(for $i=1,\ldots,m$) can be used, whilst maintaining the guarantee that the
support for the true return distributions lie within this interval. We thus
additionally report results for versions of DCFP and CDP that use environment-
specific atom locations, to investigate what practical improvements can be
gained with such additional information. Specifically, for the high random and
low random environments, we use the tighter bounds on support given by known
$r_{\text{min}}$ and $r_{\text{max}}$, as described above. In the chain
environment, we consider using the advanced knowledge that only the transition
into the terminal state is rewarding, yielding a sub-interval for possible
returns of $[0,1]$. In the two-state case, the range of possible returns is
the full interval $[0,(1-\gamma)^{-1}]$, and so we do not investigate an
environment-specific set of atoms in this case. However, it is clear from the
Monte Carlo approximations to the return distributions in this environment in
Figure 4 that these distributions do have the vast majority of their
probability mass in a much smaller interval than the worst case interval
$[0,(1-\gamma)^{-1}]$.

We ran all DP methods with 30,000 iterations. For all categorical DP methods,
we verify that the Wasserstein distance between the last iterate to the Monte
Carlo ground truth is almost identical to the Wasserstein distance between the
categorical fixed-point to the ground truth.

###  F.3 Efficient implementation of CDP and DCFP

A straightforward implementation of CDP and DCFP is to vectorise $F$, treating
it as a vector indexed by state-index pairs $(x,i)$, and correspondingly treat
$T_{\hat{P}}$ as a matrix whose rows and columns are indexed similarly.
Iterations of CDP can then be performed by simple matrix-vector
multiplications with this representation of $T_{\hat{P}}$, and the solution of
the linear system appearing in Equation (11) that forms the core of DCFP can
be implemented with a call to a standard linear system solver, such as
numpy.linalg.solve (Harris et al., 2020).

However, the operator $T_{\hat{P}}$ typically has some specific structure that
can be exploited in implementations. In particular, we highlight here the
sparse structure of $T_{\hat{P}}$, allowing for potential speed-ups in
implementation by making use of sparse linear solvers, such as
scipy.sparse.linalg.spsolve (Virtanen et al., 2020). Recall that, viewing
$T_{\hat{P}}$ as a matrix (c.f. Equation (9)), the element corresponding to
the $(x,i)$ row and $(y,j)$ column is given by

$\displaystyle\hat{P}(y|x)(H^{x}_{i,j}-H^{x}_{i,j+1})\,.$ (34)

In an MRP with sparse transition matrix $P$, the empirical estimate $\hat{P}$
inherits this sparsity, inducing sparsity in $T_{\hat{P}}$. In addition, there
is also sparsity induced by the atom index components of the row and column
indices, as we now describe. Recall from Equation (8) that
$H^{x}_{i,j}=\sum_{l\leq i}h_{l}(r(x)+\gamma z_{j})$; see Figure 5 for an
illustration the function $z\mapsto\sum_{l\leq i}h_{l}(z)$.

Figure 5: The function $z\mapsto\sum_{l\leq i}h_{l}(z)$ (grey), and a possible
configuration for $r(x)+\gamma z_{j}$, $r(x)+\gamma z_{j+1}$ in the event of a
non-zero $H^{x}_{i,j}-H^{x}_{i,j+1}$ term.

Now, suppose that for some $x\in\mathcal{X}$ and $1\leq i,j\leq m-1$,
$H^{x}_{i,j}-H^{x}_{i,j+1}$ is non-zero. This says that the function
$z\mapsto\sum_{l\leq i}h_{l}(z)$ takes on different values at $r(x)+\gamma
z_{j}$ and $r(x)+\gamma z_{j+1}$, and since the distance between these
arguments is $\gamma$ times the grid width $(1-\gamma)^{-1}m^{-1}$, it follows
that at least one of these two arguments must lie in the interval
$[z_{i},z_{i+1}]$; see Figure 5.

From this observation, we deduce two forms of sparsity for the elements given
in Equation (34). First, since one of $r(x)+\gamma z_{j}$ and $r(x)+\gamma
z_{j+1}$ must lie in $[z_{i},z_{i+1}]$, we deduce that neither of these points
can lie in any interval $[z_{i^{\prime}},z_{i^{\prime}+1}]$ with
$i^{\prime}<i-1$ or $i^{\prime}>i+1$, and hence
$H^{x}_{i^{\prime},j}-H^{x}_{i^{\prime},j}=0$ for such $i^{\prime}$. From this
reasoning, it follows that ranging over $i$ in Equation (34), there are at
most 2 non-zero elements. For large $m$, this means that $T_{\hat{P}}$ is very
sparse.

Similarly, we can deduce row sparsity by noting that at most $\lceil
2/\gamma\rceil$ indices $j$ can have the property that $r(x)+\gamma z_{j}$ or
$r(x)+\gamma z_{j+1}$ can lie in $[z_{i},z_{i+1}]$, and it is only for these
indices $j$ that we can have $H^{x}_{i,j}-H^{x}_{i,j+1}$ non-zero. So for
$\gamma\approx 1$, this also implies sparsity of the elements in Equation (34)
as we range over $j$.

###  F.4 Results

For each setting, we repeat the experiment 30 times with different sampled
transitions. We display trade-off plots (supremum-Wasserstein-1 error and
wallclock time) for each of the four environments described above, in Figures
6, 7, 8, and 9, for the cases of $m\in\\{30,100,300,1000\\}$ atoms and using
$N=10^{6}$ sample transitions from each state to estimate transition matrix.
These curves are obtained by averaging across the 30 repetitions. Some central
themes emerge from the results.

In the case when the categorical methods use the environment-independent,
standard atom locations $z_{i}=\tfrac{i-1}{m-1}(1-\gamma)^{-1}$ for
$i=1,\ldots,m$, we find that for a given atom count $m$, QDP often achieves
the lowest asymptotic Wasserstein-1 error. A notable exception is the two-
state environment; Rowland et al. (2023) observed that such environments, in
which there is a short, high-probability path from a state to itself, can
cause high approximation error for QDP, which we believe is the cause of the
inaccuracy observed here, particularly at high discounts. However, the
categorical approaches often deliver better performance as judged by wallclock
time. This is owing to the efficient implementations of the linear operator,
and solution methods for the linear system, that are associated with these
algorithms. By contrast, the QDP operator is non-linear, and requires a call
to a sorting method. We tend to observe the greatest benefits of the sparse
implementation for large atom counts, as expected, and also observe the
greatest benefits of DCFP over the iterative CDP algorithm in settings with
high discount factors. This is also to be expected, since the discount factor
controls the rate of convergence of the DP algorithms. Note additionally that
the use of environment-specific atom locations generally improves the results
obtained with categorical approaches; the improvements in the chain
environment are particularly strong, since the narrow return range allows for
a denser packing of atom locations, by a factor of $(1-\gamma)^{-1}$, for a
given atom count $m$.

We also compare how the supremum-Wasserstein distance changes while the number
of samples used to estimate $\hat{P}$ increases from $10^{2}$ to $10^{6}$.
Since the categorical methods all converge to the same fixed point, we only
show the result of DCFP and QDP, using $m\in\\{30,100,300,1000\\}$ atoms.
Figure 10(a) shows that the distance decreases as the number of samples
increase. One exception is the QDP with 30 atoms applied to high random at
high $\gamma$, where $N=100$ produced smallest distance on average. Figure
10(b) shows the results when the environment-specific return range is given.
This substantially reduced the supremum-Wasserstein distance, especially when
using a small number of atoms.

(a) Global return range

(b) Environment-specific return range

Figure 6: Distance vs. run time results for the chain environment, for
$\hat{P}$ estimated from $N=10^{6}$ sample transitions.

(a) Global return range

(b) Environment-specific return range

Figure 7: Distance vs. run time results for the low random environment, for
$\hat{P}$ estimated from $N=10^{6}$ sample transitions.

(a) Global return range

(b) Environment-specific return range

Figure 8: Distance vs. run time results for the high random environment, for
$\hat{P}$ estimated from $N=10^{6}$ sample transitions. Figure 9: Distance vs.
run time results for the two-state environment, for $\hat{P}$ estimated from
$N=10^{6}$ sample transitions. The return range of this environment coincides
with the global return range.

(a) Global return range

(b) Environment-specific return range

Figure 10: Supremum-Wasserstein distance on convergence. Error envelope
indicates 95% confidence interval by bootstrapping. The return range of the
two-state environment coincides with the global return range.


McLure et al. (2013) | $0.56$ | $0.24$ | $1.3\leq z\leq 1.5$ | Spec | ¡41 | Formation time (old) |  10.8 $-$ 11.7
McLure et al. (2013) | $0.56$ | $0.24$ | $1.3\leq z\leq 1.5$ | Spec | ¡41 | Formation time (young) |  10.8 $-$ 11.7
McLure et al. (2013) | $0.56$ | $0.23$ | $1.3\leq z\leq 1.5$ | Spec | ¡37 |  $n$/Formation time (old) |  10.8 $-$ 11.7
McLure et al. (2013) | $0.56$ | $0.26$ | $1.3\leq z\leq 1.5$ | Spec | ¡37 |  $n$/Formation time (young) |  10.8 $-$ 11.7
Miller et al. (2023)232323 MSRs using $r_{mass}$ are available (see their Tab. 2).  | $0.62\pm 0.05$ | $0.40\pm 0.01$ | $1.0\leq z<1.2$ | Phot | 279 | UVJ |  10.3 $-$ 11.4
Miller et al. (2023) | $0.57\pm 0.05$ | $0.32\pm 0.01$ | $1.2\leq z<1.4$ | Phot | 252 | UVJ |  10.3 $-$ 11.4
Miller et al. (2023) | $0.44\pm 0.07$ | $0.30\pm 0.02$ | $1.4\leq z<1.6$ | Phot | 184 | UVJ |  10.3 $-$ 11.4
Miller et al. (2023) | $0.51\pm 0.06$ | $0.28\pm 0.02$ | $1.6\leq z<1.8$ | Phot | 343 | UVJ |  10.3 $-$ 11.4
Miller et al. (2023) | $0.55\pm 0.10$ | $0.26\pm 0.03$ | $1.8\leq z\leq 2.0$ | Phot | 205 | UVJ |  10.3 $-$ 11.4
Mosleh et al. (2020)242424 There are 1363 quiescent galaxies in the sample. The slope and intercept were retrieved for the high-mass end considering 1+($M_{*}/M_{p})\approx(M_{*}/M_{p}$), leading to $\alpha=\beta$ and log(A)=log($r_{p}$)-$\beta$log($M_{p}$)+($\beta-\alpha/\delta$)log(1/2) (see their Eq. 3). MSRs for $R_{20}$ and $R_{90}$ are also available (see their Tab. 1).  | $0.680\pm 0.053$ | $0.45\pm 0.01$ | $0.3\leq z<0.7$ | Phot/Spec |  $\sim$303 | UVJ |  10.2 $-$ 11.4
Mosleh et al. (2020) | $0.770\pm 0.016$ | $0.36\pm 0.01$ | $0.7\leq z<1.0$ | Phot/Spec |  $\sim$303 | UVJ |  10.6 $-$ 11.4
Mosleh et al. (2020) | $0.81\pm 0.14$ | $0.320\pm 0.004$ | $1.0\leq z<1.3$ | Phot/Spec |  $\sim$605 | UVJ |  10.5 $-$ 11.4
Mosleh et al. (2020) | $1.08\pm 2.24$ | $0.220\pm 0.002$ | $1.3\leq z\leq 2.0$ | Phot/Spec |  $\sim$152 | UVJ |  10.9 $-$ 11.5
Mowla et al. (2019b) | $0.48\pm 0.03$ | $0.60\pm 0.02$ | $0.0\leq z<0.5$ | Phot | ¡788 | UVJ |  9 $-$ 12
Mowla et al. (2019b) | $0.58\pm 0.04$ | $0.47\pm 0.02$ | $0.5\leq z<1.0$ | Phot | ¡788 | UVJ |  9 $-$ 12
Mowla et al. (2019b) | $0.73\pm 0.02$ | $0.33\pm 0.01$ | $1.0\leq z<1.5$ | Phot | ¡788 | UVJ |  9 $-$ 12
Mowla et al. (2019b) | $0.63\pm 0.05$ | $0.21\pm 0.02$ | $1.5\leq z<2.0$ | Phot | ¡203 | UVJ |  9 $-$ 12
Mowla et al. (2019b) | $0.48\pm 0.12$ | $0.06\pm 0.06$ | $2.0\leq z<2.5$ | Phot | ¡203 | UVJ |  9 $-$ 12
Mowla et al. (2019b) | $0.59\pm 0.23$ | $0.15\pm 0.11$ | $2.5\leq z\leq 3.0$ | Phot | ¡203 | UVJ |  9 $-$ 12
Nadolny et al. (2021) | $0.38\pm 0.03$ | $0.59\pm 0.05$ | $0.0\leq z\leq 2.0$ | Phot | 122 | SED |  6.0 $-$ 11.1
Nadolny et al. (2021) | $0.37\pm 0.07$ | $0.51\pm 0.01$ | $0.0\leq z<0.5$ | Phot | 37 | SED |  6.0 $-$ 10.8
Nadolny et al. (2021) | $0.38\pm 0.06$ | $0.50\pm 0.08$ | $0.5\leq z<1.0$ | Phot | 68 | SED |  8.5 $-$ 11.1
Nadolny et al. (2021) | $0.56\pm 0.32$ | $0.59\pm 0.36$ | $1.0\leq z\leq 2.0$ | Phot | 17 | SED |  10 $-$ 10.9
Nedkova et al. (2021) | $0.68\pm 0.04$ | $0.67\pm 0.01$ | $0.2\leq z<0.5$ | Phot/Spec | 253 | UVJ |  10.3 $-$ 11.4
Nedkova et al. (2021) | $0.64\pm 0.03$ | $0.50\pm 0.01$ | $0.5\leq z<1.0$ | Phot/Spec | 539 | UVJ |  10.3 $-$ 11.7
Nedkova et al. (2021) | $0.63\pm 0.04$ | $0.32\pm 0.01$ | $1.0\leq z<1.5$ | Phot/Spec | 430 | UVJ |  10.3 $-$ 11.6
Nedkova et al. (2021) | $0.61\pm 0.05$ | $0.22\pm 0.01$ | $1.5\leq z\leq 2.0$ | Phot/Spec | 469 | UVJ |  10.3 $-$ 11.7
Newman et al. (2012)252525 The radius is estimated such as $R_{h}$=a(1+(b/a))/2. Quiescent galaxies are characterized by $\textrm{sSFR}<0.02$ Gyr-1 and no MIPS 24 $\mu$m detection.  | $0.59\pm 0.07$ | $0.46\pm 0.02$ | $0.4\leq z<1.0$ | Phot |  $\sim$193 |  sSFR/no 24 $\mu$m |  10.7 $-$ 11.9
Newman et al. (2012) | $0.62\pm 0.09$ | $0.30\pm 0.02$ | $1.0\leq z<1.5$ | Phot |  $\sim$139 |  sSFR/no 24 $\mu$m |  10.7 $-$ 11.9
Newman et al. (2012) | $0.63\pm 0.11$ | $0.21\pm 0.02$ | $1.5\leq z<2.0$ | Phot |  $\sim$108 |  sSFR/no 24 $\mu$m |  10.7 $-$ 11.7
Newman et al. (2012) | $0.69\pm 0.17$ | $0.04\pm 0.04$ | $2.0\leq z\leq 2.5$ | Phot |  $\sim$43 |  sSFR/no 24 $\mu$m |  10.7 $-$ 11.5
Newman et al. (2014)262626 The MSR is parametrized by the usual linear fit but with an additional dependence on redshift through an additional term ($-0.26(z-1.8)$).  | $0.61\pm 0.07$ | $0.07\pm 0.06$ | $1.7\leq z\leq 1.9$ | Phot |  $\sim$200 | UVJ |  10.7 $-$ 11.9
Saracco et al. (2009)272727 The MSR parameters are estimated based on their Table 2.  | $1.19\pm 0.47$ | $0.47\pm 0.09$ | $1.015\leq z\leq 1.921$ | Spec | 32 | Spectra |  10 $-$ 12
Saracco et al. (2011)282828 The MSR parameters are estimated based on their Table 1.  | $1.10\pm 0.72$ | $0.70\pm 0.28$ | $0.964\leq z\leq 1.921$ | Spec | 62 |  Spectra/Morph/$n$ |  9.77 $-$ 11.8
Saracco et al. (2014)292929 MSR for the cluster RDCS J0848+4453. The MSR parameters are estimated based on their Tables 3 and 4.  | $1.92\pm 0.99$ | $0.91\pm 0.34$ | $z=1.27$ | Phot/Spec | 16 | Morph |  9.7 $-$ 11.3
Saracco et al. (2017)303030 MSR obtained using a least-square fit with cluster and field galaxies, which is consistent with an orthogonal fit within 1$\sigma$. The MSR is better fitted with a broken power-law over the entire mass range with $R_{e}=26\times M_{*}^{-0.13\pm 0.2}$ and $R_{e}=2.77\times 10^{-7}M_{*}^{-0.64\pm 0.09}$ below and above $2.5\times 10^{10}$ $M_{\odot}$. | $0.50\pm 0.06$ | $0.50\pm 0.07$ | $1.0\leq z\leq 1.45$ | Phot/Spec | 489 | Morph |  10.5 $-$ 21
Shen et al. (2003) | $0.56$ | $0.62$ | $0.05\leq z\leq 0.15$ | Spec |  $\sim$35923 | $n$ |  10 $-$ 12
Suess et al. (2019)313131 The MSR is obtained by combining log$(r_{\textrm{mass}}/r_{\textrm{light}})\propto\textrm{log(}M_{*})$ and log$(r_{\textrm{mass}}/r_{\textrm{light}})\propto\textrm{log(}r_{light})$.  | $0.4077\pm 0.3890$ | $0.33\pm 0.25$ | $1.0\leq z<1.5$ | Phot/Spec |  $\sim$250 | UVJ |  10.5 $-$ 12
Suess et al. (2019) | $0.757\pm 0.37$ | $0.71\pm 0.49$ | $1.5\leq z<2.0$ | Phot/Spec |  $\sim$200 | UVJ |  10.5 $-$ 12
Suess et al. (2019) | $0.775\pm 1.02$ | $0.09\pm 0.11$ | $2.0\leq z\leq 2.5$ | Phot/Spec |  $\sim$100 | UVJ |  10.5 $-$ 12
Sweet et al. (2017)323232 MSR for the cluster SPT-CLJ0546-5345. There is no ETG selection but the median Sersic index is in agreement with ETGs ($n=3.8\pm 0.5$).  | $0.74\pm 0.06$ | $0.49\pm 0.04$ | $z=1.067$ | Phot/Spec | 49 | $n$ |  9.9 $-$ 11.7
Sweet et al. (2017) | $0.45\pm 0.04$ | $0.53\pm 0.05$ | $z=1.067$ | Phot/Spec | 48 | $n$ |  10 $-$ 11.9
Sweet et al. (2017) | $0.44\pm 0.08$ | $0.46\pm 0.05$ | $z=1.067$ | Phot/Spec | 43 | $n$ |  9.9 $-$ 11.9
Sweet et al. (2017) | $0.84\pm 0.06$ | $0.48\pm 0.03$ | $z=1.067$ | Phot/Spec |  $\sim$48 | $n$ |  9.9 $-$ 11.9
Sweet et al. (2017) | $1.77\pm 0.21$ | $0.15\pm 0.02$ | $z=1.067$ | Phot/Spec |  $\sim$26 | $n$ |  9.9 $-$ 11.9
Toft et al. (2012) | $0.59$ | $0.07$ | $1.8\leq z\leq 2.2$ | Spec | 4 | Post-starburst/SFR |  11 $-$ 11.7
van der Wel et al. (2014)333333 The sample (LTGs+ETGs) contains 9130 ($0\leq z\leq 1$), 16639 ($1<z\leq 2$), and 5189 ($2<z\leq 3$) galaxies.  | $0.75\pm 0.06$ | $0.68\pm 0.02$ | $0.0\leq z<0.5$ | Phot/Spec |  $\sim$500 | UVJ |  9 $-$ 11.3
van der Wel et al. (2014) | $0.71\pm 0.03$ | $0.49\pm 0.01$ | $0.5\leq z<1.0$ | Phot/Spec |  $\sim$2540 | UVJ |  9 $-$ 12
van der Wel et al. (2014) | $0.76\pm 0.04$ | $0.30\pm 0.01$ | $1.0\leq z<1.5$ | Phot/Spec |  $\sim$1400 | UVJ |  9 $-$ 11.6
van der Wel et al. (2014) | $0.76\pm 0.04$ | $0.17\pm 0.02$ | $1.5\leq z<2.0$ | Phot/Spec |  $\sim$1400 | UVJ |  9.3 $-$ 11.8
van der Wel et al. (2014) | $0.76\pm 0.04$ | $0.03\pm 0.01$ | $2.0\leq z<2.5$ | Phot/Spec |  $\sim$300 | UVJ |  9.7 $-$ 11.8
van der Wel et al. (2014) | $0.79\pm 0.07$ | $0.03\pm 0.02$ | $2.5\leq z\leq 3.0$ | Phot/Spec |  $\sim$650 | UVJ |  10 $-$ 11.3
Watkins et al. (2022) | $0.51$ | $0.59$ | Nearby galaxies |  | 126 | Morph |  10.7 $-$ 11.5
Williams et al. (2010)343434 Quiescent galaxies have $\textrm{sSFR}<0.3/t_{H}$ where $t_{H}$ is the age of the Universe at redshift $z$. The number of galaxies is not indicated.  | $0.54\pm 0.06$ | $0.46\pm 0.02$ | $0.5\leq z<1.0$ | Phot |  | sSFR |  10.6 $-$ 11.6
Williams et al. (2010) | $0.56\pm 0.06$ | $0.35\pm 0.01$ | $1.0\leq z<1.5$ | Phot |  | sSFR |  10.6 $-$ 11.6
Williams et al. (2010) | $0.50\pm 0.07$ | $0.25\pm 0.01$ | $1.5\leq z\leq 2.0$ | Phot |  | sSFR |  10.6 $-$ 11.6
Yang et al. (2021)353535 The slope is fixed from van der Wel et al. (2014). We choose the MSR derived using the Bradac lens model, which is similar within uncertainties to MSR using other lens models.  | $0.76$ | $0.37\pm 0.11$ | $1.0\leq z\leq 1.5$ | Phot/Spec |  $\sim$17 | UVJ |  10.3 $-$ 11.2
Yoon et al. (2017)363636 ETGs are defined so that $c<0.43$, red $u-r$ color, slightly negative $\Delta(g-i)$, and refined with a visual classification (see Parker et al. 2005; Choi et al. 2010). There are 73116 ETGs in the sample.  | $0.621$ | $0.66$ | $0.1\leq z\leq 0.15$ | Spec |  $\sim$55000 | urgi/concentration/Morph |  10.7 $-$ 11.2
Yoon et al. (2017) | $0.851$ | $0.59$ | $0.1\leq z\leq 0.15$ | Spec |  $\sim$18000 | urgi/concentration/Morph |  11.3 $-$ 11.9
Zanella et al. (2016)373737 The MSR is estimated by eye.  | $0.62$ | $0.34$ | $1.05\leq z<1.7$ | Spec | 22 | sSFR |  10.7 $-$ 11.8
Zanella et al. (2016) | $0.62$ | $0.17$ | $1.7\leq z\leq 2.05$ | Spec | 10 | sSFR |  10.7 $-$ 11.7
444


11institutetext:  Max Planck Institute for the Physics of Complex Systems,
Nöthnitzer Str. 38, 01187 Dresden, Germany
Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187
Dresden, Germany
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
04510 Ciudad de México, México

Kinetic theory Phases: geometric; dynamic or topological Electrochemical
properties

# Electrochemical transport in Dirac nodal-line semimetals

R. Flores-Calderón 1122 Leonardo Medel 33 A. Martín-Ruiz 33112233

###### Abstract

Nodal-line semimetals are topological phases where the conduction and the
valence bands cross each other along one-dimensional lines in the Brillouin
zone, which are symmetry protected by either spatial symmetries or time-
reversal symmetry. In particular, nodal lines protected by the combined
$\mathcal{PT}$ symmetry exhibits the parity anomaly of 2D Dirac fermions. In
this Letter, we study the electrochemical transport in a
$\mathcal{PT}$-symmetric Dirac nodal line semimetals by using the
semiclassical Boltzmann equation approach. We derive a general formula for the
topological current that includes both the Berry curvature and the orbital
magnetic moment. We first evaluate the electrochemical current by introducing
a small $\mathcal{PT}$-breaking mass term (which could be induced by
inversion-breaking uniaxial strain, pressure, or an external electric field)
and apply it to the hexagonal pnictide CaAgP. The electrochemical current
vanishes in the zero-mass limit. Introducing a tilting term that does not
spoil $\mathcal{PT}$ symmetry that protects the nodal ring, we obtain a finite
electrochemical current in the zero-mass limit, which can be regarded as a
direct consequence of the parity anomaly. We show that the parity anomaly
induced electrochemical transport is also present at nonzero temperatures.

###### pacs:

05.20.Dd

###### pacs:

03.65.Vf

###### pacs:

65.40.gk

##  1 Introduction

Topological semimetals are new states of matter characterized by momentum
space invariants over the Fermi surface [1], rather than in the Brillouin zone
as in topological insulators [2]. They exhibit band crossings at point or
lines in the Brillouin zone, and the band degeneracy at the contact is
protected by symmetries such as crystalline and time-reversal [3].
Specifically, in Weyl semimetals (WSMs), the bands closest to the Fermi level
cross each other at a discrete set of points [1], while intersect along closed
loops for Dirac nodal-line semimetals (DNLSMs) [4, 5, 6].

DNLSMs can be protected by time-reversal and non-symmorphic symmetries, by
mirror planes, or by the combined symmetry of inversion $\mathcal{P}$ and
time-reversal $\mathcal{T}$ [6], such as the hexagonal pnictides CaAgP, CaAgAs
and Ca3P2. In the latter case, they are endowed with a topological
$\mathbb{Z}_{2}$ invariant, which is given by the phase winding of the Bloch
states around the nodal loop [7, 8, 9, 10]. It is also known that, in the
absence of spin-orbit coupling, the combined parity and time-reversal symmetry
$\mathcal{PT}$ provides the symmetry protection of the nodal line [4].
Therefore, any perturbation that preserves $\mathcal{PT}$ symmetry cannot
remove the band crossing and open a gap. Anomalous transport induced by
quantum anomalies has attracted great interest in the last decade. For
example, the (3+1)-dimensional chiral anomaly in WSMs, which is intimately
linked to the nontrivial topology of the band structure, leads to a negative
longitudinal magnetoresistence [11, 12, 13] and the planar Hall effect [14,
15, 16]. In the case of $\mathcal{PT}$-symmetric DNLSMs, the (2+1)-dimensional
parity anomaly has shown to determine linear and nonlinear topological
currents. The parity anomaly is also realized in graphene [17] and leads to an
interesting anomalous transport: the valley Hall effect [18], which has
attracted great attention due to its potential applications in valleytronic
devices [19]. In the case of DNLSMs, which are (3+1)-dimensional systems, the
low energy fermionic excitations can be described by a family of
(2+1)-dimensional quantum field theories with a parity anomaly [8, 9]. In
short, one can consider the DNLSM as a collection of graphene sheets.
Therefore, although the parity anomaly occurs only in (2+1) dimensions, it
also appears in DNLSMs, which are (3+1)-dimensional systems. Interestingly,
since graphene is $\mathcal{PT}$ symmetric and its Dirac points have
codimension $d_{c}=1$, it belongs to the same topological classification of
DNLSMs [20].

So far, magnetotransport experiments are the only probe of the topological
$\mathbb{Z}_{2}$ invariant of DNLSMs [21, 22, 23]. Therefore it is natural to
ask us for other anomalous transport properties behind the parity anomaly in
DNLSMs. This is precisely the question we address in this Letter for a
$\mathcal{PT}$ symmetric DNLSM, where we use semiclassical Boltzmann equation
approach to investigate the anomalous electrochemical transport, i.e. the
Berry curvature current response induced by an electric field ${\bf{E}}$ and a
gradient of a local chemical potential $\nabla\mu(\boldsymbol{r})$. In Weyl
semimetals, the electrochemical conductivity tensor is nearly quantized (in
terms of fundamental constants and the scattering time) when the Weyl nodes
lie at different Fermi levels, being this a direct consequence of the chiral
anomaly [24]. In a similar vein, recently nonuniform chemical potential
profiles have been used to theoretically predict interesting phenomena in 2D
and 3D materials. For example, they have been proposed as a mechanism to
manipulate Majorana zero-modes in graphene armcharir nanoribbons [25] and
quantum wires situated in proximity to a $s$-wave superconductor [26, 27, 28].
Furthermore, as shown in Ref. [29] within the semiclassical Boltzmann
formalism, gradients of chemical potential produce additional driving forces
in the depletion region in a metal-semiconductor junction. Following these
ideas, in this Letter we assume that nonuniform chemical potential profiles
can be used to test the parity anomaly in $\mathcal{PT}$ symmetric DNLSMs.

In the relaxation time approximation, we derive a general formula for the
nonlinear conductivity tensor $\sigma_{ijk}$, defined by
$J_{i}=\sigma_{ijk}(\partial_{j}\mu)E_{j}$. Introducing a small
$\mathcal{PT}$-breaking mass term, we evaluate analytically the conductivity
tensor, which nevertheless vanishes in the $\mathcal{PT}$-symmetric limit. We
apply our results to the hexagonal pnictide CaAgP. In order to obtain a direct
fingerprint of the parity anomaly, we introduce a tilting term which does not
spoil the $\mathcal{PT}$ symmetry which guarantees the topological protection
of the nodal ring. We show that for in-plane chemical gradient and electric
field, an anomalous current is induced in the direction perpendicular to the
nodal loop. Also, we verify that temperature effects do not destroy the
signature of the parity anomaly.

##  2 The model

In the continuum approximation, the minimal Hamiltonian for a Dirac nodal-line
semimetal with a single Dirac ring is given by [30]:

$H({\boldsymbol{k}})=\frac{1}{\Lambda}\left(k_{0}^{2}-k_{\rho}^{2}\right)\sigma_{z}+v_{z}k_{z}\sigma_{y}+m\sigma_{x}\,,$
(1)

where the Pauli matrices $\sigma_{i}$ represent an effective orbital basis
(not necessarily the spin degree of freedom) and
$k_{\rho}^{2}=k_{x}^{2}+k_{y}^{2}$. The parameters $\Lambda$ and $v_{z}$ arise
from any particular lattice realization of the
${\boldsymbol{k}}\cdot{\boldsymbol{p}}$ model (1). For later use we have
introduced a small $\mathcal{PT}$-breaking mass $m\sigma_{x}$ which could be
generated, for example, by inversion-breaking uniaxial strain, pressure, or an
external electric field [31, 32]. The eigenenergies of the Hamiltonian are
given by

$E_{s}({\boldsymbol{k}})=s\,\mathcal{E}({\boldsymbol{k}})\equiv
s\sqrt{m^{2}+v_{z}^{2}k_{z}^{2}+\frac{1}{\Lambda^{2}}\left(k_{0}^{2}-k_{\rho}^{2}\right)^{2}}.$
(2)

where $s=\pm 1$ is the band index. In the $\mathcal{PT}$-symmetric case, when
$m$ is zero, the model consists in two bands that touch each other at a one-
dimensional ring of radius $k_{0}$ in momentum space (located in the $k_{z}=0$
plane).

The symmetry protection of the Dirac ring (2) is guaranteed by a quantized
$\mathbb{Z}_{2}$ topological charge, which is given by the phase winding of
the Bloch states around the nodal loop. Therefore, any $\mathcal{PT}$
symmetry-preserving perturbation cannot remove the nodal ring and open a gap.
In the presence of spin-orbit coupling, several combinations of symmetries can
stabilize the nodal lines [4]. From a quantum field theory perspective, it is
interesting that the $\mathbb{Z}_{2}$ invariant can be understood as a
manifestation of the parity anomaly which occurs for fermions in (2+1)
dimensions. This is possible because the low-energy excitations of the DNLSM,
being a metallic system in (3+1) dimensions, can be described by a family of
(2+1)-dimensional massless Dirac systems. The parity anomaly emerges since any
gauge symmetric regularization of the quantum theory necessarily breaks
$\mathcal{PT}$ symmetry. As a consequence, a parity-breaking Chern-Simons term
is induced [33, 34], that gives rise to a finite quantized anomalous Hall
current for each (2+1) subsystem. However, when summed over the whole nodal
ring, the total current vanishes. This is so because each point at the nodal
line has a partner related by inversion symmetry which contributes with the
opposite sign. Therefore, in the presence of an in-plane electric field,
fermions at opposite sides of the nodal line flow to opposite directions (as
in the valley Hall effect of graphene, where electrons from different valleys
flow to opposite transverse edges). The only way to measure this effect is by
filtering electrons by their momentum, as dumbbell filter devices do.

DNLSMs exhibit various anomalous transport responses due to the nontrivial
structure of the Fermi surface, which includes linear [8, 35] and nonlinear
[36] Hall effects, the Kerr effect [37, 38] and giant nonlinear response in
the presence of magnetic fields [39]. All of these can be regarded as direct
signatures of the $\mathbb{Z}_{2}$ topological invariant.

##  3 Electrochemical transport in kinetic theory

We study the nonlinear electrochemical transport using the semiclassical
formalism. In short, we investigate the current response of a DNLSM subjected
to a spatially varying chemical potential $\mu({\boldsymbol{r}})$ and an
external homogeneous and static electric field ${\bf{E}}$. We start with the
corresponding semiclassical equations of motion for an electron wave packet in
a metal [40]:

$\dot{{\boldsymbol{r}}}_{s}={\bf{v}}_{s}({\boldsymbol{k}})-\dot{{\boldsymbol{k}}}\times\boldsymbol{\Omega}_{s}({\boldsymbol{k}}),\quad\dot{{\boldsymbol{k}}}=-(e/\hbar){\bf{E}},$
(3)

where
${\bf{v}}_{s}({\boldsymbol{k}})=\frac{1}{\hbar}{\nabla}_{{\boldsymbol{k}}}E_{s}({\boldsymbol{k}})$
is the band velocity, ${\boldsymbol{k}}$ is the crystal momentum and
$\boldsymbol{\Omega}_{s}({\boldsymbol{k}})=i\left<{\nabla}_{{\boldsymbol{k}}}u_{s}({\boldsymbol{k}})|\times|{\nabla}_{{\boldsymbol{k}}}u_{s}({\boldsymbol{k}})\right>$
is the Berry curvature defined in terms of the Bloch eigenstates
$\left|u_{s}({\boldsymbol{k}})\right>$, i.e.
$H({\boldsymbol{k}})\left|u_{s}({\boldsymbol{k}})\right>=E_{s}({\boldsymbol{k}})\left|u_{s}({\boldsymbol{k}})\right>$.
The presence of the Berry curvature in the equations of motion (3) gives rise
to anomalous transport perpendicular to the applied electric field. Evaluation
of the Hall current from this term reproduces the Karplus-Luttinger formula
for the anomalous Hall conductivity [41].

The charge current is defined by

${\bf{J}}=-e\sum_{s=\pm
1}\int\frac{d^{3}{\boldsymbol{k}}}{(2\pi)^{3}}\,\left[\dot{{\boldsymbol{r}}}_{s}-\frac{1}{e}\nabla_{{\boldsymbol{r}}}\times\boldsymbol{m}_{s}({\boldsymbol{k}})\right]\,f_{s}({\boldsymbol{r}},{\boldsymbol{k}}),$
(4)

where $f_{s}({\boldsymbol{r}},{\boldsymbol{k}})$ is the nonequilibrium
quasiparticle distribution function which satisfies the Boltzmann equation. In
the relaxation time approximation, the Boltzmann equation reads

${\bf{v}}_{s}\cdot{\nabla}_{{\boldsymbol{r}}}\,f_{s}+\frac{e}{\hbar}{\bf{E}}\cdot(\boldsymbol{\Omega}_{s}\times{\nabla}_{{\boldsymbol{r}}}-{\nabla}_{{\boldsymbol{k}}})\,f_{s}=-\frac{f_{s}-f_{s}^{(0)}}{\tau},$
(5)

where we have omitted all the dependence on ${\boldsymbol{r}}$ and
${\boldsymbol{k}}$ for simplicity. The parameter $\tau$ is the transport time
and $f_{s}^{(0)}$ is the equilibrium Fermi-Dirac local distribution defined by
local temperature $T({\boldsymbol{r}})$ and local chemical potential
$\mu({\boldsymbol{r}})$. The second term in Eq. (4) is a contribution of the
magnetization current, which is defined in terms of the orbital magnetic
moment
$\boldsymbol{m}_{s}({\boldsymbol{k}})=-i\frac{e}{2\hbar}\left<{\nabla}_{{\boldsymbol{k}}}u_{s}({\boldsymbol{k}})|\times\left[H({\boldsymbol{k}})-E_{s}({\boldsymbol{k}})\right]|{\nabla}_{{\boldsymbol{k}}}u_{s}({\boldsymbol{k}})\right>$,
which generically describes the rotation of a wave packet around its center of
mass [42, 43].

Since we are interested in the nonlinear response, we recursively solve the
Boltzmann equation (5) assuming that
$f_{s}=f_{s}^{(0)}+f_{s}^{(1)}+f_{s}^{(2)}$, where
$f_{s}^{(1)}\sim\mathcal{O}(E_{i})+\mathcal{O}(\partial_{i}\mu)$ and
$f_{s}^{(2)}\sim\mathcal{O}(E_{i}\,\partial_{j}\mu)$ contain the linear and
the nonlinear terms, respectively. One finds

$\begin{split}f_{s}^{(1)}&=\tau\,{\bf{v}}_{s}\cdot(-e{\bf{E}}+{\nabla}_{{\boldsymbol{r}}}\,\mu)\;\frac{\partial
f_{s}^{(0)}}{\partial E_{s}},\\\
f_{s}^{(2)}&=\frac{e\tau}{\hbar}({\bf{E}}\times\boldsymbol{\Omega}_{s}\cdot{\nabla}_{{\boldsymbol{r}}}\,\mu)\;\frac{\partial
f_{s}^{(0)}}{\partial E_{s}}.\end{split}$ (6)

It is worth mentioning that there are two more terms not listed in
$f_{s}^{(2)}$ which are independent of the Berry curvature. In a real
experimental situation these could be relevant and therefore one should be
able to distinguish between the topological and nontopological contributions.
In the problem at hand, for $\mathcal{PT}$-symmetric DNLSMs, they do not
contribute to the electrochemical current and hence in the following we will
restrict ourselves to the analysis of the geometrical contribution.

##  4 Electrochemical transport in DNLSM

After computing all the relevant components of the nonequilibrium distribution
function $f_{s}$, one then plugs it and the velocity
$\dot{{\boldsymbol{r}}}_{s}$ defined in Eq. (3) into the quasiparticle current
(4) to get the electrochemical current. For a DNLSM as modelled by the
Hamiltonian (1) the orbital magnetic moment and the Berry curvature are
related by
$\boldsymbol{m}_{s}({\boldsymbol{k}})=-s\frac{e}{\hbar}\mathcal{E}({\boldsymbol{k}})\,\boldsymbol{\Omega}_{s}({\boldsymbol{k}})$.
All in all, the nonlinear current response can be expressed as
$J_{i}=\sigma_{ijk}(\partial_{j}\mu)E_{k}$ (latin indices span the cartesian
components $\\{\ \\!\\!x,y,z\\}\ \\!\\!$), where the nonlinear conductivity
tensor $\sigma_{ijk}$ is given by

$\sigma_{ijk}(\mu)=\sigma_{0}\,\left[\epsilon_{lik}W_{jl}(\mu)+\epsilon_{ljk}W_{il}(\mu)+\epsilon_{lij}M_{kl}(\mu)\right],$
(7)

where $\sigma_{0}\equiv e^{2}\tau/h^{2}$ is the conductance quantum and we
have defined the dimensionless integrals

$\begin{split}W_{ij}(\mu)&=-\frac{\hbar}{2\pi}\sum_{s=\pm 1}\int
v_{si}({\boldsymbol{k}})\Omega_{sj}({\boldsymbol{k}})\frac{\partial
f^{(0)}_{s}({\boldsymbol{k}})}{\partial E_{s}}d^{3}{\boldsymbol{k}},\\\
M_{ij}(\mu)&=-\frac{\hbar}{2\pi}\sum_{s=\pm 1}\int
v_{si}({\boldsymbol{k}})\Omega_{sj}({\boldsymbol{k}})\mathcal{E}({\boldsymbol{k}})\frac{\partial^{2}f^{(0)}_{s}({\boldsymbol{k}})}{\partial
E_{s}^{2}}d^{3}{\boldsymbol{k}}.\end{split}$ (8)

Physically, the function $W_{ij}$ is determined by the anomalous velocity (3)
and the second-order correction to the quansiparticle distribution (6), whilst
the function $M_{ij}$ arises from the magnetization current in Eq. (4). We now
focus on the evaluation of the above integrals for the DNLSM. The geometry of
the nodal line advises the use of the cylindrical coordinates $\\{\
\\!\\!k_{\rho},\varphi,k_{z}\\}\ \\!\\!$. The band velocity and Berry
curvature for the model Hamiltonian (1) are

$\displaystyle{\bf{v}}_{s}({\boldsymbol{k}})=s\frac{2k_{\rho}(k_{\rho}^{2}-k_{0}^{2})}{\hbar\Lambda^{2}\mathcal{E}({\boldsymbol{k}})}\hat{\boldsymbol{e}}_{\rho}+s\frac{v_{z}^{2}k_{z}}{\hbar\mathcal{E}({\boldsymbol{k}})}\hat{\boldsymbol{e}}_{z}$
(9)

and

$\boldsymbol{\Omega}_{s}({\boldsymbol{k}})=s\frac{mv_{z}k_{\rho}}{\Lambda\mathcal{E}^{3}({\boldsymbol{k}})}\hat{\boldsymbol{e}}_{\varphi},$
(10)

respectively. Here,
$\hat{\boldsymbol{e}}_{\rho}=\cos\varphi\,\hat{\boldsymbol{e}}_{x}+\sin\varphi\,\hat{\boldsymbol{e}}_{y}$
and
$\hat{\boldsymbol{e}}_{\varphi}=\hat{\boldsymbol{e}}_{z}\times\hat{\boldsymbol{e}}_{\rho}$.
Besides, we will work at zero temperature, so the derivative of the
equilibrium distribution function $f^{(0)}_{s}$ will be strongly peaked at the
Fermi level $\mu$. This means that the nonlinear transport coefficients are
properties of the Fermi surface. Without loss of generality, we will assume
that the Fermi level $\mu$ crosses the conduction band, i.e. $\mu>m$. The
axial symmetry of the nodal line implies that the only nonzero components of
the functions (8) are $W_{xy}=-W_{yx}$ and $M_{xy}=-M_{yx}$. Substituting the
velocity (9) and Berry curvature (10) into Eqs. (8) we obtain

$W_{xy}(\mu)=\frac{mv_{z}}{\Lambda^{3}\mu^{4}}\int_{-\infty}^{+\infty}\\!\\!\int_{0}^{\infty}\\!\\!k_{\rho}^{3}(k_{\rho}^{2}-k_{0}^{2})\,\delta(\mu-\mathcal{E})\,dk_{\rho}dk_{z},$
(11)

and for the function $M_{yx}$ one can spot the identity
$M_{xy}=-\frac{\partial}{\partial\mu}(\mu W_{xy})$. To evaluate the integral
in Eq. (11) we use the properties of the Dirac delta in composition with an
arbitrary continuously differentiable function $f(x)$, i.e.
$\delta(f(x))=\sum_{i}\frac{\delta(x-x_{i})}{|f^{\prime}(x_{i})|}$, where the
sum extends over all roots $x_{i}$ of the function $f(x)$. In the present case
we have $\mu=\mathcal{E}(k_{\rho},k_{z})$, wherefrom we obtain the roots

$k_{z\pm}=\pm
k_{z}^{\ast}\equiv\pm\frac{1}{v_{z}}\sqrt{(\mu^{2}-m^{2})-\frac{1}{\Lambda^{2}}\left(k_{0}^{2}-k_{\rho}^{2}\right)^{2}},$
(12)

such that
$\delta(\mu-\mathcal{E})=\frac{\mu}{v_{z}^{2}k_{z}^{\ast}}\big{[}\delta(k_{z}-k_{z}^{\ast})+\delta(k_{z}+k_{z}^{\ast})\big{]}\Theta(k_{z}^{\ast
2})$, where the Heaviside function $\Theta$ guarantees that the root
$k_{z}^{\ast}$ is real valued. Inserting this result into the integral
expression (11) and integrating with respect to $k_{z}$ we obtain

$W_{xy}(\mu)=\frac{2m}{\Lambda^{3}\mu^{3}}\int_{0}^{\infty}\frac{k_{\rho}^{3}(k_{\rho}^{2}-k_{0}^{2})\,\Theta(k_{z}^{\ast
2})}{\sqrt{(\mu^{2}-m^{2})-\frac{1}{\Lambda^{2}}\left(k_{0}^{2}-k_{\rho}^{2}\right)^{2}}}dk_{\rho}.$
(13)

The constraint imposed by the step function $\Theta(k_{z}^{\ast 2})$ defines
the limits of integration over the variable $k_{\rho}$. In fact, it defines
the region

$\mbox{Re}\left[\sqrt{k_{0}^{2}-\Lambda\sqrt{\mu^{2}-m^{2}}}\,\right]\leq
k_{\rho}\leq\sqrt{k_{0}^{2}+\Lambda\sqrt{\mu^{2}-m^{2}}},$ (14)

which motivates the introduction of a critical Fermi level
$\mu_{c}=\sqrt{m^{2}+(k_{0}^{2}/2\Lambda)^{2}}$ and the radii

$k_{\rho\pm}^{\ast}=\sqrt{\Lambda\left(\sqrt{\mu_{c}^{2}-m^{2}}\pm\sqrt{\mu^{2}-m^{2}}\right)},$
(15)

such that the condition (14) renders: (I) $k_{\rho}\in[0,k_{\rho+}^{\ast}]$
for $\mu>\mu_{c}$ and (II) $k_{\rho}\in[k_{\rho-}^{\ast},k_{\rho+}^{\ast}]$
for $\mu<\mu_{c}$. Now we are ready to perform the integral (13). Changing
variable to $x=k_{\rho}/k_{0}$, Eq. (13) becomes

$\begin{split}W_{xy}(\mu)&=\frac{2mk_{0}^{6}}{\Lambda^{3}\mu^{3}}\left[\Theta(\mu_{c}-\mu)\int_{x_{-}}^{x_{+}}+\Theta(\mu-\mu_{c})\int_{0}^{x_{+}}\right]\\\
&\times\frac{x^{3}(x^{2}-1)\,dx}{\sqrt{(\mu^{2}-m^{2})-(\mu_{c}^{2}-m^{2})\left(1-x^{2}\right)^{2}}},\end{split}$
(16)

where $x_{\pm}\equiv k_{\rho\pm}^{\ast}/k_{0}$. This integral can be evaluated
in an analytical fashion. Defining the ratio
$\gamma\equiv\frac{\mu^{2}-m^{2}}{\mu_{c}^{2}-m^{2}}$ we finally obtain

$\begin{split}W_{xy}(\mu)&=\frac{\pi
m}{2\mu}\left[1-\left(\frac{m}{\mu}\right)^{2}\right]\bigg{\\{}\
\\!\Theta(\mu_{c}-\mu)\,+\,\Theta(\mu-\mu_{c})\\\
&\times\left[\frac{1}{2}+\frac{\sqrt{\gamma-1}}{\pi\gamma}+\frac{1}{\pi}\mbox{arccot}\left(\sqrt{\gamma-1}\right)\right]\bigg{\\}}\
,\end{split}$ (17)

which fully determines the nonlinear conductivity tensor (7). In Fig. 1 we
plot $W_{xy}$ as a function of the (dimensionless) chemical potential $\mu/m$.
The vertical dotted line marks the critical value $\mu_{c}/m$ and the vertical
dashed line corresponds to the critical value $\mu^{\ast}/m$ which maximizes
the function $W_{xy}$. We observe that this function vanishes for $\mu=m$,
increases monotonically up to a maximum located at
$\mu^{\ast}=\sqrt{3}m<\mu_{c}$ (for which $W_{xy}(\mu^{\ast})=\pi/3\sqrt{3}$)
and then decreases monotonically for $\mu>\mu^{\ast}$. Note that it is
continuous at the critical Fermi level $\mu=\mu_{c}>\mu^{\ast}$.

Figure 1: Plot of the function $W_{xy}$, given by Eq. (17), as a function of
the (dimensionless) chemical potential $\mu/m$. The vertical dotted line
represents the critical value $\mu_{c}/m$, while the vertical dashed line
corresponds to the critical value $\mu^{\ast}/m$ which maximizes the function.

To discuss the electrochemical response in a realistic DNLSM, it is convenient
to consider the precise values of the parameters appearing in the low-energy
model (1). From a lattice model for the hexagonal pnictide CaAgP, ab-initio
calculations give $\Lambda=-0.083\textup{\r{A}}^{-2}\mbox{eV}^{-1}$,
$k_{0}=0.207\textup{\r{A}}^{-1}$ and $v_{z}=-2.2\textup{\r{A}}$eV [44, 23].
For $m=0.05$eV, the critical Fermi energy is found to be $\mu_{c}\approx
0.94$eV. However, the Dirac nodal line is robust against weak short-range
correlated disorder for moderate chemical potentials ($\mu<0.5$eV). So, only
the region $\mu<\mu_{c}$ in Eq. (17) would be relevant in a realistic
situation.

The electrochemical current $J_{i}=\sigma_{ijk}(\partial_{j}\mu)E_{k}$ is an
experimentally observable signature of the nontrivial topology of the DNLSM.
Figure 2 schematically illustrates the Berry curvature (blue arrows) and the
plane $k_{z}=0$. By choosing in-plane chemical potential gradient
$\nabla\mu=\mathcal{\boldsymbol{F}}_{\mu}$ (brown arrow) and electric field
$\boldsymbol{E}$ (pink arrow), i.e.
$\mathcal{\boldsymbol{F}}_{\mu}\cdot\hat{\boldsymbol{e}}_{z}=\boldsymbol{E}\cdot\hat{\boldsymbol{e}}_{z}=0$,
the local electrochemical current points along the $z$ direction (green
arrow). We get

$J_{z}(\mu)=\sigma_{0}\frac{\pi
m}{2\mu}\left[1-3\left(\frac{m}{\mu}\right)^{2}\right]\,\mathcal{\boldsymbol{F}}_{\mu}\cdot\boldsymbol{E}.$
(18)

We emphasize that for the $\mathcal{PT}$-symmetric DNLSM considered in this
work, the nontopological contributions to the electrochemical current vanish
by symmetry. However, this may not be true for general DNLSMs protected by
other symmetries such as crystalline and time-reversal.The current response
(18) is quite interesting. As discussed in Ref. [8], the anomalous Hall
current vanishes for a DNLSM. The reason behind this cancellation is that the
nodal ring can be foliated into a family of $(2+1)$-dimensional subsystems
parametrized by the polar angle $\varphi$ in momentum space, such that each
point along the nodal line has another point related by inversion symmetry
which contributes oppositely to the current, so the total transverse current
vanishes. The only possibility to measure the anomalous Hall current is by
means of a dumbbell device which filters electrons by their momentum. In this
sense, the predicted electrochemical current (18) can be clearly distinguished
from the anomalous Hall effect and it can be measured with simple contacts.

It is interesting that from the field theoretical point of view, the anomalous
Hall effect is a fingerprint of the parity anomaly in DNLSMs, i.e. the Hall
current survives to the limit $m\to 0$ [8]. As pointed out in Ref. [36], the
nonlinear response of DNLSMs is also related to the parity anomaly. In the
present case, Fig. 1 reveals that the current response vanishes in the limit
$m\to 0$. Therefore, the electrochemical current (18) is nonzero only for
$\mathcal{PT}$ symmetry-breaking DNLSMs, which is quite realistic since the
mass term can be induced by inversion-breaking uniaxial strain, pressure, or
an external electric field [31, 32]. It is natural to ask whether is possible
to obtain an imprint of the parity anomaly in the electrochemical transport in
DNLSMs, i.e. in the $\mathcal{PT}$ symmetric case ($m\to 0$). We answer this
question in the affirmative in the next Section.

Figure 2: The blue arrows indicate the Berry curvature, given by Eq. (10), and
the red circle marks the nodal ring. The in-plane electric and electrochemical
fields are represented by the pink and brown arrows, respectively. The green
arrows represent the topological electrochemical current.

##  5 Parity anomaly induced transport

As showed in the previous section, the electrochemical current vanishes in the
limit $m\to 0$, so, it cannot be regarded as a signature of the parity anomaly
in DNLSMs. In order to search for a direct signature of the anomaly, it is
mandatory to consider a $\mathcal{PT}$ symmetric model, since the stability of
the nodal line comes from the invariance of the combined symmetry. This
suggests that a tilting term of the form
$\hbar{\boldsymbol{\upsilon}}\cdot{\boldsymbol{k}}\sigma_{0}$ will do the
work. Clearly, this term breaks separately $\mathcal{P}$ and $\mathcal{T}$,
but does not spoil the combined $\mathcal{PT}$ symmetry of the model. Tilted
nodal loops have been predicted to occur in alkaline-earth stannides,
germanides, and silicides [45] and in other materials displaying non-symmorfic
symmetries [46, 47]. All in all, let us consider the $\mathcal{PT}$-symmetric
model

$H({\boldsymbol{k}})=\hbar{\boldsymbol{\upsilon}}\cdot{\boldsymbol{k}}\sigma_{0}+\frac{1}{\Lambda}\left(k_{0}^{2}-k_{\rho}^{2}\right)\sigma_{z}+v_{z}k_{z}\sigma_{y}\,,$
(19)

where ${\boldsymbol{\upsilon}}$ is the tilting velocity. The corresponding
energy dispersion is
$E_{s}=\hbar{\boldsymbol{\upsilon}}\cdot{\boldsymbol{k}}+s\sqrt{v_{z}^{2}k_{z}^{2}+\frac{1}{\Lambda^{2}}\left(k_{0}^{2}-k_{\rho}^{2}\right)^{2}}$.
In the following, we consider tilt only in the in-plane direction, because it
produces electron and hole pockets. Without loss of generality we set
${\boldsymbol{\upsilon}}=\upsilon\,(\cos\alpha\,\hat{{\boldsymbol{e}}}_{x}+\sin\alpha\,\hat{{\boldsymbol{e}}}_{y})$
such that
${\boldsymbol{\upsilon}}\cdot{\boldsymbol{k}}=\upsilon\,k_{\rho}\cos(\varphi-\alpha)$,
where $\varphi$ is the polar angle that defines the nodal line in momentum
space. In the $m=0$ limit, the Berry curvature (10) is zero in the entire
Brillouin zone, except at the nodal ring, where it becomes singular, i.e.
$\boldsymbol{\Omega}_{s}({\boldsymbol{k}})=s\pi\delta(k_{z})\delta(k_{\rho}-k_{0})\hat{\boldsymbol{e}}_{\varphi}$
while the band velocity along the nodal ring is simply
${\bf{v}}_{s}={\boldsymbol{\upsilon}}$. Substituting these expressions into
Eq. (8) we get

$W_{ij}(\mu)=\hbar\upsilon_{i}k_{0}\int_{0}^{2\pi}d\varphi\,\hat{\boldsymbol{e}}_{\varphi
j}\,\delta(\mu-\hbar\upsilon k_{0}\cos(\varphi-\alpha)).$ (20)

where we have used that the zero-temperature equilibrium distribution
functions for electrons and holes become $f^{(0)}_{s}=\Theta\\{\
\\!\\!s[\mu-\hbar\upsilon k_{0}\cos(\varphi-\alpha)]\\}\ \\!\\!$ [36]. To
evaluate the integral (20) we use the properties of the Dirac delta in
composition with an arbitrary continuously differentiable function. The
argument of the delta function determines the angles
$\varphi_{\chi}=\alpha+\chi\arccos\left(\mu/\mu_{c}\right)$, where $\chi=\pm
1$ and $\mu_{c}\equiv\hbar\upsilon k_{0}$ is a critical Fermi level. So Eq.
(20) becomes

$W_{ij}(\mu)=\frac{\hat{\upsilon}_{i}}{\sqrt{1-\left(\mu/\mu_{c}\right)^{2}}}\sum_{\chi=\pm
1}\int_{0}^{2\pi}d\varphi\,\hat{\boldsymbol{e}}_{\varphi
j}\,\delta(\varphi-\varphi_{\chi}),$ (21)

for $\mu<\mu_{c}$ and vanishes for $\mu>\mu_{c}$. Here
$\hat{\boldsymbol{\upsilon}}=\boldsymbol{\upsilon}/\upsilon$ is the unit
vector along the tilting. This integral is quite simple. The final result is

$W_{ij}(\mu)=Q_{0}(\mu/\mu_{c})\,\hat{\upsilon}_{i}(\hat{\boldsymbol{e}}_{z}\times\hat{\boldsymbol{\upsilon}})_{j}.$
(22)

where $Q_{0}(y)=\frac{2y}{\sqrt{1-y^{2}}}$. For in-plane chemical potential
gradient and electric field, the electrochemical current in the direction
perpendicular to the nodal loop becomes

$J_{z}(\mu)=-\sigma_{0}\frac{2(\mu/\mu_{c})}{\left[1-(\mu/\mu_{c})^{2}\right]^{3/2}}\,(\hat{\boldsymbol{\upsilon}}\cdot\mathcal{\boldsymbol{F}}_{\mu})(\hat{\boldsymbol{\upsilon}}\cdot\boldsymbol{E}).$
(23)

This is a direct fingerprint of the parity anomaly in the nonlinear response
of DNLSMs. Assuming a tilting of the order of the Fermi velocity, which is
typically $v_{F}\approx 10^{8}$cm/s, and $k_{0}=0.207\textup{\r{A}}^{-1}$ as
for CaAgP, we find that $\mu_{c}\sim 1$eV. Since the nodal ring is robust
against weak disorder for moderate chemical potential ($\mu<0.5$eV), hence our
predictions would been well-testable for an appropriate range of chemical
potentials.

##  6 Summary and discussion

We have studied the electrochemical transport in Dirac nodal-line semimetals
in the absence of spin-orbit coupling, where the combined $\mathcal{PT}$
symmetry stabilizes the Dirac ring. We first introduce a small
$\mathcal{PT}$-breaking mass term, which could be induced by inversion-
breaking uniaxial strain, pressure, or an external electric field [31, 32]. In
this case, we compute the electrochemical current and find, for in-plane
chemical potential gradient $\mathcal{\boldsymbol{F}}_{\mu}$ and electric
field $\mathcal{\boldsymbol{E}}$, an anomalous current in the direction
perpendicular to the Dirac ring when the Fermi level crosses the conduction
band, i.e.
$J_{z}\sim\mathcal{\boldsymbol{F}}_{\mu}\cdot\mathcal{\boldsymbol{E}}$. It is
interesting the emergence of a critical Fermi level, which reflects in the
nonlinear conductivity tensor (see Fig. 1). Interestingly, when applied to the
hexagonal pnictide CaAgP, the critical Fermi level is large as compared with
the range of chemical potentials for which the linear response is stable under
moderate perturbations. One can also observe the existence of a critical Fermi
level at $\sqrt{3}m$ which maximizes the anomalous current. Although
interesting, this result cannot be regarded as a direct consequence of the
parity anomaly, since the current response vanishes in the $m=0$ limit.

In pursuit of a direct signature of the parity anomaly, we consider the
$\mathcal{PT}$ symmetric model, given by Eq. (1) with $m=0$, supplemented by a
tilting term of the form which does not spoil the symmetry. For in-plane
tilting both the conduction and valence bands contribute to the transport. We
find that for in-plane chemical potential gradient
$\mathcal{\boldsymbol{F}}_{\mu}$ and electric field $\boldsymbol{E}$, a finite
anomalous current in the direction perpendicular to the Dirac ring is induced,
i.e.
$J_{z}\sim(\hat{\boldsymbol{\upsilon}}\cdot\mathcal{\boldsymbol{F}}_{\mu})(\hat{\boldsymbol{\upsilon}}\cdot\boldsymbol{E})$.

One may wonder whether the signature of the parity anomaly appears only at
zero temperature, or it takes place also for nonzero temperatures. We answer
this question in the affirmative: the parity anomaly induced electrochemical
transport appears also for $T>0$. This can be seen by generalizing the
discussion of the previous section for nonzero temperatures. In this case, the
function (22) generalizes to
$W_{ij}(\mu,T)=Q(\mu_{c}/k_{B}T,\mu/\mu_{c})\,\hat{\upsilon}_{i}(\hat{\boldsymbol{e}}_{z}\times\hat{\boldsymbol{\upsilon}})_{j}$,
where we have defined the dimensionless function

$Q(x,y)=y\frac{\partial}{\partial
y}\int_{0}^{2\pi}\frac{1}{1+e^{x(\cos\phi-y)}}d\phi.$ (24)

In Fig. 3 we plot $Q(x,y)$ as a function of the (dimensionless) chemical
potential $y=\mu/\mu_{c}$ for different values of $x=\mu_{c}/k_{B}T$. The
vertical dashed line marks the value $y=1$. We observe that this function
vanishes at $y=0$. For a finite value of $x$, it increases monotonically up to
a maximum value and then decreases asymptotically to zero for $y\to\infty$.
Interestingly, as the temperature decreases (i.e. increasing $x$), the
function $Q(x,y)$ confines progressively to the region $y<1$. In the zero-
temperature limit (i.e. $x\to\infty$), the function (24) reduces exactly to
the function $Q_{0}(y)$ which defines the coefficient appearing in Eq. (22),
i.e. $\lim\limits_{x\to\infty}Q(x,y)\to Q_{0}(y)$, and collapses to zero for
$y>1$, as expected. The two most peaked curves shown in Fig. 3, the gray and
black lines, correspond to the temperatures $T\approx 230$K and $T\approx
140$K, respectively. All in all, the parity anomaly manifests in the
electrochemical transport for both zero and nonzero temperatures.

Although experimental realizations of 3D Dirac materials are rather few and
recent, one can establish the best candidate materials to test our
predictions. The hexagonal pnictides CaAgP, CaAgAs and Ca3P2 are particularly
promising, because they are available in single crystal form, possess a single
nodal ring near the Fermi energy protected by the combined $\mathcal{PT}$
symmetry and the spin-orbit coupling is very weak. In particular, numerical
estimates along this Letter have been done for the optimal nodal-line CaAgP.

Finally, a precise control of the local chemical potential would be of great
importance for realizing the topologically protected electrochemical current
we predict. As discussed above, from the theoretical side, nonuniform chemical
potential profiles have been proposed as a mechanism to manipulate Majorana
fermions in graphene [25] and quantum wires in proximity to a superconductor
[26, 27, 28]. In Weyl semimetals, for example, it has been theoretically
predicted that the local chemical potential can be controlled by doping or
having inhomogeneous impurities along the sample [48, 49]. Surprisingly, the
local control of the chemical potential has been routinely implemented in
experiments by an array of gates along the sample. For example, in Refs. [50,
51], this method has been used to find signatures of Majorana fermions in
superconductor-semiconductor nanowire devices. Also, as experimentally shown
in Ref. [52] for the topological insulator Bi2-xSbxSe3, the active
manipulation of trapped charges offers a powerful way to engineer spatial
distributions of the chemical potential (significantly more efficient than
conventional gating), for instance, by creating n${}^{+}-$n and p$-$n
junctions along the nanoribbon devices. In closing, as far as we know, no
experiments have been yet carried out in nodal-lines by using nonuniform
profiles of the chemical potential, but undoubtedly, it will be achievable in
the near future.

Figure 3: Plot of the function $Q(x,y)$, given by Eq. (24), as a function of
the (dimensionless) chemical potential $y=\mu/\mu_{c}$ for different values of
$x=\mu_{c}/k_{B}T$. The vertical dashed line marks the value $y=1$.

###### Acknowledgements.

L.M.O. was supported by the CONACyT PhD fellowship No. 834773. A.M.R has been
partially supported by DGAPA-UNAM Project No. IA102722 and by Project CONACyT
(México) No. 428214.

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# On the H$\alpha$ faintness of the North Polar Spur

Yoshiaki Sofue1, Jun Kataoka2 and Ryoji Iwashita2
1\. Institute of Astronomy, The University of Tokyo, Mitaka, Tokyo 181-0015,
Japan
2\. Faculty of Science and Engineering, Waseda University, Shinjyuku, Tokyo,
169-8555, Japan E-mail<EMAIL_ADDRESS>
###### Abstract

The ratio of H$\alpha$ intensity to 1.4 GHz radio continuum intensity in the
North Polar Spur (NPS) is measured to be $\lesssim 50$, two orders of
magnitude smaller than the values of $\sim 10^{4}$ observed in the typical
shell-type old supernova remnants, Cygnus Loop and S147. The extremely low
H$\alpha$-to-radio intensity ratio favours the Galactic-Centre explosion model
for NPS, which postulates a giant shock wave at a distance of several kilo
parsecs in the hot and low-density Galactic halo with low hydrogen
recombination rate, over the local supernova(e) remnant model.

###### keywords:

ISM: individual objects: (North Polar Spur) – ISM: shock waves – ISM: bubbles
– Galaxy: centre – galaxies: individual: objects (the Milky Way)

##  1 Introduction

The North Polar Spur (NPS) forms the northeastern edge of the giant Galactic
bubble, composing the Loop I of radio continuum [Haslam et al. 1982] and X-ray
emissions [Snowden et al. 1997, Predehl et al. 2020] with the brightest ridge
at $(l,b)\sim(30^{\circ},20^{\circ})$ [Sofue & Reich 1979]. Due to the sharp-
edged shell-like morphology, the NPS is interpreted as a spherical shock wave
driven by an explosive energy release at the loop centre.

There are two ideas to explain the origin of the explosion. One is the
explosion(s) of nearby supernova(e) [Hanbury Brown et al. 1960, Berkhuijsen et
al. 1971, Egger & Aschenbach 1995, Aschenbach & Leahy 1999, Wolleben 2007,
Dickinson 2018] in the Sco-Cen OB Associations at a distance of $\sim 140$ pc
[de Zeeuw et al. 1999]. This hypothesis assumes supernova remnant(s) expanding
in the low temperature, high density ISM at $kT\sim 0.01-1$ eV ($T\sim
10^{2}-10^{4}$ K), $n\sim 1$ H cm-3, and height $|z|\lesssim 30$ pc inside the
Galactic disc. However, when most of nearby shell-type SNRs with angular
diameters $\gtrsim 1^{\circ}$ were optically identified by red
(H$\alpha$)-sensitive emulsions [van den Bergh et al. 1973], the absence of
H$\alpha$ counterpart to NPS, if it is the closest SNR(s), has been a mystery
for over half a century [Sofue et al. 1974]. In fact, this problem was already
pointed out in the earliest paper which suggested the SNR idea for the first
time [Hanbury Brown et al. 1960].

The other idea is an explosion in the Galactic nucleus or a starburst in the
Galactic Centre (GC) [Sofue 1977, Sofue et al. 2016, Kataoka et al. 2018],
which postulates a shock wave propagating in the hot, low-density halo with
$kT\sim 0.2$ keV ($\sim 10^{6.3}$ K) and $n\sim 10^{-3}$ H cm-3 at $z\sim 3-8$
kpc. Because of the high temperature the gas is almost perfectly ionized and
the hydrogen recombination is limited, so that the H$\alpha$ absence may not
contradict this idea.

In the present paper, we revisit this classical issue of NPS’s optical
dimness, which appears to have not been explored by quantitative analysis
based on the observational data. We examine the ratio of H$\alpha$ intensity
to radio continuum intensity in the NPS and in the most typical shell-type
SNRs, Cygnus Loop and S147 [van den Bergh et al. 1973], and focus on the
difference in the radiation processes in the shock fronts expanding into the
Galactic halo and into the disc.

$T_{\rm b}$ (K)

Figure 1: All-sky map of the 408 MHz brightness temperature ($T_{\rm b}$) from
the Bonn-Parkes survey (Haslam et al. 1982) overlaid by contours of the
H$\alpha$ intensity (at 5, 12, 19, 27, 39, 56R, …) from the WHAM survey
(Haffner et al. 2003, 2010). Inserted box and lines are used for analyses in
Fig. 3 and 4. Angular resolution of the radio map is $0^{\circ}.85$ and
H$\alpha$ contours are smoothed to $0^{\circ}.45$. Background filtering is not
applied here.

##  2 Optical vs radio continuum emissions

###  2.1 Data and measured quantities

The radio continuum data were taken from the 408 MHz Bonn-Parkes [Haslam et
al. 1982], 1420 MHz Stockert-Villa Elisa [Reich et al. 2001] all-sky survey,
and 1420 MHz Bonn 100-m Galactic plane survey [Reich et al. 1990, Reich et al.
1997]. Also, radio surveys at 22 [Roger et al. 1999], 150 [Landecker &
Wielebinski 1970], and 820 [Berkhuijsen 1971]. We use the brightness
temperature $T_{\rm b}$ in K which is related to the surface brightness by
$\Sigma=2kT_{\rm b}/\lambda^{2}$ in erg cm-2 s-1 sr-1 Hz-1. We also use the
integrated intensity at $\nu=1.4$ GHz defined by

$I_{\nu}=\nu\Sigma$ (1)

in unit of erg cm-2 s-1 sr-1 in order to compare with the H$\alpha$ intensity.
The H$\alpha$ data were taken from the Wisconsin H$\alpha$ all-sky map (WHAM)
[Haffner et al. 2003, Haffner et al. 2010], and the intensity unit is Rayleigh
(R) defined by

${\rm 1R}=\frac{10^{6}}{4\pi}\frac{\rm photons}{\rm cm^{2}\ s\ sr}=2.4\times
10^{-7}\frac{\rm erg}{\rm cm^{2}\ s\ sr}$ (2)

at H$\alpha$, and 1 R corresponds to emission measure of ${\rm EM}=2.25$ cm-6
pc for gas at a temperature of 8000 K [Haffner et al. 2003].

As we are interested in the H$\alpha$-to-radio continuum intensity ratio, we
introduce a parameter, $\mathcal{Q}$, defined by

$\mathcal{Q}=I_{\rm H\alpha}/I_{\rm 1.4\ GHz}.$ (3)

Here, the radio brightness at $T_{\rm b}=1$ K corresponds to $I_{\rm 1.4\
GHz}=8.76\times 10^{-10}$ erg cm-2 s-1 sr-1. So, a region with $I_{\rm
H\alpha}=1$ R and $T_{\rm b}=1$ K has the intensity ratio of
$\mathcal{Q}=274$.

###  2.2 All-sky maps

Figure 1 shows the all-sky radio brightness ($T_{\rm b}$) map at 408 MHz in
the $(l,b)$ coordinates obtained from the Bonn-Parkes survey [Haslam et al.
1982] overlaid with contours of the H$\alpha$ intensity ($I_{\rm H\alpha}$)
map from the WHAM survey [Haffner et al. 2003]. The FWHM resolution of the 408
MHz map is $0^{\circ}.85$. The H$\alpha$ data had a resolution of
$6^{\prime}$, while the contours here are drawn after smoothing to the same
resolution as radio. We immediately notice that Loop I, including NPS, is
invisible or very weak in H$\alpha$ despite the high radio brightness even
toward the most prominent ridge at $l\sim 30^{\circ}$ and $b\sim
20-70^{\circ}$.

###  2.3 Perpendicular optical filaments in the Aquila Rift

Figure 2 enlarges the radio map at 1420 MHz [Reich et al. 2001] around the
brightest region of the NPS by contours superposed on the H$\alpha$ intensity
map. The figure shows several H$\alpha$ filaments running parallel to the
Aquila Rift, which is the dark absorption belt from $(35^{\circ},0^{\circ})$
to $(20^{\circ},13^{\circ})$, with $I_{\rm H\alpha}\sim 2-5$ R at position
angle of PA$\sim 120^{\circ}$, but they cross the NPS at right angle. The
perpendicular orientation indicates that these H$\alpha$ filaments are not
related to NPS. We also stress that the darkest part of the Rift at
$(l,b)\sim(27^{\circ},8^{\circ})$ shows no enhancement across NPS as shown by
a comparison of the cross sections of radio and H$\alpha$ intensities in the
lower panel of Fig. 2. This means that no H$\alpha$ feature associated with
NPS exists in front of the Aquila Rift whose distance is $\sim 430$ pc [Sofue
2015, Sofue & Nakanishi 2017], which contradicts the local origin model for
Loop I. However, it does not necessarily deny the possibility that there is
H$\alpha$ associated with NPS, if it is located behind the Rift. In either
case, it is difficult to determine how much of the H$\alpha$ feature is
associated with the NPS. We therefore exclude this region from quantitative
measurements, and restrict our analysis to the spur at higher latitudes than
$b\sim 30^{\circ}$.

Figure 2: [Top] The brightest ridge of NPS at 1420 MHz by contours (interval
0.1 K; maximum contour $T_{\rm b}=8$ K) from the Stockert survey (Reich et al.
2001) overlaid on the H$\alpha$ map (scale is non linear by the bar). Many
H$\alpha$ filaments run parallel to the Aquila Rift (PA$\sim 120^{\circ}$),
but they are perpendicular to the NPS (PA$\sim 30^{\circ}$). No background
subtraction is applied in this figure. [Bottom] Cross section of 21-cm and
H$\alpha$ along line R-R’. Here, the smooth emission has been removed by the
background-filtering (BGF) technique (Sofue and Reich 1979). (See section 3.2
for the description of BGF.)

###  2.4 Intensity profiles

Fig. 3 shows horizontal cross sections of the 408 MHz radio brightness and
H$\alpha$ intensity across the NPS and NPS West, or across the Loop I, at
$b=61^{\circ}$ and $30^{\circ}$ along lines AA’ and BB’ in Fig. 1,
respectively. The radio profiles exhibit a typical double-horn structure
indicative of the intensity variation across a spherical shocked shell. On the
other hand, the H$\alpha$ intensity distribution is almost flat except for the
broad enhancements by local HII regions ($l\sim-25^{\circ}$ at $b=61^{\circ}$
and $l\sim 0^{\circ}$ at $b=30^{\circ}$), which are unrelated to the NPS.

A - A’

B - B’

Figure 3: Horizontal cross sections of the 408 MHz radio continuum (black
line) and H$\alpha$ (thin red line) along the lines A-A’ and B-B’ in Fig. 1 at
$b=30^{\circ}$ and $61^{\circ}$. Note the lack of H$\alpha$ emission
corresponding to the radio peaks of NPS and NPS-west. Note that H$\alpha$
enhancements around $l\sim-30^{\circ}$ along A-A’ and $l\sim 0^{\circ}$ along
B-B’ are foreground or background diffuse HII regions not related to the NPS.

###  2.5 Intensity-intensity plots

The faintness of H$\alpha$ emission in NPS can be more quantitatively
displayed by plotting the H$\alpha$ intensity against radio intensity in and
around the objects using the so-called $TT$ plots. Fig. 4 shows H$\alpha$
intensity in the squared area ’T’ of Fig. 1 across the NPS-E (east) and NPS-W
(west) plotted against 408 MHz $T_{\rm b}$ after subtracting the 2.7 K cosmic
background emission. Open circles are running means around individual centres
of the $T_{\rm b}$ bins with the bars indicating the dispersion in H$\alpha$
intensity.

Figure 4: $TT$ Plot of the H$\alpha$ intensity against 408 MHz brightness
temperature $T_{\rm b}$ after subtracting 2.73 K in the area ’T’ of Fig. 1
(from $l\sim 270^{\circ}$ to $90^{\circ}$ and $b=+30^{\circ}$ to
$+45^{\circ}$). The blue and red arrows indicate the on-NPS and off-NPS
emissions, respectively. The background-filtering is not applied here.

The mean H$\alpha$ intensity in the OFF NPS region with $T_{\rm b}\lesssim 35$
K is measured to be $I_{\rm H\alpha}\sim 1.22$ R, representing the mean
Galactic emission [Haffner et al. 2003] as indicated by the left side arrow
marking the region in Fig. 4. The NPS shows up as the high-$T_{\rm b}$
extension at $35-60$ K in the ON-NPS region marked by the right side arrow.
So, we here define the NPS in this plot as the area with $T_{\rm b}(408{\rm
MHz})\gtrsim 35$ K, or $T_{\rm b}(1420{\rm MHz})\gtrsim 1$ K for spectral
index of $\beta=-2.7$. We, then, measure the mean H$\alpha$ intensity in this
ON-NPS region to be $I_{\rm H\alpha}\sim 1.3$ R. Although the excess of ON-NPS
$I_{\rm H\alpha}$ over OFF-NPS value is $\sim 0.1$ R, we here estimate the
excess to be $\Delta I_{\rm H\alpha}\sim 0.2$R, considering the scatter and
dispersion of the plotted values. This yields the H$\alpha$-to-radio intensity
ratio of $\mathcal{Q}\lesssim 50$ at 1.4 GHz.

The H$\alpha$ intensity yields the upper limit to the emission measure as
${\rm EM}\lesssim 0.5$ cm-6 pc for assumed electron temperature $T_{\rm
e}=10^{4}$ K. The values may be compared with those estimated for the SNR,
Cygnus Loop, where $T_{\rm b}\sim 1$ K and $I_{\rm H\alpha}\sim 20$ R,
yielding $\mathcal{Q}\sim 6\times 10^{3}$, and ${\rm EM}\sim 50$ cm-6 pc for
$10^{4}$ K as obtained from individual H$\alpha$ observations [Hester et al.
1986]. We list the estimated values in table 1.

##  3 Comparison with SNRs

###  3.1 Maps

Fig. 5 shows overlays of radio continuum brightness at 1420 MHz of the Cygnus
Loop and S147 on H$\alpha$ intensity maps, where the radio maps were taken
from the Bonn-100m Galactic plane survey [Reich et al. 1990, Reich et al.
1997]. The SNRs are visible both in radio and H$\alpha$ emissions. Such
H$\alpha$-radio association is often observed in Galactic radio SNRs [Uyanıker
et al. 2004, Xiao et al. 2008], and seems also to happen in the spiral galaxy
M31 [Braun & Walterbos 1993]. On the other hand, H$\alpha$ is hardly visible
in the NPS despite the clear and sharp radio ridge as shown in Figs 1 and 2.
The NPS is, thus, extraordinarily fainter in H$\alpha$ compared with the usual
SNRs.

Figure 5:  1420 MHz maps of SNRs (top: S147, bottom: Cygnus Loop) from the
Bonn-100 m Galactic plane survey (Reich et al. 1990, 1997) by contours at an
interval of 0.1 K overlaid on H$\alpha$ intensity maps from 0 to 50R for S147
and 0 to 80R for Cygnus Loop as indicated by the bars. Compare these maps with
Figs. 1 and 2, where NPS is not visible in H$\alpha$ despite the much lower
intensity levels.

###  3.2 Spectra (SED)

Fig. 6 shows variation of peak intensities at various frequencies along the
NPS ridge as obtained using background-filtered (BGF) maps [Sofue & Reich
1979] of the radio and H$\alpha$ sky surveys. The BGF subtracts background
emission and creates a nearly zero-adjusted intensity distribution of sources
with scale sizes greater than a smoothing beam width111The ’background’ is
defined as a smoothed map after iterative clipping of peaky sources, so that
it traces the valleys of the intensity distribution. . We used a box-shaped
one-dimensional smoothing beam with full width of $10^{\circ}$ in the
longitude direction at each latitudinal grid. The one-dimensional smoothing
was so chosen that it avoids unnecessary smearing effect by the steep
intensity gradient perpendicular to the Galactic disc. Although the angular
resolutions are different at different frequencies, the NPS is sufficiently
resolved, and we used the peak intensities read on the thus obtained BGF maps
for the spectral analysis in this section.

Figure 6: [Top] Intensity profiles along the NPS ridge using the BGF maps at
22 to 2300 MHz and H$\alpha$ maps taken from the surveys cited in section 2.1.
[Bottom] Same, but enlarged for 1420 MHz and H$\alpha$ profiles.

Fig. 7 shows a spectral energy distribution (SED) of the NPS at
$b=30^{\circ}$, $45^{\circ}$ and $60^{\circ}$ as obtained from the intensity
plot along the NPS ridge shown in Fig. 6. We also plot SEDs of the northern
shell edge of Cygnus Loop and eastern edge of S147. The background emissions
around SNRs are subtracted by measuring averaged brightness in a small area
without significant emission features about half a shell radius outside each
shell edge at the same Galactic latitude. The H$\alpha$ extinction has been
corrected by $A_{\rm H\alpha}=0.2$ and 0.6 mag., respectively, for Cygnus Loop
and S147, as described in section 3.4. However, the correction is not applied
to the NPS, because we here compare the spectra, when they are assumed to be
the same type (SNR-type) objects, and so the NPS is assumed to be located at a
distance of $\sim 140$ pc with negligible extinction.

The radio spectra of NPS and SNRs are consistent with those from more accurate
analyses [Iwashita et al. 2023, Xiao et al. 2008, Uyanıker et al. 2004], and
show that the radio intensities of the NPS and SNRs are comparable. However, a
significant difference is found at H$\alpha$, where the NPS is fainter than
SNRs by two orders of magnitude. The upper limit to the H$\alpha$ intensity of
the NPS as obtained from the $TT$ analysis is indicated by the arrow.

Figure 7: Spectral energy distributions (SED) of the NPS at $b=30^{\circ}$
(dots), $45^{\circ}$ (diamonds), and $60^{\circ}$ (grey circles) made from the
peak-intensity profiles in Fig. 6. Intensities of the northern shell edge of
the Cygnus Loop (red triangles) and eastern shell edge of S147 (blue squares)
are also plotted. Interstellar extinction of H$\alpha$ is corrected for
according to their distances, while no correction is made for NPS here for its
assumed distance of 140 pc in this diagram. The arrow indicates the estimated
upper limit of $I_{\rm H\alpha}$ from the $TT$ analysis.

###  3.3 TT

In Fig. 8 we display $TT$ plots in linear and logarithmic scaling for the SNRs
shown in Fig. 5 in comparison with that of the NPS in region T of Fig. 1. The
lowest values of H$\alpha$ and radio intensities in each $TT$ plot
representing the background emission are subtracted. The plots for the SNRs
indicate that the H$\alpha$ intensity is well correlated with the radio
intensity by a relation $I_{\rm H\alpha}\propto T_{\rm b}$. Using the gradient
of the linear plot in the upper panel we obtain the H$\alpha$-to-radio
intensity ratio of $\mathcal{Q}\sim 1.1\times 10^{4}$ both for Cygnus Loop and
S147. On the other hand, the NPS’s plot shows no clear correlation between
H$\alpha$ and radio intensities.

Figure 8: Linear and logarithmic $TT$ plots of H$\alpha$ against radio
intensities for the NPS (black dots) converted to 1.4 GHz compared with those
at 1.4 GHz for Cygnus Loop (red triangles) and S147 (blue squares).
Approximate base-line (background) intensities in each object’s area are
subtracted. The dashed line represents the H$\alpha$-to-radio intensity ratio
of $\mathcal{Q}\sim 1.1\times 10^{4}$. Table 1: Intensities of the NPS and SNR
Cygnus Loop

Object | Quantity |  | EM (cm-6 pc ) |  $T_{\rm e}$(K) | Remark
---|---|---|---|---|---
NPS | X ray (3/4 keV) |  $I_{\rm X}\sim 2\times 10^{-4}$ cts/s/amin2 | ${\rm EM}\sim 0.1$ |  $T_{\rm e}\sim 10^{6.5}$ (0.3 keV) | 1, 2, 3
| H$\alpha$ for X-ray ${\rm EM}$ |  $I_{\rm H\alpha}\sim 4\times 10^{-5}$ R | ibid ($\sim 0.1$) | for $10^{6.5}$ K |
| H$\alpha$ observed here |  $I_{\rm H\alpha}\lesssim 0.2$ R | ${\rm EM}\lesssim 1.4\times 10^{2}$ | for $10^{6}$ K |
| H$\alpha$ observed here |  $I_{\rm H\alpha}\lesssim 0.2$ R | ${\rm EM}\lesssim 0.4$ | for $10^{4}$ K |
| Radio 1.42 GHz |  $T_{\rm b}\sim 1$ K |  |  |
| H$\alpha$/radio intensity ratio | $\mathcal{Q}\lesssim 50$ |  |  |
Cygnus Loop | Xray (3/4 keV) |  $I_{\rm X}\sim 10^{-2}$ cts/s/amin2 | ${\rm EM}\sim 1$ |  $T_{\rm e}\sim 6\times 10^{6}$ (0.5 keV) | 4
| H$\alpha$ |  $I_{\rm H\alpha}\sim 20-40$ R | ${\rm EM}\sim 50$ | for $10^{4}$ K | 5
| Radio 1.42 GHz |  $T_{\rm b}\sim 1$ K |  |  |
| H$\alpha$/radio intensity ratio | $\mathcal{Q}\sim 1.1\times 10^{4}$ |  |  | Fig. 8.

1\. X-ray intensity was read from ROSAT all-sky map at 3/4 keV (R4 band)
[Snowden et al. 1997] (1 cts/s/arcmin2 = 400 Jy sr-1 $=9.6\times 10^{-4}$ erg
cm-2 s-1 keV-1); 2,3:[Kataoka et al. 2013, Yamamoto et al. 2022]; 4. [Uchida
et al. 2008]; 5.[Hester et al. 1986].

###  3.4 Interstellar extinction

The radio continuum emissions are absorption free in the circumstances
discussed here regardless of the distance. In the SNR hypothesis, which
assumes a distance to NPS less than $\sim 140$ pc, the interstellar dust
extinction of the H$\alpha$ emission is negligible in the entire Loop I even
at low or zero latitudes, because it is located in front of the Aquila Rift at
a distance of $\sim 0.4$ kpc [Sofue 2015, Sofue & Nakanishi 2017]. This
assumption is used in the figures where NPS and SNRs are compared.

The SNR Cygnus Loop is observed to have $A_{v}=0.25$ mag, or $A_{\rm
H\alpha}\sim 0.2$ [Fesen et al. 2018]. S147 has foreground extinction of
$A_{v}=0.7$ mag ($A_{\rm H\alpha}\sim A_{r}\sim 0.6$) [Fesen et al. 1985], and
$A_{v}\sim 1.2$ mag or $A_{r}\sim 0.9$ including the internal extinction [Chen
et al. 2017]. Therefore, the H$\alpha$ intensities observed toward Cygnus Loop
and S147 in Fig. 5 and 8 are under-estimated by about factors of 1.2 and 1.7,
respectively. These factors for the SNRs are corrected in Fig. 7 in order to
compare with NPS as a local object with negligible extinction.

In the GC explosion hypothesis of NPS, which assumes a distance of $\sim 8$
kpc, the optical (visual) extinction can be estimated by the general law
relating it to HI column density, $N_{\rm HI}$, by

$A_{v}=N_{\rm HI}/1.79\times 10^{21}{\rm HI\ cm^{-2}}$ (4)

[Predehl & Schmitt 1995]. Measuring the HI column density from the all-sky
integrated HI intensity map [Kalberla et al. 2007], we obtain $N_{\rm HI}\sim
1.1\times 10^{21}$ H cm-2 at $b\sim 15^{\circ}$, $\sim 0.7\times 10^{21}$ at
$30^{\circ}$, and $\sim 0.15\times 10^{21}$ at $\sim 60^{\circ}$. Then,
assuming [Gordon et al. 2003]

$A_{\rm H\alpha}\simeq A_{\rm r}=0.8A_{v},$ (5)

we obtain $A_{\rm H\alpha}\sim 0.49$, 0.31 and $\sim 0.07$ mag. at $b\sim
15^{\circ}$, $30^{\circ}$, and $60^{\circ}$, respectively. Or, the H$\alpha$
intensities are under-estimated by a factor of 1.6, 1.3 and 1.07,
respectively, at these latitudes. The region closer to the low galactic
latitudes of NPS suffers from heavier extinction by the dust lane of the
Aquila Rift, where we cannot give a conclusive discussion.

##  4 Summary and Discussion

###  4.1 Summary

Analysis has shown that the H$\alpha$-to-1.4 GHz intensity ratio for NPS
($\mathcal{Q}\lesssim 50$) is more than two orders of magnitude smaller than
that for typical shell-type SNRs, Cygnus Loop and S147 ($\mathcal{Q}\sim
1.1\times 10^{4}$). No evidence of H$\alpha$ association was found along NPS,
even towards the brightest and sharpest ridge at $b\sim 8-20^{\circ}$. The low
H$\alpha$ intensity favours the GC explosion model, which postulate the
distance of $\sim 7$ kpc, over the local supernova explosion model.

Below we discuss the implication of the results for the two models about the
origin of NPS. In either model of local or GC origin, it should be noted that
NPS is much larger in size than the other known SNRs or bubbles. This may lead
to various differences in environments and physical conditions where the NPS
is situated. The $\mathcal{Q}$ value would be useful to distinguish such
differences from each other regardless of the size and distance.

###  4.2 On the local bubble model

It is difficult to interpret the H$\alpha$-dark NPS as an ordinary supernova
remnant such as the Cygnus Loop or S147 that exploded in the dense ($\rho\sim
1$ H cm-3) and cold ($T\lesssim 10^{4}$ K) Galactic disc, where the hydrogen
recombination is high and the shock compressed shells efficiently emit
H$\alpha$ line at $\sim 20-40$ R. If NPS is a remnant of multiple supernovae
exploded in a local OB association, much stronger shock waves would cause
brighter H$\alpha$. If it is a similar object to the Orion-Eridanus super
bubble [Pon et al. 2016] as seen in Fig. 1 around
$(l,b)\sim(200^{\circ},-20^{\circ})$, it should be as bright as $\sim 10-20$
R. If it is associated with high-latitude HI spurs at local velocities [Heiles
et al. 1980], then the shocked area in touch with the NPS should emit
H$\alpha$ by the same mechanism as above. Thus, the local origin models have
difficulty explaining the H$\alpha$ faintness of NPS.

###  4.3 On the GC explosion model

The H$\alpha$ faintness can be naturally explained by the GC explosion model.
The model postulates a shock wave propagating in the Galactic halo at $T\sim
10^{6.3}$ K. The temperature in NPS is observed to be much higher at $\sim
10^{6.5}$ K, while the emission measure is rather small at ${\rm EM}\sim 0.1$
cm-6 pc [Kataoka et al. 2013, Kataoka et al. 2018, Yamamoto et al. 2022]. In
such a hot plasma the hydrogen recombination rate is $10^{-2}$ times that in
the ISM at $T\sim 8000$ K, $\alpha(10^{6.5}{\rm K})\sim 10^{-2}\alpha(8000{\rm
K})$ [Hummer 1994]. Knowing that 1 R at this temperature corresponds to 2.25
cm-6 pc [Haffner et al. 2003], it leads to $I_{\rm H\alpha}\sim
10^{-2}EM/2.25\sim 4\times 10^{-3}$ R, which is consistently below the
observed upper limit of $I_{\rm H\alpha}({\rm NPS})\lesssim 0.2$ R. Thus, the
GC bubble model seems plausible to explain the observed H$\alpha$ faintness of
the NPS.

The H$\alpha$ property of NPS gives further constraint on the GC origin model.
A galactic-scale wind such as observed in starburst galaxy NGC 3079 [Cecil et
al. 2002] or M82 [Lehnert et al. 1999] presumes a large H$\alpha$ shell in the
halo. However, such an H$\alpha$ shell is not observed in NPS. So, NPS may be
a more spherical bubble directly exposed to the halo’s hot plasma. Even so,
the root region might emit H$\alpha$ associated with the 1-kpc conical wind of
HI [Sofue 2022, Sofue & Kataoka 2021] and X-ray [Bland-Hawthorn & Cohen 2003],
but it is hidden behind the Aquila Rift. In this case the NPS will be a giant
bubble similar to that observed in NGC 253, where 1-kpc scale H$\alpha$ wind
blows near the nucleus [Westmoquette et al. 2011] and a giant X-ray and radio
bubbles are expanding into the halo [Sofue & Vogler 2001].

## Acknowledgements

The data analysis was carried out on the computer system at the Astronomy Data
Center of the National Astron. Obs. of Japan. The authors are indebted to the
groups of the Bonn-Stockert radio (Dr. P. and W. Reich) and WHAM (Dr. L. M.
Haffner) surveys for the archival data base. A part of this research was
supported by Japan Science and Technology Agency (JST) ERATO Grant Number
JPMJER2102 and JSPS Kakenhi Grant Number 20K20923, Japan.

## Data availability

The radio and H$\alpha$ fits data were downloaded from the URL:
https://lambda.gsfc.nasa.gov/product/foreground/ fg_diffuse.html, and
http://www3.mpifr-bonn.mpg.de/survey.html.

## Conflict of interest

The authors declare that there is no conflict of interest.

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