# Geometric pumping and dephasing at topological phase transition

B.Q. Song  Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  J.D.H. Smith  Ames Laboratory, Iowa State University, Ames, Iowa 50011,
USA  Department of Mathematics, Iowa State University, Ames, Iowa 50011, USA
L. Luo  J. Wang  Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA

###### Abstract

A measure-preserving formalism (MPF) is constructed and applied to spin/band
models, which yield observations about pumping. It occurs at topological phase
transition (TPT), i.e., a $Z_{2}$-flip, suggesting that $Z_{2}$ can imply bulk
effects. The model’s asymptotic behavior is analytically solved via MPF. The
pumping probability is geometric, fractional, and has a ceiling of
$\frac{1}{2}$. Intriguingly, theorems are proved about occurrence conditions,
which are linked to the system’s dimension and the distinction between
rational and irrational numbers. Experimental detection is discussed.

###### pacs:

As witnessed in the past decades, geometry has intertwined with physics Moody
; Niu  primarily via a framework outlined by Berry Berry  and SimonSimon ,
which has two crucial ingredients: one is adiabaticity, vital for the analogy
to “parallel transport”; the other is Berry curvature which makes anholonomy
occur. The two points are believed necessary, serving the foundation of
recognizing the slow/fast variables, deducing topological invariants, etc. as
seen in quantizatization of adiabatic pumping Thouless ; Fu  or transports
TKNN , theory of electric polarization Vanderbilt , topological gaped TISC ;
Symm  or nodal states Weyl .

Our first result is to incorporate geometry without adiabaticity and Berry
curvatures – the two ingredients are proved dispensable. Our scheme relies on
making quantum evolution mimic a classical trajectory via a measure-preserving
formalism (MPF) MPF ; Statistics , which originates from efforts in unifying
formalisms for quantum and classical mechanics Wigner ; Flow ; Gro46 ; Moy49 .
In Berry’s scheme (and its development Samuel ; AA ), local curvature is the
key; here, the “trajectory” in phase space is at heart, entailing different
geometric intuitions (e.g., ergodicity Ebook ) and math apparatus. Moreover,
MPF helps deduce exact solving of a spin model to establish a rigorous concept
from which one can set out to examine general situations.

The second result is about an observable, an inter-band pumping, namely
geometric pumping. Its rate purely depends on an angle parameter, which, in
certain instances, is linked to $Z_{2}$ index FuInv ; FieldTI . Since the
pumping is for bulk, it challenges the wisdom that $Z_{2}$ only implies
surface/interface, known as bulk-edge correspondence Bernevig . It relies on
the flipping of $Z_{2}$ (not a specific state of $Z_{2}$) thus is a genuine
consequence of topological phase transitions (TPT). Its detection perfectly
matches the pump-probe laser technique Chirag ; Liang : “pump” is to excite
phonons that can distort the bands and may induce electronic TPT, and “probe”
is to detect charge pumping. Suitable systems include narrow-gap topological
insulators or semi-metals, which are liable to undergo TPT. A candidate
recently studied is ZrTe5, whose topological states can be altered by A1g
Chirag , B1u modes Liang .

The work spans several subjects: topological band theory, MPF, ergodic theory,
measure theory, number theory. It is fascinating that such seemingly
irrelevant topics appear together without a planner. We protrude their
relations in the main text and leave backgrounds, derivation, and proof in
Appendix and references.

Figure 1: (color online): (a) 1D. $Z_{2}:0{\rightarrow 0}$, no pumping;
$0{\rightarrow 1}$, pumping. (b) 2D. B$(t)$ follows a sector loop with angle
$\Theta$. Dimensions refer to the spatial ones. For example, if
$\textbf{B}(t)$ is restricted along the $z$-axis, it is 1D; if allowed for
rotation in a plane, e.g., the $x$-$z$ plane, it is 2D.

We set out from a spin model to introduce a generic MPF. Consider a
spin-$\frac{1}{2}$ in 1D cyclic magnetic field
$\textbf{B}(t)=B_{0}(a+\text{cos}({\omega}t)){\cdot}{\hat{z}}$, two levels
with a gap $\Delta(t)=2{\mu}_{B}B_{z}(t)$ which will momentarily close if
$|a|{\leq}1$ (Fig. 1(a)). Unlike previous models constantly with gaps Berry ;
Thouless ; Fu ; Zee , we allow gap-closing (thus non-adiabatic) for
investigating TPT. $Z_{2}$ index can be defined as
$\frac{1}{\pi}\text{cos}^{-1}(\text{sgn}({B}_{z}{\cdot}\sigma_{z}))$, valued 0
or 1 depending the relative orientations of B and spin. Flipping $Z_{2}$
requires B passing zero, i.e., gap closing.

The 1D setting is to introduce the concept of “pumping,” which refers to the
spin being excited to the higher eigenstate ${|n_{1}{\rangle}}$
(${|n_{0}{\rangle}}$ is ground state) under the influence of $\textbf{B}(t)$.
When generalized to band models, this corresponds to inter-band pumping,
different from pumping across real space onto edges Thouless ; Fu ; Moore .
For 1D, the exact solution is available, say starting with spin up:

$\begin{split}|{\varphi}(t){\rangle}=e^{-i{\int}^{t}E_{\uparrow}(\tau)d{\tau}}|{\uparrow}{\rangle},\end{split}$
(1)

where $|{\uparrow}{\rangle}$ stands for the state of spin up and
$E_{\uparrow}(\tau)={\langle}{\uparrow}|H(\tau)|{\uparrow}{\rangle}=-{\mu}_{B}B_{z}(t)$.
So $|{\varphi}(t){\rangle}$ sticks to $|\uparrow{\rangle}$. But
$|\uparrow{\rangle}$ might switch between $|n_{0}{\rangle}$ and
$|n_{1}{\rangle}$, when gap is closed (Fig. 1(a)). If level-inversion happens
($Z_{2}$ is flipped), pumping occurs; if $Z_{2}$ does not change, e.g.,
$a{\geq}1$, no pumping takes place.

Despite its simplicity, 1D clearly shows geometric characteristics: (1)
pumping only depends on the flipping of $Z_{2}$, insensitive to eigenvalues
unless the gap is closed; (2) Eq. 1 holds for arbitrary $\omega$ and lacks
energy scales (e.g., ${\Delta}/{\omega}$). Eq. 1 indicates, after two cycles,
the spin will be de-pumped back to the original eigenstate up to a phase. Such
returnable behavior is liable to be generic for 1D, as it is also observed in
1D $Z_{2}$ spin pump Fu . In that case, a spin chain under cyclic changing
potentials is considered, and after one cycle, the spin is pumped from one
edge to the other, but in the next cycle, the pumped spin will return (i.e.,
net pumping in two cycles is zero). Thus, external dephasing (e.g., coupled
with leads) is believed necessary to crush coherence and make pumping happen
Fu ; Zhou . However, when the dimension is ${\geq}2$, as we find below,
“internal” dephasing occurs.

Consider $\textbf{B}(t)$ in $x$-$z$ plane (Fig. 1(b)). The arc section with
large $|\textbf{B}|$ ensures ideal adiabatic evolution. Two straight sections
with exact solutions are to handle the gap closing. Average pumping rate over
$n$ cycles is

$\begin{split}p_{n}=\frac{1}{n}{\sum}_{j=1}^{n}|{\langle}n_{1}|\mathcal{U}^{j}|{\varphi}(t=0)\rangle|^{2}.\end{split}$
(2)

Evolution operator $\mathcal{U}({\Theta},{\Omega},{\Phi})$ is found (Appx. A),
where $\Theta,\Omega$ are polar and azimuthal angles of $\textbf{B}(t)$;
$\Phi$ contains dynamic phases from $B_{0}$ and $\omega$.

$\begin{split}\mathcal{U}=\begin{pmatrix}\text{cos}({\Theta}/{2})e^{-i{\Phi}}&-\text{sin}({\Theta}/{2})e^{-i({\Omega}-{\Phi})}\\\
\text{sin}({\Theta}/{2})e^{i({\Omega}-{\Phi})}&\text{cos}({\Theta}/{2})e^{i{\Phi}}\end{pmatrix}.\end{split}$
(3)

Here, $\mathcal{U}$ will not determine a state midst a loop but will give the
resultant state after $n$ loops by $\mathcal{U}^{n}$. $\mathcal{U}$ being
SU(2) rather than U(2) is due to time-reversal
$\mathcal{T}H(t)\mathcal{T}^{-1}=-H(t)$ (Appx. A). $\mathcal{U}$ is
independent of gauge, as required by an evolution operator. Pumping takes
place at the pathway corner (gap-closing point), but phases accumulated
elsewhere influence. Previous results about pumping Thouless ; Fu ; Moore ;
Zhou  rely on the adiabatic limit $\omega{\ll}\Delta$. Our result is
interesting on account of being valid for
$\omega{\gg}{\Delta}_{\text{min}}=0$, and error in polynomials
$\mathcal{U}^{n}$ only coming from arc sections, which can be exponentially
suppressed by raising $B_{0}$.

We seek $p_{n{\to}{\infty}}$, limiting behaviors after a number of cycles.
Later we will show $p_{n{\to}{\infty}}$ converges almost everywhere (a.e.) in
$\Theta$-$\Phi$ space (also $\Theta$-$\Phi$-$\Omega$ for 3D). It is
straightforward to show $p_{n}(\Theta)=p_{n}(-\Theta)$ and
$p_{n}(\Phi)=p_{n}(\Phi+\pi)$, so we adopt $\Phi{\in}[-\pi/2,\pi/2]$ and
$\Theta{\in}[0,\pi]$. For 1D, $\Phi$ is the variable and $\Theta=\pi$ or 0,
and $p_{n}$ is integer for arbitrary $\Phi$. For 2D ($0<\Theta<\pi$),
“fraction” pumping emerges, e.g., $\frac{1}{2}$ “shot” of spin is pumped when
$x$\- is projected to $z$-axis. Projections occur in every cycle, subject to
computation of $p_{n}$ with Eq. 2, which becomes difficult as $n$ getting
large. Fortunately, an ingenious method of MPF finds $p_{\infty}$ converge to
analytic expressions.

MPF is built for both classical and quantum, but differently. For classical,
preservation of measure $dp~{}{\wedge}~{}dq$ is directly deduced from
Hamilton’s equations Statistics , thus MPF is established on rigor. For
quantum, however, measure preservation demands all higher-order ($>$2)
derivatives of potentials must be vanishing Wigner ; Flow . Thus, the validity
of MPF relies on stringent conditions. This motivates us to construct MPF on
an equal footing: Just as Liouville’s Theorem follows from Hamilton’s
equations, can measure-preservation result directly from Schrödinger equation
(SE)? One feature of SE (perhaps unique) that ensures measure preservation is
unitarity of the evolution. A unitary operator is an endomorphism of Hilbert
space, and continuous endomorphism of a compact group will preserve its Haar
measure Ebook . Accordingly, two modifications are made. First, we leave the
${\\{}p,q{\\}}$ space and turn to the group’s parameter space. Second, Haar
measure (Appx. B) defined for the symmetry group on Hilbert space
$\mathcal{H}$ is to replace the measure $dp~{}{\wedge}~{}dq$.

In Hamilton mechanics, evolution operator $T_{t}$ updates particles’ momenta
and positions Statistics ; Ebook ; formally, it is a set
${\\{}T_{t}|t\in\mathbb{R}{\\}}$ that transform the space ${\\{}p,q{\\}}$,
preserving the symplectic measure $dp~{}{\wedge}~{}dq$ (Liouville theorem),
and satisfying semigroup law $T_{t+s}=T_{t}{\circ}T_{s}$. Taking discrete time
steps $s$, the semigroup law yields $T_{t}=T^{n}_{s}$ for $t=ns$. In
comparison, quantum $\mathcal{U}$ represents the evolution during
$\textbf{B}(t)$ completing a loop, serving as a “time step”. $\mathcal{U}$ is
a matrix of SU(2) group, satisfying
$\mathcal{U}^{l+n}=\mathcal{U}^{l}{\circ}~{}\mathcal{U}^{n}$. It will rotate
$\mathcal{H}$ and preserve SU(2)’s Haar measure
$m(\Theta,\Omega,\Phi)=2\text{sin}(2\Theta){\cdot}d{\Phi}{\cdot}d{\Theta}{\cdot}d{\Omega}$
(also its subgroups’). We observe the correspondence:

$\begin{split}T_{s}^{n}(p,q)\sim\mathcal{U}^{n}(\Theta,\Omega,\Phi);~{}~{}dp~{}{\wedge}~{}dq\sim
m(\Theta,\Omega,\Phi).\end{split}$ (4)

The step number $n$ is “time”. The phase-space coordinates $(p,q)$ correspond
to 3-dimensional state coordinates $(\Theta,\Omega,\Phi)$, referring to the
spin orientation and phases. The symplectic measure $dp~{}{\wedge}~{}dq$
corresponds to the Haar measure $m$ (Appx. B). The correspondence for the time
averages for an observable $f$ is given by

$\begin{split}\bar{f}=\frac{1}{t}\operatorname*{\sum}_{i=0}^{n}f(T_{s}^{i}(p,q))~{}{\sim}~{}\bar{f}=\frac{1}{n}\operatorname*{\sum}_{i=0}^{n}f_{n}p_{n},\end{split}$
(5)

where $f$ is certain observable to be averaged. By the substitution of Eq. 4
and 5, we define a MPF.

The valid conditions for MPFs are summarized in Table 1. There is an elegant
correspondence between the Liouville theorem and the present scheme. Liouville
requires “Hermitian” in a classical sense, i.e, the system’s dynamics obey
Hamilton equations $\dot{q}={\partial}_{p}H,\dot{p}=-{\partial}_{q}H$
Statistics . However, if dissipative terms exist (e.g.,
$\dot{p}=-{\partial}_{q}H-p$), MPF is invalid. For this work, it requires
“Hermitian” in a quantum sense, i.e., Hamiltonian is a Hermitian operator
$H^{\dagger}(t)=H(t)$, which is a fundamental feature of quantum mechanics
that ensures evolution $\mathcal{U}$ being unitary, deducing robust MPF. In
both MPFs, $H$ can be time-dependent, thus energy is allowed to flow in or
out. On the other hand, Wigner’s scheme requires perfect harmonic potentials
Wigner , which are often not satisfied even at an approximation level.

Table 1:  Comparion of different MPFs: Liouvile theorem Statistics , Wigner flow Wigner  and this work, in terms of applied scopes (classical or quantum), measure functions, and valid conditions. MPF | Scope | Measure | Condition
---|---|---|---
Liouville | C | $dp~{}{\wedge}~{}dq$ |  $\dot{q}={\partial}_{p}H,\dot{p}=-{\partial}_{q}H$ (Robust)
Wigner | Q | $dp~{}{\wedge}~{}dq$ |  $\partial_{q}^{n}H=0$ for $n>2$ (Fragile)
This work | Q | Haar |  $H^{\dagger}=H$ (Robust)

To proceed, we need a crucial concept: ergodicity (intuitive interpretation
seen in Appx. B). A system being ergodic means it can reach every region in
the space (either ${\\{}p,q{\\}}$ space or others) after sufficient time.
Formally, it is a property of the evolution operator.

Definition 1. Let ($X$,$\mathfrak{B}$,$m$) be a probability space. A measure-
preserving transformation $T$ of ($X$,$\mathfrak{B}$,$m$) is called ergodic if
the only members of $B$ of $\mathfrak{B}$ with $T^{-1}B=B$ satisfy $m(B)=0$ or
$m(B)=1$.

Here, $\mathfrak{B}$ denotes a $\sigma$-algebra of set $X$. The $m$ is a
measure function that is equipped to depict “probability” and has been
normalized to 1 (Appx. B). If $T$ is ergodic, we have the following theorems
(p. 30, 34 of Ebook ). Theorem 1.2 is known as Birkhoff Ergodic theorem.

Theorem 1.1. Let $G$ be a compact group and $T(x)=ax$ a rotation of $G$. Then
$T$ is ergodic iff ${\\{}a^{n}{\\}}^{\infty}_{-\infty}$ is dense in $G$. In
particular, if $T$ is ergodic, then $G$ is abelian.

Theorem 1.2. Let
$T$:($X$,$\mathfrak{B}$,$m$)$\rightarrow$($X$,$\mathfrak{B}$,$m$) be measure
preserving and $f{\in}L^{1}(m)$. Then $(1/n){\Sigma}_{i}^{n-1}f(T^{i}(x))$
converges a.e. to a function $f^{*}{\in}L^{1}(m)$. Also $f^{*}{\circ}T=f^{*}$
a.e. and if $m(X)<\infty$, then ${\int}f^{*}dm={\int}fdm$.

Theorem 1.1 implies that the non-abelian SU(2) cannot be ergodic but can admit
an ergodic abelian sub-group, when ${\\{}T^{n}{\\}}_{\infty}$ is dense, i.e.,
$T^{n}{\neq}\mathbb{I}$ for $n{\in}\mathbb{N}$. Theorem 1.2 implies that if
$T$ is ergodic, an auxiliary function $f^{*}$ can be introduced to evaluate
the (Lebesgue) integration of the original $f$. The $f^{*}$ gives the
probabilities of occupying a region in phase space. $f^{*}{\circ}T=f^{*}$
renders a property similar to translation invariant, except for the set
${\\{}T^{n}{\\}}_{\infty}$ is dense rather than continuous. Accordingly,
$f^{*}=\rho$ a.e. rather than everywhere. Theorem 1.2 premises $m<\infty$.
Thus, finite $m$ is indispensable.

The ergodic subgroup is explicitly found (Appx. C) and has a geometric
interpretation (Fig. 2(a)): a spin vector rotates in a “trajectory” around a
fixed axis, and pumping is simply the projection to $-z$. Integration with
$f^{*}=\rho$ over the trajectory (Fig. 2(a)) gives the below analytic result,
which perfectly matches numerical evaluations with Eq. 2 (Fig. 2(b)).

$\begin{split}p_{\infty}=\frac{1}{2}\frac{\text{sin}^{2}(\frac{\Theta}{2})}{1-\text{cos}^{2}(\frac{\Theta}{2})\text{cos}^{2}(\Phi)}.\end{split}$
(6)

Figure 2: (color online): (a) Geometric pumping casued by “spin rotation”.
Note the rotation is not smooth, but jumping by angle ${\Delta}\eta=\delta$
each loop, and $\delta$ is given by Eq. 17. With ergodicity (unstable points),
$\delta/\pi$ is irrational, and no points are coinciding. With $n\to\infty$,
it will form a quasi-continuous trajectory composed by dense pionts. (b)
Numerical results (colored) against analytic solution (black).

What is the efficient way to pump spin? Large $\Theta$ pumps more each “shot”,
but de-pumping is also more; small $\Theta$ pumps little each time, but
accumulation of $n{\to}{\infty}$ is unclear. Eq. 6 gives exact answers to this
wondering. In above, $\Theta$ is a pure angle, but $\Phi$ contains dynamic
phases. Why Eq. 6 is geometric? Note that $p_{\infty}$ curves get flat as
$\Theta{\to}0$ or $\pi$, finally constant with $\Phi$ (Fig. 2(b)). Later we
will see $\pi$ and 0 are the physically achievable values for $\Theta$. Thus,
pumping is insensitive to dynamic details (e.g., $\textbf{B}(t)$ or $\omega$),
unless gap-closing is touched, at which $\Theta$ hops from 0 to $\pi$ or
reversely, leaving a purely geometric effect. It is more evident when
$p_{G}=\frac{1}{\pi}\int_{-\pi/2}^{\pi/2}p_{\infty}{\cdot}d\Phi$ is defined.
Since $B_{0}$ is huge, even a small fluctuation will cause a drastic change in
phases, making $\Phi$ random statistically. Thus, $p_{G}$ is the quantity
practically linked to observables. If $\Phi$ evenly distributes, analyticity
survives

$p_{G}=\frac{1}{2}\text{sin}(\Theta/2).$ (7)

In this case, instead of parallel transport Moore ; Berry , it is ergodicity
that generates a quasi-continuous trajectory (Fig. 2(a)) and lets geometry
come in. Eq. 7 rises from “0/0”, i.e., both $\omega$ and
$\Delta_{\text{min}}{\to}0$, where $\omega/{\Delta}_{\text{min}}$ is ill-
defined. This reminds us of quantum criticality Sachdev , a situation of both
$\hbar{\omega}{\to}0$ and $k_{B}T_{c}{\to}0$, where characteristic energy
fails and scale invariance emerges Sachdev ; Son . Here, geometry emerges. The
$\frac{1}{2}$ factor in Eq. 7 is noteworthy. First, it suggests geometric
pumping is fractional, different from pumps driven by photons that allow a
“whole” particle to be excited once energy quanta are matched. Second, $p_{G}$
has a ceiling of $\frac{1}{2}$, a reminiscent of the principle of maximum
entropy, because $\frac{1}{2}$ makes $S{\to}S_{\text{max}}=\text{ln}2$ for two
levels. Besides, the pumping will be quantized if $\Theta$ is quantized (as
shortly seen in the band model). A major discovery here is that the returnable
behavior is fragile against dimension perturbation, which is much unnoticed
for restriction to perfect 1D Thouless ; Fu ; Zhou  or cyclic evolution AA . A
cyclic spin-rotation considered by Aharonov and Anandan AA  corresponds to a
special case $\Theta=\pi$ here.

Definition 2: The point $({\Theta},{\Phi},\Omega)$ is stable of order $N$ for
a natural number $N$, if $\mathcal{U}^{N}(\Theta,\Phi,\Omega)$ is diagonal. If
the point is not stable for arbitrary $N$, it is unstable. Note that pumping
only takes place at unstable points, because the Hamiltonian specified by
stable $({\Theta},{\Phi},{\Omega})$ allows spin to return to the original
state within finite cycles. This is reflected by that $p_{\infty}$ diverges at
a stable point. Because, if the sequence includes every $n\in\mathbb{N}$,
$p_{\infty}$ is given by Eq. 6; if $n=mN(m\in{\mathbb{N}})$ and $N$ is the
order of stable point, we obtain a subsequence ${\\{}p_{n}{\\}}$ to make
$p_{\infty}=0$, a distinct result from Eq. 6. In fact, $p_{\infty}$ can yield
many different values by choosing ${\\{}p_{n}|n~{}\text{mod}(N)=l{\\}}$. and
$l=0,1,2...$ Thus, Eq. 6 only converges for unstable points (ergodic). The
existence of the limit follows from Theorem II.11 of Funct . For stable points
of order $N$, spin hops among $N$ states and is non-ergodic.

We prove two theorems (Appx. D) about the physical conditions for pumping to
occur.

Figure 3: (color online): (a) Stable points up to order $N=17$ for
$\Theta=\pi/2$. If $N\to\infty$, stable points form a dense subset over
$\Phi$. (b) Global view of stable points in $(\Theta,\Phi)$ phase space. The
plotted lines are a portion of the stable curves, which are actually super-
dense. (a) can be generated by a cutting line at $\Theta=\pi/2$. The points on
boundaries (i.e., $\Phi={\pm}\pi/2$ and $\Theta=0,\pi$) are stable.

Theorem 2.1. Let $A$ (resp. $\bar{A}$) denotes the set of stable (resp.
unstable) points $(\Theta,\Phi)$. (a) Points with $\Theta\in\\{0,\pi\\}$ or
$\Phi=\pm\pi/2$ are stable. (b) Fix $\Theta$ in the open interval $(0,\pi)$.
Then: (i) $\Set{\Phi}{(\Theta,\Phi)\in A}$ is dense in $[-\pi/2,\pi/2]$. (ii)
$\Set{\Phi}{(\Theta,\Phi)\in\bar{A}}$ is dense in $[-\pi/2,\pi/2]$. (c) Fix
$\Phi$ in the open interval $(-\pi/2,\pi/2)$. Then:
(i)$\Set{\Theta}{(\Theta,\Phi)\in A}$ is dense in $[0,\pi]$.
(ii)$\Set{\Theta}{(\Theta,\Phi)\in\bar{A}}$ is dense in $[0,\pi]$.

Theorem 2.2. Let $(\Phi)$, $(\Theta,\Phi)$ and $(\Theta,\Omega,\Phi)$ denote
the points in 1D, 2D and 3D phase space.

(a) In 1D ($\Theta=0$, $\pi$), $\bar{A}={\emptyset}$, i.e., $m(\bar{A})=0$,
$m(A)=1$. (b) In 2D or 3D, almost every $({\Theta},{\Phi})$ or
$({\Theta},{\Omega},{\Phi})\in\bar{A}$, i.e., for Lebesgue measure
$m(\bar{A})=1$, $m(A)=0$. The theorems can be translated into three physical
conditions/features for geometric pumping: (i) Gap closing and TPT.

(ii) Robustness to energetic/dynamic details, e.g., band gap sizes, driving
frequencies $\omega$.

(iii) System’s dimension $D>1$. Reasoning is as follow. For (i), the vertex of
angle $\Theta$ is at the degeneracy point $\textbf{B}=0$ (Fig. 1(b)), thus gap
closing is required. Then, from Eq. 7, $\Theta=\pi$ Note  gives the maximum
$p_{G}=1/2$, while $\Theta=0$ leads to $p_{G}=0$. In band models (where
$\Theta$ take discrete values $\pi$ or $0$), $\Theta=\pi$ corresponds to band
inversion, altering topological states. Thus, TPT is a necessary condition.

For (ii), the energetic/dynamic information is encoded in $\Phi$ and $\Theta$,
i.e., given a $H(t)$, it will project an image in $(\Theta,\Phi)$ space.
Pumping occurrence depends on whether the model’s regime (it is a finite
region, for real systems must have a spread) can touch the distributed areas
of $\bar{A}$. Theorem 2.1(b)(c) reveals an interesting pattern: Any finite
(compact) regime must contain both $A$ and $\bar{A}$, as they are both dense
subsets. This resembles the distribution of rational $\mathbb{Q}$ and
irrational numbers $\bar{\mathbb{Q}}$ on the real axis: any finite interval
$\mathbb{R}$ must contain $\mathbb{Q}$ and $\bar{\mathbb{Q}}$. Such
distribution entails robustness for pumping, because no matter how to adjust
the model’s parameter (e.g., by changing gap sizes, or the way approach gap-
close), no matter what size or shape of the regime the model occupies in the
$(\Theta,\Phi)$ space, encountering $\bar{A}$ is unavoidable - all because
$\bar{A}$ is so densely embedded. Remarkably, such robustness is endowed by an
unprecedented source of math. We already know the protection for a physical
effect can be established on symmetry group or topology theories TISC ; Symm ;
Weyl . Here the protection is rendered by facts in number theory. In fact,
this robustness is independent of symmetry, which is a rare virtue since rigor
often breaks down for lacking required symmetries. For example, the Mermin-
Wagner theorem (the absence of long-range order in low-dimensional systems at
finite temperatures) relies on isotropic presumption and only leads to a
tentative argument in crystals, where the needed continuous symmetry is
absent.

Moreover, the distribution of $A$ and $\bar{A}$ can be solved and analytically
expressed. In Appx. C, we find $A$ ($\bar{A}$) is a family of curves
$\text{cos}(\delta/2)=\text{cos}(\Theta/2)\text{cos}(\Phi)$ with every
$\delta/\pi\in\mathbb{Q}~{}(\bar{\mathbb{Q}})$ and plot regions of $A$ in Fig.
3(b). The diagram indicates the valid range of Eq. 6, which only converges in
$\bar{A}$ when the system is ergodic. Thus, curves in Fig. 2(b) should be
discontinuous everywhere in $\Phi$ (except for $\Theta=0,\pi$). They are a
class of bizarre functions that are integrable (areas below the curve is well-
defined), but derivatives diverge everywhere.

For (iii), it is to break the illusion that $A$ and $\bar{A}$ are of “equal”
status, since (ii) has stated that they are both infinitely dense. In fact,
$A$ and $\bar{A}$ correspond to different probability weight, namely measure.
When they are simultaneously encountered, only one subset is dominant in
probability. If $D=1$, the measure of $\bar{A}$ is vanishing, i.e.,
probability of pumping is 0 (theorem 2.2(a)). If $D{\geq}2$, oppositely, the
stable $m(A)=0$ and pumping surely occurs (theorem 2.2(b)).

The “dephasing” associated with geometric pumping is a different type. First,
the non-returning is not from “real” dissipation, but from the journey
becoming infinitely long. Second, non-returnable trajectories are
statistically more than returnable ones, which has a geometric origin that
non-close trajectories allowed by geometry will explode in number as dimension
increases ($D>1$). Since the cardinality of $A$ proves equal to $\mathbb{Q}$
(Appx. C), one can also say dephasing is due to $\mathbb{Q}$ being sparser
than $\bar{\mathbb{Q}}$. Thus, without fixing physical quantities precisely on
$\mathbb{Q}$ or $\bar{\mathbb{Q}}$, two properties of $\mathbb{Q}$ and
$\bar{\mathbb{Q}}$ still impose an influence: in arbitrary finite interval (1)
$m(\mathbb{Q})=0$; (2) both $\mathbb{Q}$ and $\bar{\mathbb{Q}}$ are dense
subsets of $\mathbb{R}$.

It is easy to extend to a band model, in which geometric pumping still occurs.
Consider two-band (spin-less)
$H=\sum_{i}d_{i}(\textbf{k};\textbf{R}(t))\sigma_{i}$, where $\sigma_{i}$
represents pseudo-spins (real spin can be restored by an expanded term
$\sigma_{i}{\otimes}s_{j}$ BandM ); $d_{i}$ is controlled by parameter R(t)
(e.g., time-dependent phonon amplitudes). Definition of topological invariants
varies with symmetries and dimensions Symm ; FieldTI ; FuInv , and here we
just use an easy model to reveal the link between $\Theta$ and TPT. We adopt
$d_{x}(k)=v+w{\cdot}\text{cos}(k{\cdot}l),d_{y}(k)=w{\cdot}\text{sin}(k{\cdot}l)$.
This model can represent a chain (lattice $l$) with two atomic sites $A,B$ in
each cell:
$H=v\sum_{i}c^{\dagger}_{i,A}c_{i,B}+w\sum_{i}c^{\dagger}_{i+1,A}c_{i,B}+h.c.$
Here, topological invariant is a $Z_{2}$ type, which is the winding number (1
or 0) of the circle spanned by $k$ around the origin of $(d_{x},d_{y})$ space
Note2 . In resemblance to spin in Fig. 1(a), we can set
$v(t)=a+\text{cos}({\omega}t),w(t)=1$. Evidently, $d_{i}$ plays the role of
$B_{i}$ and the winding number will change at $\Theta=\pi$, and the trivial
gap-close happens at $\Theta=0$. In general, a different choice of $v(t),w(t)$
(details of phonon’s effects on bands) will change “orientation” of B fields,
but not alter the angle $\Theta$. Consequently, $\Theta$ in band models is
quantized, taking values in two infinitesimal regions around 0 and $\pi$ Note3
. Basically, we have neglected inelastic phonon scattering and taken each $k$
as an independent spin model. Since gap close/opening is for a specific $k$,
pumping may apply to node change in semi-metals. The time-reversal
transformation $\mathcal{T}H\mathcal{T}^{-1}=-H$ used in derivation of Eq. 3
should be replaced by particle-hole symmetry ${\Xi}H{\Xi}^{-1}=-H$. The finite
$m$ still satisfies in the new context.

Figure 4: (color online): Geometric pumping casued by band evolution in (a)
topological insulator, (b) semi-metal. The pumping regions are highlighted.

We suggest observing geometric pumping in the vicinity of various TPT TPT
(e.g., narrow-gap topological insulator Chirag ; Liang  or semi-metals Weyl ).
Under the influence of phonons, the gap is closed and opened, causing periodic
TPT; or a semi-metal’s node position is changed (Fig. 4). Recently, a mystique
of charge pumping has been detected in ZrTe5 Chirag , where carriers hop into
upper bulk bands with below-gap pumping. The proposed geometric pumping
provides a compelling implication to establish a general framework for light-
topology quantum control experiments Luo ; Yang ; Photo ; Liu ; 1 ; 2 ; 3 .

This work reports a pumping caused by TPT, challenging the wisdom that $Z_{2}$
only causes surface observables. The pumping probability is exactly solved
(Eqs. 6, 7), showing a upper limit $\frac{1}{2}$, matching the numerical
results (Fig. 2(b)). Its geometric feature (Eq. 7) is salient. The conditions
of pumping are proved (theorems 2.1 2.2). The measure-preserving formalism
defined by Eqs. 4, 5 represents a route to incorporate geometry other than
Berry’s scheme. The rational/irrational numbers have entered physics when
dimension $D>1$, and an intriguing question is if entropy can be defined
linked to $\mathbb{Q}$ or $\bar{\mathbb{Q}}$.

Acknowledgement. This work was supported by the Ames Laboratory, the US
Department of Energy, Office of Science, Basic Energy Sciences, Materials
Science and Engineering Division under contract No. DEAC02-07CH11358.
Terahertz instrument was supported in part by the National Quantum Information
Science Research Center; Superconducting Quantum Materials Systems Center.

##  I Appendix

###  I.1 A. Derivation of evolution matrix $\mathcal{U}$.

Although $\mathcal{U}$ is not adiabatic, evolution $U_{c}$ (the arc section)
is. In the adiabatic limit, spin is rigorously aligned with B field, thus
$U_{c}$ can be expressed in rotation angles of B field.
$U_{c}=\mathcal{R}{\cdot}{\Lambda}$, where $\mathcal{R}(\alpha,\beta,\delta)$
are rotations of $\delta$ with an axis of polar and azimuthal angles $\alpha$,
$\beta$.

$\begin{split}\begin{pmatrix}\text{cos}(\frac{\delta}{2})-i\text{sin}(\frac{\delta}{2})\text{cos}(\alpha)&-i\text{sin}(\frac{\delta}{2})\text{sin}\alpha{\cdot}e^{-i\beta}\\\\[4.0pt]
-i\text{sin}(\frac{\delta}{2})\text{sin}\alpha{\cdot}e^{i\beta}&\text{cos}(\frac{\delta}{2})+i\text{sin}(\frac{\delta}{2})\text{cos}(\alpha)\end{pmatrix}\end{split}$
(8)

$\mathcal{R}$ will align spin with B. Based on the setting of Fig. 1(b),
$\mathcal{R}$ has $\alpha=\pi/2,\beta=\Omega+\pi/2,\delta=\Theta$. ${\Lambda}$
is a diagonal matrix due to the undetermined phase.

$\Lambda=\begin{pmatrix}e^{-i\Gamma(\alpha,\beta,\delta)}&0\\\
0&e^{i{\Gamma}^{\prime}(\alpha,\beta,\delta)}\end{pmatrix},$ (9)

where $\Gamma$ and $\Gamma^{\prime}$ are functions of rotation angles. In
general, $\mathcal{R}{\cdot}{\Lambda}$ is a U(2) matrix. In this instance, a
constraint narrows it down to SU(2). Time-reversal $\mathcal{T}=-i\sigma_{y}K$
and $K$ is complex conjugation. Then $\mathcal{T}U_{c}\mathcal{T}^{-1}$ is

$\begin{split}\mathcal{T}&[\mathfrak{T}~{}e^{-i{\int}^{t}_{0}H(\tau)d\tau}]\mathcal{T}^{-1}=\mathbb{I}+i\int_{0}^{t}\mathcal{T}H(\tau)\mathcal{T}^{-1}d\tau\\\
&+i^{2}\int_{0}^{t}\int_{0}^{\tau}\mathcal{T}H(\tau)\mathcal{T}^{-1}\mathcal{T}H(\tau^{\prime})\mathcal{T}^{-1}d{\tau}d{\tau}^{\prime}+...\\\
&=\mathfrak{T}~{}\text{exp}(i{\int}^{t}_{0}\mathcal{T}H(\tau)\mathcal{T}^{-1}d\tau).\end{split}$
(10)

Here, $\mathfrak{T}$ denotes time-ordering; $H(t)=-{\sum}B_{i}(t)\sigma_{i}$,
we have $\mathcal{T}H(t)\mathcal{T}^{-1}=-H(t)$. Using this property and also
$U_{c}=\mathcal{R}{\cdot}{\Lambda}$, we deduce

$\begin{split}\mathcal{T}U_{c}\mathcal{T}^{-1}=U_{c},~{}\Gamma=\Gamma^{\prime}.\end{split}$
(11)

$\Gamma=\Gamma^{\prime}$ will narrow down $\mathcal{U}$ to a SU(2). Then, the
loop evolution operator $\mathcal{U}$ is the product of straight sections
$U_{1},U_{2}$ and the arc section $U_{c}$

$\begin{split}\mathcal{U}=U_{2}U_{c}U_{1}=\mathcal{R}\mathcal{R}^{-1}U_{2}\mathcal{R}\mathcal{R}^{-1}(\mathcal{R}{\Lambda})U_{1}.\end{split}$
(12)

Note $\mathcal{R}^{-1}...\mathcal{R}$ will transform to bases
$|n_{0,1}{\rangle}$ of
$(\text{cos}(\frac{\Theta}{2}),\text{sin}(\frac{\Theta}{2})e^{i{\Omega}})^{T}$
and
$(-\text{sin}(\frac{\Theta}{2})e^{-i{\Omega}},\text{cos}(\frac{\Theta}{2}))^{T}$
(the inevitable stringent points are put on the S-pole for both branches).
With the new bases, $U_{2}^{\prime}=\mathcal{R}^{-1}U_{2}\mathcal{R}$ is
diagonalized. ($U_{1}$ is already diagonal)

$\begin{split}U_{2}^{\prime}=\begin{pmatrix}e^{-i\Phi_{2}}&0\\\
0&e^{i\Phi_{2}}\end{pmatrix},~{}U_{1}=\begin{pmatrix}e^{-i\Phi_{1}}&0\\\
0&e^{i\Phi_{1}}\end{pmatrix},\end{split}$ (13)

Define variables $\Phi_{c}=\Gamma$, $\Phi=\Phi_{1}+\Phi_{2}+\Phi_{c}$. Plug
them in to Eq. 12, we obtain

$\begin{split}\mathcal{U}=\mathcal{R}U_{2}^{\prime}{\Lambda}U_{1}=\mathcal{R}(\frac{\pi}{2},\Omega+\frac{\pi}{2},\Theta)\begin{pmatrix}e^{-i\Phi}&0\\\
0&e^{i\Phi}\end{pmatrix},\end{split}$ (14)

which is just Eq. 3.

###  I.2 B. Ergodicity, probability space, Haar measure

Here we give a physical view about definition 1. we can interpret $B$ with
$m(B)>0$ as the initial setting of an ensemble. For each step of evolution
$T$, we add the newly covered region to $B$, obtaining
${\bigcup}_{i=0}~{}T^{-i}B$. After sufficient many steps, an invariant
$T^{-1}B=B$ indicates the maximum region of the phase space that may be
reached. Therefore, $m(B)=1$ implies the system can eventually cover every
part of the phase space, i.e., either right on or arbitrarily close to any
point. An equivalent definition of ergodicity is, for instance (Theorem 1.5 of
Ebook ): If $T:X{\to}X$ is a measure-preserving transformation of the
probability space $(X,\mathfrak{B},m)$, being ergodic is that for every $A$,
$B{\in}\mathfrak{B}$ with $m(A)>0$, $m(B)>0$, there exists $n>0$ with
$m(T^{-n}A~{}{\cap}~{}B)>0$. Since $A$, $B$ are arbitrary, starting from any
region $A$, one can reach arbitrary region $B$ with sufficient steps $n$. Thus
the choice of the initial state makes no difference to an ergodic system when
distant past and future are included. “Reach” means arbitarily close to. Thus
it requires $m(A)>0$ and $m(B)>0$ for some error tolerance (allowing
unreachable points, but not areas). If $A=B$, it is called “reccurence”, a
weaker property enjoyed by all measure-preserving transformations, known as
Poincaré recurrence theorem (Theorem 1.4 of Ebook ). Approximate or
conditional measure preservation, like Wigner flow, will prevent applying
theorems developed in ergodic theory. Thus, it is crucial to establish
rigorous MPF on a suitable measure space.

The pair $(X,\mathfrak{B})$ is called a measurable space. $\mathfrak{B}$ is a
$\sigma$-algebra of subsets of $X$ satisfying (i) $X{\in}\mathfrak{B}$, (ii)
if $B{\in}\mathfrak{B}$, $X/B{\in}\mathfrak{B}$, (iii) if
$B_{n}{\in}\mathfrak{B}$ then
${\bigcup}_{n=1}^{\infty}B_{n}{\in}{\mathfrak{B}}$. Measure function on
$(X,\mathfrak{B})$ is defined as a mapping $m:\mathfrak{B}{\to}\mathbb{R}^{+}$
satisfying $m(\emptyset)=0$ and $m({\bigcup}_{n}B_{n})={\sum}_{n}m(B_{n})$,
where $B_{n}$ is a sequence of members of $\mathfrak{B}$ which are pairwise
disjoint subsets of $X$. A measure space is a triple $(X,\mathfrak{B},m)$,
where $(X,\mathfrak{B})$ is a measurable space and $m$ is a finite measure
function on it. $(X,\mathfrak{B},m)$ is a probability space if $m(X)=1$, and
$m$ is called a probability measure. Haar measure is a probability measure on
a compact group $G$ which ties with the group structure on $G$ (Sec. 0.6 of
Ebook ). Simply speaking, it is a probability measure invariant under all
group transformations and can be proved unique (Theorem 0.3 of Ebook ).
Intuitively, one can imagine a measure function as a density field, and Haar
measure is just a “uniform” density. It can be conveniently represented in a
differential form. For SU(2) group parameterized by Euler angles,
$m(\phi,\theta,\psi)=\text{sin}(2\theta){\cdot}d{\phi}{\cdot}d{\theta}{\cdot}d{\psi}$.
The extra factor “2” in the main text arises from variable substitution. For
1D rotation with a fixed axis, Haar measure is simply $m(\eta)=d{\eta}$ for
Fig. 2(a). Formally, Haar measure is defined as $m(E^{\prime})=m(E)$ for
$E^{\prime}=xE$ and $E^{\prime}=Ex$, ${\forall}x{\in}G$,
${\forall}E{\in}\mathfrak{B}$. Rigorously, $E$ belongs to the $\sigma$-algebra
$\mathfrak{B}$, but from loose point view, $E$ is just a subset of group $G$,
and $E^{\prime}=xE$ or $Ex$ is just rotating $E$ by an operation $x$ in $G$.
Then $m$ remains invariant under all these rotations.

In this case, $x$ corresponds to unitary evolution operator $\mathcal{U}{\in}$
SU(2). By the defining features of Haar measure, we have
$m(E)=m(\mathcal{U}E)=...=m(\mathcal{U}^{n}E)$, where $E$ can either stand for
a single initial state or a collection of initial states (when dealing with an
ensemble). $m(E)=m(\mathcal{U}^{n}E)$ is to replace its classical counterpart
$m(E)=m(T_{s}^{n}(p,q)E)$. The current MPF setting is not limited to SU(2),
making for additional options of other continuous groups, as it merely rests
on a math fact that continuous surjective endomorphism of a compact group will
preserve its Haar measure. We will not give proof here but emphasize, in terms
of physics application, most groups of interest are compact, such as SU$(n)$,
SO$(n)$, U$(n)$, etc. The evolution operator is bijective (physically,
bijectivity corresponds to the fact that the state’s future and past must be
uniquely determined by Schrödinger equation), thus surjectivity is satisfied
too.

###  I.3 C. Ergodic subgroup of SU(2)

An ergodic group must be abelian (Theorem 1.1). First, we find an abelian
subgroup of SU(2), which is just the group formed by rotations with a fixed
axis $\mathcal{R}(\alpha,\beta,\delta)$ with $\alpha$, $\beta$ specifying the
axis orientation and $\delta$ being the rotation angle. It satisfies
$\mathcal{R}(\alpha,\beta,\delta)\mathcal{R}(\alpha,\beta,{\delta}^{\prime})=\mathcal{R}(\alpha,\beta,{\delta}+{\delta}^{\prime})$.
For example, if we rotate with $x$-axis by $\pi/3$ counter-clockwise, we have
$\mathcal{R}(\frac{\pi}{2},0,\frac{\pi}{3})$.

In addition, to be ergodic, ${\\{}\mathcal{R}^{n}|n{\in}\mathbb{N}{\\}}$
should be dense (Theorem 1.1), which is true iff
$2{\pi}/{\delta}{\in}\bar{\mathbb{Q}}$ or
${\delta}/{\pi}{\in}\bar{\mathbb{Q}}$. Thus, the ergodic subgroup $G$ is

$G={\\{}\mathcal{R}(\alpha,\beta,\delta)|~{}{\delta}/{\pi}{\in}\bar{\mathbb{Q}}{\\}}.$
(15)

Comparing with SU(2) matrices parameterized by Euler angles
$(\phi,\theta,\psi)$

$R(\phi,\theta,\psi)=\begin{pmatrix}\text{cos}(\frac{\theta}{2})~{}e^{-i\frac{\phi+\psi}{2}}&-i{\cdot}\text{sin}(\frac{\theta}{2})~{}e^{i\frac{\psi-\phi}{2}}\\\
-i{\cdot}\text{sin}(\frac{\theta}{2})~{}e^{i\frac{\phi-\psi}{2}}&\text{cos}(\frac{\theta}{2})~{}e^{i\frac{\phi+\psi}{2}}\end{pmatrix},$
(16)

we obtain the mapping
$(\alpha,\beta,\delta){\leftrightarrow}(\phi,\theta,\psi)$:

$\begin{split}&\text{cos}({\delta}/2)=\text{cos}(\theta/2)\text{cos}(\frac{\phi+\psi}{2}),\\\
&\text{cos}^{2}{\alpha}=\frac{\text{cos}^{2}(\theta/2)\text{sin}^{2}(\frac{\phi+\psi}{2})}{1-\text{cos}^{2}(\theta/2)\text{cos}^{2}(\frac{\phi+\psi}{2})},\\\
&\beta=\frac{(\phi-\psi)}{2}+n\pi,n=0,1.\end{split}$ (17)

Comparing $R(\phi,\theta,\psi)$ with $\mathcal{U}$, we obtain the mapping of
$(\phi,\theta,\psi){\leftrightarrow}(\Theta,\Phi,\Omega)$:

$\begin{split}\phi=2{\Phi}+{\Omega}-\frac{\pi}{2},~{}\theta={\Theta},~{}\psi=-\Omega+\frac{\pi}{2}.\end{split}$
(18)

Then, we can deduce two results. The first is Eq. 6. In Fig. 2(a), spin is
located by angles $\eta$ and $\alpha$. Denote the angle between spin and
$z$-axis is $\zeta(\eta,\alpha)$, which can be found by a geometry relation
$\text{sin}(\zeta/2)=\text{sin}(\alpha){\cdot}\text{sin}(\eta/2)$. The
projection (pump) probability at a given orientation is
$\text{sin}^{2}(\zeta/2)$. $p_{\infty}$ is the following integral over the
trajectory (Fig. 2(a)), and probability density $\rho$ should be constant by
Theorem 1.2.

$\begin{split}p_{\infty}=\frac{{\int}_{0}^{2\pi}\text{sin}^{2}(\zeta/2){\cdot}{\rho}~{}d{\eta}}{\int_{0}^{2\pi}{\rho}~{}d{\eta}}=\frac{1}{2}\text{sin}^{2}(\alpha).\end{split}$
(19)

Plug Eq. 17, 18 into Eq. 19, we obtain Eq. 6.

The second result is the distribution of $A$ in phase space (Fig. 3(b)). With
Eq. 17, 18, we can express $\delta$ in terms of $\Phi$, $\Theta$. Since
$\delta{\in}\mathbb{Q}$, the stable set $A$ is

$\begin{split}{\\{}(\Theta,\Phi)|~{}\text{cos}(\Theta/2){\cdot}\text{cos}({\Phi})=\text{cos}({\delta/2}),{\delta}/{\pi}{\in}\mathbb{Q}{\\}}\end{split}$
(20)

From Above, we immediately realize that the $A$ has the same cardinality as
$\mathbb{Q}$ (thus, $\bar{A}$ has the same cardinality as $\bar{\mathbb{Q}}$).
That means the lines in Fig. 3(b) should be as many as rational number, a set
of supper dense curves.

###  I.4 D. Proof of theorems 2.1, 2.2

Proof. By the Cayley–Hamilton theorem, the matrix $\mathcal{U}$ is a root of
its characteristic polynomial
$X^{2}-X\mathrm{tr}\,\mathcal{U}+\det\mathcal{U}$ in matrix space. Thus we
have

$\begin{split}\mathcal{U}^{2}=2\cos(\Theta/2)\cos(\Phi)\cdot\mathcal{U}-\mathbb{I}.\end{split}$
(21)

Substituting $\mathcal{U}=i~{}\tilde{\mathcal{U}}$ yields

$\begin{split}\tilde{\mathcal{U}}^{2}=\tilde{\lambda}~{}\tilde{\mathcal{U}}+\mathbb{I},~{}~{}\tilde{\mathcal{U}}^{n+2}=\tilde{\lambda}~{}\tilde{\mathcal{U}}^{n+1}+\tilde{\mathcal{U}}^{n}\end{split}$
(22)

with

$\tilde{\lambda}=-i\,\mathrm{Tr}(\mathcal{U})=-2i\cos(\Theta/2)\cos(\Phi)\,,$
(23)

Taking an indeterminate $X$, the _Fibonacci polynomials_ $F_{n}(X)$ and
$E_{n}(X)$ are defined recursively by

$P_{n+2}(X)=XP_{n+1}(X)+P_{n}(X)$ (24)

with respective recursion bases

$F_{0}=0\,,\ F_{1}=1\ \mbox{ and }\ E_{0}=1\,,\ E_{1}=0\,.$

Thus $E_{n}=F_{n-1}$ for $n\in\mathbb{N}$. For example,
$F_{2}(\tilde{\lambda})=\tilde{\lambda}$,
$F_{3}(\tilde{\lambda})=\tilde{\lambda}^{2}+1$,
$F_{4}(\tilde{\lambda})=\tilde{\lambda}^{3}+2\tilde{\lambda}$, etc. Induction
with Eq. 24 confirms the expression

$\tilde{\mathcal{U}}^{n}=F_{n}(\tilde{\lambda})\cdot\tilde{\mathcal{U}}+E_{n}(\tilde{\lambda})\cdot\mathbb{I}$
(25)

Being stable ${\Leftrightarrow}~{}(\mathcal{U}^{n})_{1,2}=0$. The off-diagonal
term is

$(\mathcal{U}^{n})_{1,2}=i^{n+1}F_{n}(\tilde{\lambda})~{}e^{i{\Phi}}\text{sin}(\Theta/2).$
(26)

Let $\mathcal{F}_{n}$ be the finite set of roots of all the Fibonacci
polynomials of degree up to $n$. Then the countable set
$\mathcal{F}=\lim_{n\to\infty}\mathcal{F}_{n}$ is dense in $2i[-1,1]$. More
formally stated: as $n$ tends to infinity, the set $\mathcal{F}_{n}$ converges
to $2i[-1,1]$ in the Hausdorff metric on sets [Th. 1.1] of J. Number Theory,
163, 89-100 (2016). Theorem 2.1(a): If $\Theta=0$ or $\pi$,
$\text{sin}(\Theta/2)=0$ and $(\tilde{\mathcal{U}}^{n})_{1,2}=0$. Thus these
points are stable, independent of $\Phi$. If $\Phi={\pm}\pi/2$, we have
$\tilde{\lambda}=0$. We also have $F_{2n}(0)=0$, they are all stable points of
order 2 independent of $\Theta$. Theorem 2.1(b): Fix $\Theta\in(0,\pi)$. In
particular, note $\cos\frac{\Theta}{2}>0$. Now consider
$\varphi\in(-\pi/2,\pi/2)$ and $\varepsilon>0$. Since the inverse cosine
function is continuous at $\cos\varphi$, there is a number $\delta>0$ such
that $|\cos\varphi-\cos\Phi|<\delta$ implies $|\varphi-\Phi|<\varepsilon$.
Since $2i\cos\frac{\Theta}{2}\cos\varphi\in 2i[-1,1]$ and the set
$\mathcal{F}$ of roots of Fibonacci polynomials is dense in $2i[-1,1]$, there
is a root $2i\cos\frac{\Theta}{2}\cos\Phi\in 2i[-1,1]$ with

$|\cos\tfrac{\Theta}{2}\cos\varphi-\cos\tfrac{\Theta}{2}\cos\Phi|<\delta\cos\tfrac{\Theta}{2}\,.$

Thus $(\Theta,\Phi)\in A$ for $|\varphi-\Phi|<\varepsilon$, as required for
(i). Finally, since $\Set{\Phi}{(\Theta,\Phi)\in A}$ is a countable subset of
$[-\pi/2,\pi/2]$, its complement $\Set{\Phi}{(\Theta,\Phi)\in\bar{A}}$ is a
dense subset, yielding (ii). Theorem 2.1(c): is similar to Theorem 2.1(b).
Theorem 2.2(a): follows from Theorem 2.1(a). Theorem 2.2(b): Since stable
points correspond to roots of Fibonacci polynomials, there are only countably
many of them, and they form a set of measure zero in the 2D or 3D phase
spaces. Note that stability is independent of $\Omega$, which trivially
increases the dimension by one. $\square$

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Conference on Autonomous Agents and Multiagent Systems (AAMAS 2024)May 6 – 10,
2024 Auckland, New ZealandN. Alechina, V. Dignum, M. Dastani, J.S. Sichman
(eds.) 2024 2024 ¡¡EasyChair submission id¿¿ Google Research India Bengaluru
India Harvard University Boston USA ARMMAN Mumbai India ARMMAN Mumbai India
Carnegie Mellon University Pittsburgh USA Harvard University, Google Research
India Boston USA Google Research India Bengaluru India

# Preliminary Study of the Impact of AI-Based Interventions on Health and
Behavioral Outcomes in Maternal Health Programs

Arpan Dasgupta<EMAIL_ADDRESS>,  Niclas Boehmer<EMAIL_ADDRESS>,
Neha Madhiwalla<EMAIL_ADDRESS>,  Aparna Hedge<EMAIL_ADDRESS>,
Bryan Wilder<EMAIL_ADDRESS>,  Milind Tambe<EMAIL_ADDRESS>and  Aparna Taneja<EMAIL_ADDRESS>
###### Abstract.

Automated voice calls are an effective method of delivering maternal and child
health information to mothers in underserved communities. One method to fight
dwindling listenership is through an intervention in which health workers make
live service calls. Previous work has shown that we can use AI to identify
beneficiaries whose listenership gets the greatest boost from an intervention.
It has also been demonstrated that listening to the automated voice calls
consistently leads to improved health outcomes for the beneficiaries of the
program. These two observations combined suggest the positive effect of AI-
based intervention scheduling on behavioral and health outcomes. This study
analyzes the relationship between the two. Specifically, we are interested in
mothers’ health knowledge in the post-natal period, measured through survey
questions. We present evidence that improved listenership through AI-scheduled
interventions leads to a better understanding of key health issues during
pregnancy and infancy. This improved understanding has the potential to
benefit the health outcomes of mothers and their babies.

###### Key words and phrases:

Maternal Health, Field Study, Intervention Allocation, Restless Multi-arm
Bandits

##  1\. Introduction

Mobile Health (mHealth) programs can make essential health information
accessible to less privileged communities. These programs utilize the large
accessibility of mobile phones to spread critical health information but often
suffer from beneficiary’s loss of interest and subsequent drops in
listenership over time. To address this, interventions such as a call or a
visit from a community health worker can be an effective tool that keeps
beneficiaries engaged in the program. However, the question of who should
receive an intervention is a non-trivial prediction and planning problem.
Previous work has established that AI can be used to schedule such
interventions effectively in some mHealth programs, resulting in a significant
increase in engagement in the program Verma et al. (2023b).

We partner with ARMMAN Armman Foundation (2024a), an India-based non-profit
organization that offers mHealth programs to increase awareness of antenatal
and postnatal health amongst mothers. We focus on their mMitra Armman
Foundation (2024b) program, which is the second-largest maternal mHealth
program in the world. In this program, weekly automated voice messages deliver
essential maternal health information to the beneficiaries. There is a limited
number of live service calls that can be conducted by health workers every
week to boost beneficiaries’ engagement due to limited support staff in the
NGO. SAHELI Verma et al. (2023b), a project developed in the context of
mMitra, is the first-ever large-scale deployment of AI in a mHealth program
that effectively allocates these limited intervention resources. We study the
effects of AI-scheduled interventions on the knowledge and behavior of the
beneficiaries enrolled in mMitra through a conducted survey.

Previous studies Hegde and Doshi (2016); Murthy et al. (2020) established that
mothers who consistently listen to mMitra’s automated voice messages have
improved infant care practices and knowledge of maternal practices. In
addition, previous studies Verma et al. (2023b); Boehmer et al. (2024); Mate
et al. (2022) have also shown that AI-scheduled interventions boost engagement
in the program, as they increase the amount of time the beneficiaries listen
to the automated voice calls in a statistically significant. As a result, AI-
scheduled interventions lead to an increased exposure of beneficiaries to
critical health information. In contrast, if interventions are scheduled
uniformly at random, studies were unable to establish a significant effect
Mate et al. (2022). This suggests an intuitive correlation between AI-
scheduled interventions and improved health practices. However, no previous
work has linked the usage of AI assistance and improved health outcomes.

In this work, we aim to establish a correlation between AI-scheduled
interventions and improved behavioral and health outcomes in the mMitra
program. For this, we conduct a survey on the beneficiaries, which aims to
assess the beneficiaries’ knowledge of good health practices discussed in the
automated voice calls. We hypothesize that the increase in listenership
resulting from the AI-scheduled interventions should lead to improved
knowledge and thus better survey outcomes for people receiving an
intervention. To test this hypothesis, we conducted a randomized control trial
with two arms of beneficiaries with one _intervention arm_ receiving
interventions as scheduled by the AI and a second _control arm_ receiving no
interventions. We compare the knowledge and behavioral outcomes of
beneficiaries in the two arms using the results of the conducted survey.

In our analysis and evaluation of the study, we establish a statistically
significant increase in listenership caused by the AI-scheduled interventions.
Moreover, we also observe generally better behavior and health practices in
the intervention arm (as measured by the survey). However, we are unable to
establish a statistically significant difference between the two arms with a
high level of certainty due to small sample sizes and large amounts of noise
in the responses. Our result calls for a modified larger-scale study to better
establish our hypothesis, which we are running at the moment.

In sum, in this work, we showcase preliminary results demonstrating the impact
of AI-scheduled interventions on the health and behavior of mothers enrolled
in mMitra, a maternal health program. We also establish a statistically
significant increase in listenership in the program. The paper is organized as
follows. In Section 2, we briefly discuss previous work in this domain. In
Section 3, we discuss the setup of the study including intervention scheduling
and the conducted survey. In Section 4, we describe our analysis method for
the survey and our obtained results.

##  2\. Related Work

The problem of how to allocate limited resources comes up in several domains
which require planning. Restless multi-arm bandits (RMABs) are a popular tool
for such sequential allocation problems in uncertain environments. In
particular, RMABs have shown to be very useful in applications such as anti-
poaching patrols Qian et al. (2016), multi-channel communication Liu and Zhao
(2010), scheduling Bagheri and Scaglione (2015); Yu et al. (2018), aerial
vehicle routing Zhao et al. (2008), and machine maintenance and sensor
monitoring Glazebrook et al. (2006). These limited resource allocation
problems naturally also appear when planning interventions in mHealth programs
Mate et al. (2020).

Past work has established that the health information provided in mHealth
programs leads to improved infant care practices and knowledge of maternal
practices among mothers Hegde and Doshi (2016); Murthy et al. (2020). In
particular, Hegde and Doshi (2016) use a randomized controlled trial to
measure the effect of tailored voice calls on mothers in mMitra. Hegde and
Doshi (2016) establish statistically significant results for improved infant
care knowledge among mothers as well as a direct impact on infant health as
measured by their birth weight.

These results motivate ARMMAN to try to boost beneficiary’s listenership
through service calls by health workers.

In collaboration with ARMMAN, Mate et al. (2022) describe an AI-based method
for scheduling intervention calls. This method decides how service calls are
allocated using the RMAB framework, where each beneficiary is modeled as a
Markov decision process. Their method was initially tested in simulations, and
subsequently in a field study before it was finally deployed at scale in
practice Verma et al. (2023b). A fundamental challenge in SAHELI has been to
learn the transition probabilities of the Markov decision processes modeling
beneficiaries. After multiple refinement steps, Wang et al. (2023) and Verma
et al. (2023a) utilized decision-focused learning (DFL) Shah et al. (2022) for
RMABs to learn transition probabilities as to improve the performance of the
program in deployment.

So far, the observable objective that is optimized by SAHELI and other
intervention scheduling programs for ARMMAN is the mother’s listenership of
automated voice calls and hence program’s performance is always measured in
terms of improvement in listenership metrics. However, no correlation has been
shown between AI-scheduled interventions and behavioral outcomes, a gap that
we investigate in our study.

##  3\. Setup of the Study

As the first step in the study, we divided the registered beneficiaries into
cohorts based on their time of enrollment. Subsequently, for each cohort, we
divide the beneficiaries into intervention and control groups. The automated
voice calls containing health information are received by everyone in both
arms throughout their enrollment in the program. Those enrolled in the
intervention group are eligible for receiving interventions. The AI algorithm
decides which beneficiaries receive interventions in the form of live service
calls from health workers in a given week. In the end, a survey is conducted
on subsets from both intervention and control groups to measure the behavioral
and health knowledge of the beneficiaries.

###  3.1. Experiment Arms

####  3.1.1. Cohorts

The study was conducted in three cohorts with a combined number of $60464$
beneficiaries.

  * •

Cohort 1: 27688 beneficiaries. Registered between 15th August 2022 to 31st of
September 2022.

  * •

Cohort 2: 13972 beneficiaries. Registered between 1st October 2022 to 31st
October 2022.

  * •

Cohort 3: 18804 beneficiaries. Registered between 1st November 2022 to 31st
November 2022.

As explained in Section 3.2.1, these cohorts were not viewed as fully
independent by the program and are instead primarily used to determine when
beneficiaries are eligible for receiving an intervention.

####  3.1.2. Division Into Arms

For each cohort, we split the beneficiaries into intervention and control arms
so that attribute distributions between arms are similar. This resembles
covariate adaptive randomization Lachin et al. (1988) where distributions of
the covariates are balanced across the groups. We balance between the
following attributes:

Engagement states:

a

  * •:

For every beneficiary and a specific automated voice call, we define its
engagement state E@T at threshold T as E@T = 1 if the beneficiary listened to
at least T seconds of the call and 0 otherwise.

  * •:

For every beneficiary, we calculate E@T_w for $w$ weeks before the expected
intervention start date for the cohort.

  * •:

We strive for achieving approximately equal values of E@T_w for
$T\in\\{1,5,10,30,100\\}$ and $w\in\\{1,2,3\\}$ between arms to ensure a
similar distribution of listenership profiles across the two arms.

Demographic Features:

We consider the gestational age of beneficiaries in terms of their current
trimester as a feature for preparing cohorts. This quantity is calculated by
dividing gestational age in weeks by $14$ which gives 4 bins of equal size. We
make sure that both groups have an equal number of beneficiaries in each of
the trimesters. This also makes sure that both groups contain an equal number
of beneficiaries who have already delivered.

To ensure a balanced distribution of these attributes between arms we first
create a vector $Y$ for each beneficiary by appending the attributes. We then
split the beneficiaries into two equal groups using $Y$ as the splitting
factor Wang et al. (2023). This is done by treating $Y$ as a class label and
using a stratified splitter to split it into two equal groups. We use the
stratified option in test_train_split in the sklearn Pedregosa et al. (2011)
package. Since we have enough beneficiaries in the cohort, we are able to get
an exact split.

###  3.2. Conducting Interventions

Interventions in mMitra are service calls made by healthcare workers that aim
to boost the future listenership of automated messages of the called
beneficiary.

####  3.2.1. Number of Interventions per week.

Interventions began on 21st November 2022. We only intervene on beneficiaries
that have been present for at least $6$ weeks in the program. In the
beginning, we are only allowed to act on beneficiaries from cohort $1$, then
after some time on beneficiaries from cohorts $1$ and $2$, and then finally on
beneficiaries from all three cohorts. This was done to simulate the deployment
which considers several months of enrollments at the same time:

  * •

21st November 2022 to 12th December 2022 (4 weeks) - consider only Cohort 1
for interventions.

  * •

19th December 2022 to 9th January 2022 (4 weeks) consider Cohort 1 + Cohort 2
for interventions.

  * •

16th January 2022 to 13th February 2022 (5 weeks) consider Cohort 1 + Cohort 2
+ Cohort 3 for interventions.

We conduct approximately $1000$ interventions per week while ensuring that
each beneficiary can be intervened on only once. We end up conducting
interventions on about $12000$ beneficiaries which accounts for about $43\%$
of the intervention arm.

####  3.2.2. Eligibility for Interventions.

Beneficiaries are eligible for interventions under the following conditions.

  1. (1)

Active status - they are still enrolled in the program and receive automated
voice messages.

  2. (2)

They picked up at least 1 voice message in the last 4 weeks before the start
of the intervention period for the respective cohort.

  3. (3)

Repeat Intervention - beneficiaries have not received a previous service call.

####  3.2.3. Conducting Interventions.

Each week, the DFL-RMAB Wang et al. (2023) algorithm which is our AI algorithm
of choice, determines the set of beneficiaries from the intervention arm who
will receive an intervention. We collect all beneficiaries from the
intervention arm that have received an intervention in some week in a list
$I_{D}$. We also simulate the AI algorithm on the control arm to determine the
set of beneficiaries that would have been selected for an intervention
(assuming we conducted the same number of interventions as in the intervention
arm). As in the intervention arm, beneficiaries in the control arm cannot be
selected multiple times. We collect the beneficiaries from the control arm
that have been selected by the algorithm in some week into a list $I_{C}$. We
create an intervention list $I$ that combines $I_{D}$ and $I_{C}$. The idea is
that we later compare the behavior of beneficiaries from $I_{D}$ and $I_{C}$,
as we can think of beneficiaries from $I_{C}$ as the counterparts of those
from $I_{D}$.

###  3.3. Health Survey

####  3.3.1. Conducting the Survey

The health survey is conducted on the beneficiaries from the intervention list
$I$ between September and November in $2023$. Since the program is oriented
towards helping beneficiaries who lack resources the most, we perform the
survey only on beneficiaries who are “high-risk”. A beneficiary is considered
“high-risk” if they satisfy at least one of the following three conditions:
low level of formal education (less than grade $10$), low-income family (less
than $10000$ INR per month) or they do not own the phone themselves. Around
$50\%$ beneficiaries enrolled in the program are high-risk.

Figure 1. Questions asked in the survey.

The subset of the ”high-risk” women who give birth between the intervention
and survey call and have been in the program for at least 3 months are called
by a health worker and asked to answer the questions from the survey. However,
the survey calls are only picked up by a fraction of beneficiaries ($3376$ out
of the $6448$ called). This provides a challenge for the evaluation of our
study, as we only have access to the ”outcome” of a subset of beneficiaries,
i.e., those who were willing to answer to the survey questions (in particular,
this group of beneficiaries is not chosen uniformly at random). This makes it
for instance necessary to re-balance the control and intervention group for
the final comparison.

####  3.3.2. Survey Questions.

Each participant was asked $20$ questions with the intent of measuring their
engagement with the program and measuring knowledge in different areas such as
health practices. This evaluation is guided by the content of the automated
voice messages and assesses how well the beneficiary received the messages.
Concretely, the categories covered in the survey are: engagement with the
program, knowledge, breastfeeding practices, communication, and health
supplements. Appendix A contains a description of the questions in each
category. Figure 1 shows the list of questions. The beneficiaries obtain a
score for their answer in each question.

Out of the $20$ questions, $8$ correspond to single choice (Yes/No) questions.
The remaining $12$ correspond to questions where scores are calculated based
on multiple possible correct answers with some answers potentially
contributing a higher score than others. Beneficiaries may choose one or more
of these answers and their score is the total score for correctly identified
answers in the question.

##  4\. Results and Analysis

###  4.1. Comparison Methodology

We compare the survey responses of beneficiaries in the intervention group and
the control group. However, in doing so, we face multiple challenges. Firstly,
not everyone from the list $I$ picks up and answers the survey call. Secondly,
some beneficiaries from the intervention list $I_{D}$ of the intervention arm
do not pick up the intervention call. To ensure that the beneficiaries from
the intervention and control arms that we compare to each other are similar in
distribution, we use a matching method to pair beneficiaries from different
arms.

####  4.1.1. Beneficiary Matching

We perform a one-to-one pairing of beneficiaries that answered the survey in
the two arms by using matching methods with feature variables Rubin (1979);
Rosenbaum and Rubin (1985). The features used for this matching are the
average listenership over the last $6$ weeks from the intended date of
intervention, gestational age, and the number of children they conceived
previously. Let $X_{i}$ be the feature vector of the $i_{\text{th}}$
beneficiary and $X$ the feature matrix consisting of stacked vectors of all
beneficiaries who responded to the survey. We define a closeness metric Stuart
(2010) between the feature vectors of two beneficiaries as their Mahalonobis
distance

$D_{ij}=(X_{i}-X_{j})^{\prime}\Sigma^{-1}(X_{i}-X_{j}),$ (1)

where $\Sigma$ is the variance matrix of $X$ in the pooled dataset.

Finally, for each beneficiary in the intervention group who responded to the
survey and picked up the intervention, we greedily pick the closest
beneficiary from the control group who responded to the survey using the
above-mentioned distance metric. Once a control beneficiary is matched, we no
longer consider it for further matching. Finally, we keep the beneficiaries
that are part of one pair to obtain two sets of beneficiaries $S_{D}\subseteq
I_{D}$ and $S_{C}\subseteq I_{C}$ for the intervention and control arms,
respectively.

####  4.1.2. Improving Beneficiaries

As we will discuss in Section 4.2.2, establishing effect sizes in the full
cohort of beneficiaries is challenging, among others, because some
beneficiaries do not show an improvement in listenership after the
intervention call. Recalling that previous work Hegde and Doshi (2016) has
identified that listening to automated calls leads to better health outcomes,
we expect that beneficiaries who experienced the greatest listenership boost
through the intervention will show the most significant improvements in survey
results.

Figure 2. Score on single choice questions for beneficiaries who gained
quartiles in listenership post-interventions and their corresponding
beneficiaries from the control arm. The subset of beneficiaries from the
intervention arm in all questions. The error bars represent the standard error
in the measurement.

To this end, we formulate a method for identifying which beneficiaries have
gained the most from the intervention. We define two listenership values for
each beneficiary in the intervention list $I$, the average listenership over
the past $6$ weeks before the scheduled intervention date, called the pre-
listenership and the average listenership over the next $12$ weeks after the
scheduled intervention date, called the post-listenership. The scheduled
intervention date refers to the week in which the beneficiary from $I$ has
been selected by the algorithm independent of whether they are in the
treatment or control arm (and thereby independent of whether they actually
received the intervention). Taking $12$ weeks of post-listenership (instead of
$6$ weeks) allows us to measure the long-term gains from the interventions.

We calculate the quartile each beneficiary belongs to when compared with the
other beneficiaries in $I$ for the pre- and post-listenership. For beneficiary
$i$, we denote these values as $q_{1i}$ and $q_{2i}$, respectively. We say
that a beneficiary _gains in quartiles_ if $q_{2i}>q_{1i}$. In our analysis,
we focus on the beneficiaries that gain in listenership, i.e., for which
$q_{2i}>q_{1i}$. Specifically, in our analysis, we restrict our attention to
the beneficiaries from the set $S_{d}$ with $q_{2i}>q_{1i}$ (called $R_{d}$)
and the beneficiaries from $S_{c}$ matched to these beneficiaries (called
$R_{c}$). Our final comparison involves $218$ pairs (i.e., $436$ beneficiaries
in total).

Our method of selecting beneficiaries who have gained in quartiles is
naturally only an approximation for the beneficiaries who benefit the most
from an intervention. Note that a larger subsection of beneficiaries who
received an intervention gain quartiles ($218$) compared to those who lose
quartiles ($119$). This implies that interventions oftentimes lead to a gain
in quartiles.

Figure 3. Score on multiple choice questions for beneficiaries who gained
quartiles in listenership post-interventions and their corresponding
beneficiaries from the control arm. Except on Question $5$, beneficiaries from
the intervention arm perform on average better.  Figure 4. Score on groupings
of questions for beneficiaries who gained quartiles in listenership and their
corresponding beneficiaries from the control arm. Average scores in all
categories are higher for beneficiaries from the intervention arm.

###  4.2. Analysis

####  4.2.1. Establishing Improved Listenership

We observe a consistent gain in listenership among beneficiaries who received
interventions. In particular, following the work of Boehmer et al. (2024), we
compare (a) the beneficiaries from the intervention group that have been
selected for an intervention (i.e., they are on $I_{D}$) and that responded to
the survey call to (b) the beneficiaries from the control group that would
have been selected for an intervention (i.e., they are on $I_{C}$) and
responded to the survey call. We compare these groups using the subgroup
estimator, see the work of Boehmer et al. (2024) for details. We find that an
intervention increases beneficiaries listenership compared to the control
group by, on average, $7.43$ seconds per call over the 12 automated voice
calls following the intervention, which leads to a summed additional
listenership of $89.16$ seconds over the next $12$ calls. The $95\%$
confidence interval for this value is $[45.516,132.936]$ and the hypothesis
that interventions have a positive non-zero effect on listenership can be
accepted with a p-value of $6.4656\text{\times}{10}^{-05}$. This allows us to
conclude a statistically significant positive effect of interventions on
beneficiary’s listenership. Notably, the positive effect of interventions has
already been established in previous studies Mate et al. (2022); Verma et al.
(2023a); Boehmer et al. (2024); however, our identified effect sizes and
achieved confidence level are advantageous.

####  4.2.2. Establishing Behavioral and Health Benefits

We now analyze the beneficiary’s performance in the survey. We will start by
focusing on the beneficiaries on which interventions have been effective.
Figures 2 and 3 show the results on single choice and multiple choice
questions, respectively. We plot the average score obtained on each question
for the groups $R_{d}$ and $R_{c}$, i.e., beneficiaries from the treatment
group whose listenership improved and who answered the survey and their
counterparts. The empirical standard error is included as error bars in the
plot to signify the possible error in the comparison. Note that independent of
the group membership beneficiaries achieve generally higher scores on the
single choice questions (arguably, because it is easier to respond to Yes/No
questions during a survey).

The average score of the intervened beneficiaries from $R_{d}$ is higher than
the average score of the control beneficiaries from $R_{c}$ on all but one
question. For two questions, the difference between average scores is also
statistically significant, i.e., single choice question $5$ and multiple
choice question $3$. For the latter question (”How does the baby respond when
you speak or talk?”), we can establish a difference with a p-value of
$0.002$111Note that this value should be interpreted with a bit of caution, as
we have some dependencies and confounding factors in our study, e.g.,
regarding the beneficiaries from which we received survey results., showing an
improved awareness of the baby’s behavior. A very high variance in answers is
observed in some questions such as Question $6$ and $7$ from the multiple
choice section, which ask about when a child should be first fed milk and how
many times a day they should be fed. This trend could be due to either a lack
of information among some mothers, or due to the noise in the way these
questions were answered during the survey.

Figure 4 plots the averaged summed scores on different groups of questions. In
particular, the leftmost bars, depict the summed average score over all
questions. We find that the difference between the two arms is larger than one
standard error here and that we can establish a difference between the two
arms with a p-value of $0.13$. While this is not enough to claim statistical
significance, it is a strong hint towards the positive impact of the AI-
scheduled interventions and motivates us to conduct a larger follow-up trial.
Figure 4 also covers the summed score in questions from different categories,
that are, engagement with the program, knowledge, breastfeeding practices,
communication, and health supplements. The questions comprising each category
are provided in the appendix (Appendix A). We notice that the engagement
category provides the strongest trends and shows the largest difference
between the arms. This is likely because the relationship between listenership
and engagement with the program is a direct one, while the other outcomes
interfere with more confounding factors.

In Figures 5, 6 and 7, we show the results for all beneficiaries that filled
out the survey from the intervention group and their counterpart from the
control group, i.e., $S_{D}$ and $S_{C}$. Compared to the above analysis, we
observe slightly weaker, yet still positive trends. In particular, the
intervention group performs on average better in $7$ out of $8$ single choice
questions and $8$ out of $12$ multiple choice questions. This leads to a
better average performance across all question categories, yet the differences
between the two groups are much smaller here than on the beneficiaries that
gained quantiles. This is to be expected recalling that the gaining
beneficiaries are the ones that respond most favorably to the intervention and
will thus also be the ones with the highest knowledge gain.

##  5\. Conclusion and Future Work

In conclusion, our study has shown that AI-scheduled interventions lead to
significantly higher listenership in automated voice messages. Moreover, we
provide some first evidence that these interventions also lead to improved
behavioral and health outcomes as measured by our survey. Together with the
NGO, we aim to redesign the study with a more focused cohort and updated
survey questions to better understand and correlate the impact of
intervention-induced improved listenership on behavioral outcomes.

We thank Suresh Chaudhary, Harshit Khaitan, and Shresth Verma for helpful
discussions and for facilitating the study.

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##  Appendix A Survey Appendix

###  A.1. Questions and their answers

Single Choice Questions and expected answers:

  * •

0\. Did you know your weight at the time of delivery : Yes

  * •

1\. Did you know at the time of delivery what was the baby’s weight? : Yes

  * •

2\. During the checkup, did you ask whether the baby is developing normally? :
Yes

  * •

3\. Do you breastfeed your baby? : Yes

  * •

4\. Do you get calls informing about Mmitra? : Yes

  * •

5\. Have you regularly listened to the calls received by mMitra? : Yes

  * •

6\. Did you ever discuss with your husband/family the information you
heard/told in the call? : Yes

  * •

7\. Do you never answer or disconnect your mMitra calls when you are around
other beneficiaries? : No

Multiple Choice Questions and expected answers with scores:

  * •

0\. What did you discuss with family? : (’weight’, 0.33), (’cough’, 0.33),
(’Other Response’, 0.33), (’baby is fussy’, 0.33), (’constant watering’,
0.33), (’breastfeeding problems’, 0.33)

  * •

1\. If a weak baby is born prematurely, how to take care of it after bringing
it home? : (’entire black part of the breast’, 0.33), (’pillow under the
baby’, 1), (’one breast to the other’, 1), (’I hold the babyś head’, 0.33)

  * •

2\. How does the baby respond when you smile or talk? : (’Smiling back’,
0.33), (’Watches us’, 0.33), (’Shouts or speaks back’, 0.34)

  * •

3\. Why do you want to listen to the call? : (’regarding diet’, 0.2),
(’changes are happening’, 0.2), (’answers to some questions’, 0.2),
(’information I dont́ get from doctors’, 0.2), (’doing the right thing’, 0.2)

  * •

4\. Does anyone in your family know that you are getting calls from mMitra
with information? : (’Husband’, 0.25), (’Mother’, 0.25), (’Relatives’, 0.25),
(’in-laws’, 0.25)

  * •

5\. When should the child be fed his first milk after birth? : (’an hour
later’, 1)

  * •

6\. How many times a day do you feed your baby? : (’10-12 times’, 1), (’every
two hours’, 1)

  * •

7\. Do you eat any special food or medicine to produce more milk? :
(’Traditional food’, 0.05), (’Milk products’, 0.15), (’Iron’, 0.4),
(’Calcium’, 0.4)

  * •

8\. What are you doing or eating for your health after delivery? :
(’Traditional food’, 0.083), (’Keeping ears/feet warm’, 0.083), (‘boiling
water and drinking hot water’, 0.083), (’Calcium tablets’, 0.25), (‘Iron
pills’, 0.25), (’Vitamin supplements’, 0.25)

  * •

9\. How should the umbilical cord be taken care of until it falls immediately
after birth? : (’Must be kept clean and dry’, 0.5), (’Nothing should be
applied’, 0.5)

  * •

10\. How do you know if the child can hear properly? : (’Turns head in that
direction when there is sound’, 1.0), (’Looks at us when called’, 0.5),
(’Doctor can tell after checking up’, 1.0)

  * •

11\. Do your family members also hear these calls? : (’Sometimes’, 0.5),
(’Always’, 1.0), (’Everyone in the family’, 1.0), (’I tell them what the call
was about’, 0.25)

###  A.2. Grouping of Questions

The following is the grouping mechanism followed (SC and MC refer to single
correct and multi correct respectively):

  * •

Engagement : SC - [4, 5, 6, 7], MC -[3, 4, 11],

  * •

Knowledge : SC - [0, 1],

  * •

Breastfeeding Practices : SC - [3], MC - [5, 6, 7],

  * •

Communication : SC - [2], MC - [0],

  * •

Supplements : MC - [7, 8]

Figure 5. Average score on single choice questions for all beneficiaries.
Figure 6. Score on multiple choice questions for all beneficiaries.  Figure 7.
Score on groupings of questions for all beneficiaries.