††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Anomalous Fano factor as a signature of Bogoliubov Fermi surfaces

Sayan Banerjee  Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse
1, D-70569 Stuttgart, Germany  Institute for Theoretical Physics III,
University of Stuttgart, D-70550 Stuttgart, Germany  Satoshi Ikegaya
Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
Andreas P. Schnyder<EMAIL_ADDRESS>Max-Planck-Institut für
Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

###### Abstract

Noise spectroscopy is a key technique to investigate the nature and dynamics
of charge carriers in superconductors. The recently discovered superconducting
hybrids with Bogoliubov Fermi surfaces exhibit a particularly intriguing and
rich charge dynamics, as their charge carriers consist of both Cooper pairs
and an extensive number of Bogoliubov quasiparticles. Motivated by this, we
compute the noise spectra of Bogoliubov Fermi surfaces and identify their key
signatures in the differential conductance and the Fano factor. Specifically,
we consider a semiconductor/superconductor hybrid device with an in-plane
magnetic field, which exhibits several Bogoliubov Fermi surfaces. The number
and orientation of the Bogoliubov Fermi surfaces in this device can be readily
controlled by the applied magnetic field, which in turn alters the noise
signal. In particular, we find that the Fano factor exhibits a reduced value,
substantially lower than two, whenever the charge dynamics is governed by a
large number of Bogoliubov quasiparticles. Using experimentally relevant
parameters, we make a number of specific predictions for the noise spectra,
that can be used as direct evidence of Bogoliubov Fermi surfaces. In
particular, we find that the Fano factor as a function of magnetic field and
spin-orbit coupling exhibits characteristic discontinuities at the transition
lines that separate phases with different number of Bogoliubov Fermi surfaces.

##  I Introduction

Since the discovery of unconventional superconductors in the 1980s Bednorz and
Muller (1986); Stewart (1984); Keimer  _et al._ (2015), the study of low-
energy Bogoliubov quasiparticles (BQPs) has been an important research topic
with both fundamental and applied interest Schrieffer and Brooks (2007).
Usually, unconventional superconductors exhibit only point nodes or line nodes
in their pairing gap, with dimension strictly smaller than the normal-state
Fermi surface, due to the loss of condensation energy. However, recent
findings have shown that there can also exist two-dimensional nodes in the
superconducting gap, which form a Fermi surface of BQPs whose dimension is
equal to the normal-state Fermi surface. For instance, Bogoliubov Fermi
surfaces (BFSs) can exist in multi-band superconductors with broken time-
reversal symmetry Volovik (1989); Agterberg  _et al._ (2017); Timm  _et al._
(2017); Brydon  _et al._ (2018); Menke  _et al._ (2019), in nodal
superconductors with external Zeeman potentials Yang and Sondhi (1998);
Gubankova  _et al._ (2005); Setty  _et al._ (2020), and in superconductors
carrying large supercurrents Fulde (1965); Zhu  _et al._ (2021). In addition,
BFSs can be created in superconducting hybrids, such as semiconductor-
superconductor heterostructures with strong spin-orbit coupling (SOC) and an
external Zeeman field Phan  _et al._ (2022); Yuan and Fu (2018). The recent
experimental observation of BFSs in superconductor hybrids Phan  _et al._
(2022); Zhu  _et al._ (2021) has further stimulated this field.

The presence of BFSs with their extensive number of BQPs leads to a large
density of states at zero energy, which can be detected by a number of
experimental probes. For example, the large zero-energy density of states can
be directly observed in tunneling conductance Setty  _et al._ (2020); Lapp
_et al._ (2020) and specific heat Setty  _et al._ (2020); Lapp  _et al._
(2020), and moreover leads to characteristic signatures in quasiparticle
interference patterns Zhu  _et al._ (2021). While these experimental probes
detect the presence of zero-energy quasiparticles, they do not provide
information about their nature and dynamics. This is where noise spectroscopy
comes into play, which can measure the fundamental charge of the
quasiparticles as well as their dynamics. For example, the current-noise ratio
(i.e., Fano factor) in the tunneling limit of a fully gapped superconductor is
twice as large as in the normal state de Jong and Beenakker (1994); Anantram
and Datta (1996), indicating that the charge carriers are Cooper pairs
composed of two electrons. However, in systems with BFSs, where the charge
carriers consist of both Cooper pairs and BQPs, the Fano factor is expected to
be smaller than two, since the effective charge of the BQPs is substantially
smaller than that of Cooper pairs. Hence, noise spectroscopy can give valuable
information about the presence of BFSs and can tell us to which degree BQPs
contribute to charge transport.

Motivated by these considerations, we investigate in this Letter the Fano
factor of a semiconductor/superconductor hybrid device which exhibits BFSs
that can be controlled by an external magnetic field. After defining the model
Hamiltonian of the considered device (Sec. II), we determine in Sec. III the
regions in parameter space of SOC and field strength, where BFSs exist.
Interestingly, we find that with increasing field strength our device can be
tuned through two phase transitions, where the number of BFSs increases from
zero to two to four (Fig. 2). In Sec. IV we present our results on the
differential conduction and noise spectroscopy. We observe that both the
differential conductance and the Fano factor exhibit characteristic
discontinuities at the phase transition lines. In the tunneling limit the
value of the Fano factor reduces from two to about one, indicating that a
substantial part of the charge is carried by BQPs rather than Cooper pairs.
These results provide key fingerprints for the experimental identification of
BFSs in the considered device.

##  II Model definition

Figure 1: Schematics of the considered semiconductor/superconductor hybrid
device that exhibits Bogoliubov Fermi surfaces. An external field $\bm{B}$ is
applied in-plane and can be rotated by the angle $\theta$. A potential barrier
at $i_{x}=0$ separates the superconducting from the semiconducting region.

We consider a thin-film semiconductor/superconductor hybrid device with an
external magnetic field applied in-plane, as illustrated in Fig. 1. The
superconductor is assumed to be of conventional nature with spin-singlet
$s$-wave pairing, while the semiconductor is non-centrosymmetric with a
substantial Rashba SOC, as realized in, e.g., Al/InAs heterostructures Phan
_et al._ (2022); Kjaergaard  _et al._ (2016). The semiconductor is separated
into two regions by a potential barrier at $i_{x}=0$, whose transparency can
be controlled experimentally by applying gate voltages. Due to proximity to
the parent superconductor, the semiconductor at $i_{x}>0$ acquires an induced
$s$-wave pair potential Phan  _et al._ (2022); Kjaergaard  _et al._ (2016);
Suominen  _et al._ (2017); Kjaergaard  _et al._ (2017); Mayer  _et al._
(2019). As a result, the hybrid device realizes an effective two-dimensional
NS junction with tunable potential barrier and in-plane field. As we will see,
the high tunability of the device allows us to study the conductance and noise
signals for different BFSs configurations and junction transparencies.

Mathematically, we describe the considered device by a Bogoliubov-de Gennes
(BdG) Hamiltonian on a three-dimensional cubic lattice with lattice sites
$\bm{r}$ labelled by $\bm{r}=i_{x}\bm{x}+i_{y}\bm{y}+i_{z}\bm{z}$, where the
three lattice vectors $\bm{x}$, $\bm{y}$, and $\bm{z}$ are assumed to have
unit length $a_{0}=1$. We can separate the BdG Hamiltonian into three parts,
$H=H_{n}+H_{s}+H_{i}$, representing the semiconductor, the superconductor, and
the coupling between them, respectively. The Hamiltonian of the two-
dimensional semiconductor at $i_{z}=0$ is given by

$\displaystyle H_{n}$
$\displaystyle=\sum_{\bar{\bm{r}}}\sum_{\bar{\bm{w}}=\pm\bm{x},\pm\bm{y}}C^{\dagger}_{\bar{\bm{r}}+\bar{\bm{w}}}\left\\{-t_{n}\hat{\sigma}_{0}-\frac{i\lambda}{2}\bm{z}\cdot(\hat{\bm{\sigma}}\times\bar{\bm{w}})\right\\}C_{\bar{\bm{r}}}$
$\displaystyle+\sum_{\bar{\bm{r}}}C^{\dagger}_{\bar{\bm{r}}}\left\\{(\epsilon_{n}+X_{b}\delta_{i_{x},0})\hat{\sigma}_{0}+\bm{V}_{n}\cdot\hat{\bm{\sigma}}\right\\}C_{\bar{\bm{r}}},$
(1)

where $\hat{\bm{\sigma}}=(\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z})$
are the three Pauli matrices in spin space, and $\hat{\sigma}_{0}$ denotes the
$(2\times 2)$ unit matrix. Here,
$C_{\bar{\bm{r}}}=[c_{\bar{\bm{r}},\uparrow},c_{\bar{\bm{r}},\downarrow}]^{\mathrm{T}}$
represents a spinor composed of operators $c_{\bar{\bm{r}},\sigma}$
($c^{\dagger}_{\bar{\bm{r}},\sigma}$) which annihilate (create) electrons with
spin $\sigma$ at sites $\bar{\bm{r}}=i_{x}\bm{x}+i_{y}\bm{y}$ within the
semiconductor. $t_{n}$ denotes the nearest-neighbor hopping amplitude and
$\epsilon_{n}=4t_{n}-\mu_{n}$ is the onsite energy, with $\mu_{n}$ the
chemical potential. The strength of the Rashba SOC and the potential barrier
at $i_{x}=0$ are denoted by $\lambda$ and $X_{b}$, respectively. The in-plane
magnetic field $\bm{B}=B(\cos\theta,\sin\theta,0)$ induces a Zeeman potential
$\bm{V}_{n}=V_{n}(\cos\theta,\sin\theta,0)$ of strength $V_{n}=g_{n}\mu_{B}B$,
where $\mu_{B}$ and $g_{n}$ deonte the Bohr magneton and the $g$-factor of the
semiconductor, respectively.

The Hamiltonian of the three-dimensional superconductor, located at $i_{x}>0$
and $0<i_{z}\leq L_{s}$, reads

$\displaystyle H_{s}=$ $\displaystyle-
t_{s}\sum_{i_{x}>0}\sum_{i_{y}}\sum_{i_{z}=1}^{L_{s}}\sum_{\bar{\bm{w}}=\bm{x},\bm{y}}\left[D^{\dagger}_{\bar{\bm{r}}+\bar{\bm{w}},i_{z}}D_{\bar{\bm{r}},i_{z}}+\mathrm{H.c.}\right]$
$\displaystyle-
t_{s}\sum_{i_{x}>0}\sum_{i_{y}}\sum_{i_{z}=1}^{L_{s}-1}(D^{\dagger}_{\bar{\bm{r}},i_{z}+1}D_{\bar{\bm{r}},i_{z}}+\mathrm{H.c.})$
$\displaystyle+\sum_{i_{x}>0}\sum_{i_{y}}\sum_{i_{z}=1}^{L_{s}}D^{\dagger}_{\bar{\bm{r}},i_{z}}\left\\{\epsilon_{s}\hat{\sigma}_{0}+\bm{V}_{s}\cdot\hat{\bm{\sigma}}\right\\}D_{\bar{\bm{r}},i_{z}}$
$\displaystyle+\sum_{i_{x}>0}\sum_{i_{y}}\sum_{i_{z}=1}^{L_{s}}\Delta(d^{\dagger}_{\bar{\bm{r}},i_{z},\uparrow}d^{\dagger}_{\bar{\bm{r}},i_{z},\downarrow}+\mathrm{H.c.}),$
(2)

where
$D_{\bar{\bm{r}},i_{z}}=[d_{\bar{\bm{r}},i_{z},\uparrow},d_{\bar{\bm{r}},i_{z},\downarrow}]^{\mathrm{T}}$
is a spinor composed of operators $d_{\bar{\bm{r}},i_{z},\sigma}$
($d^{\dagger}_{\bar{\bm{r}},i_{z},\sigma}$) which annihilate (create)
electrons with spin $\sigma$ at sites $\bm{r}=\bar{\bm{r}}+i_{z}\bm{z}$ within
the superconductor. $t_{s}$ denotes the hopping amplitude in the
superconductor, $\Delta$ represents the spin-singlet $s$-wave pair potential,
and $\epsilon_{s}=6t_{s}-\mu_{s}$ is the onsite energy, with $\mu_{s}$ the
chemical potential. The Zeeman potential in the superconductor is given by
$\bm{V}_{s}=V_{s}(\cos\theta,\sin\theta,0)$, with strength
$V_{s}=g_{s}\mu_{B}B$ and $g$-factor $g_{s}$. Finally, the coupling between
the semiconductor and the superconductor is given by

$\displaystyle
H_{i}=-t_{i}\sum_{i_{x}>0}\sum_{i_{y}}(D^{\dagger}_{\bar{\bm{r}},i_{z}=0}C_{\bar{\bm{r}}}+\mathrm{H.c.}),$
(3)

with amplitude $t_{i}$. This coupling induces a finite superconducting pair
potential in the semiconductor segment $i_{x}>0$, which we compute using
lattice Greens function techniques Lee and Fisher (1981); Ando (1991).

For the numerical calculations, we set the parameters to $t_{n}=10t_{s}$,
$\mu_{n}=0.5t_{s}$, $\mu_{s}=3t_{s}$, $\Delta=0.01t_{s}$, $t_{i}=0.2t_{s}$ and
$L_{s}=20$. In addition, we fix the ratio of the $g$-factors to
$g_{n}/g_{s}=5$ and express the strength of the potential barrier by the
normalized value $Z=X_{b}/(2t_{n}\sin k_{F})$, with
$k_{F}=\arccos[1-(\mu_{n}/2t_{n})]$ the Fermi momentum of the semiconductor.
We note that the choice of $\Delta$ and $g_{n}/g_{s}$ corresponds to the
experimentally relevant regime, as observed, e.g., in Al/InAs heterostructures
Phan  _et al._ (2022); Kjaergaard  _et al._ (2016). In the following, we
calculate the differential conductance and the Fano factor as a function of
junction transparency $Z$, SOC strength $\lambda$, and Zeeman field $V_{s}$,
for different field orientations $\theta$. We note that with this choice
$V_{n}$ is given by $V_{n}=5V_{s}$.

##  III Phase diagram

Before calculating the conductance and noise spectra, we first determine the
parameter regions where BFSs emerge in the considered device. For this
purpose, we remove the semiconductor segment $i_{x}\leq 0$ and apply periodic
boundary conditions in both $x$\- and $y$-directions. By Fourier transforming
the spinors $C_{\bar{\bm{r}}}$ and $D_{\bar{\bm{r}},i_{z}}$, as
$C_{\bar{\bm{r}}}=\sum_{\bar{\bm{k}}}e^{\bar{\bm{k}}\cdot\bar{\bm{r}}}C_{\bar{\bm{k}}}$
and
$D_{\bar{\bm{r}},i_{z}}=\sum_{\bar{\bm{k}}}e^{\bar{\bm{k}}\cdot\bar{\bm{r}}}D_{\bar{\bm{k}},i_{z}}$,
the BdG Hamiltonian is simplified to

$\displaystyle
H=\frac{1}{2}\sum_{\bar{\bm{k}}}\Psi^{\dagger}_{\bar{\bm{k}}}H(\bar{\bm{k}})\Psi_{\bar{\bm{k}}},$
(4)

where $H(\bar{\bm{k}})$ is given in the Supplemental Material (SM) sm_  and
$\Psi_{\bar{\bm{k}}}=[\mathcal{C}_{\bar{\bm{k}}},\mathcal{D}_{\bar{\bm{k}},1},\cdots\mathcal{D}_{\bar{\bm{k}},L_{s}}]^{\mathrm{T}}$,
with
$\mathcal{C}_{\bar{\bm{k}}}=[C_{\bar{\bm{k}}},C^{\ast}_{-\bar{\bm{k}}}]^{\mathrm{T}}$
and
$\mathcal{D}_{\bar{\bm{k}},i_{z}}=[D_{\bar{\bm{k}},i_{z}},D^{\ast}_{-\bar{\bm{k}},i_{z}}]^{\mathrm{T}}$.
In the presence of time-reversal symmetry (TRS) and inversion, the emergence
of BFSs can be detected by use of a $\mathbb{Z}_{2}$ Pfaffian index, as
discussed in Refs. [Agterberg  _et al._ , 2017; Timm  _et al._ , 2017; Brydon
_et al._ , 2018; Zhao  _et al._ , 2016]. In the case of Hamiltonian (4),
however, the Pfaffian index is no longer well defined, since TRS and inversion
are broken by the magnetic field and Rashba SOC, respectively. Instead, we
introduce for a given wave vector $\bar{\bm{k}}=(k_{x},k_{y})$ of the two-
dimensional Brillouin zone (BZ) a $\mathbb{Z}_{2}$ determinant index $\nu$,
which can be used even in the absence of TRS and inversion. It is defined by

$\displaystyle(-1)^{\nu}=\mathrm{sgn}\left[\mathrm{det}H(\bar{\bm{k}})\times\mathrm{det}H(\bar{\bm{k}}_{0})\right],$
(5)

where $\bar{\bm{k}}_{0}$ is the (arbitrarily chosen) origin of the BZ. We note
that $\nu$ is defined modulo $2$, with $\nu=1$ indicating that the energy gap
of the BdG Hamiltonian $H(\bar{\bm{k}})$ closes at a momentum between
$\bar{\bm{k}}$ and the BZ origin $\bar{\bm{k}}_{0}$. Hence, two-dimensional
regions with $\nu=1$ in the BZ are bounded by a gapless line that forms a BFS.
This is demonstrated in Figs. 2(b)-2(d), where we plot $\nu$ in the two-
dimensional BZ for three different parameter choices indicated in Fig. 2(a),
namely $(\lambda/\mu_{n},V_{s}/\Delta,\theta/\pi)=(1.0,0.2,0)$,
$(1.0,0.2,0.5\pi)$, and $(1.0,0.3,0)$, respectively. We observe pairs of thin
banana-shaped regions where $\nu=1$, which are bounded by BFSs.

The orientation of these BFSs is controlled by the field angle $\theta$. That
is, with increasing $\theta$ the BFSs rotate counterclockwise from the 12’o
clock and 6’o clock positions for $\theta=0$ [Fig. 2(b)] to the 3’o clock and
9’o clock positions for $\theta=\pi/2$ [Fig. 2(c)]. The length of the banana-
shaped BFSs can be tuned by the Zeeman field $V_{s}$. I.e., with increasing
$V_{s}$ the BFSs become longer and longer. For sufficiently large $V_{s}$ a
phase transition point is reached, at which two more point-like BFSs appear,
which are inflated into banana shapes, by further increasing $V_{s}$. As a
function of Zeeman field $V_{s}$ and SOC $\lambda$ we observe three regions
that exhibit BFSs [phases II, III, and IV in Fig. 2(a)]. Phase II has two
BFSs, while phases III and IV have four BFSs. We note that phases III and IV
are separated by a thin line, where the two pairs of BFSs touch each other,
cf. dashed line in Fig. 2(a) and Figs. S1(b) and S1(e) in the SM. In the next
section we discuss the conductance and noise spectra of phases I, II, and III,
and the transitions between them. The transition between phases III and IV is
investigated in the SM sm_ .

Figure 2:  (a) Phase diagram of the device shown in Fig. 1 as a function of
Rashba SOC strength $\lambda$ and Zeeman field $V_{s}$. Phase II has two BFSs,
while phases III and IV have four BFSs. Panels (b), (c), and (d) show the
determinant index $\nu$, Eq. (5), in the two-dimensional Brillouin zone for
$(\lambda/\mu_{n},V_{s}/\Delta,\theta/\pi)=(1.0,0.2,0)$, $(1.0,0.2,0.5\pi)$,
and $(1.0,0.3,0)$, respectively, as indicated in panel (a). The colored
regions are bounded by BFSs.

##  IV Differential conductance and noise spectroscopy

Figure 3: Panels (a) and (b) show the normalized zero-bias differential
conductance $G/G_{N}$ as a function of Zeeman field $V_{s}$ for the field
angles $\theta=0$ and $\theta=\pi/2$, respectively, with three different
junction transparencies $Z$. Panels (c)-(h) display the differential
conductance as a function of bias voltage for field angle $\theta=0$ (top row)
and $\theta=\pi/2$ (bottom row) and junction transparency $Z=0$ (dashed lines)
and $Z=5$ solid lines. The first column [(c) and (f)] corresponds to phase I
(no BFSs) with $V_{s}=0.1\Delta$, the second column [(d) and (g)] represents
phase II (two BFSs) with $V_{s}=0.2\Delta$, while the third column [(e) and
(h)] is in phase III (four BFSs) with $V_{s}=0.3\Delta$.

Let us now turn to the transport properties, which we use to characterize the
different phases I, II, and III of Fig. 2. In order to compute the
differential conductance $G$ and the current-noise ratio $F$, we impose
periodic boundary conditions in the $y$-direction of the considered device
(Fig. 1). For concreteness we fix the value of the Rashba SOC to
$\lambda/\mu_{s}=1.0$ and study the transport properties as a function of
Zeeman field $V_{s}$, with $V_{n}=5V_{s}$. By examining Fig. 2, we find that
for $\lambda/\mu_{s}=1.0$, phase I (no BFSs) occurs for $V_{s}/\Delta<0.1405$,
phase II (two BFSs) is realized for $0.1405<V_{s}/\Delta<0.2388$, while phase
III (four BFSs) exists for $V_{s}/\Delta>0.2388$. These three phases are
marked in Fig. 3 and Fig. 4 by white, green, and grey color, respectively.

We first examine the differential conductance at zero temperature, which we
compute within the Blonder–Tinkham–Klapwijk formalism (BTK) Blonder  _et al._
(1982). For this purpose we apply a bias voltage $eV$ to our device, which
injects electrons from $i_{x}=-\infty$. The differential conductance is
evaluated by

$\displaystyle
G(eV)=\frac{e^{2}}{h}{\sum_{k_{y}}}^{\prime}\mathrm{Tr}\left[\hat{\sigma}_{0}-\hat{R}_{e}+\hat{R}_{h}\right]_{E=eV},$
(6a)
$\displaystyle\hat{R}_{\alpha}=\hat{r}_{\alpha}\hat{r}_{\alpha}^{\dagger},\quad\left(\hat{r}_{\alpha}\right)_{\sigma,\sigma^{\prime}}=r^{\alpha
e}_{\sigma\sigma^{\prime}}(k_{y},E),$ (6b)

where $r^{\alpha e}_{\sigma\sigma^{\prime}}(k_{y},E)$ for $\alpha=e$
($\alpha=h$) represents the normal (Andreev) reflection coefficient from an
electron with spin $\sigma^{\prime}$ to an electron (hole) with spin $\sigma$
at momentum $k_{y}$ and energy $E$. $\sum_{k_{y}}^{\prime}$ denotes the
summation over $k_{y}$ of the propagating channels. The reflection
coefficients are calculated using lattice Green’s function techniques Lee and
Fisher (1981); Ando (1991). In the figures we show the normalized differential
conductance $G/G_{N}$, which is obtained by dividing by the normal conductance
$G_{N}$ calculated by setting $\Delta=eV=V_{s}=V_{n}=0$.

In Figs. 3(c)-(h) we show $G/G_{N}$ as a function of bias voltage for the
three phases of Fig. 2(a) and the field angles $\theta=0$ [top row, panels
(c)-(e)] and $\theta=\pi/2$ [bottom row, panels (f)-(h)]. Figs. 3(a) and 3(b)
display $G/G_{N}$ at zero bias as a function of Zeeman field $V_{s}$ for
$\theta=0$ and $\theta=\pi/2$, respectively. Let us first focus on the
tunneling limit with low junction transparency $Z=5$ (solid lines). For the
tunneling limit we observe in all panels (c)-(h) two peaks symmetrically
located around $\pm\Delta$. These double peaks are the coherence peaks of the
parent superconductor, which are split by the applied Zeeman field (cf. Fig.
S2 of the SM sm_ ). In addition there are smaller peaks (mostly at smaller
bias), which originate from the proximity induced superconductivity of the
semimetal. Around zero-bias voltage we observe clear signatures of the BFSs,
i.e., in the tunneling limit there is residual conductance at zero bias
originating from the finite zero-energy density of states of the BFSs. The
conductance around zero bias increases with increasing $V_{s}$ and shows kink
singularities at the phase transitions points, where the number of BFSs jumps
from zero to two and from two to four (see solid lines in Figs. 3(a) and
3(b)]. Interestingly, the zero-bias conductance is larger for $\theta=\pi/2$
than for $\theta=0$. This is due to the fact that for $\theta=0$ most of the
incident electrons (i.e., those with $|k_{y}/\pi|\lesssim 0.05$) cannot couple
to the BFSs, while for $\theta=\pi/2$ almost all incident electrons can tunnel
into the BFSs [see Figs. 1(b)-(d)].

Also in the ballistic limit with a completely transparent junction ($Z=0$)
distinct signatures of the BFSs can be observed in $G/G_{N}$ around zero bias
(dashed lines in Fig. 3). That is, in the absence of BFSs all incident
electrons show perfect Andreev reflection yielding $G/G_{N}=2$, while in the
presence of BFSs only parts of the electrons Andreev reflect leading to a
suppression of $G/G_{N}$, with $1<G/G_{N}<2$. As in the tunneling limit, we
observe distinct kink singularities at the transition points that separate
phases with different numbers of BFSs [see dashed lines in Figs. 3(a) and
3(b)]. In passing, we note that the conductance spectra (particularly in the
ballistic limit $Z=0$) are not symmetric around zero bias. The reason for this
is the broken time-reversal and broken inversion symmetry, as discussed in
more detail in the SM sm_ .

Figure 4:  Fano factor $F$ within the linear response regime, Eq. (8), as a
function of Zeeman field $V_{s}$ for six different junction transparencies
$Z$. Panels (a) and (c) correspond to field angle $\theta=0$, while in panels
(b) and (d) the field angle is set to $\theta=\pi/2$. The white, green, and
grey regions correspond to phase I (no BFS), phase II (two BFSs) and phase III
(four BFSs), respectively, see Fig. 2(a).

Let us now turn to the noise spectra of the BFSs. For this purpose we compute
the current-noise ratio (Fano factor) within the linear response regime, i.e.,
for $eV\ll\Delta$, where the BFSs show a dominant contribution. The current-
noise ratio is given by the ratio $F=S/e\bar{I}$ between the zero-frequency
noise power

$\displaystyle S=2\int^{\infty}_{-\infty}\overline{\delta I(0)\delta I(t)}dt$
(7)

and the time averaged current at zero temperature $\bar{I}$. Here, $\delta
I(t)=I(t)-\bar{I}$ denotes the deviation of the current at time $t$ from the
averaged current $\bar{I}$. Within the linear response regime, the Fano factor
at zero bias simplifies to

$\displaystyle F=\frac{P(eV=0)}{eG(eV=0)},$ (8)

where the noise power $P(eV=0)$ de Jong and Beenakker (1994); Anantram and
Datta (1996) is evaluated within the BTK formalism Blonder  _et al._ (1982),
as

$P(0)=\frac{e^{3}}{h}{\sum_{k_{y}}}^{\prime}\mathrm{Tr}\left[\sum_{\alpha=e,h}\hat{R}_{\alpha}(\hat{\sigma}_{0}-\hat{R}_{\alpha})+2\hat{R}_{e}\hat{R}_{h}\right]_{E=0}.$
(9)

As before, we compute the reflection coefficients entering in the above
formula by use of lattice Greens function techniques Lee and Fisher (1981);
Ando (1991).

In Fig. 4 we present the zero-bias Fano factor, Eq. (8), as a function of
Zeeman field $V_{s}$ for six different junction transparencies $Z$. The left
column [(a) and (c)] corresponds to the field angle $\theta=0$, while the
right column [(b) and (d)] shows the results for the field angle
$\theta=\pi/2$. The different phases of Fig. 2(a), with different numbers of
BFSs, are indicated by the white, green, and grey colors, respectively. In the
tunneling limit with low junction transparency $Z=5$ the Fano factor gives us
information about the fundamental charge of the quasiparticles that tunnel
across the junction. In phase I without BFSs (white regions), the Fano factor
takes the value of two, since the tunneling current is carried exclusively by
Cooper pairs composed of two electrons. In phases II and III, however, which
exhibit a large number of BQPs forming BFSs (green and grey regions), the Fano
factor decreases to a value substantially lower than two. This indicates that
the current is carried by both Cooper pairs and BQPs, which have an effective
charge substantially smaller than that of Cooper pairs. Remarkably, if we set
the field angle to $\theta=\pi/2$, such that almost all incident electrons can
couple to the BFSs, the Fano factor drops below one [solid line in Fig. 4(c)],
implying that almost all of the current is carried by the BQPs.

In the ballistic limit with transparent junction $Z=0$, we can obtain
information about the types of scattering processes that occur at the junction
barrier. In phase I without BFSs we observe that the Fano factor is almost
zero [dotted lines in Figs. 4(a) and 4(b)], as almost all incident electrons
undergo Andreev reflection such that the noise power $P$ is almost zero. In
phases II and III with BFSs, however, the Fano factor increases, indicating
that some electrons are transformed into BQPs of the BFSs, while others
Andreev reflect. By setting the field angle to $\theta=\pi/2$, we can maximize
the coupling between the lead electrons and the BFS, such that more electrons
transmit into the BFSs, thereby increasing the Fano factor [cf. dotted lines
in Figs. 4(a) and 4(b)]. As in Fig. 3 we observe distinct kinks in the Fano
factor at the transition points that separate the different phases.

##  V Conclusions

In summary, we have investigated the differential conductance and noise
spectroscopy of a NS junction with BFSs. We find that the current-noise ratio
(Fano factor) provides key fingerprints of the presence of BFSs (see Fig. 4):
In the tunneling limit the Fano factor drops from two to about one, whenever
there are BFSs present. In the ballistic limit, on the other hand, the Fano
factor jumps from zero to a finite value, due to the presence of BFSs.
Moroever, the Fano factor as a function of Zeeman field exhibits kink
discontinuities at the transition lines that separate phases with different
number of BFSs. We hope that these predictions will help experimentalists to
identify the presence of BFSs in their superconducting hybrid devices.

We note that the creation of BFSs in semiconductor/superconductor hybrid
devices requires Zeeman fields larger than the proximity-induced pair
potential in the semiconductor, but smaller than the critical field of the
parent superconductor. This condition is realized for hybrid devices, such as
Al/InAs heterostructures Phan  _et al._ (2022); Kjaergaard  _et al._ (2016),
where the $g$ factor of the semiconductor is large, while the proximity-
induced pair potential is small compared to the parent superconductor. While
we have focused in this Letter on superconducting hybrids with BFSs, the
obtained results are of relevance also to other systems with BFSs, for
example, multi-band superconductors with broken time-reversal symmetry
Agterberg  _et al._ (2017); Timm  _et al._ (2017); Brydon  _et al._ (2018);
Menke  _et al._ (2019).

In closing, we mention several promising directions for future research.
First, it would be interesting to investigate other types of junctions, e.g.,
NSN and SNS junctions, involving superconductors with BFSs, and to determine
the signatures of BFSs in the Josephson current. Experimentally, it should be
straightforward to fabricate NSN and SNS junctions using existing thin film
technology. Second, it would be of interest to study thermal transport
properties of BFSs. While conventional superconductors are prefect thermal
insulators at low temperature, superconductors with BFSs are expected to show
substantial thermal transport carried by the large density of BQPs near zero
energy. Third, the interplay between BFSs and Andreev surface bound states (or
interface bound states) and its effects on the transport properties would be a
worthwhile topic for future research.

###### Acknowledgements.

The authors thank O. Maistrenko and P. Bonetti for useful discussions. S.I. is
supported by the Grant-in-Aid for JSPS Fellows (JSPS KAKENHI Grant No.
JP21J00041) and the JSPS Core-to-Core Program (No. JPJSCCA20170002). S.B.
thanks the MPI-FKF for financial support.

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Supplemental Material for
“Anomalous Fano factor as a signature of Bogoliubov Fermi surfaces”

Authors: Sayan Banerjee, Satoshi Ikegaya, and Andreas P. Schnyder

In this supplemental material we provide the expression for the momentum space
Hamiltonian $H(\bar{\bm{k}})$ in Eq. (4) of the main text (Sec. S1) and
discuss the transition between phases III and IV (Sec. S2). In addition we
present in Sec. S3 two additional figures of the differential conductance $G$,
one without Zeeman field, and one comparing the two field angles
$\theta=-0.5\pi$ and $\theta=0.5\pi$.

##  S1 Momentum space Hamiltonian $H(\bar{\bm{k}})$

The momentum space Hamiltonian $H(\bar{\bm{k}})$ in Eq. (4) of the main text
is given by

$\displaystyle
H(\bar{\bm{k}})=\left[\begin{array}[]{cccccc}\check{H}_{n}(\bar{\bm{k}})&\check{T}_{i}&&&&\mbox{\Large
0}\\\ \check{T}_{i}&\check{H}_{s}(\bar{\bm{k}})&\check{T}_{s}&&\\\
&\check{T}_{s}&\check{H}_{s}(\bar{\bm{k}})&\check{T}_{s}&&\\\
&&\ddots&\ddots&\ddots&\\\
&&&\check{T}_{s}&\check{H}_{s}(\bar{\bm{k}})&\check{T}_{s}\\\ \mbox{\Large
0}&&&&\check{T}_{s}&\check{H}_{s}(\bar{\bm{k}})\\\ \end{array}\right],$ (S7)
$\displaystyle\check{H}_{n}(\bar{\bm{k}})=\left[\begin{array}[]{cc}\hat{h}_{n}(\bar{\bm{k}})&0\\\
0&-\hat{h}^{\ast}_{n}(-\bar{\bm{k}})\end{array}\right],\quad\check{H}_{s}(\bar{\bm{k}})=\left[\begin{array}[]{cc}\hat{h}_{s}(\bar{\bm{k}})&\Delta(i\hat{\sigma}_{y})\\\
-\Delta(i\hat{\sigma}_{y})&-\hat{h}^{\ast}_{s}(-\bar{\bm{k}})\end{array}\right],\quad\check{T}_{i(s)}=\left[\begin{array}[]{cc}-t_{i(s)}\hat{\sigma}_{0}&0\\\
0&t_{i(s)}\hat{\sigma}_{0}\end{array}\right],$ (S14)
$\displaystyle\hat{h}_{n}(\bar{\bm{k}})=\left[\begin{array}[]{cc}\xi_{n}(\bar{\bm{k}})&\lambda(\bar{\bm{k}})+V_{n}e^{-i\theta}\\\
\lambda^{\ast}(\bar{\bm{k}})+V_{n}e^{i\theta}&\xi_{n}(\bar{\bm{k}})\end{array}\right],\quad\hat{h}_{s}(\bar{\bm{k}})=\left[\begin{array}[]{cc}\xi_{s}(\bar{\bm{k}})&V_{s}e^{-i\theta}\\\
V_{s}e^{i\theta}&\xi_{s}(\bar{\bm{k}})\end{array}\right],$
$\displaystyle\xi_{n(s)}(\bar{\bm{k}})=-2t_{n(s)}(\cos k_{x}+\cos
k_{y})-\epsilon_{n(s)},\quad\lambda(\bar{\bm{k}})=\lambda(\sin k_{y}+i\sin
k_{x}),$

where $t_{n(s)}$ denotes the hopping amplitude in the semiconductor
(superconductor), and $\epsilon_{n}=4t_{n}-\mu_{n}$
($\epsilon_{s}=6t_{s}-\mu_{s}$) with $\mu_{n(s)}$ representing the chemical
potential in the semiconductor (superconductor), and the Zeeman potential in
the semiconductor (superconductor) is given by
$\bm{V}_{n(s)}=V_{n(s)}(\cos\theta,\sin\theta,0)$. The strength of the Rashba
spin-orbit coupling potential in the semiconductor is represented by
$\lambda$, and the pair potential in the superconductor is denoted by
$\Delta$. The hopping integral between the semiconductor and the
superconductor is given by $t_{i}$. The Pauli matrices in spin space are given
by $\hat{\bm{\sigma}}=(\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z})$,
and $\hat{\sigma}_{0}$ denotes the $(2\times 2)$ unit matrix.

##  S2 Transition between phases III and IV of Fig. 2

Figure S1: Panels (a), (b), and (c) show the determinant index $\nu$, Eq. (5),
in the two-dimensional Brillouin zone for $\lambda/\mu_{n}=0.04$, $0.083$, and
$0.12$, respectively. The Zeeman field and the field angle are fixed to
$V_{s}/\Delta=0.24$ and $\theta=0$. The colored regions are bounded by BFSs.
Panels (d), (e), and (f) show the energy dispersion of $H(\bar{\bm{k}})$, Eq.
(S7), for the same parameters as in (a), (b), and (c), respectively, with
$k_{y}=0$ and as a function of $k_{x}$. Note that panels (a)/(d) and (c)/(f)
correspond to phases IV and III, respectively, while panels (b)/(e) are at the
phase boundary, cf. Fig. 1(a) of the main text.

In this section, we study the phase transition between the phases III and IV
in Fig. 2(a) of the main text. In what follows, we fix $V_{s}/\Delta=0.24$ and
$\theta=0$, and vary the strength of the Rashba SOC $\lambda$. Other
parameters are chosen to be the same as those in the main text. The phase
boundary between the phases III and IV is located at $\lambda/\mu_{n}\simeq
0.083$. In Figs. S1(a), (b), and (c), we show the determinant index $\nu$, Eq.
(5), in the two-dimensional Brillouin zone for the parameters
$\lambda/\mu_{n}=0.04$ (phase IV), 0.083 (phase boundary), and 0.12 (phase
III), respectively. We observe that with increasing $\lambda/\mu_{n}$, the two
pairs of BFSs approach each other, until they merge at $\lambda/\mu_{n}\simeq
0.083$, and then separate again.

In Figs. S1(d), (e), and (f), we show the energy dispersion $E(\bar{\bm{k}})$
of the momentum space Hamiltonian $H(\bar{\bm{k}})$, Eq. (S7), for the same
parameters as in Figs. S1(a), (b), and (c), respectively, for fixed $k_{y}=0$
and as a function of $k_{x}$. Here, we focus on the pair of BFSs at 3’o clock,
i.e., near $k_{x}=0.074\pi$. We observe two parabolas, one with positive
curvature and one with negative curvature, whose crossing points with the
Fermi level define the BFSs. This is indicated by the shaded regions, which
correspond to the momenta inside the BFSs. Interestingly, in phase IV the
positive-curvature and negative-curvature parabolas intersect each other
[panel (a)], while in phase III they avoid each other [panel (f)]. At the
phase transition the two parabolas touch each other [panel (a)], such that the
two pairs of BFSs merge together. Hence, the transition from phase III to
phase IV represent a Lifshitz-type transition, where the topology of the BFSs
changes.

##  S3 Additional figures of the differential conductance

In Fig. S2(a) we show the normalized differential conductance $G/G_{N}$ as a
function of bias voltage in the absence of a Zeeman field, i.e., for
$V_{s}=V_{n}=0$. In this case the spectra of the BdG Hamiltonian are symmetric
with respect to zero energy for any fixed $k_{y}$, due to time-reversal and
particle-hole symmetry. This leads to completely symmetric differential
conductance spectra, i.e., $G/G_{N}$ does not change upon reversing the bias
voltage. In the device of Fig. 1 the parent superconductor with pairing gap
$\Delta$ induces a smaller gap $\Delta_{\textrm{ind}}$ in the semiconductor,
i.e., $\Delta_{\textrm{ind}}<\Delta$. This leads to three different voltage
regimes with different conductance characteristics. Let us first discuss this
for the case of perfect junction transparency $Z=0$ (dashed line): (i) For
bias voltage larger than $\Delta$ there is no Andreev reflection possible
leading to $G/G_{N}=1$, (ii) for bias voltage smaller than $\Delta$ but larger
than $\Delta_{\textrm{ind}}$, Andreev reflection occurs only for electrons
with suitable momenta, mostly via the parent superconductor, leading to
$1<G/G_{N}<2$, and (iii) for bias voltage smaller then
$\Delta_{\textrm{ind}}$, perfect Andreev reflection is possible for all
incident electrons, yielding $G/G_{N}=2$. Accordingly, we observe in the
tunneling limit $Z=5$ (solid line) sharp peaks at the boundary of these
regimes.

In Fig. S2(b) we show the normalized differential conductance $G/G_{N}$ for
opposite field orientations, i.e., for $\theta=-0.5\pi$ and $\theta=+0.5\pi$
with $V_{s}=0.2\Delta$ (see solid and dashed lines, respectively). In the
presence of a Zeeman field $V_{s}$ with $\theta\neq 0$ the BdG spectra acquire
an energy shift which scales linearly with $k_{y}$ and $V_{s}$ with an
opposite sign for the two BdG bands Yuan and Fu (2018). Thus, the BdG spectra
are no longer symmetric with respect to zero-energy for fixed $k_{y}$, in line
with the broken time-reversal symmetry. As a consequence, the three voltage
regimes discussed above shift in an asymmetric fashion with respect to zero
energy, as can be seen in Fig. S2(b) and Figs. 3(f)-(h). Note that upon
flipping the sign of the Zeeman potential (i.e., by letting
$\theta\to\theta+\pi$) the energy shifts of the BdG spectra change sign,
leading to a reflection of the differential conductance spectra with respect
to zero-energy (compare solid and dashed lines).

Figure S2:  (a) Differential conductance in the absence of a Zeeman field,
i.e., $V_{s}=V_{n}=0$, for $\lambda/\mu_{s}=1.0$. The solid and dashed lines
represent the tunneling limit ($Z=5$) and ballistic limit ($Z=0$),
respectively. (b) Differential conductance for oppositely oriented fields for
$V_{s}=0.2\Delta$, $\lambda/\mu_{s}=1.0$, and $Z=0$. The solid and dashed
lines show to the results for $\theta=-0.5\pi$ and $\theta=0.5\pi$,
respectively.


Toeplitz $S$-adic shifts in Section 6.2, and we lastly consider measurable
eigenvalues in Section 6.3.

###  6.1. Eigenvalues and constant coboundaries

We first start with the following remark, easily providing continuous
eigenvalues.

######  Remark 6.1.

Let $(\sigma_{n})$ be a recognizable and constant-length directive sequence
with sequence of lengths $(q_{n})$. Note that for any $m$,
$\lambda=e^{\frac{2\pi i}{q_{0}\dots q_{m}}}$ is an eigenvalue of the
recognizable $S$-adic system of constant length $(X_{\bm{\sigma}},T)$. For, as
$h_{m+1}(a)=q_{0}q_{1}\dots q_{m}$, for each letter $a$, we can use the
partition $\mathcal{P}_{m+1}=\\{T^{k}\sigma_{[0,m+1)}([a\cdot
b]):\,a,b\in\mathcal{A},ab\in\mathcal{L}_{\bm{\sigma}}^{(m+1)},0\leq
k<q_{0}q_{1}\dots q_{m}\\}$ to build an eigenfunction. Namely, let
$f(x)=\lambda^{k}$ if $x\in T^{k}\sigma_{[0,m+1)}([a\cdot b])$. Then $f$ is a
continuous eigenfunction for the eigenvalue $\lambda$.

We can translate the previous remark to information about equicontinuous
factors of $(X_{\bm{\sigma}},T)$. Indeed, we first recall that if
$\exp(2i\pi/r)$ is an eigenvalue of $(X,T)$, then the addition of $1$ on the
finite group ${\mathbb{Z}}/r{\mathbb{Z}}$ is a factor of $(X,T)$. We now
extend this fact to the case where $\exp(2i\pi/q_{n})$ is an eigenvalue of
$(X,T)$ for each $n$. We first recall the definition of an odometer associated
to a sequence $(q_{n})_{n}$ of positive integers, where we assume $q_{n}\geq
2$. Given a sequence $(q_{n})_{n\geq 1}$ of positive integers, define the
group

${\mathbb{Z}}_{(q_{n})}:=\prod_{n\geq 1}{\mathbb{Z}}/q_{n}{\mathbb{Z}},$

where the group operation of addition is performed with carry. For a detailed
exposition of equivalent formal definitions, we refer the reader to [Dow05].
Endowed with the product topology over the discrete topology on each
${\mathbb{Z}}/q_{n}{\mathbb{Z}}$ , the group ${\mathbb{Z}}_{(q_{n})}$ is a
compact metrizable topological group, where the multiples of the unit
$z=\dots..001$, which we simply write as $z=1$, are dense. We write elements
$(z_{n})_{n\geq 1}$ of ${\mathbb{Z}}_{(q_{n})}$ as left-infinite sequences
$\dots z_{2}z_{1}$ where $z_{n}\in{\mathbb{Z}}/q_{n}{\mathbb{Z}}$, so that
addition in ${\mathbb{Z}}_{(q_{n})}$ has the carries propagating to the left
in the usual way as in ${\mathbb{Z}}$. If $q_{n}=p$ is constant, then
${\mathbb{Z}}_{(q_{n})}={\mathbb{Z}}_{p}$ is the classical ring of $p$-adic
integers, and the addition of $1$ on the $p$-adic group ${\mathbb{Z}}_{p}$ is
a factor if and only if $\exp(2i\pi/p^{n})$ is an eigenvalue for every $n\geq
1$ (se e.g. [Fog02]). Given $(q_{n})$ and $\tilde{h}\in{\mathbb{N}}$, we use
${\mathbb{Z}}_{\tilde{h},(q_{n})}$ to denote the group defined by the sequence
$\tilde{h},q_{1},q_{2},\dots$. Finally recall that an odometer is a dynamical
system $(Z,+1)$ where $Z={\mathbb{Z}}_{(q_{n})}$ for some sequence $(q_{n})$
and $+1$ denotes the homeomorphism of adding the unit $1$.

The following arithmetic lemma will be essential when finding either
measurable or topological eigenvalues of a constant-length $S$-adic shift. It
can be considered as an analogue of the discussion in Section (2.2) in [Hos86]
(see also Lemma 4.16). We stress the fact that in the constant-length case,
coboundaries $h$ are constant (meaning that they do not depend on letters),
since being given by limits of the form $h=\lim_{n}\lambda^{q_{0}\dots
q_{n}}$. The following statement is motivated by the fact that taking the
quotient of two successives terms yields $\lim_{n}\lambda^{q_{0}\dots
q_{n}(q_{n+1}-1)}=1$, from which we can deduce crucial information in the
finitary case.

######  Lemma 6.2.

Let $\lambda\in S^{1}$, let the sequence $(q_{n})$ of integers take on
finitely many values, and suppose that $q_{n}\geq 2$ for all $n$. If

(6.1) $\lim_{n}\lambda^{q_{0}\dots q_{n}(q_{n+1}-1)}=1,$

then $\lambda$ is rational, and more precisely, there exist integers $m$ and
$k$ such that

$\lambda=e^{2\pi i\frac{k}{q_{0}\dots q_{m}(q_{m+1}-1)}}.$

###### Proof.

Let $\lambda=e^{2\pi it}$. By assumption

(6.2) $q_{0}\dots q_{n}(q_{n+1}-1)t=k_{n}+\varepsilon_{n}$

where $k_{n}\in{\mathbb{Z}}$ and $\varepsilon_{n}\rightarrow 0$. Let

$\displaystyle r_{n}:=q_{n+1}\frac{q_{n+2}-1}{q_{n+1}-1}.$

Note that the sequence $(r_{n})$ takes on finitely many values. We have that

$\displaystyle k_{n+1}-r_{n}k_{n}$ $\displaystyle=q_{0}\dots
q_{n+1}(q_{n+2}-1)t-\varepsilon_{n+1}-r_{n}\left(q_{0}\dots
q_{n}(q_{n+1}-1)t-\varepsilon_{n}\right)$ $\displaystyle=q_{0}\dots
q_{n+1}(q_{n+2}-1)t-r_{n}(q_{0}\dots
q_{n}(q_{n+1}-1)t)-\varepsilon_{n+1}+r_{n}\varepsilon_{n}$
$\displaystyle=-\varepsilon_{n+1}+r_{n}\varepsilon_{n}.$

Since the sequence $(r_{n})_{n}$ takes on finitely many values and
$\varepsilon_{n}\rightarrow 0$, we get $k_{n+1}-r_{n}k_{n}\rightarrow 0.$
Since in addition $(k_{n})$ is integer valued, we have
$k_{n+1}-r_{n}k_{n}=-\varepsilon_{n+1}+r_{n}\varepsilon_{n}=0$ for $n$ large
enough. But then recursively we get $\varepsilon_{n+\ell}=r_{n+\ell-1}\dots
r_{n+1}r_{n}\varepsilon_{n}$ for all $\ell\geq 1$. Since $r_{j}>1$ for each
$j$, we conclude that $\varepsilon_{n}=0$ for all $n$ large. Let
$m\in{\mathbb{N}}$ be such that for all $n\geq m$, $k_{n+1}-r_{n}k_{n}=0$
(i.e., $\varepsilon_{n}=0$). Recursively, we obtain that for all $n\geq 1$

$\displaystyle k_{m+n}=r_{m+n-1}r_{m+n-2}\dots r_{m}k$

where $k:=k_{m}$. Thus for all $n\geq 1$ we have by (6.2)

(6.3) $\displaystyle t$ $\displaystyle=\frac{r_{m+n-1}r_{m+n-2}\dots
r_{m}k}{q_{0}\dots q_{m+n}(q_{m+n+1}-1)},$

where

$\displaystyle\frac{r_{m+n-1}r_{m+n-2}\dots r_{m}k}{q_{0}\dots
q_{n+m}(q_{n+m+1}-1)}$
$\displaystyle=\frac{q_{m+n}\frac{q_{m+n+1}-1}{q_{m+n}-1}q_{m+n-1}\frac{q_{m+n}-1}{q_{m+n-1}-1}\dots
q_{m+1}\frac{q_{m+2}-1}{q_{m+1}-1}k}{q_{0}\dots q_{m+n}(q_{m+n+1}-1)}$
$\displaystyle=\frac{q_{m+1}q_{m+2}\dots q_{m+n}(q_{m+n+1}-1)k}{q_{0}\dots
q_{m+n}(q_{m+n+1}-1)(q_{m+1}-1)}$ $\displaystyle=\frac{k}{q_{0}\dots
q_{m}(q_{m+1}-1)},$

and this last expression does not depend on $n$. The statement of the lemma
follows. Writing $k$ and $q_{0}\dots q_{m}(q_{m+1}-1)$ in their unique prime
factorization and simplifying the common factors, we even get an irreducible
quotient $\frac{p}{q}$, where the prime factors of $q$ are in those of
$\\{q_{0},\cdots,q_{m},q_{m+1}-1\\}$.

∎

######  Example 6.3.

We give an example to show that the finitary assumption (i.e., of finitely
many values $q_{n}$) in Lemma 6.2 cannot be relaxed without additional
qualification. Take $\lambda=e^{2\pi i\sum_{k=1}^{\infty}\frac{1}{10^{k!}}}$.
The exponent $t=\sum_{k=1}^{\infty}\frac{1}{10^{k!}}$ is the Liouville number,
known to be transcendental. Take $(q_{n})$ such that $q_{0}q_{1}\dots
q_{n}=10^{n!}$ for $n\geq 1$. Then $q_{0}\dots
q_{n}\sum_{k=1}^{\infty}\frac{1}{10^{k!}}\mod{\mathbb{Z}}\equiv
10^{n!}\sum_{k=n+1}^{\infty}\frac{1}{10^{k!}}\rightarrow_{n}0$, so (6.1) is
satisfied.

######  Remark 6.4.

If $\lambda$ is rational, i.e., if $\lambda=e^{2i\pi t}$ and
$h(a)=\lim_{n\rightarrow\infty}\lambda^{h_{n}(a)}$ for some $a$, then $h(a)$
is rational. Note though that $h(a)$ rational does not necessarily imply that
$\lambda$ is rational: take e.g. the case of a trivial coboundary $h=1$ with
$\sigma$ being the Fibonacci substitution $\sigma:a\mapsto ab,b\mapsto a$. The
topological eigenvalues of this Pisot irreducible substitution belong to
${\mathbb{Z}}[\varrho]+{\mathbb{Z}}$, where $\varrho=\frac{1+\sqrt{5}}{2}$
(see Section 4.4). However, if we are in the finitary constant-length case,
Lemma 6.2 implies that $\lambda$ has to be rational.

In the case where the directive sequence consists of constant-length
substitutions, inspection of the proof of Theorem 4.4 yields that a coboundary
$h$ associated to a continuous function is constant, as in the constant-length
substitution case, and also that we can lighten the hypothesis concerning
fully essential words. Indeed, in Theorem 4.4 we require that for each
$a\in\mathcal{A}$, there exists $\ell\in\mathcal{A}$ such that $a\ell$ is a
fully essential word. Below, with Theorem 6.6, we just need that there exists
a fully essential word of length 2. In this statement we can also relax
Theorem 4.5 in the sense that we do not need each substitution to occur
infinitely ofter here. Note that in the majority of cases, we can boundedly
telescope to obtain a fully essential word of length two.

Before stating Theorem 6.6, let us add a few words concerning recognizability.
Recall that two-sided recognizability does not in general imply one-sided
recognizability. In fact, as we are concerned with dynamical recognizability,
and not Mossé’s combinatorial version [Mos92], there even exist substitutions
that are injective on letters, but not one-sided recognizable. Consider for
instance the following example provided to us by Dominique Perrin (see also
[BBPR22])

$\displaystyle a$ $\displaystyle\stackrel{{\scriptstyle\sigma}}{{\mapsto}}ba$
$\displaystyle b$ $\displaystyle\stackrel{{\scriptstyle\sigma}}{{\mapsto}}aa.$

Since $\sigma$ is a primitive, aperiodic and constant-length substitution
which is injective on the letters, it is one-sided recognizable in the
combinatorial sense of Mossé. However, the sequence
$x=a\sigma(a)\sigma^{2}(a)\sigma^{3}(a)\cdots\in\tilde{X}_{\sigma}$ has two
different centered $\sigma$-representations in $\tilde{X}_{\sigma}$, namely
$x=T(\sigma(ax))=T(\sigma(bx))$, thus it is not dynamically one-sided
recognizable.

Furthermore, while primitive aperiodic substitutions of constant length are
always two-sided recognizable, this is not the case for directive sequences of
constant-length substitutions, as illustrated by [BSTY19, Example 4.3].
Nevertheless some conclusions about recognizability are easier to achieve in
the case where the directive sequence consists of constant-length
substitutions, as the following lemma illustrates.

######  Lemma 6.5.

Let $\bm{\sigma}=(\sigma_{n})_{n\geq 0}$ be a primitive, constant-length
directive sequence on $\mathcal{A}$, with length sequence $(q_{n})_{n\geq 0}$.
Suppose that each $\sigma_{n}$ is injective on letters, right-permutative, and
that the two-sided shift $(X_{\bm{\sigma}},T)$ is recognizable. Then the one-
sided shift $(\tilde{X}_{\bm{\sigma}},T)$ is recognizable.

###### Proof.

First we claim that for each $n$,
$\sigma_{n}:\tilde{X}_{\bm{\sigma}}^{(n+1)}\rightarrow\sigma_{n}(\tilde{X}_{\bm{\sigma}}^{(n)})$
is injective. For suppose not. Then there is an $n$,
$\tilde{x}\in\sigma_{n}(\tilde{X}_{\bm{\sigma}}^{(n+1)})$ and
$\tilde{y}\neq\tilde{z}$, both elements of $\tilde{X}^{(n+1)}$, such that
$\tilde{x}=\sigma_{n}(\tilde{y})=\sigma_{n}(\tilde{z})$. Let $\ell$ be the
first index such that $\tilde{y}_{\ell}\neq\tilde{z}_{\ell}$. Then
$\tilde{x}_{[q_{n}\ell,q_{n}(\ell+1))}=\sigma_{n}(\tilde{y}_{\ell})=\sigma_{n}(\tilde{z}_{\ell})$,
contradicting injectivity of $\sigma_{n}$. By shifting, and using the
assumption that each $\sigma_{n}$ is right-permutative, we conclude that
$T^{j}\sigma_{n}:\tilde{X}_{\bm{\sigma}}^{(n+1)}\rightarrow
T^{j}\sigma_{n}(\tilde{X}^{(n)})$ is also injective for each $0\leq j\leq
q_{n}-1$. Suppose that for some $n$, a point
$x\in\tilde{X}_{\bm{\sigma}}^{(n)}$ has at least two centred representations
$x=T^{j}\sigma_{n}(y)=T^{k}\sigma_{n}(z)$ with
$y,z\in\tilde{X}_{\bm{\sigma}}^{(n+1)}$ and $0\leq j,k<q_{n}$. By the remarks
above, $k\neq j$, and we can assume $j=0$. The assumption of two-sided
recognizability implies that all sufficiently long words in
$\mathcal{L}_{\bm{\sigma}}^{(n)}$ can be uniquely desubstituted away from the
left and right border up to some constant $c=c(\sigma_{n})$. But then, if $N$
is some sufficiently large multiple of $q_{n}$, then
$x_{[N,2N)}=\sigma_{n}(y_{[\frac{N}{q_{n}},\frac{2N}{q_{n}})})=T^{k}\sigma_{n}(z_{[\frac{N}{q_{n}},\frac{2N}{q_{n}})})$,
a contradiction since we have desubstitued $x_{[N+c,2N-c)}$ in two ways. ∎

We can now state the following theorem.

######  Theorem 6.6.

Let $\bm{\sigma}=(\sigma_{n})_{n\geq 0}$ be a primitive, constant-length
directive sequence on $\mathcal{A}$, with length sequence $(q_{n})_{n\geq 0}$,
where each $q_{n}\geq 2$. Suppose that $\bm{\sigma}$ has a fully essential
word of length 2. If $\lambda\in S^{1}$ is a continuous eigenvalue of
$(X_{\bm{\sigma}},T)$, then

(6.4) $h:=\lim_{n\rightarrow\infty}\lambda^{q_{0}\dots q_{n}}$

exists and defines a constant coboundary. If in addition
$\bm{\sigma}=(\sigma_{n})_{n\geq 0}$ is finitary, and either

  * •

$\bm{\sigma}$ is prefix-straight and one-sided recognizable, or

  * •

$\bm{\sigma}$ is straight and recognizable,

then $\lambda$ is a continuous eigenvalue if and only if

(6.5) $h:=\lim_{n\rightarrow\infty}\lambda^{q_{0}\dots q_{n}}$

exists and is a coboundary. Moreover, each continuous eigenvalue $\lambda$ is
rational, and there exists $\tilde{h}\in{\mathbb{N}}$

  * •

which is coprime to each $q_{n}$, and

  * •

which divides $p:=\prod_{q\in\\{q_{n}:n\geq 0\\}}{(q-1)}$, where the product
is over the set of distinct values of elements in $\\{q_{n}:n\geq 0\\}$,

such that the maximal equicontinuous factor of $(X_{\bm{\sigma}},T)$ is
$({\mathbb{Z}}_{\tilde{h},(q_{n})},+1)$.

###### Proof.

To prove the first statement, we repeat the proof of Theorem 4.4, except
working only with one fully essential word $ab$. We obtain

$\lim_{n\to\infty}\lambda^{q_{0}q_{1}\dots
q_{n}}f({\bm{a}})=\lim_{n\to\infty}\lambda^{h_{n}(a)}f({\bm{a}})=f({\bm{b}}),$

so that (6.4) holds. The rest of the proof is as in Theorem 4.4.

If ${\bm{\sigma}}$ is finitary and
$h=\lim_{n\rightarrow\infty}\lambda^{q_{0}\dots q_{n}}$ exists, then by Lemma
6.2, $\lambda=e^{2\pi it}$ where $t$ is rational; write $t=p/q$ with $q\geq 1$
and where $p$ and $q$ are non-zero coprime integers. Now continue as in the
proof of Theorem 4.5 (or 4.11), noting there that if we have
$|w_{n_{k}}|\equiv 0\mod q$ for large $k$, and the morphisms are all of
constant length, then $h_{n_{k}}(w_{n_{k}})=|w_{n_{k}}|q_{0}\dots
q_{n_{k}-1}\equiv 0\mod q$.

Finally, Lemma 6.2 tells us that $\frac{p}{q}=\frac{k}{q_{0}\dots
q_{m}(q_{m+1}-1)}$ for some $m$. Taking $m$ to be minimal so that all distinct
lengths $q_{n}$ appear in $q_{0},q_{1},\dots q_{m}$, the result follows. ∎

The following definition of height in the $S$-adic case is a generalisation of
Definition 3.19. It is motivated by the close relation between eigenvalues and
height.

######  Definition 6.7 ($S$-adic height).

Let $\bm{\sigma}=(\sigma_{n})_{n\geq 0}$ be a finitary, primitive, constant-
length directive sequence on $\mathcal{A}$, with length sequence
$(q_{n})_{n\geq 0}$, where $q_{n}\geq 2$ for all $n\geq 2$. Suppose that
$\bm{\sigma}$ has a fully essential word of length two. Then by Theorem 6.6,
if either $\bm{\sigma}=(\sigma_{n})_{n\geq 0}$ is prefix-straight and one-
sided recognizable, or straight and recognizable, then

$\displaystyle\tilde{h}$ $\displaystyle:=lcm\\{q\,:\,\lambda\mbox{ is an
eigenvalue which is a $q$-th root of unity, with $q$ coprime to each $q_{n}$
}\\}$

is finite. We call $\tilde{h}$ the height of ${\bm{\sigma}}$.

As in the case of constant-length substitutions, the factors of $\tilde{h}$
define the set of eigenvalues of $(X_{\bm{\sigma}},T)$ not determined by the
sequence of lengths $(q_{n})$ (as expressed in Theorem 6.6).

######  Example 5 (continues=ex:first).

Here we consider the constant-length directive sequence in Example 1 written
as

$\sigma,\tau\circ\sigma,\sigma,\tau\circ\sigma,\sigma,\sigma,\tau\circ\sigma,\sigma\dots$

so that it is recognizable and straight as we previously discussed. Since $ba$
is a fully essential word, Theorem 6.6 tells us that if $\lambda$ is a
continuous eigenvalue, then $h:=\lim_{n}\lambda^{3^{n}}$ exists. It is
constant since the directive sequence is constant-length. As we computed
earlier in Example 1, both ${\bm{b}}$ and ${\bm{a}}$ consist of the single
limit word $u=\lim_{n\rightarrow\infty}\sigma_{[0,n)}(a)$. Therefore if $f$ is
the non-zero eigenfunction associated to $\lambda$, this gives

$f({\bm{b}})h=f({\bm{a}}),$

so that we must have $h=1$.

Lemma 6.2 applies, with $q_{n}\in\\{3,9\\}$ for each $n$, so that if $\lambda$
is an eigenvalue, then it is rational. Inspection of the proof of Lemma 6.2
tells us that $\lambda=e^{2\pi it}$, where $t=\frac{k}{q_{1}\dots
q_{m}(q_{m+1}-1)}$ for some $m$; $\lambda$ is thus an $\tilde{h}$-th root of
unity where the prime divisors of $\tilde{h}$ lie in $\\{2,3\\}$. But since
$\lim_{n}\lambda^{3^{k_{n}}}=1$ for some sequence $(k_{n})$ and since $t$ is
rational, $\lambda^{3^{k_{n}}}=1$ for large $n$ (by considering the $3$-adic
expansion of $t$); this implies that $\tilde{h}=3^{\ell}$ for some $\ell$.
Hence the only continuous eigenvalues of $(X_{\bm{\sigma}},T)$ are those given
by the heights of the towers, i.e., those in Remark 6.1. This $S$-adic system
therefore has trivial height, and so its maximal equicontinuous factor is
$({\mathbb{Z}}_{3},+1)$.

Recall the definition of height in Definition 3.19. From Theorem 6.6 we
recover its role in the case of a constant-length substitution, as seen in
[Kam72, Dek78], [Que10, Theorem 6.1].

######  Corollary 6.8.

Let $\sigma$ be a primitive straight substitution of constant-length $q$. Then
the continuous eigenvalues of $(X_{\sigma},T)$ are generated by
$\\{e^{\frac{2\pi i}{q^{n}}}:n\in{\mathbb{N}}\\}\cup\\{e^{\frac{2\pi
i}{\tilde{h}}}\\}$, where $\tilde{h}$ is the height of $\sigma$. Furthermore
$\tilde{h}$ divides $q-1$.

Similarly, if a finitary directive sequence satisfies the conditions of
Theorem 6.6, then its continuous eigenvalues of $(X_{\bm{\sigma}},T)$ are
generated by $\\{e^{\frac{2\pi i}{q_{0}\cdots
q_{n}}}:n\in{\mathbb{N}}\\}\cup\\{e^{\frac{2\pi i}{\tilde{h}}}\\}$, where
$\tilde{h}$ is the height of $\bm{\sigma}$.

From Theorem 6.6 we also obtain a version of Cobham’s theorem. For more on its
classical substitutive version, see [Dur11].

######  Corollary 6.9.

Let $\bm{\sigma}=(\sigma_{n})_{n\geq 0}$ and
$\bm{\tilde{\sigma}}=(\tilde{\sigma}_{n})_{n\geq 0}$ be two finitary,
primitive, constant-length directive sequences on $\mathcal{A}$, with length
sequences $(q_{n})_{n\geq 0}$ and $(\tilde{q}_{n})_{n\geq 0}$, where each
$q_{n}\geq 2$ and $\tilde{q}_{n}\geq 2$, and where each length occurs
infinitely often. Suppose that each directive sequence possesses a fully
essential word of length 2. Suppose that either

  * •

$\bm{\sigma}$ and $\bm{\tilde{\sigma}}$ are prefix-straight and one-sided
recognizable, or

  * •

$\bm{\sigma}$ and $\bm{\tilde{\sigma}}$ are straight and recognizable.

If there is a prime factor of some $q\in\\{q_{n}:n\geq 0\\}$ that is not a
prime factor of any $\tilde{q}\in\\{\tilde{q}_{n}:n\geq 0\\}$, and if
$(X_{\bm{\sigma}},T)$ and $(X_{\bm{\tilde{\sigma}}},T)$ have the same maximal
equicontinuous factor, then they are both finite.

In particular, if $(X_{\bm{\sigma}},T)$ is infinite, then it cannot be both
$\bm{\sigma}$-adic and $\bm{\tilde{\sigma}}$-adic. To see why the requirement
that each $q_{n}$, and $\tilde{q}_{n}$ appears infinitely often is necessary,
consider the situation where $q_{n}=3$ for each $n$, and
$(\tilde{q}_{n})=2,3,3,3\dots$; then it can happen that both
$(X_{\bm{\sigma}},T)$ and $(X_{\bm{\tilde{\sigma}}},T)$ are infinite and have
${\mathbb{Z}}_{2,(3)}$ as maximal equicontinuous factor, in particular if
$(X_{\bm{\sigma}},T)$ has height 2.

The following example gives a recipe to find constant-length directive
sequences which have a non-trivial height. We formulate it to generate
straight, recognizable directive sequences; by modifying the first two
ingredients appropriately (e.g. using Lemma 6.5 to obtain one-sided
recognizability), we can also generate one-sided examples.

######  Example 6.10.

Take any finite set $\mathcal{Q}$ of odd numbers, and let $\mathcal{S}$ be any
set of constant-length substitutions on $\mathcal{A}=\\{a,b,c,d\\}$ where

  1. (i)

for each $\alpha\in\mathcal{A}$, $\sigma(\alpha)$ begins with the same letter
for all $\sigma\in\mathcal{S}$, and each $\sigma(\alpha)$ ends with the same
letter for all $\sigma\in\mathcal{S}$,

  2. (ii)

each substitution in $\mathcal{S}$ is rotationally conjugate to a left- or
right- permutative substitution,

  3. (iii)

there is a word $w$ of length two such that for each $\sigma\in\mathcal{S}$
there is $\alpha\in\mathcal{A}$ where $w$ occurs in $\sigma(\alpha)$,

  4. (iv)

for each $\sigma\in\mathcal{S}$, any occurrence of a letter in $\\{a,b\\}$ in
the image of letter by a substitution is always followed by a letter in
$\\{c,d\\}$, and any occurrence of a letter in $\\{c,d\\}$ is always followed
by a letter $\\{a,b\\}$, and

  5. (v)

each substitution in $\mathcal{S}$ has length belonging to $\mathcal{Q}$.

Then we claim that $-1$ is an eigenvalue.

Indeed, condition (i) ensures that any directive sequence from $\mathcal{S}$
is straight. Condition (ii) ensures that any directive sequence is
recognizable, using Theorem 2.5. Condition (iii) ensures that there is a fully
essential word of length two. Therefore Theorem 6.6 applies. Finally,
Conditions (iv) and (v) ensure that -1 is an eigenvalue, as

$\mathcal{P}=\\{[a]\cup[b],[c]\cup[d]\\}$

is then a clopen partition which forms a Rokhlin tower of height 2. Take for
example $S=\\{\sigma,\tau\\}$ with

$\displaystyle a\stackrel{{\scriptstyle\sigma}}{{\mapsto}}bda,\ \ $
$\displaystyle a\stackrel{{\scriptstyle\tau}}{{\mapsto}}bcada$ $\displaystyle
b\stackrel{{\scriptstyle\sigma}}{{\mapsto}}bcb,\ \ $ $\displaystyle
b\stackrel{{\scriptstyle\tau}}{{\mapsto}}bdadb$ $\displaystyle
c\stackrel{{\scriptstyle\sigma}}{{\mapsto}}cac,\ \ $ $\displaystyle
c\stackrel{{\scriptstyle\tau}}{{\mapsto}}cacbc$ $\displaystyle
d\stackrel{{\scriptstyle\sigma}}{{\mapsto}}cbd,\ \ $ $\displaystyle
d\stackrel{{\scriptstyle\tau}}{{\mapsto}}cbdad.$

The above general remarks tell us that $-1$ is an eigenvalue. According to
Lemma 6.2, we could in principle have an eigenvalue which is a fourth root of
unity. We show that this does not occur.

Assume first that $\tau$ appears infinitely often in the directive sequence.
Now the fact that $\tau(b)$ appears infinitely often and contains the subword
$dad$ tells us that $\bar{f}(d)h^{2}=\bar{f}(d)$, and this means that $h$
cannot be a fourth root of unity, and hence the eigenvalue -1 is the only
eigenvalue that appears in addition to those in Remark 6.1. Finally if $\tau$
appears only finitely often, then, since $\bm{a}=\bm{b}$, we can repeat the
argument above, but with the essential word $bda$ to arrive at the same
conclusion.

Note that any directive sequence chosen from the specific
$S=\\{\sigma,\tau\\}$ will give a strongly prefix-straight directive sequence
(see Definition 5.6); in general though examples satisfying (i) above are not
necessarily strongly prefix-straight.

###  6.2. Toeplitz $S$-adic shifts and discrete spectrum

In this section we focus on a special family of constant-length $S$-adic
shifts which has been extensively studied, namely those that are almost-
automorphic. We apply our previous results to this family and then compare our
results to existing results in the literature.

We first start with a few definitions. The factor $(Z,S)$ of $(X,T)$ via the
map $\pi:X\rightarrow Z$ is almost one-to-one if the set $\\{x\in
X:\pi^{-1}(\\{\pi(x)\\})=\\{x\\}\\}$ is dense in $X$. In this case, we call
the system $(X,T)$ an almost one-to-one extension of $(Z,S)$. A system $(X,T)$
is almost automorphic if it is a somewhere one-to-one extension of a minimal
equicontinuous system. Almost automorphic systems are necessarily minimal, and
for minimal systems $\\{x\in X:\pi^{-1}(\\{\pi(x)\\})=\\{x\\}\\}$ is a dense
$G_{\delta}$ if it is non-empty. A Toeplitz shift is a symbolic shift
$(X,\sigma)$, $X\subset\mathcal{A}^{{\mathbb{Z}}}$ with $\mathcal{A}$ finite,
which is an almost automorphic extension of an odometer $(Z,+1)$, via
$\pi:(X,\sigma)\rightarrow(Z,+1)$. Noe te that Gjerde and Johansen [GJ02] give
a nice characterisation of Toeplitz shifts in terms of the equal path number
property for their Bratteli-Vershik representations.

Given a substitution $\sigma_{n}:\mathcal{A}\rightarrow\mathcal{A}^{q_{n}}$,
we write it as $\sigma_{n}=\sigma_{n,0}\sigma_{n,1}\dots\sigma_{n,q_{n}-1}$, a
concatenation of $q_{n}$ maps
$\sigma_{n,i}:\mathcal{A}\rightarrow\mathcal{A}$. We say that $\sigma_{n}$ has
a coincidence at i if $|\sigma_{n,i}(\mathcal{A})|=1$. We say that the
directive sequence $\bm{\sigma}$ has a coincidence if each of of the
substitutions $\sigma_{n}$ has a coincidence at some $i$. We have the
following result, a straightforward generalisation of a result due to Kamae
and Dekking [Kam72, Theorem 7]; see also [Dek78, Theorem 7].

######  Theorem 6.11.

Let $\bm{\sigma}=(\sigma_{n})_{n\geq 0}$ be a finitary, primitive, constant-
length directive sequence on $\mathcal{A}$, with length sequence
$(q_{n})_{n\geq 0}$, $q_{n}\geq 2$. Suppose that
$\bm{\sigma}=(\sigma_{n})_{n\geq 0}$ has a fully essential word of length 2,
and that infinitely many of the substitutions $\sigma_{n}$ have a coincidence.
If either

  * •

$\bm{\sigma}$ is prefix-straight and one-sided recognizable, or

  * •

$\bm{\sigma}$ is straight and recognizable,

then $(X_{\bm{\sigma}},T)$ is Toeplitz, the measure preserving system
$(X_{\bm{\sigma}},T,\mu)$ has metric discrete spectrum, and the maximal
equicontinuous factor of $(X_{\bm{\sigma}},T)$ is
$({\mathbb{Z}}_{\tilde{h},(q_{n})},+1)$ where $\tilde{h}$ is the height of
$\bm{\sigma}$.

###### Proof.

In this proof we assume that $\bm{\sigma}$ is straight and recognizable; the
proof of the one-sided case is similar. We will assume that each $\sigma_{n}$
has a coincidence; this does not change the generality of our arguments. By
Theorem 6.6, the maximal equicontinuous factor of $(X_{\bm{\sigma}},T)$ is
$({\mathbb{Z}}_{\tilde{h},(q_{n})},+1)$ for some $\tilde{h}$ which divides
$\prod_{j=1}^{N}(q_{j}-1)$ for some $N$. First suppose that $\tilde{h}=1$, so
$({\mathbb{Z}}_{\tilde{h},(q_{n})},+1)=({\mathbb{Z}}_{(q_{n})},+1)$.
Recognizability means that for each $n$ and each $x\in X_{\bm{\sigma}}$, there
are a unique $m_{i}$ with $0\leq m_{i}\leq q_{i}$ and a unique $x^{(n)}\in
X_{\bm{\sigma}}^{(n)}$ such that
$x=T^{m_{0}}\sigma_{0}T^{m_{1}}\sigma_{1}\dots
T^{m_{n-1}}\sigma_{n-1}(x^{(n)})$, and furthermore the data
$m_{0},\dots,m_{n+1}$, which let us desubstitute $x$ to
$X_{\bm{\sigma}}^{(n+1)}$, agree with the data $m_{0},\dots,m_{n}$, which let
us desubstitute $x$ to $X_{\bm{\sigma}}^{(n)}$. This allows us to define a
maximal equicontinuous factor map
$\pi:X_{\bm{\sigma}}\rightarrow{\mathbb{Z}}_{(q_{n})}$, where
$\pi(x)=\dots,m_{2},m_{1},m_{0}$. Let $\lambda$ be the Haar probability
measure on ${\mathbb{Z}}_{(q_{n})}$, and let $\mathcal{Z}$ be the set of
points in ${\mathbb{Z}}_{(q_{n})}$ which are not invertible under $\pi$.
Denote by $C_{n}$ the set of indices $i$ such that $\sigma_{n}$ has a
coincidence, i.e., $|\sigma_{n,i}(\mathcal{A})|=1$. We have that
$z\not\in\mathcal{Z}$ if $z_{n}\in C_{n}$ infinitely often. In other words, if
$[C_{n}]^{c}:=\\{z\in{\mathbb{Z}}_{(q_{n})}:z_{n}\not\in C_{n}\\}$, then

$\mathcal{Z}\subset\bigcup_{k=1}^{\infty}\bigcap_{n\geq k}[C_{n}]^{c},$

so that

$\displaystyle\lambda(\mathcal{Z})$
$\displaystyle\leq\lim_{k\rightarrow\infty}\lambda\left(\bigcap_{n\geq
k}[C_{n}]^{c}\right)\leq\lim_{k\rightarrow\infty}\prod_{n\geq
k}\frac{q_{n}-1}{q_{n}}=0,$

the last equality following since the sequence $(q_{n})$ takes on a finite
number of values.

To show that $(X_{\bm{\sigma}},T,\mu)$ has metric discrete spectrum, we must
show that $\mu(\pi^{-1}(\mathcal{Z}))=0$. Define
$D_{n}:=\\{x:(\pi(x))_{n}\not\in C_{n}\\}$. Then as above,
$\pi^{(-1)}(\mathcal{Z})\subset\bigcup_{k=1}^{\infty}\bigcap_{n\geq k}D_{n}$,
and as for each $n$, $\sigma_{n}$ defines a $q_{n}$-cyclic partition of
$X_{\bm{\sigma}}^{(n+1)}$, we have that $\mu(\bigcap_{n\geq
k}D_{n})\leq\prod_{n\geq k}\frac{q_{n}-1}{q_{n}}=0$. Therefore
$\mu(\pi^{-1}(\mathcal{Z}))=0$.

For minimal systems $\\{x\in X:\pi^{-1}(\pi(x))=\\{x\\}\\}$ is a dense
$G_{\delta}$ set if it is non-empty. The fact that it is Toeplitz follows from
the above.

Finally, if $\tilde{h}$ is non-trivial, it gives us an additional
decomposition of $X_{\bm{\sigma}}$ as a tower of height $\tilde{h}$, so we
find a map $\pi_{0}:X_{\bm{\sigma}}\rightarrow{\mathbb{Z}}/h{\mathbb{Z}}$
reflecting this. We define an equicontinuous factor map
$\pi_{1}:X_{\bm{\sigma}}\rightarrow{\mathbb{Z}}_{(q_{n})}$ as before, using
recognizability; then,
$\pi=\pi_{0}\times\pi_{1}:X_{\bm{\sigma}}\rightarrow({\mathbb{Z}}/\tilde{h}{\mathbb{Z}},+1)\times({\mathbb{Z}}_{(q_{n})},+1)\cong({\mathbb{Z}}_{\tilde{h},(q_{n})},+1)$
is a maximal equicontinuous factor map. We have that $\pi_{1}$ is injective on
a $G_{\delta}$ set of full mass. Since $\tilde{h}$ is coprime to each $q_{n}$,
then $\pi$ inherits the same injectivity properties.

∎

######  Example 6 (continues=ex:first).

We have shown that the directive sequence

$\sigma,\tau,\sigma,\sigma,\tau,\sigma,\sigma,\sigma,\tau,\sigma,\sigma\dots$

is recognizable, and also straight if we re-write it as

$\sigma,\tau\circ\sigma,\sigma,\tau\circ\sigma,\sigma,\sigma,\tau\circ\sigma,\sigma\dots$

The $S$-adic shift $(X_{\bm{\sigma}},T)$ has a unique ergodic measure $\mu$,
see for example Proposition 5.3 and the comment below it. The substitution
$\tau\circ\sigma$ is

$\displaystyle a\stackrel{{\scriptstyle\sigma}}{{\mapsto}}acb$
$\displaystyle\stackrel{{\scriptstyle\tau}}{{\mapsto}}\,abcaacacb$
$\displaystyle b\stackrel{{\scriptstyle\sigma}}{{\mapsto}}bab$
$\displaystyle\stackrel{{\scriptstyle\tau}}{{\mapsto}}acbabcacb$
$\displaystyle c\stackrel{{\scriptstyle\sigma}}{{\mapsto}}cbc$
$\displaystyle\stackrel{{\scriptstyle\tau}}{{\mapsto}}\,aacacbaac$

which has a coincidence at index $0$. Since $\tau\circ\sigma$ appears
infinitely often in the directive sequence, Theorem 6.11 tells us that
$(X_{\bm{\sigma}},T)$ is Toeplitz and that $(X_{\bm{\sigma}},T,\mu)$ has
discrete spectrum.

We compare our results on constant-length $S$-adic shifts to those in the
literature for Toeplitz shifts, of whose spectrum there is an extensive study.
The maximal equicontinuous factor of a Toeplitz shift is always an odometer
[Wil84]. In other words, each continuous eigenvalue of a Toeplitz shift is
rational. Furthermore, it was known that there could exist rational continuous
eigenvalues that are not given by the period sequence $(p_{n})_{n\geq 0}$
[Dow05]. Our contribution in the case of Toeplitz shifts is to quantify the
eigenvalues if the Toeplitz shift is given as a constant-length $S$-adic
shift, as given by Theorem 6.6, and to describe them as heights, as in the
constant-length substitution case, and finitely extending Cobham’s theorem to
this setting.

In the measurable setting, Downarowicz and Lacroix showed that if $K\subset
S^{1}$ is a countable subgroup containing infinitely many rationals, then
there exists a minimal, uniquely ergodic Toeplitz shift whose measurable
eigenvalues equals $K$ [DL96]. Therefore we see that the set of measurable
eigenvalues of a Toeplitz shift can be much larger than the set of continuous
eigenvalues. This is further studied by Durand, Frank and Maass [DFM15], who
use the characterisation of Toeplitz shifts as expansive symbolic systems
which have proper Bratteli-Vershik representations defined by a sequence of
constant-length morphisms [GJ02]. In this setting recognizability is built
into the representation, but the morphisms need to be assumed proper, which is
a restriction we do not impose, and automatically implies that the $S$-adic
shifts considered in [DFM15] are Toeplitz, whereas constant-length $S$-adic
shifts are generally not. Finally, in moving to the setting of proper
Bratteli-Vershik representations, the notion of height disappears. If one
considers a Toeplitz shift with a proper Bratteli-Vershik representation in
which the morphisms are taken from a finite set and all of whose incidence
matrices are positive, then these systems are linearly recurrent [Dur03], and
all their measurable eigenvalues are continuous [BDM05].

Mentzen studies a family of measure preserving systems on the unit interval,
those of exact uniform rank [Men91]; these are the closest dynamical systems
to constant-length $S$-adic shifts, as they can be seen as a geometric
realisation of such symbolic systems. We study this next.

###  6.3. Measurable spectrum

The notion of uniform exact rank is investigated in [Men91] \- these are
measure preserving finite rank dynamical systems on the unit interval which
have a sequence of generating partitions $(\mathcal{R}_{n})$, where
$\mathcal{R}_{n}$ consists of a fixed number of Rokhlin towers all of which
are the same height, and where this representation is of exact finite rank. By
Theorem 2.7, uniform exact systems correspond to $S$-adic shifts
$(X_{\bm{\sigma}},T)$ which are defined using a recognizable, exact finite
rank directive sequence of constant length. Mentzen shows that such a system
must necessarily have rational measurable spectrum, i.e., translating his
result to the $S$-adic setting, we have the following which should be compared
with Theorem 6.6; his proof techniques are different.

######  Theorem 6.12.

[Men91, Corollary 1] Let $\bm{\sigma}=(\sigma_{n})_{n\geq 0}$ be a straight,
recognizable, constant-length directive sequence on $\mathcal{A}$, with length
sequence $(q_{n})_{n\geq 0}$. If $\bm{\sigma}$ is of exact finite rank, then
$(X_{\bm{\sigma}},T)$ is measurably isomorphic to a uniform finite rank
dynamical system, and every measurable eigenvalue is rational.

We would like to use Theorems 6.12 and 4.5 to conclude that every measurable
eigenvalue is continuous. Corollary 6.14 is a first step in this direction.

If we specialise to constant-length directive sequences the statement of
Theorem 5.7, we may relax the condition that the directive sequence be
strongly prefix-straight in a variety of ways. We illustrate with one that we
will use. Note that the condition that there is a unique right-infinite limit
word ${\bm{a}}$ and that $\sigma_{n}(a)$ starts with $a$ for each $n$ is
weaker than the condition that there is a letter $a$ such that for each $n$
and each $\alpha$, $\sigma_{n}(\alpha)$ starts with $a$, which is the required
condition of properness in [CDHM03]. For example, recall the substitutions
$\sigma$ and $\tau$ from Example 1, and note that any directive sequence
${\bm{\sigma}}$ selected from $\\{\sigma,\tau\\}$ satisfies this condition,
provided that $\tau$ appears infinitely often.

######  Theorem 6.13.

Let $\bm{\sigma}=(\sigma_{n})_{n\geq 0}$ be a finitary, one-sided
recognizable, strongly primitive constant-length directive sequence on
$\mathcal{A}$, with length sequence $(q_{n})_{n\geq 0}$. Suppose that there is
a unique right-infinite limit word ${\bm{a}}$, and moreover that
$\sigma_{n}(a)$ starts with $a$ for each $n$. If $\lambda\in S^{1}$ is a
measurable eigenvalue of $(X_{\bm{\sigma}},T,\mu)$, then

$\sum_{n=1}^{\infty}|\lambda^{q_{0}\dots q_{n}}-1|^{2}<\infty.$

###### Proof.

We follow the proof of Theorem 5.7, to the step where

(6.6) $\sum_{n=1}^{\infty}\max_{a,b}\max_{j\in
t_{n}(a,b)}|\lambda^{j}v_{n}(a)-v_{n-1}(b)|^{2}<\infty.$

Since $\sigma_{n}(a)$ starts with $a$ for each $n$, then $0\in t_{n+1}(a,a)$,
letting $j=0$ in (5.2), we get

(6.7) $\sum_{n=1}^{\infty}|v_{n}(a)-v_{n-1}(a)|^{2}<\infty.$

Since $\bm{\sigma}$ is of exact finite rank, the measure of each tower
$\mathcal{T}_{n}(a)$ has mass at least $\delta$, and since
$||f_{n}||_{2}\rightarrow||f||_{2}$, we must have that
$\min_{a\in\mathcal{A}}|v_{n}(a)|\rightarrow 1$. Therefore from (6.7), we
obtain

$\sum_{n=1}^{\infty}|\lambda^{j}-1|^{2}<\infty.$

∎

######  Corollary 6.14.

Let $\bm{\sigma}=(\sigma_{n})_{n\geq 0}$ be a finitary, one-sided recognizable
and strongly primitive constant-length directive sequence on $\mathcal{A}$,
with length sequence $(q_{n})_{n\geq 0}$. Suppose that either

  * •

${\bm{\sigma}}$ is strongly prefix-straight, or

  * •

there is a unique right-infinite limit word ${\bm{a}}$, and such that
$\sigma_{n}(a)$ starts with $a$ for each $n$.

Then every measurable eigenvalue is continuous.

###### Proof.

By Proposition 5.3 and Theorem 6.12, every measurable eigenvalue $\lambda$ is
rational. As the sum

$\sum_{n=1}^{\infty}\max_{a,b:{\bm{a}}={\bm{b}}}|\lambda^{h_{n}(w_{n}(a,b))}-1|^{2}$

is finite by Theorem 5.7 (with the assumption that ${\bm{\sigma}}$ is strongly
prefix-straight), or, by Theorem 6.13,

$\sum_{n=1}^{\infty}|\lambda^{j}-1|^{2}<\infty$

(with the second set of assumptions), we conclude that
$\lambda^{h_{n}(w_{n}(a,b))}-1=0$ (or $\lambda^{j}-1=0$ ) for large $n$. Now
Proposition 4.6 (or Proposition 4.5) implies that $\lambda$ is a continuous
eigenvalue. ∎

######  Example 7 (continues=ex:first).

We conclude by recapping the eigenvalues, measurable and topological of this
$S$-adic shift. We have shown that the directive sequence

$\sigma,\tau,\sigma,\sigma,\tau,\sigma,\sigma,\sigma,\tau,\sigma,\sigma,\sigma\dots$

is recognizable, and also straight if we re-write it as

$\sigma,\tau\circ\sigma,\sigma,\tau\circ\sigma,\sigma,\sigma,\tau\circ\sigma,\sigma,\sigma\dots.$

Note that $\tau\circ\sigma$ has a positive incidence matrix, as do
$\sigma\circ\sigma$ and $\sigma\circ\tau\circ\sigma$. We can boundedly
telescope the above straight direct sequence as

$\sigma\circ\tau\circ\sigma,\sigma\circ\tau\circ\sigma,\sigma\circ\sigma,\tau\circ\sigma,\sigma\circ\sigma\dots$

where this directive sequence takes values in
$\\{\sigma\circ\tau\circ\sigma,\sigma\circ\sigma,\tau\circ\sigma\\}$, all of
whose matrices are positive. We conclude by Proposition 5.3 that
$(X_{\bm{\sigma}},T)$ is of exact finite rank, although it is not linearly
recurrent [Dur03]. From Example 1 above, its maximal equicontinuous factor is
$({\mathbb{Z}}_{3},+1)$. Finally, while it is not strongly prefix-straight, it
satisfies the second conditions of Corollary 6.14, so $({\mathbb{Z}}_{3},+1)$
is also its Kronecker factor, i.e., the measure-theoretic factor generated by
the measurable eigenvalues.

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