# Fast Exact NPN Classification with Influence-aided Canonical Form ††thanks:
1These authors contributed equally to this work;
${}^{\text{\Letter}}$Corresponding authors.

Yonghe Zhang1,1, Liwei Ni2,4,1, Jiaxi Zhang${}^{3,\text{\Letter}}$, Guojie
Luo3, Huawei Li4,5 and Shenggen Zheng${}^{2,\text{\Letter}}$ 1Shenzhen
University, Shenzhen, China  2Peng Cheng Laboratory, Shenzhen, China  3School
of Computer Science, Peking University, Beijing, China  4University of Chinese
Academy of Sciences, Beijing, China  5Institute of Computing Technology,
Chinese Academy of Sciences, Beijing, China  Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

NPN classification has many applications in the synthesis and verification of
digital circuits. The canonical-form-based method is the most common approach,
designing a canonical form as representative for the NPN equivalence class
first and then computing the transformation function according to the
canonical form. Most works use variable symmetries and several signatures,
mainly based on the cofactor, to simplify the canonical form construction and
computation. This paper describes a novel canonical form and its computation
algorithm by introducing Boolean influence to NPN classification, which is a
basic concept in analysis of Boolean functions. We show that influence is
input-negation-independent, input-permutation-dependent, and has other
structural information than previous signatures for NPN classification.
Therefore, it is a significant ingredient in speeding up NPN classification.
Experimental results prove that influence plays an important role in reducing
the transformation enumeration in computing the canonical form. Compared with
the state-of-the-art algorithm implemented in ABC, our influence-aided
canonical form for exact NPN classification gains up to 5.5x speedup.

###### Index Terms:

NPN Classification, Canonical From, Influence

##  I Introduction

Boolean function NPN (Negation-Permutation-Negation) classification groups
functions into NPN equivalent classes. Each function in the same equivalent
class can be obtained from the other by three transformations: Negating the
inputs, Permuting the inputs, or Negating the output. NPN classification is
one of the critical steps in many applications, such as logic synthesis [1,
2], technology mapping [3], and verification [4]. The speed of classification
will directly affect the efficiency of these tools because of the exploding
search space of the Boolean function.

Due to its great importance, NPN classification has been well studied in the
past decades. The canonical-form-based approach is one of the most commonly
used methods for exact NPN classification. First, it designs a complete and
unique representation for a Boolean function called canonical form. This
canonical form is taken as the representative, and two functions fall into the
same NPN equivalent class when their canonical forms are the same. Then a
computation algorithm decides the output polarity, the phase assignment, and
the input order of each variable for a function to get the minimum canonical
form.

Truth table is a basic canonical form, and the exhaustive transformation
enumeration is the most primitive computation method. However, $2^{n+1}n!$
transformations ($2^{n}$ for input negation, $n!$ for input permutation, $2$
for output negation) should be enumerated for an $n$ input function, which is
impractical when $n$ is large. Many optimization methods have been proposed to
improve the NPN classification efficiency, including constructing new
canonical forms [5, 6, 7, 8] and designing better computation methods [9, 10,
11, 12, 13, 14, 15]. Pruning transformation enumeration is the key to reducing
computation time in the canonical-form-based approach. Various works exploited
several signatures [5, 9, 14, 15] and variable symmetry [9, 10, 11, 12, 13,
14, 15] to prune the enumeration space. Signatures are compact representations
that characterize a Boolean function’s properties. However, these signatures
also need to be computed, introducing additional runtime. A co-design of
canonical form and computation algorithm [15] is required to achieve better
speedup.

In Boolean functions analysis, the concept of influence is defined to measure
the probability that one input variable affects the value of the function [16,
17]. The value of influence of a variable $x_{i}$ can be computed easily from
Boolean difference [18]. This concept has been applied to hardware security
[19, 20] since inputs with high influence are of more interest in critical
applications. Existing work has proved that Boolean influence can be regarded
as a type of signature for Boolean matching [21]. But this work does not
provide an exact matching solution for the functions in the same NPN classes,
posing challenges for its practical application. This paper explores Boolean
influence’s great potential in canonical-form-based exact NPN classification.

The main contributions are summarized as the following:

  * •

We redefine the concept of Boolean influence and highlight that computing the
redefined influence using commonly utilized cofactor signatures is
straightforward. Consequently, incorporating influence into canonical form-
based method requires only a modest amount of additional computation.

  * •

We analyze some properties of influence in NPN classification scenarios,
including that it is phase-independent and permutation-dependent. Such
properties can help reduce the enumeration times of permutation in canonical
form computation.

  * •

We devise two influence-aided canonical forms based on the existing canonical-
form-based NPN classification method. One directly introduces Boolean
influence into the canonical form, while the other replaces some signatures of
high computation complexity with influence.

  * •

The experiments demonstrate that both proposed two influence-aided canonical
forms in this paper effectively reduce the number of final transformation
enumerations and achieve significant speedup compared to the state-of-the-art
method.

##  II Background

In this section, we first introduce several commonly used signatures and
techniques in NPN classification. Subsequently, we illustrate a state-of-the-
art canonical form-based classification method.

###  II-A Frequently Used Signatures

We first list some basic notations before introducing the definitions of
signatures. An $n$-input and single-output Boolean function, $f(X):B^{n}\to
B$, $B$=$\\{0,1\\}$, where $X$=$(x_{1},x_{2},...,x_{n})$,$x_{i}\in B$,$1\leq
i\leq n$ is a binary vector with size $n$, representing the input variables.
$X$ can be considered as a binary number $(x_{n}x_{n-1}...x_{1})_{2}$, whose
value is $m$, denoted as $X_{(m)}$. Besides, we denote $X^{i}$ as negating the
$i$-th variable in $X$. The truth table
$T(f)=(f(X_{2^{n}-1}),...,f(X_{(1)}),f(X_{(0)}))$ is a bit vector with size
$2^{n}$.

The _satisfy count_ of $f$ is the number of all minterms for which $f$
evaluates to 1, denoted as $|f|$, where $|f|=\sum_{X\in B^{n}}f(X)$. A
_cofactor_ of $f$ with respect to a literal $x_{i}$($\overline{x_{i}}$) is the
function derived by setting $x_{i}$($\overline{x_{i}}$) to 1 and denoted by
$f_{x_{i}}$($f_{\overline{x_{i}}}$). The 0th-order cofactor signature is the
satisfy count, and the 1st-order cofactor signature is the satisfy counts of
the cofactors with respect to each variable, denoted as
$S_{cof}(f)=\\{|f|,|f_{x_{1}}|,...,|f_{x_{n}}|\\}$. Moreover, higher-order
cofactor signature is the satisfy counts of the cofactors with respect to
multiple variables [12].

Boolean functions can be represented as the disjunction of minterms. In this
representation, _row sums_ (RS) is a sorted vector composed of the number of
positive variables in each minterm. A single value can be computed by adding
the square of each value in row sums, called _sum of squared row sums_ (SSRS).
Another efficient value _sum of exponential row sums_ (SERS) can be obtained
from RS [8, 15]. The SERS of a Boolean function is the 0th-order _shifted-
cofactor_ signature, denoted as $S^{0}_{scc}(f)=\|f\|$; the 1st-order shifted-
cofactor signatures are SERS of the cofactors with respect to each variable,
denoted as $S^{1}_{scc}(f)=\\{\|f_{x_{1}}\|,...,\|f_{x_{n}}\|\\}$.

###  II-B Variable Symmetries

Two variables $x_{i}$ and $x_{j}$ are symmetric in $f$ if the function value
does not change when $x_{i}$ and $x_{j}$ are swapped. A symmetric group is
composed of symmetric variables. The higher-order symmetry is defined between
two or more symmetry groups if they can be swapped without changing the
function value. In some cases, one variable in a symmetry group is not
symmetric with each variable in another, but these two symmetry groups are
higher-order symmetric.

Symmetry variables and higher-order symmetry groups can be permuted and
negated like a single variable during the canonical form computation, thus
significantly reducing the number of transformation enumerations. Exploiting
variable symmetries and higher-order symmetries are adopted in many NPN
classification methods [7, 12, 13, 11, 14, 15].

###  II-C Canonical-Form-based Method

The key of the canonical-form-based method lies in determining the output
polarity, phase assignment, and input order of a Boolean function to get the
canonical form of its equivalent class. The output polarity is associated with
output negation, while the input phase assignment corresponds to input
negation. Furthermore, the input order corresponds to the permutation of
inputs.

The transformation space is vast ($2^{n+1}n!$), and signatures can help
effectively prune the space. If a signature fulfills the following
requirements, it can help to determine partially the three types of
transformations. The following three requirements can be met: 1)_Phase
requirement_. The signature value of a single variable is determined solely by
its own phase assignment and remains independent of the phase assignment of
other variables. 2)_Permutation requirement_. Applying a permutation to the
variables leads to the same permutation of the signature values for each
variable. 3)_Polarity requirement_. The signature value of the function is
primarily determined by the polarity of the output and remains independent of
the permutation and phase assignment of input variables.

Cofactor signatures fulfill the three requirements. They can be used to
determine the output polarity and partially decide the input order and the
phase of input variables. This means that cofactor signatures can reduce part
of negation and permutation enumeration. In addition, the computational cost
of the cofactor signatures is low. Therefore, many works used cofactor
signatures to build their NPN classification methods [9, 12, 11, 14]. Since
the order of the variables does not affect the number of positive variables in
the minterm, RS and SSRS are permutation-independent, which means these two
values do not fulfill the polarity requirement and permutation requirement.
Therefore, these two signatures cannot distinguish polarity and permutation.
Some works used these two signatures to determine phase assignments of input
variables [9, 14].

Pruning permutation is more critical because permutation enumeration space
($n!$) is more significant than phase assignment enumeration ($2^{n}$) when
$n$ is larger than 4. Shifted-cofactor signatures fulfill the permutation
requirement and have proved more effective than cofactor in distinguishing the
permutation [8]. Thus, the integration of cofactor and shifted cofactor in the
canonical form [15] outperforms the utilization of cofactor alone with SSRS in
terms of efficacy [14]. Furthermore, the state-of-the-art work [15] introduces
cost signature $C_{p}$ to assess the overhead of permutations under different
phase assignments, aiming to select more promising phase assignments for
permutation enumeration and ultimately achieve better speedup.

(a) Original enumeration space.

(b) After Variable grouping.

(c) After symmetry detection.

(d) After cost estimation.

Figure 1: Schematic of the remaining enumerations at different pruning stages,
grey points are pruned enumerations. (a) All phase assignments and
permutations are candidates to obtain the canonical form at the beginning. (b)
After variable grouping, candidate enumerations are separated into different
groups. (c) Variable symmetry detection further prunes the candidate
enumerations. (d) After cost estimation, only three transformations need to be
enumerated to finish the final exhaustive enumeration.

Figure 1 shows the schematic of the remaining enumeration space at different
pruning stages of the cost-aware hybrid signature canonical form proposed in
previous work [15]. First, they decide the output polarity and group variables
by cofactor signatures. Certain enumerations of phase assignments can be
pruned through this step. Subsequently, symmetry detection within each group
will be conducted to eliminate certain phase assignments and permutations.
Next, for the variables that have yet to determine their phase assignments and
permutations, the cost signature is utilized to identify phase assignments
with lower permutation costs. Then, the shifted-cofactor signature is employed
to further prune permutations. After all pruning strategies, an exhaustive
enumeration of residual transformations is used to achieve exact NPN
classification. However, the effectiveness of pruning permutations using the
shifted-cofactor signature remains limited. This paper discovers a more
effective signature to prune permutations, Boolean influence.

##  III The Power of Influence

In this section, we provide an in-depth examination of Boolean influence in
the context of NPN classification, including its redefined definition, facile
computation, and some properties. For primitive definitions and proofs of
difference and influence, one may refer to [17, 18].

###  III-A Redefined Influence and Facile Computation

In canonical form-based NPN classification methods, some signatures can be
used to prune the search space of transformation enumerations. But the
computation of these signatures cannot be too complicated. Otherwise, the
additional computational overhead will diminish the benefit. To better explain
the computation of Boolean influence, we redefine it using difference [18].

######  Definition 1.

(_difference_) [18]. Given a Boolean function $f(X)$, the Boolean difference
with respect to a basic variable $x_{i}$ is defined as follows:

$\frac{\delta f}{\delta x_{i}}=f(x_{1},\cdots,x_{i},\cdots,x_{n})\oplus
f(x_{1},\cdots,\overline{x_{i}},\cdots,x_{n}).$

######  Definition 2.

(_redefined influence_). The influence of the variable $x_{i}$ on $f$ is
defined as follows:

$inf_{i}(f)=\sum\limits_{X\in B^{n}}\frac{\delta f}{\delta x_{i}}(X).$

We drop a global coefficient of $1/2^{n}$ in the above definition compared to
the original definition in [17]. The influence of a variable $x_{i}$ is the
number of functions that flipping the variable $x_{i}$ flips the value of the
function. In some sense, one can see the influence of a variable is a measure
of the importance of the variable in all input variables. From the above
definition, influence can be computed for each input variable. It has the
potential to be a signature of a Boolean function to distinguish variables for
NPN classification.

The 1st-order cofactor is easy to compute, and many canonical-form-based
methods have adopted it. The characteristics of cofactor and influence seem to
be quite different from each other. However, they are relevant to each other
in some respect. We will show some detailed relationships between these
concepts as followings. We can get the following Lemma from previous
definitions [18].

######  Lemma 1.

(_difference vs. cofactor_). Difference can be computed by cofactor as
follows:

$\frac{\delta f}{\delta x_{i}}(X)=f_{x_{i}}(X)\oplus f_{\overline{x_{i}}}(X).$

From Definition 2 and Lemma 1, we can easily derive a facile computation
method for the redefined influence.

######  Lemma 2.

(_influence computation_). The influence can be obtained from the cofactor
signature as follows:

$inf_{i}(f)=\sum\limits_{X\in B^{n}}\frac{\delta f}{\delta
x_{i}}(X)=\sum\limits_{X\in B^{n}}f_{x_{i}}(X)\oplus f_{\overline{x_{i}}}(X).$

Lemma 2 demonstrates that the influence can be intuitively computed using
cofactors. This also implies that if we define a canonical form using
influence, the additional computation overhead is low. The above two lemmas
also show that influence further explores cofactors. It explores the
difference between cofactor signature pairs and has extra structural
information. An $n$-input Boolean function can be seen as an $n$-dimensional
hypercube. Cofactor focuses on the statistical characteristics of a face
$x_{i}$ of the hypercube, while influence cares about the difference between
two opposite faces $x_{i}$ and $\overline{x_{i}}$ of the hypercube. These two
features are complementary in some sense.

###  III-B Understanding Boolean Influence in NPN Classification

To prune transformation enumerations by incorporating influence into the
canonical form, it is necessary to first understand the relationship between
influence signatures and transformations. It can be seen from the following
theorem that the influence signature changes only with permutation (P)
transformation, and any input phase assignment and polarity assignment will
not change the influence.

######  Theorem 1.

(_phase-independent_). Variable negation does not change Boolean influence.

###### Proof.

Supposing $g$ is derived from $f$ through the phase assignment of any input
$i$, $g(X)=f(X^{i})$. According to definition 1 and definition 2, we have
$\frac{\delta f}{\delta x_{i}}(X)=f_{x_{i}}(X)\oplus
f_{\overline{x_{i}}}(X)=g_{\overline{x_{i}}}(X)\oplus
g_{x_{i}}(X)=\frac{\delta g}{\delta x_{i}}(X)$. Thus,
$inf_{i}(g)={\Sigma}_{X\in B^{n}}\frac{\delta g}{\delta
x_{i}}(X)={\Sigma}_{X\in B^{n}}\frac{\delta f}{\delta x_{i}}(X)=inf_{i}(f)$. ∎

######  Theorem 2.

(_polarity-independent_). Output negation does not change Boolean influence.

###### Proof.

Supposing $g$ is derived from $f$ through the polarity assignment,
$g(X)$=$\overline{f(X)}$. According to definition 1 and definition 2, we have
$\frac{\delta g}{\delta x_{i}}(X)$=$g_{x_{i}}(X)\oplus
g_{\overline{x}_{i}}(X)$=$\overline{f_{x_{i}}(X)}\oplus\overline{f_{\overline{x_{i}}}(X)}$
= $f_{x_{i}}(X)\oplus f_{\overline{x_{i}}}(X)$=$\frac{\delta f}{\delta
x_{i}}(X)$. Thus, $inf_{i}(g)$ =$inf_{i}(f)$. ∎

######  Theorem 3.

(_permutation-dependent_). The influence of variable changes synchronously
with the permutation transformation.

###### Proof.

Assuming that $g(X)=f(\pi(X))$, where $\pi$ represents a permutation
transformation and $k=\pi(i)$, we observe the following: $g(X)=f(\pi(X))$ and
$g(X^{k})=f(\pi(X^{k}))=f(\pi(X)^{i})$.

According to definition 1 and definition 2, we have

$\displaystyle inf_{k}(g)$ $\displaystyle=\sum\limits_{X\in B^{n}}\frac{\delta
g}{\delta x_{k}}(X)=\sum\limits_{X\in B^{n}}g(X)\oplus g(X^{k})$
$\displaystyle=\sum_{X\in B^{n}}f(\pi(X))\oplus f(\pi(X)^{i}).$

An essential observation is that the set $\\{X|X\in B^{n}\\}=\\{\pi(X)|X\in
B^{n}\\}$, which leads us to:

$\displaystyle inf_{k}(g)$ $\displaystyle=\sum_{X\in B^{n}}f(\pi(X))\oplus
f(\pi(X)^{i})$ $\displaystyle=\sum_{X\in B^{n}}f(X)\oplus
f(X^{i})=inf_{i}(f).$

∎

From the above three theorems, it can be found that influence signature can
determine the permutation, which is very helpful for pruning permutation
enumeration.

###  III-C An Example of Influence

In order to reduce permutations, one common way used in previous work [14, 15]
is to group variables according to signatures. Variable grouping can separate
the permutation space into isolated ones, and permutations only appear in the
same group. For a 6-bit Boolean function:

$\displaystyle f=$
$\displaystyle\overline{x_{3}}\overline{x_{5}}x_{6}+\overline{x_{3}}x_{4}x_{6}+x_{3}\overline{x_{5}}\overline{x_{6}}+x_{3}x_{4}\overline{x_{6}}+$
$\displaystyle
x_{2}x_{3}\overline{x_{4}}x_{5}+x_{1}\overline{x_{3}}\overline{x_{4}}x_{5}$
(1) $\displaystyle=$ $\displaystyle(\mathrm{5DAE51AE5DA251A2})_{16}$ (2)

The cofactor signatures, shifted-cofactor signatures, and influence signatures
of each variable are listed in Table I. And Figure 2 also shows the logic
graph of $f$ with the support set of inputs in And-Inverter Graph format. If
the signatures (combination) of two variables are the same, then this
signature (combination) can not distinguish these two variables. These two
variables belong to the same group using such signature (combination). Table
II shows the variable grouping results using the different signature
combinations. For this Boolean function, the combination of cofactor and
influence signature splits two more groups than the combination of cofactor
and shifted-cofactor signature. This shows that the influence signature prunes
more permutation enumeration than the shifted-cofactor signature. This also
shows that the influence signature has great potential for permutation pruning
in the canonical form-based NPN classification method.

Figure 2: AND-Inverter graph for $f$=$(\mathrm{5DAE51AE5DA251A2})_{16}$ TABLE I: Signatures results. Signatures | $x_{1}$ | $x_{2}$ | $x_{3}$ | $x_{4}$ | $x_{5}$ | $x_{6}$
---|---|---|---|---|---|---
cofactor | 16 | 16 | 16 | 16 | 18 | 18
shifted-cofactor | 81 | 135 | 81 | 135 | 159 | 159
influence | 24 | 8 | 28 | 4 | 4 | 4
TABLE II: Variable grouping results. Signatures (Combination) | Variable Groups
---|---
cofactor |  $(x_{1},x_{2},x_{3},x_{4})$, $(x_{5},x_{6})$
cofactor+shifted-cofactor |  $(x_{1},x_{3})$, $(x_{2},x_{4})$, $(x_{5},x_{6})$
cofactor+influence |  $(x_{1})$, $(x_{2})$, $(x_{3})$, $(x_{4})$, $(x_{5},x_{6})$
All three signatures |  $(x_{1})$, $(x_{2})$, $(x_{3})$, $(x_{4})$, $(x_{5},x_{6})$

##  IV Hybrid Influence-aided Canonical Form

In the previous section, we pointed out that influence signature is
permutation-dependent and fulfills the permutation requirement, which means
influence has excellent potential for pruning permutation enumerations. In
this section, we introduce the hybrid influence-aided canonical form and its
computation algorithm.

###  IV-A Canonical Form Definition

######  Definition 3.

(_hybrid influence-aided signature vector_). The _hybrid influence-aided
signature vector_ $S_{hi}(f)$ of a Boolean function $f$ is denoted as a vector
composed of cofactor signatures $S_{cof}(f)$, influence signatures
$S_{inf}(f)$, the permutation cost signature $C_{p}(f)$, the 0th-order and
1st-order shifted-cofactor signatures $S^{0}_{scc}(f)$ and $S^{1}_{scc}(f)$,
followed by the truth table $T(f)$, that is,

$\displaystyle S_{hi}(f)=$ $\displaystyle
S_{cof}(f)S_{inf}(f)C_{p}(f)S^{0}_{scc}(f)S^{1}_{scc}(f)T(f)$ $\displaystyle=$
$\displaystyle\Big{(}\lvert f\rvert,|f_{x_{1}}|,|f_{x_{2}}|,...,|f_{x_{n}}|,$
$\displaystyle inf_{x_{1}}(f),inf_{x_{2}}(f),...,inf_{x_{n}}(f),$
$\displaystyle C_{p}(f),\|f\|,\|f_{x_{1}}\|,\|f_{x_{2}}\|,...,\|f_{x_{n}}\|,$
$\displaystyle f(X_{(2^{n}-1)}),...,f(X_{(1)}),f(X_{(0)})\Big{)},$

where the hybrid signatures are concatenated as a larger vector.

######  Theorem 4.

(_The uniqueness of $S_{hi}(f)$_). For two Boolean functions $f$ and $g$:
$f\neq g$ if and only if $S_{hi}(f)\neq S_{hi}(g)$.

###### Proof.

(1)Sufficiency: If $f\neq g$, then $T(f)\neq T(g)$. Thus, $S_{hi}(f)\neq
S_{hi}(g)$.

(2)Necessity: If $S_{hi}(f)\neq S_{hi}(g)$, then there is at least one part
$s\in\\{S_{cof},S_{inf},C_{p},S_{scc},T\\}$ s.t. $s(f)\neq s(g)$. If $s$ =
$S_{inf}$, according to the definition of $S_{inf}$, there is at least one
variable $x_{i}$ s.t. $inf_{i}(f)\neq inf_{i}(g)$. However, if $f$=$g$, then
$inf_{x_{i}}(f)=inf_{x_{i}}(g)$. By contradiction, it is evident that $f\neq
g$. If $s\neq S_{inf}$, then $s$ must belongs to
$\\{S_{cof},C_{p},S_{scc},T\\}$. According to the existing works [8, 14, 15],
regardless of which part $s$ belongs to, there must be $g\neq f$. Therefore,
Theorem 4 holds. ∎

Theorem 4 shows that $S_{hi}(f)$ can uniquely and completely represent $f$,
and a canonical form can be derived from this vector.

######  Definition 4.

(_hybrid influence-aided canonical form_). The _hybrid influence-aided
canonical form_ of a Boolean function $f$, denoted by $\kappa_{hi}(f)$, is
defined as the function in $[f]$ with the minimum $S_{hi}(f)$, i.e.
$\kappa_{hi}(f)=min_{S_{hi}}([f])$, where $[f]$ represents the NPN equivalent
class of $f$.

######  Theorem 5.

(_$\kappa_{hi}(f)$ is an NPN canonical form_.) $\kappa_{hi}(g)=\kappa_{hi}(f)$
if and only if $g\in[f]$. That is, $f$ and $g$ belong to the same NPN
equivalent class.

###### Proof.

Let $h$=$\kappa_{hi}(f)$=$argmin(S_{hi}(k)|k\in[f])$, and $h$=$\pi_{f\to
h}(f)$. Then, we have $h\in[f]$.

(1)Sufficiency: $h\in[g]$ due to $h$=$\kappa_{hi}(g)$, thus $h$ can be
obtained from $g$ after NPN transformations; $h\in[f]$ due to
$h$=$\kappa_{hi}(f)$, thus $f$ can be obtained from $h$ after NPN
transformations; Thus, $f$ can be obtained from $g$ after NPN transformations
due to the transitivity of NPN equivalence, that is, $g\in[f]$.

(2)Necessity: If $g\in[f]$, then $[g]$=$[f]$. Thus,
$\kappa_{hi}(g)$=$min_{S_{hi}}([g])$=$min_{S_{hi}}([f])$=$\kappa_{hi}(f)$.

Therefore, Theorem 5 holds. ∎

Since $S_{hi}(f)$ can uniquely and completely represent $f$, the equation
$\kappa_{hi}(f)=min_{S_{hi}}([f])$ has a only unique value. Theorem 5
guarantees that each Boolean function has a unique _hybrid influence-aided
canonical form_.

###  IV-B Canonical Form Computation

The proposed algorithm in Algorithm 1 computes $\kappa_{hi}(f)$ for a given
function $f$. The basic idea of the algorithm is to prune the undetermined
phase assignment and permutation transformations based on $S_{hi}(f)$ and then
performing exhaustive enumeration to decide the undetermined transformations.
After exhaustive enumeration, we can get $\kappa_{hi}(f)$. Supposing that the
total number of Boolean functions is $m$ and each function has $n$ bits. The
size of truth table for each function is $2^{n}$. The time complexity to
compute cofactor is $O(mn2^{n})$. According to the computational method that
we give in Lemma 1 and 2, the time complexity to compute influence is also
$O(mn2^{n})$. Therefore, the complexity of Algorithm 1 is the same of the one
in [15].

Algorithm 1 Computing the Hybrid Influence-aided Canonical Form

1:Boolean function $f$ with $n$ input variables

2:Canonical form $\kappa_{hi}(f)$

3:Initialize $G$, $U_{phase}$ and $U_{perm}$

4:Decide output polarity by $S_{cof}(f)$ $\triangleright$ Pruning by
$S_{cof}(f)$

5:Update $G$ and $U_{phase}$ by $S_{cof}(f)$

6:Update $U_{perm}$ according to $G$

7:for group in $G$ do

8: Detect symmetry

9: Update $U_{phase}$ and $U_{perm}$

10:end for

11:Update $G$ using $S_{inf}(f)$ $\triangleright$ Pruning by $S_{inf}(f)$

12:Update $U_{perm}$ according to $G$

13:Generate all phase assignment $A_{phase}$ based on $U_{phase}$

14:for phase in $A_{phase}$ do $\triangleright$ Pruning by $S_{scc}(f)$

15: compute $C_{p}$ according to $S_{scc}^{0}(f)$ and $S_{scc}^{1}(f)$

16: if $C_{p}$ is minimum then

17: Refresh $C_{phase}$

18: end if

19:end for

20:for candidate in $C_{phase}$ do $\triangleright$ Exhaustive enumeration

21: Exhaustive enumeration according to $U_{perm}$

22: Record $f_{best}$ with the minimum truth table $T(f)$

23:end for

24:return $\kappa_{hi}(f)$=$f_{best}$.

Firstly, we initialize three empty sets to preserve groups ($G$), phase-
undetermined variables ($U_{phase}$), and permutation-undetermined variables
($U_{perm}$). The output polarity can be determined by $|f|$ (line 2). The
$S_{cof}(f)$ can help group variables and update $U_{phase}$ (line 3). Then we
update $U_{perm}$ based on the rule that variables in different groups cannot
be permuted (line 4). Next, symmetry variables are detected in all groups to
reduce variables in $U_{phase}$ and $U_{perm}$ (line 5 to line 8). The above
steps prune enumerations mainly based on cofactor signatures and variable
symmetries, which have been extensively studied by previous works [12, 11,
14]. After pruning by $S_{cof}(f)$, we update $G$ and $U_{perm}$ using
$S_{inf}(f)$ (line 9 to line 10). For each phase assignment in all possible
phase assignment set $A_{phase}$ generated based on $U_{phase}$, its
permutation cost signature $C_{p}$ is computed by the pre-trained weighted
function [15] with $S^{0}_{scc}(f)$ and $S^{1}_{scc}(f)$ (line 11 to line 13).
If the cost signature is the lowest of the phase assignments, refresh the
candidate phase assignment set $C_{phase}$ (line 14 to line 16). The cost-
aware enumeration estimation method using $S_{scc}(f)$ is proposed by [15].
For each candidate phase assignment in $C_{phase}$, exhaustively enumerate the
undetermined input order to get permutation transformations (line 17 to line
18). Thus, the output polarity, phase assignment, and input order of each
variable have already been decided, and we can get the minimum truth table as
the canonical form.

Here we give an example to explain the algorithm. For a function

$\displaystyle f$
$\displaystyle=x_{1}x_{2}+x_{1}(\overline{x_{5}+x_{6}(\overline{x_{3}}+x_{4})})+x_{2}(x_{5}+x_{6}(\overline{x_{3}}+x_{4}))$
$\displaystyle=(\mathrm{FFFF3777C8880000})_{16},$

the $S_{cof}(f)$=$(|f|,|f_{x_{1}}|,...,|f_{x_{6}}|)$=$(32,16,16,16,16,27,21)$,
and variables are separated into
$G$=$\\{(x_{1},x_{2},x_{3},x_{4}),(x_{5}),(x_{6})\\}$ (line 3). The
$U_{phase}$=$\\{x_{1},x_{2},x_{3},x_{4}\\}$ because $|f_{x_{1}}|$ to
$|f_{x_{4}}|$ are half of $|f|$ (line 3). The
$U_{perm}$=$\\{x_{1},x_{2},x_{3},x_{4}\\}$ since their cofactor signatures are
the same (line 4). Next, we only need to detect symmetry in
$G_{1}$=$\\{x_{1},x_{2},x_{3},x_{4}\\}$. Because variables $x_{3}$ and $x_{4}$
are symmetric, $x_{4}$ can be purged in $G_{1}$, $U_{perm}$ and $U_{phase}$
(line 5 to line 8), that is
$G_{1}$=$U_{perm}$=$U_{phase}$=$\\{x_{1},x_{2},x_{3}\\}$. After these steps,
the total number of residual enumerations is pruned to only $2^{3}$ $\times$
$3!$=48. Next, the $S_{inf}(f)$=$(6,10,2,2,10,22)$, so $G$ can be further
separated into five groups, $\\{(x_{1}),(x_{2}),(x_{3}),(x_{5}),(x_{6})\\}$
(line 9). The $U_{perm}$=$\varnothing$ because permutation only appears in the
same group (line 10). Since the influence signature is negation-independent,
$U_{phase}$ does not change. After the influence-aided permutation pruning,
input order is determined, and the residual enumerations only consist of phase
assignments, the number of which is $2^{3}$=8. According to the cost
estimation method described in [15], phase assignment
$\overline{x_{1}}\overline{x_{2}}x_{3}$ has the lowest permutation cost among
all 8 phase enumerations (line 12 to line 17). For this candidate phase
assignment, $U_{perm}=\varnothing$; thus, no exhaustive permutation is
required (line 18 to line 19). Then the algorithm gets the truth table with
the determined phase assignment and input order (line 20).

Figure 3 depicts the pruning results. It is a heat map of the remaining
enumerations. We divide the pruning stages into three parts, called pruning by
$S_{cof}(f)$ (line 2 to line 8), pruning by $S_{inf}(f)$ (line 9 to line 10),
and pruning by $S_{scc}(f)$ (line 12 to line 17). After each pruning stage, we
will increase its activity if an enumeration has not been pruned. The darkest
points in the figure represent the enumerations with the highest activity,
which means that these are irreducible. From this heat map, we can see more
clearly that influence is a vital ingredient for pruning permutations.

Figure 3: The heatmap of the remaining enumerations at different pruning
stages. The horizontal axis represents the permutation enumeration space,
encoded according to the lexicographical order from 1 to n; the vertical axis
represents the phase assignment enumeration space, according to binary
encoding. A point represents an NPN transformation, and all points constitute
the whole NPN transformation space.

##  V Optimized Influence-aided Canonical Form

###  V-A Runtime Analysis and Trade-off

Cost signature $C_{p}$ is proposed to estimate permutation cost and help
choose the phase assignment with the lowest permutation cost [15]. The
permutation cost is computed according to $S^{1}_{scc}$, which will bring
runtime overhead. In Section III-C, we have shown that influence signature
usually gains more groups for further permutation reduction than shifted-
cofactor signatures. Although using influence signature and cost signature
simultaneously can further prune the permutation space in some cases, the
runtime overhead of computing $C_{p}$ will reduce the benefits. In other
words, using only influence signature to prune permutation space may gain
shorter runtime.

###  V-B Optimized Influence-aided Canonical Form

######  Definition 5.

(_optimized influence-aided signature vector_). The _optimized influence-aided
signature vector_ of $f$ , denote by $S_{oi}(f)$, is a vector composed of
cofactor signatures $S_{cof}(f)$, influence signatures $S_{inf}(f)$, the 0th-
order shifted-cofactor signature $S^{0}_{scc}(f)$, followed by the truth table
$T(f)$, that is,

$\displaystyle S_{hi}(f)=$ $\displaystyle
S_{cof}(f)S_{inf}(f)C_{p}(f)S^{0}_{scc}(f)T(f)$ $\displaystyle=$
$\displaystyle\Big{(}\lvert f\rvert,|f_{x_{1}}|,|f_{x_{2}}|,...,|f_{x_{n}}|,$
$\displaystyle
inf_{x_{1}}(f),inf_{x_{2}}(f),...,inf_{x_{n}}(f),C_{p}(f),\|f\|,$
$\displaystyle f(X_{(2^{n}-1)}),...,f(X_{(1)}),f(X_{(0)})\Big{)},$

where the hybrid signatures are concatenated as a larger vector.

######  Definition 6.

(_optimized influence-aided canonical form_). Similarly, we can define
_optimized influence-aided canonical form_ $\kappa_{oi}(f)$ as the function in
$[f]$ with the minimum $S_{oi}(f)$, i.e. $\kappa_{oi}(f)=min_{S_{oi}}([f])$,
where $[f]$ represents the NPN equivalent class of $f$.

TABLE III: Comparison of exact NPN classification methods. |  n |  #Funcs |  #Classes |  Runtime (s)  |  Speedup
---|---|---|---|---|---
|  |  |  |  SOTA [15] |  INF |  INF+ |  SOTA/INF |  SOTA/INF+ |  INF/INF+
_EPFL_ |  5  |  678376  |  370  |  0.142  |  0.136  |  0.138  |  1.04 |  1.03  |  0.99
|  6  |  1054270  |  2339  |  0.579  |  0.459  |  0.471  |  1.26 |  1.23  |  0.98
|  7  |  730338  |  8824  |  1.542  |  1.037  |  1.014  |  1.49  |  1.52 |  1.02
|  8  |  1028501  |  27779  |  5.285  |  3.248  |  2.975  |  1.63  |  1.78 |  1.09
|  9  |  177974  |  26731  |  3.123  |  2.397  |  1.952  |  1.30  |  1.60 |  1.23
|  10  |  193905  |  50409  |  8.782  |  6.854  |  5.188  |  1.28  |  1.69 |  1.32
|  11  |  207345  |  80282  |  24.047  |  18.816  |  13.683  |  1.28  |  1.76 |  1.38
|  12  |  159741  |  87272  |  50.429  |  37.800  |  26.631  |  1.33  |  1.89 |  1.42
|  13  |  107069  |  74111  |  97.417  |  68.540  |  47.591  |  1.42  |  2.05 |  1.44
|  14  |  117825  |  85911  |  293.753  |  185.671  |  135.869  |  1.58  |  2.16 |  1.37
|  15  |  123699  |  94106  |  1142.931  |  513.366  |  385.223  |  2.23  |  2.97 |  1.33
|  16  |  127102  |  97283  |  5979.359  |  1378.776  |  1082.368  |  4.34  |  5.52 |  1.27
_MCNC_ |  5  |  1137438  |  398  |  0.219  |  0.211  |  0.215  |  1.04 |  1.02  |  0.98
|  6  |  2461375  |  2558  |  0.969  |  0.820  |  0.830  |  1.18 |  1.17  |  0.99
|  7  |  816538  |  8884  |  1.816  |  1.146  |  1.138  |  1.58  |  1.60 |  1.01
|  8  |  1171120  |  28031  |  6.402  |  3.680  |  3.394  |  1.74  |  1.89 |  1.08
|  9  |  92188  |  18394  |  2.769  |  1.980  |  1.578  |  1.40  |  1.75 |  1.25
|  10  |  113205  |  32796  |  9.366  |  6.870  |  5.334  |  1.36  |  1.76 |  1.29
|  11  |  96452  |  32153  |  17.997  |  13.859  |  10.299  |  1.30  |  1.75 |  1.35
|  12  |  125937  |  45546  |  61.001  |  46.436  |  34.720  |  1.31  |  1.76 |  1.34
|  13  |  137135  |  58921  |  230.835  |  148.387  |  116.286  |  1.56  |  1.99 |  1.28
|  14  |  143417  |  72602  |  843.348  |  479.178  |  384.648  |  1.76  |  2.19 |  1.25
|  15  |  167385  |  85417  |  3154.421  |  1458.433  |  1258.687  |  2.16  |  2.51 |  1.16
|  16  |  124425  |  47035  |  5485.760  |  1731.770  |  1679.458  |  3.17  |  3.27 |  1.03

Similar to Theorem 4 and Theorem 5, we can prove that the uniqueness of
$S_{oi}(f)$, and $\kappa_{oi}(f)$ is an NPN canonical form. The computation of
$\kappa_{oi}(f)$ is also similar to the computation method of $\kappa_{hi}(f)$
shown in algorithm 1, except that the part of computing $S^{1}_{scc}(f)$ is
correspondingly deleted. Candidate phase assignments are selected through the
single value $S^{0}_{scc}(f)$.

##  VI Evaluation

###  VI-A Experiment Setup

We implement the new canonical form and computation method based on Berkeley
ABC [22]. The state-of-the-art NPN classification method [15] is also
integrated into ABC as command testnpn -A 11. The procedure runs on an Intel
Xeon Gold 6252 CPU 24-core computer with 128GB RAM.

We use MCNC [23] and EPFL benchmarks [24] to test the effectiveness of our
algorithm on real synthesis applications. We enumerate the $K$-cuts in the
circuits of the benchmark suites to obtain the $K$-input Boolean functions and
extract the truth tables.

###  VI-B Comparison with State-of-the-Art Classification Method

Table III compares the influence-aided canonical form method (INF/INF+) with
the state-of-the-art (SOTA) [15] method. Among them, INF corresponds to the
hybrid influence-aided canonical form described in Section IV, and INF+
corresponds to the optimized influence-aided canonical form described in
Section V. All three methods can achieve exact NPN classification.

Table III shows the number of functions (#Funcs) with different input bits
(n), the number of exact equivalence classes (#Classes), the overall running
time of the three methods and the speedup ratio between them. Both of our two
influence-aided canonical form methods run faster than the SOTA for all NPN
classification tasks. Among them, the hybrid influence-aided canonical form
method gains up to 4.34x speedup than the SOTA while the optimized influence-
aided canonical form method achieves up to 5.52x speedup. Notably, our
influence-aided canonical form methods achieve better speedup as input
increases. Compared to INF, INF+ achieves better speedup when the input bit is
greater than 6. This further demonstrates that influence has superior pruning
efficacy over shifted-cofactor. Detailed analysis will be conducted in the
next subsection.

###  VI-C Discussion

This subsection explains why our influence-aided method can run faster by
showing some intermediate results.

TABLE IV: Comparison between the size of variable groups for SODA and INF/INF+ on EPFL benchmark. n |  #Total |  var-[1](%)  |  var-[2](%)  |  var-[3](%)  |  var-[4,5](%)  |  var-[6,7,8](%)  |  var-[$\geq$9](%)
---|---|---|---|---|---|---|---
| SODA |  INF | SODA |  INF | SODA |  INF | SODA |  INF | SODA |  INF | SODA |  INF | SODA |  INF
5  | 2675 | 2230  | 54.505 | 84.619  | 25.944 | 13.543  | 10.953 | 1.839  | 8.598 | 0  | 0.112 | 0  | 0 | 0
6  | 21532 | 17658  | 56.154 | 86.023  | 17.885 | 11.400  | 11.453 | 1.597  | 14.509 | 0.980  | 4.575 | 0  | 0 | 0
8  | 235466 | 245195  | 69.375 | 89.617  | 14.313 | 8.481  | 5.553 | 1.518  | 8.971 | 0.352  | 3.770 | 0.027  | 0 | 0
10  | 334259 | 359487  | 85.115 | 95.474  | 10.627 | 4.299  | 2.040 | 0.176  | 1.560 | 0.038  | 0.633 | 0.003  | 0 | 0
12  | 619120 | 679659  | 88.739 | 96.451  | 8.255 | 3.369  | 1.428 | 0.138  | 0.843 | 0.041  | 0.290 | 0.003  | 0.0305 | 0
13  | 556337 | 616669  | 90.002 | 96.974  | 7.238 | 2.891  | 1.283 | 0.121  | 0.715 | 0.014  | 0.298 | 0.001  | 0.0521 | 0
15  | 810309 | 904187  | 90.778 | 97.171  | 6.654 | 2.674  | 1.226 | 0.128  | 0.520 | 0.023  | 0.172 | 0.003  | 0.2968 | 0
16  | 884091 | 986068  | 91.165 | 97.206  | 6.400 | 2.625  | 1.168 | 0.138  | 0.497 | 0.028  | 0.159 | 0.006  | 0.3595 | 0
TABLE V: Details of #phase, #permutation, final exhaustive enumeration, and runtime of Cp on EPFL benchmark. n |  #Phase |  #Perm |  #Enum |  Time${}_{\textbf{cp}}$(sec)
---|---|---|---|---
| SODA | INF |  INF+ | SODA | INF |  INF+ | SODA | INF |  INF+ | SODA | INF |  INF+
5  | 1149 | 555 | 649  | 3541 | 1740 | 2703  | 1238 | 712 | 712  | 0.006914 | 0.002327 | 0.001208
6  | 10110 | 4296 | 5471  | 31678 | 13304 | 39425  | 9111 | 4727 | 4727  | 0.039637 | 0.024064 | 0.014824
8  | 66800 | 44128 | 73050  | 354438 | 162677 | 2673064  | 69353 | 49049 | 49049  | 0.409565 | 0.937477 | 0.588323
10  | 27317 | 24705 | 41901  | 291860 | 167821 | 345734  | 63186 | 57773 | 57773  | 1.646048 | 2.157659 | 0.811257
12  | 33884 | 31871 | 79205  | 988519 | 222595 | 422534  | 95719 | 92352 | 92352  | 6.400379 | 14.529945 | 5.719692
13  | 26620 | 25324 | 81517  | 2290221 | 181371 | 371319  | 78696 | 77051 | 77051  | 21.638766 | 28.091138 | 12.774357
15  | 34212 | 33170 | 212008  | 28520852 | 249233 | 19550823  | 98681 | 97393 | 97393  | 230.843127 | 225.61318 | 151.732984
16  | 35280 | 34417 | 322081  | 110090650 | 263786 | 37806293  | 101292 | 100238 | 100238  | 866.437428 | 615.558448 | 463.198676

Section III-C briefly mentioned that introducing influence can increase the
number of groups, thereby reducing the number of variables within the same
group. From the introduction of the canonical form method in Section IV, it is
evident that the more variables there are within the same group, the greater
the number of enumerations required subsequently. Table IV presents the total
number of groups and the statistics of different variable groupings for both
the SOTA and the INF/INF+ methods. The notation var-[x] represents the number
of groups with x variables, expressed as a percentage of the total number of
groups. This table shows that incorporating INF features for variable grouping
increases the number of groups when the input is larger than five while
simultaneously reducing the number of variables within each group. This also
implies that INF/INF+ will prune more transformations.

Figure 4: #Permutation enumerations of different methods.

Benefiting from the permutation-dependent property of influence, we can use
influence with cofactor and cost signature to further reduce the permutation
space. We collected the number of remaining permutation enumerations
separately pruned by $S_{inf}$, $C_{p}$, and $S_{inf}$+$C_{p}$. Fig. 4 shows
the results. The y-axis includes the proportion of #permutation of each among
the three above. The total permutation enumerations can be reduced
significantly when incorporating influence signatures. Thus, the INF and INF+
can gain speedup compared with the SOTA.

Table V provides a detailed summary of the EPFL benchmark, showcasing the
total number of transformations and the cumulative runtime for the three
canonical form methods. The headings of the columns are as follows: #Perm
denotes the remaining permutation transformations after symmetry detection;
#Phase represents the remaining phase assignment transformations after
symmetry detection; #Enum indicates the total number of final exhaustive
transformations after all pruning steps, and Time${}_{\textbf{cp}}$ denotes
the runtime for cost signature computation. This table shows that both INF and
INF+ can realize less transformation for phase, permutation, and final
exhaustive enumeration compared to SOTA. And it is also evident that computing
$C_{p}$ needs a long time. Even though it could further prune the permutation
enumeration, the time overhead for computing $C_{p}$ outweighs the enumeration
reduction. Thus, INF+ gains a better speedup than INF when $n$ is larger.
Considering the trade-off in selecting pruning signatures is essential when
developing a fast NPN classification method.

##  VII Conclusion and Future Work

This paper describes a novel canonical form and its computation algorithm with
consideration of Boolean influence. We highlight that influence signature is
negation independent and permutation dependent and design two influence-aided
canonical form methods for exact NPN classification. Experimental results
prove that influence plays a significant role in reducing the transformation
enumeration in computing the canonical form, and our influence-aided canonical
forms gain up to 5.5x speedup than the state-of-the-art NPN classification
method. In the future, we will try different combinations of signatures in the
canonical form and attempt to apply similar methods to other applications like
NPNP matching.

## Acknowledgments

This work is partly supported by the Major Key Project of PCL (No.
PCL2023AS2-3), the National Natural Science Foundation of China (No.
62090021), the National Key R&D Program of China (No.2022YFB4500500) and (No.
2022YFB4500403), the Strategic Priority Research Program of Chinese Academy of
Sciences (No. XDA0320300), the Ministry of Education of China (No.
20YJA880001) and the Innovation Program for Quantum Science and Technology
(No. 2021ZD0302900).

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# Graded components of Local cohomology modules II

Tony J. Puthenpurakal<EMAIL_ADDRESS>and  Sudeshna Roy
<EMAIL_ADDRESS>
###### Abstract.

Let $A$ be a commutative Noetherian ring containing a field $K$ of
characteristic zero and let $R=A[X_{1},\ldots,X_{m}]$. Consider $R$ as
standard graded with $\deg A=0$ and $\deg X_{i}=1$ for all $i$. We present a
few results about the behavior of the graded components of local cohomology
modules $H_{I}^{i}(R)$ where $I$ is an arbitrary homogeneous ideal in $R$. We
mostly restrict our attention to the Vanishing, Tameness and Rigidity
problems.

###### Key words and phrases:

local comohology, graded local cohomology, Weyl algebra, generalized Eulerian
modules

##  1\. introduction

######  1.1.

Standard assumption: Throughout this paper $A$ is a commutative Noetherian
ring containing a field $K$ of characteristic zero. Let
$R=A[X_{1},\ldots,X_{m}]$ be standard graded with $\deg A=0$ and $\deg
X_{i}=1$ for all $i$. We also assume $m\geq 1$. Let $I$ be a homogeneous ideal
in $R$ and $M=H_{I}^{i}(R)$. It is well-known that $M$ is a graded $R$-module.
Set $M=\bigoplus_{n\in\mathbb{Z}}M_{n}$.

In paper [TP2], the first author studied graded components of local cohomology
modules when $A$ is a regular ring containing a field $K$ of characteristic
zero and he proved that they satisfy many good properties. In this paper we
show that few of the results proved in [TP2] (in particular Theorems 1.2, 1.3
and 1.6) hold true in considerably general setting.

In this paper we show that under assumptions as in 1.1 the following results
hold true.

We first show

######  Theorem 1.2.

(with hypotheses as in 1.1). If $M_{n}=0$ for all $|n|\gg 0$ then $M=0$.

In view of Theorem 1.2, it follows that if
$M=H^{i}_{I}(R)=\bigoplus_{n\in\mathbb{Z}}M_{n}$ is _non-zero_ then either
$M_{n}\neq 0$ for infinitely many $n\ll 0$, OR, $M_{n}\neq 0$ for infinitely
many $n\gg 0$. We show that $M$ is _tame_. More precisely

######  Theorem 1.3.

(with hypotheses as in 1.1). Then we have

  1. (a)

The following assertions are equivalent:

    1. (i)

$M_{n}\neq 0$ for infinitely many $n\ll 0$.

    2. (ii)

$M_{n}\neq 0$ for all $n\leq-m$.

  2. (b)

The following assertions are equivalent:

    1. (i)

$M_{n}\neq 0$ for infinitely many $n\gg 0$.

    2. (ii)

$M_{n}\neq 0$ for all $n\geq 0$.

Surprisingly non-vanishing of a single graded component of $M=H^{i}_{I}(R)$ is
very strong. We prove the following rigidity result:

######  Theorem 1.4.

(with hypotheses as in 1.1). The we have

  1. (a)

The following assertions are equivalent:

    1. (i)

$M_{r}\neq 0$ for some $r\leq-m$.

    2. (ii)

$M_{n}\neq 0$ for all $n\leq-m$.

  2. (b)

The following assertions are equivalent:

    1. (i)

$M_{s}\neq 0$ for some $s\geq 0$.

    2. (ii)

$M_{n}\neq 0$ for all $n\geq 0$.

  3. (c)

(When $m\geq 2$.) The following assertions are equivalent:

    1. (i)

$M_{t}\neq 0$ for some $t$ with $-m<t<0$.

    2. (ii)

$M_{n}\neq 0$ for all $n\in\mathbb{Z}$.

We also give examples to show that Theorems 1.7, 1.8, 1.9, 1.13, 1.14 from
[TP2] are false if we don’t assume that $A$ is regular.

_Techniques used to prove our results:_ The main idea in [TP2] is that if
$A=K[[Y_{1},\ldots,Y_{d}]]$ then graded local cohomology modules over
$R=A[X_{1},\ldots,X_{m}]$ become holonomic over an appropriate ring of
differential operators, see [TP2, Theorem 4.2]. Furthermore local cohomology
modules are “generalized Eulerian” see [TP2, Theorem 3.6] (we will discuss
this property later). One of the observation we first made was that in
Theorems 1.2, 1.3 and 1.6 in [TP2] the generalized Eulerian property of local
cohomology modules was used more crucially. Furthermore in [TP2] the
generalized Eulerian property of local cohomology modules was proved very
generally.

We now discuss the techniques used to prove our results:
Let $A$ be a Noetherian ring containing a field of characteristic zero. Let
$R=A[X_{1},\ldots,X_{m}]$, graded with $\deg A=0$ and $\deg X_{i}=1$ for all
$i$. Let $A_{m}(A)$ be the $m^{th}$ Weyl-algebra on $A$. We consider it a
graded ring with $\deg A=0$, $\deg X_{i}=1$ and $\deg\partial_{i}=-1$. We note
that $R$ is a graded subring of $A_{m}(A)$. If $E$ is a graded
$A_{m}(A)$-module and $e$ is a homogeneous element of $E$ then set $|e|=\deg
e$.

Consider the Eulerian operator
${\mathcal{E}}=\sum_{i=1}^{m}X_{i}\partial_{i}$. If $f\in R$ is homogeneous
then it is easy to check that ${\mathcal{E}}f=|f|f$. We say a graded
$A_{m}(A)$-module $W$ is Eulerain if ${\mathcal{E}}w=|w|w$ for each
homogeneous element $w$ of $W$. Notice $R$ is an Eulerian $A_{m}(A)$-module.
We say $W$ is generalized Eulerian if for each homogeneous $w$ of $W$ there
exists $a$ depending on $w$ such that $({\mathcal{E}}-|w|)^{a}w=0$.

The notion of Eulerian modules was introduced in the case $A$ is a field $K$
by Ma and Zhang [LW] (they also defined the notion of Eulerian $D$-modules in
characteristic $p>0$, where $D$ is the ring of $K$-linear differential
operators on $R=K[X_{1},\ldots,X_{m}]$). Unfortunately however the class of
Eulerian $D$-modules is not closed under extensions (see 3.5(1) in [LW]). To
rectify this, the first author introduced the notion generalized Eulerian
$D$-modules (in characteristic zero), see [TP6].

The next technique that we use is the technique of de Rham cohomology, Koszul
homology of generalized Eulerian modules. We generalize several properties
which were proved when $A=K$, a field and the relevant module was holonomic
over $A_{m}(K)$.

The final technique we use to prove vanishing and tameness property of local
cohomology modules is that local cohomology modules is countably generated
over $R=A[X_{1},\ldots,X_{m}]$. We can exploit this fact if $A$ contains an
uncountable field. This was observed first in [TP4]. By base change one can
always assume that $A$ contains an uncountable field, see LABEL:uncountable.

######  Remark 1.5.

Note that practically all the proofs in this paper use techniques which were
first developed by several researches (including the first author) for
holonomic modules over some appropriate ring of differential operators. In the
generality we consider, we do not have the notion of holonomic modules.
However we have the notion of generalized Eulerian modules. After carefully
rewriting several proofs we were able to deduce our results.

The paper is organized as follows. In Section 2, we discuss a few preliminary
results that we need. In Section 3, we define generalized Eulerian modules on
$A_{m}(A)$, the $m^{th}$-Weyl algebra over $A$ and talk about some basic
properties of these modules. In Section 4, some properties of
$A_{m}(A)$-modules are discussed. In Section 5, we show that $H_{I}^{i}(M)$
has some nice properties due to its countable generation as an
$R=A[X_{1},\ldots,X_{m}]$-module. In Section 6 and Section 7 we prove our main
results regarding vanishing, tameness and rigidity of graded local cohomology
modules. In the last Section we give examples to show that Theorems 1.7, 1.8,
1.9, 1.13, 1.14 from [TP2] are false if we don’t assume that $A$ is regular.

Convention: Throughout modules considered are left modules over some ring
(commutative or not). Usually $A,B,R,S$ will denote commutative rings. The
only non-commutative ring we will consider will be $A_{m}(A)$, the $m^{th}$
Weyl algebra over a commutative ring $A$.

##  2\. Preliminaries

In this section we define Lyubeznik functor $\mathcal{T}$ and discuss a few
preliminary results that we need.


# Exact Mobility edges and topological Anderson insulating phase in a slowly
varying quasiperiodic model

Zhanpeng Lu 11 Zhihao Xu 11 Yunbo Zhang 111Corresponding author E-mail:
<EMAIL_ADDRESS>22 Institute of Theoretical Physics and State Key
Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University,
Taiyuan 030006, China  Key Laboratory of Optical Field Manipulation of
Zhejiang Province and Physics Department of Zhejiang Sci-Tech University,
Hangzhou 310018, China

###### Abstract

We uncover the relationship of topology and disorder in a one-dimensional Su-
Schrieffer-Heeger chain subjected to a slowly varying quasi-periodic
modulation. By numerically calculating the disorder-averaged winding number
and analytically studying the localization length of the zero modes, we obtain
the topological phase diagram, which implies that the topological Anderson
insulator (TAI) can be induced by a slowly varying quasi-periodic modulation.
Moreover, unlike the localization properties in the TAI phase caused by random
disorder, mobility edges can enter into the TAI region identified by the
fractal dimension, the inverse participation ratio, and the spatial
distributions of the wave functions, the boundaries of which coincide with our
analytical results.

###### category:

Original Paper

###### keywords:

mobility edge, quasi-periodic disorder, topological Anderson insulator

##  1 Introduction

Topological phases of matter have attracted broad interest over the past
decades [1, 2, 3, 4, 5, 6]. A topological insulator displays the gapped bulk
states and the gapless edge modes lied in the bulk gap, which can be
characterized by a non-trivially topological number. The Su-Schrieffer-Heeger
(SSH) model is the simplest two-band topological system initially introduced
to research the polyacetylene that exhibits rich physical phenomena [7], such
as fractional charge soliton excitation [8, 9]and nontrivial edge modes [10].
Its chiral symmetry leads to nontrivial topology confirmed by a non-zero
winding number and the emergence of the zero-energy edge modes under open
boundary conditions (OBCs) [11, 12, 13].

On the other hand, Anderson localization in a disordered medium [14, 15, 16,
17] was first proposed in 1958, which has been realized in various
experimental platforms such as cold atoms [18, 19] and microwaves cavity [20,
21]. Compared with traditional Anderson models, where even an infinitesimal
random potential leads to localization in one- and two-dimensional cases, the
quasi-periodic systems display distinctive localization properties. The one-
dimensional Aubry-André-Harper (AAH) model as one of the paradigmatic examples
undergoes a metal-insulator transition characterized by the self-duality
property when the strength of the quasi-periodic potential exceeds a finite
critical value [22, 23, 24, 25, 26]. Remarkably, the presence of the mobility
edge, one of the essential concepts in disordered systems seen only in the
traditional three-dimensional Anderson model [27], has been demonstrated in
various generalized AAH models even in one dimensional systems. Some
generalized AAH models displaying the mobility edges in compact analytic forms
are proposed when the so-called self-dual symmetry is broken [28, 29, 30, 31,
32, 33], or the system is in the form of slowly varying modulations [34, 35,
36, 37]. Recent advances in the mobility edges have been extended to the
anomalous mobility edges separating the localized states from the critical
ones in one-dimensional disordered systems [38], which is different from the
traditional mobility edges.

Recently, great effort has been devoted to understand the interplay of
topology and disorder, which brings new perspectives, such as the topological
phase transition can be introduced by the quasi-periodic disorder in the one-
dimensional SSH chain under the intercell hopping strength exceeds the
intracell hopping strength [37], and topological phase with critical
localization consisting of only critically localized states [13, 39, 42]. One
of the hallmark characteristics of a topological insulator is the robustness
of the nontrivial edge states against weak disorder in the underlying lattice
[2, 40]. When the amplitude of disorder is large enough, the topological
features eventually disappear, accompanied by collapse of the nontrivial
topological number [41]. Conversely, a modest disorder added to a trivial
system can lead to the emergence of protected edge modes and quantized
topological charges. Such disorder-driven topological phase is named the
topological Anderson insulator (TAI) [43, 44, 45, 46, 47, 48, 49, 50, 51, 52].
The experimental observation of the disorder-induced TAI has been reported in
two-dimensional(2D) photonics [53, 54], one-dimensional(1D) engineering
synthetic 1D chiral symmetric wires with a precisely controllable random
disorder [55]. Recent advances of the TAI induced by a random disorder have
been extended to non-Hermitian systems [56, 57] and spin-orbit coupled
superconductors [58, 59]. On the other hand, the TAI phase is also shown to
appear in the SSH model with a quasi-periodic modulated intercell hopping term
with all the eigenstates being localized [60].

Some interesting questions arise here: whether a slowly varying modulation can
lead to a TAI phase, and what the localization properties in such case are. To
address these, we consider a toy model, which is the SSH model subjected to a
slowly varying intracell hopping and open boundary condition, focusing on the
topology and localization features in the system. The topological phase
diagrams will be obtained by numerically calculating the topological
invariants in real space, i.e., the disorder-averaged winding number, and the
topological phase transition points can be determined by the localization
length of the zero modes. Moreover, the localization properties of the system
will be explored by some proven means such as the inverse participation ratio,
the fractal dimension, and the spatial distributions of the eigenstates.
Specifically, we try to obtain analytic expressions for the mobility edges, if
they exist.

The arrangement of the paper is as follows: In Sec. 2, we briefly introduce
the Hamiltonian of the SSH model with a slowly varying intracell hopping term.
In Sec. 3, we obtain the topological phase diagram and discuss the fate of
topological zero-energy modes. Furthermore, we study the localization
properties of the system in Sec. 4. Finally, a conclusion is presented in Sec.
5.

##  2 Model and Hamiltonian

We consider a toy model with a slowly varying quasi-periodic modulation, which
can be described by the Hamiltonian

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{j}v_{j}\left(c^{{\dagger}}_{j,B}c_{j,A}+h.c\right)+w\sum_{j}\left(c^{{\dagger}}_{j+1,A}c_{j,B}+h.c\right).$
(1)

Here $c^{{\dagger}}_{j,\alpha}$($c_{j,\alpha}$) creates (annihilates) a
particle on the sublattice site $A(B)$ in the $j$-th lattice cell, and $L$ is
the length of the system. As shown in Figure 1a, the intercell hopping
strength $w$ is denoted by the black dashed line, which is set as the unit
energy, _i.e._ , $w=1$, and the $j$-dependent intracell hopping

$\displaystyle v_{j}$ $\displaystyle=$ $\displaystyle v+\Delta_{j},$ (2)

with

$\displaystyle\Delta_{j}$ $\displaystyle=$
$\displaystyle{\Delta}\cos(2\pi\beta j^{u}+\phi),$ (3)

is denoted by the red solid line. Here $v$ is the site-independent hopping
inside the unit cell, $\Delta$ is the strength of incommensurate modulation,
$\beta$ is the modulation frequency, and $\phi\in[0,2\pi]$ is an arbitrary
phase. In the thermodynamic limit, we have

$\displaystyle\lim_{j\to\infty}\left|\frac{d\Delta_{j}}{dj}\right|$
$\displaystyle=$ $\displaystyle\lim_{j\to\infty}2\pi\Delta\beta
u\frac{\left|\sin(2\pi\beta j^{u}+\phi)\right|}{j^{1-u}}\to 0,$ (4)

with the slowly varying parameter $0<u<1$. The result indicates the difference
of $\Delta_{j}$ tends to 0 when $j$ is large enough as shown in Figure 1b,
which corresponds to a slowly varying modulation [34, 35]. For $\Delta=0$ or
$u=0$, the system reduces to a standard SSH model [7] with a uniform intracell
hopping amplitude. When the intercell hopping strength exceeds the intracell
hopping strength, system undergoes a topological phase transition
characterized by the emergence of zero-energy edge modes and nontrivial
winding number. For $\Delta\neq 0$ and $u=1$, $\Delta_{j}$ corresponds to an
AA-type modulation illustrated in Figure 1c. The emergence of the TAI in SSH
model with an AA-type modulation has been reported [60]. In the following, we
testify the existence of the TAI in the SSH model with a slowly varying
intracell modulation and discuss the corresponding localization properties. We
fix $\beta=\frac{\sqrt{5}-1}{2}$ and $u=0.7$ for our discussion.

Figure 1: (Color online) (a) Schematic of the SSH model of $L$ unit cells with
a modulated intracell hopping term. The dotted box represents the unit cell.
$w$ and $v+\Delta_{j}$ represent the uniform intercell hopping and the
$j$-dependent intracell hoppping, respectively. (b) Sketch of a slowly varying
modulation $\Delta_{j}=\Delta\cos(2\pi\beta j^{u}+\phi)$ for $u=0.7$. (c)
Sketch of an AA-type modulation with $u=1$. Here,
$\beta=\frac{\sqrt{5}-1}{2}$, $\phi=0$, and $\Delta=0.8$.

##  3 Topological phase diagram

A TAI is characterized by the emergence of the protected edge states and the
quantized topological charge induced by the addition of sufficient disorder or
incommensurate modulation to a trivial band structure. To detect the TAI
phase, we first utilize the open-bulk winding number to characterize the
topological properties for our slowly varying SSH model. For a given
modulation configuration, we can diagonalize the open-chain Hamiltonian as
$H|\psi^{n}\rangle=E_{n}|\psi^{n}\rangle$ and
$H|\tilde{\psi}^{n}\rangle=-E_{n}|\tilde{\psi}^{n}\rangle$ to obtain a pair of
chiral-symmetric partners $|\psi^{n}\rangle$ and $|\tilde{\psi}^{n}\rangle$
with the relation $|\tilde{\psi}^{n}\rangle=S|\psi^{n}\rangle$, where the
entries of $S$ are
$S_{i\alpha,j\gamma}=\delta_{ij}(\sigma_{z})_{\alpha\gamma}$ with $i,j$
referring to the unit cell and $\alpha,\gamma$ to the sublattice. We introduce
an open-boundary $Q$ matrix given by

$Q=\sum_{n}\left(|\psi^{n}\rangle\langle\psi^{n}|-|\tilde{\psi}^{n}\rangle\langle\tilde{\psi}^{n}|\right),$
(5)

where $\sum_{n}$ is the sum over the eigenstates in the bulk spectrum without
the edge modes. The open-bulk winding number in real space is defined as [11]

$W_{c}=\frac{1}{2L^{\prime}}\mathrm{Tr}^{\prime}(SQ[Q,X]).$ (6)

Here, $X$ is the coordinate operator, namely
$X_{i\alpha,j\gamma}=j\delta_{ij}\delta_{\alpha\gamma}$. The length of the
system $L$ can be divided into three intervals with length $l$, $L^{\prime}$
and $l$, _i.e._ , $L=L^{\prime}+2l$. The symbol $\mathrm{Tr}^{\prime}$
represents the trace over the middle interval with length $L^{\prime}$. When
$L\to\infty$, the open-bulk winding number method can well serve for the
system deviated from periodicity, and $W_{c}$ is quantized to an integer,
while a modest size is enough in the practical calculation. We also define the
disorder-averaged winding number $\overline{W}=1/N_{c}\sum_{c=1}^{N_{c}}W_{c}$
with the configuration number $N_{c}$. Here, we take $N_{c}=100$ disorder
realizations for different $\phi=[0,2\pi]$ throughout the work.

Figure 2: (Color online) (a) Disorder-averaged winding number $\overline{W}$
as a function of the modulation strength $\Delta$ and the amplitude of the
intracell hopping $v$ with $L=300$ and $N_{c}=100$ disorder realizations. The
colorbar shows the value of disorder-averaged winding number $\overline{W}$.
Two red dot lines represent the analytic critical lines Equation (11) for the
divergence of localization length $\lambda$. (b) $\lambda^{-1}$ as a function
of $\Delta$ and $v$ for $L=600$ and $N_{c}=100$ disorder realizations. The
colorbar shows the value of $\lambda^{-1}$. (c)(d) Two disorder-averaged
energies $\overline{E}_{L}$ and $\overline{E}_{L+1}$ in the center of the
spectrum, and the disorder-averaged winding number $\overline{W}$ as a
function of the modulation strength $\Delta$ under OBCs with $v=0.5$(c) and
$v=1.1$(d), respectively.

Figure 2a shows the topological phase diagram characterized by $\overline{W}$
versus the modulation strength $\Delta$ and the amplitude of the intracell
hopping $v$. In the absence of the modulation $\Delta=0$, the standard SSH
model exhibits a topological phase transition point at $v=1$ corresponding to
a jump of $\overline{W}$ from $1$ to $0$ with the increase of $v$. When we
turn on the slowly varying modulation $\Delta_{j}$ to the nontrivial band
structure with $v<1$, the zero-energy modes are robust against the slowly
varying modulation. With the increase of $\Delta\lesssim 2$, the two disorder-
averaged zero modes $\overline{E}_{L}$ and $\overline{E}_{L+1}$ start to break
into non-zero pairs, accompanied by the jump of the winding number to zero, as
shown in Figure 2c for $v=0.5$ under OBCs. Here,
$\overline{E}_{n}=1/N_{c}\sum_{c=1}^{N_{c}}E_{n}^{c}$ with $E_{n}^{c}$ being
the $n$-th eigenenergy for a given modulation configuration. As shown in
Figure 2a, we find that the topological transition point is independent of $v$
at $\Delta=2$. It indicates that the topologically nontrivial phase will
collapse as long as the strength of the slowly varying modulation reaches a
critical value. When $v\in(1,2)$ the system remains a topologically trivial
phase as in a standard SSH model for $\Delta>0$ and the TAI phase emerges in a
finite regime between $v=\Delta^{2}/4+1$ and $\Delta=2$, which is
characterized by the nontrivial winding number and the zero-energy edge modes.
The deviation from the non-zero integer of the disorder-averaged winding
numbers $\overline{W}$ near the phase transition boundaries comes from the
instability of $W_{c}$ for a given modulation configuration near the boundary,
though the one-shot $W_{c}$ is always an integer in the thermodynamic limit.
We take $v=1.1$ as an example shown in Figure 2d, the moderate slowly varying
modulation $\Delta$ leads to a change of the nontrivial winding number in the
region $\Delta\in(1,2)$, meanwhile the disorder-averaged zero modes are also
detected. When $v>2$, the TAI phase disappears, no matter how strong the
modulation amplitude is, as can be seen in Figure 2a. Our numerical results
show that the slowly varying intracell modulation will evoke the TAI phase in
a finite region for $v>1$.

According to Refs. [49, 55, 56, 60], the topological transition is accompanied
by the divergence of the localization length of the zero modes. Hence, one can
obtain the phase diagram by studying the localization length of the zero
modes. For the zero modes, the Schrödinger equation of the SSH model (1)
$H\psi=0$ reads:

$\displaystyle w\psi_{j,B}+v_{j+1}\psi_{j+1,B}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle v_{j}\psi_{j,A}+w\psi_{j+1,A}$
$\displaystyle=$ $\displaystyle 0,$ (7)

where $\psi_{j,A}$($\psi_{j,B}$) is the probability amplitude of the zero mode
on the sublattice site $A(B)$ in the $j$-th lattice cell. By solving the
coupled equations, one has
$\psi_{n+1,A}=(-1)^{n}\prod_{j=1}^{n}(v_{j}/w)\psi_{1,A}$, leading to the
localization length $\lambda$ of the zero modes given by [60]

$\displaystyle\lambda^{-1}$ $\displaystyle=$
$\displaystyle-\lim_{L\to\infty}\frac{1}{L}\ln\left|\frac{\psi_{L+1,A}}{\psi_{1,A}}\right|$
(8) $\displaystyle=$
$\displaystyle\lim_{L\to\infty}\frac{1}{L}\left|\sum_{j=1}^{L}\ln{|v+\Delta\cos(2\pi\beta
j^{u})|}\right|.$

According to Weyl’s equidistribution theorem [61, 62], we can use the ensemble
average to evaluate the last expression

$\lambda^{-1}=\left|\frac{1}{2\pi}\int^{\pi}_{-\pi}dq\ln{|v+{\Delta}\cos
q|}\right|.$ (9)

The integration can be performed straightforwardly as

$\lambda^{-1}=\begin{cases}\ln{\frac{v+\sqrt{v^{2}-\Delta^{2}}}{2}}\quad&v>\Delta,\\\
\ln{\frac{\Delta}{2}}\quad&v<\Delta.\\\ \end{cases}$ (10)

The divergence of this localization length $\lambda$, i.e $\lambda^{-1}\to 0$,
gives the two critical lines

$\displaystyle v$ $\displaystyle=\Delta^{2}/4+1\quad$ $\displaystyle
v>\Delta,$ $\displaystyle\Delta$ $\displaystyle=2\quad$ $\displaystyle
v<\Delta.$ (11)

The localized critical points match the topological phase transition points
shown in Figure 2a. Numerically the value of $\lambda^{-1}$ for the $L$-th
eigenstate is computed by [30]

$\displaystyle\lambda^{-1}$ $\displaystyle=$
$\displaystyle\lim_{L\to\infty}\frac{1}{L}||T||,$ (12)

where $\|\cdot\|$ denotes the norm of the total transfer matrix
$T=\prod_{j=2}^{L}T_{j}T_{1}$ with

$\displaystyle T_{j}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\frac{E_{L}^{2}-v_{j}^{2}-1}{v_{j}}&-\frac{v_{j-1}}{v_{j}}\\\
1&0\\\ \end{array}\right]$ (15)

and

$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\frac{(E_{L}^{2}-v_{1}^{2})}{v_{1}}&-1\\\
1&0\\\ \end{array}\right].$ (18)

The numerical result of $\lambda^{-1}$ as a function of $v$ and $\Delta$ by
means of the transfer matrix method [49, 63] is shown in Figure 2b, and the
two diverging critical lines match with the analytic solutions Equation (11)
pretty well. Moreover, for $u=1$, the modulation is reduced to the AA-type,
the topological phase diagram is the same (see Appendix A).

For $\Delta=0$, the system reduces to the original SSH model. When the
intracell hopping strength $v$ exceeds the intercell hopping strength $w$, the
system undergoes a topological phase transition from a topologically
nontrivial phase into a trivial phase accompanied by a jump of the winding
number from 1 to 0. However, when one introduces a slowly varying modulation
for $u\neq 0$ and $\Delta\neq 0$, the topologically nontrivial phase, i.e. the
TAI phase, emerges in the trivial regime of the original SSH model, which is
shown in Figure 2a for non-zero $\Delta$.

##  4 Localization transition and mobility edges

Figure 3: (Color online) (a) The disorder-averaged fractal dimension
$\overline{\mathrm{\Gamma}}$ of different eigenstates as a function of the
corresponding $\overline{E}$ and the modulation strength $\Delta$ with
$N_{c}=100$ disorder realizations. The colorbar indicates the magnitude of
$\overline{\mathrm{\Gamma}}$. The black and red solid lines represent the
mobility edges given in Equation (19). The shaded regime represents the TAI
regime. (b) The distribution of the ${\mathrm{IPR}}$ as a function of the
${E}$ with different $\Delta$ with $\phi=0$. The black dashed lines correspond
to the mobility edges given in Equation (19). The eigenstates spatial
distributions with $\phi=0$ and $\Delta=0.5$ are shown in (c) for ${E}=1.9695$
satisfying $E>E_{c1}$ and (d) for ${E}=1.1108$ satisfying $E_{c2}<E<E_{c1}$,
respectively, where $E_{c1}=1.6$ and $E_{c2}=0.6$ correspond to the mobility
edges shown in Equation (19) with $\Delta=0.5$. Here, only ${E}>0$ states are
shown for $\beta=(\sqrt{5}-1)/2$, $v=1.1$ and $L=600$. Figure 4: (Color
online) (a) The disorder-averaged fractal dimension
$\overline{\mathrm{\Gamma}}$ of different eigenstates as a function of the
corresponding $\overline{E}$ and $\Delta$ with $N_{c}=100$ disorder
realizations. The colorbar indicates the magnitude of
$\overline{\mathrm{\Gamma}}$. The black and red solid lines represent the
mobility edges given in Equation (21). (b) The distribution of the IPR as a
function of the eigenenergy $E$ with different $\Delta$ with $\phi=0$. The
black dashed lines correspond to the positions of mobility edges given in
Equation (21). Here, only $E>0$ states are considered, $\beta=(\sqrt{5}-1)/2$,
$v=0.3$ and $L=600$.

A common opinion is that when the system is in the TAI phase, all states are
localized, showing Anderson’s localization phenomenon. In this section, we
study another aspect of the localized nature of the TAI phase, i.e. the access
of mobility edges to the TAI phase in the SSH model under a slowly varying
incommensurate modulation. We follow the method in Ref. [64] to derive the
expression of mobility edges for our slowly varying model. Due to the chiral
symmetry, the system’s spectrum is symmetric about $E=0$. To simplify, we only
focus on the upper band($E>0$), and the lower one displays identical
behaviors.

The localization in the slowing varying model exhibits different features for
two cases of the site-independent intracell hopping strength $v$. As following
results imply, the delocalization-localization transition is not connected to
a topological phase transition in both regimes.

(a) For $v>1$, the mobility edges are given by (see Appendix B for details):

$E_{c}=v\pm\left(\Delta-1\right).$ (19)

Here and in Equation (21) $+$($-$) denotes the upper(lower) mobility edge
marked by the black(red) solid line shown in Figure 3a, respectively. We find
that $\Delta_{c}=1$ is the localization transition point. When
$\Delta>\Delta_{c}$, all eigenstates become localized, and there exist
mobility edges in the regime $0<\Delta<\Delta_{c}$. According to the
topological phase diagram Figure 2a, we find that the TAI phase is compatible
with the regime hosting mobility edges. To characterize the mobility edges, we
define the fractal dimension of the wave function as following:

${\mathrm{\Gamma}}=-\lim_{L\to\infty}\frac{{\ln{{\mathrm{IPR}}}}}{\ln{(2L)}},$
(20)

where the inverse participation ratio
${\mathrm{IPR}}=\sum_{j=1}^{L}\sum_{\alpha}|\psi_{j,\alpha}^{n}|^{4}$. Further
we can define the disorder-averaged fractal dimension
$\overline{\mathrm{\Gamma}}=1/N_{c}\sum_{c=1}^{N_{c}}\mathrm{\Gamma}^{c}$. It
is known that $\Gamma\to 1$ for extended states and $\Gamma\to 0$ for
localized ones [30]. In Figure 3a, we show the disorder-averaged fractal
dimension $\overline{\Gamma}$ of the eigenstates as a function of the
modulation strength $\Delta$ for $v=1.1$. The solid lines represent the
mobility edges defined by Equation (19), and the shaded regime denotes the
region with TAI. One can see that there exists an overlap between the TAI
regime and the extended states regime with mobility edges. To clarify the
mobility edges more clearly, Figure 3b plots the ${\mathrm{IPR}}$ as a
function of the eigenenergy ${E}$ for different $\Delta$ with $v=1.1$. The
magnitude of the ${\mathrm{IPR}}$ jumps from the order of magnitude of
$10^{-2}$(a typical value for the localized states [36]) to $10^{-3}$(a
typical value scales as $1/L$ for the extended states [36]), when the
eigenenergy ${E}$ crosses some critical values for $\Delta=0.3$ and $0.6$.
This clearly signals the transition between localized and extended states, and
these critical values correspond to the positions of the mobility edges in the
spectrum marked by the black dotted lines in Figure 3b defined by Equation
(19). When $\Delta=1.2>\Delta_{c}$, the ${\mathrm{IPR}}$ magnitude for all
eigenstates is up to $10^{-2}$, which indicates there is no mobility edges in
the spectrum. Furthermore, the mobility edges can also be confirmed
intuitively by the spatial distributions of the wave functions in Figure 3c
and Figure 3d. The wave functions are localized or extended when their
eigenvalues satisfy ${E}>v-\left(\Delta-1\right)$ and
$v+\left(\Delta-1\right)<{E}<v-\left(\Delta-1\right)$, respectively.

(b) For $0<v<1$, the mobility edges are instead

$E_{c}=1\pm\left(v-\Delta\right).$ (21)

Figure 4a shows the disorder-averaged fractal dimension $\overline{{\Gamma}}$
of the eigenstates as a function of disorder-averaged eigenenergy
$\overline{{E}}$ and modulation strengthen $\Delta$ for $v=0.3$. In the regime
$0<\Delta<v$, as expected from the analytical results, $\overline{\Gamma}$
approximately exhibits sharp jumps from $1$ to $0$ for energies upper or lower
lines defined by Equation (21). When $\Delta>v$, the mobility edges vanish,
and all the states are localized. Figure 4b plots the IPR as a function of the
eigenenergy for different $\Delta$ with $v=0.3$, where the jumping points
denote the positions of the mobility edges.

##  5 Conclusion

In summary, we study the topological phase transition and localization
transition of a SSH model with a slowly incommensurate modulation. We
numerically and analytically obtain the topological phase diagram, and our
results imply that a slowly vary incommensurate modulation can induce the TAI.
Different from the random disorder induced the TAI, the mobility edges can
enter into the TAI regime in our case.

Note added: on submission of this manuscript, we notice a very recent preprint
arXiv:2201:00988 entitled ’Topological Anderson insulators with different bulk
states in quasiperiodic chains’ [65]. The authors studied a similar problem,
and main results of the TAI regime with the intermediate phase were obtained.
In this paper, we derive the exact expression of mobility edges in this TAI
phase.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
(12074340), the NSFC (Grants No. 11604188 and No. 12147215), Fundamental
Research Program of Shanxi Province (Grant No. 20210302123442), Beijing
National Laboratory for Condensed Matter Physics, and STIP of Higher Education
Institutions in Shanxi under Grant No. 2019L0097. This work is also supported
by NSF for Shanxi Province Grant No. 1331KSC.

## Conflict of Interest

The authors declare no conflict of interest.

##  Appendix A

## AA-type disorder

Figure A1: (Color online) Disorder-averaged winding number $\overline{W}$ as a
function of the modulation strength $\Delta$ and the amplitude of the
intracell hopping $v$ with $L=300$ and 100 disorder realization. Two red dot
lines represent the analytic critical lines Equation (11). Other parameters:
$\beta=(\sqrt{5}-1)/2$ and $u=1$.

When $u=1$, $\Delta_{j}$ is reduced to the AA-type modulation. Figure A1 shows
the topological phase diagram characterized by $\overline{W}$ versus the
disorder strength $\Delta$ and intracell hopping $v$, which is the same with
the result in Figure 2a. As shown in Figure A1, the TAI phase also can be
introduced by AA-type modulation and topological critical points still can be
described by Equation (11), which labeled by two red dot lines, shown in
Figure A1.

Figure A2: (Color online)(a) Fractal dimension $\Gamma$ of different
eigenstates as a function of the corresponding eigenvalues and $\Delta$. (b)
The $\Gamma$ associated to the eigenstate number indices as a function
$\Delta$. Other parameters: $\beta=(\sqrt{5}-1)/2$, $u=1$, $\phi=0$, $v=1.1$
and $L=600$.

To figure out the localization property in the TAI phase, we display the
Fractal dimension ${\Gamma}$ of different eigenstates as a function of the
corresponding ${E}$ and $\Delta$ for $v=1.1$ in Figure A2a. One can see that
some eigenstates are localized and the rest are extended, which indicates the
mobility edges exist in system. By increasing $\Delta$, we can find an
intermediate region consisting of both extended, localized, and even critical
eigenstates exists in spectrum. For a clear comparison, the ${\Gamma}$
associated to eigenstates as a function of $\Delta$ is shown in Figure A2b.
With the increase of $\Delta$, more extended states become localized, shown in
Figure A2a and b. And an intermediate region with a mixture of extended,
localized, and critical eigenstates emerge in $1.1<\Delta<2$. When the AA-type
modulation is introduced in the system, the localization feature becomes very
complex. So we can’t get the analytic expression of mobility edges.

## B Derivation of the analytical expression for mobility edges

| $v$ | $\Delta$ | $\left(v,\Delta\right)$ | the smallest energy overlaps
---|---|---|---|---
I |  $v$ <1 |  $\Delta$<1 |  $\Delta$<$v$ | $\left(|{E}_{3}|,|{E}_{2}|\right)$
|  |  |  $\Delta$>$v$ | $\left(|{E}_{2}|,|{E}_{3}|\right)$
II |  $v$>1 |  $\Delta$<1 |  | $\left(|{E}_{1}|,|{E}_{2}|\right)$
|  |  $\Delta$>1 |  | $\left(|{E}_{2}|,|{E}_{1}|\right)$
III |  $v$<1 |  $\Delta$<1 |  $\Delta$>$v$ | $\left(|{E}_{2}|,|{E}_{3}|\right)$
|  |  |  $\Delta$<$v$ | $\left(|{E}_{3}|,|{E}_{2}|\right)$
|  $v$<1 |  $\Delta$>1 |  $\Delta$>$v$ | $\left(|{E}_{1}|,|{E}_{3}|\right)$
|  $v$>1 |  |  | $\left(|{E}_{3}|,|{E}_{1}|\right)$
|  $v$>1 |  $\Delta$<1 |  $\Delta$<$v$ | $\left(|{E}_{1}|,|{E}_{2}|\right)$
|  |  $\Delta$>1 |  | $\left(|{E}_{2}|,|{E}_{1}|\right)$
IV |  $v$<1 |  $\Delta$>1 |  | $\left(|{E}_{1}|,|{E}_{3}|\right)$
|  $v$>1 |  |  | $\left(|{E}_{3}|,|{E}_{1}|\right)$
Table BI: The smallest energy overlaps of extended states in ${\rm{I}}$,
${\rm{II}}$, ${\rm{III}}$ and ${\rm{IV}}$ regions.  Figure B1: (Color
online)The sign of ${E}_{1}$, ${E}_{2}$ and ${E}_{3}$ in the $v-\Delta$ plane.

In this Appendix, we present a detailed derivation of the expression of
mobility edges Equations (19) and (21) in details following the method
proposed in Ref. [64]. To do this, we approximate the slowly varying
modulation model with an ensemble of different periodic models $M_{a}$. And
the extended state regime in the spectrum of the slowly varying modulation
model corresponds to the range of energy bands of these periodic models
$M_{a}$. Thus, the mobility edges can be determined as the smallest overlaps
of these ranges.

We first assume the slowly varying incommensurate modulation $\Delta_{j}$ may
take a series of constants $\Delta_{a}(\Delta_{a}\in[-\Delta,\Delta])$ in the
thermodynamic limit, and the system becomes an ensemble of periodic models
$M_{a}$. Under periodic boundary conditions(PBCs) with
$c^{\dagger}_{L+1,\alpha}=c^{\dagger}_{1,\alpha}$, the energy spectra of
$M_{a}$ can be obtained by the tight-binding approximation as follows:

$\displaystyle E$ $\displaystyle=$
$\displaystyle\pm\sqrt{(v+\Delta_{a})^{2}+1+2(v+\Delta_{a})\cos k},$ (B1)

where $k\in(0,2\pi]$ is the wave number. Here we only focus on the upper
band($E>0$) for positive $v$ and $\Delta$.

The extended state regime for $M_{a}$ is just the range of the energy band:

$\displaystyle|v+\Delta_{a}-1|<E<|v+\Delta_{a}+1|$ (B2)

or

$\displaystyle|v+\Delta_{a}+1|<E<|v+\Delta_{a}-1|.$ (B3)

There exists two limit situations for $\Delta_{a}$. When $\Delta_{a}=\Delta$,
we have

$\displaystyle|{E}_{1}|<E<|{E}_{4}|,$ (B4)

with $E_{1}=v+\Delta-1$ and $E_{4}=v+\Delta+1$. The other limit is
$\Delta_{a}=-\Delta$, for that we have

$\displaystyle|E_{2}|<E<|E_{3}|$ (B5)

or

$\displaystyle|E_{3}|<E<|E_{2}|,$ (B6)

with $E_{2}=v-\Delta+1$ and $E_{3}=v-\Delta-1$. To determine the smallest
overlaps of energy bands of $M_{a}$, one shall first sort the values of
$|E_{1}|$, $|E_{2}|$, and $|E_{3}|$. We first determine the sign of $E_{1}$,
$E_{2}$, and $E_{3}$ in the $v$-$\Delta$ plane. As shown in Figure B1, the
$v$-$\Delta$ plane is divided into four regions by three lines corresponding
to $E_{1}=0$, $E_{2}=0$, and $E_{3}=0$, respectively, each labeled with sign
combinations of $E_{1}$, $E_{2}$, and $E_{3}$. In region ${\rm{I}}$, for
instance, we have $|{E}_{1}|<|E_{3}|$ and $|E_{1}|<|E_{2}|$. When $\Delta<v$,
$|E_{1}|<|E_{3}|<|E_{2}|<|E_{4}|$, and the smallest energy overlaps for the
extended states is $E\in(|{E}_{3}|,|{E}_{2}|)$. Thus, the mobility edges reads

$\displaystyle E_{c}=1\pm(v-\Delta).$ (B7)

When $\Delta>v$, $|{E}_{1}|<|{E}_{2}|<|{E}_{3}|<|E_{4}|$. The smallest energy
overlaps of the extended states is given by $E\in(|{E}_{2}|,|{E}_{3}|)$, and
the expression of mobility edges is

$\displaystyle E_{c}=1\pm(v-\Delta).$ (B8)

For the regions ${\rm{II}}$, ${\rm{III}}$ and ${\rm{IV}}$, the smallest energy
overlaps can be obtained by the same procedure. Table BI lists the smallest
energy overlaps for different $v$ and $\Delta$ in four regions.

According to Table BI, there are three forms for mobility edges, i.e.,
$E_{c1}=|{E}_{1}|$, $E_{c2}=|{E}_{2}|$, and $E_{c3}=|{E}_{3}|$.

Figure B2: (Color online)The distribution of the smallest energy overlaps for
different $v$ and $\Delta$ in different regions. The shaded area indicates the
regime of the system with mobility edges.

In Figure B2, we show the region of the smallest energy overlaps of the
extended states in the $v$-$\Delta$ plane for $M_{a}$. We find that the
smallest energy overlaps of the extended states change from $(E_{c3},E_{c2})$
to $(E_{c2},E_{c3})$, when $\Delta$ goes beyond $v$ in the region $\rm{I}$. At
$E_{c2}=E_{c3}$, the extended state regime shrinks to a point with $\Delta=v$.
In the regions $\rm{II}$ and $\rm{III}$, the localization transition occurs at
$\Delta=1$ and $\Delta=v$. The shaded area in Figure B2 indicates the regime
of the system where the mobility edges actually exist. Outside this region,
all states are kept localized. As a conclusion, when $v>1$,
$E_{c}=v\pm\left(\Delta-1\right)$, and the localization transition occurs at
$\Delta=1$. And for $0<v<1$, $E_{c}=1\pm\left(v-\Delta\right)$, and the
localization transition occurs at $\Delta=v$.

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