# on canonical number systems and monogenity of pure number fields

Hamid Ben Yakkou and Brahim Boudine  Faculty of Sciences Dhar El Mahraz, P.O.
Box 1874 Atlas-Fes , Sidi mohamed ben Abdellah University, Morocco
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract.

Let $m$ be a rational integer $(m\neq 0,\pm 1)$ and consider a pure number
field $K=\mathbb{Q}(\sqrt[n]{m})$ with $n\geq 3$. Most papers discussing the
monogenity of pure number fields focus only on the case where $m$ is square-
free. In this paper, based on a classical theorem of Ore on the prime ideal
decomposition [27, 29], we study the monogenity of $K$ with $m$ is not
necessarily square-free. As an application, several examples of the
application of monogenic fields to CNS (Canonical Number System) are shown. In
this way, our results extend some parts of previously established results [3,
5, 23].

###### Key words and phrases:

Canonical number system, Newton polygon, Theorem of Ore, prime ideal
factorization, power integral basis, monogenity

###### 2020 Mathematics Subject Classification:

11A63, 11R04, 11R16, 11R21, 11Y40

##  1\. Introduction

Consider a monic irreducible polynomial $F(x)\in{\mathbb{Z}}[x]$ of degree
$n$, $K:={\mathbb{Q}}(\alpha)$ a number field with $\alpha$ a roof of $F(x)$
and $\mathbb{Z}_{K}$ the ring of integers of $K$. A primitive element of
$\mathbb{Z}_{K}$ is an element $\theta$ verifying $K=\mathbb{Q}(\theta)$. Let
$\widehat{\mathbb{Z}}_{K}$ be the set of all primitive elements of
$\mathbb{Z}_{K}$. We say that $K$ is monogenic if there exists
$\theta\in\widehat{\mathbb{Z}}_{K}$ such that
$(1,\theta,\theta^{2},...,\theta^{n-1})$ is a basis of the $\mathbb{Z}$-module
$\mathbb{Z}_{K}$. In this case, we say that $\theta$ is a generator of a power
integral basis of ${\mathbb{Z}}_{K}$ (shortly, GPIB of ${\mathbb{Z}}_{K}$).

The monogenity of number fields is among the most active classical problems in
algebraic number theory (cf. [1, 5, 8, 6, 10, 11, 17, 18, 24, 31, 32]). By
Hensel’s discovery in 1894, the monogenity of $K$ is encoded by the
solvability of an index form equation

$I(x_{2},...,x_{n})=\pm 1\,\,\mbox{in}\,\,x_{2},...,x_{n}\in\mathbb{Z}$ (1.1)

associated with an integral basis $\\{1,\omega_{2},...,\omega_{n}\\}$ of K
(see [8] and [28, 2.2, Section 6, p. 64]).

In [17, 18, 20], Győry provided general algorithms to decide whether $K$ is
monogenic or not and to determine all power integral bases in
${\mathbb{Z}}_{K}$ and gave effective bounds for the solutions of
$(\ref{IFE})$ and gave algorithms for solving index form equations. He also
studied in [19, 21] the monogenity of relative extensions. For detailed
surveys on discriminant and index form theory and its applications, including
related Diophantine equations and monogenity of number fields, see the book
[8] by Evertse and Győry.

For quartic number fields see [7] by Davis and Spearman and [13] by Gaál,
Pethő and Pohst. In [11], Gaál and Győry described an algorithm to solve index
form equations in quintic number fields and made very extensive computations
for power integral bases in totally real quintic number fields with Galois
group $S_{5}$. In [6], Bilu, Gaál and Győry studied the monogenity of sextic
number fields. In [32], Pethő and Ziegler gave an efficient criterion to
decide whether the maximal order of a biquadratic field has a unit power
integral basis or not. For multiquadratic number fields, see [31] by Pethő and
Pohst. For a survey on monogenity with a focus on efficient algorithms for
several classes of number fields, see the books [8, 10]. For another
bibliography of monogenity, see Narkiewicz’s book [28].

Recently, the interest in the study of monogenity of pure number fields
${\mathbb{Q}}(\sqrt[6]{m})$ increased clearly. Despite the lack of a standard
classification, there has been some progress in this direction. There are
several works that consider the case when $m$ is a square-free rational
integer: we remain for example that Gassert [14] investigated when
${\mathbb{Z}}_{K}={\mathbb{Z}}[\alpha]$, Ahmad et al. [1, 2] explored the case
$n=6$, Hameed and Nakahara [22] studied the case $n=8$, Gaál and Remete
studied the cases $3\leq n\leq 9$. In [3, 4, 5], Ben Yakkou et al. studied
several classes of pure number fields with variables exponents.

In the present paper, we study the monogenity of any pure number field
$K=\mathbb{Q}(\sqrt[n]{m})$ without the condition that $m$ is square-free. In
this way, our results extend some part of several of previously published
works.

##  2\. Main results

Let $p$ be a prime. Throughout this paper, ${\mathbb{F}}_{p}$ denotes the
finite field with $p$ elements. For $m\in{\mathbb{Z}}$, $\nu_{p}(m)$ stands
for the $p$-adic valuation of $m$ and $PD(m)$ stands for the set of prime
divisors of $m$. For two positive rational integers $d$ and $u$, we denote by
$N_{p}(d)$ the number of monic irreducible polynomials of degree $d$ in
${\mathbb{F}}_{p}[x]$ and by $N_{p}(d,u,m)$ the number of monic irreducible
factors of degree $d$ of the polynomial $x^{u}-\overline{m}$ in
${\mathbb{F}}_{p}[x]$.

Now, let us give our results. We start by the following theorem which gives
infinite families of non-monogenic pure number fields; that is
${\mathbb{Z}}_{K}$ has no power integral basis.

######  Theorem 2.1.

Let $F(x)=x^{n}-m\in{\mathbb{Z}}[x]$ be an irreducible polynomial,
$K:={\mathbb{Q}}(\alpha)$ a pure number field with $\alpha$ a root of $F(x)$.
Assume that $n=u\cdot p^{r}$, where $p$ is an odd prime not dividing $u\cdot
m$ and let $\nu=\nu_{p}(m^{p-1}-1)$. If $\min\\{r+1,\nu\\}N(d,u,m)>N_{p}(d)$
for some positive integer $d$, then $K$ is not monogenic.

######  Remark 2.2.

Note that the condition $m$ is square free is not required for the above
theorem.

######  Remark 2.3.

The above theorem implies [5, Theorem 2.4] and [3, Theorem 2.2], where the
special cases $n=p^{r}$ and $n=2^{5}\cdot 5^{s}$ with $m$ is square-free are
previously and respectively studied.

As a consequence of the above result, the following three corollaries provide
infinite parametric families of non-monogenic pure number fields with variable
exponents.

######  Corollary 2.4.

For $F(x)=x^{5^{r}\cdot 7^{s}}-m$. If any one of the following conditions
holds:

  1. (1)

$r\geq 1,s\geq 7$ and $m^{6}\equiv 1\ \mbox{\rm(mod }{7^{8}})$.

  2. (2)

$r\geq 5,s\geq 1$ and $m^{4}\equiv 1\ \mbox{\rm(mod }{5^{6}})$,

then $K$ is not monogenic.

######  Corollary 2.5.

For $F(x)=x^{3^{r}\cdot 11^{s}}-m$. If any one of the following conditions
holds:

  1. (1)

$r\geq 1,s\geq 11$ and $m^{10}\ \mbox{\rm(mod }{11^{12}})$.

  2. (2)

$r\geq 2,s\geq 1$ and $m^{2}\equiv 1\ \mbox{\rm(mod }{27})$,

then $K$ is not monogenic.

######  Corollary 2.6.

For $F(x)=x^{5^{r}\cdot 11^{s}}-m$. If any one of the following conditions
holds:

  1. (1)

$r\geq 1,s\geq 2,m\equiv-1\ \mbox{\rm(mod }{11})$ and $m^{10}\equiv 1\
\mbox{\rm(mod }{1331})$.

  2. (2)

$r\geq 6,s\geq 1$ and $m^{4}\equiv 1\ \mbox{\rm(mod }{5^{6}})$,

then $K$ is not monogenic.

Recall that a monic polynomial $F(x)\in{\mathbb{Z}}[x]$ is called monogenic if
it is irreducible over ${\mathbb{Q}}$ and
${\mathbb{Z}}_{K}={\mathbb{Z}}[\alpha]$, where $K={\mathbb{Q}}(\alpha)$ with
$F(\alpha)=0$. But, it may happen that the field $K$ is monogenic despite its
defining polynomial $F(x)$ is not monogenic (see e.g [24] by Jones). In the
following result, we identify new infinite parametric families of monogenic
pure number fields defined by non-monogenic binomials.

######  Theorem 2.7.

Let $F(x)=x^{n}-a^{u}\in{\mathbb{Z}}[x]$ such that $u\geq 2$, $\gcd(u,n)=1$,
$PD(n)\subseteq PD(a)$ and $a$ is square-free. Then $F(x)$ is irreducible over
${\mathbb{Q}}$. Let $K:={\mathbb{Q}}(\alpha)$ a pure number field with
$\alpha$ a root of $F(x)$ and $\mathbb{Z}_{K}$ the ring of integers of $K$.
Then, we have the following:

  1. (1)

${\mathbb{Z}}[\alpha]\neq{\mathbb{Z}}_{K}$.

  2. (2)

There exist $\theta\in\widehat{\mathbb{Z}}_{K}$ such that
${\mathbb{Z}}_{K}={\mathbb{Z}}[\theta]$; $K$ monogenic.

######  Remark 2.8.

  1. (1)

Note that in above theorem $a^{u}$ is not square-free.

  2. (2)

Recall that in [14], Gassert gave a necessary and sufficient condition so that
${\mathbb{Z}}[\alpha]={\mathbb{Z}}_{K}$ when $u=1$. In the above theorem, we
have $u\geq 2$, $\alpha$ is not a GPIB of ${\mathbb{Z}}_{K}$ but
${\mathbb{Z}}_{K}$ admits a GPIB $\theta$ different to $\alpha$.

##  3\. Preliminaries

In what follows, $K={\mathbb{Q}}(\alpha)$ is a number field with ring of
integers $\mathcal{O}_{K}$ with $\alpha\in{\mathbb{Z}}_{K}$ a root of a monic
irreducible polynomial $F(x)\in{\mathbb{Z}}[x]$ of degree $n$. Recall that for
any $\theta$ in $\widehat{\mathbb{Z}}_{K}$, ${\mathbb{Z}}[\theta]$ is a sub-
${\mathbb{Z}}$-module of $\mathbb{Z}_{K}$ with rank $n$. Thus, the index of
$\mathbb{Z}[\theta]$ in $\mathbb{Z}_{K}$ is finite. This index is denoted by
$(\mathbb{Z}_{K}:\mathbb{Z}[\theta])$. So, $K$ is monogenic if
$(\mathbb{Z}_{K}:\mathbb{Z}[\theta])=1$ for some primitive element $\theta$ of
$K$. But, if $K$ does not have any such $\theta$, then $K$ is not monogenic.

The greatest common divisor of the indices of all integral primitive elements
of $K$ is called the index of K and is denoted by $i(K)$, that is

$i(K)=\gcd\\{({\mathbb{Z}}_{K}:{\mathbb{Z}}[\theta])\,|\,\,\theta\in\widehat{\mathbb{Z}}_{K}\\}.$

A prime $p$ dividing $i(K)$ is called a prime common index divisor of $K$. If
$K$ is monogenic, then $i(K)=1$. Therefore a field having a prime common index
divisor is not monogenic.

For a positive integer $d$, we denote $L_{p}(d)$ the number of distinct prime
ideals of $\mathbb{Z}_{K}$, that are factors of $p\mathbb{Z}_{K}$ with residue
degree $d$ and $N_{p}(d)$ denote the number of monic and irreducible
polynomial having degree $d$ in $\mathbb{Z}/p\mathbb{Z}[x]$. Then, we have the
following lemma which provides a sufficient condition for a prime $p$ to be a
common index divisor of a number field:

######  Lemma 3.1.

[28, Theorems 4.33 and 4.34] With the above notation, if $L_{p}(d)>N_{p}(d)$
for some positive integer $d$, then $p$ is a common index divisor of $K$. In
particular, $K$ is not monogenic.

To prove our result, we shall use the index theorem of Ore which gives
$\nu_{p}(({\mathbb{Z}}_{K}:{\mathbb{Z}}[\theta]))$ and the factorization of
$p{\mathbb{Z}}_{K}$. This theorem allows us to apply the above lemma ( for
non-monogenity) as well as to test if
$\nu_{p}(({\mathbb{Z}}_{K}:{\mathbb{Z}}[\theta]))=1$ or not (for monogenity).
Ore’s Theorem is based on Newton’s polygon techniques. So, let us briefly
describe the related notions and the use of this technique. For more details,
we refer to [29] by Ore, [27] by Montes and Nart, and to [15, 16] by Guàrdia,
Montes and Nart.

Let $p$ be a prime, $\nu_{p}$ the $p$-adic valuation For any polynomial
$F(x):=\sum\limits_{i=0}^{n}a_{i}x^{i}\in\mathbb{Z}_{p}[x]$, the $p$-adic
valuation of $F(x)$ is $\nu_{p}(F(x))=\min\\{\nu_{p}(a_{i})\mid 0\leq i\leq
n\\}$. Let $\phi(x)$ be a monic polynomial in $\mathbb{Z}[x]$, where
$\overline{\phi(x)}$ is irreducible in $\mathbb{F}_{p}[x]$. Let

$F(x)=a_{l}(x)\phi(x)^{l}+a_{l-1}(x)\phi(x)^{l-1}+\cdots+a_{1}(x)\phi(x)+a_{0}(x)$

be the $\phi$-adic development of $F(x)$: $\deg(a_{j}(x))<\deg(\phi(x))$ for
each $j\in\\{0,1,...,l\\}$. We consider the set
$\mathfrak{S}_{\\{F,\phi\\}}=\\{(j,\nu_{p}(a_{j}(x)))\mid a_{j}(x)\neq 0\emph{
and }0\leq j\leq l\\}$. The lower boundary convex envelope of
$\mathfrak{S}_{\\{F,\phi\\}}$ is called the $\phi$-Newton polygon of $F(x)$,
denoted by $N_{\phi}(F)$. It is made up of some sides $S_{k}$ with different
slopes $\lambda_{k}\in\mathbb{Q}$, then we write
$N_{\phi}(F)=S_{1}+S_{2}+...+S_{g}$, where $g$ is the number of its sides with
different slopes. In particular, the $\phi$-principal Newton polygon of
$F(x)$, denoted by $N_{\phi}^{+}(F)$, is the polygon formed by the sides of
$N_{\phi}(F)$ which have negative slopes. Every side $S$ of $N_{\phi}^{+}(F)$
joins two points: 1) the beginning point $(s,\nu_{p}(a_{s}(x)))$, and 2) the
final point $(r,\nu_{p}(a_{r}(x)))$, where $0\leq s<r\leq n$. In this case,
the length of $S$ is $l(S):=r-s$, its height is
$h(S):=\nu_{p}(a_{s}(x))-\nu_{p}(a_{r}(x))$, its slope is
$-\lambda=-\frac{h(S)}{l(S)}<0$, its degree is $d(S):=\gcd(l(S),h(S))$, and
its ramification index is $e(S)=\frac{l(S)}{d(S)}$. Let $d=d(S)$ and $e=e(S)$.
Then, the residual polynomial of $F(x)$ associated to $S$ is

$R_{\lambda}(F)(y)=c_{s+de}y^{d}+c_{s+(d-1)e}y^{d-1}+\cdots+c_{s+e}y+c_{s}\in\mathbb{F}_{\phi}[y]:=(\mathbb{F}_{p}[x]/\langle\overline{\phi(x)}\rangle)[y],$

where

$c_{i}=\left\\{\begin{array}[]{ll}0,&\emph{if }(i,v_{p}(a_{i}(x)))\emph{ lies
strictly above }N_{\phi}^{+}(F),\\\
\frac{a_{i}(x)}{p^{v_{p}(a_{i}(x))}}\mod{(p,\phi(x))},&\emph{if
}(i,v_{p}(a_{i}(x)))\emph{ lies on }N_{\phi}^{+}(F).\end{array}\right.$

The polynomial $F(x)$ is called $\phi$-regular, if for each side $S_{\lambda}$
of $N_{\phi}^{+}(F)$, $R_{\lambda}(F)(y)$ is separable in
$\mathbb{F}_{\phi}[y]$. Further, $F(x)$ is called $p$-regular, if it is
$\phi$-regular, for every monic polynomial $\phi(x)\in\mathbb{Z}[x]$ such that
$\overline{\phi(x)}$ is a monic irreducible factor of $\overline{F(x)}$ in
$\mathbb{F}_{p}[x]$.

Now, we have all the necessary tools to give Theorem of Ore (see [16, 27,
29]).

######  Theorem 3.2 (Theorem of Ore).

With the above notations, suppose that

  * •

$\overline{F(x)}=\prod\limits_{i=1}^{t}\overline{\phi_{i}(x)}^{l_{i}}$ is the
factorization of $\overline{F(x)}$ into a product of powers of distinct monic
irreducible polynomials in $\mathbb{F}_{p}[x]$,

  * •

$N_{\phi_{i}}^{+}(F)=S_{i1}+S_{i2}+...+S_{ir_{i}}$ and $\lambda_{ij}$ is the
slope of $S_{ij}$, for each $i\in\\{1,2,...,t\\}$ and
$j\in\\{1,2,...,r_{i}\\}$,

  * •

$R_{\lambda_{ij}}(F)(y)=\prod\limits_{s=1}^{s_{ij}}\psi_{ijs}(y)^{n_{ijs}}$ is
the factorization of $R_{\lambda_{ij}}(F)(y)$ into a product of powers of
distinct irreducible polynomials in $\mathbb{F}_{\phi_{i}}[x]$, for each
$i\in\\{1,2,...,t\\}$ and $j\in\\{1,2,...,r_{i}\\}$.

Then, we have

  1. (1)

$\nu_{p}(({\mathbb{Z}}_{K}:{\mathbb{Z}}[\alpha]))\geq\sum_{i=1}^{t}\mbox{ind}_{\phi_{i}}(F).$
The equality holds if $F(x)$ is $p$-regular.

  2. (2)

If $F(x)$ is $p$-regular ($n_{ijs}=1$), then

$p{\mathbb{Z}}_{K}=\prod_{i=1}^{t}\prod_{j=1}^{r_{i}}\prod_{s=1}^{s_{ij}}\mathfrak{p}^{e_{ij}}_{ijs},$

where $e_{ij}$ is the ramification index of the side $S_{ij}$, and
$\mathfrak{p}_{ijs}$ is a prime ideal of $\mathbb{Z}_{K}$ its residue degree
over $p$ is $f_{ijs}=\mbox{deg}(\phi_{i})\times\mbox{deg}(\psi_{ijs})$.

######  Example 3.3.

Consider the quartic pure field $K={\mathbb{Q}}(\sqrt[4]{17})$. In this case,
we have $\alpha=\sqrt[4]{17}$ and $F(x)=x^{4}-17$. Let $p=2$. Reducing modulo
$2$, we get $\overline{F(x)}=\overline{\phi_{1}(x)}$ in ${\mathbb{F}}_{2}[x]$.
The $\phi_{1}$-adic development of $F(x)$ is

$F(x)=-16+4\phi_{1}(x)+6\phi_{1}(x)^{2}+4\phi_{1}(x)^{3}+\phi_{1}(x)^{4}.$

Here, we have
$\nu_{0}=\nu_{2}(-16)=4,\nu_{1}=\nu_{2}(4)=2,\nu_{2}=\nu_{2}(6)=1,\nu_{3}=\nu_{2}(4)=2$,
and $\nu_{4}=\nu_{2}(1)=0$. It follows that
$N_{\phi_{1}}^{+}(F)=S_{11}+S_{12}+S_{13}$ has three sides of degree $1$ each
joining the points $(0,4),(1,2),(2,1)$ and $(4,0)$ (see FIGURE 1). Moreover,
for $j=1,2,3$, the residual polynomial $R_{\lambda_{1j}}(F)(y)$ has degree
$1$. So, it is separable. Thus, $F(x)$ is $2$-regular. By Theorem 3.2, we have

$2\mathbb{Z}_{K}=\mathfrak{p}_{11}\cdot\mathfrak{p}_{12}\cdot\mathfrak{p}_{13}^{2}\,\mbox{
with the same residue degree}\,f(\mathfrak{p}_{1j}/2)=1\,\mbox{for}\,j=1,2,3,$

and

$\nu_{2}((\mathbb{Z}_{K}:\mathbb{Z}[\alpha]))=\mbox{ind}_{\phi_{1}}(F)=3.$

In this case, by Lemma 3.1, the prime $2$ divides $i(K)$. Hence, $K$ is not
monogenic.

$1$$2$$3$$4$$1$$2$$3$$4$$S_{1}$$S_{2}$$S_{3}$ Figure 1.  $N_{\phi_{1}}^{+}(F)$
with respect to $\nu_{2}$.

##  4\. Proofs of the main results

In this section, we prove our results. To prove Theorem 2.1, we will use Lemma
4.1 below. Let $p$ be a prime. Assume that $p$ divides $n$ and does not divide
$m$, and set $n=u\cdot p^{r}$. Let $\phi(x)\in{\mathbb{Z}}[x]$ be a monic
irreducible polynomial. Then $\overline{\phi(x)}$ is a monic irreducible
factor of $\overline{F(x)}$ in ${\mathbb{F}}_{p}[x]$ if and only if
$\overline{\phi(x)}$ is a monic irreducible factor of $\overline{x^{u}-m}$ in
${\mathbb{F}}_{p}[x]$. Let

  * •

$x^{u}-m=\phi(x)U(x)+pT(x)$, where $U(x)$ and $T(x)\in{\mathbb{Z}}[x]$ two
polynomials such that $\overline{\phi(x)}$ does not divide
$\overline{U(x)T(x)}$.

  * •

$H(x)=m^{p^{r}-1}T(x)+\frac{1}{p^{r+1}}\sum_{j=0}^{p^{r}-2}\binom{p^{r}}{j}m^{j}(pT(x))^{p^{r}-j}.$

  * •

Let $V(x)$ and $R(x)$ be the quotient and the remainder upon the Euclidean
division of $H(x)$ by $\phi(x)$, respectively.

  * •

$A_{0}(x)=p^{r+1}R(x)+m^{p^{r}}-m$, and $\nu_{0}=\nu_{p}(A_{0}(x))$.

In [4], Ben Yakkou and Didi proved the following result:

######  Lemma 4.1.

Let $F(x)$, $p$, and $\phi(x)$ be as in above. Then $N_{\phi}^{+}(F)$ is the
Newton polygon joining the points
$\\{(0,\nu_{0})\\}\cup\\{(p^{j},k-j)\,|\,0\leq j\leq r\\}$ in the euclidean
plane.

We recall also that if a prime integer $p$ does not divide a non-zero rational
integer $m$, then for every positive integer $k$,
$\nu=\nu_{p}(m^{p^{u}}-m)=\nu_{p}(m^{p^{u}-1}-1)=\nu_{p}{(m^{p-1}-1)}$ (see
[3]).

###### Proof of Theorem 2.1.

With the assumptions of Theorem 2.1, let $\phi(x)$ be a monic irreducible
factor of $F(x)$ modulo $p$ of degree $d$. We distinguish two cases:

  * •

If $\nu\geq r+1$, then $\nu_{0}\geq\min\\{\nu,r+1\\}\geq r+1$. It follows by
Lemma 4.1 that the principal Newton polygon
$N_{\phi}^{+}(F)=S_{1}+\cdot+S_{r+1}$ has $r+1$ sides of degree $1$ each. More
precisely, $N_{\phi}^{+}(F)$ is the lower convex hull of the points
$(0,\nu_{0})$, $(1,r)$, $(p,r-1)$, $\ldots$, and $(p^{r},0)$ (see FIGURE 2).
Thus, for $j=1,\ldots,r+1$, the residual polynomial $R_{\lambda_{j}}(F)(y)$ is
separable in ${\mathbb{F}}_{\phi}[y]$ since it is of degree $1$. By Ore’s
Theorem (Theorem 3.2), the factor $\phi(x)$ provides $r+1$ distinct prime
ideals of ${\mathbb{Z}}_{K}$ of residue degree equals
$\deg(R_{\lambda_{j}}(F)(y))\times\deg(\phi(x))=1\times d=d$ lying above
$p{\mathbb{Z}}_{K}$. Since the number of monic irreducible factors of
$\overline{F(x)}$ equals $N_{p}(d,u,m)$, there exist $(r+1)\times
N_{p}(d,u,m)$ distinct prime ideals of ${\mathbb{Z}}_{K}$ which divide
$p{\mathbb{Z}}_{K}$.

  * •

If $r\geq\nu$, then $\nu_{0}\geq\min\\{\nu,r+1\\}=\nu$. Thus, by Lemma 4.1,
$N_{\phi}^{+}(F)=S_{1}+\cdot+S_{\nu}$ has $\nu$ sides of degree $1$ each.
Explicitly, $N_{\phi}^{+}(F)$ is the lower convex hull of the points
$(0,\nu)$, $(p^{r-\nu+1},\nu-1)$, $(p^{r-\nu+2},\nu-2)$, $\ldots$ and
$(p^{r},0)$. Thus, $R_{\lambda_{j}}(F)(y)$, $j=1,\ldots,r+1$ is separable.
Thus, for $k=1,\ldots,N_{p}(d,u,m)$, $F(x)$ is $\phi_{k}$-regular, for every
factor $\phi_{k}(x)$ of $F(x)$ of degree $d$. The ramification indices of the
sides of $N_{\phi}^{+}(F)$ are $e_{1}=e(S_{1})=p^{r-\nu+1}$,
$e_{j}=e(S_{j})=(p-1)\cdot p^{r-\nu+j-1}$, for $j=2,\ldots,\nu$. By Theorem
3.2, we get

$p{\mathbb{Z}}_{K}=\prod_{k=1}^{N_{p}(d,s,b)}\prod_{j=1}^{\nu}\mathfrak{p}_{kj}^{ej}\mathfrak{a},$

where $\mathfrak{a}$ is a proper ideal of ${\mathbb{Z}}_{K}$, and
$\mathfrak{p}_{kj}$ is a prime ideal of ${\mathbb{Z}}_{K}$ with residue degree
$f(\mathfrak{p}_{j}/p)=d$, for $k=1,\ldots,N_{p}(d,s,b)$ and $j=1,\ldots,\nu$.
So there are at least $\nu\times N_{p}(d,s,b)$ prime ideals of residue degree
$d$ lying above $p{\mathbb{Z}}_{K}$. By Lemma 3.1, if $\nu\times
N_{p}(d,s,b)>N_{p}(d)$, then $p$ divides $i(K)$, and so $K$ is not monogenic.

∎

$0$$1$$p$$p^{r-2}$$p^{r-1}$$p^{r}$$1$$2$$r$$\nu_{0}\geq\nu\geq
r+1$$S_{1}$$S_{2}$$S_{r}$$S_{r+1}$ Figure 2.  $N_{\phi}^{+}(F)$ when $\nu\geq
r+1$.

The proofs of Corollaries 2.4, 2.6, 2.5 are similar. Since they are obtained
by a direct application of Theorem 2.1, we propose only the proof of Corollary
2.4.

###### Proof of Corollary 2.4.

  1. (1)

$r\geq 1,s\geq 7$, and $m^{6}\equiv 1\ \mbox{\rm(mod }{7^{8}})$. Here, we have
$F(x)=x^{5^{r}\cdot 7^{s}}-m\equiv(x^{5^{r}}-m)^{7^{s}}\ \mbox{\rm(mod
}{7})\equiv(\phi_{1}(x)V(x))^{7^{s}}\ \mbox{\rm(mod }{7})$, with
$\phi_{1}(x)=x-b_{m}$ and $V(b_{m})\not\equiv 0\ \mbox{\rm(mod }{p})$ (since
$x^{5^{r}}-m$ is square free in ${\mathbb{F}}_{7}[x]$, because $7$ does not
divide $5^{r}$). Precisely, modulo $7$, we have $b_{1}=1$, $b_{2}=4$,
$b_{3}=5$, $b_{4}=2$, $b_{5}=3$, $b_{6}=6$. So, $\phi_{1}$ is an irreducible
factor of $x^{2^{r}}-m$ modulo $7$. According to the notations of Theorem 2.1,
for $d=1$, we have $N_{7}(1,2^{r},m)=1$. Notice also that $N_{7}(1)=1$. By
applying Theorem 2.1, $K$ is not monogenic if the condition
$\min\\{s+1,\nu_{7}(m^{6}-1)\\}\times 1>7$ holds, which is equivalent to
$s\geq 7$, and $m^{6}\equiv 1\ \mbox{\rm(mod }{7^{8}})$.

  2. (2)

$r\geq 5$, $s\geq 1$, and $m^{4}\equiv 1\ \mbox{\rm(mod }{5^{6}})$. In this
case, for $p=5$ and $d=1$, we have $N_{5}(1,7^{s},m)=1$. Then, similarly to
the above case, by changing the role of $r$ and $s$, and applying Theorem 2.1,
we obtain the desired result.

∎

Now, let us prove Theorem 2.7.

###### Proof of Theorem 2.7.

Let $p$ be a prime dividing $a$. Then $F(x)\equiv\phi_{1}(x)^{n}\
\mbox{\rm(mod }{p})$, where $\phi_{1}(x)=x$. Note that $\nu_{p}(a^{u})=u$,
since $a$ is square-free. As $\gcd(u,n)=1$, $N_{\phi_{1}}^{+}(F)=S_{1}$ has
only one side of degree $1$ joining the points $(0,u)$ and $(n,0)$. Thus,
$\deg(S_{1})=\gcd(u,n)=1$, and so the degree of the residual polynomial
$R_{\lambda_{1}}(F)(y)$ equals to $1$. By Theorem of the residual polynomial,
$F(x)$ is irreducible in ${\mathbb{Q}}_{p}(x)$. So, it is irreducible in
${\mathbb{Q}}(x)$.

  1. (1)

Since $\deg(R_{\lambda_{1}}(F)(y))=1$, the polynomial $R_{\lambda_{1}}(F)(y)$
is separable in ${\mathbb{F}}_{\phi_{1}}[y]$. Thus, $F(x)$ is
$\phi_{1}$-regular. So, it is $p$-regular. By using Theorem 3.2(2), we get

$\nu_{p}({\mathbb{Z}}_{K}:{\mathbb{Z}}[\alpha])\geq
ind_{\phi_{1}}(F)=\frac{(n-1)(u-1)}{2}\geq 2.$

It follows that ${\mathbb{Z}}[\alpha]\subsetneq{\mathbb{Z}}_{K}$.

  2. (2)

Let $(t,s)\in(\mathbb{N}^{*})^{2}$ be the positive solution of the Diophantine
equation $ux-ny=1$, and $\theta=\frac{\alpha^{t}}{a^{s}}$. We have

$\theta^{n}=(\frac{\alpha^{t}}{a^{s}})^{n}=\frac{(\alpha^{n})^{t}}{a^{ns}}=\frac{a^{ut}}{a^{ns}}=a^{ut-
ns}=a$

Thus, $G(\theta)=0$, where $G(x)=x^{n}-a\in{\mathbb{Z}}[x]$. Since $m$ is
square free, $F(x)$ is irreducible over ${\mathbb{Q}}$ (it is a $q$-Eisenstein
type polynomial with respect to any prime $q$ dividing $a$). Then, we consider
the field $K^{{}^{\prime}}={\mathbb{Q}}(\theta)$. Obviously
$K^{{}^{\prime}}\subseteq K$. On the other hand, $\alpha^{t}=a^{s}\theta\in
K^{{}^{\prime}}$. It follows that

$\alpha=\alpha^{1}=\alpha^{ut-
ns}=\frac{\alpha^{ut}}{\alpha^{ns}}=\frac{(\alpha^{t})^{u}}{(\alpha^{n})^{s}}=\frac{(\alpha^{t})^{u}}{a^{s}}\in
K^{{}^{\prime}}.$

So $K={\mathbb{Q}}(\alpha)={\mathbb{Q}}(\theta)$. By [28, Propositions 2.9 and
2.13], one has

$\displaystyle\Delta(G)=n^{n}\cdot
a^{n-1}={({\mathbb{Z}}_{K}:{\mathbb{Z}}[\theta])}^{2}\cdot D_{K},$ (4.1)

where $D_{K}$ the field discriminant of $K$. Let $q$ be a prime. As
$PD(n)\subseteq PD(a)$, by the above relation, if $q$ divides the index
$({\mathbb{Z}}_{K}:{\mathbb{Z}}[\theta])$, then $q$ divides $a$. So to show
${\mathbb{Z}}_{K}={\mathbb{Z}}[\theta]$ we need to prove that
$q\nmid({\mathbb{Z}}_{K}:{\mathbb{Z}}[\theta])$, for every $q\in PD(a)$. Let
$q\in PD(a)$. We have $G(x)\equiv\phi_{1}(x)^{n}\ \mbox{\rm(mod }{q})$, where
$\phi_{1}(x)=x$. As $a$ is square free, $\nu_{q}(a)=1$. It follows that
$N_{\phi_{1}}^{+}(F)=S_{1}$ with respect to $q$, has only one side of degree
$1$ joining the points $(0,1)$ and $(n,0)$. So, by applying Ore’s Theorem with
the polynomial $G(x)$, we see that
$\nu_{q}(({\mathbb{Z}}_{K}:{\mathbb{Z}}[\theta]))=0$. This completes the
proof.

∎

##  5\. Application to the existence of Canonical number system

Let $N_{K/{\mathbb{Q}}}$ be the norm function over $K$. The pair
$(\theta,N_{\theta})$, where
$N_{\theta}=\\{0,1,\ldots,|N_{K/{\mathbb{Q}}}(\theta)|-1\\}$ is called a
canonical number system (CNS) in ${\mathbb{Z}}_{K}$ for an integer $\theta$ if
each element of ${\mathbb{Z}}_{K}$ can be represented uniquely as
$a_{0}+a_{1}\theta+\cdots+a_{l}\theta^{l}$, where $\pm a_{i}\in N_{\theta}$
for every $i=0,\ldots,l$ and $a_{l}\neq 0$. We call $\theta$ the base of the
number system, $N_{\theta}$ the set of digits, and the minimal polynomial
$F(x)$ of $\theta$ is called the CNS polynomial. For more details and some
applications, we refer to [9, 30]. The most important connection between CNS
and monogenity of is in the following result (see [25] by Kov$\acute{a}$cs,
and [26] by Kov$\acute{a}$cs and Pethő).

######  Theorem 5.1.

Let $K$ is a number field of degree $n\geq 3$, $\theta\in{\mathbb{Z}}_{K}$,
and $F(x)=x^{n}+a_{n-1}+\cdots+a_{0}$ such that $1\leq a_{n-1}\leq\ldots\leq
a_{0}$, $a_{0}\geq 2$, and $|N_{K/{\mathbb{Q}}}(\theta)|>2$. Then

$(\theta,N_{\theta})\,\mbox{is a CNS in
}{\mathbb{Z}}_{K}\Leftrightarrow{\mathbb{Z}}_{K}={\mathbb{Z}}[\theta].$

Using the above theorem, Hameed, Nakahara, and Husnine identify in Theorems
2.1 and 3.1 of [23] CNS polynomials of type $x^{2^{r}}-m$ and $x^{n}-m$ with
the condition $m$ is square free. By the above theorem and our results, we
deduce that:

  * •

Under the hypotheses of Theorem 2.7, ${\mathbb{Z}}_{K}$ admits a CNS, and such
system is explicitly given (we combine Theorem 5.1 and Proof of Theorem 2.7).
In this direction, our results extend those given in [23].

  * •

Under the hypotheses of Theorem 2.1, ${\mathbb{Z}}_{K}$ has no CNS.

We propose the following examples.

######  Example 5.2.

Let $F(x)=x^{6}-m^{5}$ with $(m\neq 0,\pm 1)$, $m$ square free, $6$ divides
$m$, and $K={\mathbb{Q}}(\sqrt[6]{m^{5}})$ (here $\alpha=\sqrt[6]{m^{5}}$).
According to the notations of Proof of Theorem 2.7, $t=5$, $s=4$, and
$\theta=\frac{(\sqrt[6]{m^{5}})^{5}}{m^{4}}$. By Theorem 2.7, $\theta$
generates a power integral basis of ${\mathbb{Z}}_{K}$. Note that the minimal
polynomial of $\theta$ is $G(x)=x^{6}-m$. By [28, Propositions 2.9 and 2.13],
we have

$6^{6}\cdot
m^{5}=\Delta(G)=D_{K/{\mathbb{Q}}}(1,\theta,\ldots,\theta^{5})=(-1)^{\frac{6\times
5}{2}}N_{K/{\mathbb{Q}}}(G^{\prime}(\theta))=-N_{K/{\mathbb{Q}}}(6\theta^{5})=-6^{6}(N_{K/{\mathbb{Q}}}(\theta))^{5}$

Thus, $|N_{K/{\mathbb{Q}}}(\theta)|=|m|$. So,
$N_{\theta}=\\{0,1,\ldots,|m|\\}$. Hence by Theorem 5.1, $(\theta,N_{\theta})$
is a CNS in ${\mathbb{Z}}_{K}$ and $G(x)=x^{6}-m$ is a CNS polynomial. As a
particular case, if $m=30$, then every element of ${\mathbb{Z}}_{K}$ can be
expressed uniquely as $\sum_{i=0}^{l}a_{i}\theta^{i}$ with
$a_{i}\in\textlbrackdbl-30,30\textrbrackdbl$ for $i=1,\ldots,l$, and
$a_{l}\neq 0$.

######  Example 5.3.

Let $F(x)=x^{2^{r}\cdot 3^{k}}-6^{w}$ with $w\geq 5$, and $w\equiv 1,5\
\mbox{\rm(mod }{6})$. Let $t,s$, and $\theta$ be as in Proof of Theorem 2.7,
then $(\theta,N_{\theta})$ is CNS ${\mathbb{Z}}_{K}$, where the set of digits
is $N_{\theta}=\\{1,\ldots,6\\}$.

## Declarations

Conflict of interest: There is no conflict of interest related to this paper.
The authors have freely chosen this journal without any consideration.

## Data availability

Not applicable.

## Authors contributions

The authors are equally contributed to this paper. All authors read and
improved the final manuscript.

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# Classical and quantum facilitated exclusion processes

Amit Kumar Chatterjee  Yukawa Institute for Theoretical Physics, Kyoto
University, Kyoto 606-8502, Japan  Adhip Agarwala  Indian Institute of
Technology Kanpur, Kalyanpur, Kanpur, India 208 016

###### Abstract

We demonstrate exciting similarities between classical and quantum many body
systems whose microscopic dynamics are composed of non-reciprocal three-site
facilitated exclusion processes. We show that the quantum analog of the
classical facilitated process engineers an interesting quantum absorbing
transition where the quantum particles transit from an unentangled direct-
product absorbing phase to an entangled steady state with a finite current at
density $\rho=1/2$. In the generalized classical facilitated exclusion
process, which includes independent hopping of particles with rate $p$, our
analytical and Monte-Carlo results establish emergence of a special density
$\rho^{*}=1/3$ that demarcates two regimes in the steady state, based on the
competition between two current carrying modes (facilitated and independent).
The corresponding quantum system also displays similar qualitative behaviors
with striking non-monotonic features in the bipartite entanglement. Our work
ties the two sub-fields of classically interacting exclusion processes, and
interacting non-Hermitian quantum Hamiltonians to show common themes in the
non-equilibrium phases they realise.

Introduction: Classical and quantum non-equilibrium phenomena has been a
cornerstone of all of physics. In the classical domain, their applicability
has ranged from understanding transport phenomena Schadschneider  _et al._
(2011), out-of-equilibrium materials Giannetti  _et al._ (2016); Tena-Solsona
_et al._ (2017), disease-modelling Kamenev and Meerson (2008), game theory
Babajanyan  _et al._ (2020), biological systems Allman and Rhodes (2003);
Blythe and McKane (2007) and even socio-economic structures Castellano  _et
al._ (2009). In the quantum regime, such systems have been studied in context
of open quantum systems (such as cold-atomsLewenstein  _et al._ (2007); Bloch
_et al._ (2008); Langen  _et al._ (2015); Liang  _et al._ (2022)), and have
been recently explored in random quantum circuits Fleckenstein  _et al._
(2022) and broadly in context of non-Hermitian physics Lieu (2019, 2018);
Panda and Banerjee (2020); Carlström (2020). While the classical and quantum
community has been working on the various aspects of such systems, their
motivations and quantities of interest have also been varied. For instance,
while in classical Markovian systems, non-equilibrium steady states Blythe and
Evans (2007); Zhang  _et al._ (2012); Ge  _et al._ (2012) as well as
relaxations towards steady states Krapivsky  _et al._ (2010); Evans  _et al._
(2020); Lu and Raz (2017) have been centre of interests for interacting many
particle systems; in quantum non-Hermitian phenomena largely the focus has
been on single-particle spectra and their topological properties Alvarez  _et
al._ (2018); Ghatak and Das (2019); Torres (2019); Ashida  _et al._ (2020);
Bergholtz  _et al._ (2021); Gong  _et al._ (2018); Kawabata  _et al._ (2019);
Lee and Thomale (2019). Very recently interacting non-Hermitian Hamiltonians
Kawabata  _et al._ (2022, 2021); Fukui and Kawakami (1998); Hatano and Nelson
(1996); Wang  _et al._ (2022); Crippa  _et al._ (2021); Xu and Chen (2020);
Panda and Banerjee (2020); Liu  _et al._ (2020); Mu  _et al._ (2020); Yamamoto
_et al._ (2019); Nakagawa  _et al._ (2018); Zhang  _et al._ (2022); Banerjee
_et al._ (2022); Zhang and Song (2021); Yang and Song (2022); Pan  _et al._
(2019); Groenendijk  _et al._ (2021); Ghosh  _et al._ (2022); Heußen  _et al._
(2021), and the long-time steady states they may realize have been studied –
however the field remains broadly unexplored.

Investigations on the common themes between the physics of such classical-
quantum problems have been far and few van Horssen  _et al._ (2015); Kavanagh
_et al._ (2022); Hruza and Bernard (2022); Berthier  _et al._ (2019), for
instance only recently the rare trajectories of a Markovian system was mapped
to the ground states of a quantum topological HamiltonianGarrahan and Pollmann
(2022). This broadly builds on idea that Markov matrices of the classical rate
processes could be mapped to a sub-class of constrained quantum many body
Hamiltonians. But apart from deriving such structural equivalence between the
mathematical structures not a lot of work has gone into finding common physics
in the non-equilibrium phases they realize.

Figure 1:  QF-TASEP: (top) The variation of steady state current as a function
of density of particles $\rho$ in the quantum version of F-TASEP (see eqn.
(2)). The dashed lines corresponds to mean-field expectations (see text).
(bottom) The bipartite entanglement entropy of the first site with the rest of
the system in the steady state as a function of density. Dashed line
corresponds to the result for a free Fermi gas.

In this work, we demonstrate intriguing resemblance in macroscopic physical
characteristics between classical and quantum many body systems, whose
dynamics are based on a three site microscopic process in one dimension. This
three site process, familiar as facilitated totally asymmetric simple
exclusion process (F-TASEP), allows unidirectional hopping of particle to a
vacant neighbor only if its other neighbor is occupied. The F-TASEP,
interestingly, is known to exhibit absorbing phase transition between active
and inactive phases with $\rho=\frac{1}{2}$ being the transition point where
$\rho$ is the conserved particle density. We introduce QF-TASEP, the quantum
version of F-TASEP, for which the bipartite entanglement and current
remarkably exhibit quantum absorbing phase transition at $\rho=\frac{1}{2}$
with active and inactive phases, similar to the classical F-TASEP. Such
active-inactive phase transitions disappear for both classical and quantum
cases as we generalize the F-TASEP by including the independent hopping (i.e.
not facilitated by neighboring particles) of particles with rate $p$,
consequently termed as $p$F-TASEP (classical) and $p$QF-TASEP (quantum)
respectively. Due to the presence of multiple current carrying modes
(facilitated and independent), we compare their individual contributions to
the total steady state particle current in the $\rho-p$ plane. Interestingly,
our exact analytical calculations for $p$F-TASEP reveals the emergence of a
special point $\rho=\frac{1}{3}$ that demarcates between two different
regimes. For the system density with any fixed value between
$0<\rho\leqslant\frac{1}{3}$, the independent current always dominates over
the facilitated current for any $p$. On the contrary, for
$\frac{1}{3}<\rho<1$, there is always a finite $p^{\ast}$ such that the
facilitated (independent) current dominates for $p<p^{\ast}$ ($p>p^{\ast}$).
Remarkably, in the quantum problem $p$QF-TASEP, the bipartite entanglement
shows striking features around $\rho=\frac{1}{3}$ indicating towards its
speciality similar to the classical case.

Model: We start with the toy model F-TASEP defined on a one dimensional
periodic lattice with the following microscopic dynamics

$110\,\,\stackrel{{\scriptstyle 1}}{{\longrightarrow}}\,\,101,$ (1)

where $1$ and $0$ denote particle and vacancy respectively and each lattice
site can be occupied by at most one particle. The model is a special instance
of the Katz-Lebowitz-Spohn model Katz  _et al._ (1984); Hager  _et al._ (2001)
broadly used to model superionic conductors Dieterich  _et al._ (1980); Katz
_et al._ (1984), vehicular traffic flow Gray and Griffeath (2001); Levine  _et
al._ (2004) etc. Notably, alongside non-equilibrium physics literature Basu
and Mohanty (2009); Gabel  _et al._ (2010); Antal and Schütz (2000); Shi  _et
al._ (2012); Hao  _et al._ (2016); Chatterjee and Mohanty (2018); Botto  _et
al._ (2018); Pelizzola  _et al._ (2019), the F-TASEP and its variations have
gained lot of recent interests in the area of probability and mathematics Baik
_et al._ (2018); Goldstein  _et al._ (2019, 2021); Blondel  _et al._ (2020,
2021); Goldstein  _et al._ (2022); Ayyer  _et al._ (2020); Barraquand  _et
al._ (2023). The F-TASEP as discussed in Basu and Mohanty (2009), shows a
finite density transition at $\rho_{c}=\frac{1}{2}$ where the system transits
from inactive and non-ergodic state to a current carrying and ergodic steady
state. This active-inactive transition is characterised by the order parameter
$\langle 110\rangle$ which actually measures the activity (density of active
sites) or current in the system. The order parameter remains zero for
$0\leqslant\rho\leqslant\frac{1}{2}$ whereas it is non-zero for
$\rho>\frac{1}{2}$ and it varies as $(\rho-\rho_{c})^{\beta}$ with the
exponent $\beta=1$.

We now study the corresponding quantum process QF-TASEP by introducing the
following non-Hermitian quantum Hamiltonian given by

$H=\sum_{i}n_{i-1}c^{\dagger}_{i+1}c_{i}$ (2)

Given a set of many body eigenvalues $E_{n}$, the quantum steady state is
given by the eigenstate whose imaginary component of the eigenvalue is maximum
such that given the time evolution of every many body state $|\psi_{n}\rangle$
with energy $E_{n}$ the state grows as a function of time
($|\psi_{n}(t)\rangle=\exp(-iE_{n}t)|\psi_{n}\rangle$) and gets chosen as the
steady state at infinite time.

Quantum Absorbing Phase Transition: Investigating the steady states of QF-
TASEP (see eqn. (2)) and measuring the current operator

$J_{Q}=\frac{1}{L}\sum_{j}-i\langle
c^{\dagger}_{j+1}c_{j}-c^{\dagger}_{j}c_{j+1}\rangle$ (3)

on the non-equilibrium quantum steady state shows a similar absorbing
transition as a function of density – where the system transits to a current
carrying steady state when $\rho>\frac{1}{2}$ (see Fig. 1). Remarkably,
although the quantum current operator defined in Eq. (3) is different from the
usual classical current $J=\langle 110\rangle$ for F-TASEP, they essentially
capture similar characteristics. For $\rho<\frac{1}{2}$, the correlated
hopping process as described above, given any initial state forms charge
density wave patches where every time a dipole is formed (01), the region is
rendered inactive. This charge-density wave patch spans the complete lattice
at $\rho=\frac{1}{2}$ – at which any additional particle leads to a finite
current steady state. This active-inactive transition can be also found in the
density of active sites $\langle n_{i-1}n_{i}(1-n_{i+1})\rangle$, which is
basically the classical order parameter. Given the quantum nature of the
state, it would be interesting to ask if we can find the signature of quantum
absorbing phase transition in a bipartite entanglement. Indeed, it shows a
discontinuous jump at $\rho=\frac{1}{2}$ as shown in Fig. 1. A trivial mean-
field where the $n_{i}\rightarrow\rho$ (average density) reduces to an
effective Hatano-Nelson model Hatano and Nelson (1996) where the hopping is
renormalized by density. This leads to a boosted Fermi sea steady state where
the finite momentum modes lead to a finite current state with
$J_{Q}=\frac{2}{\pi}\sin(\rho\pi)$, as seen in the dashed line in Fig. 1(a)
where the current matches the analytical result quite well. Near $\rho\sim
0.5$, we have effective carriers as $\tilde{\rho}=\rho-\frac{1}{2}$ which
matches the current profile near $\rho\sim\frac{1}{2}$ where
$J_{Q}\sim\frac{2}{\pi}\sin(2\tilde{\rho}\pi)$. It is interesting that this
dynamical transition has no symmetry broken order parameter but current itself
can be thought of as an order parameter, with the critical exponent $\beta=1$
[$J_{Q}\sim(\rho-\rho_{c})^{\beta}$] same as that of the classical case.
Interestingly, the bipartite entanglement entropy of a site with rest of the
system, follows $S(\rho)=-\Big{(}\rho\log(\rho)+(1-\rho)\log(1-\rho)\Big{)}$
(similar to a free Fermi gas) (see the analytical curve in Fig. 1(b)) as soon
as $\rho>\frac{1}{2}$, showing a discrete jump in the entanglement entropy
characteristic of this quantum transition.

Figure 2: $p$F-TASEP phase diagram: Analytically obtained phase diagram for
the $p$F-TASEP (see eqn. (4)) with both facilitated and independent hopping.
The figure compares the steady state contributions of the two different
current modes $J_{fa}$ and $J_{in}$. The point $\rho=\frac{1}{3}$
distinguishes between two different regimes. For $0<\rho\leqslant\frac{1}{3},$
the independent current is always dominant irrespective of the value of $p$.
In sharp contrast, for $\frac{1}{3}<\rho<1$, there exists a finite
$p^{\ast}(\rho)$ (see eqn. (10)) that decides which mode (facilitated or
independent) will be dominant for a fixed density.

Generalization to $p$F-TASEP.- Both the classical F-TASEP and quantum QF-TASEP
exhibit absorbing phase transition which for $\rho\leqslant\frac{1}{2}$, is
identified by the non-ergodicity in the system due to the presence of multiple
absorbing or inactive configurations. However, such absorbing phase transition
vanishes as we generalize the dynamics of F-TASEP [Eq. (1)] by initiating
another mode of hopping, namely the independent hopping of particles which
occurs without any facilitation from the neighbor. This generalized classical
process, which we call $p$F-TASEP, is defined as

$110\,\,\stackrel{{\scriptstyle 1}}{{\longrightarrow}}\,\,101,\hskip
28.45274pt010\,\,\stackrel{{\scriptstyle p}}{{\longrightarrow}}\,\,001.$ (4)

Clearly, the limits $p=0$ and $p=1$ of $p$F-TASEP correspond to F-TASEP and
TASEP respectively. The independent hopping of particles with rate $p$
restores ergodicity in the system for any density. The model has been studied
previously as a cooperative exclusion process, in context of time evolution of
a step initial condition leading to rich phenomena like formation of shocks
and compression or rarefaction waves Gabel and Redner (2011). Our focus for
$p$F-TASEP would be to make a comparative study of the individual
contributions to steady state current produced by two different modes,
facilitated current $J_{fa}$ and independent current $J_{in}$. Analysis of
multiple current carrying modes become naturally important for wide range of
scenarios e.g. multiple chemical reagents diffusing through narrow channels
and undergoing specific reactions among selective pairs, different types of
vehicles moving through same road, decision making in teams with sub-groups
etc. We reformulate the Ising model like pair-factorized steady state of
$p$F-TASEP as a non-equilibrium matrix product state by obtaining the
particles and vacancies as two non-commuting matrices obeying certain matrix
algebra (see sup  for details). These matrix algebra help us to analytically
calculate the observables of interest, $J_{fa}=\langle 110\rangle$ and
$J_{in}=p\langle 010\rangle$. The exact expressions are obtained to be

$\displaystyle J_{fa}(\rho,p)$ $\displaystyle=$
$\displaystyle\frac{z(2\rho-1)}{(1-p)(z-p)}\frac{z(1-\rho)-p\rho}{z\rho-p(1-\rho)},$
(5) $\displaystyle J_{in}(\rho,p)$ $\displaystyle=$
$\displaystyle\frac{p(1-\rho)}{(1-p)}\frac{z(1-\rho)-p\rho}{z\rho-p(1-\rho)},$
(7) $\displaystyle\mathrm{where}\hskip 5.69046ptz(\rho,p)$ $\displaystyle=$
$\displaystyle\frac{1}{2\rho(1-\rho)}[1+2\rho(1-\rho)(p-2)$ (9)
$\displaystyle-(1-2\rho)\sqrt{1-4\rho(1-\rho)(1-p)}].$

To extract the competition between the two current carrying modes in $\rho-p$
plane, one has to simply equate $J_{fa}$ and $J_{in}$ from Eq. (9) and
consequently find the curve that separates the region of dominance between the
facilitated mode and the independent mode in the $\rho-p$ plane. This leads us
to the following non-trivial solution

$p^{\ast}(\rho)=\frac{3\rho-1}{1-\rho}.$ (10)

Figure 3:  Facilitated and Independent steady state currents: The behavior of
facilitated steady state current ($J_{fa}$) and independent current ($J_{in}$)
as a function of $p$ at $\rho=0.2$ ($<\frac{1}{3}$) in (a) and for $\rho=0.4$
($>\frac{1}{3}$) in (b). Analytical results are shown with a line and Monte
Carlo results (ensemble averages are performed over $10^{6}$ configurations)
are shown in dots. For $\rho>1/3$ there is a finite $p$ below which
$J_{fa}>J_{in}$.

The interesting fact evident from Eq. (10) is that no physical solution for
$p^{\ast}(\rho)$ exists for $\rho\leqslant\frac{1}{3}$. This implies that, for
$p$F-TASEP with conserved density $0\leqslant\rho\leqslant\frac{1}{3}$, one
mode has always higher contribution to current than the other irrespective of
how small or large $p$ is. We identify this dominant mode as the independent
mode in Fig. 2 where we present the comparison between two current carrying
modes in the $\rho-p$ plane. Contrary to $0\leqslant\rho\leqslant\frac{1}{3}$,
for any conserved system density $\frac{1}{3}\leqslant\rho\leqslant 1$, there
is a finite $p^{\ast}$ such that $J_{fa}$ dominates over $J_{in}$ for
$p<p^{\ast}$ whereas $J_{in}$ acts as the higher current carrying mode for
$p>p^{\ast}$. Clearly, as shown in Fig. 2, $\rho=\frac{1}{3}$ emerges as a
special density which demarcates between two regimes– one with $p^{\ast}=0$
(for $0\leqslant\rho\leqslant\frac{1}{3}$) and the other with $p^{\ast}>0$
(for $\frac{1}{3}\leqslant\rho\leqslant 1$). Note that the demarcator point
$\rho=\frac{1}{3}$ for $p$F-TASEP is different from the absorbing transition
point $\rho=\frac{1}{2}$ for F-TASEP and QF-TASEP.

To show the individual behaviors of $J_{fa}$ and $J_{in}$ explicitly, as
functions of the parameter $p$, we present Fig. 3 (a)-(b). In both figures, we
observe that our exact analytical results (solid lines) agree with Monte Carlo
simulation results (dots). The Fig. 3(a) shows that the independent current is
always higher than the facilitated current as expected for $\rho<\frac{1}{3}$.
In contrast, we observe in Fig. 3(b) ($\rho>\frac{1}{3}$) that $J_{fa}$ and
$J_{in}$ cross each other at $p=p^{\ast}$.

Figure 4: $p$F-TASEP and $p$QF-TASEP: Comparison between the facilitated
activity and independent activity in the classical and quantum facilitated
processes for different values of $p$ ($0.001,0.01,0.5,1.0$). (Analytical
results from MPS in classical and numerical ED for $L=14$ in the Quantum case)

Quantum analogue of $p$F-TASEP.- In order to develop $p$QF-TASEP, the quantum
process corresponding to classical $p$F-TASEP, we study the following non-
Hermitian Hamiltonian

$H=\sum_{i}n_{i-1}c^{\dagger}_{i+1}c_{i}+p(1-n_{i-1})c^{\dagger}_{i+1}c_{i}.$
(11)

As observables, we analyze $\langle 110\rangle$ and $\langle 010\rangle$ which
are directly related to the facilitated and independent currents that have
played crucial roles in the discussions in the previous section. In connection
to the activity $\langle 110\rangle$ for F-TASEP and QF-TASEP, we can term
$\langle 110\rangle$ and $\langle 010\rangle$ as facilitated activity and
independent activity, respectively, in cases of $p$F-TASEP and $p$QF-TASEP.
For these expectation values, we show the analogy between the classical and
quantum cases in Fig. 4. Since at $p=1$ the $p$F-TASEP reduces to the usual
TASEP that has uncorrelated steady state, classically both these expectation
values behave as $\rho^{2}(1-\rho)$ and as $(1-\rho)^{2}\rho$ as shown via the
dashed curves in Fig. 4 (top). With decreasing $p$, the independent hopping
decreases, thereby increasing $\langle 010\rangle$ for some fixed density. On
the other hand, $\langle 110\rangle$ decreases with decreasing $p$ since the
production rate of nearest neighbor particle pair $\langle 11\rangle$ through
the process $0101\stackrel{{\scriptstyle p}}{{\longrightarrow}}0011$ reduces.
The situation becomes most dramatic at $p\rightarrow 0$ where $\langle
110\rangle$ shows the near-critical behavior reflective of the absorbing phase
transition. Interestingly, similar qualitative behaviors are observed for
$p$QF-TASEP (see Fig. 4 (bottom)), where we study the expectation value of the
two operators $\langle n_{i}n_{i+1}(1-n_{i+2})\rangle$ and
$\langle(1-n_{i})n_{i+1}(1-n_{i+2})\rangle$. Thereby, the quantum steady state
fascinatingly shows a behavior equivalent to the classical non-equilibrium
steady state.

Figure 5: Enhanced bipartite entanglement: The behavior of the bipartite
entanglement entropy of half of the system with the other half, as a function
of density $\rho$ for the pQF-TASEP (see eqn. (11)) for different values of
$p$ for $L=16$.

We next investigate the bipartite entanglement entropy of this system. The
corresponding behavior is shown in Fig. 5. Interestingly at $p=1$ when the
system is just the quantum Hatano-Nelson model, the entanglement behaves
characteristic of a equilibrium Fermi sea where
$S(\rho)\sim\log(L\rho(1-\rho))$ since the steady state can be thought of as
the boosted Fermi sea. Similarly at $p=0$ when the system shows a quantum
active-inactive transition at $\rho\sim\frac{1}{2}$ one finds that
$\rho<\frac{1}{2}$, $\langle S\rangle$=$0$ as is expected for an absorbing
state. What is interesting is the rise in entanglement at small values of $p$
for less than half filling in the system. We suspect that this rise is due to
the formation of doublons in the system. For instance at $p\rightarrow 0$, the
doublon state $|110\rangle$ disperses with a second order process with an
effective non-Hermitian Hamiltonian $\sim pe^{ik}$. Notably, the entanglements
for sufficiently small values of $p$ exhibit maximums near $\rho=\frac{1}{3}$.
The striking behaviors of the bipartite entanglement entropy for $p$QF-TASEP
and the relation between $p$F-TASEP and $p$QF-TASEP in context of
$\rho=\frac{1}{3}$ being a special point, will be explored further in the
future work.

Conclusion.-In this work we study similarities in macroscopic behaviors of
classical and quantum many body systems, based on a three site microscopic
process, the facilitated exclusion process. Remarkably, analogous to the
absorbing phase transition shown by classical F-TASEP, we find that the
quantum facilitated process Q-FTASEP exhibits quantum absorbing phase
transition characterized by the bipartite entanglement entropy and current.
Notably, the absorbing transition for both the classical and quantum models
occur at $\rho=\frac{1}{2}$ and the current, acting as the order parameter,
varies as $(\rho-\rho_{c})^{\beta}$ with $\beta=1$ for both cases. The
absorbing transition vanishes as we generalize the classical F-TASEP to
$p$F-TASEP by including the independent hopping of particles with rate $p$.
Intriguingly, the comparison between the two sources of current (facilitated
and independent) reveals the existence of a special density $\rho=\frac{1}{3}$
which demarcates between two regimes, one where the independent current always
dominates irrespective of $p$ value and the other where a finite $p$
distinguishes between the region of dominance of the two currents. The
corresponding quantum model $p$QF-TASEP displays similar qualitative behaviors
as the classical one and particularly the bipartite entanglement manifests
striking features including sudden rise and maximal values near
$\rho=\frac{1}{3}$. It would be interesting to further explore these
classical-quantum connections for deeper understandings. Some immediate future
directions include the investigations of classical-quantum connections with
reciprocal hopping, open boundary conditions and relaxation phenomena. We
believe that our present study paves paths to explore interesting connections
for non-equilibrium phenomena overlapping between generic many-body classical
systems and interacting non-Hermitian quantum Hamiltonians.

Acknowledgements.- We thank Diptarka Das, Subrata Pacchal, Abhisodh Prakash,
Arghya Das, Urna Basu, Hisao Hayakawa, Diptiman Sen for discussions. AA
acknowledges support from IITK Initiation Grant (IITK/PHY/2022010) and
workstation Wigner at IITK. A.K.C. gratefully acknowledges postdoctoral
fellowship from the YITP and partial support from Grants-in-Aid for Scientific
Research (Grant No. JP21H01006). We acknowledge use of open-source
QuSpinWeinberg and Bukov (2017, 2019) for exact diagonalisation calculations.
Several numerical calculations and simulations have been done on the cluster
Yukawa-21 at YITP. Both of us acknowledge hospitality at ICTS, Bangalore.

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Supplemental Material for “Classical and quantum facilitated exclusion
processes”

In this supplementary material, we briefly describe the matrix product steady
state of $p$F-TASEP, defined in the main text as

$110\,\,\stackrel{{\scriptstyle 1}}{{\longrightarrow}}\,\,101,\hskip
28.45274pt010\,\,\stackrel{{\scriptstyle p}}{{\longrightarrow}}\,\,001.$ (12)

The model obeying the microscopic dynamics Eq. (12) evolves according to the
Master equation $\frac{\mathrm{d}|P\rangle}{\mathrm{d}t}=M|P\rangle$ where the
vector $|P\rangle$ includes probabilities of all possible configurations
$P(\left\\{s_{1},\dots,s_{L}\right\\})$. The Markovian rate matrix $M$ can be
decomposed as

$M=\sum_{i=1}^{L}I\otimes\dots I\otimes h_{i-1,i,i+1}\otimes I\dots\otimes I,$
(13)

where $h_{i-1,i,i+1}$ is the local three site matrix with dimension $8\times
8$ and $I$ is $2\times 2$ identity matrix sitting at every site except the
triad $(i-1,i,i+1)$. The off-diagonal elements of $h_{i-1,i,i+1}$ contain the
transition rates between two different local three site configurations,
whereas the diagonal terms carry the total outward rates from a three site
configuration. The explicit form of the matrix $h_{i-1,i,i+1}$ for dynamics in
Eq. (12) is

$h_{i-1,i,i+1}=\left(\begin{array}[]{cccccccc}0&0&0&0&0&0&0&0\\\
0&-1&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&-p&0&0\\\ 0&0&0&0&0&p&0&0\\\ 0&0&0&0&0&0&0&0\\\ \end{array}\right).$
(14)

The steady state, by definition, implies $M|P\rangle=0$. We make the following
matrix product ansatz, assuming that any configuration in the steady state can
be represented as a product of matrices and the corresponding probability
would be determined as

$\displaystyle P\left(\left\\{s_{i}\right\\}\right)$ $\displaystyle\propto$
$\displaystyle Tr\left[X_{i}\right],$ (15) $\displaystyle X_{i}$
$\displaystyle=$ $\displaystyle D\delta_{s_{i},1}+E\delta_{s_{i},0},$ (16)

with $D$ and $E$ representing particle and vacancy respectively. A sufficient
condition to ensure the steady state is to consider the following local
(probability) flux cancellation scheme

$\displaystyle h_{i-1,i,i+1}\left(\begin{array}[]{c}D\\\ E\\\
\end{array}\right)\otimes\left(\begin{array}[]{c}D\\\ E\\\
\end{array}\right)\otimes\left(\begin{array}[]{c}D\\\ E\\\ \end{array}\right)$
(23) $\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}D\\\ E\\\
\end{array}\right)\otimes\left(\begin{array}[]{c}\tilde{D}\\\ \tilde{E}\\\
\end{array}\right)\otimes\left(\begin{array}[]{c}D\\\ E\\\
\end{array}\right)-\left(\begin{array}[]{c}D\\\ E\\\
\end{array}\right)\otimes\left(\begin{array}[]{c}D\\\ E\\\
\end{array}\right)\otimes\left(\begin{array}[]{c}\tilde{D}\\\ \tilde{E}\\\
\end{array}\right),$ (36)

where $\tilde{D}$ and $\tilde{E}$ are known as auxiliary matrices that have to
be chosen suitably such that the correct steady state is achieved. Such a
suitable pair of choices for the present model turns out to be $\tilde{D}=-1$
and $\tilde{E}=0$. Along with these choices, from Eq. (LABEL:eq:s4), we find
the matrix algebra to be satisfied the matrices $D$ and $E$, are

$\displaystyle DDE$ $\displaystyle=$ $\displaystyle DE,$ (38) $\displaystyle
pEDE$ $\displaystyle=$ $\displaystyle EE.$ (39)

A possible set of explicit matrix representations satisfying Eq. (39) are
obtained to be

$\begin{array}[]{ccc}D=\left(\begin{array}[]{cc}0&1\\\ 0&1\\\
\end{array}\right),&\hskip 28.45274ptE=\left(\begin{array}[]{cc}p&0\\\ 1&0\\\
\end{array}\right).\end{array}$ (40)

If we put $p=0$ (i.e. the F-TASEP) in the matrix algebra in Eq. (39) and
matrix representations in Eq. (40), the matrix algebra ($D^{2}=D$ and
$E^{2}=0$) and matrix representations for the active phase in F-TASEP are
correctly recovered Basu and Mohanty (2009). With the explicit steady state
matrix representations being obtained [Eq. (40)], we can proceed to calculate
observables, in particular the currents of the facilitated mode and
independent mode, $J_{fa}$ and $J_{in}$ respectively. These quantities can be
calculated analytically as

$\displaystyle J_{fa}$ $\displaystyle=$ $\displaystyle\langle
110\rangle=\frac{Tr[DDET^{L-3}]}{Tr[T^{L}]},$ (41) $\displaystyle J_{in}$
$\displaystyle=$ $\displaystyle p\langle
010\rangle=p\frac{Tr[EDET^{L-3}]}{Tr[T^{L}]},\ $ (42)

where $T$ is the transfer matrix, translating which over the lattice sites
generates all possible configurations, so that the partition function becomes
$Tr[T^{L}]$. The matrix $T$ is defined as

$T=zD+E=\left(\begin{array}[]{cc}p&z\\\ 1&z\\\ \end{array}\right),$ (43)

where $z$ acts as the fugacity associated with the particles. As usual, the
transfer matrix can be brought into diagonal form by $T_{d}=U^{-1}TU$ where
the unitary matrix $U$ has eigenvectors of $T$ as its columns

$U=\left(\begin{array}[]{cc}\lambda_{-}-z&\lambda_{+}-z\\\ 1&1\\\
\end{array}\right),$ (44)

with $\lambda_{\pm}$, the eigenvalues of $T$, are given by

$\lambda_{\pm}=\frac{1}{2}\left(p+z\pm\sqrt{(p-z)^{2}+4z}\right).$ (45)

We perform the calculations for the classical $p$-FASEP in the thermodynamic
limit $L\rightarrow\infty$. Correspondingly, the partition function simplifies
to $Tr[T^{L}]\approx\lambda_{+}^{L}$. In this limit, it is straightforward to
show that the currents for two different modes from Eq. (42) result in

$\displaystyle J_{I}$ $\displaystyle=$
$\displaystyle\frac{z^{2}}{\lambda_{+}^{3}\Delta U}(U_{11}-U_{21})U_{12},$
(46) $\displaystyle J_{II}$ $\displaystyle=$
$\displaystyle\frac{pz}{\lambda_{+}^{3}\Delta U}(U_{11}-pU_{21})U_{12},$ (47)

where $U_{jk}$ correspond to the entry of matrix $U$ [Eq. (44)] in $j$-th row
and $k$-th column, $\Delta U$ is the determinant of the matrix $U$. Following
few simple intermediate steps, we can arrive at the expressions of the
currents given in Eq. (9). The fugacity $z$ is determined from the density-
fugacity relation $\rho=\frac{z}{\lambda_{+}}\frac{d}{dz}\lambda_{+}$, as a
function of the input parameters $\rho$ and $p$. The corresponding solution
$z(\rho,p)$ is provided in Eq. (9) of the main text.