based on Abhyankar-type constructions) to handle the $n\geq p,\,n\neq p+1$
case; Theorem 4.11 (proved in Section 4.2 based on the tamely ramified
rational function construction) to show $r_{\mathbb{F}_{q}(T)}(G)\leq 2$
whenever $p>n$ or $n=p+1$; Theorem 4.18 (combined with Proposition 4.15 which
is used to ensure the requisite $H(q,d,e)$ conditions hold; both proved in
Section 4.3) to eliminate the ramification at infinity and obtain
$r_{\mathbb{F}_{q}(T)}(G)=1$ in the $p>n$ and $n=p+1$ cases, when applicable;
a few remaining cases are handled ad hoc (some of the realizations were found
by a computer search).

######  Theorem 6.1 (Main results for $S_{n}$).

Let $n\geq 2$ and $q=p^{\nu}$ a prime power. Then
$r_{\mathbb{F}_{q}(T)}(S_{n})\leq 2$, and $r_{\mathbb{F}_{q}(T)}(S_{n})=1$ in
each of the following cases:

  1. (1)

$p<n-1$ or $p=n$ or $p=2$

  2. (2)

$q\equiv 1\mbox{ mod }4$

  3. (3)

$q>(2n-3)^{2}$

  4. (4)

The function field analogue of Schinzel’s hypothesis H (Conjecture 4.13) holds
for $\mathbb{F}_{q}(T)$.

###### Proof.

The case $n=2$ is very easy and follows for example from Theorem 3.3, so
assume from now on that $n\geq 3$. If $p<n-1$, $p=n$ or $(n,p)=(3,2)$, then
$r_{\mathbb{F}_{q}(T)}(S_{n})=1$ by Theorem 1.8, so suppose that $3\leq p=n-1$
or $p>n$.

For the rest of this proof, we call an $S_{n}$-extension of
$\mathbb{F}_{q}(U)$ a $(q,n,m)$-realization if it is the splitting field of
$f-Uc$ with $f,c\in\mathbb{F}_{q}[X]$, ${\rm deg}(f)=n$, ${\rm deg}(c)=m<n$,
and $\Delta_{X}(f-Uc)$ a prime power.

We first prove $r_{\mathbb{F}_{q}(T)}(S_{n})\leq 2$ in the remaining cases,
for which it suffices to exhibit a $(q,n,m)$-realization for some $m$: For
$n=5$ or $n\geq 7$, Theorem 4.11 with $m=n-4$ gives a $(q,n,n-4)$-realization
(note that $q\geq p\geq n-1>(2n-7)^{2/3}$ for all $n\geq 4$). For $n=3,4,6$,
Proposition 4.4 gives a $(q,n,0)$-realization.

We now prove $r_{\mathbb{F}_{q}(T)}(S_{n})=1$ in cases (2)-(4), for which it
suffices by Lemmas 4.16 and 4.17 to exhibit a $(q,n,m)$-realization such that
$(p,n-m)=1$ and $H(q,n+m-1,n-m)$ holds:

If (4) holds, then $H(q,d,e)$ always holds by Proposition 4.15(1), so we can
take the realizations from above.

If (3) holds and $n\neq 4$, Theorem 4.11 with $m=n-2$ gives a
$(q,n,n-2)$-realization (as $q>(2n-3)^{2}=(2m+1)^{2}$), and $H(q,2n-3,2)$
always holds by Proposition 4.15(3) (as $q$ is odd). If (3) holds and $n=4$,
we take the $(q,4,0)$-realization from above, and Proposition 4.15(5) gives
$H(q,3,4)$ for every $q\geq(2^{1}\cdot 3-1)^{2}=25=(2n-3)^{2}$.

If (2) holds, then $H(q,d,4)$ holds for every odd $d$ by Proposition 4.15(3);
if $n=5$ or $n\geq 7$, we obtained a $(q,n,n-4)$-realization above, and
$H(q,2n-5,4)$ holds; if $n=4$ we take the $(q,4,0)$-realization from above,
and $H(q,3,4)$ holds; if $n=3$ or $n=6$ we are in case (3) as soon as $q>9$
resp. $q>81$; for $n=6$ and $q\equiv 1\mbox{ mod }3$, $H(q,5,6)$ holds by
Proposition 4.15(3) so we can take the $(q,6,0)$-realization from above. In
each of the remaining cases, the following table provides
$f,c\in\mathbb{F}_{p}[X]$ with $\deg f=n$, $\deg c=m<n$ and
$h\in\mathbb{F}_{p}[T]$ with $\deg h=n-m$ such that $f^{\prime}c-cf^{\prime}$
is irreducible and $\mathcal{F}:=\Delta_{X}(f(X)-Tc(X))$ is such that
$\mathcal{F}(h)$ is irreducible, so that by the arguments in the proof of
Theorem 4.18(i) (where now $h$ is given explicitly instead of applying
$H(q,n+m-1,n-m)$, and $f$ is given explicitly instead of applying Proposition
4.10(i)), the splitting field of $f(X)-h(T)c(X)$ over $\mathbb{F}_{p}(T)$ is
geometric with Galois group $S_{n}$ and is ramified only over the prime
$\mathcal{F}(h)$.

$n$ | $p$ | $f$ | $c$ | $h$
---|---|---|---|---
$3$ | $5$ | $X^{3}+1$ | $X+2$ | $T^{2}$
$6$ | $5$ | $X^{6}+1$ | $X^{2}+X$ | $T^{4}+1$
$6$ | $17$ | $X^{6}+X^{2}+X$ | $1$ | $T^{6}+T+2$
$6$ | $29$ | $X^{6}+X^{2}+X$ | $1$ | $T^{6}+T+6$
$6$ | $41$ | $X^{6}+X$ | $1$ | $T^{6}+T+1$
$6$ | $53$ | $X^{6}+X^{2}+13X$ | $1$ | $T^{6}+T$

∎

######  Remark 6.2.

Note that instead of $q\equiv 1\mbox{ mod }4$ we could also treat other
arithmetic progressions: For any prime number $\ell$, if $q\equiv 1\mbox{ mod
}2\ell$, $n\geq 2\ell+3$, and $2n\not\equiv 1\mbox{ mod }\ell$, let
$m=n-2\ell$. Then, in the language of the previous proof, Theorem 4.11 gives a
$(q,n,m)$-realization, and $H(q,n+m-1,n-m)=H(q,2n-2\ell-1,2\ell)$ holds by
Proposition 4.15(3).

######  Theorem 6.3 (Main results for $A_{n}$).

Let $n\geq 3$ and $q=p^{\nu}$ a prime power. If $p>2$ or
$\mathbb{F}_{q}\supseteq\mathbb{F}_{4}$, then
$r_{\mathbb{F}_{q}(T)}(A_{n})\leq 2$, and $r_{\mathbb{F}_{q}(T)}(A_{n})=1$ in
each of the following cases:

  1. (1)

$2<p<n-1$ or $p=n$ or $p=n-1,\mathbb{F}_{q}\supseteq\mathbb{F}_{p^{2}}$ or
$p=3,n=4$.

  2. (2)

$p=2$ and $\mathbb{F}_{q}\supseteq\mathbb{F}_{4}$ or $10\neq n\geq 8,n\equiv
0,1,2,6,7\pmod{8}$

  3. (3)

$12\neq n\geq 10$ and the function field analogue of Schinzel’s hypothesis H
(Conjecture 4.13) holds for $\mathbb{F}_{q}(T)$.

###### Proof.

The cases $p=2$ and $p=3$ follow from Theorem 1.8, so henceforth we will
assume $p>3$.

If $p\leq n,p\neq n-1$ or $p=n-1,\mathbb{F}_{q}\supseteq\mathbb{F}_{p^{2}}$,
all assertions follow from Theorem 1.8.

If $p=n-1$ and $n\geq 14$ we may use Theorem 4.11 with $m=3$ to obtain
$r_{\mathbb{F}_{q}(T)}(A_{n})\leq 2$ in this case. Assuming Conjecture 4.13
allows us to apply Theorem 4.18 combined with Proposition 4.15(1) (once again
with $m=3$) to conclude $r_{\mathbb{F}_{q}(T)}(A_{n})=1$ in this case.

If $p=n-1$ and $n\in\\{6,8,12\\}$, the following table provides irreducible
$f\in\mathbb{F}_{p}[T,X]$ with $\deg f=n$ and $\Delta_{X}(f)$ a square in
$\mathbb{F}_{p}(T)$ with only one prime divisor $T$. Computer verification
shows that ${\rm Gal}(f/\mathbb{F}_{p}(T))$ contains a $3$-cycle and an
$(n-1)$-cycle, hence by Theorem 2.3 contains $A_{n}$. Thus the splitting field
of $f$ over $\mathbb{F}_{q}(T)$ is ramified at most over $T$ and the infinite
prime, and ${\rm Gal}(f/\mathbb{F}_{q}(T))=A_{n}$ as $A_{n}$ is simple.

$n$ | $p$ | $f$ | $\Delta_{X}(f)$
---|---|---|---
$6$ | $5$ | $X^{6}+X^{5}T-2X^{3}T^{3}+XT+T^{2}$ | $4T^{18}$
$8$ | $7$ | $X^{8}+3X^{2}+XT-2$ | $4T^{2}$
$12$ | $11$ | $X^{12}+5XT^{3}-5X^{2}-2$ | $4T^{6}$

Now assume that $p>n$. If $n\geq 13$ or $n=11$ we apply Theorem 4.11 with
$m=2$ or $m=3$ chosen so that $m\not\equiv n\pmod{2}$ and thus obtain
$r_{\mathbb{F}_{q}(T)}(A_{n})\leq 2$ in this case. If $n=10$ we apply Theorem
4.11 with $m=1$ (note that
$\left(\frac{q}{n-m}\right)=\left(\frac{q}{3}\right)^{2}=1$). Assuming
Conjecture 4.13 allows us to apply Theorem 4.18 combined with Proposition
4.15(1) (with $m$ as above) to conclude $r_{\mathbb{F}_{q}(T)}(A_{n})=1$ in
these cases. In all other cases, Theorem 4.21 gives that
$r_{\mathbb{F}_{q}(T)}(A_{n})\leq 2$. ∎

######  Remark 6.4.

The proofs of Theorems 6.1 and 6.3 above rely on Theorems 4.11 and 5.7, which
use the Classification of Finite Simple Groups (CFSG) for some of the cases.
By Remark 4.6, the CFSG is not necessary in the proof of Theorem 4.18 provided
$n-m\leq 15$. The proof of Theorem 5.7 uses the CFSG only in the cases $n=p+2$
and $G=A_{n},p=2|n$. Hence Theorems 6.1, 6.3, and by implication Theorems 1.6,
1.7, 1.8, 1.10, require the CFSG only in the following cases:

  * •

$n=p+2$

  * •

$G=A_{n}$ and ($p>n$ or ($n=p+1$ and
$\mathbb{F}_{q}\not\supset\mathbb{F}_{p^{2}})$ or $p=2|n$).

The upper bound $r_{\mathbb{F}_{q}(T)}(A_{n})\leq 2$ in the case $p>n$ follows
from Theorem 4.21, which does not require the CFSG.

###  6.2. Application to the minimal ramification problem over $\mathbb{Q}$

Finally, we apply our results for $S_{n}$ over $\mathbb{F}_{q}(T)$ to give a
conditional proof of Conjecture 1.1 for $S_{n}$ over $\mathbb{Q}$:

######  Lemma 6.5.

Let $n\in\mathbb{N}$, $h_{0}\in\mathbb{Z}$, and $S$ a finite set of primes
numbers, and for each $p\in S$ let $f_{p}\in\mathbb{Z}[X]$ be monic of degree
$n$. There exist $f\in\mathbb{Z}[X]$ monic of degree $n$ and $c\in\mathbb{Z}$
such that $f\equiv f_{p}\pmod{p}$ and $c\equiv 1\pmod{p}$ for every $p\in S$,
and $f-c^{n}h_{0}$ has $n$ roots in $\mathbb{R}$.

###### Proof.

By the Chinese Remainder Theorem there exists a monic $f_{0}\in\mathbb{Z}[X]$
with $f_{0}\equiv f_{p}\pmod{p}$ for every finite $p\in S$. Let
$\pi=\prod_{p\in S}p$, write $f_{0}=\sum_{i=0}^{n}a_{i}X^{i}$ and let
$f_{\infty}\in\mathbb{Z}[X]$ be any monic polynomial of degree $n$ with $n$
roots in $\mathbb{R}$. Since fractions of the form $\frac{x\pi}{y\pi+1}$ with
$x,y\in\mathbb{Z}$ are dense in $\mathbb{R}$, we can choose
$x_{i},y_{i}\in\mathbb{Z}$ such that
$\tilde{f}:=X^{n}+\sum_{i=0}^{n-1}(a_{i}+\frac{x_{i}\pi}{y_{i}\pi+1})X^{i}$ is
arbitrarily close to $f_{\infty}+h_{0}$, in particular so close that also
$\tilde{f}-h_{0}$ has $n$ roots in $\mathbb{R}$. Then with
$c=\prod_{i=0}^{n-1}(y_{i}\pi+1)$, the polynomial
$f(X):=c^{n}\tilde{f}(c^{-1}X)\in\mathbb{Z}[X]$ satisfies the claim. ∎

######  Theorem 6.6.

Schinzel’s hypothesis H (Conjecture 4.12) implies that
$r_{\mathbb{Q}}(S_{n})=1$ for every $n\geq 2$, i.e. Conjecture 1.1 holds for
symmetric groups.

###### Proof.

In case $n=2$, for example $L=\mathbb{Q}(\sqrt{2})$ does the job, so assume
that $n\geq 3$. Let $S_{0}$ denote the set of prime numbers $\ell<d:=n(n-1)$
and for $\ell\in S_{0}$ fix a separable polynomial
$f_{\ell}\in\mathbb{F}_{\ell}[X]$. By Proposition 4.2, if $p$ is sufficiently
large with respect to $n$, there exists a monic $f_{p}\in\mathbb{F}_{p}[X]$ of
degree $n$ such that the splitting field of $f_{p}-U$ over $\mathbb{F}_{p}(U)$
is geometric with Galois group $S_{n}$, and
$\mathcal{F}=\Delta_{X}(f_{p}-U)\in\mathbb{F}_{p}[U]$ is irreducible of degree
$n-1$ (cf. (2.4) and Lemma 2.8). By Proposition 4.15(2,c), if $p$ is
sufficiently large with respect to $n$, there exists a monic
$h_{p}\in\mathbb{F}_{p}[T]$ with ${\rm deg}(h_{p})=n$ such that
$\mathcal{F}(h_{p}(T))$ is irreducible. Fix such a sufficiently large $p\notin
S_{0}$ and $f_{p},h_{p}$ as above, and let $g_{p}:=f_{p}(X)-h_{p}(T)$. Then
$\Delta_{X}(g_{p})=\mathcal{F}(h_{p}(T))\in\mathbb{F}_{p}[T]$ is irreducible,
and Lemma 4.16 implies that the splitting field of $g_{p}$ over
$\mathbb{F}_{p}(T)$ is geometric with Galois group $S_{n}$.

Choose $h\in\mathbb{Z}[T]$ monic of degree $n$ with $h\equiv h_{p}\pmod{p}$
and $h\equiv T^{n}\pmod{\ell}$ for $\ell\in S_{0}$. By Lemma 6.5 there exist a
monic $f\in\mathbb{Z}[X]$ and $c\in\mathbb{N}$ such that $f\equiv
f_{\ell}\pmod{\ell}$ and $c\equiv 1\mbox{ mod }\ell$ for every $\ell\in
S_{0}\cup\\{p\\}$, and such that $f-c^{n}h(0)$ has $n$ roots in $\mathbb{R}$.
Let

$g(X,T)=f(X)-c^{n}h(c^{-1}T)$

and note that $g\in\mathbb{Z}[X,T]$ is monic of degree $n$ both in $X$ and in
$T$, $g\equiv g_{p}\pmod{p}$ and $g\equiv f_{\ell}-T^{n}\pmod{\ell}$ for
$\ell\in S_{0}$. So as ${\rm Gal}(g_{p}/\overline{\mathbb{F}}_{p}(T))=S_{n}$
and ${\rm deg}_{X}(g)={\rm deg}_{X}(g_{p})=n$, we get that the splitting field
$L$ of $g$ over $\mathbb{Q}(T)$ is geometric with Galois group $S_{n}$:
Reduction modulo $p$ extends to a place from $\overline{\mathbb{Q}}$ to
$\overline{\mathbb{F}}_{p}$ by Chevalley’s theorem, and further to a place
from $\overline{\mathbb{Q}}(T)$ to $\overline{\mathbb{F}}_{p}(T)$ mapping $T$
to $T$, so ${\rm Gal}(g_{p}/\overline{\mathbb{F}}_{p}(T))=S_{n}$ embeds into
${\rm Gal}(g/\overline{\mathbb{Q}}(T))$ (cf. [FJ08, Lemma 16.1.1(a)]), which
is itself a subgroup of $S_{n}$. Similarly, also
$\Delta_{X}(g)\equiv\Delta_{X}(g_{p})\mbox{ mod }p$, so as $\Delta_{X}(g_{p})$
is irreducible and ${\rm deg}_{T}(\Delta_{X}(g))=d={\rm
deg}_{T}(\Delta_{X}(g_{p}))$, we get that $\Delta_{X}(g)$ is irreducible. In
particular, $\Delta_{X}(f-U)$ is irreducible. Identifying
$U=c^{n}h(c^{-1}T)\in\mathbb{Q}(T)$, since $\Delta_{X}(f-U)$ is irreducible,
the splitting field of $f-U$ over $\mathbb{Q}(U)$ is ramified at most over the
finite prime $\mathcal{F}_{0}:=\Delta_{X}(f-U)$ and the infinite prime, so by
Lemma 4.17, $L/\mathbb{Q}(T)$ is ramified only at the prime
$\mathcal{F}:=\mathcal{F}_{0}(c^{n}h(c^{-1}T))=\Delta_{X}(g)$.

In geometric terms, the extension $L/\mathbb{Q}(T)$ corresponds to a branched
covering of geometrically irreducible smooth curves $\phi\colon
C\rightarrow\mathbb{P}^{1}_{\mathbb{Q}}$ with ${\rm
Gal}(\mathbb{Q}(C)/\mathbb{Q}(\mathbb{P}^{1}))=S_{n}$. We wish to apply [BS20,
Proposition 1.11] to $\phi$, for which we use the notation and definitions
from there. As $\phi$ has only the one ramified prime divisor $\mathcal{F}$,
its branch locus is the zero locus of an irreducible homogeneous polynomial
$D(X,Y)\in\mathbb{Z}[X,Y]$ of degree $d$. As in [BS20, p. 923], for
$\xi\in\mathbb{P}^{1}(\mathbb{Q})$ we denote by $A^{\phi}_{\xi}$ the
specialized algebra at $\xi$, which for
$\xi=[a:1]\in\mathbb{A}^{1}(\mathbb{Q})$ is
$A^{\phi}_{[a:1]}=R\otimes_{\mathbb{Q}[T]_{(T-a)}}\mathbb{Q}=R/(T-a)$, where
$R$ denotes the integral closure of $\mathbb{Q}[T]_{(T-a)}$ in
$\mathbb{Q}(C)$. As $\xi$ is unramified in $\mathbb{Q}(C)$, $A^{\phi}_{\xi}$
is an étale $\mathbb{Q}$-algebra, i.e.
$A^{\phi}_{\xi}\cong\prod_{i=1}^{r}F_{i}$ with $F_{i}/\mathbb{Q}$ a field
extension, and a prime of $\mathbb{Q}$ is called unramified in
$A^{\phi}_{\xi}$ if it is unramified in each $F_{i}$. We now show that the set
of universally ramified primes $U(\phi)$ as defined in [BS20, p. 924] is
empty, i.e. every prime of $\mathbb{Q}$ is unramified in some
$A^{\phi}_{\xi}$:

As $g(X,0)=f-c^{n}h(0)$ is separable, the integral closure $R$ of
$\mathbb{Q}[T]_{(T)}$ in $\mathbb{Q}(C)$ is generated by the roots of $g$ (cf.
[FJ08, Lemma 6.1.2]), hence $A^{\phi}_{[0:1]}$ is generated by elements
$a_{1},\dots,a_{n}$ with $g(a_{i},0)=0$. In particular, since $g(X,0)$ splits
into linear factors over $\mathbb{R}$, the infinite prime is unramified in
$A^{\phi}_{[0:1]}$. For prime numbers $\ell<d$, $g(X,0)\equiv f_{\ell}\mbox{
mod }\ell$ is separable, so $\ell$ is unramified in each $\mathbb{Q}(a_{i})$
(see e.g. [Neu99, Prop. I.8.3]) and hence in $A^{\phi}_{[0:1]}$. Finally, as
$\Delta_{X}(g)$ is of degree $d$ with leading coefficient $\pm n^{n}$, it is
also not the zero function modulo any prime number $\ell>d$, so for every such
$\ell$ there exists $a\in\mathbb{Z}$ such that $g(X,a)\mbox{ mod }\ell$ is
separable, and thus the same argument shows that $\ell$ is unramified in
$A^{\phi}_{[a:1]}$.

Thus in the language of [BS20, Definition 1.10], $G=S_{n}$ has a
$(U;\mathbf{d})$-realization with $U=\emptyset$ and $\mathbf{d}=(d)$.
Therefore, [BS20, Proposition 1.11] gives a realization of $G$ over
$\mathbb{Q}$ with at most $B(\mathbf{d})+\\#U$ many ramified primes, and under
Schinzel’s hypothesis H, $B(\mathbf{d})\leq 1$, cf. [BS20, (13) on p. 923]. ∎

######  Remark 6.7.

Plans [Pla04, Remark 3.10] sketches how to use Schinzel’s hypothesis H
(Conjecture 4.12) to construct $S_{n}$-extensions $L$ of $\mathbb{Q}$ ramified
in only one finite prime. However, he constructs $L$ as the splitting field of
an irreducible trinomial $f=X^{n}+aX^{i}+b$, and therefore $L$ is not totally
real if $n\geq 5$: Indeed, replacing $X$ by $X^{-1}$ if necessary we may
assume that $i\geq\frac{n}{2}$, so since the derivative
$f^{\prime}=X^{i-1}(nX^{n-i}+ia)$ has at most $n-i+1$ distinct roots, in
particular at most $n-i+1$ distinct roots in $\mathbb{R}$, Rolle’s theorem
shows that $f$ has at most $n-i+2\leq\frac{n}{2}+2<n$ roots in $\mathbb{R}$.

## Acknowledgements

The authors would like to thank Stephen Cohen for helpful discussion and for
pointing out the results in [Coh10], David Harbater and Robert Lemke-Oliver
for some helpful discussions, Joachim König for pointing out the observation
in Remark 1.5 and providing a reference for Lemma 3.1, and Danny Neftin for
helpful discussions on [KNS10].

The computations in Section 4.4 were carried out with SageMath. The
computations in Section 6.1 were carried out with SageMath and verified using
Magma.

L. B.-S. was partially supported by a grant of the Israel Science Foundation
no. 702/19. A. E. was partially supported by a grant of the Israel Science
Foundation no. 2507/19. Some of this research was conducted while A. F. was
visiting Tel Aviv University, and he would like to thank Zeév Rudnick and the
Shaoul Fund for financial support.

## References

  * [Abh92] S. S. Abhyankar.  Galois theory on the line in nonzero characteristic.  Bull. Amer. Math. Soc. 27(1):68–133, 1992.
  * [Abh93] S. S. Abhyankar.  Alternating group coverings of the affine line for characteristic greater than two.  Math. Ann. 296:63–68, 1993.
  * [Abh95] S. S. Abhyankar.  Mathieu group coverings and linear group coverings.  In Recent developments in the inverse Galois problem, Contemp. Math. 186:293–319, Amer. Math. Soc., Providence, RI, 1995.
  * [Abh01] S. S. Abhyankar.  Resolution of Singularities and modular Galois theory.  Bull. Amer. Math. Soc. 38(2):131–169, 2001.
  * [AOS94] S. S. Abhyankar, J. Ou, and A. Sathaye.  Alternating group coverings of the affine line for characteristic two.  Disc. Math. 133:25–46, 1994.
  * [AY94] S. S. Abhyankar and I. Yie.  Small degree coverings of the affine line in characteristic 2.  Disc. Math., 133:1–23, 1994.
  * [Adr17] N. Adrianov.  Primitive monodromy groups of rational functions with one multiple pole.  J. Math. Sci. 226(5):548–560, 2017.
  * [AZ15] N. Adrianov and A. Zvonkin.  Weighted trees with primitive edge rotation groups.  J. Math. Sci. 209(2):160–191, 2015.
  * [Bae64] R. Baer.  Der reduzierte Rang einer Gruppe.  J. Reine Angew. Math. 214/215:146-173, 1964.
  * [Bar12] L. Bary-Soroker.  Irreducible values of polynomials.  Adv. Math. 229:854–874, 2012.
  * [BE21] L. Bary-Soroker and E. Entin.  Explicit Hilbert’s irreducibility theorem in function fields.  In: Jarden, Shaska (eds.), Abelian Varieties and Number Theory. Contemporary Mathematics 767, pp. 125–134, 2021.
  * [BS20] L. Bary-Soroker and T. Schlank.  Sieves and the minimal ramification problem.  J. Inst. Math. Jussieu 19(3):919–945, 2020.
  * [BM09] N. Boston and N. Markin.  The fewest primes ramified in a $G$-extension of $\mathbb{Q}$.  Ann. Sci. Math. Québec 33(2):145–154, 2009.
  * [CHLPT15] A. Castillo, C. Hall, R. J. Lemke Oliver, P. Pollack, and L. Thompson.  Bounded gaps between primes in number fields and function fields.  Proc. Amer. Math. Soc. 143:2841–2856, 2015.
  * [Coh10] S. D. Cohen.  Primitive elements on lines in extensions of finite fields.  In: McGuire, Mullen, Panario and Shparlinski (eds.), Finite Fields: Theory and Applications, Contemporary Mathematics 518, pp. 113–127, 2010
  * [Con05] K. Conrad.  Irreducible Values of Polynomials: a Non-analogy.  In: van der Geer, Moonen, Schoof (eds.), Number Fields and Function Fields: Two Parallel Worlds, 71–85, Birkhäuser, 2005.
  * [CCG08] B. Conrad, K. Conrad, and R. Gross.  Prime specialization in genus 0.  Trans. Amer. Math. Soc. 360(6):2867–2908, 2008.
  * [DeW14] M. De Witt.  Minimal ramification and the inverse Galois problem over the rational function field $\mathbb{F}_{p}(t)$.  J. Number Theory 143:62–81, 2014.
  * [DM96] J. D. Dixon and B. Mortimer.  Permutation groups.  1st ed., GTM, vol. 163, Springer, 1996.
  * [Ent16] A. Entin.  On the Bateman-Horn Conjecture for Polynomials over Large Finite Fields.  Compos. Math. 152(2), 2016.
  * [Ent19] A. Entin.  Monodromy of Hyperplane Sections of Curves and Decomposition Statistics over Finite Fields.  Int. Math. Res. Not. rnz120, 2019.
  * [Fri70] M. D. Fried.  On a conjecture of Schur.  Michigan Math. J. 17:41–55, 1970.
  * [FJ08] M. D. Fried and M. Jarden.  Field Arithmetic.  Third Edition. Springer, 2008.
  * [FM69] M. D. Fried and R. E. MacRae. On the invariance of chains of fields.  Illinois J. Math. 13:165–171, 1969.
  * [Gey99] W.-D. Geyer.  Appendix to: M. Jarden and A. Razon, Skolem density problems over large Galois extensions of global fields.  In: Denef, Lipshitz, Pheidas, Van Geel (eds.), Hilbert’s tenth problem: relations with arithmetic and algebraic geometry, American Mathematical Society, 2000.
  * [Gor80] D. Gorenstein.  Finite groups. Second Edition. Chelsea Pub. Co., 1980.
  * [GR71] A. Grothendieck and A. Raynaud.  Revêtements étales et groupe fondamental (SGA 1).  Lecture Notes in Mathematics 224, Springer, 1971.
  * [GMPS16] S. Guest, J. Morris, C. E. Praeger, and P. Spiga.  Finite primitive permutation groups containing a permutation having at most four cycles.  J. Algebra, 454:233–251, 2016.
  * [Har94] D. Harbater.  Abhyankar’s conjecture on Galois groups over curves.  Invent. Math. 117(1):1–25, 1994.
  * [HP02] D. Harbater and M. van der Put.  Valued fields and covers in characteristic $p$. With an appendix by R. Guralnick.  Fields Inst. Commun. 32:175-204, 2002.
  * [HOPS18] D. Harbater, A. Obus, R. Pries, and K. Stevenson.  Abhyankar’s conjectures in Galois theory: Current status and future directions.  Bull. Amer. Math. Soc. 55(2):239–287, 2018.
  * [Hay74] D. R. Hayes.  Explicit class field theory for rational function fields.  Trans. Amer. Math. Soc. 189, 1974.
  * [Jon14] G. A. Jones.  Primitive permutation groups containing a cycle.  Bull. Australian Math. Soc. 89(1):159–165, 2014.
  * [KNS10] H. Kisilevsky, D. Neftin, and J. Sonn.  On the minimal ramification problem for semiabelian groups.  Algebra Number Theory 4(8):1077–1090, 2010.
  * [KS06] H. Kisilevsky and J. Sonn.  Abelian extensions of global fields with constant local degrees.  Math. Research Letters 13(4):599–605, 2006.
  * [KS10] H. Kisilevsky and J. Sonn.  On the minimal ramification problem for $\ell$-groups.  Compos. Math. 146(3):599–606, 2010.
  * [Kut76] P. Kutzko.  On groups of finite weight.  Proc. Amer. Math. Soc. 55(2):279-280, 1976.
  * [Lan02] S. Lang.  Algebra.  Springer, 2002.
  * [LW54] S. Lang and A. Weil.  Number of points of varieties in finite fields.  Amer. J. Math., 76:819–827, 1954.
  * [May15] J. Maynard.  Small gaps between primes.  Ann. Math., 181:1–31, 2015.
  * [Mil80] J. Milne.  Étale Cohomology. Princeton Math. Series 33, Princeton Univ. Press, 1980.
  * [Mül93] P. Müller.  Primitive monodromy groups of polynomials.  Contemp. Math. 186:385–398, 1993.
  * [Mül94] P. Müller.  Monodromiegruppen rationaler Funktionen und Irreduzibilität von Polynomen mit variablen Koeffizienten.  Thesis, 1994.
  * [Neu99] J. Neukirch.  Algebraic Number Theory. Springer, 1999.
  * [Pla04] B. Plans.  On the minimal number of ramified primes in some solvable extensions of $\mathbb{Q}$.  Pacific J. Math. 215(2), 2004.
  * [Pol08] P. Pollack.  Simultaneous prime specializations of polynomials over finite fields.  Proc. Lond. Math. Soc. 97(3):545–567, 2008.
  * [Ray94] M. Raynaud.  Revêtements de la droite affine en caractéristique $p>0$ et conjecture d’Abhyankar.  Invent. Math. 116(1-3):425–462, 1994.
  * [Ros02] M. I. Rosen.  Number Theory in Function Fields.  Graduate Texts in Math. vol. 210, Springer-Verlag, 2002.
  * [SS20] W. Sawin and M. Shusterman.  Möbius cancellation on polynomial sequences and the quadratic Bateman-Horn conjecture over function fields.  arXiv:2008.09905 [math.NT], 2020.
  * [Ser73] J.-P. Serre  A course in arithmetic.  Graduate Texts in Math. vol. 7, Springer-Verlag, 1973.
  * [Ser79] J. P. Serre.  Local fields.  Graduate Texts in Math. vol. 67, Springer, 1987.
  * [Ser08] J.-P. Serre.  Topics in Galois theory.  Second edition. Springer, 2008.
  * [Sti09] H. Stichtenoth.  Algebraic Function Fields and Codes.  Springer, 2009.
  * [Völ96] H. Völklein.  Groups as Galois groups.  Cambridge University Press, 1996.
  * [Wie64] H. Wielandt.  Finite permutation groups. Academic Press, 1964.


# Topological Anderson insulator phases in one dimensional quasi-periodic
mechanical SSH chains

Sayan Sircar  Tata Institute of Fundamental Research, Hyderabad 500046, India

(July 29, 2024)

###### Abstract

In this paper, we investigate the transition between topological phases in a
Su-Schrieffer-Heeger (SSH) model composed of springs and masses in which the
intracellular Aubry-André disorder modulates the spring constants. We observe
the presence of a topological Anderson insulating (TAI) phase within a
specific range of quasi-periodic modulation strength and calculate the phase
transition boundary analytically. We examine the localization properties of
normal modes using their inverse participation ratio (IPR) of eigenstates of
the dynamical matrix, and the corresponding fractal dimension associated with
quasiperiodic modulation. We also examine the stability of the TAI phase
across a range of modulation strengths and comments on the presence of
mobility edge that separate localized modes from non-localized ones. We
demonstrate the fact that special analytical techniques are needed to compute
an exact expression for mobility edges in such scenarios.

##  I INTRODUCTION

The study of topological phases of matter Thouless  _et al._ (1982) has
gardened significant research interest in the last decades. Among this,
topological insulators Hasan and Kane (2010); Qi and Zhang (2011) is a
fascinating phase characterized by bulk states with a gap at the Fermi level,
while hosting conducting edge modes in the bulk gap, Armitage  _et al._ (2018)
characterized by a non-trivial winding number, as per bulk-edge correspondence
Prodan and Schulz-Baldes (2016). The Su–Schrieffer–Heeger (SSH) model Su  _et
al._ (1979) is a fundamental system consisting of two bands and showing non-
trivial band topology , has been introduced to examine conductivity and
transport properties of polyacetylene chain, particularly focusing on
fractional charge soliton excitations, Heeger  _et al._ (1988); Jackiw and
Rebbi (1976) and non-trivial conducting edge modes Ganeshan  _et al._ (2013)
in open boundary condition (OBC). Conducting edge states are maintained by
internal symmetries and are resilient to specific types of disorder. Disorder
in translational invariant systems disrupts the translational invariance of
the system, leading to the assumption that it could undermine the topological
properties. The emergence of topological Anderson insulators (TAIs) indicates
that disorder can trigger the transition from topologically trivial to non-
trivial phases, Li  _et al._ (2009); Jiang  _et al._ (2009); Groth  _et al._
(2009); Song and Prodan (2014); Mondragon-Shem  _et al._ (2014). Anderson
localization was proposed in a model of disordered system Anderson (1958); Lee
and Ramakrishnan (1985) first in 1958, which since then has been performed in
various experimental settings Pradhan and Sridhar (2000). Typically, real
materials exhibit Anderson-type disorder Anderson (1958), characterized by
random potentials generated by limited concentration of contaminants dispersed
through out the material. Depending on spatial dimensions, potential intensity
$\delta$ has a critical value for example in three dimensions Abrahams  _et
al._ (1979). In contrast, in one dimension, for an infinitely large system,
most of the states remain localized for arbitrary finite value of $\delta$.

The emergence of twisted bilayer-graphene Gonçalves  _et al._ (2021)
introduces a novel form of disorder known as quasi-periodic disorder. The best
known model to track quasi-periodicity in condensed matter systems is Aubry-
Andre-Harper (AAH) model. The 1D Aubry-Andre-Harper (AAH) model Aubry and
Andre (1980) illustrates a metal-insulator transition governed by self-duality
point Domínguez-Castro and Paredes (2019). This transition occurs when the
quasi-periodic modulation strength surpasses a critical threshold. Aubry and
Andre introduced a one dimensional hopping model in tight-binding
approximation, in which electrons experience a sinusoidal electrostatic
potential that is incommensurate with the lattice. They demonstrated that the
transition occurs at $\delta^{AA}=2J$, where $J$ represents the nearest-
neighbour electron hopping amplitude. In this scenario, all eigenmodes become
exponentially localized for $\delta^{AA}>2J$ or extend as plane waves for
$\delta^{AA}<2J$. These results differ from Anderson localization phenomenon
that occurs in one dimensional systems. These conclusions have also been
extended to non-Hermitian systems Longhi (2019a, b). The presence of mobility
edges in disordered systems, particularly in 3D Anderson insulators, has been
demonstrated even in the 1D generalized AAH model. Compact analytic results
for mobility edges under broken self-duality have been obtained Biddle and Das
Sarma (2010); Liu  _et al._ (2020a); Wang  _et al._ (2020); Zhu  _et al._
(2023) and for slowly varying modulation incommensurate with the lattice Das
Sarma  _et al._ (1988, 1990); Liu and Guo (2018).These works focus on
calculating mobility edges that differentiate localized from non-localized
modes. Recent studies have expanded this concept to anomalous mobility edges,
which separate localized states from critical ones in one-dimensional
disordered systems, differing from conventional mobility edges, Liu  _et al._
(2022a). Disorder-induced topological Anderson insulators (TAI) have been
experimentally observed in 2D photonics Liu  _et al._ (2020b), 1D wires with
controllable random disorder Meier  _et al._ (2018) protected by chiral
symmetry, superconductors with spin-orbit coupling Zangeneh-Nejad and Fleury
(2020); Liu  _et al._ (2020b). The TAI phase also emerges in this model (SSH)
modulated with quasi-periodic intercell hopping, where most of the eigenstates
are localized Longhi (2020); Sircar (2024). Recently there are interest in
exploring similar phases in several acoustic systems and elastic Yang  _et
al._ (2015); Vila  _et al._ (2017). Hence its a natural urge to search for a
similar phase in various mechanical settings Shi  _et al._ (2021) as well.

In this article, starting from the model as shown in Ref. Lu  _et al._ (2022),
we apply quasi-periodic modulation in on-site intra-cellular spring stiffness
in a mechanical SSH chain, and calculate the topological phase transition
boundary, localization properties with modulation $\delta^{AA}\cos(2\pi\beta
j^{u}+\phi)$, where $\delta^{AA}$ is the quasi-periodic modulation strength
and $u=1$.

(a) (b)
(c) (d)

Figure 1: Plot of energy versus mode number for a 1D spring mass chain under
following conditions: (a) $K_{a}<K_{b}$, (b) $K_{a}>K_{b}$ for
$\delta^{AA}=1.4$, (c) $K_{a}>K_{b}$ for $\delta^{AA}=1.6$, (d) $K_{a}>K_{b}$
for $\delta^{AA}=2$. For all cases we calculate the eigenvalues using the
dynamical matrix from Eq. (3). For cases (a), (b), (c) the spectrum shows a
gap near the Fermi level with mid-gap modes. For those cases, the insets
displays the probability distribution of the mid-gap states, which are
localized at the edges. The color will indicate the chirality of two edge
states, highlighting the topologically non-trivial phase. In case (d), the
plot reveals an absence of mid-gap modes in the spectrum. The inset
illustrates the probability distribution of mid-gap states extending across
the bulk without any presence of edge modes. This phenomenon accompanies lack
of chiral symmetry, which indicates that our system is indeed in a phase,
which has a non-zero winding number. The scenarios in cases (b) and (c) falls
under the TAI phase range, which is shown in Fig. 3, that reveals the presence
of conducting edge states protected by the symmetries of the dynamical matrix.
In case (d), we verify that as $\delta^{AA}=2$, the TAI phase disappears,
leading to a trivial Anderson insulator phase.

##  II Mechanical spring-mass model

This section provides introduction to the SSH mechanical model used, and
examines its topological properties in real space, akin to Ref. Sircar (2024).

A$K_{a}$B…$K_{b}$A$K_{a}$B Figure 2: Schematic representation of a mechanical
SSH chain, where dashed lines represent intra-cellular springs ($K_{a}$) and
solid lines denote inter-cellular springs ($K_{b}$). Each unit cell consists
of $A$ and $B$ masses, with the black solid circles at the ends representing
fixed points.

###  II.1 Dynamical matrix of the spring-mass chain

We are studying a one-dimensional finite chain whose ends are held fixed
(FBC), and is made up of N unit cells Shi  _et al._ (2021); Miranda  _et al._
(2024); Sircar (2024).

Figure 3: Plot of topological invariant defined in Eq. (7) in the phase space
of $K_{a}$ and $\delta^{AA}$. The phase plots clearly shows that at the
critical strength of quasi-periodic strength $\delta^{AA}=2$, the topological
phase transition is independent of the value of $K_{a}$.

One can identify two masses (having same mass) in each unit cell by a sub-
lattice labelling $\alpha={A,B}$ along with a unit cell labelling $j\in[1,N]$.
There are two types of spring stiffness constants: $K_{a}$ for intra-cellular
and $K_{e}$ for inter-cellular interactions, analogous to the hopping
parameters in the original SSH chain. By tuning those parameters in this
mechanical setting, different topological phases, akin to the original quantum
SSH model, can be realized. The equation of motion for each mass can be
formulated using Newton’s second law or, equivalently, the Euler-Lagrange
equations, which are commonly employed in normal modes analysis. The equations
can be compactly expressed in matrix notation, assuming each mass undergoes
harmonic displacement over time,

$u_{j}(t)=u_{j}e^{-i\omega_{j}t}.$ (1)

The second-order equations of motion can be converted into a linear matrix
equation.

Figure 4: Evolution of topological invariant defined in Eq. (7) versus the
quasi-periodic modulation strength $\delta^{AA}$ . The plot shows emergence of
TAI phase in some intermediate value of modulation strength $\delta^{AA}$, the
exact value is calculated in Eq. (23) and Eq. (24). The black arrows does not
indicate the range of TAI phase, it only indicates the values where the LTM
does not converge to winding number due to finite size effects.

This matrix, known as the dynamical matrix, contains the eigenvectors and
eigenvalues that represent the normal modes and their frequencies,

$DU=\omega^{2}U,$ (2)

where $D$ is the dynamical matrix of the system having dimensions $2N\times
2N$, and $U$ is displacement vector associated with the displacement of each
mass
$U=[u_{1}^{A},u_{1}^{B},u_{2}^{A},u_{2}^{B},\cdot\cdot\cdot,u_{N}^{A},u_{N}^{B}]^{T}$
with $u_{j}^{\alpha}$ is the displacement of mass with unit cell index $j$ and
sub-lattice index $\alpha=A/B$. The explicit form for the matrix is

$D=\begin{bmatrix}K_{a}+K_{b}&-K_{a}&\dots&0\\\
-K_{a}&K_{a}+K_{b}&-K_{b}&\dots\\\ \vdots&\vdots&\ddots&-K_{a}\\\
\dots&\dots&-K_{a}&K_{a}+K_{b}\\\ \end{bmatrix}.$ (3)

Figure 5: The plot of the Lyapunov exponent in the phase space of $K_{a}$ and
$\delta^{AA}$ indicates that a divergence in localization length is marked by
the Lyapunov exponent approaching $0$. This boundary of localization length
divergence aligns precisely with our phase plot of the topological invariant,
as illustrated in Fig. 3.

The matrix has a tri-diagonal structure. Unlike the initial SSH Hamiltonian,
changes in inter-cellular hopping only affect the off-diagonal elements, while
in our mechanical scenario, adjustments to the inter-cellular stiffness
constant also impact the diagonal elements. However, since all diagonal
elements are identical, the system maintains chiral symmetry; these diagonal
elements merely shift the eigenvalue spectrum to $\omega_{0}^{2}=K_{a}+K_{b}$.
We measure all other eigenvalues relative to this value. We can define the
matrix $D^{{}^{\prime}}=D-\omega_{0}^{2}I$, that anti-commutes with the chiral
operator $\Gamma$, centering the eigenvalue spectrum around
$\omega_{0}^{2}=0$, as shown below:

$\Gamma(D-\omega_{0}^{2}I)+(D-\omega_{0}^{2}I)\Gamma=0,$ (4)

where $I$ is the identity matrix having dimension $2N\times 2N$, and $\Gamma$
is the chiral operator, whose matrix representation is:

$\Gamma=\begin{bmatrix}1&0&\dots&0\\\ 0&-1&0&\dots\\\
\vdots&\vdots&\ddots&0\\\ \dots&\dots&0&1\\\ \end{bmatrix}.$ (5)

###  II.2 Characterizing the topology in real space

In quantum tight-binding hopping models, we compute the winding number in
momentum space (where translational invariance is preserved) for topological
characterization Chen  _et al._ (2018). In large systems, translational
symmetry holds in the bulk but is broken near the boundary. In the clean limit
(without disorder), our system maintains this symmetry in the bulk, but
disorder disrupts it. We will conduct the analysis in real space Shi  _et al._
(2021); Mondragon-Shem  _et al._ (2014) and characterize the topology using
local topological marker (LTM) Meier  _et al._ (2018); Sircar (2024). An LTM
is defined for each unit cell, which converges to the winding number computed
using translational symmetry in the periodic system, when averaged away from
the boundaries. The primary advantage of the LTM is its ability to handle
disordered systems Mondragon-Shem  _et al._ (2014); Shi  _et al._ (2021). For
computing LTMs for a chain composed of $N$ unit cells, we need to create two
$N\times N$ matrices of eigenvectors from the dynamical matrix:
$U_{-}=[U_{1},U_{2},\dots,U_{N}]$ for eigenvectors below the Fermi level, and
$U_{+}=[U_{N+1},U_{N+2},\dots,U_{2N}]$ for those above it. Using these
matrices, we can build the projectors for the bands as $P_{-}=U_{-}U_{-}^{T}$
and $P_{+}=U_{+}U_{+}^{T}$, allowing us to construct a flat-band Hamiltonian
Shi  _et al._ (2021). The flat-band Hamiltonian, topologically equivalent to
$D$, is expressed as $Q=P_{+}-P_{-}$. This can be spectral decomposed into
sub-lattices $A$ and $B$ as
$Q=Q_{AB}+Q_{BA}=\Gamma_{A}Q\Gamma_{B}+\Gamma_{B}Q\Gamma_{A}$, with

$\Gamma_{A/B}=\begin{bmatrix}1/0&0&0&0&\dots\\\ 0&0/1&0&0&\dots\\\
0&0&1/0&0&\dots\\\ \dots&\dots&\dots&\dots&\dots\\\ \end{bmatrix},$ (6)

are the spectral decomposition operators and $\Gamma=\Gamma_{A}-\Gamma_{B}$ is
the chiral operator. The mathematical definition of LTM Meier  _et al._
(2018); Shi  _et al._ (2021); Miranda  _et al._ (2024); Sircar (2024) is

$\nu(k)=\frac{1}{2}\sum_{\alpha=A,B}{(Q_{BA}[X,Q_{AB}])_{k\alpha,k\alpha}+(Q_{AB}[Q_{BA},X])_{k\alpha,k\alpha}},$
(7)

$X$ is defined as the position operator, whose matrix representation has
dimensions $2N\times 2N$. The specific form of $X$ is given by
$X=[-N,-N,-(N-1),-(N-1),\dots,(N-1),(N-1)]$, all arranged in as diagonal
elements with off-diagonal components being set to $0$. The local topological
marker (LTM) converges to the winding number and can have only two values:
$\nu=0$, characterizes topologically trivial phase, while $\nu=1$
characterizes topologically non-trivial phase, respectively. Each phase is
tuned to a specific ratio between the parameters $K_{a}$ and $K_{b}$, as shown
in Fig. 1. The ratio $\frac{K_{b}}{K_{a}}>1$ corresponds to the non-trivial
phase, which is characterised by the system possessing conducting edge states,
while $\frac{K_{b}}{K_{a}}<1$ corresponds to the trivial phase, that does not
host conducting edge states.

Figure 6: Plot of local density of states with respect to Fermi energy. The
plot shows a region where the local density of states vanishes indicating a
gap in the bulk spectrum.

##  III MECHANICAL MODEL MODULATED WITH INTRACELL AUBRY-ANDRE SPRING CONSTANTs

###  III.1 Dynamical matrix of the modulated spring-mass system

Our model is a nearest-neighbour mechanical model Shi  _et al._ (2021);
Miranda  _et al._ (2024); Sircar (2024) that is influenced by $AA$ modulation.
The $AA$ modulation is defined by a sinusoidal term whose periodic
characterisation is in-commensurate with the underlying lattice structure,
because of the irrational number $\beta$, defined in our case as the inverse
of golden ratio, $g^{-1}=\frac{2}{1+\sqrt{5}}$.

(a) (b)
(c) (d)

Figure 7: Plot of $\ln(IPR)$ as a function of the eigenvalue spectrum
$\omega^{2}-\omega_{o}^{2}$ for $N=1597$ unit cells, $\phi=0$. In real space,
the transition of eigenstates take place from extended to localized, and from
localized to extended in momentum space. For $K_{a}>K_{b}$, one can observe
the presence of critical states with in the parameter regime of $\delta^{AA}$
between $1$ and $2$. In this case the eigenstates are either localized or in
critical phase for real space, while extended or in critical phase for
momentum space.This provides an indication of the presence of a TAI phase, in
this parameter regime, which we have quantitatively verified in Fig. 8, Fig.
9. (a): $K_{a}<K_{b}$ (real space), (b): $K_{a}>K_{b}$ (real space), (c):
$K_{a}<K_{b}$ (momentum space), (d): $K_{a}>K_{b}$ (momentum space).

The equations of motion of our model for type $A$ mass in $j$th unit cell will
be

$\begin{split}\ddot{u}^{A}_{j}&=-[K_{a}+K_{b}+\delta^{AA}\cos(2\pi\beta
j+\phi)]u^{A}_{j}\\\ &\quad+[K_{a}+\delta^{AA}\cos(2\pi\beta
j+\phi)]u^{B}_{j}+[K_{b}]u^{B}_{j-1}.\end{split}$ (8)

Similarly for type $B$ mass in $j$th unit cell,

$\begin{split}\ddot{u}^{B}_{j}&=-[K_{a}+K_{b}+\delta^{AA}\cos(2\pi\beta(j)+\phi)]u^{B}_{j}\\\
&\quad+[K_{a}+\delta^{AA}\cos(2\pi\beta
j+\phi)]u^{A}_{j}+[K_{b}]u^{A}_{j+1}.\end{split}$ (9)

where $\delta^{AA}$ is the $AA$ amplitude (with $j\in[1,N]$ as unit cell
index), the spring constants are defined as

$k_{j}=k_{j}+\delta_{j}^{AA}=K_{a}+\delta^{AA}\cos(2\pi\beta j+\phi),$ (10)

for $j$ even. Different values of $\phi$ are possible; however, in this
article, we will fix $\phi$ at $0$ when discussing incommensurate modulation.
One can also use an ensemble of different $\phi$ values and then compute a
disorder average of all relevant quantities. In this article, as we have
already fixed the value of $\phi$, hence no disorder averaging is needed.
Perturbing the stiffness of the springs introduces new diagonal terms, which
do not appear in the original SSH model. This perturbation alters both
diagonal and off-diagonal terms, hence disrupting the chirality of the
dynamical matrix. To maintain the chiral symmetry, we introduce local springs
to each mass. The equations of motion of our model for type $A$ mass in $j$th
unit cell will be

$\begin{split}\ddot{u}^{A}_{j}&=-[K_{a}+K_{b}+\delta^{AA}\cos(2\pi\beta
j+\phi)+K^{0,A}_{j}]u^{A}_{j}\\\ &\quad+[K_{a}+\delta^{AA}\cos(2\pi\beta
j+\phi)]u^{B}_{j}+[K_{b}]u^{B}_{j-1}.\end{split}$ (11)

(a) (b)
(c) (d)

Figure 8: Plot of fractal dimension as a function of $\delta^{AA}$ for
$\phi=0$ for various system sizes. (a): $K_{b}>K_{a}$ (real space), (b):
$K_{a}>K_{b}$ (real space), (c): $K_{b}>K_{a}$ (momentum space), (d):
$K_{a}>K_{b}$ (momentum space). In cases (a) and (b) the eigenstates of the
system starts from being in an extended phase and make transition very fast
into a localized phase as $\delta^{AA}$ increases. In cases (c) and (d), the
eigenstates of the system starts from a localized phase and make transitions
in a slow manner into an extended phase. In case (c), the kind of critical
states (where the fractal dimension $d_{f}(N)$ does not converge to either $0$
or $1$) exist well some below $\delta^{AA}<1$ and extends for some
$\delta^{AA}>2$, while in case (d) the value of $d_{f}(N)$ converge neither to
$0$ or $1$ in the parameter range $\delta^{AA}>1$ and $\delta^{AA}<2$, which
well matches with the parameter range of the TAI phase.

Similarly for type $B$ mass in $j$th unit cell will be

$\begin{split}\ddot{u}^{B}_{j}&=-[K_{a}+K_{b}+\delta^{AA}\cos(2\pi\beta(j)+\phi)+K^{0,B}_{j}]u^{B}_{j}\\\
&\quad+[K_{a}+\delta^{AA}\cos(2\pi\beta
j+\phi)]u^{A}_{j}+[K_{b}]u^{A}_{j+1},\end{split}$ (12)

where $K^{0,A}_{j}=\delta^{AA}[1-\cos(2\pi\beta j+\phi)]=K^{0,B}_{j}$ . This
selection of on-site spring stiffness preserves the chirality of the matrix
and guarantees that all eigenvalues are positive. Assuming a normal mode
solution, the two equations can be expressed as follows:

$\displaystyle\omega^{2}u^{A}_{j}$
$\displaystyle=(K_{a}+K_{b}+\delta^{AA})u^{A}_{j}$ (13)
$\displaystyle\quad-(K_{a}+\delta^{AA}\cos(2\pi\beta(j)+\phi))u^{B}_{j}$
$\displaystyle\quad-(K_{b})u^{B}_{j-1},$

similarly for $B$ type mass one has

$\displaystyle\omega^{2}u^{B}_{j}$
$\displaystyle=(K_{a}+K_{b}+\delta^{AA})u^{B}_{j}$ (14)
$\displaystyle\quad-(K_{a}+\delta^{AA}\cos(2\pi\beta(j)+\phi))u^{A}_{j}$
$\displaystyle\quad-(K_{b})u^{A}_{j+1}.$

The two equations can be compactly expressed in matrix form using a dynamical
matrix, similar to Eq. (3) as

$D=\begin{bmatrix}K_{a}+K_{b}+\delta^{AA}&-K_{a}-\delta^{AA}\cos(2\pi\beta+\phi)&\dots&-K_{b}\\\
-K_{a}-\delta^{AA}\cos(2\pi\beta+\phi)&K_{a}+K_{b}+\delta^{AA}&-K_{b}&\dots\\\
\vdots&\vdots&\ddots&-K_{a}-\delta^{AA}\cos(2\pi\beta(N-1)+\phi)\\\
-K_{b}&\dots&-K_{a}-\delta^{AA}\cos(2\pi\beta(N-1)+\phi)&K_{a}+K_{b}+\delta^{AA}\\\
\end{bmatrix}.$ (15)

(a) (b)

Figure 9: Plot of $-\ln(\overline{IPR})$ versus $\ln(2N)$ for $\phi=0$ in the
cases of (a) real space and (b) momentum space. The slope of the curve
indicates the value of $d_{f}(N)$, which as $N$ increases, converges to the
fractal dimension. In case (b), the violet and blue curves indicate that the
eigenstates of the system are in a critical phase, while violet curve is
particularly interesting due to the TAI phase and provides a quantitative
estimation of the presence of critical states in the TAI phase.

In order to achieve genuine quasi-periodicity, it is important to work with an
infinite system. To simulate this, we apply periodic boundary conditions
within the lattice and find a suitable rational approximation
$\tilde{\beta}=\frac{F_{n-1}}{F_{n}}$, where the numerator and denominator are
successive terms in the Fibonacci sequence.

###  III.2 Topological characterisation

We characterize the topology by computing the LTM, as indicated in Eq. (7),
for each $K_{a}$ and $\delta^{AA}$ while fixing $K_{b}=1$. This approach leads
to convergence with the winding number. Specifically, we set $\phi=0$, use the
rational approximation $\tilde{\beta}=\frac{377}{610}$, include $N=610$ unit
cells, and vary $K_{a}\in[0,5]$ and $\delta^{AA}\in[0,6]$. The topological
phase transition in clean systems i.e. with $\delta^{AA}=0$ occurs when
$K_{a}=K_{b}$ Asbóth  _et al._ (2016). Secondly, we will examine the behavior
of localization length, which is known to diverge at the Fermi level during
topological phase transitions in electronic systems Mondragon-Shem  _et al._
(2014); Liu  _et al._ (2022b); Altland  _et al._ (2014). This divergence
aligns with the closing of the band gap. To validate this, we analytically
calculate the localization length $\Lambda$ and verify its divergence in the
transition region. We measure the entire eigenvalue spectrum relative to the
middle frequency $\omega_{0}^{2}=K_{a}+K_{b}+\delta^{AA}$, so that the Fermi
level can be centered at zero energy level i.e. $\omega^{2}-\omega_{0}^{2}=0$.
Hence the equation of motions for states at the Fermi level are:

$(K_{a}+\delta^{AA}\cos(2\pi\beta j+\phi))u^{B}_{j}+(K_{b})u^{B}_{j-1}=0,$
(16) $(K_{a}+\delta^{AA}\cos(2\pi\beta
j+\phi))u^{A}_{j}+(K_{b}+\delta^{AA}\cos(2\pi\beta(j)+\phi))u^{A}_{j+1}=0.$
(17)

Simplifying the equation for $A$ sub-lattice we have

$u^{A}_{j+1}=\left(\frac{-(K_{a}+\delta^{AA}\cos(2\pi\beta
j+\phi))}{K_{b}}\right)u^{A}_{j}.$ (18)

The above equation leads to recursive relation as

$u^{A}_{N}=(-1)^{N-1}\prod_{j=1}^{j=N-1}\left(\frac{-(K_{a}+\delta^{AA}\cos(2\pi\beta
j+\phi))}{K_{b}}\right)u^{A}_{1}.$ (19)

We can derive the Lyapunov exponent $\gamma$, which is the inverse of the
localization length Mondragon-Shem  _et al._ (2014); Scales and Van Vleck
(1997),

$\gamma=-\lim\limits_{N\to\infty}\frac{1}{N}\ln\left|\frac{u^{A}_{N}}{u^{A}_{1}}\right|,$
(20)
$\gamma=-\lim\limits_{N\to\infty}\frac{1}{N}\sum_{j=1}^{N-1}\ln\left|\frac{K_{b}}{K_{a}+\delta^{AA}\cos(2\pi\beta
j+\phi)}\right|.$ (21)

Due to the AA modulation being incommensurate with the underlying lattice, the
summation can be expressed as a Riemann integral. Utilizing the properties of
irrational rotation Choe (1993), integrating it over angular variable
$\theta\in[0,2\pi]$, one has

$\gamma=-\frac{1}{2\pi}\int_{0}^{2\pi}\ln\left|\frac{K_{b}}{K_{a}+\delta^{AA}\cos(\theta)}\right|d\theta.$
(22)

The Eq. (22) can be evaluated to

$\gamma=\ln\left(\frac{K_{a}+\sqrt{K_{a}^{2}-(\delta^{AA})^{2}}}{2K_{b}}\right),$
(23)

for $K_{a}>\delta^{AA}$. And for $K_{a}<\delta^{AA}$ the expression evaluate
to

$\gamma=\ln\left(\frac{\delta^{AA}}{2K_{b}}\right).$ (24)

Setting the Lyapunov exponent to $0$ computes the boundary of the topological
phase transition, indicating a divergent localization length. Equations for
the critical diverging line are

$K_{a}=\frac{(\delta^{AA})^{2}}{4K_{b}}+K_{b},$ (25)

for $K_{a}>\delta^{AA}$ and

$\delta^{AA}=2K_{b}$ (26)

for $K_{a}<\delta^{AA}$. The initial phase $\phi$ in the modulation does not
need to vary, as its information is lost during integration with
incommensurate modulation. The localization length at the central frequency
can be numerically computed using the transfer matrix method Scales and Van
Vleck (1997) as a function of relevant parameters.

###  III.3  Calculation of inverse participation ratio (IPR)

To gain insight into the localization properties of the model, we can analyze
the inverse-participation ratios ($IPR$). The $IPR$ for a specific eigenstate
is defined as

$IPR(\psi\rangle)=\sum_{j=1}^{2N}|u_{j}|^{4},$ (27)

where $u_{j}$ are the components of the normalized eigenstate $|\psi\rangle$
in a space of dimension $2N$. For extended eigenmodes, the inverse
participation ratio (IPR) scales as $\frac{1}{N}$, where $N$ is the system
size. In contrast, fully localized eigenmodes exhibit an IPR that scales as
$1$. Eigenmodes that do not fit these patterns define the critical phase of
the phase transition Evers and Mirlin (2008); Roy  _et al._ (2021); Zhang and
Zhang (2022); Duncan (2024). The $IPR$ of eigenstates can be plotted against
energy eigenvalues and Aubry-André strength $\delta^{AA}$. These plots
facilitate the visualization of transitions among non-localized, localized,
and critical phases in both real and momentum spaces. When studying
localization with $AA$ modulation, a crucial parameter to calculate is fractal
dimension $D_{f}$, which does not depend on the size of the system, and
influences the behavior of $IPR$ (or average of all state $IPR$) as size $N$
varies. The scaling relation is $\overline{IPR}\approx(2N)^{-D_{f}}$. To
derive $D_{f}$ from finite size systems, we use the size-dependent quantity
$d_{f}(N)$, which approaches the fractal dimension $D_{f}$ as $N$ increases.
The expression for $d_{f}(N)$ is as follows:

$d_{f}(N)=-\frac{\ln(\overline{IPR})}{\ln(2N)}.$ (28)

In the context where $2N$ denotes the system size, $D_{f}$ equals $1$ in the
non-localized phase and $0$ in the localized phase Lu  _et al._ (2022);
Miranda  _et al._ (2024); Sircar (2024). Numerical computations as shown in

(a) (b)

Figure 10: Plot of $\ln(IPR)$ versus the eigenvalue spectrum $\omega^{2}$ for
$N=1597$ unit cells and $\phi=0$ in the original non-chiral model, to capture
the complete eigenvalue dependence for subsequent cases: (a) $K_{a}<K_{b}$,
(b) $K_{a}>K_{b}$. In case (b) one can observe the presence of critical states
bounded by mobility edges with in the TAI phase regime, also shown in Fig. 7.
We try to fit the energy dependence of mobility edge using trial and error
method. The fitting curve can be expressed as
$\omega^{2}=K_{a}+(\delta^{AA}-\alpha)$ for the lower mobility edge and
$\omega^{2}=K_{a}-(\beta\delta^{AA}-\eta)$ for the upper mobility edge.
Through trial and error, one can find that the values of $\alpha=1$,
$\beta=3$, and $\eta=3.7$ match well with the mobility edges. These
expressions are exact for a slowly varying quasi-periodic modulation. This
article does not provide an exact expression for the mobility edge in the case
of on-site Aubry-André quasi-periodic modulation.

Fig. 9, shows that as $N$ increases, the value of $d_{f}(N)$ approaches
$D_{f}$. The value of $d_{f}(N)$ converges to $1$ in case of the non-localized
phase and to $0$ in case of the localized phase, where a finite portion of
states remains localized. In the critical region, $d_{f}(N)$ stabilizes at
value between $0$ and $1$.

##  IV CONCLUSION

In this study, we expanded the exploration of topological phase transitions in
a mechanical SSH model, as described in Sircar (2024), by applying Aubry-André
modulation only to the intra-cellular spring constants, rather than the
intercellular ones. This approach resulted in distinct localization properties
and the emergence of mobility edges, which could not be analytically derived
from Avila’s global theory Avila (2015). We preserved the chiral symmetry of
the dynamical matrix throughout our calculations. Consequently, we utilized a
topological invariant from class-AIII or DBI, along with its real-space
covariant form, to analyze and characterize the topological phases. We
developed an analytical expression for the Lyapunov exponent, enabling us to
predict the boundary of the topological phase transition based on system
parameters ($K_{a}$, $K_{b}$, $\delta^{AA}$). We examined the localization
properties of the eigenmodes by calculating the inverse participation ratio
($IPR$) and fractal dimensions. We have used fractal dimension Lu  _et al._
(2022); Miranda  _et al._ (2024); Sircar (2024) to quantitatively estimate the
localized and non-localized phases as well as to detect the presence of
critical phase.

The article emphasized that the global theory Avila (2015) did not enable the
exact calculation of mobility edges in such a scenario. We planned to develop
novel analytical techniques for calculating the exact expression of mobility
edge in our future work.

## References

  * Thouless  _et al._ (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * Armitage  _et al._ (2018) N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018).
  * Prodan and Schulz-Baldes (2016) E. Prodan and H. Schulz-Baldes,  _Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics_ (Springer International Publishing, 2016).
  * Su  _et al._ (1979) W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
  * Heeger  _et al._ (1988) A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su, Rev. Mod. Phys. 60, 781 (1988).
  * Jackiw and Rebbi (1976) R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976).
  * Ganeshan  _et al._ (2013) S. Ganeshan, K. Sun, and S. Das Sarma, Phys. Rev. Lett. 110, 180403 (2013).
  * Li  _et al._ (2009) J. Li, R.-L. Chu, J. K. Jain, and S.-Q. Shen, Phys. Rev. Lett. 102, 136806 (2009).
  * Jiang  _et al._ (2009) H. Jiang, L. Wang, Q.-f. Sun, and X. C. Xie, Phys. Rev. B 80, 165316 (2009).
  * Groth  _et al._ (2009) C. W. Groth, M. Wimmer, A. R. Akhmerov, J. Tworzydło, and C. W. J. Beenakker, Phys. Rev. Lett. 103, 196805 (2009).
  * Song and Prodan (2014) J. Song and E. Prodan, Phys. Rev. B 89, 224203 (2014).
  * Mondragon-Shem  _et al._ (2014) I. Mondragon-Shem, T. L. Hughes, J. Song, and E. Prodan, Phys. Rev. Lett. 113, 046802 (2014).
  * Anderson (1958) P. W. Anderson, Phys. Rev. 109, 1492 (1958).
  * Lee and Ramakrishnan (1985) P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
  * Pradhan and Sridhar (2000) P. Pradhan and S. Sridhar, Phys. Rev. Lett. 85, 2360 (2000).
  * Abrahams  _et al._ (1979) E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
  * Gonçalves  _et al._ (2021) M. Gonçalves, H. Z. Olyaei, B. Amorim, R. Mondaini, P. Ribeiro, and E. V. Castro, 2D Materials 9, 011001 (2021).
  * Aubry and Andre (1980) S. Aubry and G. Andre, Annals of the Israel Physical Society 3, 133 (1980).
  * Domínguez-Castro and Paredes (2019) G. A. Domínguez-Castro and R. Paredes, European Journal of Physics 40, 045403 (2019).
  * Longhi (2019a) S. Longhi, Phys. Rev. Lett. 122, 237601 (2019a).
  * Longhi (2019b) S. Longhi, Phys. Rev. B 100, 125157 (2019b).
  * Biddle and Das Sarma (2010) J. Biddle and S. Das Sarma, Phys. Rev. Lett. 104, 070601 (2010).
  * Liu  _et al._ (2020a) Y. Liu, X.-P. Jiang, J. Cao, and S. Chen, Phys. Rev. B 101, 174205 (2020a).
  * Wang  _et al._ (2020) Y. Wang, X. Xia, L. Zhang, H. Yao, S. Chen, J. You, Q. Zhou, and X.-J. Liu, Phys. Rev. Lett. 125, 196604 (2020).
  * Zhu  _et al._ (2023) B. Zhu, L.-J. Lang, Q. Wang, Q. J. Wang, and Y. D. Chong, Phys. Rev. Res. 5, 023044 (2023).
  * Das Sarma  _et al._ (1988) S. Das Sarma, S. He, and X. C. Xie, Phys. Rev. Lett. 61, 2144 (1988).
  * Das Sarma  _et al._ (1990) S. Das Sarma, S. He, and X. C. Xie, Phys. Rev. B 41, 5544 (1990).
  * Liu and Guo (2018) T. Liu and H. Guo, Physics Letters A 382, 3287 (2018).
  * Liu  _et al._ (2022a) T. Liu, X. Xia, S. Longhi, and L. Sanchez-Palencia, SciPost Physics 12, 027 (2022a).
  * Liu  _et al._ (2020b) G.-G. Liu, Y. Yang, X. Ren, H. Xue, X. Lin, Y.-H. Hu, H.-x. Sun, B. Peng, P. Zhou, Y. Chong, and B. Zhang, Physical Review Letters 125 (2020b).
  * Meier  _et al._ (2018) E. J. Meier, F. A. An, A. Dauphin, M. Maffei, P. Massignan, T. L. Hughes, and B. Gadway, Science 362, 929–933 (2018).
  * Zangeneh-Nejad and Fleury (2020) F. Zangeneh-Nejad and R. Fleury, Disorder-induced signal filtering with topological metamaterials (2020).
  * Longhi (2020) S. Longhi, Optics Letters 45, 4036 (2020).
  * Sircar (2024) S. Sircar, Disorder driven topological phase transitions in 1d mechanical quasicrystals (2024), arXiv:2407.17883 [cond-mat.stat-mech] .
  * Yang  _et al._ (2015) Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, Phys. Rev. Lett. 114, 114301 (2015).
  * Vila  _et al._ (2017) J. Vila, R. K. Pal, and M. Ruzzene, Phys. Rev. B 96, 134307 (2017).
  * Shi  _et al._ (2021) X. Shi, I. Kiorpelidis, R. Chaunsali, V. Achilleos, G. Theocharis, and J. Yang, Phys. Rev. Res. 3, 033012 (2021).
  * Lu  _et al._ (2022) Z. Lu, Z. Xu, and Y. Zhang, Annalen der Physik 534, 2200203 (2022).
  * Miranda  _et al._ (2024) D. A. Miranda, T. V. C. Antão, and N. M. R. Peres, Phys. Rev. B 109, 195427 (2024).
  * Chen  _et al._ (2018) H. Chen, H. Nassar, and G. Huang, Journal of the Mechanics and Physics of Solids 117, 22 (2018).
  * Asbóth  _et al._ (2016) J. K. Asbóth, L. Oroszlány, and A. Pályi, Lecture notes in physics 919 (2016).
  * Liu  _et al._ (2022b) S.-N. Liu, G.-Q. Zhang, L.-Z. Tang, and D.-W. Zhang, Physics Letters A 431, 128004 (2022b).
  * Altland  _et al._ (2014) A. Altland, D. Bagrets, L. Fritz, A. Kamenev, and H. Schmiedt, Phys. Rev. Lett. 112, 206602 (2014).
  * Scales and Van Vleck (1997) J. A. Scales and E. S. Van Vleck, Journal of Computational Physics 133, 27 (1997).
  * Choe (1993) G. H. Choe, Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences 93A, 193 (1993).
  * Evers and Mirlin (2008) F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008).
  * Roy  _et al._ (2021) S. Roy, T. Mishra, B. Tanatar, and S. Basu, Phys. Rev. Lett. 126, 106803 (2021).
  * Zhang and Zhang (2022) Y.-C. Zhang and Y.-Y. Zhang, Phys. Rev. B 105, 174206 (2022).
  * Duncan (2024) C. W. Duncan, Phys. Rev. B 109, 014210 (2024).
  * Avila (2015) A. Avila, Acta Mathematica 215, 1 (2015).