8282470/files/PostTranslationalModification.tar.gz.

  * •

Licensing We have released a preprocessed version of the dataset under a
Creative Commons Attribution 4.0 International license. The original dataset
is available under a Creative Commons Attribution 4.0 International license at
https://zenodo.org/record/7655709.

  * •

Maintenance We will announce any errata discovered in or changes made to the
dataset using the benchmark’s GitHub repository at
https://github.com/a-r-j/ProteinWorkshop.

  * •

Uses This dataset can be used for multilabel node classification tasks where
each residue is mapped to a label $y\in\\{1,\dots,13\\}$ distinguishing
between modifications on different amino acids (e.g., phosphorylation on S/T/Y
and N-linked glycosylation on N).

  * •

Metric ROC-AUC.

###  E.5 FOLD - Fold Prediction

This is a multiclass graph classification task where each protein,
$\mathcal{G}$, is mapped to a label $y\in\\{1,\dots,1195\\}$ denoting the fold
class.

  * •

Motivation The utility of fold prediction is that it serves as a litmus test
for the ability of a model to distinguish different structural folds. It
stands to reason that models that perform poorly on distinguishing fold
classes likely learn limited or low-quality structural representations.

  * •

Collection We adopt the fold classification dataset originally curated from
SCOP 1.75 by (Hou et al., 2017).

  * •

Composition This dataset provides three different test sets stratified based
on topological similarity: Fold, in which proteins originating from the same
superfamily are absent during training; Superfamily, in which proteins
originating from the same family are absent during training; and Family, in
which proteins from the same family are present during training.

  * •

Hosting A preprocessed version of the dataset can be downloaded from the
benchmark’s Zenodo data record at
https://zenodo.org/record/8282470/files/FoldClassification.tar.gz.

  * •

Licensing We have released a preprocessed version of the dataset under a
Creative Commons Attribution 4.0 International license. The original dataset
is available under a Creative Commons Attribution 4.0 International license at
https://academic.oup.com/bioinformatics/article/34/8/1295/4708302.

  * •

Maintenance We will announce any errata discovered in or changes made to the
dataset using the benchmark’s GitHub repository at
https://github.com/a-r-j/ProteinWorkshop.

  * •

Uses This dataset can be used for multilabel graph classification tasks where
each protein, $\mathcal{G}$, is mapped to a label $y\in\\{1,\dots,1195\\}$
denoting the fold class.

  * •

Metric Micro-averaged accuracy.

###  E.6 GO - Gene Ontology Prediction

This is a multilabel classification task, assigning functional Gene Ontology
(GO) annotation to structures. GO annotations are assigned within three
ontologies: biological process (BP), cellular component (CC) and molecular
function (MF).

  * •

Motivation Predicting protein function in the form of functional annotations
such as gene ontology (GO) terms has important applications in protein
analysis and engineering, providing researchers with the ability to cluster
functionally-related structures or to guide protein generation methods to
design new proteins with desired functional properties.

  * •

Collection We adopt the dataset of experimental structures originally curated
from the PDB by Gligorijević et al. (2021).

  * •

Composition We retain the original multi-cutoff based dataset splits proposed
by (Gligorijević et al., 2021), with cutoff at 30% sequence similarity.

  * •

Hosting A preprocessed version of the dataset can be downloaded from the
benchmark’s Zenodo data record at
https://zenodo.org/record/8282470/files/GeneOntology.tar.gz.

  * •

Licensing We have released a preprocessed version of the dataset under a
Creative Commons Attribution 4.0 International license. The original dataset
is available under a Creative Commons Attribution 4.0 International license at
https://www.nature.com/articles/s41467-021-23303-9.

  * •

Maintenance We will announce any errata discovered in or changes made to the
dataset using the benchmark’s GitHub repository at
https://github.com/a-r-j/ProteinWorkshop.

  * •

Uses This dataset can be used for multilabel graph classification tasks,
assigning a functional Gene Ontology (GO) annotation to protein structures.

  * •

Metric $\mathrm{F}_{\mathrm{max}}$ score.

###  E.7 EC Reaction - Reaction Class Prediction

This is a multiclass graph classification task where each protein,
$\mathcal{G}$, is mapped to a label $y\in{\\{1,...,384\\}}$ denoting which
class of reactions a given protein catalyzes; all four levels of the EC
assignment are employed to define the reaction class label.

  * •

Motivation As proteins’ reaction classifications are based on their enzyme-
catalyzed reaction according to all four levels of the standard Enzyme
Commission (EC) number, methods that predict such classifications can help
elucidate the function of newly-designed proteins as they are developed.

  * •

Collection We adopt the reaction class prediction dataset originally curated
from the PDB by Hermosilla et al. (2020).

  * •

Composition The dataset is split on the basis of sequence similarity using a
50% threshold.

  * •

Hosting A preprocessed version of the dataset can be downloaded from the
benchmark’s Zenodo data record at
https://zenodo.org/record/8282470/files/ECReaction.tar.gz.

  * •

Licensing We have released a preprocessed version of the dataset under a
Creative Commons Attribution 4.0 International license. The original dataset
is available under an MIT license at
https://github.com/phermosilla/IEConv_proteins/blob/master/LICENSE.

  * •

Maintenance We will announce any errata discovered in or changes made to the
dataset using the benchmark’s GitHub repository at
https://github.com/a-r-j/ProteinWorkshop.

  * •

Uses This dataset can be used for multilabel graph classification tasks where
each protein, $\mathcal{G}$, is mapped to a label $y\in{\\{1,...,384\\}}$
denoting which class of reactions a given protein catalyzes.

  * •

Metric Accuracy.

###  E.8 TDC - Antibody Developability Prediction

Therapeutic antibodies must be optimised for favourable physicochemical
properties in addition to target binding affinity and specificity to be viable
development candidates. Consequently, we frame prediction of antibody
developability as a binary graph classification task indicating whether a
given antibody is developable.

  * •

Motivation From a benchmarking perspective, predicting the developability of a
given antibody is important as it enables targeted performance assessment of
models on a specific (immunoglobulin) fold, providing insight into whether
general-purpose structure-based encoders can be applicable to fold-specific
tasks.

  * •

Collection We adopt the antibody developability dataset originally curated
from SabDab (Dunbar et al., 2014) by Chen et al. (2020).

  * •

Composition This dataset contains 2,426 antibodies that have both sequences
and PDB structures available, where each example contains both a heavy chain
and a light chain with resolution < 3 (Å). Labels are based on thresholding
the developability index (DI) ((Lauer et al., 2012)) as computed by BIOVIA’s
platform ((Systèmes, 2016)), which relies on an antibody’s hydrophobic and
electrostatic interactions.

  * •

Hosting A preprocessed version of the dataset can be downloaded from the
benchmark’s Zenodo data record at
https://zenodo.org/record/8282470/files/AntibodyDevelopability.tar.gz.

  * •

Licensing We have released a preprocessed version of the dataset under a
Creative Commons Attribution 4.0 International license. The original dataset
is available under a Creative Commons Attribution 3.0 Unported license at
https://tdcommons.ai/single_pred_tasks/develop/#sabdab-chen-et-al.

  * •

Maintenance We will announce any errata discovered in or changes made to the
dataset using the benchmark’s GitHub repository at
https://github.com/a-r-j/ProteinWorkshop.

  * •

Uses This dataset can be used for binary graph classification tasks indicating
whether a given antibody is developable.

  * •

Metric AUPRC.


# Random Cantor sets and mini-bands in local spectrum of quantum systems

B. L. Altshuler  V. E. Kravtsov

###### Abstract

In this paper we give a physically transparent picture of singular-continuous
spectrum in disordered systems which possess a non-ergodic extended phase. We
present a simple model of identically and independently distributed level
spacing in the spectrum of local density of states and show how a fat tail
appears in this distribution at the broad distribution of eigenfunction
amplitudes. For the model with a power-law local spacing distribution we
derive the correlation function $K(\omega)$ of the local density of states and
show that depending on the relation between the eigenfunction fractal
dimension $D_{2}$ and the spectral fractal dimension $D_{s}$ encoded in the
power-law spacing distribution, a singular continuous spectrum of a random
Cantor set or that of an isolated mini-band may appear. In the limit of an
infinite number of degrees of freedom the function $K(\omega)$ in the non-
ergodic extended phase is singular at $\omega=0$ with the branch-cut
singularity for the case of a random Cantor set and with the $\delta$-function
singularity for the case of an isolated mini-band. For an absolutely
continuous spectrum $K(\omega)$ tends to a finite limit as $\omega\rightarrow
0$. For an arbitrary local spacing distribution function we formulated a
criterion of fractality of local spectrum and tested it on simple examples.

###### keywords:

non-ergodic extended states, multifractality of wave functions, singular-
continuous spectrum, Cantor set, mini-bands

††journal: Annals of Physics

[1]organization=Department of Physics, Columbia University, addressline=116th
and Broadway, city=New York, postcode=10027, state=NY, country=U.S.A.
[2]organization=The Abdus Salam International Centre for Theoretical
Physics,addressline=P.O.B. 586, city=Trieste, postcode=34100, country=Italy

Non-ergodic extended states and a singular-continuous (s.c.) spectrum

Two types of s.c. spectrum: Cantor set and isolated mini-band

Model for s.c. spectrum: i.i.d. power-law spacing distribution

Correlation function of LDoS $K(\omega)$ as a measure for s.c. spectrum

Singularity of $K(\omega)$ at $\omega=0$ for s.c. spectrum in the infinite
system

##  1 Preamble

This paper is dedicated to memory of our friend, collaborator and teacher
Kostya Efetov. His scientific fearlessness, persistent innovation and human
qualities made an unforgettable and crucial impact on our scientific careers,
as well as on the development of our scientific community as a whole.

##  2 Introduction

Recently there has been a vigorous discussion on the existence of non-ergodic
extended (NEE) states in a finite range of parameters (NEE phase) of various
disordered systems, both free-particle [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] and
interacting [11, 12, 13, 14, 15, 16, 17, 18, 19]. By extended states one
denotes the states that occupy an extensive part of the Hilbert space, while
the lack of ergodicity implies that not all the Hilbert space is occupied by a
wave function. The most well-known example of such NEE states are the
multifractal states at the critical point of the Anderson localization
transition [20]. They, however, do not form a phase. Another example of such
states are the states in the Critical Power-Law Banded Random Matrices [21]
which can be viewed as a one-dimensional systems with long-range hopping. In
this case, however, all the states have an NEE, multifractal character. The
first and the simplest system which possesses the NEE phase separated by the
ergodic transition from the ergodic states and by the Anderson localization
transition from the localized states was found in the generalized Gaussian
Rosenzweig-Porter (RP) model [22, 3] and further elaborated in [4, 7, 23] to
finally bring it to the status of a mathematical theorem [5].

Remarkably, in Ref.[4] the question of the structure of the local spectrum in
the Rosenzweig-Porter model was raised and it was shown that this spectrum has
a non-trivial form of a mini-band. This peculiarity of the spectrum determines
the local dynamics of this model, e.g. the return, or survival, probability,
which was studied in detail in Ref.[24]. Furthermore, a modification of the RP
model, the log-normal RP model [7], was shown [25] to exhibit a sub-difusive,
stretch-exponential relaxation of the averaged survival probability, similar
to that in the extended phase of the Anderson localization model on a Random
Regular Graph [26, 27] and in interacting spin-chains [28, 29, 30, 31, 32, 33,
13]. Since the character of local spectrum determines local dynamics and the
latter is important for manipulation of the chains of qubits (which are
equivalent to interacting spin-chains), the problem of local spectra has a
wide application in quantum computing.

There is, however, another, fundamental dimension in this problem. From the
mathematical literature [34] it is known that the spectrum of any Hamiltonian
can be decomposed into a sum of three distinctly different types of spectra:
the pure-point, absolutely continuous and singular-continuous. Their loose
definitions are the following:

  * 1.

$(a)$ pure point: This type of spectrum corresponds to $L^{2}$ integrable
eigenfunctions, e.g. the localized wave functions with a finite localization
radius. Finite-amplitude delocalized wave functions (e.g. the plane waves) and
those localized in one single point [e.g. $\psi\sim\delta(p-p_{0})$ in the
momentum space] are not $L^{2}$-integrable in an infinite continuous system
and, therefore, they cannot correspond to a pure point spectrum according to
the mathematical literature. By spectrum we understand the set of real numbers
$\\{E_{n}\\}$ such that the operator $\hat{H}-E_{n}$ is not invertible.

  * 2.

$(b)$ absolutely-continuous: In this case the wave functions are not $L^{2}$
integrable (not normalizable). These maybe the delocalized wave functions in
an infinite coordinate space, or the corresponding non-normalizable in the
continuous space functions divergent strongly enough at some point $p$ in the
momentum space. For this type of spectrum the set of $E_{n}$ must be dense,
i.e. for any given real $E$ inside the spectrum there is a $E_{n}$ such that
$|E-E_{n}|$ is arbitrary small.

  * 3.

$(c)$ singular-continuous: Loosely speaking, this is the type of spectrum that
does not fall in neither of the above two types $(a)$ and $(b)$. This means
that the corresponding wave functions are extended in some basis (otherwise
they belong to $(a)$) but the set of $E_{n}$ is not dense. The archetypical
example of such a spectrum is the regular Cantor set. Indeed, in the Cantor
set there is a hierarchy of gaps (nearest neighbor spacings)
$\delta_{m}=3^{-m}$ ($m=1,2,...,\ln N/\ln 3$) which values in the limit
$1/N\rightarrow 0$ are accumulated near $\delta_{\infty}=0$. Therefore any
point $E$ inside such a spectrum falls inside some gap $\delta_{n}$ with a
fixed finite $n$ which does not shrink to zero as $1/N\rightarrow 0$. Instead,
with increasing $N$ more small gaps of the higher generations $m\gg n$ appear
and accumulate near $\delta_{\infty}=0$. Hence, the distance $|E-E_{n}|$ with
$E\in\delta_{n}$ remains finite in this limit, and the spectrum is nowhere
dense albeit contains an infinite number $N\rightarrow\infty$ of levels.

The above definitions make sense in the continuous systems, while the most of
physical systems are defined on discrete sites of some lattices or graphs. In
such systems at an infinite number of sites $N$ the non-normalizability can
happen only for extended wave functions, while any localized wave function is
always normalizable. However, the Fourier transform of the extended wave
function is a localized one and vice versa. Thus the normalizability in the
infinite discrete systems is not basis-invariant, while the global spectrum
is. This means that normalizability in discrete systems cannot be used for
classification of basis-invariant global spectra like in the above definitions
for the continuous systems.

However, even in discrete systems one may classify the spectra but in a non-
invariant way associated with a certain basis. In order to accomplish this
goal we consider the local density of states (LDoS) and its correlation
functions where the spectral $\delta$-functions $\delta(E-E_{n})$ are weighted
by the amplitudes $|\psi_{n}(r)|^{2}$ of the normalized eigenfunctions at a
position $r$ in a chosen basis. However, if one adopts this rout one should
inevitably consider a finite dimension of the Hilbert space $N$, as otherwise
the amplitude of an extended eigenfunction will be zero. In such a discrete
finite-$N$ systems the spectrum is always discrete and the solutions
$\psi_{n}(r)$ of the Schroedinger equation are always normalizable. The limit
$N\rightarrow\infty$ should be implemented only in the final results for the
correlation functions of the LDoS.

The question is: Which are the analogues of the pure point, absolutely
continuous and singular continuous for the local spectra (e.g. the spectra
seen in LDoS) and how to distinguish statistically between them? The simple
solution we propose is to study the behavior of the local density of states
correlation function $K(\omega)$ in the limit $N\rightarrow\infty$, where
$\omega$ is the difference in the energy. We show that for a local analogue of
the absolutely continuous spectrum (referred to as $l$-absolutely continuous
spectrum) this correlation function has a finite limit at $\omega\rightarrow
0$, while for the $l$-singular continuous spectrum $K(\omega)$ is singular at
$\omega=0$. For a $l$-pure point spectrum $K(\omega\rightarrow 0)=0$, the
limit $N\rightarrow\infty$ should be taken first in all three cases.

A crucial step in doing this limit is to discriminate eigenstates according to
their amplitude at the observation point $r$ and eliminate the states which
amplitude $|\psi_{i}(r)|^{2}$ is lower than the discriminant level $\sim 1/N$.
In the non-ergodic phases, this operation diminishes drastically the number of
states remaining in the spectrum of LDoS (the ”local levels”) and it also
changes qualitatively statistics of spacing between the corresponding energy
levels leading to the fat tails in the level spacing distribution.

We propose the following classification of the local spectra:

  * 1.

$l$-pure point spectrum $(a)$: This type of local spectrum emerges in the
localized systems. The local spectrum consists of a finite number of levels
with the typical spacing that remains finite as $N\rightarrow\infty$. The
corresponding states are all localized in the same localization volume.

  * 2.

$l$-absolutely continuous spectrum $(b)$: The local spectrum is dense in the
limit $N\rightarrow\infty$ in a certain spectral interval that remains finite
in this limit. This type of spectrum corresponds to the ergodic delocalized
states.

  * 3.

$l$-singular continuous spectrum $(c)$: These are the local spectra that do
not fall in neither of the above types (a) and (b). Two important examples
are: ${\bf(i)}$ The local spectrum contains an infinite number of levels in a
certain finite interval of energies in the limit $N\rightarrow\infty$.
However, the spectrum is not dense in this interval (e.g. it is a random
Cantor set). Alternatively, ${\bf(ii)}$ the local spectrum is all concentrated
within a mini-band that contains an infinite number of levels in the
$N\rightarrow\infty$ limit but the width of the mini-band shrinks to zero in
this limit.

We will show below that in the case ${\bf(i)}$
$K(\omega)\sim\omega^{-\alpha}$, $0<\alpha<1$ in the limit
$N\rightarrow\infty$, while in the case ${\bf(ii)}$
$K(\omega)\sim\delta(\omega)$ in this limit.

In order to substantiate this classification we consider a simple model of
local levels with independent identically distributed (i.i.d.) spacing which
distribution is a power-law. We show that such a simple model is capable of
reproducing the two principle cases of $l$-singular continuous spectra: the
random Cantor set and the isolated mini-band. We show that upon a proper
choice of parameters such a model gives a correct behavior of $K(\omega)$ both
for the Power-Law Banded Random Matrices and for the Gaussian Rosenzweig-
Porter random matrix ensemble. This allows us to conclude that the local
spectrum of the former is a random Cantor set, while that of the latter is an
isolated mini-band with the dense spectrum inside a mini-band.

However, our model gives many more possibilities of singular continuous
spectra which Hamiltonian realizations are not yet found. Thus it can be
viewed as a guide for a further search of such models.

Finally, we extend our model of i.i.d. local spacings to an it arbitrary
distribution of spacings and formulate a general criterion of fractality of
local spectrum. This criterion may be used to characterize complex statistics
of local spectra in random systems.

The paper is organized as follows. In Introduction (Section 2) we review the
relevant literature and discuss the difference between the classification of
global spectra in continuous systems in mathematical literature and
classification of local spectra in descrete systems in this work. In Section 3
we describe the procedure of discrimination of eigenstates which explains a
qualitative difference between the global and the local spectral statistics.
The two principle classes of the $l$-singular continuous spectra and the
corresponding Hamiltonian realizations are also described in this section. In
Section 4 a toy model of local spectrum of the form of the textbook Cantor set
is considered, the power-law spacing distribution $P(s)$ in this model is
derived and the model local density of states correlation function $K(\omega)$
is computed. In Section 5 a random Cantor set is constructed from the i.i.d.
power-law local spacing distributions and the correlation function $K(\omega)$
is computed. In Section 6 from the same model of i.i.d. local spacing
distributions we obtain an isolated mini-band. In Section 7 a general
criterion of fractality of local spectrum is formulated for an arbitrary
$P(s)$. This formalism is tested in Appendix A for a power-law $P(s)$ for a
choice of parameters that corresponds to a random Cantor set and to an
isolated mini-band in the NEE phases and to a dense spectrum in ergodic
phases. In Conclusion we present the table of all the principle results for
the $\omega$\- and $N$\- dependence of the local DoS correlation function
$K(\omega)$ and for the scaling of the mini-band with the dimension $N$ of the
Hilbert space. Finally, in Conclusion we present an expression for $K(\omega)$
in $N\rightarrow\infty$ limit which distinguishes between the absolutely
continuous and the two principal classes of the singular continuous spectra
that emerge in our model.

##  3 Discrimination of eigenstates

###  3.1 The global and the local density of states

We start by reviewing the differences between the global an the local spectrum
of a quantum system with the broad distribution of the eigenfunction
amplitudes $|\psi_{n}(r)|^{2}$. In this section we introduce the notion of a
mini-band in the local spectrum and the peculiarities of the local spacing
distribution compared to its global counterpart.

First of all we remind the readers the definitions of the global, $\rho(E)$,
and the local, $\rho(E;r)$, densities of states (DoS) in terms of the
eigenvalues $E_{n}$ and eigenfunctions $\psi_{n}(r)$ :

$\rho(E)=\frac{1}{N}\sum_{n}\delta(E-E_{n}),$ (1)
$\rho(E;r)=\sum_{n}|\psi_{n}(r)|^{2}\,\delta(E-E_{n}).$ (2)

In disordered systems both $\rho(E)$ and $\rho(E;r)$ are random quantities. By
normalization of the eigenfunctions:

$\sum_{r}|\psi_{n}(r)|^{2}=1,$ (3)

the mean eigenfunction amplitude is given by:

$\langle|\psi_{n}(r)|^{2}\rangle=N^{-1},$ (4)

where $\langle...\rangle$ denotes the ensemble average and $N$ is the
dimension of the Hilbert space (the total number of eigenfunctions).

If the distribution of the eigenfunction amplitudes is narrow
$|\psi_{n}(r)|\sim N^{-1}$ (as it happens e.g. for the Wigner-Dyson random
matrices) then there is no qualitative difference between the statistics of
the global and the local DoS. On the other hand, for the localized
eigenfunctions only few of them have an appreciable amplitude in the
observation point $r$. If one discards the states with exponentially small
amplitudes in the observation point, the local spectrum will consist of a
finite number of the $\delta$-functions with a finite separations between them
in the thermodynamic limit $N\rightarrow\infty$ (an $l$-pure point spectrum).
In contrast, the global spectrum in most of the cases has a dense set of
eigenvalues with the mean level spacing $\delta\sim N^{-1}$ that vanishes in
the limit $N\rightarrow\infty$. In what follows we will always assume that the
global spectrum is dense, i.e. in the thermodynamic limit in the arbitrary
close vicinity $\epsilon$ of any point $E$ inside the spectral band there is
an eigenvalue $E_{n}$ such that $|E-E_{n}|<\epsilon$.

The two limiting examples mentioned above: (i) the absolutely continuous local
and global spectra of the Wigner-Dyson random matrices [35] and (ii) the pure
point local spectrum concomitant with absolutely continuous global spectrum of
localized eigenstates, do not cover all the possible cases.

Figure 1: (Color online) Eigenstates before (a) and after (b) discarding
states with the amplitude at the observation point $r$ smaller than the
discrimination level $\sim N^{-1}$ (red dashed lines). As the result of
discrimination large gaps appear in the local spectrum (e.g. in the yellow
shadowed areas).

Below we describe other possibilities of the $l$-singular continuous spectra
and present a transparent physical picture as to why and where they may arise.

###  3.2 Discrimination of eigenstates and the fat tail in the local level
spacing distribution

The main cause for a drastic difference between the statistics of the local
and the global spectrum is best illustrated by the example of the localized
eigenstates. It is the broad distribution of the eigenfunction amplitudes. In
the case of localized eigenstates there are rare large amplitudes if the
localization center is close to the observation point and the range of
exponentially small amplitudes if the observation point is beyond the
localization volume. This allows to introduce a discriminant such that the
eigenstates with amplitudes smaller than the discriminant level are discarded
from the local spectrum (see Fig.1). In practice the discriminant level should
be chosen to be much lower than the mean eigenfunction amplitude
$\langle|\psi_{n}(r)|^{2}\rangle=N^{-1}$. The obvious consequence of that is
the emergence in the local spectrum of level spacings much larger than those
in the global one (see Fig.1). Statistically, this exhibits itself in the fat
tails in the local spacing distribution function.

An important special case is when after the discrimination the number of
states in the local spectrum $M$ is infinite in the thermodynamic limit but it
scales with $N$ like a fractional power of $N$:

$M\sim N^{D_{s}},\;\;\;0<D_{s}<1$ (5)

In this case the mean level spacing $\delta_{{\rm av}}$ in the local spectrum
scales like $M^{-1}\sim N^{-D_{s}}$, while the typical level spacing
$\delta_{{\rm typ}}\sim\delta$ is supposed to be the same as in the global one
111In general, the discrimination procedure insures only that $\delta_{{\rm
typ}}\gtrsim\delta$. However, it seems natural that for the extended states
the amplitudes are correlated at a scale of $\delta$. Therefore if a state
survives the discrimination, the neighboring in energy state is likely to do
so.:

$\delta_{{\rm typ}}\approx\delta=1/N,\;\;\;\delta_{{\rm av}}\sim N^{-D_{s}}.$
(6)

Here and everywhere throughout the paper we assume that the energy units are
chosen so that the global spectral band-width is equal to 1.

Figure 2: (Color online) (Left panel): Random Cantor set in the local spectrum
with $D_{s}=0.63$. (Right panel): Dense global spectrum as a unification of
random Cantor sets (shown by different colors) seen in different observation
points.

###  3.3 Random Cantor set in Power-Law banded random matrices

A well-known example of such a situation is a random Cantor set (see Fig.2)
which will be considered in detail in the next Section. This type of local
spectrum is realized in the Critical Power-Law Banded Random Matrix (CPLBRM)
ensemble [21] defined as follows. It is an ensemble of random Hermitean
matrices $\hat{H}$ with real or complex entries $H_{nm}=\Re H_{nm}+i\Im
H_{nm}$ with statistically independent real and imaginary parts, $\Re
H_{nm}=\Re H_{mn}$ and $\Im H_{nm}=-\Im H_{mn}$, which are the Gaussian
distributed random variables with zero mean and the variance:

$\langle|H_{nm}|^{2}\rangle=\frac{1}{1+(n-m)^{2}/b^{2}},$ (7)

where the only control parameter is the band-width $0<b<\infty$. It is well-
known [20] that all eigenstates $\psi_{i}(r)$ of such matrices are
multifractal, i.e. they are extended but non-ergodic with

$\sum_{r}|\psi_{i}(r)|^{2q}\sim N^{-D_{q}(q-1)},$ (8)

where the $q$-dependent $0<D_{q}<1$ is the eigenfunction multifractal
dimension that varies from $0$ to $1$ as the parameter $b$ increases from
$b=0$ to $b=\infty$.

###  3.4 Mini-bands in the local spectrum: Rosenzweig-Porter Random matrix
ensemble

Figure 3: (Color online) Typical local spectrum of the Rosenzweig-Porter
random matrix ensemble in its non-ergodic extended phase. States depicted in
blue survive discrimination (set by the red dashed line) and form a mini-band
with the Lorentz envelope of $|\psi_{n}(r)|^{2}$ (red solid line). The states
shown in gray are discarded. The number of levels in the mini-band $M\sim
N^{D}$ is extensive but at $0<D<1$ is much less then the total number of
states $N$. The width of the Lorentzian envelope is $M\delta\sim N^{-(1-D)}$
and tends to zero in the thermodynamic limit.

The difference between the global and the local DoS in the delocalized systems
is especially drastic when most of the states are discarded in the
discrimination process and the remaining ones form a compact mini-band (see
Fig.3). Such a situation emerges in the Rosenzweig-Porter (RP) random matrix
ensemble [22, 3, 4, 36, 23]. As in the case of CPLBRM the RP random matrices
have statistically-independent random Gaussian entries with zero mean. But the
variance is bi-modal:

$\langle|H_{nn}|^{2}\rangle=1,\;\;\;\;\langle|H_{n\neq
m}|^{2}\rangle=N^{-\gamma},$ (9)

where $0<\gamma<\infty$ is the effective disorder parameter, and $N$ is the
size of a matrix.

In the non-ergodic extended phase $1<\gamma<2$ the mini-band in this model
contains $M\sim N^{D}$ fractal states of the typical amplitude $|\psi|^{2}\sim
N^{-D}$ characterized by the fractal dimension:

$D_{q}=D=2-\gamma,$ (10)

which is independent of $q$ for all $q>1/2$. It is important that inside the
mini-band the statistics of local spectrum is Wigner-Dyson [35, 37, 24, 38].
Thus in the thermodynamic limit $N\rightarrow\infty$ the width of the mini-
band $M\delta\sim N^{D-1}$ shrinks to zero but the number of levels in it
tends to infinity.

##  4 Regular Cantor set as a spectral counting function

In order to better understand the properties of the random Cantor set and its
connection with the local spectrum of quantum systems such as CPLBRM, let us
consider the toy model of local spectrum in a form of a regular textbook
Cantor set. Namely, let us consider a set of $M=N^{D_{s}}$ delta-functions
$\delta(E-E_{n})$ with the equal weight $|\psi_{n}(r)|^{2}=N^{-D_{s}}$. Such a
model obeys the completeness condition:

$\sum_{n=1}^{M}|\psi_{n}(r)|^{2}=1.$ (11)

The points $E_{n}$ in the segment $[0,1]$ which represent the eigenvalues in
this toy model are organized in generations as follows (see Eq.(12)). The
first generation consists of two points $1/3$ and $2/3$ which are the ends of
the central $1/3$-segment. The second generation are the ends of the central
$1/3$-segment of the remaining two segments. This procedure is repeated until
the minimal spacing $3^{-n}$ becomes $N^{-1}$.

$\displaystyle\frac{1}{3}\;$ $\displaystyle\;\frac{2}{3}$ (12)
$\displaystyle\frac{1}{9}\;\;\frac{2}{9}\;\;$
$\displaystyle\;\;\;\frac{7}{9}\;\;\frac{8}{9}$
$\displaystyle\frac{1}{27}\;\frac{2}{27}\;\;\frac{7}{27}\,\frac{8}{27}$
$\displaystyle\frac{19}{27}\;\frac{20}{27}\;\;\frac{25}{27}\;\frac{26}{27}$
$\displaystyle................$ $\displaystyle................$

The entire spectrum is the unification of all $n$ generations. The total
number of levels in the spectrum is the sum over generations:

$M=\sum_{i=1}^{n}2^{i}=2(2^{n}-1)\approx 2^{n+1}=2N^{\frac{\ln 2}{\ln 3}}.$
(13)

According to Eq.(5) this implies:

$D_{s}=\frac{\ln 2}{\ln 3}$ (14)

Next, we notice that the number of times the spacing $s_{m}=3^{-m}$ occurs is
$2^{m-1}$. Thus the probability $p(s_{m})$ to have spacing $s_{m}$ appears to
be a power-law:

$p(s_{m})=\frac{2^{m-1}}{2^{n}}=\frac{1}{2}\left(\frac{\delta}{s_{m}}\right)^{D_{s}}.$
(15)

This corresponds to the power-law probability density, or the spacing
distribution function $P(s)$, for the regular Cantor set:

$P(s)=\left\\{\matrix{A\,\delta^{D_{s}}\,s^{-(D_{s}+1)},&(1>s>\delta=N^{-1})\cr
0,&\mbox{otherwise}}\right.,$ (16)

where $A$ is the normalization constant and $D_{s}$ is given by Eq.(14). For
$s<\delta$ and for $s>1$ $P(s)$ is zero. One can easily see that Eq.(6) is
valid for this distribution.

This form of $P(s)$ will be the key starting point to define the random Cantor
set in the next Section.

###  4.1 Construction of a regular Cantor counting function

Now let us construct the counting function $C(E)$ corresponding to the regular
Cantor set defined by Eq.(12):

$C(E)=M^{-1}\sum_{i=1}^{M}\theta(E-E_{i}),$ (17)

where $\theta(x)$ is the Heaviside step function, $E_{i}$ is the ordered set
of ’eigenvalues’ $E_{n}$ defined by Eq.(12) and $M\sim N^{D_{s}}$ is the total
number of levels, Eq.(13). The so defined Cantor counting function is plotted
in Fig.4.

Figure 4: (Color online) Cantor counting function

###  4.2 Convolution of two Cantor counting functions and the LDoS correlation
function

From the definition of Cantor counting function it follows that the derivative
$\partial_{E}C(E)$ is the set of $\delta$-functions that mimics the LDoS. For
random systems one is interested in the correlation function of two LDoS
defined by Eq.(2):

$K(\omega)=\langle\rho(E+\omega/2;r)\,\rho(E-\omega/2;r)\rangle.$ (18)

In terms of the counting functions:

$C(E;r)=\sum_{n}|\psi_{n}(r)|^{2}\,\theta(E-E_{n}),$ (19)

we have:

$K(\omega)=-4\frac{d^{2}}{d\omega^{2}}\,\langle
C(E+\omega/2;r)\,C(E-\omega/2)\rangle$ (20)

In our toy model with the regular Cantor counting function $C(E)$ instead of
the random $C(E;r)$ we replace the ensemble averaging by the averaging over
the spectrum:

$K_{{\rm
Cantor}}(\omega)=-4\frac{d^{2}}{d\omega^{2}}\,\int_{\frac{\omega}{2}}^{1-\frac{\omega}{2}}C(E+\omega/2)C(E-\omega/2)\,dE,$
(21)

where $C(E)$ is given by Eq.(17).

Figure 5: (Color online) $C(E+\omega/2)C(E-\omega/2)-C(E)^{2}$ integrated over
$E$ as in Eq.(21) for the Cantor set with $n=12$ generations.

In Fig.5 the averaging over the spectrum for
$F(\omega)=C(E+\omega/2)C(E-\omega/2)-C(E)^{2}$ is done numerically for the
Cantor set with $n=12$ generations. One can see that the result is well
described by a power law $F(\omega)\propto\omega^{1+D_{s}}$. We conjecture
that in the limit $N=3^{n}\rightarrow\infty$ this power law becomes exact, so
that down to $\omega=0$ we have:

$K_{{\rm
Cantor}}(\omega)\propto-4\frac{d^{2}F(\omega)}{d\omega^{2}}\sim\left(\frac{1}{\omega}\right)^{1-D_{s}},\;\;\;\;\left(D_{s}=\frac{\ln
2}{\ln 3}\right).$ (22)

The regular Cantor set is the prototype of the singular continuous spectrum
defined rigorously in mathematical literature [34]. We see, therefore, that
the ’smoking gun’ of the local analogue of such a spectrum, the $l$-singular
continuous spectrum, in the limit $N\rightarrow\infty$, is the singular
behavior of the LDoS correlation function, Eq.(18), at $\omega\rightarrow 0$.

##  5 Random Cantor set

###  5.1 Random Cantor set and its counting function

Figure 6: (Color online)The counting function $C(E)$ for a set of $M=N^{D}$
levels with the power-law spacing distribution, Eq.(16), characterized by: (a)
$D=D_{s}=0.63$; (b) $D=D_{s}=1$; (c) $D=1,D_{s}=2$; (d) $D=0.5,D_{s}=2$. In
the random Cantor set (a) the gaps (plateaus) between the mini-bands (green
strips) at any scale $\Delta$ in the hierarchy (shown by increasing intensity
of green strips) are of the order of the typical width of mini-bands. In the
case (b) the gaps between the strings of levels shown by green strips, are
smaller than the typical width of the strings (a sub-dense spectrum). In this
case the local spectrum lacks a hierarchy of mini-bands. In the case (c) the
spectrum is dense with no large gaps at all. In the case (d) an isolated
narrow mini-band appears with the zoomed counting function of the type (c)
shown in the inset. A similar isolated mini-band appears in all the cases when
$D<D_{s}$ with the spectrum inside the mini-band being a Cantor set (a) if
$D_{s}<1$ or it is sub-dense as in (b) if $D_{s}=1$.

The power law, Eq.(16) for the spacing distribution in the regular Cantor set
can be used to define a random Cantor set which is a natural generalization of
a regular Cantor set to describe the singular continuous spectrum of random
Hamiltonians.

Let us consider a model of local spectrum in which all $s_{n}=E_{n+1}-E_{n}$
are:

  * 1.

independent random quantities,

  * 2.

distributed identically according to the power-law Eq.(16) with some
$1>D_{s}>0$.

An example of the counting function $C(E)$ drawn from such a model of random
spectra with $D_{s}=0.63$ is shown in Fig.6(a). Like in the regular Cantor
set, this counting functions consists of the hierarchy of plateaus and
staircases between them (denoted by green stripes of increasing intensity as
the scale decreases). The length of the staircases (the width of the green
stripes) at all scales is of the order of the length of the plateaus at the
corresponding scales.

###  5.2 Correlation function of two random Cantor counting functions

Now we compute the ensemble average of the product of a random Cantor counting
function and the same counting function shifted by $\omega$ in the energy
space. Such an average corresponds to the LDoS correlation function, provided
that the local spectrum (after discrimination) is a random Cantor set. We
demonstrate that the result of such averaging for a model with $D_{s}<1$
reproduces the LDoS correlation function in many critical systems (including
the Anderson models at the localization transition and CPLBRM) just
corroborating the conjecture that the local spectrum of the corresponding
quantum critical systems is a random Cantor set.

Because of independently fluctuating and identically distributed spacings
$s_{j}$ in our model of a random Cantor set one can easily compute the
correlation function $K(\omega)$ given by Eq.(20). First we note that since
the averaged LDoS and its correlation function $K(\omega)$ for the random
Cantor set are independent of $E$ everywhere in the interval $[0,1]$ one may
integrate Eq.(20)[where $C(E;r)$ is given by Eq.(17) with random $E_{i}$] over
$E$ in this interval and represent $K(\omega)$ as follows:

$K(\omega)=M^{-2}\sum_{i,i^{\prime}=1}^{M}\delta(\omega-
E_{i}+E_{i^{\prime}}),$ (23)

where $M\leq N$ is the total number of levels $E_{i}$ in our model of local
spectrum, and

$E_{i}-E_{i^{\prime}}=\sum_{j=i^{\prime}}^{i}s_{j}.$ (24)

The function $K(\omega)$ is normalized:

$\int K(\omega)\,d\omega=1.$ (25)

Furthermore, for $M\gg 1$ its regular part corresponding to $i=i^{\prime}$ in
Eq.(23) also obeys Eq.(25) with the accuracy $O(1/M)$.

Next, using $\langle e^{-i\tau\sum_{j=i^{\prime}}^{i}s_{j}}\rangle=\langle
e^{-i\tau\,s}\rangle^{i-i^{\prime}}$ that follows from the statistical
independence of $s_{j}$, and summing the geometric series one obtains for the
Fourier transform of $K(\omega)$:

$\displaystyle\tilde{K}(\tau)$ $\displaystyle=$ $\displaystyle
M^{-2}\left[M+2\sum_{i^{\prime}=1}^{M}\sum_{i=i^{\prime}+1}^{M}\,\Re\langle
e^{-i\tau\sum_{j=i^{\prime}}^{i}s_{j}}\rangle\right]$ $\displaystyle=$
$\displaystyle
M^{-1}+2M^{-2}\Re\left[\left(M-1-\frac{g-g^{M}}{1-g}\right)\,\frac{g}{1-g}\right].$

In Eq.(5.2) we denote $g\equiv g(t=i\tau)$ where on a real axis $t>0$:

$g(t)=\langle e^{-t\,s}\rangle=\int_{0}^{\infty}e^{-t\,s}\,P(s)\,ds.$ (27)

By normalization of the probability distribution $P(s)$ one has on the real
axis of $t$:

$g(0)=1,\;\;\;\;\;\;0<g(t)\leq 1.$ (28)

The function $\tilde{K}(\tau)$ gives the mean return/survival probability:

$\tilde{K}(\tau)=\langle|\Psi(\tau;r)|^{2}\rangle,$ (29)

where initially at $\tau=0$ the wave function $\Psi(t=0,r)$ was non-zero only
at a point $r$.

The spectral form-factor is given by an inverse Fourier transform of
$\tilde{K}(\tau)$:

$K(\omega)=\int_{-\infty}^{+\infty}\tilde{K}(\tau)\,e^{i\omega\tau}\,\frac{d\tau}{2\pi}.$
(30)

The function $g(t)$ that corresponds to the distribution Eq.(16) at
$0<D_{s}<1$ has a brunch-cut singularity at $t=0$:

$g(t)=1-c\,(\bar{t})^{D_{s}},$ (31)

where $\bar{t}=t\delta=t/N<<1$ and $c$ is a positive constant of order 1. At
$D_{s}=1$ one obtains:

$g(t)=1+\bar{t}\,\ln\bar{t},$ (32)

while for $2>D_{s}>1$ a linear in $\bar{t}$ term appears:

$g(t)=1-c_{1}\bar{t}-c_{2}\bar{t}^{D_{s}}.$ (33)

In general a regular $\bar{t}^{m}$ expansion with an integer $m$ holds for the
first $m=[D_{s}]$ terms, where $[...]$ denotes an integer part.

At large enough $M$ the term $\propto g^{M}$ in Eq.(5.2) (i.e. the boundary
term in summation of the geometric series) can be omitted. In this case one
obtains:

$\tilde{K}(\tau)\approx 2M^{-1}\Re\,(1-g(i\tau))^{-1},$ (34)

which for a random Cantor set ($D_{s}<1$ and $M=N^{D_{s}}$) results in:

$K(\omega)\sim\frac{M^{-1}\delta^{-D_{s}}}{{\bf\omega}^{{\bf
1-D_{s}}}}\sim\frac{1}{{\bf\omega}^{{\bf 1-D_{s}}}},$ (35)

i.e. the same power law, Eq.(22), as for the regular Cantor set but with an
arbitrary $0<D_{s}<1$.

It is remarkable that Eq.(35) at $D_{s}=D_{2}=d_{2}/d$ (where $d_{2}$ is the
multi-fractal dimension of the critical wave function and $d$ is the
dimensionality of the lattice) reproduces exactly the Chalker’s ansatz first
established by Chalker and Daniel for the Integer Quantum Hall effect at the
center of Landau band [39] and later on found true for the $d$-dimensional
Anderson model at the mobility edge $E=E_{c}$ [40]222The critical region near
the mobility edge or the center of Landau band $E=E_{c}$ is defined so as the
critical length $\xi\propto|E-E_{c}|^{-\nu}$ is larger than the system size
$L=N^{\frac{1}{d}}$. Thus in the critical energy window
$|E-E_{c}|<N^{-\frac{1}{\nu d}}$ there are $N^{1-\frac{1}{\nu d}}$ critical
states, which by Harris criterion [41] $\nu d>2$ tends to infinity in the
thermodynamic limit.  and for the Power-Law Banded Random Matrix ensemble [21,
42, 43, 44]. Thus we can argue that in all these cases the local spectrum
(after discrimination described in Sec.3 ) is a random Cantor set.

However, at $D_{s}=1$ one obtains from Eqs.(34),(32) a slowly decreasing

$K(\omega)\sim M^{-1}\delta^{-1}\,\left(\ln\frac{\omega}{\delta}\right)^{-1},$
(36)

rather than a constant. Indeed, we see from Fig.6(b) that the corresponding
counting function does contain the plateaus and staircases, albeit the length
of the staircases is now larger than that of the plateaus. The corresponding
local spectrum is no longer a random Cantor set, neither it is dense as for
ergodic systems. The model, Eq.(16), with $D_{s}=1$ represents a border-line
case in between of a random Cantor set and a dense spectrum (which we refer to
as a sub-dense spectrum). Its Hamiltonian realization is not known so far to
the best of our knowledge.

It is only at $D_{s}>1$ where the constant $K(\omega)$ at $\omega\gg\delta$
emerges. The point is that with taking the boundary term $g^{M}$ into account
in Eq.(5.2) one should modify Eq.(34) as follows:

$\tilde{K}(\tau)\approx 2M^{-1}\Re\,(1-g(i\tau)+\eta)^{-1},$ (37)

where $\eta=M^{-1}\rightarrow+0$.

Then at $D_{s}>1$ one obtains from Eq.(33):

$\tilde{K}(\tau)\sim\Re\,\frac{M^{-1}}{i\tau\delta+\eta+c_{2}\,(i\tau\delta)^{D_{s}}}.$
(38)

At $D_{s}>1$, in the leading approximation in $\tau\delta\ll 1$ Eq.(38)
results in:

$\tilde{K}(\tau)\sim\Re\,\frac{M^{-1}}{i\tau\delta+\eta}\rightarrow\pi\,M^{-1}\delta^{-1}\,\delta(\tau),$
(39)

which implies a constant

$K(\omega)\sim M^{-1}\delta^{-1},$ (40)

at $\omega\gg\delta$, like for a dense local spectrum in the Wigner-Dyson RMT.
The corresponding typical counting function is shown in Fig.6(c) and it does
not exhibit plateaus which correspond to anomalously large gaps in the local
spectrum. This example shows that the case $D_{s}>1$ in Eq.(16) is essentially
the same as for a fast decreasing $P(s)$ [like the Poisson or the Wigner-Dyson
$P(s)$ in the global spectrum] without fat tails. Thus this local spectrum is
dense, the same as the global spectrum.

##  6 An isolated mini-band

For a random Cantor set we considered the model in which the number of local
levels $M\sim N^{D_{s}}$, where $D_{s}$ controls the power-law distribution
Eq.(16). However, this is not the only possibility but a very particular case.
Indeed, comparing Eqs.(17) and (19) we see that the number of levels $M$ is in
fact related with the multifractal dimension of an eigenfunction. Indeed, in
the simplified case (realized in the Rosenzweig-Porter RMT) when all
multifractal dimensions $D_{q}=D$ are identical for $q\geq 1/2$, one may
replace, by the order of magnitude, $|\psi_{n}(r)|^{2}\rightarrow N^{-D}$, so
that $M$ in Eq.(17) must be $M\sim N^{D}$ in order to reproduce Eq.(19), while
$D_{s}$ is yet another independent parameter of the model. In a generic
multifractal case we have in $K(\omega)$:

$M\sim N^{D_{2}},$ (41)

as the correlation function $K(\omega)$ is second order in $|\psi|^{2}$.

The question is: What will happen if $D_{2}<D_{s}$? The answer is known for
the Rosenzweig-Porter RMT in its fractal phase where $K(\omega)$ has a simple
Lorenzian shape [3, 24, 36, 23]. It is nearly independent of $\omega$ for
$\delta\ll\omega\ll\Gamma$, where $\Gamma\sim N^{-1+D}\gg\delta$ is the width
of the mini-band (see Fig.3), and falls down as $\Gamma/\omega^{2}$ at
$\omega\gg\Gamma$. Can we reproduce this behavior and find the correct scaling
of $\Gamma\sim N^{-(1-D)}$ with $N$ from the above model of independently
fluctuating spacings with $D_{2}<D_{s}$? As a matter of fact, the answer to
this question is affirmative.

Indeed, consider the case $D_{2}<D_{s}$ in Eqs.(41),(16) of our model of
independent identically distributed spacings. Then there is a region of
$\omega$ where $M\ln(g(t\sim 1/\omega))\lesssim 1$:

$\displaystyle N^{-1+\frac{D_{2}}{D_{s}}}\lesssim\omega\ll
1,\;\;\;\;\;D_{s}<1,$ (42) $\displaystyle
N^{-1+D_{2}}\ln(N^{D_{2}})\lesssim\omega\ll 1,\;\;\;\;\;D_{s}=1,$
$\displaystyle N^{-1+D_{2}}\lesssim\omega\ll 1,\;\;\;\;\;D_{s}\geq 1.$

In this region the boundary term $g^{M}$ in the summation of geometric series,
Eq.(5.2), cannot be neglected.

A thorough inspection of Eq.(5.2) in the region Eq.(42) gives the following
result:

$\tilde{K}(\tau)\approx 2\left(1-\frac{M}{3}\,\Re(1-g(i\tau))\right).$ (43)

The $\tau$-independent term in Eq.(43) results in a delta-function in $\omega$
which corresponds to a fast exponentially decreasing with $\omega$ term in the
exact $K(\omega)$ which should be neglected in the considered region, Eq.(42).
The term proportional to $g(i\tau)$ after Fourier-transforming gives a tail in
$K(\omega)$ in the region, Eq.(42), which is proportional to the power-law
spacing distribution $P(s)$:

$K(\omega)=\frac{2N^{D_{2}}}{3}\,P(\omega)\sim\frac{N^{-(D_{s}-D_{2})}}{\omega^{1+D_{s}}}.$
(44)

For $\omega$ smaller than the lower bound of Eq.(42) the boundary term $g^{M}$
can be neglected and the results, Eqs.(35),(36),(40), of the previous section
hold, if one sets $M\sim N^{D_{2}}$:

$K(\omega)=\left\\{\matrix{N^{D_{s}-D_{2}}\,\omega^{-(1-D_{s})},&$if$\;\;\;D_{2}<D_{s}<1\cr
N^{1-D_{2}}\,\ln^{-1}(\omega/\delta),&$if$\;\;\;D_{2}<D_{s}=1\cr
N^{1-D_{2}},&$if$\;\;\;D_{2}\leq 1,D_{s}>1}\right.$ (45)

The function $K(\omega)$ at different relationship between $D_{2}\equiv D$ and
$D_{s}$ is plotted in Fig.7.

Eqs. Eq.(44) and (45) show that in order to reproduce the known Lorenzian
$K(\omega)$ in the Rosenzweig-Porter RMT one should set
$D_{s}=1+\epsilon\equiv 1_{+}$, where $\epsilon\rightarrow+0$:

$K(\omega)\sim\left\\{\matrix{N^{1-D_{2}},&\delta\ll\omega\ll\Gamma=N^{-(1-D_{2})}\cr\frac{N^{-(1-D_{2})}}{\omega^{2}},&\omega\gg\Gamma=N^{-(1-D_{2})}}\right..$
(46)

The constant $K(\omega)$ at $\delta\ll\omega\ll\Gamma$ (see Fig.7(c)) which is
similar to that for the Wigner-Dyson RMT, signals of the dense spectrum of
levels inside a mini-band (see Fig.3 and the inset in Fig.6(d)). According to
our classification presented in Introduction, such type of the spectrum is
neither $l$-pure point, since the number of levels $M\rightarrow\infty$ in the
limit $N\rightarrow\infty$, nor it is $l$-absolutely continuous, as the width
of a mini-band $\Gamma\rightarrow 0$ in this limit. The singularity of such a
spectrum is corroborated by the fact that in the limit of an infinite system
$N\rightarrow\infty$ the correlation function $K(\omega)$, Eq.(46), reduces to
a singular $\delta(\omega)$ function.

Remarkably, Eq.(46) reproduces both the behavior of $K(\omega)$ for the Rosen-
zweig-Porter RMT [3] in all limiting cases and the scaling of the width of a
mini-band $\Gamma\sim N^{-(1-D_{2})}$ with the system size $N$. Generalizing
this result we conclude from the lower bound in Eq.(42) that the width of a
mini-band (seen in $K(\omega)$) in a generic case $D_{2}<D_{s}$ is given by:

$\Gamma\sim\left\\{\matrix{N^{-1+\frac{D_{2}}{D_{s}}},&$if$\;\;\;D_{s}<1\cr
N^{-1+D_{2}}\ln(N^{D_{2}}),&$if$\;\;\;D_{s}=1\cr
N^{-1+D_{2}},&$if$\;\;\;D_{s}>1}\right..$ (47)

Notice that Eq.(44) and (45) do not match at $\omega\sim\Gamma$ if $D_{s}\geq
1$. This happens because of the narrow interval around $\omega=\Gamma$ of the
width $\sim\Gamma N^{-D(D_{s}-1)}\ll\Gamma$ where a sharp drop of $K(\omega)$
by a factor of $N^{-D(D_{s}-1)}$ occurs before a power-law tail, Eq.(44),
becomes the leading contribution. At large $N$ the amplitude of the power-law
tail tends to zero at any $D_{s}>1$, so that the edge of a mini-band is sharp
on the logarithmic scale. This region of a sharp drop is encoded in the
$\tau$-independent term in Eq.(43) and its detailed description is only
possible by a full expression, Eq.(5.2). In Fig.7(c) it is shown that this
drop is in fact linear in $\omega$: $K(\omega)\sim
N^{1-D}\,(1-\omega/\Gamma)$.

Figure 7: (Color online) The model LDoS correlation function computed exactly
from Eqs.(16),(5.2) for: $(a)$ a random Cantor set with $D=D_{s}=0.5$,
$N=10^{6}$; $(b)$ a mini-band with the Cantor set inside, $D=0.5$,
$D_{s}=0.7$, $N=10^{6}$; a mini-band with the dense spectrum inside, $D=2/3$,
$D_{s}=1.5$, $N=10^{6}$. The red dashed lines represent the power-laws,
Eqs.(45),(44), expected at $N\rightarrow\infty$. Insets: $(a)$ a power-law
survival probability $\tilde{K}(\tau)$; $(b)$ a slow dynamics of survival
probability $\tilde{K}(\tau)$ at $\tau<1/\Gamma$ and a power-law tail; $(c)$ a
quasi-linear behavior of $K(\omega)$ for $\delta<\omega<\Gamma$.

Concluding this section we would like to emphasize that in general, the
exponent $D$ in the definition of the number of levels in the local spectrum:

$M\equiv N^{D}<N^{D_{s}},\;\;\;\;\;(D<D_{s}),$ (48)

may depend on the way the discrimination of states is implemented and on the
type of the local measure. In a particular case when the local measure is
$K(\omega)$ we have $D=D_{2}$ but for a generic local measure in a
multifractal phase one may have a different $0<D<1$ which depends on the
spectrum of multifractal dimensions $D_{q}$.

If $D<D_{s}$, a special type of the local singular-continuous spectrum, the
isolated mini-band, emerges. For $D<D_{s}<1$ the levels inside a mini-band
form a random Cantor set with a hierarchical structure of higher-order mini-
bands (see Fig.6(a)), while at $D_{s}>1$ the spectrum inside a mini-band is
dense (see Fig.6(d)). At $D_{s}=1$ a special case of non-hierarchical and not
completely dense spectrum (the sub-dense spectrum) shown in Fig.6(b) emerges
inside a mini-band.

Note that the case $D_{s}<D\leq 1$ in Eq.(16) is inconsistent with the
assumption that the typical level spacing $\delta_{{\rm typ}}$ in the local
spectrum is the same as in the global one $\delta_{{\rm typ}}\sim\delta$.
Indeed, with this assumption the total local spectral band-width is given by
Eq.(47): $W_{{\rm local}}\sim N^{\frac{D}{D_{s}}-1}$. By the discrimination
procedure that eliminates levels from the global spectrum in order to obtain a
local one, it must be equal or smaller than the total global spectral band-
width $W_{{\rm glob}}=1$. This leaves only two options $D=D_{s}$ (random
Cantor set) or $D<D_{s}$ (isolated mini-band). For the same reason the case
$D=D_{s}=1$ is marginally inconsistent with the above assumption, since in
this case $W_{{\rm loc}}\sim\ln(N)>1$.

##  7 Hierarchy of mini-bands

Figure 8: (Color online) A hierarchy of mini-bands. At each scale $\Delta_{i}$
$(i=1,2,3...)$ one can identify a mini-band of the width $E_{mb}^{(i)}$ such
that $E_{mb}^{(i)}\leq\Delta_{i}$ and the spacings $s_{>}^{(i)},s_{<}^{(i)}$
on the right and on the left of the mini-band obey ${\rm
min}\\{s_{>}^{(i)},s_{<}^{(i)}\\}\geq\Delta_{i}$. A mini-band that at a scale
$\Delta_{1}$ looks like a single level at a scale $\Delta_{2}$ (after a zoom)
appears to be a mini-band containing yet a narrower mini-band at a scale
$\Delta_{3}$.

As it is shown in Fig.4 and Fig.6(a) the regular and a random Cantor set can
be considered as a hierarchy of mini-bands.

By a hierarchy of mini-bands we understand a spectrum such that at each of
progressively decreasing scales $\Delta_{i}$ ($i=1,2.3...$) one finds a mini-
band, i.e a set of consecutive levels obeying the following conditions (see
Fig.8):

  * 1.

all the spacings between them are smaller than $\Delta_{i}$

  * 2.

their total sum, the width of a mini-band $E_{mb}^{(i)}$, obeys
$E_{mb}^{(i)}\leq\Delta_{i}$,

  * 3.

the spacings on the right and on the left of a mini-band $s_{>}^{(i)}$ and
$s_{<}^{(i)}$ obey the inequality ${\rm
min}\\{s_{>}^{(i)},s_{<}^{(i)}\\}\geq\Delta_{i}$.

Indeed, one can easily see that the regular Cantor set is such a hierarchy of
mini-bands at any legitimate scale $\Delta_{m}=3^{-m}$ bounded only by the
minimal scale $\Delta=\delta$ from below and by the total spectral band-width
1 from above. Indeed, at the scale $\Delta_{1}=1/3$ the width of the two mini-
bands $[0,1/3]$ and $[2/3,1]$ with the maximal spacing inside them
$1/9<\Delta_{1}$, are equal to $\Delta_{1}$; at the next scale
$\Delta_{2}=1/9$ one has 4 mini-bands of the width $\Delta_{2}=1/9$ with the
maximal spacing $1/27<\Delta_{2}$ and so on. Notice that the width of a mini-
band at each scale $\Delta_{i}$ is equal to $\Delta_{i}$. The same property
(by the order of magnitude) has a random Cantor set (see Fig.6(a) and
corresponding figure captions).

Figure 9: (Color online) More complicated local spectra. Fractality at small
scales $\Delta<\Delta_{{\rm max}}$ and a non-hierarchical set of mini-bands at
large scales $\Delta>\Delta_{{\rm max}}$ $[(a),(c)]$. A dense spectrum at
small scales $\Delta<\Delta_{{\rm min}}$ and a hierarchical set of mini-bands
at large scales $\Delta>\Delta_{{\rm min}}$ $[(b),(d)]$. The region of a
hierarchical spectrum in $(a)$ and $(b)$ is shown by a texture.

However, the hierarchy of mini-bands is not necessarily extended throughout
the entire local spectrum. It may terminate from below prior to reaching
$\Delta=\delta$. This happens if below some scale $\Delta_{{\rm min}}$ the
local spectrum is dense, i.e. $\delta_{{\rm av}}\sim\delta_{{\rm
typ}}\sim\delta$ (see Fig.9(b),(d)). It can also terminate from above at
$\Delta_{\rm max}<1$ if a mini-band of the width $\Gamma\sim\Delta_{{\rm
max}}$ is isolated or if the gaps greater than $\Delta_{{\rm max}}$ between
the mini-bands are all of the same order (see Fig.9((a),(c)). In particular,
an isolated mini-band of Fig.3 with the dense spectrum inside realized in the
Rosenzweig-Porter RMT, does not meet the above criteria of a hierarchy at any
scale $\Delta$, as in this case $\Delta_{{\rm min}}=\Delta_{{\rm
max}}=\Gamma$.

In order to be able to describe such more complicated than a simple Cantor set
cases we present the above conditions for the hierarchy of mini-bands in a
form of a mathematical expression in terms of an arbitrary i.i.d. spacing
distribution $P(s)$.

###  7.1 Criterion for existence of a hierarchy of mini-bands

In this section we derive a quantitative criterion of existence of a hierarchy
of mini-bands and compute $\Delta_{{\rm min}}$ and $\Delta_{{\rm max}}$ for an
arbitrary distribution $P(s)$ of independent identically distributed local
spacings. We start by the expression for the probability for a string of $n$
consecutive levels to obey the following conditions:

  * 1.

all $(n-1)$ independently fluctuating spacings between them are smaller than
some $\Delta$,

  * 2.

the spacing $s_{>}$ between the last level in a string and its nearest
neighbor on the right is greater or equal than $\Delta$. 333There is no need
to impose the condition that the spacing $s_{<}$ between the first level in a
string and its nearest neighbor on the left is greater or equal than $\Delta$.
This condition is automatically fulfilled, as the neighboring string on the
left must have a gap $s_{>}^{\prime}=s_{<}$ which is greater than $\Delta$.

This probability is given by:

$p_{\Delta}(n)=Q\,(1-Q)^{n-1}\approx Q\,e^{-n\,Q},$ (49)

where $Q\ll 1$ is the probability for a given spacing to be greater than
$\Delta$:

$Q\equiv Q(\Delta)=\int_{\Delta}^{1}P(s)\,ds.$ (50)

Now let us define the conditional probability ${\cal P}(E|n)$ that the string
of $n$ levels has a width $E$:

${\cal P}(E|n)=\prod_{k=1}^{n-1}\int
ds_{k}\,P(s_{k})\,\delta\left(\sum_{i=1}^{n-1}s_{i}-E\right).$ (51)

For independently fluctuating spacings $s_{k}$ this probability distribution
function (PDF) is easy to calculate in terms of its (simple) Laplace
transform:

$\tilde{{\cal P}}(t|n)=\int_{0}^{\infty}{\cal
P}(E|n)\,e^{-t\,E}=[g(t)]^{n-1},$ (52)

where $g(t)$ is defined by Eq.(27).

The PDF $P_{\Delta}(E)$ of the width $E$ of a string of $n$ levels with all
spacings $s_{i}<\Delta$ separated from the rest of the spectrum by two gaps of
the size larger than (or equal to) $\Delta$ is then given by:

$P_{\Delta}(E)=\sum_{n=1}^{M}{\cal P}(E|n)\,p_{\Delta}(n).$ (53)

Now we impose the last and the most important condition to be fulfilled for
such a string of $n$ levels to be a mini-band. It is the condition that the
typical width of the string $E_{{\rm typ}}(\Delta)$ is smaller than or equal
to $\Delta$:

$E_{{\rm typ}}(\Delta)={\rm
exp}\left[\int\ln(E)\,P_{\Delta}(E)\,dE\right]\lesssim\Delta.$ (54)

Eq.(54) is one of the principal results of this paper. Together with
Eqs.(49)-(53) it defines the region where the hierarchy of mini-bands, or the
fractality of local spectrum, is present (see Fig.9). The boundaries
$\Delta_{{\rm min}}$ and $\Delta_{{\rm max}}$ of this region are given by the
solutions for $\Delta$ of the equation at an equality sign in Eq.(54).

###  7.2 Examples of a random Cantor set and the dense and sub-dense spectrum

In order to test our approach we compute in Appendix A the typical width of a
mini-band $E_{{\rm typ}}(\Delta)$ for the case of a power-law $P(s)$, Eq.(16),
and show that for $\Delta\gg\delta$:

$\displaystyle\frac{\Delta}{E_{{\rm typ}}(\Delta)}\sim\left\\{\matrix{1,&$for
a random Cantor
set$\;\;D=D_{s}<1\cr\ln^{-1}\left(\frac{\Delta}{\delta}\right)\ll 1,&$for a
sub-dense spectrum$\;\;D=D_{s}=1\cr\left(\delta/\Delta\right)^{D_{s}-1}\ll
1,&$for a dense spectrum$\;\;D=1,\;D_{s}>1}\right.$ (55)

As expected, this calculation showed that only the case of a random Cantor set
obeys the criterion of fractality, Eq.(54), while the dense and sub-dense
spectra violate it. More complicated examples are illustrated in Fig.9

##  8 Conclusion

The goal of this paper was to explain in simple terms what does the singular-
continuous spectrum mean from the physics point of view. In contrast to the
mathematical literature where a continuous limit is always assumed, we
consider a discrete system of a finite (but large) number of degrees of
freedom $N$ assuming a coarse-graining at small distances (e.g. a lattice with
a fixed lattice constant $a=1$) and taking the limit $1/N\rightarrow 0$ only
in the final results.

Starting from a regular textbook Cantor set we increased complexity of the
model of singular continuous spectrum by considering first a random Cantor set
and then generalizing it to the hierarchy of mini-bands.

The key ingredient of such constructions was the power-law tailed distribution
function $P(s)$ of independent identically distributed spacings between levels
seen in the local density of states (LDoS). We have shown how such a tailed
distribution emerges in the limit $N\rightarrow\infty$ as the result of
discrimination that discards states with the amplitude in the observation
point smaller than $N^{-1}$. This mechanism explains a qualitative difference
between the fractal local spectrum (seen in LDoS) and the dense global
spectrum (seen in the global DoS)in the non-ergodic extended phases of
disordered systems.

We have demonstrated that within our simple model there are two classes of the
local singular-continuous spectra in the non-ergodic extended phases: (i) the
random Cantor set and (ii) the isolated mini-band. They are realized in the
critical Power-Law Banded Random Matrices (CPLBRM) [21, 42, 43, 44] and in the
Rosenzweig-Porter (RP) RMT [3, 4, 7], respectively. Using the model of
independent identically distributed (i.i.d.) local level spacings we
reproduced in both cases (i) and (ii) the correlation functions $K(\omega)$ of
the local densities of states which was previously known from the analytical
and numerical studies [42, 43, 44, 3, 45, 36, 23]. This confirms our
conjecture that the simplified picture of i.i.d. local level spacings with the
power-law distribution is a correct qualitative description of the above two
limiting cases and that the CPLBRM and RP are characterized by the local
singular continuous spectrum of the type of the Cantor set and of the type of
the isolated mini-band, respectively.

Our model is characterized444besides the typical level spacing $\delta=1/N$
which in extended phases is supposed to be of the same order both in the
global and in the local spectrum by two fractal dimensions: the eigenfunction
fractal dimension $D$ that describes the number of states $N^{D}$ (with the
typical amplitude of wave function $|\psi|^{2}\sim N^{-D}$) seen in the local
spectrum after the discrimination, and the spectral fractal dimension $D_{s}$
which determines the power-law distribution of local spacing, Eq.(16).
Depending on the relationship between them, different types of local spectra
emerge from our model. All our results are summarized in Table 1.

Table 1: The character of local spectrum, $K(\omega)$ and width of a mini-band
$\Gamma$

condition | local spectrum | $K(\omega)$ | $\Gamma$
---|---|---|---
$D=D_{s}<1$ | random Cantor set | $\omega^{-(1-D_{s})},\delta<\omega<1$ | 1
$D=1,D_{s}>1$ | dense set of levels | $\sim 1$ | 1
| isolated mini-band |  $\Gamma^{-D_{s}}\omega^{-(1-D_{s})}$, $\delta<\omega<\Gamma$, |
$D<D_{s}<1$ | random Cantor set inside |  $\Gamma^{D_{s}}\omega^{-(1+D_{s})}$, $\Gamma<\omega<1$ | $N^{-(1-D/D_{s})}$
| isolated mini-band |  $\Gamma^{-1}\ln(N^{D})\ln^{-1}(\omega/\delta)$, $\delta<\omega<\Gamma$, |
$D<D_{s}=1$ | sub-dense spectrum inside |  $\Gamma\omega^{-2}\,\ln^{-1}(N^{D})$, $\Gamma<\omega<1$ | $N^{-(1-D)}\ln(N^{D})$
| isolated mini-band |  $\Gamma^{-1}$, $\delta<\omega<\Gamma$ |
$D<1,D_{s}>1$ | dense spectrum inside |  $\Gamma^{-1}(\Gamma/\omega)^{1+D_{s}}\,N^{-D(D_{s}-1)}$, $\Gamma<\omega<1$ | $N^{-(1-D)}$
$D=D_{s}=1$ | inconsistent with |  |
$D_{s}<D\leq 1$ | $\delta_{{\rm typ}}\sim\delta$ |  |

Three of these cases shown in the rows $1$ [Fig.6(a)], $2$ [Fig.6(c)] and $5$
[Fig.6(d)] have well-known Hamiltonian realizations: the multifractal states
of the critical Power-Law Banded Random Matrices and those at the critical
points of the Anderson localization transitions (row 1), the ergodic states of
the Wigner-Dyson RMT (row 2) and the fractal states in the Gaussian
Rosenzweig-Porter RMT (row 5 with $D_{s}=1+\epsilon$, $\epsilon\rightarrow
0$). We are confident that the cases presented in row $3$ [a mini-band with a
random Cantor set, Fig.6(a), inside ] and row $4$ [a mini-band with a sub-
dense spectrum of Fig.6(b) inside] also have their Hamiltonian realizations.
To find them is an interesting problem of both fundamental and applied
significance 555The two cases presented in the row $6$ are inconsistent with
the assumption that the typical level spacing in the local spectrum
$\delta_{{\rm typ}}\sim\delta$ is of the same order as that in the global one
(see footnote 1). .

It is important that in all the cases when the eigenfunction fractal dimension
$D<1$ (non-ergodic extended states) our model gives a singular continuous
spectrum of the type of a random Cantor set or an isolated mini-band. An
absolutely continuous local spectrum of Fig.6(a) emerges only if $D=1$.

In the limit $N\rightarrow\infty$ we obtain from Table 1:

$\lim_{N\rightarrow\infty}K(\omega)\sim\left\\{\matrix{\omega^{-1+D_{s}},&$for
a random Cantor set$\cr\delta(\omega),&$for an isolated mini band$\cr 1,&$for
an absolutely continuous spectrum $}\right.$ (56)

Eq.(56) expresses a very general important result: The LDoS correlation
function in the limit $N\rightarrow\infty$ is a measure that distinguishes
between the absolutely continuous and the singular continuous spectra in
delocalized phases. While the former results in a finite limit for
$\lim_{N\rightarrow\infty}K(\omega)$ at $\omega\rightarrow 0$, the latter
implies the singularity at $\omega=0$. Furthermore, the type of the
singularity unambiguously identifies the type of a singular-continuous
spectrum. For the localized eigenstates the typical level spacing in the local
spectrum is finite in $N\rightarrow\infty$ limit and, hence,
$K(\omega\rightarrow 0)\rightarrow 0$ in this limit.

Extending the simple picture described above to an arbitrary distribution of
local level spacing, we formulated a general criterion, Eq.(54), of fractality
of local spectra and tested it by considering the examples of the random
Cantor set, the isolated mini-band and the dense spectrum. The criterion
Eq.(54) is one of the main results of the paper which can be used to
characterize more complicated local spectra in the non-ergodic extended
phases.

Acknowledgment. We are grateful to E. Bogomolny, M. V. Feigel’man, I. M.
Khaymovich, P. Nosov, A. Scardicchio and M. Zirnbauer for stimulating
discussions and useful comments on the manuscript. V.E.K. also appreciates
collaboration with L. B. Ioffe and M. Pino during the work on Ref.[11] where
some preliminary results of this paper were communicated. The support from the
Google Quantum Research Award “Ergodicity Breaking in Quantum Many-Body
Systems” (V.E.K.) is gratefully acknowledged.

##  Appendix A A test of Eq.(54) for a power-law $P(s)$ in Eq.(16)

###  A.1 Example of a random Cantor set

In order to illustrate a general formalism Eqs.(49)-(54) we consider an
example of a random Cantor set with $D=D_{s}<1$, where $D$ is defined in
Eq.(48).

Consider Eq.(52) in the limit $n\gg 1$ with $g(t)$ from Eq.(31):

$[g(t)]^{n-1}\approx{\rm exp}[-n(t\,\delta)^{D_{s}}].$ (57)

Then the PDF ${\cal P}(E|n)$ is given by the Laplace transform:

${\cal P}(E|n)=\int_{B}{\rm exp}\left[-n(t\,\delta)^{D_{s}}+Et\right]\,dt,$
(58)

where the integral runs over the Bromwich contour $t\in(-i\infty+0,i\infty+0)$
as customary for the inverse Laplace transform 666As a matter of fact, Eq.(58)
defines a PDF of the Levy $\alpha$-stable distribution with $\alpha=D_{s}$. .

Next we do summation over $n$ according to Eqs.(53),(49):

$P_{\Delta}(E)=Q\int_{B}\frac{e^{Et}}{Q+(t\,\delta)^{D_{s}}}\,\frac{dt}{2\pi
i},$ (59)

using the smallness of $Q$ and $t\delta$ to approximate
$1-e^{-(Q+(t\delta)^{D_{s}})}\approx Q+(t\delta)^{D_{s}}$ and large values of
$M(Q+(t\delta)^{D_{s}})\gg 1$ to extend summation over $n$ to infinity.
Integrating $e^{Et}$ over the Bromvich contour first to obtain $\delta(t)$,
one can easily check that the distribution function $P_{\Delta}(E)$ is
correctly normalized.

The Bromvich contour in Eq.(59) can be deformed to go along the branch cut
$(-\infty-i0,0)\cup(0,-\infty+0)$. Then switching $t\rightarrow-t$ we obtain:

$P_{\Delta}(E)=\frac{\sin\left(\pi
D_{s}\right)}{\pi}\,Q\int_{0}^{\infty}dt\,e^{-Et}\,\frac{(t\delta)^{D_{s}}}{Q^{2}+(t\delta)^{2D_{s}}+2Q(t\delta)^{D_{s}}\,\cos(\pi
D_{s})}.$ (60)

The integral in this expression is controlled by a parameter $E/E_{*}$, where:

$E_{*}=\frac{\delta}{Q^{\frac{1}{D_{s}}}},$ (61)

and in the limiting cases $E\ll E_{*}$ and $E\gg E_{*}$ it is equal to:

$P_{\Delta}(E)=\frac{\sin\left(\frac{\pi
D_{s}}{2}\right)}{\pi}\,\left\\{\matrix{\frac{\Gamma(1-D_{s})}{E}\,\left(\frac{E}{E_{*}}\right)^{D_{s}},&(E\ll
E_{*})\cr\frac{\Gamma(1+D_{s})}{E}\,\left(\frac{E_{*}}{E}\right)^{D_{s}},&(E\gg
E_{*})}\right.$ (62)

Figure 10: (Color online) The distribution of local spacings in the log-log
coordinates (Left panel) and the corresponding distribution of the width $E$
of a mini-gap at a scale $\Delta$ (Right panel) for the random Cantor set. The
main contribution to the normalization integral of $P_{\Delta}(E)$ is given by
$E_{*}=E_{{\rm typ}}(\Delta)$.

The sketch of the distribution function $P_{\Delta}(E)$ and the corresponding
spacing distribution is shown in Fig.10. As the integral of $P_{\Delta}(E)$
over $E<E_{*}$ (the normalization integral) is dominated by the upper limit
$E\sim E_{*}$, and the corresponding integral over $E>E_{*}$ is dominated by
the lower limit $E\sim E_{*}$, the main contribution to the normalization
integral of $P_{\Delta}(E)$ comes from the vicinity of $E\approx E_{*}$ given
by Eq.(61). By definition $E_{*}$ is the typical width $E_{{\rm typ}}(\Delta)$
of a mini-band at a scale $\Delta$.

The last step is to substitute $Q=Q(\Delta)$ in Eq.(61). For the local spacing
distribution function $P(s)$, Eq.(16), one readily finds:

$Q(\Delta)\sim\left(\frac{\delta}{\Delta}\right)^{D_{s}},$ (63)

so that

$E_{*}=E_{{\rm typ}}(\Delta)\sim\Delta.$ (64)

We conclude, therefore, that for the random Cantor set, like for its regular
counterpart, the typical width of a mini-band at a scale $\Delta$ is of the
order of $\Delta$. This saturates the criterion, Eq.(54) which is therefore
fulfilled for any $\Delta>\delta$. This result was expected but it serves as
an important test for the consistency of the formalism Eqs.(49)-(54) developed
in Section 7. Since the random Cantor set is considered as an archetypical
example of a fractal spectrum, the fulfillment of the criterion Eq.(54) in
general signals on the fractality of the local spectrum.

###  A.2 Example of an isolated mini-band

Now we consider the case $D<D_{s}$ in Eqs.(48),(16) which corresponds to an
isolated mini-band. First of all we note that Eq.(59) is only valid if one can
neglect the boundary term in the summation of geometric series. This
necessarily requires $MQ=N^{D}Q\gg 1$, which by the virtue of Eq.(63) implies:

$\Delta\ll\Delta_{{\rm
max}}=\Gamma\sim\left\\{\matrix{N^{-1+\frac{D}{D_{s}}},&D_{s}<1\cr
N^{-1+D},&D_{s}\geq 1}\right..$ (65)

Within this restriction let us consider the character of spectrum inside a
mini-band of the width $\Gamma$. If $D_{s}<1$ the results of the previous
subsection [Eq.(35) with $M\sim N^{D}$] show that the spectrum inside a mini-
band is a random Cantor set with the hierarchy of mini-bands.

If $D<1$ and $D_{s}=1$ we have using Eq.(32):

$P_{\Delta}(E)\approx\frac{Q}{2}\int_{0}^{\infty}dt\,e^{-Et}\,\frac{(t\delta)}{(Q-(t\delta)\ln(t\delta))^{2}}=\left\\{\matrix{\frac{Q}{2\delta\,\ln^{2}(E/\delta)},&\delta\ll
E\ll E_{*}\cr\frac{\delta}{2QE^{2}},&\Gamma\gg E\gg E_{*}}\right.,$ (66)

where $Q=\delta/\Delta$ and

$E_{*}=\Delta\ln(\Delta/\delta).$ (67)

As in the previous subsection, one can see that the main contribution to the
normalization integral for $P_{\Delta}(E)$ is given by $E\approx E_{*}$, so
that $E_{{\rm typ}}(\Delta)=E_{*}$. Therefore,

$E_{{\rm typ}}(\Delta)>\Delta$ (68)

in the entire region $\delta\ll\Delta<\Gamma$. The failure to fulfill the
criterion Eq.(54) implies that for $D<D_{s}=1$ the spectrum inside the mini-
band does not have a hierarchical structure. At the same time it has large
gaps and thus is not completely dense. The similar spectrum appears in the
entire band for the case $D=D_{s}=1$ (see Fig.6(b)).

Finally, let us consider the case $D<1$ but $2>D_{s}>1$. In this case it
follows from Eq.(33) that:

$P_{\Delta}(E)\approx
Q\int_{B}\frac{e^{Et}}{Q+c_{1}(t\delta)+c_{2}(t\,\delta)^{D_{s}}}\,\frac{dt}{2\pi
i},$ (69)

where $c_{1}$ and $c_{2}$ are coefficients of order 1.

The Bromwich contour $B$ of integration can be deformed to encircle the cut
$(-\infty,0)$. Then at $E/\delta\gg 1$ the main contribution comes from
$t\delta\ll 1$, so that the term $c_{2}(t\,\delta)^{D_{s}}$ is a correction.
The leading $t$-dependence leads to a pole at $c_{1}t\delta=Q$ with the
result:

$P_{\Delta}(E)|_{{\rm pole}}=\frac{c_{1}}{E_{*}}\,e^{-(E/E^{*})}$ (70)

where

$E_{*}=\frac{c_{1}\delta}{Q}.$ (71)

This pole contribution is almost constant for $E\ll E_{*}$ and it is
exponentially small at $E\gg E_{*}$. In this latter region one should take
into account a correction $c_{2}(t\,\delta)^{D_{s}}$ which takes different
values at $t=-x+i0$ and $t=-x-i0$, $(x>0)$:

$P_{\Delta}(E)|_{{\rm corr}}=\frac{c_{2}\,\sin(\pi
D_{s})}{\pi}\int_{0}^{\infty}dx\,e^{-Ex}\,\frac{(x\delta)^{D_{s}}}{Q}$ (72)

Collecting the results at $E\ll E_{*}$ and $E\gg E_{*}$ we finally obtain:

$P_{\Delta}(E)\sim\left\\{\matrix{\frac{e^{-E/E_{*}}}{E_{*}},&\delta\ll E\ll
E_{*}\ln[c(E_{*}/\delta)^{D_{s}-1}]\cr\left(\frac{\delta}{E}\right)^{D_{s}+1}\frac{E_{*}}{\delta^{2}},&\Gamma\gg
E\gg E_{*}\ln[c(E_{*}/\delta)^{D_{s}-1}]}\right.,$ (73)

where $c\sim 1$.

One can see again that the main contribution to the normalization integral
comes from $E\sim E_{*}$. So, from Eqs.(63),(71) we obtain:

$E_{{\rm typ}}(\Delta)\sim
E_{*}\sim\Delta\,\left(\frac{\Delta}{\delta}\right)^{D_{s}-1}\gg\Delta,$ (74)

for all $\Delta\gg\delta$. This result remains valid for all $D_{s}>1$.

In this case the length of the string of levels with all spacing between them
smaller than $\Delta$ is larger by a polynomially large parameter
$(\Delta/\delta)^{D_{s}-1}$ than the gaps between such strings. This makes
large gaps extremely rare which means dense typical local spectrum inside a
mini-gap at $D<1$ and that in the entire spectral band for $D=1$ (see
Fig.6(c)).

###  A.3 Example of a dense spectrum

This case is easy to obtain from the results of the previous subsection at
$D=1$ and $D_{s}>1$. The only modification needed is to replace $\Delta_{{\rm
max}}=\Gamma$ given by Eq.(65) by the entire band-width $\Delta_{{\rm
max}}=\Gamma=1$. After this modification Eq.(73) remains valid in the entire
region $\delta<E<1$, and Eq.(74) gives:

$\frac{\Delta}{E_{{\rm
typ}}(\Delta)}\sim\left(\frac{\delta}{\Delta}\right)^{D_{s}-1}\ll 1.$ (75)

Thus the criterion of fractality of local spectrum, Eq.(54), is strongly
violated as it should be for the dense spectrum.

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