$\displaystyle=\lim_{m\rightarrow\infty}\left\|{\displaystyle\sum\limits_{n=0}^{m}}P_{k}\left(n\right)\left(M^{n}\mu-\pi\right)\right\|_{1}\leq\lim_{m\rightarrow\infty}{\displaystyle\sum\limits_{n=0}^{m}}P_{k}\left(n\right)\left\|M^{n}\mu-\pi\right\|_{1}$
$\displaystyle\leq\lim_{m\rightarrow\infty}{\displaystyle\sum\limits_{n=0}^{m}}P_{k}\left(n\right)2\left(1-\varepsilon\right)^{n}={\displaystyle\sum\limits_{n=0}^{\infty}}P_{k}\left(n\right)2\left(1-\varepsilon\right)^{n}$

The sequence $\left\\{a_{k}\right\\}_{k\in\mathbb{N}}$ of functions $a_{k}$
$:\mathbb{N\rightarrow}\left[0,\infty\right)\ $given by

$a_{k}\left(n\right)=P_{k}\left(n\right)2\left(1-\varepsilon\right)^{n}$

is bounded above by the function
$a:\mathbb{N\rightarrow}\left[0,\infty\right)$ given by

$a\left(n\right)=2\left(1-\varepsilon\right)^{n}$

The latter is summable with respect to the counting measure $\gamma$ on
$\mathbb{N}$. In addition, $\lim_{k\rightarrow\infty}a_{k}\left(n\right)=0$
for all $n\in\mathbb{N}$. By the Lebesgue Dominated Convergence Theorem,

$\lim_{k\rightarrow\infty}{\displaystyle\sum\limits_{n=0}^{\infty}}P_{k}\left(n\right)2\left(1-\varepsilon\right)^{n}=\lim_{k\rightarrow\infty}\int_{\mathbb{N}}a_{k}\left(n\right)\mathrm{d}\gamma\left(n\right)=0$

Therefore, $\lim_{k\rightarrow\infty}\left\|p_{k}-\pi\right\|_{1}=0$.
$\blacksquare$

####  B.3.2 Proof of Theorem 15

Given a menu $A$, with typical elements $i$, $j$ and $k$, we denote by
$P=\left[P\left(i\mid j\right)\right]_{i,j\in A}$ a
$\left|A\right|\times\left|A\right|$ a __ stochastic matrix __ such that
$P\left(i\mid j\right)$ is interpreted as the probability with which a system
moves from state $j$ to state $i$. Clearly, $P\left(\cdot\mid
j\right)\in\Delta\left(A\right)$ for all $j\in A$.

###### Definition 13

A stochastic matrix $P$ is _transitive_ if

$P\left(j\mid i\right)P\left(k\mid j\right)P\left(i\mid k\right)=P\left(k\mid
i\right)P\left(j\mid k\right)P\left(i\mid j\right)\qquad\forall i,j,k\in A$
(59)

Transitivity is known as the _Kolmogorov criterion_ in the Markov chains
literature (see, e.g., Kelly, 1979, p. 24) and as the _product rule_ in the
stochastic choice literature (Luce and Suppes, 1965, p. 341).

Transitivity is automatically satisfied if at least two of the three states
$i$, $j$, and $k$ in $A$ coincide. In fact,

  * •

if $i=j$, then

$\displaystyle P\left(j\mid i\right)P\left(k\mid j\right)P\left(i\mid
k\right)$ $\displaystyle=P\left(i\mid i\right)P\left(k\mid
i\right)P\left(i\mid k\right)$ $\displaystyle P\left(k\mid
i\right)P\left(j\mid k\right)P\left(i\mid j\right)$
$\displaystyle=P\left(k\mid i\right)P\left(i\mid k\right)P\left(i\mid
i\right)$

  * •

if $i=k$, then

$\displaystyle P\left(j\mid i\right)P\left(k\mid j\right)P\left(i\mid
k\right)$ $\displaystyle=P\left(j\mid i\right)P\left(i\mid
j\right)P\left(i\mid i\right)$ $\displaystyle P\left(k\mid
i\right)P\left(j\mid k\right)P\left(i\mid j\right)$
$\displaystyle=P\left(i\mid i\right)P\left(j\mid i\right)P\left(i\mid
j\right)$

  * •

if $j=k$, then

$\displaystyle P\left(j\mid i\right)P\left(k\mid j\right)P\left(i\mid
k\right)$ $\displaystyle=P\left(j\mid i\right)P\left(j\mid
j\right)P\left(i\mid j\right)$ $\displaystyle P\left(k\mid
i\right)P\left(j\mid k\right)P\left(i\mid j\right)$
$\displaystyle=P\left(j\mid i\right)P\left(j\mid j\right)P\left(i\mid
j\right)$

Therefore, transitivity can be restated as

$P\left(j\mid i\right)P\left(k\mid j\right)P\left(i\mid k\right)=P\left(k\mid
i\right)P\left(j\mid k\right)P\left(i\mid j\right)$

for all distinct$\ i$, $j$ and $k$ in $A$.303030This argument applies to any
function $P:A\times A\rightarrow\mathbb{R}$ and is independent of its
“diagonal” values $P\left(i\mid i\right)$.

The next result, which relates reversibility and transitivity, builds upon
Kolmogorov (1936) and Luce and Suppes (1965).

###### Proposition 24

Let $P$ be a positive stochastic matrix. The following conditions are
equivalent:

  1. (i)

$P$ is reversible under some $\pi\in\Delta\left(A\right)$;

  2. (ii)

$P$ is transitive.

In this case, given any $i\in A$, it holds

$\pi\left(j\right)=\frac{r\left(j\mid
i\right)}{{\displaystyle\sum\limits_{k\in A}}r\left(k\mid
i\right)}\qquad\forall j\in A$

where $r\left(j\mid i\right)=P\left(j\mid i\right)/P\left(i\mid j\right)$. In
particular, $\pi$ is unique.

Proof Assume that there exists $\pi\in\Delta\left(A\right)$ such that
$P\left(i\mid j\right)\pi\left(j\right)=P\left(j\mid
i\right)\pi\left(i\right)$ for all distinct $i,j\in A$ (note that this is
weaker that reversibility in that $\pi$ is not assumed to be positive), then

$P\left(i\mid j\right)\pi\left(j\right)=P\left(j\mid
i\right)\pi\left(i\right)\qquad\forall i,j\in A$ (60)

If $\pi\left(i^{\ast}\right)=0$ for some $i^{\ast}\in A$, then (being $P$
positive)

$\pi\left(j\right)=\frac{P\left(j\mid i^{\ast}\right)}{P\left(i^{\ast}\mid
j\right)}\pi\left(i^{\ast}\right)=0\qquad\forall j\in A$ (61)

But, this is impossible since ${\displaystyle\sum\limits_{j\in
A}}\pi\left(j\right)=1$. Hence, $\pi$ is positive. Moreover, by (61) we have

$\frac{\dfrac{P\left(j\mid i^{\ast}\right)}{P\left(i^{\ast}\mid
j\right)}}{{\displaystyle\sum\limits_{k\in A}}\dfrac{P\left(k\mid
i^{\ast}\right)}{P\left(i^{\ast}\mid k\right)}}=\frac{\dfrac{P\left(j\mid
i^{\ast}\right)}{P\left(i^{\ast}\mid
j\right)}\pi\left(i^{\ast}\right)}{{\displaystyle\sum\limits_{k\in
A}}\dfrac{P\left(k\mid i^{\ast}\right)}{P\left(i^{\ast}\mid
k\right)}\pi\left(i^{\ast}\right)}=\frac{\pi\left(j\right)}{{\displaystyle\sum\limits_{k\in
A}}\pi\left(k\right)}=\pi\left(j\right)\qquad\forall j\in A$

irrespective of the choice of $i^{\ast}\in A$. Hence, $\pi$ is unique.
Finally, given any $i,j,k\in A$, by (60) we have:

$\displaystyle\frac{\pi\left(j\right)}{\pi\left(i\right)}\frac{\pi\left(k\right)}{\pi\left(j\right)}\frac{\pi\left(i\right)}{\pi\left(k\right)}$
$\displaystyle=1\implies\frac{P\left(j\mid i\right)}{P\left(i\mid
j\right)}\frac{P\left(k\mid j\right)}{P\left(j\mid k\right)}\frac{P\left(i\mid
k\right)}{P\left(k\mid i\right)}=1$ $\displaystyle\implies\frac{P\left(j\mid
i\right)P\left(k\mid j\right)P\left(i\mid k\right)}{P\left(k\mid
i\right)P\left(j\mid k\right)P\left(i\mid j\right)}=1$ $\displaystyle\implies
P\left(j\mid i\right)P\left(k\mid j\right)P\left(i\mid k\right)=P\left(k\mid
i\right)P\left(j\mid k\right)P\left(i\mid j\right)$

So, transitivity holds.

Conversely, if transitivity holds, choose arbitrarily $i^{\ast}\in A$ and set

$\pi^{\ast}\left(j\right):=\frac{\dfrac{P\left(j\mid
i^{\ast}\right)}{P\left(i^{\ast}\mid
j\right)}}{{\displaystyle\sum\limits_{k\in A}}\dfrac{P\left(k\mid
i^{\ast}\right)}{P\left(i^{\ast}\mid k\right)}}=\dfrac{P\left(j\mid
i^{\ast}\right)}{P\left(i^{\ast}\mid j\right)}\zeta\qquad\forall j\in A$ (62)

where $1/\zeta={\displaystyle\sum\limits_{k\in A}}P\left(k\mid
i^{\ast}\right)/P\left(i^{\ast}\mid k\right)>0$. By transitivity, for all
$i,j\in A$,

$P\left(j\mid i\right)P\left(i^{\ast}\mid j\right)P\left(i\mid
i^{\ast}\right)=P\left(i^{\ast}\mid i\right)P\left(j\mid
i^{\ast}\right)P\left(i\mid j\right)$

and, since $P$ is positive,

$P\left(j\mid i\right)\frac{P\left(i\mid i^{\ast}\right)}{P\left(i^{\ast}\mid
i\right)}=P\left(i\mid j\right)\frac{P\left(j\mid
i^{\ast}\right)}{P\left(i^{\ast}\mid j\right)}$

Thus, for all $i,j\in A$,

$P\left(j\mid i\right)\frac{P\left(i\mid i^{\ast}\right)}{P\left(i^{\ast}\mid
i\right)}\zeta=P\left(i\mid j\right)\frac{P\left(j\mid
i^{\ast}\right)}{P\left(i^{\ast}\mid j\right)}\zeta$

In view of (62), reversibility with respect to $\pi^{\ast}$ holds (note that
$\pi^{\ast}$ is strictly positive). $\blacksquare\vskip 12.0pt plus 4.0pt
minus 4.0pt$

Proof of Theorem 15 To ease notation we write $\rho$ in place of
$\rho_{\mathrm{C}}$.

“If.” By Lemma 22, since $Q$ is quasi-positive, then $M$ is positive. By
assumption $M$ is reversible. But then, by Proposition 24, $M$ is transitive,
thus

$M\left(j\mid i\right)M\left(k\mid j\right)M\left(i\mid k\right)=M\left(k\mid
i\right)M\left(j\mid k\right)M\left(i\mid j\right)$

for all distinct$\ i$, $j$ and $k$ in $A$. By definition of $M$,

$\displaystyle Q\left(j\mid i\right)\rho\left(j\mid i\right)Q\left(k\mid
j\right)\rho\left(k\mid j\right)Q\left(i\mid k\right)\rho\left(i\mid k\right)$
$\displaystyle=Q\left(k\mid i\right)\rho\left(k\mid i\right)Q\left(j\mid
k\right)\rho\left(j\mid k\right)Q\left(i\mid j\right)\rho\left(i\mid j\right)$

for all distinct$\ i$, $j$ and $k$ in $A$. By symmetry and quasi-positivity of
$Q$, this implies

$\rho\left(j\mid i\right)\rho\left(k\mid j\right)\rho\left(i\mid
k\right)=\rho\left(k\mid i\right)\rho\left(j\mid k\right)\rho\left(i\mid
j\right)$

for all distinct$\ i$, $j$ and $k$ in $A$. Therefore, $\rho$ is transitive,
and by Theorem 4 it admits a binary value representation, so that the Neural
Metropolis Algorithm is value based.

“Only if.” If the Neural Metropolis Algorithm is value based, by definition,
$\rho$ admits a binary value representation, and by Theorem 4 it is
transitive. Thus

$\rho\left(j\mid i\right)\rho\left(k\mid j\right)\rho\left(i\mid
k\right)=\rho\left(k\mid i\right)\rho\left(j\mid k\right)\rho\left(i\mid
j\right)$

for all distinct$\ i$, $j$ and $k$ in $A$. Since $Q$ is symmetric and quasi-
positivite, then

$\displaystyle Q\left(j\mid i\right)\rho\left(j\mid i\right)Q\left(k\mid
j\right)\rho\left(k\mid j\right)Q\left(i\mid k\right)\rho\left(i\mid k\right)$
$\displaystyle=Q\left(k\mid i\right)\rho\left(k\mid i\right)Q\left(j\mid
k\right)\rho\left(j\mid k\right)Q\left(i\mid j\right)\rho\left(i\mid j\right)$

for all distinct$\ i$, $j$ and $k$ in $A$. By definition of $M$,

$M\left(j\mid i\right)M\left(k\mid j\right)M\left(i\mid k\right)=M\left(k\mid
i\right)M\left(j\mid k\right)M\left(i\mid j\right)$

for all distinct$\ i$, $j$ and $k$ in $A$. By Lemma 22, since $Q$ is quasi-
positive, then $M$ is positive. But then, by Proposition 24, $M$ is
reversible. $\blacksquare$

###  B.4 Section 5

Proposition 16 is a consequence of the following result.

###### Proposition 25

Given any positive $\rho_{\mathrm{C}}$ and any irreducible exploration matrix
$Q$, the transition matrix $M$ is primitive. Moreover, denoting by $\pi$ the
stationary distribution of $M$, it follows that

$\lim_{t\rightarrow\infty}p_{N_{t}}\left(j\right)=\frac{\pi\left(j\right)\bar{\tau}_{j}}{{\displaystyle\sum\limits_{k\in
A}}\pi\left(k\right)\bar{\tau}_{k}}\qquad\forall j\in A$

provided the distribution of $\mathrm{RT}_{i,j}$ has (strictly) positive
expectation, is continuous at $0$, and has no singular part for all
$\left(i,j\right)\in A^{2}$.

Proof of Proposition 25 To ease notation we write $\rho$ in place of
$\rho_{\mathrm{C}}$. The stochastic process $\left(I,J,T\right)$ produces
sequences

$\left(\underset{\text{state
}x_{0}}{\underbrace{j_{-1},i_{0}}},t_{0},\underset{\text{state
}x_{1}}{\underbrace{j_{0},i_{1}}},t_{1},...,\underset{\text{state
}x_{n}}{\underbrace{j_{n-1},i_{n}}},t_{n},\underset{\text{state
}x_{n+1}}{\underbrace{j_{n},i_{n+1}}},...\right)$

it then can be seen as a semi Markov chain with state space

$\mathcal{X}=\left\\{\left(j,i\right)\in A^{2}:Q\left(i\mid
j\right)>0\right\\}$

where state $x=\left(j,i\right)\in\mathcal{X}$ represents the comparison
between incumbent $j$ and proposal $i$. Since the comparison between $j$ and
$i$ produces incumbent $k=i$ with probability $\rho\left(i\mid j\right)$,
$k=j$ with probability $1-\rho\left(i\mid j\right)$, and all other incumbents
with probability $0$, then the probability of switching from comparison
$\left(j,i\right)$ to comparison $\left(k,h\right)$ is given by

$\mathbb{P}\left[X_{n+1}=\left(k,h\right)\mid
X_{n}=\left(j,i\right)\right]=\left(\delta_{i}\left(k\right)\rho\left(i\mid
j\right)+\delta_{j}\left(k\right)\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid k\right)$

In fact,

  * •

if $i=j$, then the comparison between $j$ and $i$ produces new incumbent $i$
for sure and

    * $\circ$

if $k=i$, then

$\displaystyle\mathbb{P}\left[X_{n+1}=\left(k,h\right)\mid
X_{n}=\left(i,i\right)\right]$ $\displaystyle=Q\left(h\mid i\right)$
$\displaystyle=\left(\underset{=1}{\underbrace{\delta_{i}\left(k\right)}}\rho\left(i\mid
j\right)+~{}\underset{=1}{\underbrace{\delta_{j}\left(k\right)}}\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid k\right)$

    * $\circ$

else $k\neq i$, and

$\displaystyle\mathbb{P}\left[X_{n+1}=\left(k,h\right)\mid
X_{n}=\left(i,i\right)\right]$ $\displaystyle=0$
$\displaystyle=\left(\underset{=0}{\underbrace{\delta_{i}\left(k\right)}}\rho\left(i\mid
j\right)+~{}\underset{=0}{\underbrace{\delta_{j}\left(k\right)}}\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid k\right)$

  * •

if $i\neq j$, then the comparison between $j$ and $i$ produces new incumbent
$k=i$ with probability $\rho\left(i\mid j\right)$ and $k=j$ with probability
$1-\rho\left(i\mid j\right)$ and

    * $\circ$

if $k=i$, then

$\displaystyle\mathbb{P}\left[X_{n+1}=\left(k,h\right)\mid
X_{n}=\left(j,i\right)\right]$ $\displaystyle=\rho\left(i\mid
j\right)Q\left(h\mid i\right)$
$\displaystyle=\left(\underset{=1}{\underbrace{\delta_{i}\left(k\right)}}\rho\left(i\mid
j\right)+~{}\underset{=0}{\underbrace{\delta_{j}\left(k\right)}}\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid k\right)$

    * $\circ$

if $k=j$, then, we have

$\displaystyle\mathbb{P}\left[X_{n+1}=\left(k,h\right)\mid
X_{n}=\left(j,i\right)\right]$ $\displaystyle=\left(1-\rho\left(i\mid
j\right)\right)Q\left(h\mid j\right)$
$\displaystyle=\left(\underset{=0}{\underbrace{\delta_{i}\left(k\right)}}\rho\left(i\mid
j\right)+~{}\underset{=1}{\underbrace{\delta_{j}\left(k\right)}}\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid k\right)$

    * $\circ$

else $k\neq i$ and $k\neq j$, thus

$\mathbb{P}\left[X_{n+1}=\left(k,h\right)\mid
X_{n}=\left(j,i\right)\right]=0=\left(\underset{=0}{\underbrace{\delta_{i}\left(k\right)}}\rho\left(i\mid
j\right)+~{}\underset{=0}{\underbrace{\delta_{j}\left(k\right)}}\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid k\right)$

Set

$\hat{M}\left(\left(k,h\right)\mid\left(j,i\right)\right)=\left(\delta_{i}\left(k\right)\rho\left(i\mid
j\right)+\delta_{j}\left(k\right)\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid
k\right)\qquad\forall\left(k,h\right),\left(j,i\right)\in\mathcal{X}$

Next we show that $\hat{M}$ is a _bona fide_ stochastic matrix. Given any
$\left(j,i\right)\in\mathcal{X}$,

$\displaystyle{\displaystyle\sum_{\left(k,h\right)\in\mathcal{X}}}\hat{M}\left(\left(k,h\right)\mid\left(j,i\right)\right)$
$\displaystyle={\displaystyle\sum_{\left(k,h\right)\in\mathcal{X}}}\left(\delta_{i}\left(k\right)\rho\left(i\mid
j\right)+\delta_{j}\left(k\right)\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid k\right)$
$\displaystyle={\displaystyle\sum_{\left(k,h\right)\in
A^{2}}}\left(\delta_{i}\left(k\right)\rho\left(i\mid
j\right)+\delta_{j}\left(k\right)\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid k\right)$
$\displaystyle={\displaystyle\sum_{h\in A}}\left({\displaystyle\sum_{k\in
A}}\left(\delta_{i}\left(k\right)\rho\left(i\mid
j\right)+\delta_{j}\left(k\right)\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid k\right)\right)$

where equality in the second line follows from the fact that if
$\left(k,h\right)\in A^{2}\setminus\mathcal{X}$ then $Q\left(h\mid
k\right)=0$. We will use this fact repeatedly. Now,

  * •

if $i=j$, then

${\displaystyle\sum_{k\in A}}\left(\delta_{i}\left(k\right)\rho\left(i\mid
j\right)+\delta_{j}\left(k\right)\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid k\right)=Q\left(h\mid i\right)$

hence

${\displaystyle\sum_{\left(k,h\right)\in\mathcal{X}}}\hat{M}\left(\left(k,h\right)\mid\left(j,i\right)\right)={\displaystyle\sum_{h\in
A}}Q\left(h\mid i\right)=1$

  * •

else

${\displaystyle\sum_{k\in A}}\left(\delta_{i}\left(k\right)\rho\left(i\mid
j\right)+\delta_{j}\left(k\right)\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid k\right)=\rho\left(i\mid
j\right)Q\left(h\mid i\right)+\left(1-\rho\left(i\mid
j\right)\right)Q\left(h\mid j\right)$

hence

$\displaystyle{\displaystyle\sum_{\left(k,h\right)\in\mathcal{X}}}\hat{M}\left(\left(k,h\right)\mid\left(j,i\right)\right)$
$\displaystyle={\displaystyle\sum_{h\in A}}\left(\rho\left(i\mid
j\right)Q\left(h\mid i\right)+\left(1-\rho\left(i\mid
j\right)\right)Q\left(h\mid j\right)\right)$ $\displaystyle=\rho\left(i\mid
j\right){\displaystyle\sum_{h\in A}}Q\left(h\mid
i\right)+\left(1-\rho\left(i\mid j\right)\right){\displaystyle\sum_{h\in
A}}Q\left(h\mid j\right)=1$

Next we show that if $\pi\in\Delta\left(A\right)$ and $M\pi=\pi$, then setting

$\hat{\pi}\left(j,i\right)=Q\left(i\mid
j\right)\pi\left(j\right)\qquad\forall\left(j,i\right)\in\mathcal{X}$

defines an element of $\Delta\left(\mathcal{X}\right)$ such that
$\hat{M}\hat{\pi}=\hat{\pi}$. Clearly

${\displaystyle\sum_{\left(j,i\right)\in\mathcal{X}}}\hat{\pi}\left(j,i\right)={\displaystyle\sum_{\left(j,i\right)\in\mathcal{X}}}Q\left(i\mid
j\right)\pi\left(j\right)={\displaystyle\sum_{\left(j,i\right)\in
A^{2}}}Q\left(i\mid j\right)\pi\left(j\right)={\displaystyle\sum_{j\in
A}}\left({\displaystyle\sum_{i\in A}}Q\left(i\mid
j\right)\pi\left(j\right)\right)=1$

Moreover, for all $\forall\left(k,h\right)\in\mathcal{X}$,

$\displaystyle\left(\hat{M}\hat{\pi}\right)_{\left(k,h\right)}$
$\displaystyle={\displaystyle\sum_{\left(j,i\right)\in\mathcal{X}}}\hat{M}\left(\left(k,h\right)\mid\left(j,i\right)\right)\hat{\pi}\left(j,i\right)=$
$\displaystyle={\displaystyle\sum_{\left(j,i\right)\in\mathcal{X}}}\left(\delta_{i}\left(k\right)\rho\left(i\mid
j\right)+\delta_{j}\left(k\right)\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid k\right)Q\left(i\mid
j\right)\pi\left(j\right)$
$\displaystyle={\displaystyle\sum_{\left(j,i\right)\in
A^{2}}}\left(\delta_{i}\left(k\right)\rho\left(i\mid
j\right)+\delta_{j}\left(k\right)\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(h\mid k\right)Q\left(i\mid
j\right)\pi\left(j\right)$ $\displaystyle=Q\left(h\mid
k\right){\displaystyle\sum_{\left(j,i\right)\in
A^{2}}}\left(\delta_{i}\left(k\right)\rho\left(i\mid
j\right)+\delta_{j}\left(k\right)\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(i\mid j\right)\pi\left(j\right)$

Next we show that, for all $k\in A$,

${\displaystyle\sum_{\left(j,i\right)\in
A^{2}}}\left(\delta_{i}\left(k\right)\rho\left(i\mid
j\right)+\delta_{j}\left(k\right)\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(i\mid
j\right)\pi\left(j\right)=\pi\left(k\right)$

obtaining $\left(\hat{M}\hat{\pi}\right)_{\left(k,h\right)}=Q\left(h\mid
k\right)\pi\left(k\right)=\hat{\pi}_{\left(k,h\right)}$. Indeed

$\displaystyle{\displaystyle\sum_{\left(j,i\right)\in
A^{2}}}\left(\delta_{i}\left(k\right)\rho\left(i\mid
j\right)+\delta_{j}\left(k\right)\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(i\mid j\right)\pi\left(j\right)$
$\displaystyle={\displaystyle\sum_{\left(j,i\right)\in
A^{2}}}\delta_{i}\left(k\right)\rho\left(i\mid j\right)Q\left(i\mid
j\right)\pi\left(j\right)+{\displaystyle\sum_{\left(j,i\right)\in
A^{2}}}\delta_{j}\left(k\right)Q\left(i\mid
j\right)\pi\left(j\right)-{\displaystyle\sum_{\left(j,i\right)\in
A^{2}}}\delta_{j}\left(k\right)\rho\left(i\mid j\right)Q\left(i\mid
j\right)\pi\left(j\right)$ $\displaystyle={\displaystyle\sum_{j\in
A}}\rho\left(k\mid j\right)Q\left(k\mid
j\right)\pi\left(j\right)+{\displaystyle\sum_{i\in A}}Q\left(i\mid
k\right)\pi\left(k\right)-{\displaystyle\sum_{i\in A}}\rho\left(i\mid
k\right)Q\left(i\mid k\right)\pi\left(k\right)$

now the central summand ${\displaystyle\sum_{i\in A}}Q\left(i\mid
k\right)\pi\left(k\right)$ is $\pi\left(k\right)$, we conclude by showing that
$M\pi=\pi$ implies that

${\displaystyle\sum_{j\in A}}\rho\left(k\mid j\right)Q\left(k\mid
j\right)\pi\left(j\right)={\displaystyle\sum_{i\in A}}\rho\left(i\mid
k\right)Q\left(i\mid k\right)\pi\left(k\right)$

in fact

$\displaystyle{\displaystyle\sum_{j\in A}}\rho\left(k\mid j\right)Q\left(k\mid
j\right)\pi\left(j\right)$ $\displaystyle={\displaystyle\sum_{j\neq
k}}\rho\left(k\mid j\right)Q\left(k\mid
j\right)\pi\left(j\right)+\rho\left(k\mid k\right)Q\left(k\mid
k\right)\pi\left(k\right)$ $\displaystyle={\displaystyle\sum_{j\neq
k}}M\left(k\mid j\right)\pi\left(j\right)+\rho\left(k\mid k\right)Q\left(k\mid
k\right)\pi\left(k\right)$ $\displaystyle=-M\left(k\mid
k\right)\pi\left(k\right)+{\displaystyle\sum_{j\in A}}M\left(k\mid
j\right)\pi\left(j\right)+\rho\left(k\mid k\right)Q\left(k\mid
k\right)\pi\left(k\right)$ $\displaystyle=-M\left(k\mid
k\right)\pi\left(k\right)+\pi\left(k\right)+\rho\left(k\mid
k\right)Q\left(k\mid k\right)\pi\left(k\right)$
$\displaystyle=\left(1-M\left(k\mid
k\right)\right)\pi\left(k\right)+\rho\left(k\mid k\right)Q\left(k\mid
k\right)\pi\left(k\right)$ $\displaystyle=\left({\displaystyle\sum_{i\neq
k}}\rho\left(i\mid k\right)Q\left(i\mid
k\right)\right)\pi\left(k\right)+\rho\left(k\mid k\right)Q\left(k\mid
k\right)\pi\left(k\right)$ $\displaystyle={\displaystyle\sum_{i\in
A}}\rho\left(i\mid k\right)Q\left(i\mid k\right)\pi\left(k\right)$

So far, we did not use the fact that $M$ is primitive, this is used next to
show that $\hat{M}$ is primitive too. By irreducibility of $M$, for all
$i,k\in A$, there exists of a finite sequence

$i=i_{0},i_{1},i_{2},...,i_{n}=k$ (63)

satisfying $i_{a+1}\neq i_{a}$ and $M\left(i_{a+1}\mid i_{a}\right)>0$, for
all $a=0,...,n-1$, by the definition of $M$, it follows that

$Q\left(i_{a+1}\mid i_{a}\right)>0$ (64)

for all $a=0,...,n-1$.

Consider any $\left(j,i\right),\left(k,h\right)\in\mathcal{X}$, and a chain
$i_{0},i_{1},i_{2},...,i_{n}$ satisfying (63) and (64). The derived chain

$\left(j,i\right)=\left(j,i_{0}\right),\left(i_{0},i_{1}\right),\left(i_{1},i_{2}\right)...,\left(i_{n-1},i_{n}\right),\left(i_{n},h\right)=\left(k,h\right)$

belongs to $\mathcal{X}$ because
$\left(j,i_{0}\right)=\left(j,i\right)\in\mathcal{X}$,
$\left(i_{n},h\right)=\left(k,h\right)\in\mathcal{X}$, and also
$\left(i_{a},i_{a+1}\right)\in\mathcal{X}$ because of (64). Now

$\hat{M}\left(\left(i_{0},i_{1}\right)\mid\left(j,i_{0}\right)\right)=\left(\delta_{i_{0}}\left(i_{0}\right)\rho\left(i_{0}\mid
j\right)+\delta_{j}\left(i_{0}\right)\left(1-\rho\left(i_{0}\mid
j\right)\right)\right)Q\left(i_{1}\mid i_{0}\right)$

which is strictly positive because, $\rho\left(i_{0}\mid
j\right)=\rho\left(i\mid j\right)>0$ (since $\rho$ is positive) and
$Q\left(i_{1}\mid i_{0}\right)>0$. Moreover,

$\hat{M}\left(\left(i_{a+1},i_{a+2}\right)\mid\left(i_{a},i_{a+1}\right)\right)=\left(\delta_{i_{a+1}}\left(i_{a+1}\right)\rho\left(i_{a+1}\mid
i_{a}\right)+\delta_{i_{a}}\left(i_{a+1}\right)\left(1-\rho\left(i_{a+1}\mid
i_{a}\right)\right)\right)Q\left(i_{a+2}\mid i_{a+1}\right)$

which is strictly positive for $a=0,...,n-2$, because $\rho\left(i_{a+1}\mid
i_{a}\right)>0$ and $Q\left(i_{a+2}\mid i_{a+1}\right)>0$. Finally,

$\hat{M}\left(\left(i_{n},h\right)\mid\left(i_{n-1},i_{n}\right)\right)=\left(\delta_{i_{n}}\left(i_{n}\right)\rho\left(i_{n}\mid
i_{n-1}\right)+\delta_{i_{n-1}}\left(i_{n}\right)\left(1-\rho\left(i_{n}\mid
i_{n-1}\right)\right)\right)Q\left(h\mid i_{n}\right)$

which is strictly positive, because $\rho\left(i_{n}\mid i_{n-1}\right)>0$ and
$Q\left(h\mid i_{n}\right)=Q\left(h\mid k\right)>0$.

This shows irreducibility of $\hat{M}$. Having proved irreducibility,
primitivity can be established by exhibiting a non-zero diagonal element in
the transition matrix $\hat{M}$. By definition, for all
$\left(j,i\right)\in\mathcal{X}$

$\hat{M}\left(\left(j,i\right)\mid\left(j,i\right)\right)=\left(\delta_{i}\left(j\right)\rho\left(i\mid
j\right)+\delta_{j}\left(j\right)\left(1-\rho\left(i\mid
j\right)\right)\right)Q\left(i\mid j\right)$

Positivity of $\rho$ guarantees that
$\delta_{j}\left(j\right)\left(1-\rho\left(i\mid j\right)\right)>0$, while
$Q\left(i\mid j\right)>0$ by definition of $\mathcal{X}$.

All the assumptions of Howard (1971) p. 713 are then satisfied by the semi-
Markov chain with embedded Markov chain
$\hat{M}\left(\left(k,h\right)\mid\left(j,i\right)\right)$ and holding times
$\hat{T}\left(\left(k,h\right)\mid\left(j,i\right)\right)=\mathrm{RT}_{i,j}$.313131Observe
that holding times are independent of the “next state” $\left(k,h\right)$
therefore “average waiting times” are just average holding times (see Howard,
1971, p. 691). Hence the limit as $t\rightarrow\infty$ of the probability
$\phi_{\left(j,i\right)}\left(t\right)$ with which comparison
$\left(j,i\right)$ is taking place at time $t$ is given by

$\displaystyle\phi_{\left(j,i\right)}$
$\displaystyle=\frac{\hat{\pi}\left(j,i\right)\tau_{\mathrm{RT}}\left(i\mid
j\right)}{{\displaystyle\sum_{\left(k,h\right)\in\mathcal{X}}}\hat{\pi}\left(k,h\right)\tau_{\mathrm{RT}}\left(h\mid
k\right)}=\frac{Q\left(i\mid
j\right)\pi\left(j\right)\tau_{\mathrm{RT}}\left(i\mid
j\right)}{{\displaystyle\sum_{\left(k,h\right)\in\mathcal{X}}}Q\left(h\mid
k\right)\pi\left(k\right)\tau_{\mathrm{RT}}\left(h\mid k\right)}$
$\displaystyle=\frac{Q\left(i\mid
j\right)\pi\left(j\right)\tau_{\mathrm{RT}}\left(i\mid
j\right)}{{\displaystyle\sum_{\left(k,h\right)\in A^{2}}}Q\left(h\mid
k\right)\pi\left(k\right)\tau_{\mathrm{RT}}\left(h\mid
k\right)}\qquad\forall\left(j,i\right)\in\mathcal{X}$

The same is true if $\left(j,i\right)\notin\mathcal{X}$, because in that case
$\phi_{\left(j,i\right)}\left(t\right)=0$ for all $t$ and $Q\left(i\mid
j\right)=0$. Thus

$\displaystyle\lim_{t\rightarrow\infty}p_{N_{t}}\left(j\right)$
$\displaystyle=\lim_{t\rightarrow\infty}{\displaystyle\sum_{i\in
A}}\phi_{\left(j,i\right)}\left(t\right)={\displaystyle\sum_{i\in
A}}\lim_{t\rightarrow\infty}\phi_{\left(j,i\right)}\left(t\right)={\displaystyle\sum_{i\in
A}}\phi_{\left(j,i\right)}$ $\displaystyle=\frac{{\displaystyle\sum_{i\in
A}}Q\left(i\mid j\right)\pi\left(j\right)\tau_{\mathrm{RT}}\left(i\mid
j\right)}{{\displaystyle\sum_{\left(k,h\right)\in A^{2}}}Q\left(h\mid
k\right)\pi\left(k\right)\tau_{\mathrm{RT}}\left(h\mid
k\right)}=\frac{\pi\left(j\right){\displaystyle\sum_{i\in A}}Q\left(i\mid
j\right)\tau_{\mathrm{RT}}\left(i\mid j\right)}{{\displaystyle\sum_{k\in
A}}\pi\left(k\right){\displaystyle\sum_{h\in A}}Q\left(h\mid
k\right)\tau_{\mathrm{RT}}\left(h\mid k\right)}$
$\displaystyle=\frac{\pi\left(j\right)\tau_{j}}{{\displaystyle\sum_{k\in
A}}\pi\left(k\right)\tau_{k}}$

as desired. $\blacksquare$

Consider, like in Cerreia-Vioglio et al. (2022, Section 2), a symmetric DDM
and an exploration matrix with off diagonal entries that are inversely
proportional to mean response times. This means that there exists $w>0$ such
that

$Q\left(i\mid j\right)=\frac{w}{\tau_{\mathrm{RT}}\left(i\mid
j\right)}\text{\qquad}\forall i\neq j$

so that

$Q\left(j\mid j\right)=1-{\displaystyle\sum\limits_{i\neq
j}}\frac{w}{\tau_{\mathrm{RT}}\left(i\mid j\right)}\text{\qquad}\forall j$

Also assume that $\tau_{\mathrm{RT}}\left(j\mid j\right)<\delta$ (small) for
all $j\in A$.323232The Drift Diffusion Model makes no predictions on the
response time for comparisons of an alternative with itself. But, it makes
sense to assume that the decision unit takes almost no time in realizing that
no actual comparison needs to be made. With this, for all $j\in A$,

$\displaystyle\bar{\tau}_{j}$ $\displaystyle={\displaystyle\sum\limits_{i\in
A}}Q\left(i\mid j\right)\tau_{\mathrm{RT}}\left(i\mid j\right)$
$\displaystyle={\displaystyle\sum\limits_{i\neq
j}}\frac{w}{\tau_{\mathrm{RT}}\left(i\mid
j\right)}\tau_{\mathrm{RT}}\left(i\mid
j\right)+\left(1-{\displaystyle\sum\limits_{i\neq
j}}\frac{w}{\tau_{\mathrm{RT}}\left(i\mid
j\right)}\right)\tau_{\mathrm{RT}}\left(j\mid
j\right)=\left(\left|A\right|-1\right)w+\delta_{j}$

with $0\leq\delta_{j}<\delta$. Since $\delta$ is small, then

$\bar{\tau}_{j}\approx\left(\left|A\right|-1\right)w$

irrespective of $j$, so that $\alpha\left(j\right)$ is approximately constant
in (36) and

$\lim_{t\rightarrow\infty}p_{N_{t}}\left(j\right)\approx\frac{e^{\lambda
v\left(j\right)}}{{\displaystyle\sum\limits_{k\in A}}e^{\lambda
v\left(k\right)}}\qquad\forall j\in A$

that is, the limit probability is approximately of the multinomial logit type.

## References

  * [1] S. P. Anderson, J. K. Goeree and C. A. Holt, Noisy directional learning and the logit equilibrium, _Scandinavian Journal of Economics_ , 106, 581-602, 2004.
  * [2] C. Baldassi, S. Cerreia-Vioglio, F. Maccheroni, M. Marinacci and M. Pirazzini, A behavioral characterization of the drift diffusion model and its multialternative extension for choice under time pressure, _Management Science_ , 66, 5075-5093, 2020.
  * [3] H. D. Block and J. Marschak, Random orderings and stochastic theories of response, in _Contributions to probability and statistics_ (I. Olkin et al., eds), Stanford University Press, Stanford, 1960.
  * [4] R. Bogacz, E. Brown, J. Moehlis, P. Holmes and J. D. Cohen, The physics of optimal decision making: a formal analysis of models of performance in two-alternative forced-choice tasks, _Psychological Review_ , 113, 700-765, 2006.
  * [5] R. Bogacz, M. Usher, J. Zhang and J. L. McClelland, Extending a biologically inspired model of choice: multi-alternatives, nonlinearity and value-based multidimensional choice, _Philosophical Transactions of the Royal Society of London B: Biological Sciences_ , 362, 1655-1670, 2007.
  * [6] S. Cerreia-Vioglio, F. Maccheroni, M. Marinacci and A. Rustichini, Multinomial logit processes and preference discovery: inside and outside the black box, _Review of Economic Studies_ , forthcoming.
  * [7] J. Ditterich, A comparison between mechanisms of multi-alternative perceptual decision making: ability to explain human behavior, predictions for neurophysiology, and relationship with decision theory, _Frontiers in Neuroscience_ , 4, 2010.
  * [8] W. Feller, _An introduction to probability theory and its applications_ , v. 1, Wiley, New York, 1968.
  * [9] P. C. Fishburn, _Utility theory for decision making_ , Wiley, New York, 1970.
  * [10] N. J. Higham, _Functions of matrices: theory and computation_ , Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2008.
  * [11] R. A. Howard. _Dynamic probabilistic systems_ , v. 2, Wiley, New York, 1971.
  * [12] N. L. Johnson, A. W. Kemp and S. Kotz,  _Univariate discrete distributions_ , Wiley, New York, 2005.
  * [13] F. P. Kelly, _Reversibility and stochastic networks_ , Wiley, New York, 1979.
  * [14] A. N. Kolmogorov, Zur theorie der Markoffschen ketten, _Mathematische Annalen_ , 112, 155-160, 1936 (trans. in _Selected Works of A. N. Kolmogorov_ , Springer, 1992).
  * [15] I. Krajbich, C. Armel and A. Rangel, Visual fixations and the computation and comparison of value in simple choice, _Nature Neuroscience_ 13, 1292-1298, 2010
  * [16] I. Krajbich and A. Rangel, Multialternative drift-diffusion model predicts the relationship between visual fixations and choice in value-based decisions, _Proceedings of the National Academy of Sciences_ , 108, 13852-13857, 2011.
  * [17] D. M. Kreps, _Notes on the theory of choice_ , Westview, Boulder, 1988.
  * [18] R. D. Luce, _Individual choice behavior: a theoretical analysis_ , Wiley, New York, 1959.
  * [19] R. D. Luce and P. Suppes, Preference, utility and subjective probability, in _Handbook of Mathematical Psychology_(R. D. Luce, R. R. Bush and E. Galanter, eds.), v. 3, Wiley, New York, 1965.
  * [20] J. Marschak, Binary-choice constraints and random utility indicators, in _Mathematical Methods in the Social Sciences_ (K. J. Arrow, S. Karlin and P. Suppes, eds.), Stanford University Press, Stanford, 1960.
  * [21] T. McMillen and P. Holmes, The dynamics of choice among multiple alternatives, _Journal of Mathematical Psychology_ , 50, 30-57, 2006.
  * [22] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines, _Journal of Chemical Physics_ , 21, 1087-1092, 1953.
  * [23] M. Milosavljevic, J. Malmaud, A. Huth, C. Koch and A. Rangel, The drift diffusion model can account for the accuracy and reaction time of value-based choices under high and low time pressure, _Judegment and Decision Making_ , 5, 437-449, 2010.
  * [24] M. Pinsky and S. Karlin, _An introduction to stochastic modeling_ , 4${}^{\text{th}}$ ed., Academic Press, Burlington, 2011.
  * [25] R. Ratcliff, A theory of memory retrieval, _Psychological Review_ , 85, 59-108, 1978.
  * [26] E. Reutskaja, R. Nagel, C. F. Camerer and A. Rangel, Search dynamics in consumer choice under time pressure: an eye-tracking study, _American Economic Review_ , 101, 900-926, 2011.
  * [27] R. F. Rinehart, The equivalence of definitions of a matric function, _American Mathematical Monthly_ , 62, 395-414, 1955.
  * [28] R. M. Roe, J. R. Busemeyer and J. T. Townsend, Multialternative decision field theory: a dynamic connectionst model of decision making, _Psychological Review_ , 108, 370, 2001.
  * [29] J. E. Russo and L. D. Rosen, An eye fixation analysis of multialternative choice, _Memory and Cognition_ , 3, 267-276, 1975.
  * [30] G. M. Stine, A. Zylberberg, J. Ditterich and M. N. Shadlen, Differentiating between integration and non-integration strategies in perceptual decision making, _eLife_ , 9, e55365, 2020.
  * [31] K. Valkanova, Markov stochastic choice, mimeo, 2020.

Powers of a matrix

$N=\left\\{1,2,...,n\right\\}$, $T$ an $n\times n$ square matrix, $T^{0}=I$,
$T^{m}=TT^{m-1}$ for all $m\geq 1$. Denoting by $t_{ij}^{\left(m\right)}$ the
$\left(i,j\right)$-th element of $T^{m}$, for $m=2$ we have

$t_{ij}^{\left(2\right)}=\sum\limits_{k=1}^{n}t_{ik}t_{kj}=\sum\limits_{k_{1}\in
N^{2-1}}t_{ik_{1}}t_{k_{1}j}\qquad\forall i,j\in N$

###### Proposition 26

By induction if $m\geq 3$,

$t_{ij}^{\left(m\right)}=\sum\limits_{\left(k_{1},k_{2},...,k_{m-1}\right)\in
N^{m-1}}t_{ik_{1}}t_{k_{1}k_{2}}\cdot\cdot\cdot t_{k_{m-1}j}\qquad\forall
i,j\in N$

Proof The result is true for $m=3$, in fact,

$\displaystyle t_{ij}^{\left(3\right)}$
$\displaystyle=\sum\limits_{h=1}^{n}t_{ih}t_{hj}^{\left(2\right)}=\sum\limits_{h\in
N}t_{ih}\sum\limits_{k\in N}t_{hk}t_{kj}=\sum\limits_{h\in N}\sum\limits_{k\in
N}t_{ih}t_{hk}t_{kj}$ $\displaystyle=\sum\limits_{\left(h,k\right)\in
N^{2}}t_{ih}t_{hk}t_{kj}=\sum\limits_{\left(k_{1},k_{2}\right)\in
N^{2}}t_{ik_{1}}t_{k_{1}k_{2}}t_{k_{2}j}$

Let $m>3$ and assume true for $m-1$, then

$\displaystyle t_{ij}^{\left(m\right)}$
$\displaystyle=\sum\limits_{h=1}^{n}t_{ih}t_{hj}^{\left(m-1\right)}=\sum\limits_{h=1}^{n}t_{ih}\sum\limits_{\left(k_{1},k_{2},...,k_{m-2}\right)\in
N^{m-2}}t_{hk_{1}}t_{k_{1}k_{2}}\cdot\cdot\cdot t_{k_{m-2}j}$
$\displaystyle=\sum\limits_{h\in
N}t_{ih}\sum\limits_{\left(h_{2},...,h_{m-1}\right)\in
N^{m-2}}t_{hh_{2}}t_{h_{2}h_{3}}\cdot\cdot\cdot t_{h_{m-1}j}=\sum\limits_{h\in
N}\sum\limits_{\left(h_{2},...,h_{m-1}\right)\in
N^{m-2}}t_{ih_{1}}t_{h_{1}h_{2}}\cdot\cdot\cdot t_{h_{m-1}j}$
$\displaystyle=\sum\limits_{\left(h_{1},h_{2},...,h_{m-1}\right)\in
N^{m-1}}t_{ih_{1}}t_{h_{1}h_{2}}\cdot\cdot\cdot t_{h_{m-1}j}$

as wanted. $\blacksquare$

Quasipositivity and primitivity

Let

$T=\left[\begin{array}[c]{cc}0&1\\\ 1&0\end{array}\right]$

then

$T^{2}=\left[\begin{array}[c]{cc}0&1\\\
1&0\end{array}\right]\left[\begin{array}[c]{cc}0&1\\\
1&0\end{array}\right]=\left[\begin{array}[c]{cc}1&0\\\
0&1\end{array}\right]=I$

Therefore $T^{2m}=I$ and $T^{2m+1}=T$ for all $m\geq 0$. $T$ is quasi-positive
but not primitive.

###### Proposition 27

If $n\geq 3$, then any quasi-positive matrix $T$ is primitive, indeed $T^{2}$
is positive.

Proof For all $i,j\in N$,

$t_{ij}^{\left(2\right)}=\sum\limits_{k\in N}t_{ik}t_{kj}$

where, by quasipositivity, all summands are non-negative and $t_{ik}t_{kj}>0$
for all $k\notin\left\\{i,j\right\\}$. $\blacksquare$

Reversible stochastic matrices are diagonalizable

###### Fact 28

If $M$ is any square matrix and
$D=\mathrm{diag}\left\\{d_{i}:i=1,...,n\right\\}$, then

$\displaystyle\left(DM\right)_{ij}$ $\displaystyle=d_{i}m_{ij}$
$\displaystyle\left(MD\right)_{ij}$ $\displaystyle=m_{ij}d_{j}$

and in particular, if $d_{i}\neq 0$ for all $i\in N$,
$D^{-1}=\mathrm{diag}\left\\{d_{i}^{-1}:i=1,...,n\right\\}$.

Now let $B$ be reversible, then

$b_{ij}p_{j}=b_{ji}p_{i}$

and since $p>0$, then

$b_{ij}=p_{i}b_{ji}p_{j}^{-1}$

Consider the diagonal matrix
$D=\mathrm{diag}\left\\{p_{i}^{1/2}:i=1,...,n\right\\}$, so that
$D^{-1}=\mathrm{diag}\left\\{p_{i}^{-1/2}:i=1,...,n\right\\}$. Then

$\left(D^{-1}B\right)_{i,j}=p_{i}^{-1/2}b_{ij}$

and

$\left(D^{-1}BD\right)_{i,j}=p_{i}^{-1/2}b_{ij}p_{j}^{1/2}$

by reversibility

$\left(D^{-1}BD\right)_{i,j}=p_{i}^{-1/2}p_{i}b_{ji}p_{j}^{-1}p_{j}^{1/2}=p_{i}^{1/2}b_{ji}p_{j}^{-1/2}=p_{j}^{-1/2}b_{ji}p_{i}^{1/2}=\left(D^{-1}BD\right)_{j,i}$

Therefore, a reversible matrix is similar to a symmetric matrix which is
similar to a diagonal matrix. But similarity is an equivalence relation.

Transitivities

An unbiased, $0$-$1$ valued stochastic choice kernel defines a strict
preference

$j\succ_{\rho}i\iff\rho\left(j\mid i\right)=1$

note that, by unbiasedness,

$j\succ_{\rho}i\iff 1-\rho\left(i\mid j\right)=1\iff\rho\left(i\mid
j\right)=0$

thus, $\succ_{\rho}$ is _total_ : for all $i,j\in A$ either $i\succ j$ or
$i=j$ or $j\succ i$. Moreover, $j\succ_{\rho}i$ if and only if
$i\nsucc_{\rho}j$.

Next we show that $\rho$ is transitive if and only if $\succ_{\rho}$ is.

If $\rho$ is transitive, and $\succ_{\rho}$ is not, then there exists $i$,
$j$, and $k$ such that $i\succ_{\rho}k\succ_{\rho}j$, but $i\nsucc_{\rho}j$.
In this case, $i$, $k$, and $j$ must be distinct and $j\succ_{\rho}i$.
Therefore, $i\succ_{\rho}k\succ_{\rho}j\succ_{\rho}i$ implying
$\rho\left(i\mid k\right)=\rho\left(k\mid j\right)=\rho\left(j\mid i\right)=1$
and $\rho\left(k\mid i\right)=\rho\left(j\mid k\right)=\rho\left(i\mid
j\right)=0$ contradicting (LABEL:eq:trans-ddm); thus must be $\succ_{\rho}$
transitive.

Conversely, if $\rho$ is not transitive, then there are three distinct
alternatives $i$, $j$ and $k$ for which (LABEL:eq:trans-ddm) does not hold;
that is,

$\rho\left(j\mid i\right)\rho\left(k\mid j\right)\rho\left(i\mid
k\right)\neq\rho\left(k\mid i\right)\rho\left(j\mid k\right)\rho\left(i\mid
j\right)$

It is then impossible that both sides contain a zero factor. Since $\rho$ is
$0$-$1$ valued, then:

  * •

either $\rho\left(j\mid i\right)\rho\left(k\mid j\right)\rho\left(i\mid
k\right)=1$, then $i\succ_{\rho}k\succ_{\rho}j\succ_{\rho}i$, so that
$i\succ_{\rho}k\succ_{\rho}j$ and $i\nsucc_{\rho}j$, thus $\succ_{\rho}$ is
not transitive;

  * •

or $\rho\left(k\mid i\right)\rho\left(j\mid k\right)\rho\left(i\mid
j\right)=1$, then $i\succ_{\rho}j\succ_{\rho}k\succ_{\rho}i$, so that
$i\succ_{\rho}j\succ_{\rho}k$ and $i\nsucc_{\rho}k$, thus $\succ_{\rho}$ is
not transitive.


# Gate voltage modulation of the superconducting state in a degenerate
semiconductor

Bikash C. Barik  Department of Physics, Indian Institute of Technology Bombay,
India  Himadri Chakraborti  Department of Physics, Indian Institute of
Technology Bombay, India  SPEC, CEA, CNRS, Universite Paris-Saclay, CEA
Saclay, 91191 Gif sur Yvette Cedex France  Buddhadeb Pal  Department of
Physics, Indian Institute of Technology Bombay, India  Indian Association for
the Cultivation of Science, Kolkata, 700032, India  Aditya K. Jain  Department
of Physics, Indian Institute of Technology Bombay, India  Department of
Physics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK
Swagata Bhunia  Department of Physics, Indian Institute of Technology Bombay,
India  Sounak Samanta  Department of Physics, Indian Institute of Technology
Bombay, India  Apurba Laha  Department of EE, Indian Institute of Technology
Bombay, India, India  Suddhasatta Mahapatra  Department of Physics, Indian
Institute of Technology Bombay, India  K. Das Gupta<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Bombay, India

###### Abstract

In this work, we demonstrate that the modulation of carrier density can alter
the superconducting transition temperature by up to $204$ mK in epitaxial
Indium Nitride on Gallium Nitride, accounting for the $10$% of the transition
temperature in ungated conditions. Our samples are likely free from strong
localization effects and significant granularity, as indicated by $k_{f}l\gg
1$, suggesting that the primary determinant of the transition temperature in
InN is carrier density, rather than disorder scattering. The observed behavior
is consistent with BCS s-wave superconductivity, corroborated by the
superconducting parameters we measured. Furthermore, we observed a $60$%
bipolar suppression of the supercurrent in our experiments.

The bare interaction between two particles in a many-body system is
necessarily modified by the presence of the other particles and excitations.
This modification or screening is a function of particle density, which in
metallic systems is very difficult to vary. Metals have carrier densities
$\sim 10^{23}$ ${\rm cm^{-3}}$ and even a monolayer of metal will have an
areal density of at least $\sim 10^{15}$ ${\rm cm^{-2}}$. These can not be
modulated by any conventional solid-state gate in a Field Effect Transistor
(FET) configuration. This limitation arises because achieving even $\sim$ 1 %
change in carrier density demands an unsustainable electric field. Even when
utilizing very thin (thickness $d$) high-$K$ dielectrics, the breakdown field
(approximately $\sim 1\rm\ V/nm$) constrains the maximum achievable areal
density modulation
$\delta{n}\approx\left(\dfrac{{\epsilon_{0}}K}{e}\right)\left(\dfrac{V_{G}}{d}\right)$
to be on the order of $\sim 5\times 10^{13}{\rm\ cm^{-2}}$, where $V_{G}$
represents the gate voltage. In contrast, non-degenerate semiconductor
channels offer the flexibility for density modulation due to their
significantly lower bulk concentration, typically ranging from ($10^{16}$ to
$10^{17}$ ${\rm cm^{-3}}$). Historically, this is the key reason
superconducting pairing, ferromagnetic exchange etc. could not be gate-
controlled because these usually occur in metals, alloys or oxides which have
carrier densities much higher than $\sim$ $10^{20}$ $\rm cm^{-3}$. In recent
years ionic liquid gating technique [1, 2, 3, 4, 5, 6, 7] has brought ferro
and antiferromagnetic interactions and potential superconducting interactions
to gate tunable regime [3, 8, 9] to a certain extent. It is interesting to ask
whether an effective attractive interaction between electrons near the Fermi
surface can persist in systems with significantly lower electron densities,
characteristic of semiconductors (typically $10^{16}$ \- $10^{20}$ $\rm
cm^{-3}$). Certain semiconductors, such as GeTe, B-doped diamond, B-doped Si
and InN do show superconductivity. For instance, InN films exhibit a
transition at temperatures around $T$ $\sim$ $2-3\rm\ K$, heavily alloyed
GeTe, a transition is seen around $T$ $<$ 0.3 K [10, 11]. Diamond and Si,
require carrier densities above $10^{21}\rm\ cm^{-3}$ for superconductivity
[12, 13]. Interestingly, the carrier density required for superconductivity in
InN is notably lower, falling within the range of ($10^{17}$ \-
$10^{18}$${\rm\ cm^{-3}}$) [14, 15, 16], which is $\sim$ $10^{3}-10^{4}$ times
less than conventional metals.

In this work, through controlled modulation of the carrier density by ionic
liquid gating, we achieved a reversible change of the superconducting
transition temperature ($T_{C}$) by $204{\rm\ mK}$ with $\pm$ $3.25$ V,
corresponding to approximately $10\%$ of the ungated $T_{C}$. In these
devices, carrier densities were successfully changed by nearly $30\%$, and the
critical current by more than $50\%$ compared to their ungated values. The
areal capacitance between the gate and the channel was corroborated through
independent methods, specifically, Hall effect measurements and
electrochemical impedance spectroscopy (EIS) [17, 6]. It is important to note
that previous investigations into the field effect on superconductors employed
solid dielectric [18] and ferroelectric [19] gates, yielding minimal effects.
More recently, a shift of approximately $80{\rm\ mK}$ ($\Delta T_{C}$) was
observed in Nb thin films subjected to applied gate voltages ranging from -$4$
V to $5$ V [20]. On NbN thin films, a $\Delta T_{C}$ of around $85{\rm\ mK}$
was achieved with applied gate voltages of $\pm 3$ V [5, 21].
Our prior experiments on angle-dependent critical field [22] revealed that
even in epitaxially grown $\rm{InN}$ films with a thickness of upto $300{\rm\
nm}$, superconducting electrons are inherently confined to a surface layer
with a thickness of $d_{s}\sim 25{\rm\ nm}$—a dimension smaller than the
coherence length $\xi_{0}$. Additionally, the significant density of surface
states pins the surface Fermi level within the conduction band, eliminating
the presence of a Schottky barrier at the surface and, concurrently, making
hole-doping of the material exceptionally challenging [23]. However, it is
precisely this combination of low bulk carrier density and the potential for a
substantial response to perturbations in the surface states that renders ${\rm
InN}$ an intriguing candidate for ionic-liquid gating where the strongest
perturbation occurs close to the surface and gradually diminishes within a few
Thomas-Fermi screening lengths as one moves into the bulk of the material.

Figure 1: Electric double-layer transistor & Output characteristics: (a)
Schematic diagram of InN-IL gated transistor. The area of the gate electrode
is much larger than the channel area. (see explanation in text). DEME-TFSI
ionic liquid is used. (b) Cross sectional view. (c) Output characteristics at
room temperature. (inset) shows a clear shift in current values with applied
gate voltages. Gate leakages current are shown in the right panel. This data
is taken from InN-B3601 sample.

The InN samples were grown on sapphire substrates by plasma-assisted molecular
beam epitaxy (PA-MBE) on c-oriented sapphire [24, 25, 26, 22] with negligible
oxygen impurity demonstrated by the absence of oxygen O1s core level
transition in XPS [24]. Patterned channels (Fig. 1(a)) on these were
lithographically fabricated using $\rm Al_{2}O_{3}$ as a hard mask layer
followed by chlorine-based dry etching and subsequent removal of the hard-mask
by $\rm{HF}$. The devices had effective channel area of 1150 $\rm\mu m$
$\times$ 100 $\rm\mu m$ and thickness of $\sim$ 300 nm (Fig. 1(b)). Au/Ti (80
nm/20 nm) contacts were used for source (S) and drain (D) contacts. The gate
electrode is much larger in area than the channel, eliminating the necessity
of a reference electrode [6]. We used DEME-TFSI as the ionic liquid (IL).
Contacts were covered with silicone gel during application of the IL. (See
supplementary for detailed fabrication steps.). The ionic conductivity of
DEME-TFSI are much higher than most other electrolytes [1]. The ionic liquid
was cured in high vacuum at $\rm 110\degree\ C$ for $\sim$ $48$ hrs. All low-
temperature measurements were carried out in a vacuum of $10^{-5}$ mbar or
better. Fig. 1(c), shows typical output characteristics (i.e. $V_{SD}$ vs
$I_{SD}$ measurements) conducted at different gate voltages ($V_{G}$), ranging
from $+0.8$ V to $-0.8$ V with a step increment of $0.2$ V. The gate leakage
remains $10^{3}$ times smaller than $I_{SD}$. The observed trends and direct
Hall voltage measurements at $300\rm K$, (as well as $4.8\rm K$), confirm that
the carrier density is stable and indeed changes in a predictable way,
allowing clear determination of the electron density ($n$) and mobility
($\mu_{e}$) from four terminal measurements. We recorded a $\sim 30\%$ change
in carrier density with applied gate voltages, a finding of considerable
relevance within the context of high carrier density systems [3, 27, 4, 28].

At a temperature of $300{\rm\ {K}}$, we observe $\mu_{e}\sim$ $220\rm\
cm^{2}V^{-1}s^{-1}$ at $V_{G}$ = 0 V, yielding a mean free path
$l\approx{10\rm\ {nm}}$ (see Table 1). The capacitance determined from our
measurements, approximately 8.8$\rm\ \mu Fcm^{-2}$ as depicted in Fig. 2(b),
surpasses what is achievable in MOSFET or HEMT-type devices. All low
temperature measurements were performed below the glass transition temperature
($T_{G}$) of the ionic liquid (refer to supplementary data for the behavior of
an ionic liquid gate). It is crucial to note that changing the gate voltage
will have the desired effect only if done at $T\gg T_{G}$. In our experiments,
we always allowed the sample to warm up to room temperature for setting
$V_{G}$.

Figure 2: Hall measurements: (a) Hall voltage data at T=4.8 K, Carrier density modulation is shown in the inset. (b) The measured capacitance from the data over a range of gate voltages ($V_{G}$) spanning from $\pm 0.5\rm\ V$. Table 1: Bulk carrier density($n_{3D}$), the density of states (DOS) at the Fermi level ($N(0)$), superconducting transition temperature ($T_{c}$), Fermi velocity ($v_{f}$), mean-free path ($l$), Ioffe-Regel parameter ($k_{f}l$), electron-phonon coupling strength ($\lambda$), coherence length ($\xi$), and Maki parameter ($\alpha$) are obtained from two devices, namely InN-B3601 and InN-B3603 respectively. $\rm B3601$ | $V_{G}(\rm V)$ |  $n_{3D}$$(m^{-3})$ |  $N(0)$($m^{-3}/eV$) | $T_{C}\rm(K)$ | $v_{f}(m/s)$ | $l(nm)$ | $k_{f}l$ | $\lambda$ | $\xi(nm)$ | $\alpha$
---|---|---|---|---|---|---|---|---|---|---
| -2 | 6.53$\times$$10^{24}$ | 7.69$\times$$10^{26}$ | 2.177 | 6.68$\times$$10^{5}$ | 9.07 | 5.24 | 0.44229 | 38.82 | 0.0767
| -1 | 6.56$\times$$10^{24}$ | 7.71$\times$$10^{26}$ | 2.174 | 6.69$\times$$10^{5}$ | 9.30 | 5.38 | 0.44215 | 39.09 | 0.0745
| 0 | 7.41$\times$$10^{24}$ | 8.02$\times$$10^{26}$ | 2.146 | 6.97$\times$$10^{5}$ | 9.84 | 5.93 | 0.44090 | 40.73 | 0.0699
| +1 | 9.00$\times$$10^{24}$ | 8.56$\times$$10^{26}$ | 2.087 | 7.44$\times$$10^{5}$ | 9.52 | 6.12 | 0.43844 | 42.76 | 0.0649
| +1.25 | 9.03$\times$$10^{24}$ | 8.57$\times$$10^{26}$ | 2.072 | 7.45$\times$$10^{5}$ | 9.35 | 6.02 | 0.43779 | 43.06 | 0.0647
$\rm B3603$ | $V_{G}(\rm V)$ |  $n_{3D}$$(m^{-3})$ |  $N(0)$($m^{-3}/eV$) | $T_{C}\rm(K)$ | $v_{f}(m/s)$ | $l(nm)$ | $k_{f}l$ | $\lambda$ | $\xi(nm)$ | $\alpha$
| -3.25 | 5.11$\times$$10^{24}$ | 7.09$\times$$10^{26}$ | 2.074 | 6.15$\times$$10^{5}$ | 6.15 | 3.27 | 0.43788 | 31.8 | 0.1214
| 0 | 6.82$\times$$10^{24}$ | 7.80$\times$$10^{26}$ | 2.012 | 6.78$\times$$10^{5}$ | 8.60 | 5.03 | 0.43519 | 37.2 | 0.0949
| +0.5 | 7.07$\times$$10^{24}$ | 7.90$\times$$10^{26}$ | 2.004 | 6.86$\times$$10^{5}$ | 8.26 | 4.90 | 0.43484 | 37.4 | 0.0937
| +3.25 |  |  | 1.870 |  |  |  | 0.42887 |  |
Figure 3: Critical temperature (Tc) measurements: (a) The normalized $R-T$ for
different applied gate voltages for InN-B3603 sample. The inset displays the
corresponding superconducting $T_{C}$ measured at various gate voltages,
revealing an estimated value of $\Delta T_{C}$ = $204\rm\ mK$ within the
voltage range of $\pm$ $3.25\rm\ V$. The dashed line (at 0.5) is the reference
where resistance becomes half of the normal resistance ($R_{N}/2$). (b)
Normalized $R-T$ for different gate voltages for InN-B3601 sample. The data
shown in the inset are not normalized. (c) Carrier density and corresponding
superconducting transition temperature for different gate voltages. Carrier
density measured at $\rm T=4.8\rm\ K$ (above the superconducting transition
temperature). (d) Estimation of density of states (DOS) from the free electron
theory relation (left axis) and the dirty limit relation (right axis).

The superconducting critical temperature was measured in InN devices subjected
to ionic liquid gating. The findings presented here stem from two such devices
i.e. B3601 and B3603 (from two different wafers), demonstrating tunable
superconducting parameters with variations in $T_{C}$ of $204$ ${\rm mK}$ and
$105$ ${\rm mK}$ shown in Fig. 3 (a) & (b) respectively. The transition
temperature of InN is primarily governed by carrier density rather than any
disorder scattering ($k_{F}l$) [29][30]. It has to be noted that we are able
to both enhance and supress the $T_{C}$ in the same device. This cannot be
explained by assuming a thin perturbed layer near the surface and a relatively
unperturbed bulk superconductor in parallel. In such a case the resistive
$T_{C}$ can be enhanced by a surface layer with a higher $T_{C}$ “short
circuiting” the bulk, but cannot be suppressed by a layer with a lower
$T_{C}$, since the bulk superconductor would then act as a shorting path.
Furthermore we see no evidence of a two step resistive transition that would
indicate the presence of two parallel conducting layers. The perturbation of
the superconducting state that we see thus is likely to be a state spanning
the entire superconducting region of the ${\rm InN-GaN}$ stack.

We find reasonable agreement of the measured variation of the superconducting
parameters with the expectations from a s-wave BCS superconductor having a
simple free electron (density $n$) Fermi surface. The zero temperature
coherence length for BCS s-wave superconductor is [31],

$\xi(0)\approx\dfrac{\hbar v_{f}}{\pi\Delta(0)}$ (1)

where, $v_{f}={\dfrac{\hbar}{m}}\left(3{\pi^{2}}n\right)^{1/3}$ is the Fermi
velocity and $\Delta(0)\propto T_{c}$ is the superconducting energy gap at
$T=0$, should vary as

$\dfrac{\delta\xi}{\xi(0)}\approx\dfrac{1}{3}\dfrac{{\delta}n}{n}-\dfrac{{\delta}T_{c}}{T_{c}}$
(2)

The value of $\xi(0)$ can be conveniently calculated from the critical field
data (Fig. 4), the carrier density and $T_{C}$ are directly measured. This
relation thus gives a good check on the consistency of the measurements, that
does not require the exact constant of proportionality between $\Delta(0)$ and
$T_{C}$ to be known. The calculated and measured values of
$\dfrac{\delta{\xi}}{\xi(0)}$ differ by about $40\%$ which can be attributed
to the disorder correction arising from a relatively small mean free path
$l\approx 10\rm\ nm$, that forces the observed $\xi$ to reduce to
$\xi_{obs}\approx 0.85\sqrt{{\xi(0)}l}$ [32] [Chap 6]. Given that we have
$l\sim{10{\rm\ {nm}}}\ll\xi(0)$ a correction of the order of what we observe
is certainly expected. However given that ${k_{f}}l\gg 1$, we do not expect
any strong localisation effects or significant effects of granularity. This is
also supported by hysteresis-free ${\rm IV}$ characteristics in the critical
current measurements as shown in Fig. 5.

We thus consider free electron estimation of the density of states of at the
Fermi level $N(0)=\dfrac{m}{{\pi^{2}}\hbar^{2}}\left(3{\pi^{2}}n\right)^{1/3}$
to be reasonable. This can be used to infer the variation of the strength of
the attractive interaction $V$, near the Fermi surface via the determination
of $\lambda=N(0)V$ by inverting the Eliashberg-McMillan eq.,

$T_{C}=\dfrac{\Theta}{1.45}\cdot\rm
exp\left(\dfrac{-1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\right)$ (3)

where $\mu^{*}\approx 0.1$ is the Coulomb pseudopotential [33], which accounts
for the electron-electron repulsive interaction and $\Theta$ $\approx$ 370 K
is the Debye temperature [34]. The coupling constant found here
($\lambda$$<$1) suggests that $\rm InN$ behaves like a weak to medium coupling
superconductor. Clearly, at lower carrier densities the pairing interaction
strength gets stronger resulting in a higher $T_{C}$ as depicted in Fig. 3(c).
The zero temperature coherence lengths $\xi(0)$ calculated from the
extrapolated values of the critical field ( Table-1), also mirror this
increase of $T_{C}$ showing the superconducting state is stronger at lower
carrier densities.

Figure 4: Critical field ($H_{C2}$) measurements: Temperature dependence of
critical field under various applied gate voltages, fitted with WHH model. WHH
model stands for Werthammer-Helfand - Hohenberg model. Figure 5: Critical
current ($I_{C}$) measurements: (a) Critical current measurements for
$V_{G}=0\rm\ V$ for an InN hall bar. (b) The measured values of the critical
current ($I_{C}$) exhibit suppression for both positive and negative applied
gate voltages for InN-B3603 sample. (c) Temperature-dependent measurements of
critical current ($I_{C}$) for InN-B3601 sample exhibits bipolar suppression
of $I_{C}$ as shown in the inset.

Furthermore, it is known that DOS is proportional to the slope of the upper
critical field [35, 36, 37], i.e.

$N(0)\propto-{\sigma}\left(\dfrac{dH_{c2}}{dT}\right)_{T_{C}}$ (4)

The calculated $N(0)$ is shown in the right panel of Fig. 3(d) and compared
with the free electron values
$N(0)=\dfrac{m}{\pi^{2}\hbar^{2}}\left(3{\pi^{2}}n\right)^{1/3}$ exhibiting a
very good agreement.

Next we looked into the temperature dependence of critical field. The magnetic
field value (normal to the surface) at which the resistance of the
superconductor attains half of its normal state value (the mean field value)
is taken as critical field ( $H_{C2}$). $H_{C2}(T)$ curves were fitted using
the well-known Werthammer-Helfand-Hohenberg (WHH) model [35, 38] to understand
the role of spin-orbit coupling (SOC), if present in the system, on the
superconducting phase. WHH model gives the eq. 5.

$\displaystyle\ln\left({\dfrac{1}{t}}\right)$
$\displaystyle=\sum_{\nu=-\infty}^{\infty}\left(\dfrac{1}{|2\nu+1|}-\left[|2\nu+1|\right.\right.$
$\displaystyle\left.\left.+\dfrac{b_{c}}{t}+\dfrac{(\alpha
b_{c}/t)^{2}}{|2\nu+1|+(b_{c}+\lambda_{so})/t}\right]^{-1}\right)$ (5)

where, $t=T/T_{C}$, $\alpha=3/2m{v_{F}}^{2}\tau$, is the Maki parameter, the
dimensionless magnetic field
$b_{c}$=$\mu_{0}h_{c}$=$\dfrac{e\hbar}{2m\pi\alpha}\dfrac{\mu_{0}H_{c2}}{k_{B}T_{C}}$
and $\lambda_{so}$ is the spin-orbit scattering constant. The analysis reveals
a small Maki parameter $\alpha<0.1$ and virtually no impact of the spin-orbit
term within the measured range of carrier densities (see Table 1). That is
further confirmed by evaluating the limiting orbital field, $H_{c2}^{orb}$,
using the expression $H_{c2}^{orb}$ (T=0) $\approx$ $0.7\lvert
T\frac{dH_{c2}}{dT}\rvert_{T_{C}}$. The calculated values closely correspond
to the experimental data, as demonstrated for a gate voltage of $V_{G}=-1\rm\
V$ (sky blue line) in the Fig. 4. In cases where spin-orbit coupling is
strong, one obtains a high value of Maki parameter ($\sim$ 2-3), as observed
in our earlier work [39].

However there is one significant aspect in which an indicator of the strength
of the superconducting state follows an opposing trend. In general the Landau-
Ginzburg calculation of the critical current ($I_{C}$) of a thin film [32]
[Chap 6] would predict a variation $I_{C}(T)\propto\dfrac{1}{\xi(T)}$ and
hence approximately increase with $\sqrt{H_{C2}}$. However we find that the
application of a gate voltage either positive or negative substantially
suppresses the critical current by as much as $60\%$ (see Fig. 5(b)). Many
recent experiments demonstrated gate controlled critical current suppression
in superconducting films [40, 41, 42, 43, 44]. A set of experiments suggest
that critical current suppression is occurring due to Cooper pair breaking due
to gate field [41, 40, 42]. Most likely the effect is due to modification of
the electron distribution in the channel in response to the additional
electric field arising from the gate. It is possible that the wavefunction as
it either gets pulled towards the surface or as it gets pushed into the non-
superconducting bulk region in fact sees regions with smaller normal-state
mean free path and subsequent surface scattering and granularity effects
leading to the suppression of the critical current.

The ability to change carrier density in a material to such an extent that the
Fermi surface runs through a large volume of the Brillouin zone, potentially
changing its shape from simple spherical or elliptical to more complex
geometries incorporating nodes, necks, nesting vectors etc. does not exist as
of now. For example in a simple tight binding on a square lattice, the Fermi
surface changes from a simple circle to a square at half filling and then
opens up. Development of techniques and materials in which potentially very
large density modulations are possible are important as electron interaction
driven phenomena inherently depend on the shape of the Fermi surface. In this
paper, we have shown an example that can be a step forward in this direction.

Acknowledgements: We acknowledge support from Department of Science and
Technology, Government of India under project: RD/0120-DSTIC01-001, Science
and Engineering Research Board (SERB), Government of India under project:
CRG/2018/001405. BCB acknowledges financial support through DST-INSPIRE
fellowship of Govt. of India. We thank Amit P Shah (TIFR, Mumbai) for his
assistance with the dry etch process and Prof. Saibal Sarkar (Department of
Energy Science and Engineering, IIT Bombay) for providing access to deposit
$\rm Al_{2}O_{3}$ films. We thank Dr. Parushottam Majhi for useful
discussions.

## References

  * Misra  _et al._ [2007] R. Misra, M. McCarthy, and A. F. Hebard, Applied Physics Letters 90, 052905 (2007).
  * Daghero  _et al._ [2012] D. Daghero, F. Paolucci, A. Sola, M. Tortello, G. Ummarino, M. Agosto, R. Gonnelli, J. R. Nair, and C. Gerbaldi, Physical review letters 108, 066807 (2012).
  * Petach  _et al._ [2014] T. A. Petach, M. Lee, R. C. Davis, A. Mehta, and D. Goldhaber-Gordon, Physical Review B 90, 081108 (2014).
  * Ueno  _et al._ [2011] K. Ueno, S. Nakamura, H. Shimotani, H. Yuan, N. Kimura, T. Nojima, H. Aoki, Y. Iwasa, and M. Kawasaki, Nature nanotechnology 6, 408 (2011).
  * Piatti  _et al._ [2017] E. Piatti, D. Daghero, G. Ummarino, F. Laviano, J. Nair, R. Cristiano, A. Casaburi, C. Portesi, A. Sola, and R. Gonnelli, Physical Review B 95, 140501 (2017).
  * Yuan  _et al._ [2010] H. Yuan, H. Shimotani, J. Ye, S. Yoon, H. Aliah, A. Tsukazaki, M. Kawasaki, and Y. Iwasa, Journal of the American Chemical Society 132, 18402 (2010).
  * Nakagawa  _et al._ [2021] Y. Nakagawa, Y. Kasahara, T. Nomoto, R. Arita, T. Nojima, and Y. Iwasa, Science 372, 190 (2021).
  * Walter  _et al._ [2020] J. Walter, B. Voigt, E. Day-Roberts, K. Heltemes, R. M. Fernandes, T. Birol, and C. Leighton, Science advances 6, eabb7721 (2020).
  * Ueno  _et al._ [2008] K. Ueno, S. Nakamura, H. Shimotani, A. Ohtomo, N. Kimura, T. Nojima, H. Aoki, Y. Iwasa, and M. Kawasaki, Nature materials 7, 855 (2008).
  * Hein  _et al._ [1964] R. Hein, J. Gibson, R. Mazelsky, R. Miller, and J. Hulm, Physical Review Letters 12, 320 (1964).
  * Smith and Stiles [1977] J. Smith and P. Stiles, Journal of Low Temperature Physics 26, 101 (1977).
  * Ekimov  _et al._ [2004] E. Ekimov, V. Sidorov, E. Bauer, N. Mel’Nik, N. Curro, J. Thompson, and S. Stishov, nature 428, 542 (2004).
  * Bustarret  _et al._ [2006] E. Bustarret, C. Marcenat, P. Achatz, J. Kačmarčik, F. Lévy, A. Huxley, L. Ortéga, E. Bourgeois, X. Blase, D. Débarre, _et al._ , Nature 444, 465 (2006).
  * Miura  _et al._ [1997] N. Miura, H. Ishii, A. Yamada, M. Konagai, Y. Yamauchi, and A. Yamamoto, Japanese journal of applied physics 36, L256 (1997).
  * Inushima  _et al._ [2001] T. Inushima, V. Vecksin, S. Ivanov, V. Davydov, T. Sakon, and M. Motokawa, physica status solidi (b) 228, 9 (2001).
  * Inushima  _et al._ [2006] T. Inushima, N. Kato, D. K. Maude, H. Lu, W. Schaff, R. Tauk, Y. Meziani, S. Ruffenack, O. Briot, W. Knap, _et al._ , physica status solidi (b) 243, 1679 (2006).
  * Yuan  _et al._ [2009] H. Yuan, H. Shimotani, A. Tsukazaki, A. Ohtomo, M. Kawasaki, and Y. Iwasa, Advanced Functional Materials 19, 1046 (2009).
  * Glover III and Sherrill [1960] R. Glover III and M. Sherrill, Physical Review Letters 5, 248 (1960).
  * Stadler [1965] H. Stadler, Physical Review Letters 14, 979 (1965).
  * Choi  _et al._ [2014] J. Choi, R. Pradheesh, H. Kim, H. Im, Y. Chong, and D.-H. Chae, Applied Physics Letters 105 (2014).
  * Piatti  _et al._ [2016] E. Piatti, A. Sola, D. Daghero, G. A. Ummarino, F. Laviano, J. R. Nair, C. Gerbaldi, R. Cristiano, A. Casaburi, and R. S. Gonnelli, Journal of Superconductivity and Novel Magnetism 29, 587 (2016).
  * Pal  _et al._ [2018] B. Pal, B. P. Joshi, H. Chakraborti, A. K. Jain, B. K. Barick, K. Ghosh, A. Laha, S. Dhar, and K. D. Gupta, in  _AIP Conference Proceedings_, Vol. 1942 (AIP Publishing, 2018).
  * Jones  _et al._ [2006] R. Jones, K. Yu, S. Li, W. Walukiewicz, J. Ager, E. Haller, H. Lu, and W. Schaff, Physical Review Letters 96, 125505 (2006).
  * Ghosh  _et al._ [2017] K. Ghosh, J. S. Rathore, and A. Laha, Superlattices and Microstructures 101, 405 (2017).
  * Barick  _et al._ [2016] B. K. Barick, N. Prasad, R. K. Saroj, and S. Dhar, Journal of Vacuum Science & Technology A 34 (2016).
  * Barick  _et al._ [2018] B. Barick, R. K. Saroj, N. Prasad, D. Sutar, and S. Dhar, Journal of Crystal Growth 490, 104 (2018).
  * Sagmeister  _et al._ [2006] M. Sagmeister, U. Brossmann, S. Landgraf, and R. Würschum, Physical review letters 96, 156601 (2006).
  * Prassides [2011] K. Prassides, Nature nanotechnology 6, 400 (2011).
  * Dubi  _et al._ [2007] Y. Dubi, Y. Meir, and Y. Avishai, Nature 449, 876 (2007).
  * Ghosal  _et al._ [1998] A. Ghosal, M. Randeria, and N. Trivedi, Physical review letters 81, 3940 (1998).
  * Tinkham [2004] M. Tinkham,  _Introduction to superconductivity_ (Courier Corporation, 2004).
  * De Gennes [2018] P.-G. De Gennes,  _Superconductivity of metals and alloys_ (CRC press, 2018).
  * McMillan [1968] W. McMillan, Physical Review 167, 331 (1968).
  * Davydov  _et al._ [1999] V. Y. Davydov, V. Emtsev, I. Goncharuk, A. Smirnov, V. Petrikov, V. Mamutin, V. Vekshin, S. Ivanov, M. Smirnov, and T. Inushima, Applied physics letters 75, 3297 (1999).
  * Werthamer  _et al._ [1966] N. Werthamer, E. Helfand, and P. Hohenberg, Physical Review 147, 295 (1966).
  * Guo  _et al._ [2004] Y. Guo, Y.-F. Zhang, X.-Y. Bao, T.-Z. Han, Z. Tang, L.-X. Zhang, W.-G. Zhu, E. Wang, Q. Niu, Z. Qiu, _et al._ , Science 306, 1915 (2004).
  * Chockalingam  _et al._ [2008] S. Chockalingam, M. Chand, J. Jesudasan, V. Tripathi, and P. Raychaudhuri, Physical Review B 77, 214503 (2008).
  * Helfand and Werthamer [1966] E. Helfand and N. Werthamer, Physical Review 147, 288 (1966).
  * Chakraborti  _et al._ [2022] H. Chakraborti, B. P. Joshi, C. K. Barman, A. K. Jain, B. Pal, B. C. Barik, T. Maiti, R. Schott, A. D. Wieck, M. Prasad, _et al._ , Applied Physics Letters 120 (2022).
  * Rocci  _et al._ [2020] M. Rocci, G. De Simoni, C. Puglia, D. D. Esposti, E. Strambini, V. Zannier, L. Sorba, and F. Giazotto, ACS nano 14, 12621 (2020).
  * De Simoni  _et al._ [2018] G. De Simoni, F. Paolucci, P. Solinas, E. Strambini, and F. Giazotto, Nature nanotechnology 13, 802 (2018).
  * Paolucci  _et al._ [2021] F. Paolucci, F. Crisá, G. De Simoni, L. Bours, C. Puglia, E. Strambini, S. Roddaro, and F. Giazotto, Nano letters 21, 10309 (2021).
  * Golokolenov  _et al._ [2021] I. Golokolenov, A. Guthrie, S. Kafanov, Y. A. Pashkin, and V. Tsepelin, Nature Communications 12, 2747 (2021).
  * Alegria  _et al._ [2021] L. D. Alegria, C. G. Bøttcher, A. K. Saydjari, A. T. Pierce, S. H. Lee, S. P. Harvey, U. Vool, and A. Yacoby, Nature Nanotechnology 16, 404 (2021).

See pages - of SM.pdf