# Locked entropy in partially coherent optical fields

Mitchell Harling  PROBE Lab, School of Engineering, Brown University,
Providence, RI 02912, USA  Varun A. Kelkar  Department of Electrical and
Computer Eng., University of Illinois at Urbana-Champaign, Urbana, IL 61801,
USA  Currently at Algorithmic Systems Group, Analog Garage, Analog Devices,
Inc., Boston, MA 02110, USA  Kimani C. Toussaint, Jr  PROBE Lab, School of
Engineering, Brown University, Providence, RI 02912, USA  Ayman F. Abouraddy
CREOL, The College of Optics & Photonics, University of Central Florida,
Orlando, FL 32816, USA

###### Abstract

We introduce a taxonomy for partially coherent optical fields spanning
multiple degrees of freedom (DoFs) based on the rank of the associated
coherence matrix (the number of non-zero eigenvalues). When DoFs comprise two
spatial modes and polarization, a fourfold classification emerges, with rank-1
fields corresponding to fully coherent fields. We demonstrate theoretically
and confirm experimentally that these classes have heretofore unrecognized
different properties. Specifically, whereas rank-2 fields can always be
rendered separable with respect to its DoFs via a unitary transformation,
rank-3 fields are always non-separable. Consequently, the entropy for a rank-2
field can always be concentrated into a single DoF (thus ridding the other DoF
of statistical fluctuations), whereas some entropy is always ‘locked’ in one
DoF of a rank-3 field.

††preprint: APS/123-QED

The study of optical coherence and the statistical fluctuations in optical
fields extends back to the pioneering work of Zernike Zernike (1938), and
subsequently reached maturation in the work of Wolf and Mandel Mandel and Wolf
(1965, 1995); Wolf (2007). Recently, new insights into optical coherence have
been brought to light Qian and Eberly (2011); Kagalwala  _et al._ (2013) by
exploiting the mathematical correspondence between the coherence matrix for
classical optical fields involving multiple degrees of freedom (DoFs) Gori
(1998); Gori  _et al._ (2006, 2007) and the density operator representing
multipartite quantum mechanical states. This correspondence has led to the
coinage of the term ‘classical entanglement’ Kagalwala  _et al._ (2013);
Spreeuw (1998, 2001); Aiello  _et al._ (2015); Forbes  _et al._ (2019); Konrad
and Forbes (2019) to describe optical fields that are not separable with
respect to their DoFs, in analogy with quantum entanglement that is intrinsic
to non-separable multipartite quantum states. The concept of classical
entanglement has helped solve problems with regards to Mueller matrices Simon
_et al._ (2010), determine the maximum achievable Young double-slit
interference visibility Abouraddy (2017), and enable the characterization of
quantum optical communications channels Ndagano  _et al._ (2017), among many
other applications Otte  _et al._ (2018); Mamani  _et al._ (2018); Kondakci
_et al._ (2019); Toninelli  _et al._ (2019); Yao-Li  _et al._ (2020); Shen
_et al._ (2021a, b); Hall and Abouraddy (2022); Aiello  _et al._ (2022).

The study of classical entanglement in optical fields is enriched by the
possibility of implementing inter-DoF (or global) unitary transformations
(‘unitaries’ henceforth for brevity Abouraddy (2017); Halder  _et al._
(2021)), including entangling and disentangling unitaries; e.g., a spatial
light modulator can entangle or disentangle polarization and spatial modes
Kagalwala  _et al._ (2017). This feature is central to the recent
demonstration of entropy swapping Okoro  _et al._ (2017); Harling  _et al._
(2022, 2023), which refers to the reversible reallocation of statistical
fluctuations from one DoF to another in a partially coherent field. For
example, starting with a polarized but spatially incoherent field (the entropy
is confined to the spatial DoF), a global unitary can convert the field to one
that is unpolarized but spatially coherent (the entropy has been swapped to
the polarization DoF with no loss of energy). A similar approach enables
entropy concentration, whereby the entropy shared among the DoFs can be
optimally transferred into a single DoF via a unitary Harling  _et al._
(2022).

Here we uncover a surprising feature of partially coherent optical fields that
places a constraint on entropy concentration under arbitrary global unitaries
Abouraddy (2017); Okoro  _et al._ (2017). For concreteness, we examine a
canonical optical field model having two binary DoFs, and introduce a fourfold
taxonomy based on the coherence rank of the associated $4\\!\times\\!4$
coherence matrix, which corresponds to the number of its non-zero eigenvalues
(from 1 to 4). While the rank-1 class embraces all coherent fields, rank-2
through rank-4 classes comprise partially coherent fields. We find that fields
of different ranks have altogether different characteristics that have not
been investigated to date. Specifically, we find that the potential for
concentrating the field entropy into a single DoF depends crucially on the
rank. Most conspicuously, the entropy of rank-2 fields – no matter how high –
can always be concentrated into a single DoF, thereby leaving the other DoF
free of statistical fluctuations [Fig. 1]. Indeed, there always exists a
global unitary that renders the field separable with respect to its DoFs, with
all the initial entropy concentrated into a single DoF. In stark contrast, it
is impossible to concentrate all the entropy of rank-3 fields – no matter how
low – into one DoF, and residual fluctuations must be retained by the other
DoF, which we call ‘locked entropy’ [Fig. 1]. This stems from the fact that
rank-3 fields possess a fundamentally non-separable structure that cannot be
eliminated unitarily. We demonstrate these effects experimentally using
optical fields defined by polarization and two spatial modes as the binary
DoFs of interest. These results open a new window onto understanding the
dynamics of optical coherence upon traversing optical systems or media that
couple multiple DoFs, and suggests new applications that may exploit the
coherence rank in optical imaging and communications.

Vector-space formulation of partially coherent optical fields. The most
general state of an optical field characterized by a binary DoF is described
by a $2\times 2$ coherence matrix. The polarization coherence matrix is
$\mathbf{G}_{\mathrm{p}}\\!=\\!\left(\\!\begin{array}[]{cc}G^{\mathrm{HH}}&G^{\mathrm{HV}}\\\
G^{\mathrm{VH}}&G^{\mathrm{VV}}\end{array}\\!\right)$, where
$G^{ij}\\!=\\!\langle E^{i}(E^{j})^{*}\rangle$,
$i,j\\!=\\!\mathrm{H},\mathrm{V}$, and $E^{i}$ is a scalar field component at
a point. Similarly, the spatial coherence matrix at two points $a$ and $b$ in
a scalar field is
$\mathbf{G}_{\mathrm{s}}\\!=\\!\left(\\!\begin{array}[]{cc}G_{aa}&G_{ab}\\\
G_{ba}&G_{bb}\end{array}\\!\right)$, where $G_{kl}\\!=\\!\langle
E_{k}E_{l}^{*}\rangle$, $k,l\\!=\\!a,b$, and $E_{k}$ is the scalar field at a
point. The polarization entropy is
$S_{\mathrm{p}}\\!=\\!-\lambda_{1}\log_{2}\lambda_{1}-\lambda_{2}\log_{2}\lambda_{2}$,
where $\lambda_{1}$ and $\lambda_{2}$ are the eigenvalues of
$\mathbf{G}_{\mathrm{p}}$; the spatial entropy $S_{\mathrm{s}}$ associated
with $\mathbf{G}_{\mathrm{s}}$ is similarly defined. In general
$0\\!\leq\\!S_{\mathrm{p}},S_{\mathrm{s}}\\!\leq\\!1$, with
$S_{\mathrm{p}},S_{\mathrm{s}}\\!=\\!0$ in the case of fully coherent fields
(no statistical fluctuations) Brosseau and Dogariu (2006). The maximum entropy
is 1 bit when the field is unpolarized or spatially incoherent
$\mathbf{G}_{\mathrm{p}},\mathbf{G}_{\mathrm{s}}\\!=\\!\tfrac{1}{2}\mathcal{I}$
(where $\mathcal{I}$ is the identity matrix).

Taking both DoFs (i.e., two points in a vector field), the first-order
coherence is described by a $4\\!\times\\!4$ coherence matrix $\mathbf{G}$
Gori  _et al._ (2006); Kagalwala  _et al._ (2013); Abouraddy (2017),

$\mathbf{G}=\left(\begin{array}[]{cccc}G_{aa}^{\mathrm{HH}}&G_{aa}^{\mathrm{HV}}&G_{ab}^{\mathrm{HH}}&G_{ab}^{\mathrm{HV}}\vspace{0.1cm}\\\
G_{aa}^{\mathrm{VH}}&G_{aa}^{\mathrm{VV}}&G_{ab}^{\mathrm{VH}}&G_{ab}^{\mathrm{VV}}\vspace{0.1cm}\\\
G_{ba}^{\mathrm{HH}}&G_{ba}^{\mathrm{HV}}&G_{bb}^{\mathrm{HH}}&G_{bb}^{\mathrm{HV}}\vspace{0.1cm}\\\
G_{ba}^{\mathrm{VH}}&G_{ba}^{\mathrm{VV}}&G_{bb}^{\mathrm{VH}}&G_{bb}^{\mathrm{VV}}\end{array}\right),$
(1)

where $G_{kl}^{ij}\\!=\\!\langle E_{k}^{i}(E_{l}^{j})^{*}\rangle$,
$i,j\\!=\\!\mathrm{H},\mathrm{V}$, and $k,l\\!=\\!a,b$. The coherence matrices
$\mathbf{G}$, $\mathbf{G}_{\mathrm{s}}$, and $\mathbf{G}_{\mathrm{p}}$ are all
Hermitian, positive semi-definite, unity-trace matrices. A $4\\!\times\\!4$
unitary $\hat{U}$ spanning both DoFs Abouraddy (2017) diagonalizes
$\mathbf{G}$:
$\mathbf{G}_{\mathrm{D}}\\!=\\!\hat{U}\mathbf{G}\hat{U}^{{\dagger}}\\!=\\!\mathrm{diag}\\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\\}$,
with $\sum_{j}\lambda_{j}\\!=\\!1$, and the field can carry up to 2 bits of
entropy $S\\!=\\!-\sum_{j=1}^{4}\lambda_{j}\log_{2}\lambda_{j}$, where
$0\\!\leq\\!S\\!\leq\\!2$ and $\mathrm{diag}\\{\cdot\\}$ refers to a diagonal
matrix with the listed elements along the diagonal. If, and only if,
$\lambda_{1}\lambda_{4}\\!=\\!\lambda_{2}\lambda_{3}$ can
$\mathbf{G}_{\mathrm{D}}$ be separated into a direct product with respect to
the two DoFs,
$\mathbf{G}_{\mathrm{D}}\\!=\\!\mathrm{diag}\\{\psi_{a},\psi_{b}\\}\otimes\mathrm{diag}\\{\gamma^{\mathrm{H}},\gamma^{\mathrm{V}}\\}$,
where each $2\\!\times\\!2$ coherence matrix corresponds to one DoF Abouraddy
_et al._ (2001). The condition
$\lambda_{1}\lambda_{4}\\!=\\!\lambda_{2}\lambda_{3}$ therefore delineates
optical fields that can – in principle – be rendered separable with respect to
their DoFs via unitaries.

We introduce the reduced coherence matrices that result from ‘tracing out’ one
DoF from $\mathbf{G}$: the reduced spatial coherence matrix
$\mathbf{G}^{\mathrm{red.}}_{\mathrm{s}}$ after tracing out polarization, and
the reduced polarization coherence matrix
$\mathbf{G}^{\mathrm{red.}}_{\mathrm{p}}$ after tracing over space. We define
entropies $S_{\mathrm{s}}$ and $S_{\mathrm{p}}$ for
$\mathbf{G}^{\mathrm{red.}}_{\mathrm{s}}$ and
$\mathbf{G}^{\mathrm{red.}}_{\mathrm{p}}$, respectively; in general,
$S\\!\leq\\!S_{\mathrm{s}}+S_{\mathrm{p}}$, with equality occurring only when
the field is separable. Crucially, whereas $S$ is invariant with respect to
global unitaries, $S_{\mathrm{s}}$ and $S_{\mathrm{p}}$ are not. Indeed,
whereas $\mathbf{G}$ suffices to completely identify the field coherence,
these reduced matrices do not Kagalwala  _et al._ (2013, 2015); Abouraddy
(2017); Okoro  _et al._ (2017).

Figure 1: Starting with a non-separable field with 1 bit of entropy
($S\\!=\\!1$, left) that is unpolarized $S_{\mathrm{p}}\\!=\\!1$ and spatially
incoherent $S_{\mathrm{s}}\\!=\\!1$, a unitary can reversibly convert it into
one of two forms depending on the rank of $\mathbf{G}$. For a rank-2 field,
the entropy can be always fully concentrated into one DoF, leaving the other
DoF free of statistical fluctuations. For a rank-3 field, entropy can never be
fully concentrated in one DoF. There always remains ‘locked entropy’ in the
other DoF.

Coherence rank and entropy concentration. We classify these optical fields
into four families according to their coherence rank, defined as the number of
non-zero eigenvalues of $\mathbf{G}$. Rank-1 fields,
$\\{\lambda\\}\\!=\\!\\{1,0,0,0\\}$, comprise fully coherent fields,
$S\\!=\\!0$ (no statistical fluctuations). It is always possible to render
rank-1 fields separable via a unitary:
$\mathbf{G}\\!\rightarrow\\!\mathbf{G}_{\mathrm{D}}\\!=\\!\mathrm{diag}\\{1,0\\}\otimes\mathrm{diag}\\{1,0\\}$,
whereupon both DoFs are fully coherent.

Partially coherent rank-2 fields,
$\\{\lambda\\}\\!=\\!\\{\lambda_{1},\lambda_{2},0,0\\}$, with entropy in the
range $0\\!<\\!S\\!\leq\\!1$, can always be transformed unitarily into the
separable form:
$\mathbf{G}_{\mathrm{D}}\\!=\\!\mathrm{diag}\\{1,0\\}\otimes\mathrm{diag}\\{\lambda_{1},\lambda_{2}\\}$.
This corresponds to a partially polarized field that is fully coherent
spatially ($S_{\mathrm{p}}\\!=\\!S$ and $S_{\mathrm{s}}\\!=\\!0$).
Alternatively, the field can be converted into a fully polarized field that is
partially coherent spatially ($S_{\mathrm{p}}\\!=\\!0$ and
$S_{\mathrm{s}}\\!=\\!S$). In general, the entropy of a rank-2 field is shared
between the two DoFs. Nevertheless, even in its highest-entropy state
$S\\!=\\!1$, $\\{\lambda\\}\\!=\\!\\{\tfrac{1}{2},\tfrac{1}{2},0,0\\}$, such
fields can always be rendered separable such that one DoF is fully coherent
(ridding it completely from statistical fluctuations), with the 1 bit of field
entropy concentrated in the other DoF Okoro  _et al._ (2017); Harling  _et
al._ (2022, 2023); see Fig. 1.

In stark contrast, the coherence matrices associated with rank-3 fields,
$\\{\lambda\\}\\!=\\!\\{\lambda_{1},\lambda_{2},\lambda_{3},0\\}$, whose
entropy is in the range $0\\!<\\!S\\!\leq\\!1.585$, cannot be expressed as a
direct product
($\lambda_{1}\lambda_{4}\\!=\\!0\\!\neq\\!\lambda_{2}\lambda_{3}$); that is,
rank-3 fields are never separable with respect to their DoFs. This fundamental
non-separability is independent of the values $\\{\lambda\\}$ and is solely a
consequence of the rank of $\mathbf{G}$. This hitherto unrecognized feature
has important consequences for entropy concentration: it prevents ridding
either DoF altogether from statistical fluctuations. Indeed, after
concentrating the entropy into one DoF, a residual amount of entropy is
retained that we refer to as locked entropy. The entropy in a rank-3 field
must always be shared between the DoFs no matter how low $S$ is. Even when
$S\\!<\\!1$, it is impossible to realize the condition
$S_{\mathrm{p}}\\!=\\!S$ and $S_{\mathrm{s}}\\!=\\!0$ (or
$S_{\mathrm{p}}\\!=\\!0$ and $S_{\mathrm{s}}\\!=\\!S$) unitarily, which is
attainable for rank-2 fields of the same entropy [Fig. 1]. Furthermore, when
$S\\!>\\!1$ one cannot concentrate 1 bit of entropy in one of the DoFs.
Defining the function
$f(x)\\!=\\!-\\!x\log_{2}{x}\\!-\\!(1\\!-\\!x)\log_{2}{(1\\!-\\!x)}$, the
minimum entropy that is locked in one DoF is
$S_{\mathrm{min}}\\!=\\!f(\lambda_{1}\\!+\\!\lambda_{2})$, in which case the
entropy concentrated into the other DoF is
$S_{\mathrm{max}}\\!=\\!f(\lambda_{1}\\!+\\!\lambda_{3})$.

Rank-4 fields,
$\\{\lambda\\}\\!=\\!\\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\\}$,
can sometimes be unitarily rendered separable with respect to their DoFs
depending on the eigenvalues, and they thus share the properties of rank-2 or
rank-3 fields. We will not examine rank-4 fields here and focus instead on
delineating the characteristics of rank-2 and rank-3 fields.

Figure 2: (a) Schematic of the OCmT measurement scheme used to measure
coherence matrices; L: spherical lens (focal length $f\\!=\\!30$ cm); F:
spectral filter; $\tfrac{\lambda}{4}$: quarter wave plate; P: linear
polarizer; C: CMOS camera. (b) Source preparation for a rank-2 field and the
measured $4\times 4$ coherence matrix $\mathbf{G}$. (c) Same as (b) for a
rank-3 field.

Experiment. We first prepare and characterize representative rank-2 and rank-3
fields [Fig. 2]. Starting from unpolarized, spatially incoherent light from an
LED (wavelength 625 nm), we select two spatial modes using slits at points $a$
and $b$ that are sufficiently separated to guarantee mutual incoherence [Fig.
2(a)]. For a rank-2 field
$\mathbf{G}\\!=\\!\tfrac{1}{2}\mathrm{diag}\\{1,0,1,0\\}$, the source
configuration along with the measured coherence matrix are shown in Fig. 2(b),
and the corresponding results for the rank-3 field with
$\mathbf{G}\\!=\\!\tfrac{1}{3}\mathrm{diag}\\{1,0,1,1\\}$ are shown in Fig.
2(c). The rank-2 field is prepared by placing a polarizer at both $a$ and $b$,
yielding $S\\!=\\!1$: the field is polarized $S_{\mathrm{p}}\\!=\\!0$ but
spatially incoherent $S_{\mathrm{s}}\\!=\\!1$. The rank-3 field is prepared by
placing a linear polarizer at $b$ only (the field at $a$ remains unpolarized)
to yield $S\\!=\\!1.585$: the field is partially polarized and partially
coherent spatially. Throughout, $\mathbf{G}$ is reconstructed via optical
coherence matrix tomography (OCmT) [Fig. 2(a)], which extends to optical
fields with multiple DoFs Abouraddy  _et al._ (2014); Kagalwala  _et al._
(2015) the analogous procedure of quantum state tomography Wootters (1990);
James  _et al._ (2001); Abouraddy  _et al._ (2002); see Supplementary for
further experimental details.

Figure 3: Unitary entropy conversion for rank-2 and rank-3 fields. (a) Source
configurations for rank-2 and (b) rank-3 fields. P: Linear polarizer oriented
along H; N: neutral density filter. (c) Setup for entropy conversion. W: Half-
wave plate; PBS: polarizing beam splitter. (d-f) From left to right:
$\mathbf{G}$ reconstructed before the entropy converter; $\mathbf{G}^{\prime}$
after the entropy converter with W2 oriented at $0^{\circ}$; and
$\mathbf{G}^{\prime}$ with W2 at $45^{\circ}$. All matrices are measurements,
and the fidelity throughout was $>\\!98\%$ with respect to theoretical
expectations (see Supplementary). (d) Rank-2 and rank-3 fields with
$S\\!\approx\\!0.75$; (e) same as (d) for $S\\!\approx\\!1$; and (f) rank-3
field with $S\\!\approx\\!1.585$.

The impact of the coherence rank on the limits of entropy concentration is
illustrated in Fig. 3. We consider iso-entropy rank-2 [Fig. 3(a)] and rank-3
[Fig. 3(b)] fields. We make use of an entropy converter that unitarily couples
the two DoFs [Fig. 3(c)], which comprises a half-wave plate (HWP) W1 in path
$a$ oriented at $45^{\circ}$ with respect to H (H$\rightarrow$V,
V$\rightarrow$H), a polarizing beam splitter (PBS) that couples modes $a$ and
$b$ and produces modes $a^{\prime}$ and $b^{\prime}$, followed by a HWP W2 in
mode $a^{\prime}$ in one of two orientations: at $0^{\circ}$ with H
(H$\rightarrow$H and V$\rightarrow-$V), and at $45^{\circ}$ with H
(H$\rightarrow$V, V$\rightarrow$H). The first orientation minimizes the
entropy in the spatial DoF (entropy concentration), while the second
orientation swaps the entropy of the spatial and polarization DoFs (entropy
swapping).

Either binary DoF (polarization or spatial modes) can support up to 1 bit of
entropy. We thus first prepare rank-2 and rank-3 fields with $S\\!=\\!0.75$
[Fig. 3(d)]. For the rank-2 field, the entire entropy can be concentrated in
the spatial DoF, $S_{\mathrm{s}}\\!=\\!0.75$ (partially coherent spatially)
and $S_{\mathrm{p}}\\!=\\!0$ (fully polarized). Using the first setting for
W2, the entropy converter minimizes the spatial entropy:
$S_{\mathrm{s}}\\!\rightarrow\\!0$ (spatially coherent) and
$S_{\mathrm{p}}\\!\rightarrow\\!0.75$ (partially polarized). The second
setting for W2 swaps the entropy between the DoFs, which yields here the same
result as that of entropy concentration with the first setting.

The corresponding results for the rank-3 field are entirely in contrast to
those for the iso-entropy $S\\!=\\!0.75$ rank-2 field. The rank-3 source
configuration yields theoretical values of $S_{\mathrm{s}}\\!=\\!0.6$
(partially coherent spatially) and $S_{\mathrm{p}}\\!=\\!0.38$ (partially
polarized); see Supplementary. The first setting minimizes the spatial entropy
but cannot concentrate all the entropy into the polarization DoF; rather, some
entropy remains locked in the spatial DoF
$S_{\mathrm{s}}\\!\rightarrow\\!0.38$. The second setting for the entropy
converter swaps the entropy between the DoFs:
$S_{\mathrm{s}}\\!\rightarrow\\!0.38$ (partially coherent spatially) and
$S_{\mathrm{p}}\\!\rightarrow\\!0.6$ (partially polarized). Similar results
are obtained when the initial field has a total of 1 bit of entropy,
$S\\!=\\!1$ [Fig. 3(e)]. Whereas the entire entropy can be concentrated in
either DoF in the case of a rank-2 field, this cannot be achieved for the iso-
entropy rank-3 field, and some entropy must remain locked in either DoF.
Finally, the entropy of rank-3 can exceed 1 bit (whereas that of rank-2 fields
cannot). In Fig. 3(f) we repeat the measurements with a maximum-entropy rank-3
field, $S\\!=\\!1.585$. Here the locked entropy in the spatial DoF is
$S_{\mathrm{s}}\\!=\\!f(\tfrac{2}{3})\\!=\\!0.92$.

The field rank can be identified by reconstructing $\mathbf{G}$, as shown in
Fig. 3. Nevertheless, information concerning the coherence rank can be deduced
by observing the visibility of the spatial interference fringes produced by
the field when the fields at $a$ and $b$ are superposed after a polarization
projection. Two theorems (see Supplementary for proofs) help establish a
strategy for this approach.

######  Theorem 1.

For a vector optical field supported on two spatial points with a coherence
matrix $\mathbf{G}$, if there exists a polarization projection along vector
$\mathbf{P}$ along which the field is spatially coherent (i.e., it can produce
spatial interference fringes with $100\%$ visibility), then
$\mathcal{R}(\mathbf{G})\leq 3$.

######  Theorem 2.

For a vector optical field supported on two points with a coherence matrix
$\mathbf{G}$, if there exist two orthogonal polarization projections
$\mathbf{P}$ and $\mathbf{Q}$ along which the field is spatially coherent
(i.e., it can produce $100\%$-visibility spatial interference fringes), then
$\mathcal{R}(\mathbf{G})\\!\leq\\!2$.

Figure 4: Identifying the coherence rank through the spatial coherence after a
polarization projection. (a,b) From left to right: the source preparation;
optimal interference fringes along the H and V polarization projections after
the entropy converter in Fig. 3(c), with W2 oriented at $0^{\circ}$ with H;
and optimal interference fringes along the H and V polarization projections
with W2 oriented at $45^{\circ}$. (a) Iso-entropy rank-2 and rank-3 fields
with $S\\!=\\!0.75$. (b) Same as (a) for $S\\!=\\!1$.

In other words, identifying an orthogonal pair of polarization projections
that both yield a spatially coherent field indicates that the field is either
rank-1 or rank-2. Identifying only a single polarization projection that
yields a spatially coherent field indicates that the field is rank-3. There is
no polarization projection for a rank-4 field that yields a spatially coherent
field.

We demonstrate these results experimentally in Fig. 4 with pairs of iso-
entropy rank-2 and rank-3 fields. After the field is prepared, it is directed
through the entropy converter shown in Fig. 3(c), and then the field is
globally projected onto a prescribed polarization. We search for pairs of
directions along which the resulting scalar field yields spatial interference
fringes with $100\%$ visibility.

We start with a pair of fields at $S\\!\approx\\!0.75$ [Fig. 4(a)]. The rank-2
field is prepared by projecting the polarization at $45^{\circ}$ with respect
to H and adjusting the amplitude of one spatial mode to obtain the targeted
entropy (see Supplementary for the full coherence matrices associated with the
fields in Fig. 4). After the entropy converter with W2 oriented at
$0^{\circ}$, no spatial interference fringes of high visibility are observed
at any polarization projection. After setting W2 at $45^{\circ}$, the
polarization projections along H and V yield high-visibility spatial
interference fringes, as expected for a rank-2 field.

We contrast these observations with those for an iso-entropy rank-3 field
$S\\!\approx\\!0.75$. This field is prepared by projecting the polarization at
$a$ alone along $45^{\circ}$ and adjusting the amplitude at $b$ to obtain the
target entropy. After the entropy converter with W2 oriented at $0^{\circ}$,
no spatial interference fringes are observed at any polarization projection.
However, after setting W2 at $45^{\circ}$, projecting the polarization along H
yields a field that produces high-visibility spatial interference fringes. The
corresponding polarization projection along V does not yield a spatially
coherent field, and no interference fringes can be observed. We increase the
field entropy for an iso-entropy pair of rank-2 and rank-3 fields to
$S\\!\approx\\!1$ (the maximum entropy for rank-2) [Fig. 4(b)], and observe
similar results to those for the lower-entropy fields [Fig. 4(a)]. Despite the
higher entropy, we can still identify a pair of polarization projections for
the rank-2 field that result in spatial coherence, whereas only a single
polarization projection is identified for the rank-3 field.

Discussion. The approach outlined here in terms of coherence matrices Fano
(1957); Gamo (1964); Perina (1985); Gori  _et al._ (2006) reveals features
that are difficult to discern otherwise when extended to multiple DoFs. The
analysis and experiments suggest a wealth of fundamental questions regarding
the statistical behavior of optical fields: How does the rank vary spatially
across a vector optical field? How does the spatial distribution of the rank
evolve with free propagation? How is the coherence rank affected by optical
nonlinearities? Although we have couched the coherence matrix here in terms of
polarization and spatial modes, this description can be extended to other
DoFs, including higher-dimensional DoFs (e.g., orbital angular momentum), and
even continuous DoFs after implementing the Schmidt decomposition to obtain an
effective finite-dimensional representation Ekert and Knight (1995); Law  _et
al._ (2000); Law and Eberly (2004); Eberly (2006); Hall and Abouraddy (2022).
This is particularly relevant in light of recent realizations of optical
fields in which the spatial, temporal, and polarization DoFs are all coupled
Diouf  _et al._ (2021); Yessenov  _et al._ (2022a, b, c). In addition to the
intrinsic interest of the coherence rank as a potential thermodynamic variable
for electromagnetic fields, it may also serve as an integer identifier of the
global properties of the field to be exploited for communications schemes
using partially coherent light Nardi  _et al._ (2022).

In conclusion, we have presented a classification scheme of partially coherent
optical fields based on the rank of the $4\times 4$ coherence matrix for two
binary DoFs. This classification unveils surprising structural distinctions:
all rank-2 fields are fundamentally separable whereas all rank-3 fields are
intrinsically non-separable. Consequently, the entropy in rank-2 fields – no
matter how high – can always be concentrated into one DoF, thereby leaving the
other DoF free of statistical fluctuations. In contrast, in a rank-3 field the
entropy – no matter how low – cannot be fully concentrated into one DoF, and
locked entropy remains associated with the other DoF.

###### Acknowledgements.

We thank C. Okoro, M. Yessenov, and A. Dogariu for useful discussions and
assistance. This work was funded by the US Office of Naval Research (ONR)
under contracts N00014-17-1-2458 and N00014-20-1-2789.

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#  Topological transitions by magnetization rotation in kagome monolayers of
ferromagnetic Weyl semimetal Co-based shandite

Kazuki Nakazawa1,2, Yasuyuki Kato1, and Yukitoshi Motome1
<EMAIL_ADDRESS>1Department of Applied Physics, The University of
Tokyo, Bunkyo, Tokyo 113-8656, Japan
2RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198,
Japan

###### Abstract

Co-based shandite Co3Sn2S2 is a ferromagnet hosting Weyl fermions in the
layered Co kagome structure. The band topology as well as the magnetism is
predicted to vary drastically in the atomically thin films depending on the
thickness and surface termination, and as an extreme case, the quantum
anomalous Hall state is expected in a monolayer of the Co kagome lattice.
Given that the bulk Weyl gap depends on the magnetization direction, here we
theoretically study how the topological nature and transport properties vary
with the magnetization direction in the systems with kagome monolayer with
both Sn and S surface terminations. By using ab initio calculations, we find
that in the Sn-end monolayer the anomalous Hall conductivity shows successive
discrete changes between different quantized values by rotating the
magnetization, indicating several topological transitions between the
anomalous quantum Hall insulators with different Chern numbers. Notably, when
the magnetization is oriented in-plane and perpendicular to the Co-Co bond,
the system exhibits a planar quantized anomalous Hall effect. We clarify that
these peculiar behaviors are due to topological changes in the band structures
associated with gap closing of the Weyl nodes. In contrast, the S-end
monolayer shows rather continuous changes in the transport properties since
the system is metallic, although the band structure contains many Weyl nodes.
Our results pave the way for controlling Weyl fermions in atomically thin
films of Co-based shandite, where the topological nature associated with the
Weyl nodes appears more clearly than the bulk.

##  I Introduction

The Weyl fermion was predicted as a massless elementary particle, which has
not yet been found in nature. In condensed matter physics, however, Weyl
fermions can appear as quasiparticles in particular crossing points of the
electronic band structure, called Weyl nodes AMV . The Weyl nodes always
appear in pairs and are regarded as emergent monopoles and antimonopoles in
momentum space, which can be characterized by chirality $+1$ and $-1$,
respectively NN . Their topological properties result in the so-called Fermi
arcs on particular surfaces of the system as a consequence of the bulk-edge
correspondence Murakami , and unconventional electromagnetic phenomena, such
as the anomalous Hall effect Murakami ; Burkov ; Suzuki , the anomalous Nernst
effect Sakai , and the negative magnetoresistance Burkov ; XHuang ; Zhang . A
pair of Weyl nodes is generated from a Dirac node by lifting the spin
degeneracy, as found in several materials with spatial-inversion symmetry
breaking Xu ; Souma ; WFFBD ; Soluyanov ; Feng  or time-reversal symmetry
breaking Suzuki ; Sakai ; WTVS ; XWWDF ; KF .

Figure 1: (a) Lattice structure of the bulk Co3Sn2S2. The black box represents
the conventional unit cell containing three primitive unit cells. The inset
shows the definitions of the polar and azimuthal angles, $\theta$ and
$\varphi$, respectively, which specify the direction of magnetization, with
respect to the lattice vectors $\bm{a}$, $\bm{b}$, and $\bm{c}$.
${\bm{a}}_{\rm B}$, ${\bm{b}}_{\rm B}$, and ${\bm{c}}_{\rm B}$ are the
primitive unit vectors for the bulk system. (b) Top view of the Co kagome
plane. The directions of $\varphi=0^{\circ}$ and $90^{\circ}$ are indicated;
the former (latter) is parallel (perpendicular) to a Co-Co bond. (c) Two-
dimensional Brillouin zone for monolayers (left). The cut of the three-
dimensional Brillouin zone for the bulk at $k_{z}=0$ is also shown (right).
[(d) and (e)] Side view of the (d) Sn-end monolayer and (e) S-end monolayer.
The crystal structures are visualized by VESTA MI .

The Co-based shandite Co3Sn2S2, which has a stacked kagome planes composed of
Co ions [Figs. 1(a) and 1(b)], has attracted attention as a candidate for
ferromagnetic Weyl semimetal with Weyl nodes associated with spontaneous
breaking of time-reversal symmetry Wang ; Liu . Indeed, this material exhibits
giant anomalous Hall and Nernst effects, large negative magnetoresistance Wang
; Liu ; Guin ; Yang ; Yanagi , and surface Fermi arcs DFLiu . Subsequently, it
was proposed that a thin film with Co kagome monolayer of this material
realizes a Chern insulator with the quantum anomalous Hall effect Muechler ,
which has stimulated the fabrication of atomically thin films Fujiwara ;
Shiogai ; Ikeda1 ; Ikeda2 ; Fujiwara2 . Furthermore, it was theoretically
revealed that the electronic and magnetic states, band topology, and transport
properties vary significantly depending on the number of kagome layers and the
surface termination in thin films NKM .

In the bulk case, Co3Sn2S2 exhibits the ferromagnetic moment perpendicular to
the kagome plane Wang ; Liu . A theoretical study based on an effective model
showed that the Weyl nodes form nodal rings when the magnetization is oriented
in-plane ON . Furthermore, a previous ab initio study demonstrated that the
positions of the Weyl nodes in the energy-momentum space can be tuned by
controlling the direction of the magnetic moment Ghimire . The magnetization
angle dependence of the anomalous Hall effect as well as the spin Hall effect
was also predicted for the effective model OKN . These results indicate that
the band topology in this bulk material is sensitive to the magnetization
direction. Given that the thin films of this material show a variety of the
electronic and magnetic properties as mentioned above, it is interesting to
clarify how these properties vary with the magnetization direction in the thin
film case. It would be important not only for further understanding the
relation between the magnetism and the band topology but also for further
stimulating the experiments on thin film fabrication.

In this paper, we study the electronic band structure and transport properties
based on the ab initio calculations for the thin films with kagome monolayer
of the Co-based shandite while changing the direction of the ferromagnetic
moment. We consider two types of monolayers with Sn and S surface
terminations, following the previous studies Muechler ; NKM . We find that the
anomalous Hall conductivity $\sigma_{\rm H}$ as well as the anomalous Nernst
conductivity $\alpha_{\rm N}$ exhibits significantly different behaviors from
the bulk system for the magnetization rotation. In particular, in the
monolayer with Sn termination, we find that $\sigma_{\rm H}$ shows several
discrete changes between different quantized values, suggesting topological
transitions between the quantum anomalous Hall insulators with different Chern
numbers. Interestingly, when the magnetization is oriented in-plane and
perpendicular to the Co-Co bond, $\sigma_{\rm H}$ is nonzero at a quantized
value, indicating a planar quantized anomalous Hall effect. In contrast, in
the monolayer with S termination, $\sigma_{\rm H}$ shows more moderate changes
without showing quantization. We explain these contrasting behaviors depending
on the tilted direction of the magnetization and the surface termination by
the analyses of the band structure and the Berry phase as well as the symmetry
argument.

This paper is organized as follows. After describing the details of ab initio
calculations and the computations of transport properties in Sec. II, we show
the magnetization angle dependences of $\sigma_{\rm H}$ and $\alpha_{\rm N}$
in Sec. III. To understand the results, we discuss the electronic band
structure and the Berry curvature in Sec. IV, and also present the symmetry
argument in Sec. V. Finally, Sec. VI is devoted to the summary.

Figure 2: $\theta$ dependences of the anomalous Hall conductivity $\sigma_{\rm
H}$ and the anomalous Nernst conductivity $\alpha_{\rm N}$ for (a) the Sn-end
monolayer, (b) the S-end monolayer, and (c) the bulk of the Co-based shandite.
$\sigma_{\rm H}$ is calculated at zero temperature, while $\alpha_{\rm N}$ is
at $5~{}{\rm meV}$. The blue and red points represents the data for
$\varphi=0^{\circ}$ and $\varphi=90^{\circ}$, respectively. The lines are
guide for the eye. The orange arrows in the inset of the upper panel of (a)
denote the Co magnetic moments.

##  II Method

The optimized lattice structures and electronic properties are computed using
the OpenMX code OpenMX ; Ozaki  within the framework of the density functional
theory. The Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation
(GGA) PBE  is employed for the exchange-correlation functional. We consider
two types of thin films each including a Co kagome monolayer with different
surface terminations: One has Sn atoms at both surfaces [Fig. 1(d)], while the
other has S atoms [Fig. 1(e)]. We call the former and latter the Sn-end and
S-end monolayer, respectively. For these monolayers, we employ the set of unit
vectors $\bm{a}$ and $\bm{b}$ indicated in the inset of Fig. 1(b), while we
use the primitive unit vectors ${\bm{a}}_{\rm B}$, ${\bm{b}}_{\rm B}$, and
${\bm{c}}_{\rm B}$ in Fig. 1(a) for the bulk calculations. Following the
previous study NKM , we perform the structural optimization for the
monolayers, starting from the lattice structures taken from the bulk and
assuming a spin-polarized state within the nonrelativistic calculations. After
the optimization, we symmetrize the structure to retain the original
$p\bar{3}m1$ symmetry. In the bulk calculations, we use the experimental
lattice structures Li .

To study the electronic states while changing the directions of the
ferromagnetic moment, we adopt the fully-relativistic calculation with a
constraint on the spin orientation OpenMX . The magnitude of the energy
constraint is taken as $5$ meV. We vary the moment direction by changing the
polar angle $\theta$ from $0^{\circ}$ to $180^{\circ}$ for the azimuth angle
$\varphi=0^{\circ}$ and $\varphi=90^{\circ}$; see the inset of Fig. 1(a) and
Fig. 1(b). Note that the cases of $\theta=0^{\circ}$ and $180^{\circ}$
correspond to the out-of-plane moments, and that the case of
$\theta=90^{\circ}$ for $\varphi=0^{\circ}$ ($90^{\circ}$) corresponds to the
in-plane moment parallel (perpendicular) to a Co-Co bond in the kagome plane.
Note also that the resultant moments are roughly oriented in the direction of
the constraint, but not perfectly. Such slight tilting leads to small
deviations from the results expected from the symmetry, as will be discussed
in Sec. V. We take the in-plane direction with $\varphi=0^{\circ}$
($90^{\circ}$) as the $x$ ($y$) axis.

To investigate the band topology and the transport properties, we construct
tight-binding models from the maximally-localized Wannier functions MV ; SMV ,
which are obtained by OpenMX WOT  to reproduce the electronic band structure
for each tilting angle of the ferromagnetic moment. Following the previous
study NKM , we employ Co $3d$, Sn $5s$ and $5p$, and S $3p$ orbitals. The
Berry curvature of the $n$th band at the momentum ${{\bm{k}}}$ is calculated
by

$\displaystyle\Omega_{n}({{\bm{k}}})=-\sum_{m\neq n}\frac{2{\rm
Im}\left[J_{x,nm}J_{y,mn}\right]}{[E_{n}({{\bm{k}}})-E_{m}({{\bm{k}}})]^{2}},$
(1)

where $m$ is the band index, $E_{n}({{\bm{k}}})$ is the eigenvalue of the
tight-binding Hamiltonian $H({{\bm{k}}})$, and
${\bm{J}}_{nm}=(J_{x,nm},J_{y,nm})$ is the current operator defined by

$\displaystyle{\bm{J}}_{nm}=\left\langle
n{{\bm{k}}}\left|\nabla_{{\bm{k}}}H({{\bm{k}}})\right|m{{\bm{k}}}\right\rangle,$
(2)

with the eigenvector of the Hamiltonian, $|n{{\bm{k}}}\rangle$. The Chern
number of the $n$th band is calculated by

$\displaystyle C_{n}=\sum_{{{\bm{k}}}\in{\rm BZ}}\Omega_{n}({{\bm{k}}}),$ (3)

where the sum is taken for the momentum ${{\bm{k}}}$ within the first
Brillouin zone [Fig. 1(c)]. Then, based on the Kubo formula, we compute the
anomalous Hall conductivity $\sigma_{\rm H}$ and the anomalous Nernst
conductivity $\alpha_{\rm N}$ as

$\displaystyle\sigma_{\rm H}$ $\displaystyle=-\frac{e^{2}}{\hbar
V_{d}}\sum_{{\bm{k}}\in{\rm
BZ}}\sum_{n}f(E_{n}({{\bm{k}}}))\Omega_{n}({{\bm{k}}}),$ (4)
$\displaystyle\alpha_{\rm N}$ $\displaystyle=\frac{ek_{\rm B}}{\hbar
V_{d}}\sum_{{\bm{k}}\in{\rm
BZ}}\sum_{n}s(E_{n}({{\bm{k}}}))\Omega_{n}({{\bm{k}}}),$ (5)

respectively, where $e$ is the elementary charge, $k_{\rm B}$ is the Boltzmann
constant, $\hbar$ is Dirac’s constant, and $V_{d}$ is a $d$-dimensional volume
of the system; $f({\varepsilon})=({{\rm e}}^{({\varepsilon}-\mu)/k_{\rm
B}T}+1)^{-1}$ is the Fermi distribution function, where $T$ and $\mu$ are the
temperature and the chemical potential, respectively, and $s({\varepsilon})$
is the entropy density given by $s({\varepsilon})=-f({\varepsilon})\ln
f({\varepsilon})-[1-f({\varepsilon})]\ln[1-f({\varepsilon})]$. In the
${{\bm{k}}}$ summations, the number of the $\bm{k}$-mesh in the first
Brillouin zone is taken as $2000\times 2000$ for monolayers and $480\times
480\times 480$ for bulk.

##  III Magnetization angle dependence of transport properties

Figure 3: Electronic band structures near the Fermi level ($E=0$) of (a) the
Sn-end monolayer, (b) the S-end monolayer, and (c) the bulk for selected
values of the tilting angle $\theta$. The upper and lower panels show the
results for $\varphi=0^{\circ}$ and $\varphi=90^{\circ}$ cases, respectively.

In this section, we discuss the angular dependences of $\sigma_{\rm H}$ and
$\alpha_{\rm N}$ while tilting the ferromagnetic moment in the
$\varphi=0^{\circ}$ and $\varphi=90^{\circ}$ planes. Figure 2 displays the
results for the Sn-end and S-end monolayers as well as the bulk.

Let us first focus on the Sn-end monolayer [Fig. 2(a)]. In both cases of
$\varphi=0^{\circ}$ and $90^{\circ}$, $\sigma_{\rm H}$ exhibits discrete
stepwise changes for $\theta$. For $\varphi=0^{\circ}$, $\sigma_{\rm H}$
changes between integer values in unit of $e^{2}/h$: $+3$, $+1$, $-1$, and
$-3$. This behavior indicates that the system undergoes successive topological
transitions between the quantum anomalous Hall insulating states with nonzero
Chern numbers $C=-3$, $-1$, $+1$, and $+3$. We can confirm this by the
plateaulike features in the chemical potential dependence of $\sigma_{\rm H}$;
see Appendix A. In this case, $\sigma_{\rm H}$ as well as $\alpha_{\rm N}$
shows almost antisymmetric dependence on $\theta$ with respect to
$\theta=90^{\circ}$, indicating that the topologically trivial state with
$C=0$ is realized for the in-plane magnetization with $\theta=90^{\circ}$. In
contrast, in the case of $\varphi=90^{\circ}$, both $\sigma_{\rm H}$ and
$\alpha_{\rm N}$ are no longer antisymmetric. In particular, $\sigma_{\rm H}$
shows stepwise changes from $+3$ to $+1$ and to $-3$ in unit of $e^{2}/h$,
without taking $-1$, indicating that the system undergoes different sequences
of topological transitions from the case of $\varphi=0^{\circ}$. Strikingly,
$\sigma_{\rm H}$ is quantized at $+1$ even for the in-plane magnetization with
$\theta=90^{\circ}$, suggesting a planar quantized anomalous Hall effect LHL ;
Ren ; ZRHZQ ; ZLiu . The contrasting behaviors between $\varphi=0^{\circ}$ and
$90^{\circ}$ will be discussed from the symmetry point of view in Sec. V.

Figure 4: $\theta$ dependences of the Berry curvature $\Omega_{n}({{\bm{k}}})$
in the first Brillouin zone of the band just below the Fermi level for the Sn-
end monolayer: (a) $\varphi=0^{\circ}$ and (b) $\varphi=90^{\circ}$. The
changes of the Chern number $C$ are also indicated. (i)-(v) represent the
topological transitions associated with the changes of $C$. The band index $n$
for the Chern number is dropped.

Next, we discuss the results for the S-end monolayer [Fig. 2(b)]. In this
case, $\sigma_{\rm H}$ is no longer quantized at any nonzero integer values
and shows rather continuous changes with $\theta$ for both $\varphi=0^{\circ}$
and $90^{\circ}$. This is because the system is always metallic, as will be
shown in Sec. IV. The complicated $\theta$ dependences with sign changes are
ascribed to the changes of the band structure with several Weyl nodes. We note
that the $\theta$ dependences of $\sigma_{\rm H}$ and $\alpha_{\rm N}$ are
again almost antisymmetric with respect to $\theta=90^{\circ}$ for
$\varphi=0^{\circ}$, while they are not for $\varphi=90^{\circ}$, similar to
the Sn-end case in Fig. 2(a).

For comparison, we also studied the bulk case [Fig. 2(c)]. We observe rather
smooth and monotonic $\theta$ dependences of $\sigma_{\rm H}$ and $\alpha_{\rm
N}$ compared with the monolayers. These behaviors are qualitatively consistent
with the previous studies ON ; Ghimire ; OKN . Also in this bulk case,
$\sigma_{\rm H}$ and $\alpha_{\rm N}$ are almost antisymmetric for
$\varphi=0^{\circ}$, but not for $\varphi=90^{\circ}$ OKN .

Consequently, our results highlight the distinctive transport behaviors while
changing the magnetization direction in the monolayer systems. In particular,
the Sn-end monolayer exhibits successive discrete changes of $\sigma_{\rm H}$,
indicating topological transitions between different anomalous quantum Hall
insulators. In addition, this behavior appears differently for
$\varphi=0^{\circ}$ and $90^{\circ}$, suggesting that the topological
transitions take place in a different manner depending on the tilting
direction. These findings will be discussed from the viewpoints of the
electronic band structure and the Berry curvature in the next section.

##  IV Electronic band structure and Berry curvature

Figure 5: Schematic pictures of (a) the in-plane ($\varphi=0^{\circ}$), (b)
in-plane ($\varphi=90^{\circ}$), and (c) out-of-plane directions of the
ferromagnetic moments, which are parallel to ${\bm{a}}$,
${\bm{c}}\times{\bm{a}}$, and ${\bm{c}}$, respectively. The orange arrows
represent the Co magnetic moments. The relevant symmetries to our analysis are
denoted in (a) and (b).

In this section, we present the electronic band structures obtained by the ab
initio calculations, and discuss the relation between their angular
dependences and the resuts in the previous section. Figure 3 shows the
electronic band structures along the symmetric lines in the Brillouin zones
shown in Fig. 1(c) near the Fermi level set at zero while changing the tilting
angle $\theta$ for (a) the Sn-end monolayer, (b) the S-end monolayer, and (c)
the bulk. The upper and lower panels display the results for
$\varphi=0^{\circ}$ and $\varphi=90^{\circ}$, respectively.

We first discuss the case of Sn-end monolayer [Fig. 3(a)]. In this case, the
system at $\theta=0^{\circ}$, namely, with the out-of-plane magnetic moment,
is an anomalous quantum Hall insulator with the Chern number $C=-3$ with the
Weyl gap opening along the $\Gamma$-K line Muechler ; NKM . While tilting the
moment by increasing $\theta$, the Weyl gap decreases and closes around
$\theta=90^{\circ}$ for $\varphi=0^{\circ}$, as shown in the upper panel of
Fig. 3(a); see also Appendix B. In this case, the change of the band structure
is almost antisymmetric with respect to $\theta=90^{\circ}$; for instance, the
results for $\theta=73^{\circ}$ and $\theta=108^{\circ}$ are almost
equivalent. Meanwhile, for $\varphi=90^{\circ}$, the Weyl gap similarly
decreases and closes around $\theta=69^{\circ}$ as shown in the lower panel of
Fig. 3(a) comm1 .

In Fig. 4, we show the angular dependence of the Berry curvature of the band
just below the Fermi level in the Sn-end monolayer. We present the Berry
curvature of the other bands near the Fermi level in Appendix B. In the case
of $\varphi=0^{\circ}$ [Fig. 4(a)], we find that the sign reversal of the
Berry curvature associated with the Weyl gap closing occurs (i) between
$\theta=76^{\circ}$ and $\theta=80^{\circ}$, (ii) at $\theta=90^{\circ}$,
(iii) between $\theta=101^{\circ}$ and $\theta=105^{\circ}$. These correspond
to the topological transitions inferred by $\sigma_{\rm H}$ in Fig. 2(a) with
the changes of the Chern number as (i) $C=-3\to-1$, (ii) $C=-1\to+1$ via $C=0$
at $\theta=90^{\circ}$, and (iii) $C=1\to 3$. We note that the Berry curvature
along the $k_{x}=0$ axis remains at $\theta=90^{\circ}$ since the very small
Weyl gap remains due to small spin canting; the gap should be closed if the
magnetic moments are perfectly parallel to the $a$ axis; see the symmetry
argument in Sec. V. In contrast, in the case of $\varphi=90^{\circ}$ [Fig.
4(b)], there are two topological transitions (iv) between $\theta=69^{\circ}$
and $\theta=79^{\circ}$ and (v) between $\theta=93^{\circ}$ and
$\theta=99^{\circ}$, where the Chern number changes as $C=-3\to-1$ and $-1\to
3$, respectively. These correspond well to the $\theta$ dependence of
$\sigma_{\rm H}$ in Fig. 2(a). The $\theta$ evolution of the Berry curvature
is asymmetric with respect to $\theta=90^{\circ}$, and the number of
topological transitions is one less than in $\varphi=0^{\circ}$, because the
two pairs of Weyl gaps close simultaneously at the transition (v). The
asymmetric change leaves the nonzero Chern number at $\theta=90^{\circ}$ (see
also Appendix B), which leads to the planar quantized anomalous Hall effect
discussed in Sec. III.

Next, we discuss the results for the S-end monolayer [Fig. 3(b)]. In this
case, the system is always metallic for both $\varphi=0^{\circ}$ and
$90^{\circ}$, which prevents $\sigma_{\rm H}$ from being quantized as found in
Fig. 2(b). Furthermore, there are a greater number of gapped Weyl nodes at low
energy than the Sn-end monolayer; see also Appendix B. These Weyl gaps close
and cause topological transitions at various magnetization angles, leading to
the complicated $\theta$ dependences in Fig. 2(b). Nevertheless, for
$\varphi=0^{\circ}$, the band structure changes almost antisymmetric with
respect to $\theta$, as represented by almost the same band structures for
$\theta=45^{\circ}$ and $\theta=135^{\circ}$. This is consistent with almost
antisymmetric behavior of $\sigma_{\rm H}$ and $\alpha_{\rm N}$ in Fig. 2(b),
similar to the Sn-end case.

Finally, we briefly discuss the bulk result [Fig. 3(c)]. The system is always
metallic, similar to the S-end monolayer case. Besides, the band structure
around the Fermi level is much simpler and the number of Weyl gaps are more
scarce than the S-end monolayer, leading to the simpler $\theta$ dependences
of $\sigma_{\rm H}$ and $\alpha_{\rm N}$. We observe the gap closing on the
M′-$\Gamma$ line at $\theta=90^{\circ}$ and the $\Gamma$-$\rm K^{\prime}$ line
at $\theta=135^{\circ}$ in the $\varphi=0^{\circ}$ case, and the M′-$\Gamma$
line at $\theta=132^{\circ}$ in the $\varphi=90^{\circ}$ case.

##  V Symmetry argument

Figure 6: Chemical potential dependences of [(a) and (c)] the anomalous Hall
conductivity $\sigma_{\rm H}$ at zero temperature and [(b) and (d)] the
anomalous Nernst conductivity $\alpha_{\rm N}$ at the temperature of $5$ meV
in the Sn-end monolayer. The data are calculated for $\varphi=0^{\circ}$ in
(a) and (b), and for $\varphi=90^{\circ}$ in (c) and (d), at several tilting
angles $\theta$. Figure 7: Berry curvature with the band structure of [(a),
(d), (g)] Sn-end monolayer, [(b), (e), (h)] S-end monolayer, and [(c), (f),
(i)] $k_{z}=0$ cut of bulk for the [(a)–(c)] out-of-plane, [(d)–(f)] in-plane
($\varphi=0^{\circ}$), and [(g)–(i)] in-plane ($\varphi=90^{\circ}$) magnetic
moments in the energy range from $-0.2$ eV to $0.2$ eV. The black lines on the
gray hexagons in each panel represent the Fermi surfaces. The small panels in
the row of S-end monolayer and bulk present the results in each band
separately.

Let us discuss the band topology from the symmetry point of view. As mentioned
in Sec. II, the lattice structures of the monolayers are symmetrized after the
optimization, which have threefold rotational symmetry around the $c$ axis,
$C_{3c}$, twofold rotational symmetry around the $a$ axis, $C_{2x}$, and
mirror symmetry perpendicular to the $a$ axis, $M_{x}$.

We first focus on the $\varphi=0^{\circ}$ case. When $\theta=90^{\circ}$ where
all the magnetic moments are perfectly oriented along the ${\bm{a}}$ axis
[Fig. 5(a)], all the above lattice symmetries are preserved. Hence, in this
case, the Berry curvature should satisfy the relations

$\displaystyle\Omega(k_{x},-k_{y})|_{\theta=90^{\circ},\varphi=0^{\circ}}=-\Omega(k_{x},k_{y})|_{\theta=90^{\circ},\varphi=0^{\circ}},$
(6)
$\displaystyle\Omega(-k_{x},k_{y})|_{\theta=90^{\circ},\varphi=0^{\circ}}=-\Omega(k_{x},k_{y})|_{\theta=90^{\circ},\varphi=0^{\circ}}.$
(7)

Note that the band index $n$ for the Berry curvature is dropped here and
hereafter. These lead to the cancellation of the Berry curvature in each band,
and consequently, the disappearance of the anomalous Hall and Nernst
conductivities. In particular, these relations indicate
$\Omega(k_{x},0)=\Omega(0,k_{y})=0$, meaning that the Berry curvature vanishes
on the $k_{x}$ and $k_{y}$ axes. These arguments are indeed approximately
satisfied in the ab initio results in the previous sections.

We can extend the argument to general $\theta$. Since the state with
$\theta=90^{\circ}-\gamma$ and $\theta=90^{\circ}+\gamma$ are related with
each other via the $C_{2x}$ or $M_{x}$ operation, we obtain the relations

$\displaystyle\Omega(k_{x},-k_{y})|_{\theta=90^{\circ}-\gamma,\varphi=0^{\circ}}=-\Omega(k_{x},k_{y})|_{\theta=90^{\circ}+\gamma,\varphi=0^{\circ}},$
(8)
$\displaystyle\Omega(-k_{x},k_{y})|_{\theta=90^{\circ}-\gamma,\varphi=0^{\circ}}=-\Omega(k_{x},k_{y})|_{\theta=90^{\circ}+\gamma,\varphi=0^{\circ}}.$
(9)

These are consistent with the results in Fig. 4(a). Equations (8) and (9) lead
to the relations of $\sigma_{\rm H}$ and $\alpha_{\rm N}$ as

$\displaystyle\sigma_{\rm
H}|_{\theta=90^{\circ}-\gamma,\varphi=0^{\circ}}=-\sigma_{\rm
H}|_{\theta=90^{\circ}+\gamma,\varphi=0^{\circ}},$ (10)
$\displaystyle\alpha_{\rm
N}|_{\theta=90^{\circ}-\gamma,\varphi=0^{\circ}}=-\alpha_{\rm
N}|_{\theta=90^{\circ}+\gamma,\varphi=0^{\circ}},$ (11)

respectively. These explain well our ab initio results in Fig. 2.

Next we consider the case of $\varphi=90^{\circ}$. In Fig. 5(b), we illustrate
the situation of in-plane ferromagnetic state with $\theta=90^{\circ}$ as a
typical example. In this case, the system remains invariant under the $\Theta
C_{2x}$ and $\Theta M_{x}$ operations, where $\Theta$ is a time-reversal
operator. This is because the magnetic moment is parallel to the mirror plane.
Since the same argument can be made for general $\theta$, the Berry curvature
should satisfy the relations

$\displaystyle\Omega(k_{x},-k_{y})|_{\varphi=90^{\circ}}=\Omega(k_{x},k_{y})|_{\varphi=90^{\circ}},$
(12)
$\displaystyle\Omega(-k_{x},k_{y})|_{\varphi=90^{\circ}}=\Omega(k_{x},k_{y})|_{\varphi=90^{\circ}},$
(13)

for all $\theta$. These are consistent with the ab initio-based analysis shown
in Fig. 4(b) and allow the appearance of the anomalous Hall and Nernst effects
Ghimire , including the planar quantized anomalous Hall effect found in Fig.
2(a).

From the above discussion, we can predict the behavior of transport properties
when the magnetization is rotated in the kagome plane ($\theta=90^{\circ}$).
Since $\sigma_{\rm H}=0$ when $\varphi=0^{\circ}$, it should vanish when
$\varphi=(60n)^{\circ}$ ($n$ is an integer). Similarly, $\sigma_{\rm
H}/(e^{2}/h)=\pm 1$ when $\varphi=(\mp 30+120n)^{\circ}$. Thus, when the
magnetization is rotated in the kagome plane, $\sigma_{\rm H}$ changes as
$0\to+1\to 0\to-1\to 0\cdots$ in unit of $e^{2}/h$ with $120^{\circ}$ period.

Finally, we comment on the case of out-of-plane moment [Fig. 5(c)]. In this
case, Eqs. (8) and (9) with $\gamma=90^{\circ}$ give the relations between the
two cases of out-of-plane moments, $\theta=0^{\circ}$ and
$\theta=180^{\circ}$. In addition, Eqs. (12) and (13) hold. These relations
are again consistent with the results in the previous sections.

##  VI Summary

To summarize, focusing on the thin films of Co-based shandite with Co kagome
monolayer, we have investigated the magnetization angle dependences of the
electronic state, band topology, and transport properties. Using the ab initio
calculations, we revealed distinctive behaviors depending on the surface
termination of the films. In the Sn-end monolayer, we found successive
topological phase transitions between the Chern insulating states with
different Chern numbers while rotating the magnetization direction, resulting
in discrete changes in the anomalous Hall conductivity $\sigma_{\rm H}$
between the corresponding quantized values. Remarkably, we discovered the
planar quantized anomalous Hall effect: $\sigma_{\rm H}$ is quantized with
nonzero Chern number even when the magnetization is oriented in the in-plane
direction perpendicular to a Co-Co bond. In contrast, in the S-end monolayer,
$\sigma_{\rm H}$ as well as the anomalous Nernst conductivity $\alpha_{\rm N}$
shows moderate changes without quantization as the system is always metallic.
We show the origins of these behaviors by careful analyses of the Berry
curvature distributions in each band and the symmetry arguments. Our results
pave the way for controlling topological properties of the Weyl semimetal
toward device applications, by leveraging various tuning methods for the
magnetic anisotropy in thin films, such as magnetostriction, substrate
effects, and thickness changes.

## Acknowledgements

The authors thank K. Fujiwara, K. Kobayashi, K. Nomura, S. Okumura, A. Ozawa,
and A. Tsukazaki for fruitful discussions. This work is supported by JST CREST
(Grant No. JPMJCR18T2) and JSPS KAKENHI (Grant No. JP21K13875 and JP22K03509).

##  Appendix A Chemical potential dependences of transport properties in the
Sn-end monolayer

In this Appendix, we present the chemical potential dependences of the
anomalous Hall conductivity $\sigma_{\rm H}$ and the anomalous Nernst
conductivity $\alpha_{\rm N}$ for the Sn-end monolayer. Figure 6 show the
results calculated for the band structures at several values of $\theta$, by
changing the chemical potential in Eqs. (4) and (5). As discussed in Secs. III
and IV, the Sn-end monolayer is in the Chern insulating states with different
quantized values of $\sigma_{\rm H}$ depending on the moment direction.
Correspondingly, we find integer plateaus around zero chemical potential:
$\sigma_{\rm H}/(e^{2}/h)=+3$, $+1$, $-1$, and $-3$ for $\varphi=0^{\circ}$
[Fig. 6(a)], and $\sigma_{\rm H}/(e^{2}/h)=+3$, $+1$, and $-3$ for
$\varphi=90^{\circ}$ [Fig. 6(c)]. These plateaus represent the Weyl gaps which
make the system being the Chern insulators. In addition, in the case of
$\varphi=0^{\circ}$, weconfirm that the relations in Eqs. (10) and (11) are
approximately satisfied at all values of the chemical potential. We also note
that $\alpha_{\rm N}$ shown in Figs. 6(b) and 6(d) are approximately given by
the chemical potential derivative of $\sigma_{\rm H}$, as expected from the
generalized Mott relation XYFN .

##  Appendix B Berry curvature in each band

In this Appendix, we show the Berry curvature in each band for Sn-end and
S-end monolayers as well as the bulk. Figures 7(a), 7(d), and 7(g) show that
the Sn-end monolayer is in the distinct Chern insulating states for the out-
of-plane, in-plane $(\varphi=0^{\circ})$, and in-plane $(\varphi=90^{\circ})$
moments, respectively. We can also observe that the gap is almost closed on
the $k_{x}$ and $k_{y}$ axes in the in-plane $x$ case from Fig. 7(d). Note
that the Berry curvatures at $\theta=90^{\circ}$ shown in Fig. 4 correspond to
the results in the lower band depicted in Figs. 7(d) and 7(g). In the S-end
cases shown in Figs. 7(b), 7(e), and 7(h), the distribution of the Weyl nodes
where the Berry curvature becomes intense is complicated as discussed in Secs.
III and IV. Figures 7(c), 7(f), and 7(i) are the similar plots for the bulk in
the $k_{z}=0$ cut, again showing the consistency with the symmetry arguments.

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