# Bases of the quantum cluster algebra of the Kronecker quiver

Ming Ding and Fan Xu  Institute for advanced study
Tsinghua University
Beijing 100084, P. R. China<EMAIL_ADDRESS>(M.Ding)
Department of Mathematical Sciences
Tsinghua University
Beijing 100084, P. R. China<EMAIL_ADDRESS>(F.Xu)

###### Abstract.

We construct bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]-$bases of the
quantum cluster algebra of the Kronecker quiver which are quantum analogues of
the canonical basis, semicanonical basis and dual semicanonical basis of the
cluster algebra of the Kronecker quiver in the sense of [14],[4] and [11]
respectively. As a byproduct, we prove the positivity of the elements in these
bases.

###### Key words and phrases:

quantum cluster algebra, $\mathbb{Z}[q^{\pm\frac{1}{2}}]-$basis, positivity

###### 2000 Mathematics Subject Classification:

Primary 16G20, 20G42; Secondary 14M17

Fan Xu was partially supported by the Ph.D. Programs Foundation of Ministry of
Education of China (No. 200800030058)

##  1\. Introduction

Cluster algebras were introduced by S. Fomin and A. Zelevinsky [9][10] in
order to study total positivity in algebraic groups and the specialization of
canonical bases of quantum groups at $q=1$. The study of $\mathbb{Z}$-bases of
cluster algebras is important. There are many results involving the
construction of $\mathbb{Z}$-bases of cluster algebras (for example, see [14]
and [4] for cluster algebras of rank 2, [3] for finite type, [7] for type
$\widetilde{A}$, [5] for $\widetilde{A}_{2}^{(1)}$, [6] for affine type and
[11] for $Q$ without oriented cycles). As a quantum analog of cluster
algebras, quantum cluster algebras were defined by A. Berenstein and A.
Zelevinsky in [1] in order to study canonical bases. A quantum cluster algebra
is generated by a set of generators called cluster variables inside an ambient
skew-field $\mathcal{F}$. Under the specialization $q=1,$ the quantum cluster
algebras are exactly cluster algebras.

Naturally, one may hope to construct $\mathbb{Z}[q^{\pm\frac{1}{2}}]-$bases
for quantum cluster algebras and further quantum analogues of bases of the
corresponding cluster algebras. In this short note, we deal with the case of
the quantum cluster algebra of the Kronecker quiver and construct various bar-
invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]-$bases by applying the
$q$-deformation of the Caldero-Chapoton formula defined in [13] and the method
in [14]. Under the specialization $q=1$, these
$\mathbb{Z}[q^{\pm\frac{1}{2}}]-$bases are exactly the canonical basis,
semicanonical basis and dual semicanonical basis of the corresponding cluster
algebra in the sense of [14],[4] and [11] respectively. As a byproduct, we
prove the positivity of the elements in these bases.

Recently, in [12], the author attached to certain element $w$ in Weyl group a
subalgebra $U_{q}^{+}(w)$ of the positive part $U_{q}(n)$ of the universal
enveloping algebra of a Kac-Moody Lie algebra of type
$\widetilde{A}_{1}^{(1)}$. The author proved that $U_{q}^{+}(w)$ is a quantum
cluster algebra in the sense of Berenstein-Zelevinsky and gave explicit
formulae for the cluster variables. Note that the cluster variables are some
elements of $q$-deformation of dual canonical basis elements of
$U_{q}^{+}(w)$. However it is not clear whether cluster monomials belong to
the dual canonical basis. Thus comparing these
$\mathbb{Z}[q^{\pm\frac{1}{2}}]-$bases constructed in this note with the dual
canonical basis of $U_{q}^{+}(w)$ becomes an interesting thing.

##  2\. Preliminaries

###  2.1. Quantum cluster algebras

We begin with some of the terminology related to quantum cluster algebras. One
can refer to [1] for more details. Let $L$ be a lattice of rank $m$ and
$\Lambda:L\times L\to\mathbb{Z}$ a skew-symmetric bilinear form. We will need
a formal variable $q$ and consider the ring of integer Laurent polynomials
$\mathbb{Z}[q^{\pm 1/2}]$. Define the based quantum torus associated to the
pair $(L,\Lambda)$ to be the $\mathbb{Z}[q^{\pm 1/2}]$-algebra $\mathcal{T}$
with a distinguished $\mathbb{Z}[q^{\pm 1/2}]$-basis $\\{X^{e}:e\in L\\}$ and
the multiplication given by

$X^{e}X^{f}=q^{\Lambda(e,f)/2}X^{e+f}.$

It is easy to see that $\mathcal{T}$ is associative and the basis elements
satisfy the following relations:

$X^{e}X^{f}=q^{\Lambda(e,f)}X^{f}X^{e},\ X^{0}=1,\ (X^{e})^{-1}=X^{-e}.$

It is known that $\mathcal{T}$ is an Ore domain, i.e., is contained in its
skew-field of fractions $\mathcal{F}$. The quantum cluster algebra will be
defined as a $\mathbb{Z}[q^{\pm 1/2}]$-subalgebra of $\mathcal{F}$.

A toric frame in $\mathcal{F}$ is a map
$M:\mathbb{Z}^{m}\to\mathcal{F}\setminus\\{0\\}$ of the form

$M({\bf c})=\varphi(X^{\eta({\bf c})})$

where $\varphi$ is an automorphism of $\mathcal{F}$ and
$\eta:\mathbb{Z}^{m}\to L$ is an isomorphism of lattices. By the definition,
the elements $M({\bf c})$ form a $\mathbb{Z}[q^{\pm 1/2}]$-basis of the based
quantum torus $\mathcal{T}_{M}:=\varphi(\mathcal{T})$ and satisfy the
following relations:

$M({\bf c})M({\bf d})=q^{\Lambda_{M}({\bf c},{\bf d})/2}M({\bf c}+{\bf d}),\
M({\bf c})M({\bf d})=q^{\Lambda_{M}({\bf c},{\bf d})}M({\bf d})M({\bf c}),$
$M({\bf 0})=1,\ M({\bf c})^{-1}=M(-{\bf c}),$

where $\Lambda_{M}$ is the skew-symmetric bilinear form on $\mathbb{Z}^{m}$
obtained from the lattice isomorphism $\eta$. Let $\Lambda_{M}$ also denote
the skew-symmetric $m\times m$ matrix defined by
$\lambda_{ij}=\Lambda_{M}(e_{i},e_{j})$ where $\\{e_{1},\ldots,e_{m}\\}$ is
the standard basis of $\mathbb{Z}^{m}$. Given a toric frame $M$, let
$X_{i}=M(e_{i})$. Then we have

$\mathcal{T}_{M}=\mathbb{Q}(q^{1/2})\langle X_{1}^{\pm 1},\ldots,X_{m}^{\pm
1}:X_{i}X_{j}=q^{\lambda_{ij}}X_{j}X_{i}\rangle.$

An easy computation shows that:

$M({\bf
c})=q^{\frac{1}{2}\sum_{i<j}c_{i}c_{j}\lambda_{ji}}X_{1}^{c_{1}}X_{2}^{c_{2}}\cdots
X_{m}^{c_{m}}=:X^{({\bf c})}\ \ \ ({\bf c}\in\mathbb{Z}^{m}).$

Let $\Lambda$ be an $m\times m$ skew-symmetric matrix and let $\tilde{B}$ be
an $m\times n$ matrix, $n\leq m$. We call the pair $(\Lambda,\tilde{B})$
compatible if $\tilde{B}^{T}\Lambda=(D|0)$ is an $n\times m$ matrix with
$D=diag(d_{1},\cdots,d_{n})$ where $d_{i}\in\mathbb{N}$ for $1\leq i\leq n$.
The pair $(M,\tilde{B})$ is called a quantum seed if the pair
$(\Lambda_{M},\tilde{B})$ is compatible. Define the $m\times m$ matrix
$E=(e_{ij})$ by

$e_{ij}=\begin{cases}\delta_{ij}&\text{if $j\neq k$;}\\\ -1&\text{if
$i=j=k$;}\\\ max(0,-b_{ik})&\text{if $i\neq j=k$.}\end{cases}$

For $n,k\in\mathbb{Z}$, $k\geq 0$, denote ${n\brack
k}_{q}=\frac{(q^{n}-q^{-n})\cdots(q^{n-r+1}-q^{-n+r-1})}{(q^{r}-q^{-r})\cdots(q-q^{-1})}$.
Let ${\bf c}=(c_{1},\ldots,c_{m})\in\mathbb{Z}^{m}$ with $c_{k}\geq 0$. Define
the toric frame $M^{\prime}:\mathbb{Z}^{m}\to\mathcal{F}\setminus\\{0\\}$ as
follows:

(2.1) $M^{\prime}({\bf c})=\sum^{c_{k}}_{p=0}{c_{k}\brack
p}_{q^{d_{k}/2}}M(E{\bf c}+p{\bf b}^{k}),\ \
M^{\prime}({\bf-c})=M^{\prime}({\bf c})^{-1}.$

where the vector ${\bf b}^{k}\in\mathbb{Z}^{m}$ is the $k-$th column of
$\tilde{B}$. Then the quantum seed $(M^{\prime},\tilde{B}^{\prime})$ is
defined to be the mutation of $(M,\tilde{B})$ in direction $k$. We say that
two quantum seeds are mutation-equivalent if they can ba obtained from each
other by a sequence of mutations. Let
$\mathcal{C}=\\{M^{\prime}(e_{i}):i\in[1,n]\\}$ where
$(M^{\prime},\tilde{B}^{\prime})$ is mutation-equivalent to $(M,\tilde{B})$.
The elements of $\mathcal{C}$ are called cluster variables. Let
$\mathcal{P}=\\{M(e_{i}):i\in[n+1,m]\\}$ and the elements of $\mathcal{P}$ are
called _coefficients_. The quantum cluster algebra
$\mathcal{A}_{q}(\Lambda_{M},\tilde{B})$ is the $\mathbb{Z}[q^{\pm
1/2}]$-subalgebra of $\mathcal{F}$ generated by $\mathcal{C}\cup\mathcal{P}$.
We associated with $(M,\tilde{B})$ the $\mathbb{Z}$-linear _bar-involution_ on
$\mathcal{T}_{M}$ by setting:

$\overline{q^{r/2}M({\bf c})}=q^{-r/2}M({\bf c}),\ \ (r\in\mathbb{Z},\ {\bf
c}\in\mathbb{Z}^{n}).$

It is easy to show that $\overline{XY}=\overline{Y}~{}\overline{X}$ for all
$X,Y\in\mathcal{A}_{q}(\Lambda_{M},\tilde{B})$ and that each element of
$\mathcal{C}\cup\mathcal{P}$ is _bar-invariant_.

###  2.2. The Kronecker quiver

Given a compatible pair $(\Lambda,\widetilde{B})$, we can associate a valued
quiver (see [13, Section 2] for more details). Now we set
$\Lambda=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)$ and
$\widetilde{B}=\left(\begin{array}[]{cc}0&2\\\ -2&0\end{array}\right)$. The
quiver $Q$ associated to this pair is the Kronecker quiver:

1$\bullet$$\bullet$2

Let $k$ be a finite field with cardinality $|k|=q^{2}$. The category $rep(kQ)$
of finite-dimensional representations can be identified with the category of
mod-$kQ$ of finite-dimensional modules over the path algebra $kQ.$ It is well-
known (see [8]) that indecomposable $kQ$-module contains (up to isomorphism)
three families: the indecomposable regular modules with dimension vector
$(nd_{p},nd_{p})$ for $p\in\mathbb{P}^{1}_{k}$ of degree $d_{p}$ (in
particular, denoted by $R_{p}(n)$ for $d_{p}=1$), the preprojective modules
with dimension vector $(n-1,n)$ (denoted by $M(n)$) and the preinjective
modules with dimension vector $(n,n-1)$ (denoted by $N(n)$). Here
$n\in\mathbb{N}$.

For $m\in\mathbb{Z}\setminus\\{1,2\\}$, set

$V(m)=\begin{cases}N(m-2)&\text{if $m\geq 3$;}\\\ M(-m+1)&\text{if $m\leq
0$.}\end{cases}$

Now, let $\mathcal{T}=\mathbb{Q}(q^{1/2})\langle X_{1}^{\pm 1},X_{2}^{\pm
1}:X_{1}X_{2}=qX_{2}X_{1}\rangle$ and ${\mathcal{F}}$ be the skew field of
fractions of $\mathcal{T}$ and thus the quantum cluster algebra of the
Kronecker quiver $\mathcal{A}_{q}(\Lambda,\tilde{B})$ (denoted by
$\mathcal{A}_{q}(2,2)$ in the following) is the
$\mathbb{Q}(q^{1/2})$-subalgebra of ${\mathcal{F}}$ generated by the cluster
variables $X_{k}$, $k\in\mathbb{Z}$, defined recursively by

(2.2) $X_{m-1}X_{m+1}=qX_{m}^{2}+1.$

The quantum Laurent phenomenon ([1]) implies that each $X_{k}$ belongs to the
subring of $\mathcal{T}$ generated by $q^{\pm 1/2}$, $X_{1}^{\pm 1},X_{2}^{\pm
1}$. The explicit Laurent expansion of each $X_{k}$ in $X_{1},X_{2}$ is given
in [12] and [13].

Let $V$ be a representation of the Kronecker quiver with dimension vector
$\underline{\text{dim}}V=(v_{1},v_{2})$. For ${\bf
e}=(e_{1},e_{2})\in\mathbb{Z}_{\geq 0}^{2}$, denote by $Gr_{\bf e}(V)$ the set
of all subrepresentations $M$ of $V$ with $\underline{\text{dim}}M={\bf e}$.
In [13], the author define the element $X_{V}$ of the quantum torus
$\mathcal{T}$ by

(2.3) $X_{V}=\sum_{{\bf e}}q^{-\frac{1}{2}d_{{\bf e}}^{V}}|Gr_{{\bf
e}}(V)|X^{(-v_{1}+2v_{2}-2e_{2},2e_{1}-v_{2})}$

where $d_{{\bf e}}^{V}=2e_{1}(v_{1}-e_{1})-2(2e_{1}-e_{2})(v_{2}-e_{2})$. This
formula is called a $q$-deformation of the Caldero-Chapoton formula ([2]).
Here and the following, we simply write $X^{\textbf{c}}$ instead of
$X^{(\textbf{c})}$ for $\textbf{c}\in\mathbb{Z}^{2}.$

######  Theorem 2.1.

[13] For any $m\in\mathbb{Z}\setminus\\{1,2\\}$, the $m$-th cluster variable
$X_{m}$ of $\mathcal{A}_{q}(2,2)$ equals $X_{V(m)}$.

##  3\. Bases of the quantum cluster algebra $\mathcal{A}_{q}(2,2)$

In this section, we will construct various bar-invariant
$\mathbb{Z}[q^{\pm\frac{1}{2}}]-$bases of quantum cluster algebra
$\mathcal{A}_{q}(2,2).$ Under the specialization $q=1$, these bases are just
bases of the cluster algebra of the Kronecker quiver.

######  Definition 3.1.

For any $(r_{1},r_{2})$ and $(s_{1},s_{2})\in\mathbb{Z}^{2}$, we write
$(r_{1},r_{2})\preceq(s_{1},s_{2})$ if $r_{i}\leq s_{i}$ for $1\leq i\leq 2$.
Moreover, if there exists some $i$ such that $r_{i}<s_{i}$, then we write
$(r_{1},r_{2})\prec(s_{1},s_{2}).$

######  Remark 3.2.

By the definition of the $q$-deformation of the Caldero-Chapoton formula and
the partial order in Definition 3.1, we obtain that the expansion of
$X_{V(m)}$ have a minimal non-zero term
$f(q^{\frac{1}{2}},q^{-\frac{1}{2}})X^{-\underline{\text{dim}}V(m)}$ where
$f(q^{\frac{1}{2}},q^{-\frac{1}{2}})\in\mathbb{Z}[q^{\frac{1}{2}},q^{-\frac{1}{2}}]$.
In fact, $f(q^{\frac{1}{2}},q^{-\frac{1}{2}})=1$ by the explicit expansion of
$X_{V(m)}$ in [12] and [13, Proposition 1.1].

######  Lemma 3.3.

Let $R_{p}(1)$ be the indecomposable regular module of degree $1$ as above.
Then

$X_{R_{p}(1)}=X^{(-1,1)}+X^{(1,-1)}+X^{(-1,-1)}.$

###### Proof.

Note that $R_{p}(1)$ contains the three submodules: $0,M(1)$ and $R_{p}(1)$.
Thus the lemma immediately follows from the $q$-deformation of the Caldero-
Chapoton formula. ∎

By Lemma 3.3, the expression of $X_{R_{p}(1)}$ is independent of the choice of
$p\in\mathbb{P}^{1}_{k}$ of degree 1. Hence, we set

$X_{\delta}:=X_{R_{p}(1)}.$

######  Definition 3.4.

(1) The n-th Chebyshev polynomials of the first kind is the polynomial
$F_{n}(x)\in\mathbb{Z}[x]$ defined by

$F_{0}(x)=1,F_{1}(x)=x,F_{2}(x)=x^{2}-2,F_{n+1}(x)=F_{n}(x)F_{1}(x)-F_{n-1}(x)\
for\ n\geq 2.$

(2) The n-th Chebyshev polynomials of the second kind is the polynomial
$S_{n}(x)\in\mathbb{Z}[x]$ defined by

$S_{0}(x)=1,S_{1}(x)=x,S_{2}(x)=x^{2}-1,S_{n+1}(x)=S_{n}(x)S_{1}(x)-S_{n-1}(x)\
for\ n\geq 2.$

It is obvious that $F_{n}(x)=S_{n}(x)-S_{n-2}(x).$ We denote
$z=X_{\delta},z_{n}=F_{n}(z),s_{n}=S_{n}(z)$ for $n\geq 0$ and $z_{n}=s_{n}=0$
for $n<0$. Set

$\mathcal{B}=\\{X^{a}_{m}X^{b}_{m+1}|m\in\mathbb{Z},(a,b)\in\mathbb{Z}^{2}_{\geq
0}\\}\cup\\{z_{n}|n\in\mathbb{N}\\}$
$\mathcal{S}=\\{X^{a}_{m}X^{b}_{m+1}|m\in\mathbb{Z},(a,b)\in\mathbb{Z}^{2}_{\geq
0}\\}\cup\\{s_{n}|n\in\mathbb{N}\\}$
$\mathcal{D}=\\{X^{a}_{m}X^{b}_{m+1}|m\in\mathbb{Z},(a,b)\in\mathbb{Z}^{2}_{\geq
0}\\}\cup\\{z^{n}|n\in\mathbb{N}\\}$

######  Remark 3.5.

(1) It is easy to check that $X^{(r,r)}X^{(s,s)}=X^{(r+s,r+s)}$ for any
$r,s\in\mathbb{Z}$, thus the expansions of $z_{n},s_{n}$ and $z^{n}$ have a
minimal non-zero term $f(q^{\frac{1}{2}},q^{-\frac{1}{2}})X^{-(n,n)}$
according to the partial order in Definition 3.1.

(2) The elements c associated to these minimal non-zero terms
$f(q^{\frac{1}{2}},q^{-\frac{1}{2}})X^{\textbf{c}}$ in the expansion of the
elements in the set $\mathcal{B}$ are different from each other. Indeed, it is
easy to compute

$\textbf{c}=\begin{cases}(b,-a)&\text{if
$X^{\textbf{c}}=X_{0}^{a}X_{1}^{b}$;}\\\ (a,b)&\text{if
$X^{\textbf{c}}=X_{1}^{a}X_{2}^{b}$;}\\\ (-b,a)&\text{if
$X^{\textbf{c}}=X_{2}^{a}X_{3}^{b}$;}\\\
a\cdot\underline{\text{dim}}V(m)+b\cdot\underline{\text{dim}}V(m+1)&\text{if
$X^{\textbf{c}}=X_{m}^{a}X_{m+1}^{b}$ for $m\neq 0,1\mbox{ or }2$;}\\\
(n,n)&\text{if $X^{\textbf{c}}=z_{n}$.}\end{cases}$

We note that there is at most one exceptional module in each dimension vector.

Now we define a ring homomorphism of the quantum cluster algebra
$\mathcal{A}_{q}(2,2)$:

$\sigma_{1}:\ \mathcal{A}_{q}(2,2)\longrightarrow\mathcal{A}_{q}(2,2)$

which sends $X_{m}$ to $X_{m+1}$ and $q^{\pm\frac{1}{2}}$ to
$q^{\pm\frac{1}{2}}$. It is obviously an automorphism which preserves the
defining relations. The following lemma is easy but important.

######  Lemma 3.6.

$\sigma_{1}(X_{\delta})=X_{\delta}.$

###### Proof.

By Theorem 2.1 and the definition of the $q$-deformation of the Caldero-
Chapoton formula, we have

$\displaystyle X_{0}=X_{V(0)}$ $\displaystyle=$ $\displaystyle
X^{(2,-1)}+X^{(0,-1)},$ $\displaystyle X_{3}=X_{V(3)}$ $\displaystyle=$
$\displaystyle X^{(-1,2)}+X^{(-1,0)},$ $\displaystyle X_{-1}=X_{V(-1)}$
$\displaystyle=$ $\displaystyle
X^{(3,-2)}+X^{(-1,-2)}+(q+q^{-1})X^{(1,-2)}+X^{(-1,0)}.$

Following these identities and Lemma 3.3, one easily confirm the relations

(3.1)
$X_{\delta}=q^{\frac{1}{2}}(X_{0}X_{3}-qX_{1}X_{2})=q^{\frac{1}{2}}(X_{-1}X_{2}-qX_{0}X_{1}).$

Thus
$\sigma_{1}(X_{\delta})=\sigma_{1}(q^{\frac{1}{2}}(X_{-1}X_{2}-qX_{0}X_{1}))=q^{\frac{1}{2}}(X_{0}X_{3}-qX_{1}X_{2})=X_{\delta}.$
∎

######  Lemma 3.7.

For any $n\in\mathbb{Z},$

$X_{n}X_{\delta}=q^{-\frac{1}{2}}X_{n-1}+q^{\frac{1}{2}}X_{n+1}.$

###### Proof.

By an easy computation, we have:

$\displaystyle X_{0}$ $\displaystyle=$ $\displaystyle X^{(2,-1)}+X^{(0,-1)}$
$\displaystyle X_{-1}$ $\displaystyle=$ $\displaystyle
X^{(3,-2)}+X^{(-1,-2)}+(q+q^{-1})X^{(1,-2)}+X^{(-1,0)}.$

Then by Lemma 3.3, it is easy to prove

$X_{0}X_{\delta}=q^{-\frac{1}{2}}X_{-1}+q^{\frac{1}{2}}X_{1}.$

Thus we can finish the proof by Lemma 3.6 and applying the automorphism
$\sigma_{1}$. ∎

######  Lemma 3.8.

$\overline{X_{\delta}}=X_{\delta}.$

###### Proof.

$\overline{X_{\delta}}=q^{-\frac{1}{2}}(\overline{X_{0}X_{3}}-q^{-1}\overline{X_{1}X_{2}})=q^{-\frac{1}{2}}(X_{3}X_{0}-q^{-1}X_{2}X_{1})=X_{\delta}.$
∎

######  Remark 3.9.

By Lemma 3.8, we can verify that
$\overline{z_{n}}=z_{n},\overline{s_{n}}=s_{n}$ and
$\sigma_{1}(z_{n})=z_{n},\sigma_{1}(s_{n})=s_{n}$.

The following proposition, which can be viewed as the quantum analogue of [14,
Proposition 5.4], plays an essential role to construct
$\mathbb{Z}[q^{\pm\frac{1}{2}}]-$bases of the quantum cluster algebra
$\mathcal{A}_{q}(2,2).$

######  Proposition 3.10.

(1) For $m>n\geq 1$:

$\displaystyle z_{n}z_{m}$ $\displaystyle=$ $\displaystyle z_{m+n}+z_{m-n}$
$\displaystyle z_{n}z_{n}$ $\displaystyle=$ $\displaystyle z_{2n}+2.$

(2) $m\geq 1$ and $n\in\mathbb{Z}$:

$\displaystyle X_{n}z_{m}$ $\displaystyle=$ $\displaystyle
q^{\frac{m}{2}}X_{n+m}+q^{-\frac{m}{2}}X_{n-m}.$

(3) For $m\geq 0\ and\ n\in\mathbb{Z}:$

$\displaystyle X_{n}X_{n+2m}$ $\displaystyle=$ $\displaystyle
q^{m}X^{2}_{n+m}+\sum_{l=0}^{m-1}q^{-m+2l+1}\sum_{k=l+1}^{m}z_{2(m-k)},$
$\displaystyle X_{n}X_{n+2m+1}$ $\displaystyle=$ $\displaystyle
q^{m}X_{n+m}X_{n+m+1}+\sum_{l=0}^{m-1}q^{-m+2l+\frac{1}{2}}\sum_{k=l+1}^{m}z_{2(m-k)+1}.$

###### Proof.

The proof of (1) follows from the inductive relations in the definition of
Chebyshev polynomials.
As for (2), we make induction on $m$. If $m=1,$ the equation in (2) is a
direct corollary of Lemma 3.7. We assume that (2) holds for $m\leq k.$ For
$m=k+1,$ we have

$\displaystyle X_{n}z_{k+1}$ $\displaystyle=$ $\displaystyle
X_{n}(z_{k}z_{1}-z_{k-1})$ $\displaystyle=$
$\displaystyle(q^{\frac{k}{2}}X_{n+k}+q^{-\frac{k}{2}}X_{n-k})z_{1}-(q^{\frac{k-1}{2}}X_{n+k-1}+q^{-\frac{k-1}{2}}X_{n-k+1})$
$\displaystyle=$ $\displaystyle
q^{\frac{k}{2}}X_{n+k}z_{1}+q^{-\frac{k}{2}}X_{n-k}z_{1}-(q^{\frac{k-1}{2}}X_{n+k-1}+q^{-\frac{k-1}{2}}X_{n-k+1})$
$\displaystyle=$ $\displaystyle
q^{\frac{k}{2}}(q^{\frac{1}{2}}X_{n+k+1}+q^{-\frac{1}{2}}X_{n+k-1})+q^{-\frac{k}{2}}(q^{\frac{1}{2}}X_{n-k+1}+q^{-\frac{1}{2}}X_{n-k-1})$
$\displaystyle-(q^{\frac{k-1}{2}}X_{n+k-1}+q^{-\frac{k-1}{2}}X_{n-k+1})$
$\displaystyle=$ $\displaystyle
q^{\frac{k+1}{2}}X_{n+k+1}+q^{-\frac{k+1}{2}}X_{n-k-1}.$

This proves (2). Now we prove (3). If $m=0,$ it is obvious. If $m=1,$ by the
recurrence relations (2.2), we have

$X_{n}X_{n+2}=qX^{2}_{n+1}+1.$

We have proved that the equation $X_{0}X_{3}=qX_{1}X_{2}+q^{-\frac{1}{2}}z$
(see (3.1)) holds, thus by Lemma 3.6, we have

$X_{n}X_{n+3}=qX_{n+1}X_{n+2}+q^{-\frac{1}{2}}z.$

Now we assume that equations in (3) hold for $m\leq k$. For $m=k+1,$ by Lemma
3.7, we have

$X_{n}X_{n+2k+2}=q^{-\frac{1}{2}}X_{n}(X_{n+2k+1}z_{1}-q^{-\frac{1}{2}}X_{n+2k}).$

Following the inductive assumption, it is equal to

$q^{-\frac{1}{2}}(q^{k}X_{n+k}X_{n+k+1}+\sum_{l=0}^{k-1}q^{-k+2l+\frac{1}{2}}\sum_{i=l+1}^{k}z_{2(k-i)+1})z_{1}-q^{-1}(q^{k}X^{2}_{n+k}+\sum_{l=0}^{k-1}q^{-k+2l+1}\sum_{i=l+1}^{k}z_{2(k-i)}).$

Using Lemma 3.7 again and (1) of this proposition, it is

$q^{k-\frac{1}{2}}X_{n+k}(q^{\frac{1}{2}}X_{n+k+2}+q^{-\frac{1}{2}}X_{n+k})+\sum_{l=0}^{k-1}q^{-k+2l}\sum_{i=l+1}^{k}z_{2(k-i)+1}z_{1}$
$-q^{-1}(q^{k}X^{2}_{n+k}+\sum_{l=0}^{k-1}q^{-k+2l+1}\sum_{i=l+1}^{k}z_{2(k-i)})$
$=q^{k+1}X^{2}_{n+k+1}+\sum_{l=0}^{k}q^{-k+2l}\sum_{i=l+1}^{k+1}z_{2(k-i+1)}.\quad\quad\text{(*)}$

Similarly, by Lemma 3.7, we have

$X_{n}X_{n+2k+3}=q^{-\frac{1}{2}}X_{n}(X_{n+2k+2}z_{1}-q^{-\frac{1}{2}}X_{n+2k+1})$

Using the equation (*) and similar proof, we obtain

$X_{n}X_{n+2k+3}=q^{k+1}X_{n+k+1}X_{n+k+2}+\sum_{l=0}^{k}q^{-k+2l-\frac{1}{2}}\sum_{i=l+1}^{k+1}z_{2(k-i+1)+1}.$

∎

######  Remark 3.11.

By Lemma 3.8 and properties of bar-invariant, we can easily obtain the similar
results for $z_{m}X_{n},X_{n+2m}X_{n},X_{n+2m}X_{n}$.

We similarly define the quantized version of the definition of positivity in
[14].

######  Definition 3.12.

A nonzero element $x\in\mathcal{A}_{q}(2,2)$ is positive if for every
$m\in\mathbb{Z},$ all the coefficients in the expansion of $x$ as a Laurent
polynomial in $\\{x_{m},x_{m+1}\\}$ belong to
$\mathbb{N}[q^{\pm\frac{1}{2}}].$

######  Corollary 3.13.

Every element in $\mathcal{B},\mathcal{S}$ and $\mathcal{D}$ is a positive
element of quantum cluster algebra $\mathcal{A}_{q}(2,2)$.

###### Proof.

By Lemma 3.3 and the fact $z_{n}(x)=s_{n}(x)-s_{n-2}(x)$, we only need to
prove every element in $\mathcal{B}$ is positive. By the definition of
$\sigma_{1}$ and Remark 3.9, it is enough to prove the positivity in
$\\{x_{1},x_{2}\\}.$ We prove it by induction. For convenience, we write down
the following equations according to Proposition 3.10 and Remark 3.11:
For $m\geq 1:$

(3.2) $\displaystyle X_{1}z_{m}$ $\displaystyle=$ $\displaystyle
q^{\frac{m}{2}}X_{1+m}+q^{-\frac{m}{2}}X_{1-m}.$

For $m\geq 0:$

$\displaystyle X_{1}X_{1+2m}$ $\displaystyle=$ $\displaystyle
q^{m}X^{2}_{1+m}+q^{-m+1}\sum_{k=1}^{m}{z_{2(m-k)}}+q^{-m+3}\sum_{k=2}^{m}{z_{2(m-k)}}$
$\displaystyle+\cdots+q^{m-3}\sum_{k=m-1}^{m}{z_{2(m-k)}}+q^{m-1};$
$\displaystyle X_{1}X_{2+2m}$ $\displaystyle=$ $\displaystyle
q^{m}X_{1+m}X_{2+m}+q^{-m+\frac{1}{2}}\sum_{k=1}^{m}{z_{2(m-k)+1}}+q^{-m+\frac{5}{2}}\sum_{k=2}^{m}{z_{2(m-k)+1}}$
$\displaystyle+\cdots+q^{m-\frac{7}{2}}\sum_{k=m-1}^{m}{z_{2(m-k)+1}}+q^{m-\frac{3}{2}}z_{1}.$

$\displaystyle X_{1}X_{1-2m}$ $\displaystyle=$ $\displaystyle
q^{-m}X^{2}_{1-m}+q^{m-1}\sum_{k=1}^{m}{z_{2(m-k)}}+q^{m-3}\sum_{k=2}^{m}{z_{2(m-k)}}$
$\displaystyle+\cdots+q^{-m+3}\sum_{k=m-1}^{m}{z_{2(m-k)}}+q^{-m+1};$
$\displaystyle X_{1}X_{-2m}$ $\displaystyle=$ $\displaystyle
q^{-m-1}X_{-m}X_{1-m}+q^{m-\frac{1}{2}}\sum_{k=1}^{m}{z_{2(m-k)+1}}+q^{m-\frac{5}{2}}\sum_{k=2}^{m}{z_{2(m-k)+1}}$
$\displaystyle+\cdots+q^{-m+\frac{7}{2}}\sum_{k=m-1}^{m}{z_{2(m-k)+1}}+q^{-m+\frac{3}{2}}z_{1}.$

It is easy to check that

$\\{X_{-2},X_{-1},X_{0},X_{1},X_{2},z_{1},z_{2}\\}$

are positive elements in $\\{x_{1},x_{2}\\}.$ Now assume that

$\\{X_{-2m},X_{-2m+1},\cdots,X_{2m-1},X_{2m},z_{1},\cdots,z_{2m}\\}$

are positive elements in $\\{x_{1},x_{2}\\}.$ Then by (3.2), (3.3) and (3.4),
we know that $X_{2m+1},X_{2m+2}$ and $X_{-1-2m}$ are positive. Thus we obtain
that $z_{2m+1}$ is positive by (3.1). Therefore by (3.5), we have that
$X_{-2m-2}$ is positive. Again by (3.2), we know that $X_{2m+3}$ is positive.
Thus we get $z_{2m+2}$ is positive by (3.1) again. Throughout the above
discussions we obtain

$\\{X_{-2m-2},X_{-2m-1},\cdots,X_{2m+1},X_{2m+2},z_{1},\cdots,z_{2m+2}\\}$

are positive elements in $\\{x_{1},x_{2}\\}.$ The proof is finished. ∎

######  Remark 3.14.

In fact, by [15][12][13], the positivity in the cluster variables is obvious,
then applying the equation
$X_{1}z_{m}=q^{\frac{m}{2}}X_{1+m}+q^{-\frac{m}{2}}X_{1-m}$, we can deduce the
positivity in the elements $z_{m}$ for any $m\in\mathbb{N}$. Here, we give an
alternative proof without needing the explicit expansions of cluster
variables.

######  Theorem 3.15.

The sets $\mathcal{B},\mathcal{S}$ and $\mathcal{D}$ are
$\mathbb{Z}[q^{\pm\frac{1}{2}}]-$bases of the quantum cluster algebra
$\mathcal{A}_{q}(2,2)$.

###### Proof.

Note that if $\mathcal{B}$ is a $\mathbb{Z}[q^{\pm\frac{1}{2}}]-$basis of the
quantum cluster algebra $\mathcal{A}_{q}(2,2)$, then $\mathcal{S}$ and
$\mathcal{D}$ are naturally $\mathbb{Z}[q^{\pm\frac{1}{2}}]-$bases of quantum
cluster algebra $\mathcal{A}_{q}(2,2)$ because there are have unipotent matrix
transformations between $\\{z_{n}\mid n\in\mathbb{N}\\}$, $\\{s_{n}\mid
n\in\mathbb{N}\\}$ and $\\{z^{n}\mid n\in\mathbb{N}\\}$. In the following, we
will focus on the set $\mathcal{B}$ and prove it is a
$\mathbb{Z}[q^{\pm\frac{1}{2}}]-$basis of the quantum cluster algebra
$\mathcal{A}_{q}(2,2)$.

By Proposition 3.10, we obtain that any element of the quantum cluster algebra
$\mathcal{A}_{q}(2,2)$ can be a $\mathbb{Z}[q^{\pm\frac{1}{2}}]-$combination
of the elemnets in the set $\mathcal{B}$. Thus we only need to prove the
elemnets in $\mathcal{B}$ are $\mathbb{Z}[q^{\pm\frac{1}{2}}]-$independent.

By Remark 3.5, we know that the elements c associated to these minimal non-
zero terms $f(q^{\frac{1}{2}},q^{-\frac{1}{2}})X^{\textbf{c}}$ in the
expansion of the elements in the set $\mathcal{B}$ are different from each
other. Now we suppose that a finite
$\mathbb{Z}[q^{\pm\frac{1}{2}}]-$combination of the elemnets in the set
$\mathcal{B}$ is equal to $0.$ Let $S\subset\mathbb{Z}^{2}$ be the set of all
$\alpha$ such that the corresponding element occurs with a non-zero
coefficient in this $\mathbb{Z}[q^{\pm\frac{1}{2}}]-$combination. If $S$ is
non-empty, pick a minimal element $\alpha\in S$, by Remark 3.2 and Remark 3.5,
we know that $X^{\alpha}$ does not occur in the expansion of any other element
in above equation which gives a contradiction. This completes the proof of the
theorem. ∎

Set

$\mathcal{B^{\prime}}=\\{q^{-\frac{ab}{2}}X^{a}_{m}X^{b}_{m+1}|m\in\mathbb{Z},(a,b)\in\mathbb{Z}^{2}_{\geq
0}\\}\cup\\{z_{n}|n\in\mathbb{N}\\}$
$\mathcal{S^{\prime}}=\\{q^{-\frac{ab}{2}}X^{a}_{m}X^{b}_{m+1}|m\in\mathbb{Z},(a,b)\in\mathbb{Z}^{2}_{\geq
0}\\}\cup\\{s_{n}|n\in\mathbb{N}\\}$
$\mathcal{D^{\prime}}=\\{q^{-\frac{ab}{2}}X^{a}_{m}X^{b}_{m+1}|m\in\mathbb{Z},(a,b)\in\mathbb{Z}^{2}_{\geq
0}\\}\cup\\{z^{n}|n\in\mathbb{N}\\}.$

Then we can obtain the following corollary.

######  Corollary 3.16.

The sets $\mathcal{B^{\prime}},\mathcal{S^{\prime}}$ and
$\mathcal{D^{\prime}}$ are bar-invariant
$\mathbb{Z}[q^{\pm\frac{1}{2}}]-$bases of the quantum cluster algebra
$\mathcal{A}_{q}(2,2)$.

##  4\. An representation-theoretic interpretation of the element $s_{n}$

Recall we denote by $R_{p}(n)$ the indecomposable regular modules with
dimension vector $(n,n)$ for $n\geq 1$ and some $p\in\mathbb{P}^{1}_{k}$ of
degree 1. In this section, we will prove that $s_{n}$ is equal to
$X_{n\delta}$ for every $n\in\mathbb{N}.$ The following proposition shows the
Laurent expansion of $X_{m}$ in $A_{q}(2,2)$.

######  Proposition 4.1.

([12],[13]) For every $n\geq 0$, we have

(4.1) $\displaystyle X_{-n}$
$\displaystyle\textstyle=X^{(n+2,-n-1)}+\sum_{p+r\leq n}{n-r\brack
p}_{q}{n+1-p\brack r}_{q}X^{(2r-n,2p-n-1)};$ (4.2) $\displaystyle X_{n+3}$
$\displaystyle\textstyle=X^{(-n-1,n+2)}+\sum_{p+r\leq n}{n-r\brack
p}_{q}{n+1-p\brack r}_{q}X^{(2p-n-1,2r-n)}.$

######  Lemma 4.2.

For every $n\in\mathbb{N}$, we have

$s_{n}=q^{\frac{n}{2}}X_{1}X_{n+3}-q^{\frac{n}{2}+1}X_{2}X_{n+2}.$

###### Proof.

It is easy to check that
$s_{1}=q^{\frac{1}{2}}X_{1}X_{4}-q^{\frac{3}{2}}X_{2}X_{3}.$ Assume that it
holds for $n\leq k,$ then

$\displaystyle s_{k+1}$ $\displaystyle=$ $\displaystyle s_{k}s_{1}-s_{k-1}$
$\displaystyle=$
$\displaystyle(q^{\frac{k}{2}}X_{1}X_{k+3}-q^{\frac{k}{2}+1}X_{2}X_{k+2})s_{1}-(q^{\frac{k-1}{2}}X_{1}X_{k+2}-q^{\frac{k+1}{2}}X_{2}X_{k+1})$
$\displaystyle=$ $\displaystyle
q^{\frac{k}{2}}X_{1}(q^{\frac{1}{2}}X_{k+4}+q^{-\frac{1}{2}}X_{k+2})-q^{\frac{k}{2}+1}X_{2}(q^{\frac{1}{2}}X_{k+3}+q^{-\frac{1}{2}}X_{k+1})$
$\displaystyle-$
$\displaystyle(q^{\frac{k-1}{2}}X_{1}X_{k+2}-q^{\frac{k+1}{2}}X_{2}X_{k+1})$
$\displaystyle=$ $\displaystyle
q^{\frac{k+1}{2}}X_{1}X_{k+4}-q^{\frac{k+1}{2}+1}X_{2}X_{k+3}.$

∎

Denote the quantum binomial coefficients ${n\choose
r}_{q}=\frac{(q^{n}-1)(q^{n-1}-1)\cdots(q^{n-r+1}-1)}{(q^{r}-1)\cdots(q-1)}$
and take ${n\choose 0}_{q}=1$ for any $n\in\mathbb{Z}$, ${n\choose l}_{q}=0$
for any $l<0,$ and ${n\choose l}_{q}=0$ for any $0\leq n<l.$ Then we have the
following theorem proved in [15]:

######  Theorem 4.3.

[15, Theorem 4.6] Let $\textbf{e}=(a,b)$ for $(a,b)\in\mathbb{Z}^{2}_{\geq
0}$. Then for $n\geq 1,$

$|Gr_{\textbf{e}}(R_{p}(n))|={n-a\choose n-b}_{q^{2}}{b\choose a}_{q^{2}}.$

######  Proposition 4.4.

$X_{R_{p}(n)}=s_{n}$ for $n\in\mathbb{N}$.

###### Proof.

By the $q$-deformation of the Caldero-Chapoton formula and Theorem 4.3, we
have

$\displaystyle X_{R_{p}(n)}$
$\displaystyle=\sum_{(a,b)}q^{-\frac{1}{2}d_{\bf(a,b)}^{R_{p}(n)}}|Gr_{(a,b)}(R_{p}(n))|X^{(n-2b,2a-n)}$
$\displaystyle=\sum_{(a,b)}q^{(a-b)(n-b)}{n-a\choose
n-b}_{q^{2}}q^{(a-b)a}{b\choose a}_{q^{2}}X^{(n-2b,2a-n)}$
$\displaystyle=\sum_{(a,b)}{n-a\brack n-b}_{q}{b\brack a}_{q}X^{(n-2b,2a-n)}$
$\displaystyle=\sum_{p+r\leq n}{n-r\brack p}_{q}{n-p\brack
r}_{q}X^{(2p-n,2r-n)}.$

On the other hand, by Proposition 4.1, we have

$\displaystyle q^{\frac{n}{2}}X_{1}X_{n+3}$
$\displaystyle=q^{\frac{n}{2}}X_{1}X^{(-n-1,n+2)}+q^{\frac{n}{2}}X_{1}\sum_{p+r\leq
n}{n-r\brack p}_{q}{n+1-p\brack r}_{q}X^{(2p-n-1,2r-n)}$
$\displaystyle=q^{n+1}X^{(-n,n+2)}+\sum_{p+r\leq n}{n-r\brack
p}_{q}{n+1-p\brack r}_{q}q^{r}X^{(2p-n,2r-n)}.$

And

$\displaystyle q^{\frac{n}{2}+1}X_{2}X_{n+2}$
$\displaystyle=q^{\frac{n}{2}+1}X_{2}X^{(-n,n+1)}+q^{\frac{n}{2}+1}X_{2}\sum_{p+r\leq
n-1}{n-1-r\brack p}_{q}{n-p\brack r}_{q}X^{(2p-n,2r+1-n)}$
$\displaystyle=q^{n+1}X^{(-n,n+2)}+\sum_{p+r\leq n-1}{n-1-r\brack
p}_{q}{n-p\brack r}_{q}q^{n-p+1}X^{(2p-n,2r+2-n)}$
$\displaystyle=q^{n+1}X^{(-n,n+2)}+\sum_{p+r\leq n}{n-r\brack p}_{q}{n-p\brack
r-1}_{q}q^{n-p+1}X^{(2p-n,2r-n)}.$

Note that it is easy to confirm the following identity:

${n+1-p\brack r}_{q}q^{r}={n-p\brack r}_{q}+{n-p\brack r-1}_{q}q^{n-p+1}.$

Then by Lemma 4.2, we have

$\displaystyle s_{n}$ $\displaystyle=$ $\displaystyle
q^{\frac{n}{2}}X_{1}X_{n+3}-q^{\frac{n}{2}+1}X_{2}X_{n+2}$ $\displaystyle=$
$\displaystyle\sum_{p+r\leq n}{n-r\brack p}_{q}{n-p\brack
r}_{q}X^{(2p-n,2r-n)}$ $\displaystyle=$ $\displaystyle X_{R_{p}(n)}.$

∎

## Acknowledgements

The authors would like to thank Professor Jie Xiao for many helpful
discussions.

## References

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# Spin-selective equilibration among integer quantum Hall edge channels

Giorgio Nicolí<EMAIL_ADDRESS>Solid State Physics Laboratory, ETH
Zürich, 8093 Zürich, Switzerland  Christoph Adam  Solid State Physics
Laboratory, ETH Zürich, 8093 Zürich, Switzerland  Marc P. Röösli  Solid State
Physics Laboratory, ETH Zürich, 8093 Zürich, Switzerland  Peter Märki  Solid
State Physics Laboratory, ETH Zürich, 8093 Zürich, Switzerland  Jan
Scharnetzky  Solid State Physics Laboratory, ETH Zürich, 8093 Zürich,
Switzerland  Christian Reichl  Solid State Physics Laboratory, ETH Zürich,
8093 Zürich, Switzerland  Werner Wegscheider  Solid State Physics Laboratory,
ETH Zürich, 8093 Zürich, Switzerland  Thomas M. Ihn  Solid State Physics
Laboratory, ETH Zürich, 8093 Zürich, Switzerland  Klaus Ensslin  Solid State
Physics Laboratory, ETH Zürich, 8093 Zürich, Switzerland

###### Abstract

The equilibration between quantum Hall edge modes is known to depend on the
disorder potential and the steepness of the edge. Modern samples with higher
mobilities and setups with lower electron temperatures call for a further
exploration of the topic. We develop a framework to systematically measure and
analyze the equilibration of many (up to 8) integer edge modes. Our results
show that spin-selective coupling dominates even for non-neighboring channels
with parallel spin. Changes in magnetic field and bulk density let us control
the equilibration until it is almost completely suppressed and dominated only
by individual microscopic scatterers. This method could serve as a guideline
to investigate and design improved devices, and to study fractional and other
exotic states.

Quantum Hall devices remain paradigmatic for research on topological systems
[1]. The Hall regime is accessed with a quantizing magnetic field
perpendicular to a two-dimensional electron gas (2DEG) [2, 3]. Dissipationless
non-equilibrium currents flow in one-dimensional chiral channels along the
edge of the system in response to an external voltage [4, 5, 6, 7],
experiencing inter-edge-state scattering in the presence of a background
disorder potential [8, 9, 10, 11, 12, 13, 14]. Equilibration phenomena among
non-equilibrium edge currents are not yet fully understood despite the rich
history of past experiments on semiconducting devices.

Haug and coworkers found length-dependent equilibration in spin-degenerate
quantum Hall systems with top gates acting as partially transmitting barriers
[15, 9, 6], but did not report about spin-related effects. Later, Müller found
that in the presence of a background disorder potential, spin-orbit
interactions mediate the equilibration between spin-polarized edge modes by
allowing charge carriers to flip their spin [11, 16]. The continuous
advancements in material technologies thus motivated a revival of
equilibration experiments [17, 18, 19, 20].

Local probe experiments by Weis et al. already showed the complexity of the
microscopic recostruction of the edge potential [21, 22, 23, 24]. Further
details on the edge could be revealed assuming that the presence of an
incompressible region of a specific filling factor between two channels
implies weak equilibration.

Quantum Hall edge state equilibration experiments gradually expanded to the
fractional regime too, often finding non-trivial edge recontructions and
current distributions [25, 26, 27]. Graphene is another mature platform for
quantum Hall experiments unraveling the role of valley and spin degrees of
freedom in equilibration phenomena [28, 29, 30, 31].

In this manuscript, we address the question of inter-edge-mode scattering in
state-of-the-art devices using electronic transport experiments. The design
that we use is inspired by historically well-known experiments, where edge
channels can be reflected and transmitted with barrier gates to obtain well-
controlled out-of-equilibrium population of edge modes [15, 32, 33, 16, 34].
We study how the excitation of selected integer edge modes is redistributed as
they co-propagate and extract the strength of pair-wise coupling among many
channels (up to 8). We find that the spin of the modes determines the
equilibration at low enough fields, spin-selectively coupling even distant
channels in contrast with many findings from the past [8, 11, 6, 12]. For
larger fields the equilibration is almost completely suppressed and mesoscopic
impurities dominate the weak equilibration between spin-split channels.

Figure 1: (a) Schematic device structure and measurement setup, not to scale.
A current $I_{\mathrm{ac}}=$500\text{\,}\mathrm{pA}$$ flows through the device
between injector and ground contacts, at potentials $\mu_{\mathrm{I}}$ and
$\mu=0$ respectively. Two top gates act as barriers downstream of the injector
(violet gate) and upstream of the detector (yellow gate) contacts. They are
controlled via two dc voltage sources $V_{\mathrm{I}}$ and
$V_{\mathrm{D}}$(not shown). A side gate (green) laterally depletes the 2DEG
along a length $L_{\mathrm{eq}}=$35\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.
(b) 2-terminal conductance measured through the device as a function of the
left barrier gate voltage $V_{\mathrm{I}}$. The right barrier gate is grounded
[35]. The magnetic field $B$ and bulk density $n_{\mathrm{B}}$, controlled
with the back gate, are stepped together to ensure constant bulk filling
factor $\nu_{\mathrm{B}}=5$. Diagonal dashed lines indicate regions of
constant conductance, corresponding to a quantized local filling factor
$\nu_{\mathrm{I}}$ below the injector gate. Figure 2:  (a) Equilibration
measurement performed at the star-shaped symbol in Fig 1(b). The red circles
indicate the data points required to calculate a set of $\mu_{i}$ following
Eq. (3). (b)–(f) Electrochemical potential $\widetilde{\mu}_{i}$ of the modes
for integer bulk filling factors $\nu_{\mathrm{B}}=$84\text{\,}$$ after the
equilibration path. Matching labels indicate the spin and index of the
channels. The back gate voltage $V_{\mathrm{BG}}$ (controlling
$n_{\mathrm{B}}$) and the magnetic field $B$ are simultaneously stepped to fix
$\nu_{\mathrm{B}}$ during each experiment, similar to Fig. 1(b). The shaded
regions in figures (a)–(d) indicate the range where coupling parameters have
been extracted [see text and Fig. 3(c)–(f)]. Figure 3: (a) Density profile at
the edge of the 2DEG. The unperturbed curve (grey dashed line) was calculated
following Ref. [36]. The reconstructed density profile is sketched on top with
colored shaded regions indicating the compressible stripes. (b) Pictorial
representation of channels flowing along one edge of the device. A vertical
axis labels the coupling terms represented as horizontal dashed arrows. Even
though in our model we consider pair-wise coupling terms $\gamma_{ij}$ between
all channels, the sketch shows only terms involving the two outermost channels
(1 in blue and 2 in orange). (c)–(f) Coupling parameters for different bulk
filling factors. The arrows indicate the spin alignment of the channels
coupled by each term. The values reported here are averaged in the grey shaded
areas in Fig. 2(b)–(e). The central magnetic field for each shaded area is
reported in the figure. The corresponding density range
$n_{\mathrm{B}}=$1.912.03\text{\,}\times\mathrm{1}\mathrm{0}^{15}\mathrm{c}\mathrm{m}^{-2}$$
is the same for each plot.

Our device is an MBE-grown [Al]GaAs heterostructure equipped with a patterned
back gate located roughly $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ below the
plane of the 2DEG [37, 38]. We lithographically defined top gates as indicated
in Fig. 1(a). While the back gate tunes the whole device, we locally control
the electron density with the injector and detector gates, creating tunable
barriers in front of the injector and detector contacts. An additional gate
located between the barriers on the side of the device pushes the 2DEG away
from the physical edge of the mesa and creates a smooth electrostatic edge. A
dilution refrigerator lowers the device temperature to
$\lesssim$30\text{\,}\mathrm{mK}$$. From previous characterization, we expect
the 2DEG to thermalize with the lattice in our setup [39, 40].

At integer filling factors of the 2DEG, an external magnetic field $B$ induces
edge conduction with the chirality indicated in Fig. 1(a). Following the
Landauer-Büttiker formalism [41, 42], the number of channels transmitted by
each barrier depends on the local filling factors

$\nu_{\mathrm{I,D}}=\frac{hn_{\mathrm{I,D}}}{eB}\leq\nu_{\mathrm{B}},$ (1)

where $\nu_{\mathrm{I}}$ and $\nu_{\mathrm{D}}$ are the filling factors of the
2DEG under the injector and detector gates respectively, when we fix the local
densities to $n_{\mathrm{I}}$ and $n_{\mathrm{D}}$. We tune the system such
that $\nu_{\mathrm{I}}$, $\nu_{\mathrm{D}}$ and the bulk filling factor
$\nu_{\mathrm{B}}$ have integer values to perform the experiments in a
controlled way and suppress bulk equilibration [19]. We can route channels
carrying different electrochemical potentials to flow along the co-propagation
path [$L_{\mathrm{eq}}$ in Fig. 1(a)]. Measuring the longitudinal resistance
across this path [6, 11] or the potential of the detector with respect to
ground (our case), will yield information about the strength of the
equilibration processes among the channels.

We measure the two-terminal conductance $G_{\mathrm{2T}}$ as a function of the
barrier voltage $V_{\mathrm{I}}$ while an ac current $I_{\mathrm{ac}}$ flows
from the injector to ground and the other barrier is fully transparent. Figure
1(b) shows the result measured at $\nu_{\mathrm{B}}=5$. Plateaus of constant
conductance matching integer multiples of $e^{2}/h$ are found as the barrier
gate voltage decreases, (white dashed lines). Each diagonal feature
corresponds to a fixed number of channels transmitted through the barrier
region. We repeat the same experiment with the detector gate and for different
bulk filling factors to observe the transmission characteristics of both
barriers (not shown).

After flowing through the injector barrier, the transmitted channels will have
a different electrochemical potential $\mu_{\mathrm{I}}$ compared to the
reflected modes on the other side of the barrier, coming from the grounded
contact [cf. red versus yellow and white lines in Fig. 1(a)]. However,
measuring the transverse voltage $V_{\mathrm{xy}}$ between the detector and
ground will reveal no details about inter-mode coupling along the path if all
channels equilibrate in the detector ($\nu_{\mathrm{D}}=\nu_{\mathrm{B}}$)
[43]. The contact settles at the electrochemical potential

$\mu_{\mathrm{D}}=\frac{1}{\nu_{\mathrm{D}}}\sum_{i=1}^{\nu_{\mathrm{D}}}\mu_{i},$
(2)

where $\mu_{i}$ is the potential of an individual channel $i$ when entering
the detector. In the integer regime, all channels have transmission of one and
contribute equally to the potential of the contact. When the detector barrier
allows only selected modes to be transmitted
($\nu_{\mathrm{D}}<\nu_{\mathrm{B}}$), measurements of the transverse
resistance will yield

$R^{(\nu_{\mathrm{D}})}_{\mathrm{xy}}=\frac{V_{\mathrm{xy}}}{I_{\mathrm{ac}}}=\frac{h}{e^{2}}\frac{\mu_{\mathrm{D}}}{\nu_{\mathrm{I}}\mu_{\mathrm{I}}}=\frac{h}{e^{2}}\frac{1}{\nu_{\mathrm{I}}\nu_{\mathrm{D}}}\displaystyle\sum_{i=1}^{\nu_{\mathrm{D}}}\frac{\mu_{i}}{\mu_{\mathrm{I}}}.$
(3)

The total equilibration between the channels does not depend on the specific
tuning of the barriers, but rather on the edge potential along the propagation
path, on the mesoscopic disorder background and on the length of co-
propagation. Equilibration among channels under the detector gate does not
affect the measurements [43]. If the external current is completely injected
in the outermost channel ($\nu_{\mathrm{I}}=1$), an out-of-equilibrium
population of spin-polarized electrons is built up. This channel can
equilibrate either with other channels of the same spin polarization, or with
channels of the opposite spin, if spin-flips are involved.

We devised a measurement protocol to extract the electrochemical potential of
the channels at the detector. We measure $R^{\nu_{\mathrm{D}}}_{\mathrm{xy}}$
for different values of $\nu_{\mathrm{D}}$ while the injector barrier is tuned
to $\nu_{\mathrm{I}}=1$. A system of equations of the form of Eq. (3) with
values $1\leq\nu_{\mathrm{D}}\leq\nu_{\mathrm{B}}$ describes the measurements
[see Fig. 2(a)]. We can solve the system to find the normalized
electrochemical potentials of the channels
$\widetilde{\mu}_{i}=\mu_{i}/\mu_{\mathrm{I}}$, with the initial conditions
$\widetilde{\mu}_{1}^{0}=1$, and $\widetilde{\mu}_{j}^{0}=0$ for $j\neq 1$

Figures 2(b)–(f) show the results of the analysis for different
$\nu_{\mathrm{B}}$ and in a range of magnetic fields and bulk densities. In
Fig. 2(b)–(d), we observe that electrons preferentially equilibrate with
states of the same spin, leaving channel 1 to occupy states in modes 3 and 5.
Channels labeled with even numbers were mostly decoupled from the only
initially excited channel and their potential is closer to the bottom of our
energy scale.

In particular, when $\nu_{\mathrm{B}}=8$ or $6$, two bundles of channels with
opposite spin are resolved and well separated in energy. Even though the clear
separation between the two bundles is not visible for the case of
$\nu_{\mathrm{B}}=7$, also here the system favors spin-selective
equilibration. The presence of reproducible fluctuations is likely due to
impurities occurring on mesoscopic length scales, modulating the coupling
between the modes [44].

If the three spin-up channels in Fig. 2(b) completely equilibrate while the
others do not participate, we expect to find
$\widetilde{\mu}_{1,3,5}=1/3\simeq 0.33$, a case nearly reached at the highest
densities. Conversely, if all channels equilibrate, then
$\widetilde{\mu}_{i}=1/8=0.125$ for all of them, which is nearly the case at
the lowest densities.

As the number of channels in the bulk decreases with increasing external
magnetic field, so does the coupling between them. Figure 2(e)
($\nu_{\mathrm{B}}=5$) shows that electrons in channels 1 and 3 are not fully
equilibrating along $L_{\mathrm{eq}}$, contrary to the cases with
$\nu_{\mathrm{B}}>5$. We observe that the coupling becomes weaker for larger
$B$, but spin-selective equilibration still remains the favored process. In
Fig. 2(f) ($\nu_{\mathrm{B}}=4$) the coupling weakens to the point where
$\widetilde{\mu}_{1}\approx\widetilde{\mu}_{1}^{0}=1$ for the whole range. Few
mesoscopic features increase the coupling between the two modes in the lowest
Landau level, which requires some spin-flip mechanism. Spin-selective
equilibration is not observed in this case. Performing the same experiments
over a distance
$L`\prime_{\mathrm{eq}}=$535\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ reveals
full equilibration irrespective of the spin alignment, although the innermost
channel remains decoupled [43].

The density profile at the edge, sketched in the spirit of Ref. [45] in Fig.
3(a), guides us in understanding the results of Fig. 2. The edge channels
represent discrete conducting regions located where the density has a non-zero
gradient. Incompressible stripes with a fixed filling factor separate
compressible regions that form at the edge as a result of screening and
interactions in the presence of an external $B$ field [45, 46, 36, 47].
Decreasing $\nu_{\mathrm{B}}$ at constant density, by increasing $B$, means
that a smaller number of channels spans the density profile, pushing the
innermost channels further into the bulk [26, 48, 21]. Increasing $B$ and
$n_{\mathrm{B}}$ while keeping $\nu_{\mathrm{B}}$ constant instead results in
wider incompressible stripes and a larger separation between the channels. The
magnetic length $\ell_{\mathrm{B}}$, and consequently the spatial extent of
the wavefunction of the edge modes, decreases for stronger fields. Since
charge transfer between channels requires wavefunction overlap, a larger
distance and stronger confinement can quickly suppress the tunneling
probability amplitude.

Electrons can in principle tunnel from one channel to any other, conserving or
flipping their spin. The energy transfer between modes can be approximated
with a system of rate equations of the form [43]

$\frac{d\mu_{i}}{dx}=-\frac{1}{2}\sum_{j\neq i}\gamma_{ij}(\mu_{i}-\mu_{j}).$
(4)

Here the potentials $\mu_{i}$ are intended to be position dependent along the
equilibration path between injector and detector. The terms
$\gamma_{ij}=\gamma_{ji}$ model a uniform coupling between channels $i$ and
$j$ [see Fig. 3(b)]. These parameters encapsulate any equilibration process in
our model, giving us a quantity related to the average equilibration lengths
$\ell_{ij}^{\mathrm{eq}}=\gamma_{ij}^{-1}$ between channels $i$ and $j$. We
can numerically calculate the whole set of $\gamma_{ij}$ by performing an
equal amount of independent measurements at the detector, each time setting
the barrier filling factors to integer values such that $\nu_{I}\leq\nu_{D}$
and $\nu_{I},\nu_{D}<\nu_{B}$ [43].

Starting with $\nu_{\mathrm{B}}=8$ in Fig. 3(c), we observe that spin-
conserving coupling terms dominate, while spin-flip terms can be more than one
order of magnitude smaller. Spin-selective tunneling couples not only the
spatially closest channels with parallel spin (channels 1 and 3), but also
terms like $\gamma_{15}$ and $\gamma_{26}$ are much larger than spin-flip
terms coupling nearest-neighbors. This shows that it is more likely for
electrons to tunnel a larger distance without flipping their spin rather than
tunneling through a thinner barrier undergoing a spin-flip event.

Increasing the field and decreasing $\nu_{\mathrm{B}}$ at constant density
progressively decouples the channels. In Fig. 3(d) and (e) we observe a
reduction of the coupling, in particular of the long-distance terms
$\gamma_{15}$ and $\gamma_{26}$. In Fig. 3(f) the trend continues and also
short-range spin-selective coefficients decrease. Finally, for
$\nu_{\mathrm{B}}\leq 4$, all the integer channels are mostly decoupled,
either too far removed towards the bulk or limited by the frequency of spin-
flip events.

In this manuscript, we analyzed our data based on the well-established edge
channel picture of the integer quantum Hall effect, finding that channels with
parallel spin selectively couple with each other, while flipping the spin of
electrons is much less likely. At low enough fields, spin-conserving tunneling
even couples modes separated by several compressible and incompressible
stripes instead of only neighboring channels with parallel spin. In general,
the equilibration process is influenced by experimental parameters, like
magnetic field and temperature [49], and by sample properties, such as
material quality and heterostructure design. Controlling the transfer of
particles between channels could lead to the use of edge modes as spin rails
to transport well-defined magnetic moments in quantum computation experiments
[50, 51, 52]. The presence of spin-selective signatures at low field would
help to integrate such a technology with others that do not tolerate or
require high fields.

In the fractional regime, a precise knowledge of the equilibration length is
sought after to improve experiments involving interferometers and other
confined systems [53, 54], anyonic statistics [55, 56], the thermal
conductance of exotic states [57, 58, 59] and the complex edge reconstruction
associated to fractional states like $2/3$ [60, 61] and $5/2$ [62, 63].
Investigating the fractional quantum Hall regime is a natural next step and we
believe that the techniques established in this paper could complement
investigations of edge reconstruction and the formation of stripes with
fractional filling factor in a variety of materials [27].

###### Acknowledgements.

We thank M. Shayegan for illuminating discussions. We acknowledge support from
the ETH FIRST laboratory and financial support from the National Center of
Competence in Research ”QSIT—Quantum Science and Technology”.

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