# Derivative based global sensitivity analysis and its entropic link

Jiannan Yang
School of Physics, Engineering and Technology
University of York
Heslington, York, YO10 5DD, UK
<EMAIL_ADDRESS>

(September 29, 2023)

###### Abstract

Distribution-based global sensitivity analysis (GSA), such as variance-based
and entropy-based approaches, can provide quantitative sensitivity
information. However, they can be expensive to evaluate and are thus limited
to low dimensional problems. Derivative-based GSA, on the other hand, require
much fewer model evaluations. It is known that derivative-based GSA is closely
linked to variance-based total sensitivity index, while its relationship with
the entropy-based measure is unclear. To fill this gap, we introduce a log-
derivative based functional to demonstrate that the entropy-based and
derivative-based sensitivity measures are strongly connected. In particular,
we give proofs that, similar to the case with variance-based GSA, there is an
inequality relationship between entropy-based and derivative-based important
measures. Both analytical and numerical verifications are provided. Examples
show that the derivative-based methods give similar variable rankings as
entropy-based index and can thus be potentially used as a proxy for both
variance-based and entropy-based distribution-type GSA.

_Keywords_ entropic sensitivity index; sensitivity inequality; conditional
entropy; exponential entropy; Ishigami function

##  1 Introduction

The most widely adopted global sensitivity analysis (GSA) methods are
derivative-based and distribution-based.

The most common distribution-based approach examines variability using the
output variance. Variance-based methods, also called Sobol’s indices,
decompose the function output into a linear combinations of input and
interaction of increasing dimensionality, and estimate contribution of each
input factor to the variance of the output [2]. As only the 2nd order moments
are considered, it was pointed out in [3] that the variance based sensitivity
measure is not well suited for heavy tailed or multimodal distributions.
Entropy is a measure of uncertainty similar to variance: higher entropy tends
to indicate higher variance (for Gaussian, entropy is proportional to log
variance). Nevertheless, it was shown in [3] that entropy-based methods and
variance-based methods can sometimes produce significantly different results.

Both variance-based and entropy-baesd global sensitivity analysis (GSA) can
provide quantitative contributions of each input variable to the output
quantity of interest. However, the estimation of variance and entropy based
sensitivity indices can become expensive in terms of number of model
evaluations. This limits the the appliation of both variance-based and
entropy-based methods to low dimensional problems.

In contrast, derivative-based methods are much more efficient as only the
average of the functional gradients across the input space is needed. It is
thus often used for screening of a large number of input variables. For
example, the Morris’ method [4] constructs a global sensitivity measure by
computing a weighted mean of the finite difference approximation to the
partial derivatives, and it requires only a few model evaluations.

It is thus interesting to find out how to use derivative-based methods to
mitigate the dimensionality issues of variance and entropy based approaches.

Previous studies have found a link between the derivative-based and variance-
based indices. In [5], a sensitivity measure $\mu*$ is proposed based on the
absolute values of the partial derivatives. It is empirically demonstrated
that for some practical problems $\mu*$ is similar to the variance based total
index. In [6], Sobol and Kucherenko have proposed the so-called derivative-
based global sensitivity measures (DGSM). This importance criterion is similar
to the modified Morris measure, except that the squared partial derivatives
are used instead of their absolute values. In addition, an inequality link
between variance based global sensitivity indices and the DGSM is established
in the case of uniform or Gaussian input variables.

This inequality between DGSM and variance-based GSA has been extended to input
variables belonging to the large class of Boltzmann probability measures in
[7]. A new sensitivity index, which is defined as a constant times the crude
derivative-based sensitivity, is shown to be a maximal bound of the variance
based total sensitivity index. Furthermore, in [8], the variance-based
sensitivity indices are interpreted as difference-based measures, where the
total sensitivity index is equivalent to taking a difference in the output
when perturbing one of the parameters with the other parameters fixed. The
similarity to partial derivatives helps to explain why the mean of absolute
elementary effects from the Morris’ method can be a good proxy for the total
sensitivity index.

However, the relationship between derivative-based and entropy-based
sensitivity indices is still unknown. In this paper, we will provide a novel
proof that, for a monotonic function, entropy-based total sensitivity index is
equivalent to the global sensitivity index based on log derivatives. More
generally, we demonstrate that the exponential of the entropy-based
sensitivity measure is bounded by the derivative-based indices, including the
modified Morris’ index and DGSM proposed in the literature. The missing link
between the derivative-based and entropy-based GSA is thus established and
that is the main contribution of this paper.

In this paper, we focus on entropy-based GSA and explore its link with
derivative-based GSA. However, it should be noted that there are many other
moment-indepdent sensitivity measures [9], which are often based on a distance
metric to measure the discrepancy between the conditional and unconditional
output probability density functions (PDFs). For example, sensitivity indices
based on the modification of the input PDFs have been proposed in [10] for
reliability sensitivity analysis, where the input perturbation is derived from
minimizing the probability divergence under constraints. [11] proposed a
moment independent $\delta$-indicator that looks at the entire input/output
distribution. The definition of $\delta$-indicator examines the expected total
shift between the conditional and unconditional output PDFs, where the shift
is conditional on one or more of the random input variables. Recently, the
Fisher Information Matrix has been proposed to examine the perturbation of the
entire joint probability density function (jPDF) of the outputs, and is
closely linked to the relative entropy between the jPDF of the outputs and its
perturbation due to an infinitesimal variation of the input distributions [12,
13].

In what follows, we will first review both derivative-based and entropy-based
global sensitivity measures in Section 2. In Section 3, the relationship
between these two indices are established, where mathematical proofs are given
for general multivariate functions. For the purpose of verification,
demonstrating examples are given in Section 4 for both monotonic and general
functions. Concluding remarks are given in Section 5.

##  2 Global sensitivity analysis

In this section, both derivative-based and entropy-based sensitivity measures
will be briefly reviewed, for the purpose of establishing a link between them
in the next section.

###  2.1 Derivative-based indices

Consider the function $y=g(\mathbf{x})$ where the function $g(\cdot)$ is
differentiable. Functionals based on $\partial g/\partial x_{i}$ have been
proposed to examine the global sensitivity with respect to the parameter
$x_{i}$.

For example, the modified Morris sensitivity index $\mu^{*}$ [14] is an
approximation of the functional:

$\mu_{i}=\mathbb{E}_{X}\left[\left|\frac{\partial g}{\partial
x_{i}}\right|\right]$ (1)

where in [6], a similar functional called DGSM has been studied:

$\nu_{i}=\mathbb{E}_{X}\left[\left|\frac{\partial g}{\partial
x_{i}}\right|^{2}\right]$ (2)

In addition, we will introduce the following functional based on log
derivatives:

$l_{i}=\mathbb{E}_{X}\left[\ln{\left|\frac{\partial g}{\partial
x_{i}}\right|}\right]$ (3)

to explore a link between entropy and derivatives based global sensitivity
indices. Note that different from $\mu_{i}$ and $\nu_{i}$, the log-derivative
based index $l_{i}$ can be negative.

Note that these there sensitivity indices are closely related as:

$e^{l_{i}}\leq\mu_{i}\leq\sqrt{\nu_{i}}$ (4)

where it is evident that $\mu_{i}\leq\sqrt{\nu_{i}}$ based on Cauchy-Schwarz
inequality. In addition, we have $e^{l_{i}}\leq\mu_{i}$ using Jensen’s
inequality as the logrithmic function is a concave function.

###  2.2 Entropy-based indices

Entropy is a measure of the average uncertainty, or the degree of non-
uniformality, represented by the distribution. Global sensitivity indices can
be formulated using entropy, analogously to the variance based sensitivity
indices [15]:

$\eta_{i}=\frac{H(Y)-\mathbb{E}[H(Y|x_{i})]}{H(Y)}=\frac{I(X_{i},Y)}{H(Y)}$
(5)

where $H(Y)$ is the entropy of $Y$, and $H(Y|X_{i})=\mathbb{E}[H(Y|x_{i})]$ is
the expected conditional entropy of $Y$ given $X_{i}$. $I(X_{i},Y)$ is the
mutual information which measures how much knowing $X_{i}$ reduces uncertainty
of $Y$ or vice versa.

The index $\eta_{i}$ in Eq 5 measures the excepted reduction in the entropy of
of $Y$ by fixing $X_{i}$. This can be regarded as the main effect contribution
of $X_{i}$ to the entropy of $Y$, in analogy to the main effect index from
variance-based indices as seen in Table 1.

Similarly, an entropy-based total sensitivity index can be defined as [16]:

$\eta_{Ti}=\frac{\mathbb{E}[H(Y|\mathbf{x}_{\sim i})]}{H(Y)}$ (6)

where the average is calculated over all possible values of $\mathbf{X}_{\sim
i}$. This measures the remaining entropy of $Y$ if the true values of
$\mathbf{X}_{\sim i}$ can be determined, in analogy to the total effect index
from variance-based indices as seen in Table 1.

Table 1: Variance-based indices vs. entropy-based indices

Variance-based indices Entropy-based indices Main Effect
$\displaystyle\frac{V(Y)-\mathbb{E}[V(Y|x_{i})]}{V(Y)}$
$\displaystyle\frac{H(Y)-\mathbb{E}[H(Y|x_{i})]}{H(Y)}$ Total Effect
$\displaystyle\frac{\mathbb{E}[V(Y|x_{\sim i})]}{V(Y)}$
$\displaystyle\frac{\mathbb{E}[H(Y|x_{\sim i})]}{H(Y)}$

##  3 Link between derivative-based and entropy-based GSA

In this section, the relationship between the derivative-based and entropy-
based sensitivity indices is explored. We will consider the function
$y=g(\mathbf{x})$ where the function $g(\cdot)$ is differentiable. Let $X$ be
a continuous random variable with probability density function (PDF) given by
$f_{X}(x)$. $Y=g(X)$ can be regarded as a transformed variable and the
transformed PDF $f_{Y}(y)$ can then be found via the Jacobian matrix. We will
start from a one-dimensional input variable and extend it to two-dimensional
and multi-dimensional cases using the method of dummy variables.

###  3.1 one dimensional variable

If $y=g(x)$ has a unique inverse $x=g^{-1}(y)$, the conditional PDF of the
output $Y$ given the input $X$ can be written as:

$f_{Y|X}(x,y)=\delta(y-g(x))=\frac{\delta(x-g^{-1}(y))}{\left|g^{\prime}\left(g^{-1}(y)\right)\right|}$
(7)

where $\delta(\cdot)$ is Dirac’s delta distribution. The marginal PDF of $Y$
is thus given as:

$f_{Y}(y)=\frac{f_{X}(g^{-1}(y))}{\left|g^{\prime}\left(g^{-1}(y)\right)\right|}=\frac{f_{X}(x)}{\left|g^{\prime}\left(x\right)\right|}$
(8)

The entropy of the continuous random variable $Y$ is thus:

$\begin{split}H(Y)&=-\int f_{Y}(y)\ln f_{Y}(y)dy\\\ &=-\int
f_{X}(x)\ln\frac{f_{X}(x)}{\left|g^{\prime}\left(x\right)\right|}dx\\\
&=H(X)+\int f_{X}(x)\ln{\left|g^{\prime}\left(x\right)\right|}dx\end{split}$
(9)

where $dy=g^{\prime}\left(x\right)dx$ for the change of variable.

$\mathbb{E}[\ln{\left|g^{\prime}\left(x\right)\right|}]=H(Y)-H(X)$ (10)

###  3.2 two dimensional variables

In the two-dimensional case, we will make use of the concept of dummy
variables to utilise the Jacobian matrix. Let
$y_{1}=g_{1}(x_{1},x_{2})=g(\mathbf{x})$ and $y_{2}=g_{2}(x_{1},x_{2})=x_{2}$
as a dummy variable. We assume that the functions $g_{1}(\cdot)$ and
$g_{2}(\cdot)$ have unique inverses (this condition will be generalised in
Section 3.4). Then the transformed joint PDF of $\mathbf{Y}=\\{Y_{1},Y_{2}\\}$
is:

$f(y_{1},y_{2})=\frac{f(x_{1},x_{2})}{\left|\det{\mathbb{J}}\right|}$ (11)

where $\mathbb{J}$ is the Jacobian matrix:

$\det{\mathbb{J}}=\begin{vmatrix}\frac{\partial g_{1}}{\partial
x_{1}}&\frac{\partial g_{1}}{\partial x_{2}}\\\ \frac{\partial g_{2}}{\partial
x_{1}}&\frac{\partial g_{2}}{\partial x_{2}}\end{vmatrix}=\frac{\partial
g_{1}}{\partial x_{1}}$ (12)

The entropy of $\mathbf{Y}$ can then be found as:

$\begin{split}H(Y_{1},Y_{2})&=-\int f_{Y}(y_{1},y_{2})\ln
f_{Y}(y_{1},y_{2})dy_{1}dy_{2}\\\ &=-\int
f_{X}(x_{1},x_{2})\ln\frac{f_{X}(x_{1},x_{2})}{\left|g^{\prime}_{1}\left(x_{1}\right)\right|}dx_{1}dx_{2}\\\
&=H(X_{1},X_{2})+\int
f_{X}(x_{1},x_{2})\ln{\left|g^{\prime}_{1}\left(x_{1}\right)\right|}dx_{1}dx_{2}\end{split}$
(13)

Assuming the input variables are independent, i.e.,
$H(X_{1},X_{2})=H(X_{1})+H(X_{2})$, and note that $y1=y$ and $y2=x2$ is a
dummy variable:

$H(Y|X_{2})=H(X_{1})+\mathbb{E}\left[\ln{\left|\frac{\partial
g(\mathbf{x})}{\partial x_{1}}\right|}\right]$ (14)

where the expectation is with respect to the jPDF of the input variables. We
have used the property of the conditional entropy, where
$H(Y,X_{2})=H(Y|X_{2})+H(X_{2})$. Note that
$H(Y|X_{2})=\mathbb{E}\left[H(Y|x_{2})\right]$ is the expected conditional
entropy.

###  3.3 multi-dimensional variables

Let $y_{1}=g_{1}(\mathbf{x})$ where $\mathbf{x}=(x_{1},x_{2},\dots,x_{n})$,
and introduce dummy variables $y_{i}=g_{i}(\mathbf{x})=x_{i}$ with
$i=2,\dots,n$. Assuming the functions $g_{i}(\cdot)$ have unique inverses, the
transformed jPDF is:

$f_{Y}(\mathbf{y})=\frac{f(\mathbf{x})}{\left|\det{\mathbb{J}}\right|}$ (15)

where $\mathbb{J}$ is the Jacobian matrix with $\mathbb{J}_{ij}=\partial
g_{i}/\partial x_{j}$.

Note that for the Jacobian matrix from the 2nd row onwards, i.e., $i\geq 2$,
$\partial g_{i}/\partial x_{j}=1$ when $i=j$ and $\partial g_{i}/\partial
x_{j}=0$ when $i\neq j$. Therefore, the Jacobian matrix in this case is a
triangular matrix. As a result, the Jacobian determinant is the product of the
diagonal entries:

$\det{\mathbb{J}}=\left|\frac{\partial g_{1}}{\partial
x_{1}}\times\underbrace{1\times\dots\times
1}_{2\,\text{to}\,n}\right|=\left|\frac{\partial g_{1}}{\partial
x_{1}}\right|$ (16)

The entropy of $\mathbf{Y}$ is thus:

$\begin{split}H(\mathbf{Y})&=-\int f_{Y}(\mathbf{y})\ln
f_{Y}(\mathbf{y})d\mathbf{y}\\\ &=-\int
f_{X}(\mathbf{x})\ln\frac{f_{X}(\mathbf{x})}{\left|\frac{\partial
g_{1}(\mathbf{x})}{\partial x_{1}}\right|}d\mathbf{x}\\\ &=H(\mathbf{X})+\int
f_{X}(\mathbf{x})\ln{\left|\frac{\partial g_{1}(\mathbf{x})}{\partial
x_{1}}\right|}d\mathbf{x}\end{split}$ (17)

where $\mathbf{Y}=\\{Y_{1},X_{2},X_{3},\dots,X_{n}\\}$.

Note that $H(\mathbf{Y})=H(Y_{1}|\mathbf{X}_{2\sim n})+H(\mathbf{X}_{2\sim
n})$, and assuming the input variables are independent, we then have:

$H(Y|\mathbf{X}_{2\sim n})=H(X_{1})+\mathbb{E}\left[\ln{\left|\frac{\partial
g(\mathbf{x})}{\partial x_{1}}\right|}\right]$ (18)

where the expectation is with respect to the jPDF of the input variables. Note
that $H(Y|\mathbf{X}_{2\sim n})=\mathbb{E}\left[H(Y|\mathbf{x}_{2\sim
n})\right]$ is the expected conditional entropy.

The reasoning above uses the first variable $x_{1}$ as an example. However,
the results hold for any variables via simple row/column exchanges, which only
affects the sign of the determinant but not its modulus.

###  3.4 summary

In summary, for a differentiable function $y=g(\mathbf{x})$, where
$\mathbf{x}=(x_{1},x_{2},\dots,x_{n})$, there exists a relationship between
the entropy and the expectation of the partial derivatives:

$\kappa_{i}=\mathbb{E}\left[H(Y|\mathbf{x}_{\sim
i})\right]-H(X_{i})=\mathbb{E}\left[\ln{\left|\frac{\partial
g(\mathbf{x})}{\partial x_{i}}\right|}\right]$ (19)

where $\sim i$ indicates the index ranges from 1 to $n$ excluding $i$. Note
that $H(Y|\mathbf{X}_{\sim i})=\mathbb{E}\left[H(Y|\mathbf{x}_{\sim
i})\right]$, where the expectation is with respect to all possible values of
$\mathbf{X}_{\sim i}$.

The above relationship holds only if the transformation $g(\cdot)$ has a
unique inverse. For a general differentiable function, we have the following
inequality instead [17]:

$\kappa_{i}\leq l_{i}$ (20)

where $l_{i}$ is used to denote $\mathbb{E}\left[\ln{\left|\frac{\partial
g(\mathbf{x})}{\partial x_{i}}\right|}\right]$ as in Eq 3.

As the exponential function is a monotonic increasing function, and taking
account of the relationship in Eq 4, we can then relate entropy to the
derivative based global sensitivity indices as:

$e^{\kappa_{i}}\leq e^{l_{i}}\leq\mu_{i}\leq\sqrt{\nu_{i}}$ (21)

where $l_{i}$, $\mu_{i}$ and $\nu_{i}$ are derivative-based sensitivity
indices as given in Section 2.1. For example, $\mu_{i}$ is used to denote
$\mathbb{E}\left[{\left|{\partial g(\mathbf{x})}/{\partial
x_{i}}\right|}\right]$, which can be regarded as the limiting value of the
modified Morris’ global sensitivity index [6] as given in Eq 1, while
$\nu_{i}$ is the DGSM sensitivity index proposed in [6]. Equation 21 thus
establises an inequality relationship between exponential entropy and the
global sensitivity measures based on partial derivatives.

Note that it is the exponential entropy that has a direct relationship with
the derivative-based global sensitivity measures. Recall that the entropy of a
random variable with a uniform distribution is $\ln(b-a)$, where $a,b$ are the
bounds of the distribution. Taking the natural exponential of the entropy in
this case results in $b-a$, which is the range of the uniform distribution.
Therefore, the exponential entropy $e^{H}$ can be regarded as a measure of the
extent, or effective support, of a distribution [5]. As $H(Y|\mathbf{X}_{\sim
i})$ measures the remaining entropy in average if the true values of
$\mathbf{X}_{\sim i}$ can be determined, the exponential entropy based index
$e^{\kappa_{i}}$ thus indicates the ratio between the range of output
distribution, conditioning on $\mathbf{X}_{\sim i}$ are known, and the range
of input $X_{i}$. And this ratio is upper bounded by derivative-based
sensitivity indices, including the modified Morris’ sensitivity index
$\mu_{i}$.

Studies in [3] already noted that an exponetial transformation of the standard
entropy-based sensitivity measures may improve its discrimination power. In
addition, as pointed out in [16], entropy for continuous random variables (aka
differential entropy) can become negative. This is a drawback of entropy-based
sensitivity method. In contrast, $e^{\kappa_{i}}$ is based on exponential
entropy and thus always positive.

For the purpose of screening and identifying non-influential inputs, the
inputs are typically assumed to have the same level of uncertainties, i.e.,
$H(X_{i})=H(X_{j})$. As the output entropy $H(Y)$ does not affect the relative
values, the new index $e^{\kappa_{i}}$ is essentially the same as the entropy-
based total index $\eta_{i}$ given in 1.

##  4 Examples

The mathematical relationship proved in Section 3, between derivative-based
and entropy-based GSA, is numerically verified in this section. The equality
between $\kappa_{i}$ and $l_{i}$ in Eq 19 is first examined for monotonic
functions, and the Ishigami function and G-function are then used to
demonstrate the inequality in Eq 21 for general functions.

All input variables are assumed to have the same uniform distribution for each
function, while Gaussian distributions are used for example 5. Having the same
distributions for all inputs, on the one hand, is for simplicity of the
demonstrating examples; on the other hand, is a commonly adopted approach for
input variable screenings, which is the main function of the entropy-based
total sensitivity index.

###  4.1 Monotonic functions

For verification purposes, all the examples in this section are chosen to have
tractable expressions for both the integral of derivatives and the conditional
entropies. For examples $1-3$, the conditional entropies are also numerically
estimated using the method given in Appendix A. This is to demonstrate that
numerical estimation of entropy-based sensitivity indices can be very
expensive. And that is the main motivation to establish a link with the
derivative-based method, so that it can be used as a potential proxy for
entropy-based GSA.

Figure 1: Surface plots, with contours shown underneath, for the monotonic
functions in examples 1-3. Table 2: Sensitivity results for the monotonic
functions in examples 1-3 (surface plots in Figure 1). Conditional entropy
results are obtained from Monte Carlo sampling with number of samples ranging
from $10^{3}$ to $10^{8}$. The results from $10^{8}$ samples are compared to
the exact results and the relative error are less than 1% for all functions.
Also given are the corresponding analytical results for derivative-based
indices.

Number of Samples $x_{i}\sim\mathbb{U}(0,1)$ $y=x_{1}+e^{x_{2}}$
$y=x_{1}\times x_{2}$ $y=x_{1}+3x_{2}$ $\kappa_{1}$ $\kappa_{2}$ $\kappa_{1}$
$\kappa_{2}$ $\kappa_{1}$ $\kappa_{2}$ 1.00E+03 0.0934 0.5251 -0.9057 -0.8468
0.2279 1.1269 1.00E+04 0.0602 0.5225 -0.9111 -0.9211 0.1043 1.1093 1.00E+05
0.0315 0.5131 -0.9528 -0.9593 0.0526 1.1047 1.00E+06 0.0148 0.5069 -0.9760
-0.9755 0.0242 1.1019 1.00E+07 0.0068 0.5034 -0.9879 -0.9878 0.0113 1.1000
1.00E+08 0.0032 0.5015 -0.9939 -0.9939 0.0052 1.0993 Exact results 0.0000
0.5000 -1.0000 -1.0000 0.0000 1.0986 error - -0.30% 0.61% 0.61% - -0.06%
$l_{1}$ $l_{2}$ $l_{1}$ $l_{2}$ $l_{1}$ $l_{2}$
$\mathbb{E}\left[\ln{\left|\frac{\partial y}{\partial x_{i}}\right|}\right]$
0.0000 0.5000 -1.0000 -1.0000 0.0000 1.0986

#### Example 1.

Consider the function $y=x_{1}+e^{x_{2}}$, where the partial derivatives are
$\partial y/\partial x_{1}=1$ and $\partial y/\partial x_{2}=e^{x_{2}}$. For
$x_{i}\sim\mathbb{U}(0,1)$, the expected value
$l_{i}=\mathbb{E}\left[\ln{\left|\frac{\partial g(\mathbf{x})}{\partial
x_{i}}\right|}\right]$ can be integrated analytically as 0 and 1/2 for $x_{1}$
and $x_{2}$ respectively. The sensitivities indices based on conditional
entropy can also be calculated analytically in this case.
$\kappa_{1}=H(Y|X_{2})-H(X_{1})=0$, where $H(Y|x_{2})=H(X_{1})=0$ as the
differential entropy remains constant under addition of a constant
($e^{x_{2}}$ is a constant for $H(Y|x_{2})$). As the transformed variable
$V=e^{x}$ has a PDF $V\sim 1/v,1<v<e$,
$\kappa_{2}=H(Y|X_{1})-H(X_{2})=\mathbb{E}_{X_{1}}\left[H(Y|x_{1})\right]-0=\mathbb{E}_{X_{1}}\left[-\int_{1}^{e}\frac{1}{y}\ln\frac{1}{y}\right]=1/2$.
Therefore, $\kappa_{1}=l_{1}$ and $\kappa_{2}=l_{2}$, as given by Eq 19.

#### Example 2.

Consider the function $y=x_{1}\times x_{2}$, where the partial derivatives are
$\partial y/\partial x_{1}=x_{2}$ and $\partial y/\partial x_{2}=x_{1}$. For
$x_{i}\sim\mathbb{U}(0,1)$, the expected value
$\mathbb{E}\left[\ln{\left|\frac{\partial g(\mathbf{x})}{\partial
x_{i}}\right|}\right]$ are $-1$ for both $x_{1}$ and $x_{2}$. Recall that the
differential entropy increases additively upon multiplication with a constant,
$\kappa_{1}=H(Y|X_{2})-H(X_{1})=\mathbb{E}_{X_{2}}\left[H(x_{1})+\ln|x_{2}|\right]=\int_{0}^{1}\ln{x_{2}}d{x_{2}}=-1$.
Therefore, $\kappa_{i}=l_{i},i=1,2$ as given by Eq 19.

#### Example 3.

Consider the function $y=x_{1}+3x_{2}$, where the partial derivatives are 1
and 3 for $x_{1}$ and $x_{2}$ respectively. For $x_{i}\sim\mathbb{U}(0,1)$,
the expected value $\mathbb{E}\left[\ln{\left|\frac{\partial
g(\mathbf{x})}{\partial x_{i}}\right|}\right]$ can be integrated analytically
as 0 and $\ln 3\simeq 1.0986$. It is straightforward to show that, as in
previous examples, $\kappa_{1}=0$ and $\kappa_{2}=\ln 3$, which are the same
as $l_{1}$ and $l_{2}$.

#### Example 4.

Consider the product function $y=x_{1}x_{2}^{r}$, where $r\geq 1$ (for $r=1$,
we recover the function in example 2). For $x_{1},x_{2}\sim\mathbb{U}(0,1)$,
it is straightforward to show that $s_{1}=-r$ and $s_{2}=\ln{r}-r$. For
entropy based indices, we have
$\kappa_{1}=H(Y|X_{2})-H(X_{1})=\mathbb{E}_{X_{2}}\left[H(x_{1})+\ln|x_{2}|\right]=\int_{0}^{1}\ln{x_{2}^{r}}d{x_{2}}=-r$,
which is the same as $l_{1}$.
$\kappa_{2}=H(Y|X_{1})-H(X_{2})=\mathbb{E}_{X_{1}}\left[H(Y|x_{1})+\ln|x_{1}|\right]-0=\mathbb{E}_{X_{1}}\left[-\int_{0}^{1}p(y|x_{1})\ln
p(y|x_{1})\right]-1=\ln{r}-r$, where $p(y|x_{1})=\frac{1}{r}y^{\frac{1}{r}-1}$
as the transformed variable $V=x^{r}$ has a PDF
$V\sim\frac{1}{r}v^{\frac{1}{r}-1},0<v<1$. Therefore, $\kappa_{1}=l_{1}$ and
$\kappa_{2}=l_{2}$ as shown by Eq 19.

#### Example 5.

Consider the linear function $y=\sum_{i=1}^{n}a_{i}x_{i}$. This function has
been used in [15] to demonstrate the equivalence between entropy based and
variance based sensitivity indices for Guassian random inputs. In the case
with independent inputs, the sensitivity index based on the conditional
entropy can be obtained as $\kappa_{i}=H(Y|\mathbf{X}_{\sim
i})-H(X_{i})=1/2\ln{2\pi ea_{i}^{2}\sigma_{i}^{2}}-1/2\ln{2\pi
e\sigma_{i}^{2}}=\ln{|a_{i}|}$, where $\sigma_{i}^{2}$ is the variance for the
random input $x_{i}$. The results are the same as the derivative based index
$l_{i}=\ln{|\partial y/\partial x_{i}|}=\ln|a_{i}|$, which demonstrates the
equality given in Eq 19.

###  4.2 General functions

Both Ishigami function and G-function are commonly used test functions for
global sensitivity analysis, due to the presence of strong interactions. These
two functions, each with three input variables, are used in this section to
demostrate the inequality relationship derived in Eq 21.

The conditional entropies, for the estimation of the sensitivity index
$\kappa_{i}$, are estimated numerically using Monte Carlo sampling as
described in Appendix A. Different number of samples are used, ranging from
$1e6$ to $1e8$. For each estimation, the computation is repeated for 20 times
and both the mean value and the standard deviation (std) are reported in Table
3 and 4.

For the estimation of derivative-based indices, $l_{i}$ and $\mu_{i}$,
Matlab’s inbuild numerical integrator "integral" is used with default
tolerance setting. Note that the index $\nu_{i}$ is not compared here, as its
relationship with $\mu_{i}$ has been studied elsewhere [6].

To rank the input variables, sensitivity indices in this section are
normalised. For example, $e^{\kappa_{i}}/\sum e^{\kappa_{i}}$ is used for the
entropy-based sensitivity results. As a result, the rankings in Table 3 and 4
are shown as percentages.

#### Example 6.

The Ishigami function, $y=\sin(x_{1})+a\sin^{2}(x_{2})+bx^{4}_{3}\sin(x_{1})$,
is often used as an example for uncertainty and sensitivity analysis. It
exhibits strong nonlinearity and nonmonotonicity, as can be seen in Figure 2.
In this case, $a=7$ and $b=0.1$ are used, and the input random variables have
uniform distributions, i.e., $x_{i}\sim\mathbb{U}(-\pi,\pi)$ for $i=1,2,3$.

The sensitivity results are listed in Table 3. The index $\kappa_{i}$ is
estimated with different number of samples, and each estimation with 20
repetitiions for variability assurance. It is clear from the small std that
the estimation is well converged. The index $e^{\kappa_{i}}$ is then estimated
using the mean value from the results with $1e8$ samples, and compared with
derivative-based indices $e^{l_{i}}$ and $\mu_{i}$.

First, from Table 3, it is clear that the inequality among $e^{\kappa_{i}}$,
$e^{l_{i}}$ and $\mu_{i}$ from Eq 21 is satisfied. Second, the ranking is
similar from these three indices, with $x_{2}$ being the most important, thus
suggesting using the derivative-based indices as a proxy for entropy-based
importance measure. Third, we note in passing that the quantitative ranking
based on $e^{\kappa_{i}}$ is very close to the results reported in [3]. This
is different from the variance-based total sensitivity results, as compared in
[3], which ranks $x_{1}$ being the most important.

Figure 2: Scatter plot for the Ishigami function,
$y=\sin(x_{1})+a\sin^{2}(x_{2})+bx^{4}_{3}\sin(x_{1})$. Table 3: Sensitivity
results for the Ishigami function. The conditional entropy based sensitivity
results are obtained for different number of samples. This is repeated for 20
times and the mean and standard deviation (std) are given. Also shown are the
three indices $e^{\kappa_{i}}$, $e^{l_{i}}$ and $\mu_{i}$, for which the
inequality in Eq 21 is clearly satisfied.

Ishigami function $y=\sin(x_{1})+a\sin^{2}(x_{2})+bx^{4}_{3}\sin(x_{1})$
Number of Samples $\kappa_{1}$ $\kappa_{2}$ $\kappa_{3}$ mean std mean std
mean std 1.00E+06 -0.4474 7.40E-04 -0.0764 1.07E-03 -0.8669 1.61E-03 3.16E+06
-0.4994 2.87E-04 -0.1079 3.61E-04 -0.9726 5.24E-04 1.00E+07 -0.5400 2.40E-04
-0.1356 1.57E-04 -1.0685 4.06E-04 3.16E+07 -0.5756 1.97E-04 -0.1582 1.17E-04
-1.1543 3.24E-04 1.00E+08 -0.6043 9.77E-05 -0.1770 8.56E-05 -1.2313 1.69E-04
$x_{1}$ $x_{2}$ $x_{3}$ sensitivity ranking sensitivity ranking sensitivity
ranking $e^{\kappa_{i}}$ 0.5464 32.6% 0.8378 50.0% 0.2919 17.4% $e^{l_{i}}$
1.0667 21.9% 3.5000 71.8% 0.3087 6.3% $\mu_{i}$ 1.8769 22.6% 4.4563 53.6%
1.9739 23.8%

#### Example 7.

Consider the so-called G-function,
$y=\prod_{i=1}^{3}(|4x_{i}-2|+a_{i})/(1+a_{i})$, which is often used for
numerical experiments in sensitivity analysis. It is a highly nonlinear
function, as can be seen in Figure 3 for a two-variable example. In this case,
$a_{i}=(i-2)/2$, for $i=1,2,3$. The input random variables have uniform
distributions, i.e., $x_{i}\sim\mathbb{U}(0,1)$ for $i=1,2,3$. A lower value
of $a_{i}$ indicates a higher importance of the input variable $x_{i}$, i.e.,
$x_{1}$ is the most important, while $x_{3}$ is the least important in this
case.

The sensitivity results for the G-function, in the same format as Table 3, are
reported in Table 4. It is clear that, not only the inequality relationship in
Eq 21 is satisfied, the three indices give the same quantitative ranking of
the three variables. Note that $e^{l_{i}}$ and $\mu_{i}$ produce the same
resutls, as the effect of logrithm and exponential operations cancels out for
this product function.

Figure 3: Surface plots, with contours shown underneath, for an example of
G-function with 2 variables. Table 4: Sensitivity results for the G-function,
where the inequality in Eq 21 is clearly satisfied. Same key as Table 3

G-function $\displaystyle y=\prod_{i=1}^{3}\frac{|4x_{i}-2|+a_{i}}{1+a_{i}}$
Number of Samples $\kappa_{1}$ $\kappa_{2}$ $\kappa_{3}$ mean std mean std
mean std 1.00E+06 0.3473 1.40E-03 -0.1383 1.48E-03 -0.3993 1.55E-03 3.16E+06
0.3431 9.71E-04 -0.1584 8.19E-04 -0.4273 7.72E-04 1.00E+07 0.3401 6.17E-04
-0.1735 5.41E-04 -0.4480 4.77E-04 3.16E+07 0.3384 4.66E-04 -0.1841 3.75E-04
-0.4630 3.20E-04 1.00E+08 0.3377 3.58E-04 -0.1917 2.67E-04 -0.4739 2.34E-04
$x_{1}$ $x_{2}$ $x_{3}$ sensitivity ranking sensitivity ranking sensitivity
ranking $e^{\kappa_{i}}$ 1.4018 49.2% 0.8255 29.0% 0.6226 21.8% $e^{l_{i}}$
4.0000 46.2% 2.6667 30.8% 2.0000 23.1% $\mu_{i}$ 4.0000 46.2% 2.6667 30.8%
2.0000 23.1%

##  5 Conclusions

  * •

A link between derivative-based and entropy-based global sensitivity measures
has been establised.

  * •

For monotonic functions, mathematical proofs are given for the equality
between the expected log-derivative sensitivities and entropy based global
measures.

  * •

For general functions, the exponential of the entropy-based sensitivity
measures is found to be upper bounded by derivative-based global sensitivity
indices.

  * •

Numerical examples show that the derivative-based methods give similar
variable rankings as entropy-based index and can thus be potentially used as a
proxy for entropy-based GSA.

## Acknowledgment

For the purpose of open access, the author has applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising.

## Data availability statement

The datasets generated during and/or analysed during the current study are
available in the GitHub repository: XXX

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##  Appendix A Numerical estimation of entropy

Adopting the approach from [18], the $xy$-plane is gridded by equal size cells
($\Delta x\times\Delta y$) with coordinates ($i$, $j$). The probability of
observing a sample in cell ($i$, $j$) is:

$p_{ij}=\iint\limits_{\text{cell}(i,j)}f(x,y)dxdy\approx f(x_{i},y_{j})\Delta
x\Delta y$ (A.1)

where $(x_{i},y_{j})$ is the centre of the cell.

Assuming the jPDF is approximately constant within a cell, the joint entropy
can be represented as:

$\begin{split}H(X,Y)&=-\int f(x,y)\ln f(x,y)dxdy\\\ &\approx-\sum
f(x_{i},y_{j})\ln f(x_{i},y_{j})\Delta x\Delta y\\\ &\approx-\sum
p_{ij}\left(\ln p_{ij}-\ln(\Delta x\Delta y)\right)\\\
&\approx-\sum\left(\frac{k_{ij}}{N}\ln\frac{k_{ij}}{N}\right)+\ln(\Delta
x\Delta y)\end{split}$ (A.2)

where $k_{ij}$ represents the number of samples observed in the cell ($i$,
$j$), and $N$ is the total number of samples.

Similarly, the conditional entropy can be approximated as:

$\begin{split}H(Y|X)&=-\int f(x,y)\frac{\ln f(x,y)}{f(x)}dxdy\\\ &\approx-\sum
p_{ij}\left(\ln p_{ij}-\ln p_{i}-\ln\Delta y\right)\\\
&\approx-\sum\left(\frac{k_{ij}}{N}\ln\frac{k_{ij}}{k_{i}}\right)+\ln\Delta
y\end{split}$ (A.3)

where $k_{i}=\sum_{j}k_{ij}$ and similar expressions can be derived when
$\mathbf{X}$ is a vector variable.


# Observation of a linked loop quantum state

Ilya Belopolski111These authors contributed equally to this work.
<EMAIL_ADDRESS>Laboratory for Topological Quantum Matter and
Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New
Jersey 08544, USA  RIKEN Center for Emergent Matter Science (CEMS), Wako,
Saitama 351-0198, Japan  Guoqing Chang∗ Division of Physics and Applied
Physics, School of Physical and Mathematical Sciences, Nanyang Technological
University, 21 Nanyang Link, 637371, Singapore  Tyler A. Cochran∗ Laboratory
for Topological Quantum Matter and Spectroscopy (B7), Department of Physics,
Princeton University, Princeton, New Jersey 08544, USA  Zi-Jia Cheng∗
Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of
Physics, Princeton University, Princeton, New Jersey 08544, USA  Xian P. Yang
Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of
Physics, Princeton University, Princeton, New Jersey 08544, USA  Cole
Hugelmeyer  Department of Mathematics, Princeton University, Princeton, New
Jersey 08544, USA  Kaustuv Manna  Max Planck Institute for Chemical Physics of
Solids, Nöthnitzer Straße 40, 01187 Dresden, Germany  Department of Physics,
Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India  Jia-
Xin Yin  Laboratory for Topological Quantum Matter and Spectroscopy (B7),
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Guangming Cheng  Princeton Institute for Science and Technology of Materials,
Princeton University, Princeton, New Jersey, 08544, USA  Daniel Multer
Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of
Physics, Princeton University, Princeton, New Jersey 08544, USA  Maksim
Litskevich  Laboratory for Topological Quantum Matter and Spectroscopy (B7),
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Nana Shumiya  Laboratory for Topological Quantum Matter and Spectroscopy (B7),
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Songtian S. Zhang  Laboratory for Topological Quantum Matter and Spectroscopy
(B7), Department of Physics, Princeton University, Princeton, New Jersey
08544, USA  Chandra Shekhar  Max Planck Institute for Chemical Physics of
Solids, Nöthnitzer Straße 40, 01187 Dresden, Germany  Niels B. M. Schröter
Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen, Switzerland
Alla Chikina  Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen,
Switzerland  Craig Polley  MAX IV Laboratory, Lund University, P.O. Box 118,
221 00 Lund, Sweden  Balasubramanian Thiagarajan  MAX IV Laboratory, Lund
University, P.O. Box 118, 221 00 Lund, Sweden  Mats Leandersson  MAX IV
Laboratory, Lund University, P.O. Box 118, 221 00 Lund, Sweden  Johan Adell
MAX IV Laboratory, Lund University, P.O. Box 118, 221 00 Lund, Sweden  Shin-
Ming Huang  Department of Physics, National Sun Yat-Sen University, Kaohsiung
804, Taiwan  Nan Yao  Princeton Institute for Science and Technology of
Materials, Princeton University, Princeton, New Jersey, 08544, USA  Vladimir
N. Strocov  Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen,
Switzerland  Claudia Felser  Max Planck Institute for Chemical Physics of
Solids, Nöthnitzer Straße 40, 01187 Dresden, Germany  M. Zahid Hasan
<EMAIL_ADDRESS>Laboratory for Topological Quantum Matter and
Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New
Jersey 08544, USA  Princeton Institute for Science and Technology of
Materials, Princeton University, Princeton, New Jersey, 08544, USA  Materials
Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA

###### Abstract

Quantum phases can be classified by topological invariants, which take on
discrete values capturing global information about the quantum state
news_Buchanan ; GeoTopoPhys_Nakahara ; Topo_ChaikinLubensky ; Nobel_Haldane ;
RevMod_Topology_Wen ; HeliumDroplet_Volovik ; HeliumRMP_Volovik ;
TopoMag_Hoffmann ; SkyrmionsNatNano_Tokura ; news_ChernQHE_Avron ;
Colloquium_Zahid ; Topo_Andrei ; ReviewSSH_Kane ; RMPTopoBandThy_Bansil ;
ReviewMagTopo_Tokura ; ARCMP_me ; Review_ZahidSuyangGuang ;
RMPWeylDirac_Armitage ; RMPClassification_ChingKai . Over the past decades,
these invariants have come to play a central role in describing matter,
providing the foundation for understanding superfluids HeliumDroplet_Volovik ;
HeliumRMP_Volovik , magnets TopoMag_Hoffmann ; SkyrmionsNatNano_Tokura , the
quantum Hall effect Nobel_Haldane ; news_ChernQHE_Avron , topological
insulators Colloquium_Zahid ; Topo_Andrei ; ReviewSSH_Kane ;
RMPTopoBandThy_Bansil ; ReviewMagTopo_Tokura , Weyl semimetals ARCMP_me ;
Review_ZahidSuyangGuang ; RMPWeylDirac_Armitage ; RMPClassification_ChingKai
and other phenomena. Here we report a remarkable linking number (knot theory)
invariant associated with loops of electronic band crossings in a mirror-
symmetric ferromagnet Co2MnGa_me ; Co2MnGa_Guoqing ;
NonAbelianLinks_WuSoluyanov ; NodalLink_ZhongboYan ; Knots_Ezawa ;
WeylLink_Yee ; PentagonC_ShengbaiZhang . Using state-of-the-art spectroscopic
methods, we directly observe three intertwined degeneracy loops in the
material’s bulk Brillouin zone three-torus, $\mathbb{T}^{3}$. We find that
each loop links each other loop twice. Through systematic spectroscopic
investigation of this linked loop quantum state, we explicitly draw its link
diagram and conclude, in analogy with knot theory, that it exhibits linking
number $(2,2,2)$, providing a direct determination of the invariant structure
from the experimental data. On the surface of our samples, we further predict
and observe Seifert boundary states protected by the bulk linked loops,
suggestive of a remarkable Seifert bulk-boundary correspondence. Our
observation of a quantum loop link motivates the application of knot theory to
the exploration of exotic properties of quantum matter.

Quantum topology is powerful in understanding condensed matter systems that
exhibit a winding news_Buchanan ; GeoTopoPhys_Nakahara ; Topo_ChaikinLubensky
; Nobel_Haldane ; RevMod_Topology_Wen ; HeliumDroplet_Volovik ;
HeliumRMP_Volovik ; TopoMag_Hoffmann ; SkyrmionsNatNano_Tokura ;
news_ChernQHE_Avron ; Colloquium_Zahid ; Topo_Andrei ; ReviewSSH_Kane ;
RMPTopoBandThy_Bansil ; ReviewMagTopo_Tokura ; ARCMP_me ;
Review_ZahidSuyangGuang ; RMPWeylDirac_Armitage ; RMPClassification_ChingKai .
Often, this winding occurs in real space. For example, in a magnetic material,
the local magnetization may exhibit a rotating pattern centered around a point
in real space, forming a magnetic vortex encoding an integer winding number
Topo_ChaikinLubensky ; TopoMag_Hoffmann . Alternatively, the winding may occur
in momentum space. For example, in a one-dimensional topological insulator,
the quantum-mechanical wavefunctions wind as the momentum scans through the
Brillouin zone Nobel_Haldane ; RevMod_Topology_Wen ; news_ChernQHE_Avron ;
Colloquium_Zahid ; Topo_Andrei ; ReviewSSH_Kane ; RMPTopoBandThy_Bansil .
These two broad paradigms—order parameters, such as magnetization, which wind
in real space and quantum wavefunctions which wind in momentum space—capture a
vast landscape of topological phases of matter, spanning decades of research
by myriad communities of physicists. Real-space order parameter winding
further encompasses disclinations in liquid crystals; vortices in
superconductors and superfluid 4He; and magnetic skyrmions, whose invariants
are proposed as the basis for next-generation computing memory and logic
GeoTopoPhys_Nakahara ; Topo_ChaikinLubensky ; HeliumDroplet_Volovik ;
HeliumRMP_Volovik ; TopoMag_Hoffmann ; SkyrmionsNatNano_Tokura . On the other
hand, momentum-space wavefunction winding is associated with emergent Dirac
fermions in two- and three-dimensional topological insulators; Weyl fermions
in topological semimetals; and the quantum Hall effect, which sets the
prevailing von Klitzing standard of electrical resistance news_ChernQHE_Avron
; Colloquium_Zahid ; Topo_Andrei ; ReviewSSH_Kane ; RMPTopoBandThy_Bansil ;
ReviewMagTopo_Tokura ; ARCMP_me ; Review_ZahidSuyangGuang ;
RMPWeylDirac_Armitage ; RMPClassification_ChingKai . Despite their importance
in modern physics, there is no indication that these two paradigms are
exhaustive. Novel paradigms for topology promise to deepen our fundamental
understanding of nature, as well as enable new quantum technologies.

Recently, there has been considerable interest in node loops, an electronic
structure where two bands cross along a closed curve in momentum space
Co2MnGa_me ; Co2MnGa_Guoqing ; PbTaSe2_Guang ; MagSpaceGroups_Vishwanath ;
WeylLoopSuperconductor_Nandkishore ; WeylLines_Kane ;
WeylDiracLoop_Nandkishore ; RMPClassification_ChingKai . Away from the
crossing curve, the two bands disperse linearly, so that the node loop
consists of a cone dispersion persisting along a loop. Within the paradigm of
momentum-space wavefunction winding, node loops are topological, with a
quantized Berry phase invariant Colloquium_Zahid ; Topo_Andrei ;
ReviewSSH_Kane ; RMPTopoBandThy_Bansil ; WeylLines_Kane ;
WeylDiracLoop_Nandkishore ; RMPClassification_ChingKai . However, in contrast
to other electronic structures studied to date news_ChernQHE_Avron ;
Colloquium_Zahid ; Topo_Andrei ; ReviewSSH_Kane ; RMPTopoBandThy_Bansil ;
ReviewMagTopo_Tokura ; ARCMP_me ; Review_ZahidSuyangGuang ;
RMPWeylDirac_Armitage ; RMPClassification_ChingKai , node loops can link each
other, encoding a linking number invariant (Fig. 1a, Extended Data Fig. 1)
NonAbelianLinks_WuSoluyanov ; NodalLink_ZhongboYan ; Knots_Ezawa ;
WeylLink_Yee ; PentagonC_ShengbaiZhang . Unlike the traditional paradigms of
winding, this linking number is associated with the composite loop structure
of quantum-mechanical band crossings of the Hamiltonian. Such linked node
loops offer the possibility of a new bridge between physics and knot theory.
It has further been proposed that these links are governed by emergent non-
Abelian node loop charges NonAbelianLinks_WuSoluyanov  and that the linking
number determines the $\theta$ angle of the axion Lagrangian in certain node
loop phases WeylLink_Yee ; ShouChengZhang_HelixNodalLine ; Biao_WilsonLoops .
Since the three-dimensional condensed matter Brillouin zone is a three-torus
$\mathbb{T}^{3}$, linked node loops also offer the rare possibility of
observing links in a space other than ordinary infinite space
$\mathbb{R}^{3}$. Moreover, the Seifert surface of the bulk link is associated
with topological boundary states, opening the possibility of a unique Seifert
bulk-boundary correspondence in quantum matter Seifert_1935 ; NonHermitian_RMP
; Seifert_LinhuLi ; NonHermitian_Carlstrom ; NonHermitian_Tidal .

Ferromagnets with crystalline mirror symmetry naturally give rise to node
loops. In this scenario, the ferromagnetic exchange interaction produces spin-
split electronic bands which are generically singly-degenerate throughout
momentum space, while mirror symmetry protects two-fold band degeneracies
along closed curves confined to the momentum-space mirror planes
MagSpaceGroups_Vishwanath . Such node loops are called Weyl loops, by analogy
with the two-fold degeneracy of a Weyl point ARCMP_me ;
Review_ZahidSuyangGuang ; RMPWeylDirac_Armitage ;
WeylLoopSuperconductor_Nandkishore ; WeylLines_Kane ;
WeylDiracLoop_Nandkishore . Weyl loops are extremely effective at
concentrating Berry curvature, giving rise to giant anomalous Hall and Nernst
effects, up to room temperature and promising for technological applications
Co2MnGa_me ; Fe3Ga_Nakatsuji ; Co2MnGa_Nakatsuji ; Co2MnGa_Felser_Nernst ;
Co2MnGa_Thomas_Nernst ; CrPt3_Anastasios . In crystallographic space groups
with multiple perpendicular mirror planes, different Weyl loops living in
different mirror planes can naturally link each other Co2MnGa_Guoqing ;
PentagonC_ShengbaiZhang . The ferromagnet Co2MnGa exhibits a crystal structure
with multiple perpendicular mirror planes and was recently observed to host
electronic Weyl loops Co2MnGa_me ; Co2MnGa_Guoqing , bringing together the key
ingredients for node loop links.

Co2MnGa crystallizes in the full Heusler structure, with face-centered cubic
Bravais lattice; space group $Fm\bar{3}m$ (No. 225); octahedral point group
$O_{h}$; and conventional unit cell lattice constant $c=5.77$ Å (Fig. 1b,
Extended Data Fig. 2). We observe that our Co2MnGa samples are ferromagnetic,
with Curie temperature $T_{\rm C}=690$ K, consistent with earlier reports
Co2MnGa_neutron ; Co2MnGa_CurieTemp . The point group includes mirror planes
normal to $\hat{x}$, $\hat{y}$ and $\hat{z}$ (defined by the conventional unit
cell lattice vectors, representative mirror plane shown in orange in Extended
Data Fig. 2a), as well as three $C_{4}$ rotation symmetries relating any one
of these mirror planes to the other two. We first perform a characterization
by atomic-level energy-dispersive X-ray spectroscopy (EDS), providing direct
structural evidence that our Co2MnGa samples are crystallographically well-
ordered, show the expected lattice constant and exhibit these mirror and
rotation symmetries (Fig. 1b). The real-space mirror planes give rise to
momentum-space mirror planes, labelled $M_{1}$ (normal to $\hat{z}$); $M_{2}$
($\hat{y}$); and $M_{3}$ ($\hat{x}$, Fig. 1c). These mirror planes contain the
time-reversal invariant momenta $X_{1}$, $X_{2}$ and $X_{3}$, sitting at the
centers of the square faces of the bulk Brillouin zone.

Motivated by the observation of mirror-symmetry-protected magnetic Weyl loops
in Co2MnGa Co2MnGa_me ; Co2MnGa_Guoqing , we explore the electronic structure
of our samples on $M_{1}$, $M_{2}$ and $M_{3}$. We perform ab initio
calculations of Co2MnGa in the ferromagnetic state, focusing on these three
mutually-perpendicular mirror planes. We find that each mirror plane hosts a
Weyl loop, and that the three Weyl loops link one another (Fig. 1c, Extended
Data Fig. 9). To experimentally investigate this ab initio prediction, we
carry out angle-resolved photoemission spectroscopy (ARPES) using soft X-ray
photons, optimized for exploring bulk electronic states ADRESS1_Strocov ;
ADRESS2_Strocov . Without loss of generality, we consider the crystal cleaving
plane in our photoemission experiments to be parallel to $M_{1}$. We first
acquire a Fermi surface at a fixed incident photon energy $h\nu=544$ eV,
chosen to fix the $k_{z}$ momentum to this ‘in-plane’ mirror plane ($M_{1}$,
Fig. 1d). We observe a diamond-shaped contour centered on $X_{1}$, which
traces out a momentum-space trajectory encircling the square top face of the
bulk Brillouin zone. We also observe a small circular feature at the corners
of the Fermi surface, which arises from an unrelated band at $\Gamma$,
irrelevant for what follows. We next perform a photon-energy dependence on the
same sample, measuring from $h\nu=500$ to $800$ eV, which allows us to access
the electronic structure on the ‘out-of-plane’ mirror $M_{2}$ (Fig. 1e). We
again observe a diamond-shaped loop contour, now centered on $X_{2}$ and
encircling the square side face of the bulk Brillouin zone. We then rotate the
sample by $90^{\circ}$ and repeat the same photon-energy dependence to capture
the electronic structure on the other ‘out-of-plane’ mirror $M_{3}$. We
observe again a similar diamond-shaped loop contour centered on $X_{3}$ (Fig.
1f). These systematic observations suggest a family of symmetry-related
diamond-shaped loop contours, consistent with the octahedral point group
$O_{h}$. Each of the three loops is related to the others by rotation symmetry
and each lives in one of the three mirror planes.

To further understand the loop electronic structures, we examine energy-
momentum photoemission spectra slicing through the $M_{1}$ loop (Fig. 2a). We
observe two bands which disperse toward each other and meet near the Fermi
level, $E_{\textrm{F}}$ ($E_{\textrm{B}}=0$), suggesting a cone dispersion.
The presence of a cone dispersion in both slices further shows that the cone
persists as we move in momentum space, following the $M_{1}$ loop Fermi
surface. Since Co2MnGa is ferromagnetic, we expect generically singly-
degenerate bands throughout the Brillouin zone Co2MnGa_neutron ;
Co2MnGa_CurieTemp . This suggests a cone dispersion consisting of singly-
degenerate branches which exhibit a double degeneracy at the touching points,
indicating a Weyl loop electronic structure. To better understand the cone
dispersions, we perform ab initio calculations of the electronic structure on
energy-momentum slices corresponding to these ARPES spectra. The calculation
shows a Weyl cone with characteristic two-fold degenerate crossing and linear
dispersion away from the crossing (Fig. 2b). Assembled together, this series
of Weyl cones forms a Weyl loop. The ARPES and ab initio calculations are in
good agreement, further suggesting that we have observed a magnetic Weyl loop
on $M_{1}$.

To characterize this Weyl loop using ARPES, we systematically track all cone
crossings observed along the full $M_{1}$ loop trajectory (Fig. 2c). We then
fit the crossings to a two-band effective $k\cdot p$ Hamiltonian for a Weyl
loop,

$H=\sum_{k,\ a,b\in\\{\pm\\}}c^{{\dagger}}_{ka}h_{ab}(k)c_{kb},\hskip
28.45274pth(k)=f(k)\sigma_{z}+v_{\rm F}q_{z}\sigma_{x}+g(k)\mathbbm{1}$ (1)

Here the $c^{{\dagger}}_{ka}$ are fermionic creation operators, $k$ is the
crystal momentum, $\sigma_{z}$ and $\sigma_{x}$ are Pauli matrices,
$\mathbbm{1}$ is the $2\times 2$ identity and $q_{z}\equiv k_{z}-2\pi/c$ is
the $\hat{z}$ component of the momentum measured relative to $M_{1}$, where
$c$ is the conventional unit cell lattice constant. This Hamiltonian exhibits
a Weyl loop on $q_{z}=0$ with trajectory given by $f(k)=0$, formed from two
bands with opposite mirror eigenvalues. From our ARPES spectra, we
experimentally extract the full Weyl loop trajectory by fitting to a low-order
expansion around $X_{1}$, consistent with the symmetries of the system,

$f(k)=\gamma\big{(}1+\alpha(k_{x}^{2}+k_{y}^{2})+\beta
k_{x}^{2}k_{y}^{2}\big{)}$ (2)

Here $\alpha$ and $\beta$ fix the Weyl loop trajectory and the scaling factor
$\gamma$ sets an energy scale. The train of crossing points observed in ARPES
is well-captured by $\alpha=-1.23\pm 0.03$ Å2 and $\beta=-31.5\pm 4.1$ Å4
(Fig. 2c). We also find that our ARPES-extracted Weyl loop trajectory agrees
well with the trajectory observed in ab initio calculations (Extended Data
Fig. 3). The energy dispersion of the Weyl loop is set by $g(k)$, well-
described by $g(k)=\delta+\eta\cos(4\theta)$, where $\delta=-75\pm 17$ meV,
$\eta=46\pm 17$ meV, and $\theta$ is the ordinary polar angle of $k$,
$\tan\theta\equiv k_{y}/k_{x}$. Away from the loop, the bands disperse
linearly. To capture this dispersion, we examine a deeper constant-energy
photoemission slice intersecting the Weyl loop (Fig. 2d). We observe that the
experimental dispersion is well-captured by $\gamma=0.15\pm 0.05$ eV. The
extracted dispersion reaches the Fermi level within experimental resolution,
suggesting that the observed Weyl loops are relevant for transport and other
low-energy response. This result is consistent with previous reports that Weyl
loops play a dominant role in the giant anomalous Hall effect and other exotic
transport properties of Co2MnGa Co2MnGa_me ; Co2MnGa_Nakatsuji ;
Co2MnGa_Felser_Nernst ; Co2MnGa_Thomas_Nernst . Our analytical model, with all
parameters extracted from photoemission spectra, allows us to achieve a
complete experimental characterization of both the momentum trajectory and
energy dispersion of the family of Weyl loops (Fig. 2e, Extended Data Fig. 3).

We next investigate the composite structure formed by the three Weyl loops,
focusing on the Weyl loop crossing itself (degeneracy curve, shown in cyan in
Fig. 2e). The loop crossing disperses in energy, so a constant-energy slice
through the electronic structure typically intersects a loop crossing only at
several discrete points (for example, the cyan dots in Fig. 2d). However,
despite this energy dispersion, we find that the typical ‘radius’ of the Weyl
loop, $|k|_{\rm avg}=0.66$ Å-1, is much larger than the typical momentum
separation of the two branches of the Weyl cone at the Fermi level,
$\eta/v_{\rm F}=0.07$ Å-1. Since $|k|_{\rm avg}\gg\eta/v_{\rm F}$, we can
treat the Weyl loop crossing as approximately flat in energy and we can
accurately capture its trajectory by examining a constant-energy slice near
the Fermi level. Therefore, to understand the composite structure of the Weyl
loops, we can focus on the $M_{1}$, $M_{2}$ and $M_{3}$ Weyl loop Fermi
surfaces. We first consider the $M_{1}$ and $M_{2}$ Weyl loops in an extended
zone scheme. We study two adjacent Brillouin zones (Fig. 3a, inset) and zoom
in on the momentum-space region around $X_{1}$. By plotting the $M_{1}$ and
$M_{2}$ Weyl loop Fermi surfaces simultaneously in this region of three-
dimensional momentum space, we observe that these two loops appear to link
each other twice (Fig. 3a). We next consider the $M_{2}$ and $M_{3}$ Weyl
loops and we shift our momentum-space field of view to a region around
$X_{2}$. We plot the $M_{2}$ and $M_{3}$ Weyl loop Fermi surfaces and we again
directly observe from our photoemission spectra that the two loops link each
other twice (Fig. 3b). Repeating the analogous procedure for $X_{3}$, we
observe that the $M_{3}$ and $M_{1}$ Weyl loops also link twice (Fig. 3c). To
estimate the stability of these links, we can measure how far apart one would
need to slide two Weyl loops in order to unlink them (Extended Data Figs. 4,
5). From the loop Fermi surfaces, we find that the typical ‘depth’ of the link
in momentum-space is $d_{\rm avg}=0.58\pm 0.08$ Å-1, of the same order of
magnitude as the radius $|k|_{\rm avg}$ of the Weyl loop. This large depth
suggests that the system lives well within a linked electronic phase. Our
three-dimensional momentum-space analysis of the photoemission spectra
suggests that each of the $M_{1}$, $M_{2}$ and $M_{3}$ Weyl loops links each
other Weyl loop twice, forming a robust linked structure.

To further explore this link, we examine all three Weyl loops simultaneously
using the experimentally-extracted loop trajectory, Eq. 2. In an extended zone
scheme, we plot the $M_{1}$, $M_{2}$ and $M_{3}$ Weyl loops around six nearby
$X$ points, so that two redundant copies of each Weyl loop are included (Fig.
4b). We find that the $M_{1}$ Weyl loop links both the $M_{2}$ and $M_{3}$
Weyl loops; the $M_{2}$ Weyl loop links both the $M_{3}$ and $M_{1}$ loops;
and similarly for the $M_{2}$ loop. This suggests that the three Weyl loops
together form a single composite linked structure. By plotting additional
redundant copies of the loops in higher Brillouin zones, we can form a Weyl
loop network extending outward to infinity. To more deeply explore this linked
structure, we examine energy-momentum photoemission slices tangential to all
three loops near their extrema (Fig. 4a). All slices exhibit a cone
dispersion, consistent with the Weyl loop electronic structure. Moreover, we
find quantitative agreement between the Weyl loop extrema expected from Eq. 2
and the locations of the photoemission cone dispersions, for all three loops.
This agreement again suggests the observation of a composite structure of
three interwoven Weyl loops. To better visualize the complete link structure,
we construct a link diagram for our Weyl loops. In such a link diagram, we
flatten the link from three to two dimensions while preserving the link
information using an over/under notation (illustrated for the example of a
Hopf link, Fig. 4c). Because the Weyl loop link lives in the periodic momentum
space of the crystal, we flatten the link into the surface Brillouin zone.
Moreover, because our analysis shows that all three Weyl loops are symmetry-
related, we choose the (111) surface Brillouin zone (Extended Data Fig. 2),
which treats $X_{1}$, $X_{2}$ and $X_{3}$ equivalently (Fig. 4d). The
resulting link diagram shows three loops straddling the edges of the surface
Brillouin zone (Fig. 4e). We observe that the link wraps around
$\mathbb{T}^{3}$ in all three momentum-space directions. This behavior
suggests that the link is geometrically essential, in the sense that it cannot
be smoothly perturbed to live entirely within a local region of the Brillouin
zone. The link diagram further shows that each loop is linked with each other
loop exactly twice. This gives $2$ for the geometric linking number, defined
as the minimal number of crossing changes between link components needed to
separate the components. The geometric linking number of the composite Weyl
loop structure can then be written as $(2,2,2)$, where the first entry in the
list corresponds to the linking number between $M_{1}$ and $M_{2}$, the second
entry between $M_{2}$ and $M_{3}$, and the third entry between $M_{3}$ and
$M_{1}$. By analogy with topological insulators and Weyl point semimetals,
this Weyl loop link is expected to be stable under arbitrary, small, symmetry-
preserving perturbations of the electronic structure.

Having systematically characterized the link structure in the bulk of Co2MnGa,
we next consider its topological surface states. Unlinked loop nodes host
conventional drumhead surface states, which fill a simply connected region of
momentum space in the surface Brillouin zone. By contrast, linked loops
exhibit an alternating pattern of topologically distinct regions where surface
states are either present or suppressed, and which are pinned together at
generic points in momentum space. This topological structure is captured by
the Seifert surface, defined as a three-dimensional surface which has the link
as its boundary Seifert_1935 . For a condensed matter system, we consider a
Seifert surface defined in $(k_{x},k_{y},k_{z})$ and bounded by the linked
loop nodes, with energy axis collapsed. For the minimal case of a Hopf link,
the Seifert surface exhibits a branched structure which ‘wraps’ around the
link (Fig. 5a, left). A two-dimensional projection of the Seifert surface then
produces alternating filled and empty regions, which meet at characteristic
touching points (Fig. 5a, right). In the case of the Weyl loop link which we
observe in Co2MnGa, the Seifert projection on the (111) hexagonal surface
Brillouin zone then predicts several alternating regions with and without
topological boundary states (blue and white regions, Fig. 5c), exhibiting
touching points along $\bar{\Gamma}-\bar{K}$. Since the Seifert surface
encodes the linking number Seifert_1935 , the topological boundary states are
associated with a Seifert bulk-boundary correspondence. In this
correspondence, the linking number of the Weyl loops in the bulk is encoded in
a Seifert surface, whose projection gives the topological boundary states. A
measurement of the bulk link determines the Seifert boundary states, while a
measurement of the Seifert boundary states allows a reconstruction of the bulk
linking number. To explore these possible surface states, we examine the (111)
surface of our Co2MnGa samples in ab initio calculation and surface-sensitive
vacuum ultraviolet (VUV) ARPES. On an energy-momentum cut through the touching
point we observe in calculation a pair of surface modes pinned together at the
Weyl loops (Figs. 5d). Moreover, our photoemission spectra are consistent with
our ab initio prediction, suggesting the observation of Seifert boundary
states approaching the Weyl loop linking point (Fig. 5e). On iso-energy
contours of the electronic structure, we expect to observe arc-like slices of
the Seifert states, stretching across the filled regions and connecting the
linked Weyl loops. Examining the Fermi surface obtained in calculation, we
observe a sharp arc of surface states connecting the linked Weyl loops,
consistent with the Seifert projection (Fig. 5f, left). At the same time, the
suppressed region exhibits no topological surface states in calculation. Our
Fermi surface obtained by VUV-ARPES matches the ab initio prediction well
(Fig. 5f, right). We observe distinct arcs of states connecting the linked
Weyl loops across the topological region, corresponding to the topological
surface states observed on the energy-momentum cuts (Figs. 5d, e) and
suggestive of Seifert states at the Fermi level in Co2MnGa. Our ab initio
calculations and photoemission spectra suggest the observation of Seifert
boundary states.

Our photoemission spectra, ab initio calculations and theoretical analysis
suggest the observation of a loop node link in a quantum magnet. On the sample
surface, we further observe Seifert boundary states protected by the bulk
link, indicating a Seifert bulk-boundary correspondence. These results
establish a new bridge between physics and knot theory, motivating further
exploration of links and knots in electronic structures. Moreover, the linked
loop state in Co2MnGa, as well as in other materials, may give rise to exotic
response quantized to the linking number, such as a link-quantized topological
magneto-electric effect WeylLink_Yee ; Biao_WilsonLoops ;
ShouChengZhang_HelixNodalLine ; Armitage_QuantizedKerr . Since high-symmetry
magnetic and correlated materials are abundant in nature, these ideas open the
way to understanding the exotic behavior of a wide class of quantum magnets
and superconductors, as well as their photonic analogs.

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##  I Acknowledgments

I.B. thanks Nikita Lvov and Zoltán Szabó for discussions on linking numbers.
The authors thank D. Lu and M. Hashimoto at Beamline 5-2 of the Stanford
Synchrotron Radiation Lightsource (SSRL) at the SLAC National Accelerator
Laboratory, CA, USA for support. I.B. and D.M. thank Takayuki Muro for
experimental support during preliminary ARPES measurements carried out at
BL25SU of SPring-8 in Hyogo, Japan. I.B. thanks Biao Lian for discussions on
the topological magneto-electric effect. I.B., T.A.C., X.P.Y. and D.M. thank
Jessica McChesney and Fanny Rodolakis for experimental support during
preliminary ARPES measurements carried out at BL29 of the Advanced Photon
Source (APS) in Illinois, USA. I.B. acknowledges discussions with Boris
Belopolski on Savitzky-Golay analysis. G.C. acknowledges the support of the
National Research Foundation, Singapore under its NRF Fellowship Award (NRF-
NRFF13-2021-0010) and the Nanyang Assistant Professorship grant from Nanyang
Technological University. T.A.C. acknowledges support by the National Science
Foundation Graduate Research Fellowship Program under Grant No. DGE-1656466.
A.C. acknowledges funding from the Swiss National Science Foundation under
Grant No. 200021-165529. The authors acknowledge synchrotron radiation
beamtime at the ADRESS beamline of the Swiss Light Source of the Paul Scherrer
Institut in Villigen, Switzerland under Proposals 20170898, 20190740 and
20191674. The authors further acknowledge use of Princeton’s Imaging and
Analysis Center (IAC), which is partially supported by the Princeton Center
for Complex Materials (PCCM), a National Science Foundation (NSF) Materials
Research Science and Engineering Center (MRSEC; DMR-2011750). This research
used resources of the Advanced Photon Source, a U.S. Department of Energy
(DOE) Office of Science User Facility operated for the DOE Office of Science
by Argonne National Laboratory under Contract No. DE-AC02-06CH11357. The
authors acknowledge beamtime at BL25SU of SPring-8 under Proposal 2017A1669
and at BL29 of the APS under Proposals 54992 and 60811. K.M. and C.F.
acknowledge financial support from the European Research Council (ERC)
Advanced Grant No. 742068 “TOP-MAT”. C.F. acknowledges the DFG through SFB
1143 (project ID. 247310070) and the Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter ct.qmat (EXC2147, project ID.
39085490). M.Z.H. acknowledges visiting scientist support at Berkeley Lab
(LBNL) during the early phases of this work. Work at Princeton University was
supported by the Gordon and Betty Moore Foundation (Grants No. GBMF4547 and
No. GBMF9461; M. Z. H.). The ARPES and theoretical work were supported by the
United States Department of Energy (US DOE) under the Basic Energy Sciences
programme (Grant No. DOE/BES DE-FG-02-05ER46200; M. Z. H.). Use of the
Stanford Synchrotron Radiation Lightsource (SSRL), SLAC National Accelerator
Laboratory, is supported by the U.S. Department of Energy, Office of Science,
Office of Basic Energy Sciences, under Contract No. DE-AC02-76SF00515. The
authors acknowledge MAX IV Laboratory for time on the BLOCH Beamline under
Proposal 20210268. Research conducted at MAX IV, a Swedish national user
facility, is supported by the Swedish Research council under contract
2018-07152, the Swedish Governmental Agency for Innovation Systems under
contract 2018-04969, and Formas under contract 2019-02496.

##  II Author contributions

I.B., G. Chang, T.A.C. and M.Z.H. initiated the project. I.B., T.A.C., Z.-J.
C. and M.Z.H. acquired and analyzed ARPES spectra with help from X.P.Y., D.M.,
J.-X.Y., M.L., N.S. and S.S.Z. ARPES measurements were supported by N.B.M.S.,
A.C., C.P., B.T., M.L., J.A. and V.N.S. G. Chang performed the first-
principles calculations. I.B. performed the $k\cdot p$ model analysis with
help from G. Chang and S.-M.H. I.B. performed the linking number analysis with
help from C.H. G. Cheng and N.Y. performed the scanning transmission electron
microscopy measurements. K.M., C.S. and C.F. synthesized and characterized the
single crystals. I.B. wrote the manuscript with contributions from all
authors.

##  III Competing Interests

The authors declare no competing interests.

##  IV Methods

Single crystal growth: Co2MnGa single crystals were grown using the Bridgman-
Stockbarger method. A polycrystalline ingot was first prepared using an
induction melt technique, with a stoichiometric mixture of Co, Mn and Ga metal
pieces of 99.99% purity. Then the powdered material was poured into an alumina
crucible and sealed in a tantalum tube. Growth temperatures were controlled
using a thermocouple attached to the bottom of the crucible. During the
heating cycle, the material was melted at temperatures above 1200∘C and then
slowly cooled below 900∘C.

Angle-resolved photoemission spectroscopy: Soft X-ray ARPES measurements were
carried out at the ADRESS beamline of the Swiss Light Source in Villigen,
Switzerland under vacuum better than $5\times 10^{-11}$ Torr and a temperature
of 16 K ADRESS1_Strocov ; ADRESS2_Strocov ; ADRESS3_Strocov . Rod-shaped
single crystals of Co2MnGa oriented along the conventional unit cell $\hat{z}$
direction were cleaved in situ at base temperature. The constant-energy cuts
were symmetrized about $M_{x}$ and $M_{xy}$ (Fig. 1d), $M_{x}$ and $M_{xz}$
(Fig. 1e) and $M_{y}$ and $M_{yz}$ (Fig. 1f). The high-symmetry energy-
momentum cuts were similarly symmetrized about $M_{x}$, $M_{y}$ or $M_{z}$, as
appropriate and consistent with the nominal symmetries of the crystal (Fig.
4a). A background was removed from the photoemission spectra by a fixed
intensity cutoff (raw, unsymmetrized data in Extended Data Figs. 6, 7, 8). For
the Fermi surfaces acquired at $h\nu=544$ eV, the nominal energy resolution
was $\delta E=75$ meV; for the photon-energy dependences, the nominal energy
resolution varied from $\delta E=75$ meV at $h\nu=500$ eV to $\delta E=125$
meV at $h\nu=800$ eV. The angular resolution was better than 0.2∘ in all
cases. The Fermi surfaces were binned in an energy window of $\pm 38$ meV
(Fig. 1d) and $\pm 25$ meV (Fig. 1e,f) around $E_{\rm F}$. Vacuum ultraviolet
ARPES measurements were carried out at Beamline 5-2 of the Stanford
Synchrotron Radiation Lightsource in Menlo Park, CA, USA at $\delta E=15$ meV
and temperature 20 K.

Ab initio calculations: The electronic structure of Co2MnGa in the
ferromagnetic phase was calculated within the density functional theory (DFT)
framework using the projector augmented wave method as implemented in the VASP
package DFT2 ; DFT3 . The generalized gradient approximation (GGA) DFT4  and a
$\Gamma$-centered $k$-point $12\times 12\times 12$ mesh were used. Ga $s,p$
orbitals and Mn, Co $d$ orbitals were used to generate a real space tight-
binding model, from which Wannier functions were determined. The Fermi level
in DFT was shifted to match the ARPES.

Scanning transmission electron microscopy: Thin lamellae for microstructure
characterization were prepared from bulk single crystals by focused ion beam
cutting using a ThermoFisher Helios NanoLab G3 UC DualBeam system (FIB/SEM).
Atomic resolution high-angle annular dark-field (HAADF) scanning transmission
electron microscopy (STEM) imaging and atomic-level energy-dispersive X-ray
spectroscopy (EDS) mapping were performed on a double Cs-corrected
ThermoFisher Titan Cubed Themis 300 scanning/transmission electron microscope
(S/TEM) equipped with an X-FEG source operated at 300 kV with a Super-X EDS
system.

Figure 1:  Signatures of linked node loops in Co2MnGa. a, Weyl loops in the
electronic structure of Co2MnGa, predicted by density functional theory (DFT).
Three distinct Weyl loops are confined to the three mirror planes $M_{1}$,
$M_{2}$ and $M_{3}$, in such a way that the loops link one another (additional
copies of the loops in higher Brillouin zones not shown). b, Element-resolved
crystal structure of Co2MnGa along the [001] direction, acquired by atomic-
level energy-dispersive X-ray spectroscopy (EDS). Atomic columns consist
either entirely of cobalt (green) or alternating manganese (red) and gallium
(blue). c, Bulk Brillouin zone (black truncated octahedron) of Co2MnGa with
three mirror planes indicated, $M_{1}$ (magenta, constant $k_{z}$), $M_{2}$
(red, constant $k_{y}$) and $M_{3}$ (gold, constant $k_{x}$). Each mirror
plane contains square faces of the Brillouin zone. The high-symmetry momentum-
space points at the center of each square are marked $X_{1}$, $X_{2}$,
$X_{3}$. d, Fermi surface acquired by angle-resolved photoemission
spectroscopy (ARPES) at incident photon energy $544$ eV, corresponding to
$M_{1}$. e, Out-of-plane Fermi surface acquired on the same Co2MnGa sample by
an ARPES photon energy dependence from $500$ eV to $800$ eV in steps of $2$
eV, corresponding to $M_{2}$. f, Analogous out-of-plane Fermi surface
corresponding to $M_{3}$, again on the same sample.

Figure 2:  Weyl loop trajectory in Co2MnGa. a, Energy-momentum photoemission
slices through the loop Fermi surface (slice locations marked by the dotted
lines in (d) and Fig. 1d). b, Energy-momentum slices through the Weyl loop
from DFT, showing a Weyl loop cone (slice locations marked in Extended Data
Fig. 3c). c, Cone locations (magenta squares) systematically extracted from
cone dispersions observed in photoemission spectra on $M_{1}$. Experimental
loop trajectory extracted by fitting to the cone locations (cyan, see main
text). The binding energy axis is collapsed. d, Constant-energy photoemission
slice with analytical model of the Weyl loop (black lines). This slice
intersects the Weyl loop at a discrete set of points (cyan dots). e,
Dispersion of an effective $k\cdot p$ Hamiltonian for the Weyl loop, capturing
the experimental loop trajectory.

Figure 3:  Linked Weyl loops in Co2MnGa. a, $M_{1}$ and $M_{2}$ loop Fermi
surfaces from adjacent bulk Brillouin zones, plotted in an extended zone
scheme, exhibiting a link structure. Inset: $M_{1}$ and $M_{2}$ plotted across
multiple Brillouin zones. b, $M_{2}$ and $M_{3}$ loop Fermi surfaces from
adjacent bulk Brillouin zones. c, $M_{3}$ and $M_{1}$ loop Fermi surfaces from
adjacent bulk Brillouin zones.

Figure 4:  Linking number $(2,2,2)$ in topological quantum matter. a, Energy-
momentum photoemission slices tangential to the $M_{1}$, $M_{2}$ and $M_{3}$
Weyl loops at their extrema. b, Weyl loops from adjacent bulk Brillouin zones,
based on the analytical model extracted in Eq. 2 and Fig. 2, exhibiting links.
Weyl cone positions (blue dots) extracted from the slices in Fig. 4a (dotted
blue lines), consistent with the analytical model (short blue line segments
indicate the error). c, Link diagrams help visualize a three-dimensional link
structure by flattening it to two dimensions while retaining the link
information, illustrated for the example of a Hopf link. d, In a crystal, it
is natural to draw link diagrams in the surface Brillouin zone, such as the
(001) surface Brillouin zone (hexagon, Extended Data Fig. 2). e, Link diagram
for the Co2MnGa Weyl loop link. There are three distinct Weyl loops and each
Weyl loop links each other Weyl loop exactly twice, giving linking number
$(2,2,2)$. The arrows indicate out-of-plane wrapping: as one follows the loop
in the direction of the arrow, the loop wraps out of the page, exiting the
Brillouin zone from the front and re-entering from the back, at the same time
reconnecting at the opposite edge of the hexagon.

Figure 5:  Seifert bulk-boundary correspondence. a, A Seifert surface is
defined as a three-dimensional surface bounded by a link, shown for the
example of a Hopf link. Its two-dimensional projection produces alternating
filled and empty regions pinned together at characteristic touching points. b,
In a condensed matter system, the Seifert surface is taken as a surface
bounded by the linked loop nodes in three-dimensional momentum space
$(k_{x},k_{y},k_{z})$, shown for the case of the link observed in Co2MnGa. c,
The projection of the Seifert surface into the surface Brillouin zone is
associated with topological boundary modes (blue regions) which touch at
points in momentum space. Energy axis collapsed for clarity. d, Ab initio
calculation of the surface states through the touching point, exhibiting pairs
of boundary modes pinned together at the Weyl loops. e, Surface-sensitive
vacuum ultraviolet (VUV) ARPES energy-momentum cut through the touching point,
exhibiting signatures of the pinned Seifert boundary modes, consistent with ab
initio calculations. Photon energy $h\nu=63$ eV. f, Fermi surface in ab initio
calculation (left) and VUV-ARPES (right) exhibiting Seifert boundary modes
that stretch across the topological regions, connecting different Weyl loops,
consistent with the Seifert projection.

Extended Data Fig. 1:  Topological invariants in physics. a, An example of an
order parameter winding in real space: a magnetic vortex. In this case, the
order parameter is the local magnetization ${\bf m}({\bf x})$, confined to a
magnetic easy plane in real space ($x,y$). It may happen that ${\bf m}({\bf
x})$ winds around a point in real space, forming a magnetic vortex
characterized by a winding number topological invariant, in this example given
by $w=1$. b, An example of a quantum wavefunction winding in momentum space:
the one-dimensional topological insulator (Su-Schrieffer-Heeger model). This
phase is described by Bloch Hamiltonian $h(k)={\bf d}(k)\cdot\sigma$, where
$k$ is the one-dimensional crystal momentum, $\sigma$ refers to the Pauli
matrices and d(k) is a two-component object confined to the ($d_{x},d_{y}$)
plane. The normalized quantity ${\bf\hat{d}}(k)\equiv{\bf d}(k)/|{\bf d}(k)|$
(orange arrow) moves around the unit circle (dotted blue) as $k$ varies. The
topological invariant is related to how many times ${\bf\hat{d}}(k)$ winds
around the origin as $k$ scans through the one-dimensional Brillouin zone. c,
Node loops linking in momentum space: a three-dimensional electronic structure
may exhibit multiple node loops (cyan and purple), characterized by ${\bf
k}_{n}(\theta)$, where $n$ indexes the loops and $\theta$ parametrizes the
loop trajectory in momentum space. The loops may link one another, encoding a
linking number topological invariant. This example shows a Hopf link.

Extended Data Fig. 2:  Crystal structure and Brillouin zone of Co2MnGa. a,
Conventional unit cell with representative crystallographic mirror plane $M$
(orange). b, The primitive unit cell (grey) includes one formula unit. c,
Brillouin zone, with reciprocal lattice basis vectors (grey). In the
reciprocal lattice basis, the $M_{1}$ plane corresponds to $(001)$, $M_{2}$
corresponds to $(010)$ and $M_{3}$ corresponds to $(100)$. d, Slice through
$\Gamma$ in an extended zone scheme.

Extended Data Fig. 3:  Energy dispersion of the Weyl loop. a, Crossing point
energies $E_{\textrm{B}}$ and b, crossing point momenta $(k_{x},k_{y})$
systematically extracted from cone dispersions observed in the ARPES spectra
(magenta squares), same dataset as Fig. 2c ($h\nu=544$ eV), with fit of the
Weyl loop momentum trajectory and energy dispersion (cyan, see main text). The
crossing point energies are parametrized by a polar angle $\theta$ defined by
$\tan\theta\equiv k_{y}/k_{x}$. c, Weyl loop trajectory from DFT, with dotted
lines indicating the DFT energy-momentum slices shown in Fig. 2b. The binding
energy axes in (b) and (c) are collapsed.

Extended Data Fig. 4:  Link ‘depth’ of the Weyl loops. a-c, Distance between
the extrema of the Weyl loops and the bulk Brillouin zone $W$ points for the
$M_{1}$, $M_{2}$ and $M_{3}$ Weyl loops. We estimate $s_{1}=0.32\pm 0.1$ Å-1,
$s_{2}=0.27\pm 0.1$ Å-1 and $s_{3}=0.29\pm 0.1$ Å-1. d, The link depth
captures how far in momentum space one would need to slide the Weyl loops in
order to unlink them, providing a measure of the stability of the link. Based
on the loop Fermi surfaces (a-c), we estimate $d_{12}=0.58\pm 0.14$ Å-1,
$d_{23}=0.55\pm 0.14$ Å-1 and $d_{31}=0.60\pm 0.14$ Å-1. The average gives a
typical link depth extracted from ARPES, $d_{\rm avg}=0.58\pm 0.08$ Å-1. e,
Energy-momentum slice along the high-symmetry path $X_{1}-X_{2}$ from DFT,
passing through two linked Weyl loops. We obtain $d_{\rm DFT}=0.68$ Å-1.

Extended Data Fig. 5:  Supplementary measurement of the link depth. a,
$M_{1}$, $M_{2}$ and $M_{3}$ Weyl loops, with trajectories obtained from the
analytical model (see main text), showing that $M_{1}$ links $M_{2}$ twice and
$M_{3}$ twice. Energy-momentum photoemission slices along the high-symmetry
paths b, $X_{1}-X_{2}$ and c, $X_{3}-X_{1}$ obtained at photon energy
$h\nu=642$ eV. We observe $d_{12}=0.56\pm 0.1$ Å-1 and $d_{31}=0.61\pm 0.1$
Å-1, consistent with Extended Data Fig. 4. d, Fermi surface acquired at
$h\nu=642$ eV, exhibiting an in-plane Weyl loop contour, $M_{1}$. We further
observe spectral weight emanating along $k_{x}$ and $k_{y}$ from the center of
$M_{1}$, corresponding to the linearly dispersive branches in (b, c), again
suggesting that $M_{1}$ is linked by $M_{2}$ and $M_{3}$.

Extended Data Fig. 6:  Unsymmetrized Fermi surfaces. a-c, Left: photoemission
spectra displayed in Fig. 1d-f, without symmetrization. Right: the same
spectra, with the experimentally-determined Weyl loop trajectory overlaid
across multiple Brillouin zones. The irrelevant $\Gamma$ pocket is
consistently observed in all unsymmetrized spectra. Signatures of Weyl loops
are observed around all $X$ points.

Extended Data Fig. 7:  Energy-momentum cuts through the Weyl loop.
Photoemission spectra used to extract Fig. 2c.

Extended Data Fig. 8:  Unsymmetrized energy-momentum cuts. Photoemission
spectra displayed in Fig. 4a, without symmetrization.

Extended Data Fig. 9:  Linked Weyl loop Fermi surface. Constant-energy slice
of the pockets (navy) making up two linked Weyl loops obtained by ab initio
calculation, at binding energy $E_{\rm B}=-10$ meV, below the experimental
Fermi level. The Fermi surface pockets touch at a set of discrete points,
where the Weyl loop disperses through this particular $E_{\rm B}$. For
reference, the full Weyl loop trajectories are indicated, collapsed in energy
(orange around $X_{3}$, magenta around $X_{1}$). We observe that the Weyl loop
Fermi surface pockets form a linked structure.

Extended Data Fig. 10:  Measured Fermi surfaces in an extended zone scheme.
The Brillouin zone corresponds to $\Gamma_{(066)}$ in the primitive reciprocal
basis.