# Density-wave tendency from a topological nodal-line perspective

Tianlun Zhao, Yi Zhang<EMAIL_ADDRESS>International Center for
Quantum Materials, School of Physics, Peking University, Beijing, 100871,
China

###### Abstract

The understanding of density waves is a vital component of our insight into
electronic quantum matters. Here, we propose an additional mosaic to the
existing mechanisms such as Fermi-surface nesting, electron-phonon coupling,
and exciton condensation. In particular, we find that certain 2D spin density-
wave systems are equivalent to 3D Dirac nodal-line systems in the presence of
a magnetic field, whose electronic structure takes the form of Dirac-fermion
Landau levels and allows a straightforward analysis of its optimal filling.
The subsequent minimum-energy wave vector varies over a continuous range and
shows no direct connection to the original Fermi surfaces in 2D. Also, we
carry out numerical calculations where the results on model examples support
our theory. Our study points out that we have yet to attain a complete
understanding of the emergent density wave formalism.

##  I I. Introduction

Density waves (DW)[1, 2, 3, 4], including charge density waves (CDW) and spin
density waves (SDW), relevant to various physical phenomena in electron and
spin quantum matter, have been a fundamental yet controversial problem in
condensed matter physics for several decades. In the Peierls theory of CDW,
the Fermi surface nesting (FSN) in the 1D chain gives rise to a spatially
periodic re-distribution of charge density [5, 6, 7, 8] with a $2\pi/{q}_{n}$
period, commonly accompanied by a distortion of the lattice structure and a
metal-insulator transition, where ${q}_{n}=2k_{F}$ is the nesting vector
between the two Fermi points. Though the Peierls transition has successfully
described the properties of various quasi-1D DW materials [9, 10], its
extensions to higher dimensions have witnessed many difficulties [5]. Other
mechanisms based upon electron-phonon coupling [11, 12] and exciton
condensation [13, 14, 15, 16] offer consistent explanations on the DW origin
and physical properties of a series of materials such as $\ce{NbSe_{2}}$,
$\ce{TaSe_{2}}$ and $\ce{CeTe_{3}}$ [17, 18, 19, 20, 21] without FSN. Besides,
Overhauser’s theory pointed out the importance of electron interaction and
correlations in the formation of DW in certain materials [22, 23].

The existing theories do well in their respective sphere of applications. Some
of them are based upon perturbative analysis and become less controlled for
stronger coupling [12, 24]. In addition, most theories cater to preferential
DW wave vectors that are discrete and special to the band structures and/or
the auxiliary degrees of freedom. Still, the origin of various CDWs, e.g., the
3D DW states in $\ce{M_{3}T_{4}Sn_{13}}$ (e.g. $\ce{Ca_{3}Ir_{4}Sn_{3}}$ and
$\ce{Sr_{3}Ir_{4}Sn_{3}}$) [25, 26], the CDW in cuprate materials [27], remain
controversial to a degree. For instance, the charge modulation in some cuprate
materials exhibits a dependence on the spectral gap [27] with its wave vector
spanning a continuous spectrum [28, 29, 30]. Therefore, our overall
understanding of the DW mechanism has not, by far, reached a complete picture
yet.

Here, we propose a novel, independent understanding of DW from a Dirac-fermion
Landau levels (LLs) energetics perspective[31]. We note that a class of models
with SDW are equivalent to higher-dimensional lattice models in the presence
of a magnetic field [32]. Therefore, to locate the optimal DW wave vector, we
can, in turn, look for the magnetic field strength that minimized the energy
of the corresponding higher-dimensional system. Such an argument is unrelated
to the FSN and remains relevant even when the DW strength is no longer weak in
comparison with the bandwidth. Here, we focus on a specific set of two-
dimensional (2D) models we find particularly illuminating: these models’
counterparts in three dimensions (3D) possess Dirac-fermion nodal lines (NLs),
whose electronic structure in a magnetic field possesses a zeroth LL with a
large degeneracy and a large gap from the rest of the LLs. Therefore, both the
optimal amplitude and direction of the magnetic field and thus the
corresponding DW wave vector in the original 2D model depend solely on the
geometry of the NL and can be estimated theoretically in a wide parameter
region. Also, without changing the Fermi surface, the NL is continuously
variable with respect to the DW parameters [22, 23, 33, 34], such as the DW
amplitude, so is the optimal DW wave vector [35, 36, 37, 38]. To solidify our
claim, we further illustrate numerical results quantitatively consistent with
our theoretical expectations.

We organize the rest of the paper as follows: In the next section, we
introduce the Dirac-fermion LLs perspective for density wave tendencies,
including the duality between a 2D DW system and a 3D system with a magnetic
field, and the optimal DW wave vector in terms of the energetics on filling
the Dirac-fermion LLs. We also summarize the approximations enlisted in our
theory and their potential impacts, whose details are also discussed further
in Appendix. In section 3, we showcase the emergent DW in a benchmark 2D model
and compare the numerical results on the optimal DW wave vectors with our
theoretical expectations. Section 4 concludes our discussions with a summary
and further implications of our theory and models. In the Appendix, we provide
results on several additional models, especially a quasi-1D example showing
the compatibility of our theory with the Fermi surface nesting picture.

##  II II. Theory

###  II.1 Equivalence between a 2D DW system and a 3D system with a magnetic
field

Consider a 2D incommensurate CDW order described by an effective
Hamiltonian[39, 32]:

$\hat{H}_{2D,\phi}=\hat{H}_{0}+\sum_{\bm{r}=(x,y)}V_{q}\cos(\bm{q}\cdot\bm{r}+\phi)\hat{c}^{\dagger}_{r}\hat{c}_{r},$
(1)

where $\hat{H}_{0}$ includes the translation-invariant terms, $\hat{c}_{r}$ is
the electron annihilation operator at site $\bm{r}$, $V_{q}$ is the DW
amplitude, and $\bm{q}=(q_{x},q_{y})$ is the DW wave vector, is equivalent to
a 3D system $\hat{H}_{3D}=\sum_{k_{z}}\hat{H}_{2D,k_{z}}$:

$\hat{H}_{2D,k_{z}}=\hat{H}_{0,k_{z}}+\sum_{\bm{r}=(x,y)}V_{q}\cos(\bm{q}\cdot\bm{r}+k_{z})\hat{c}^{\dagger}_{r,k_{z}}\hat{c}_{r,k_{z}},$
(2)

with a magnetic field $\bm{B}=(q_{y},-q_{x},0)$ corresponding to the vector
potential $\bm{A}=(0,0,\bm{q}\cdot\bm{r})$, and $k_{z}=\phi$ is a good quantum
number. We have applied the convention that the lattice spacing is $1$ and
$e=\hbar=1$. We aim to determine the optimal $\bm{q}$ with the lowest energy
given the rest of the model parameters, in analogy to the search for the
Peierls transition in a mean-field treatment of the electron-phonon couplings,
etc. A thorough study of a specific model starts from the pristine model with
the interactions instead of the density-wave term, which emerges from
spontaneous symmetry breaking with minimal total energy. For generality and
simplicity, we do not specify an interaction, e.g., electron-electron
interaction or electron-boson coupling, focusing on the favorable density-wave
scenarios from only the electrons’ perspective[40, 41], neglecting or
identifying the (energy) contributions from any auxiliary degrees of freedom.

For an incommensurate $\bm{q}$, the bulk physics of $\hat{H}_{2D,\phi}$ is
independent of $\phi=k_{z}$, which is sometimes denoted as a ‘sliding
symmetry.’[42, 32] Therefore, $\hat{H}_{2D,\phi}$ is equivalent to
$\hat{H}_{3D}$ upto a constant factor of $N_{z}$, the number of sites of the
3D system in the $\hat{z}$ direction, and we can analyze the former with the
help of the latter (or vice versa). We emphasize that such a mapping seems
between a 2D system and a “$k_{z}$-slice” of a 3D system for an incommensurate
$\bm{q}$; however, as we analyze the 3D system in the absence of the magnetic
field, all $k_{z}$ values and the entirety of the 3D system matter. The above
equivalence relation offers a clear physical picture when the zero-field
electronic structure around the Fermi energy takes the form of NLs in the 3D
system, which we will discuss next.

Figure 1: (a) The 3D model in Eq. 5 possesses two overlapping nodal lines in
the $k_{x}$-$k_{y}$ planes at $k_{z}=\pm\frac{\pi}{2}$ for
$h(\bm{k}_{xy})=2t[\cos(k_{x})+\cos(k_{y})]$,
$h^{\prime}(k_{z})=2t_{\hat{z}}\cos(k_{z})$, $h(k_{z})=\epsilon_{0}=0$,
$\epsilon_{1}=-1$, $t=1$, $t_{\hat{z}}=1$. In a magnetic field along the
$\hat{y}$ direction, $k_{y}$ is a good quantum number that decouples the
system into cross-sections intersecting the nodal lines, as the dashed lines.
(b) The low-energy massless or massive Dirac fermions in the $k_{x}$-$k_{z}$
plane for different $k_{y}$ values. The colors and labels highlight the
correspondence between (a) and (b).

###  II.2 Dirac-fermion LLs for a NL system

Let’s consider a class of 3D nodal line models in the presence of a magnetic
field:

$\begin{split}&\hat{H}_{3D}=\hat{H}_{xy}+\hat{H}_{z}+\hat{H}_{\epsilon}\\\
&\hat{H}_{xy}=\sum_{\bm{r},\bm{r}_{xy}}t_{r_{xy}}e^{iA(\bm{r},\bm{r}+\bm{r}_{xy})}\hat{c}^{\dagger}_{r,s}\sigma^{l}_{s,s^{\prime}}\hat{c}_{r+r_{xy},s^{\prime}}+h.c.\\\
&\hat{H}_{z}=\sum_{\bm{r},\bm{r}_{z},p\in\\{l,m\\}}t_{r_{z}}e^{iA(\bm{r},\bm{r}+\bm{r}_{z})}\hat{c}^{\dagger}_{r,s}\sigma^{p}_{s,s^{\prime}}\hat{c}_{r+r_{z},s^{\prime}}+h.c.\\\
&\hat{H}_{\epsilon}=\sum_{\bm{r}}\hat{c}_{r,s}^{\dagger}(\bm{\epsilon}\cdot\bm{\sigma})_{s,s^{\prime}}\hat{c}_{r,s^{\prime}},\end{split}$
(3)

where $\hat{H}_{xy}$ ($\hat{H}_{z}$) is the intra-layer (inter-layer) hopping
between the $x-y$ planes, $\hat{H}_{\epsilon}$ is the onsite potential and
$\bm{\epsilon}\cdot\bm{\sigma}=\epsilon_{1}\sigma^{l}+\epsilon_{0}\sigma^{m}$,
$\sigma^{l}$ and $\sigma^{m}$ are Pauli matrices on the spin or pseudo-spin
$s,s^{\prime}$, $l,m=x,y,z,l\neq m$. $\bm{A}=(0,0,q_{x}x+q_{y}y)$ is the
magnetic vector potential of a magnetic field $\bm{B}=(q_{y},-q_{x},0)$. In
this gauge, if we only consider the nearest-neighbor hopping in the
$\hat{z}$-direction, the Hamiltonian can be diagonalized in the $k_{z}$ basis
as:

$\begin{split}&\hat{H}_{3D}(k_{z})=\hat{H}_{xy,k_{z}}+\hat{H}_{\epsilon,k_{z}}+\hat{H}_{z,k_{z}}\\\
&\hat{H}_{z,k_{z}}=\sum_{\bm{r}^{\prime},p\in\\{l,m\\}}V(\bm{r}^{\prime},k_{z})\hat{c}^{\dagger}_{r^{\prime},k_{z},s}\sigma^{p}_{s,s^{\prime}}\hat{c}_{r^{\prime},k_{z},s^{\prime}}\\\
&V(\bm{r}^{\prime},k_{z})=2t^{\prime}_{\hat{z}}\cos(\bm{q}\cdot\bm{r}^{\prime}-k_{z})-2t^{\prime\prime}_{\hat{z}}\sin(\bm{q}\cdot\bm{r}^{\prime}-k_{z}),\\\
\end{split}$ (4)

where $t^{\prime}_{\hat{z}}$ ($t^{\prime\prime}_{\hat{z}}$) is the real
(imaginary) part of $t_{\hat{z}}$, and $\bm{r}^{\prime}=(x,y)$.
$\hat{H}_{2D}\propto\hat{H}_{3D}(k_{z})$ reflects a 2D SDW system and is our
focus in this work. To understand the preferential DW wave vector $\bm{q}$ in
such a 2D system, we can analyze the preferential magnetic field amplitude and
direction in the equivalent 3D system in Eq. 3.

Without the magnetic field, the 3D Hamiltonian $H_{3D}$ can be diagonalized in
the $\bm{k}$ space given its fully restored translation symmetry:

$\begin{split}h_{3D}(\bm{k})&=\left[h(\bm{k}_{xy})+h(k_{z})+\epsilon_{1}\right]\sigma^{l}+\left[h^{\prime}(k_{z})+\epsilon_{0}\right]\sigma^{m},\\\
\end{split}$ (5)

where $\bm{k}_{xy}=(k_{x},k_{y})$, and $h({\bm{k}_{xy}})$ represents the in-
plane terms in Eq. 3. The Hamiltonian has nodal lines wherever
$h(\bm{k}_{xy})+h(k_{z})+\epsilon_{1}=0$ and
$h^{\prime}(k_{z})+\epsilon_{0}=0$, which is illustrated in Fig. 1a. We note
that all the nodes on the nodal line are at the same energy in this specific
model example, which simplifies our upcoming discussion but can be relaxed to
some extent as long as there is little mixing between the zeroth and higher
LLs once the magnetic field is present.

In the presence of the magnetic field $\bm{B}$, the momentum $k_{\parallel}$
parallel to the magnetic field is a good quantum number that labels the
different perpendicular cross-sections. Each cross-section may possess pairs
of Dirac nodes, as shown in Fig. 1b, which develop into discrete LLs
$\epsilon_{n}\propto\pm\sqrt{nB},n=0,1,2\dots$ in the presence of the magnetic
field - the zeroth Landau level exists at the energy of the Dirac nodes, while
the rest of the LLs are either above or below with a gap $\propto\sqrt{B}$.
Summing over $k_{\parallel}$, we obtain a large zeroth LL degeneracy
proportional to the number of Dirac nodes intersected - the NLs’ projection
along $k_{\parallel}$, see Fig. 2. The counting works for both strong (large
$t_{\hat{z}}$) and weak (small $t_{\hat{z}}$) DWs.

###  II.3 Magnetic field for optimal filling

For a system with a fixed electron density $n_{e}=1+\delta n_{e}$, $\delta
n_{e}\ll 1$, the optimal filling is to fill the zeroth LLs and leave all the
higher LLs empty. When $|\bm{B}|<|\bm{B}|_{opt}$, the electrons will be forced
into the higher LLs, leading to an excitation in an incompressible system and
an increase in the systematic energy; when $|\bm{B}|>|\bm{B}|_{opt}$, on the
other hand, while the zeroth LLs fully accommodating the electrons above half-
filling provide no further energy reduction, the Fermi sea sees an
uncompensated energy rise due to the larger magnetic field. Such energy
dependence versus the external magnetic field or LL filling constitutes the
premise of quantum oscillations [43, 44], e.g., the dHvA effect. Therefore,
quantitatively, the electron density above half-filling should match half of
the zeroth LL degeneracy:

$\begin{split}(n_{e}-1)L_{\parallel}S_{\perp}&=\frac{\int
n_{D}(k_{\parallel})dk_{\parallel}}{2\pi/L_{\parallel}}\frac{|\bm{q}_{opt}|S_{\perp}}{2\pi}\frac{1}{2}\\\
\Rightarrow n_{e}-1&=\frac{\bar{n}_{D}}{2}\frac{|\bm{q}_{opt}|}{2\pi},\\\
\end{split}$ (6)

where $L_{\parallel}$ ($S_{\perp}$) is the length (area) of the system
parallel (perpendicular) to the magnetic field, $|\bm{B}|/2\pi=|\bm{q}|/2\pi$
is the LL degeneracy over unit space, $n_{D}(k_{\parallel})$ denotes the
number of Dirac nodes in the cross-section at $k_{\parallel}$, and
$\bar{n}_{D}=\int n_{D}(k_{\parallel})dk_{\parallel}/2\pi$ is its average over
all $k_{\parallel}$. With $\bar{n}_{D}$, the expression for optimal
$|\bm{q}_{opt}|$ resembles the one-dimensional case in Ref. [45], yet
$\bar{n}_{D}$ is a continuous variable instead of an integer just like
$n_{D}(k_{\parallel})$.

Figure 2: The optimal direction of the magnetic field $\bm{B}$ for an
anisotropic nodal line in the $k_{x}$-$k_{y}$ plane is to maximize the range
$|\bm{k}_{1}-\bm{k}_{2}|$ parallel to $\bm{B}$, which yields the largest
$\bar{n}_{D}=|\bm{k}_{1}-\bm{k}_{2}|/\pi$.

A similar analysis yields the favorable $\bm{q}\equiv\bm{B}$ direction: we
note that under the circumstance of the filling of the zeroth Landau level,
$\bm{B}$, and thus the energy penalty to the Fermi sea is minimal when
$\bar{n}_{D}$ is maximum, which depends on the geometry of the nodal lines and
favors the direction with the largest projection. In Fig. 2, we illustrate an
example of a simple-loop NL and the direction to maximize the span of the
projection $|\bm{k}_{1}-\bm{k}_{2}|$ parallel to $\bm{B}$ and thus
$\bar{n}_{D}=|\bm{k}_{1}-\bm{k}_{2}|/\pi$.

###  II.4 Our approximations

Our arguments are valid when the magnetic field is small enough to treat the
Dirac nodes of the same $k_{\parallel}$ independently. Otherwise, quantum
tunneling kicks in and gaps the Dirac nodes out, which happens if the
separation between them $|\Delta{k}_{\perp}|\lesssim|l_{B}^{-1}|$, where
$l_{B}=\sqrt{\hbar/eB}$ is the magnetic length [46, 47]. Those zeroth LLs
split beyond the nonzero LLs should not count in Eq. 6. As we can see in Fig.
1a, there are always regions where pairs of Dirac nodes get arbitrarily close
and violate the condition $|\Delta{k}_{\perp}|\gg|l_{B}^{-1}|$. Fortunately,
the splitting $\propto B$ is generally smaller than the spacing between the
zeroth and nonzero LLs $\propto\sqrt{B}$ for small $B$.

On the other hand, even the Dirac nodes have pairwise-annihilated and
developed a mass, as the points $i$ and $j$ in Fig. 1b, the original zeroth
LLs may have not yet shifted beyond the first LLs, leading to an
underestimation (over-estimation) of $\bar{n}_{3D}$ ($\bm{q}_{opt}$) following
Eq. 6. We find the latter effect more dominant in our examples and provide a
detailed analysis of these effects in the Appendix A.

Also, we have assumed that the energy of the Fermi sea depends monotonically
on the strength of the magnetic field and is insensitive to its direction; see
supporting materials in the Appendix B.

Figure 3: (a) The dispersion of $\hat{H}_{0}$ in Eq. III.1 shows shows a
$4\epsilon_{0}$ gap between the two bands. $t=1$ ,$t^{\prime}=0.1$,
$\epsilon_{0}=0.5$, $\epsilon_{1}=-2.15$. (b) The Fermi surface of (a) at
Fermi energy $E_{F}=1.3$ slightly above half filing shows no sign of FSN.

##  III Numerical results

###  III.1 2D model example

For demonstration purpose, let’s consider a specific effective SDW model
$\hat{H}_{2D}=\hat{H}_{0}+\hat{H}_{DW}$ which comes from Eq.4 as follows:

$\displaystyle\hat{H}_{0}$ $\displaystyle=$
$\displaystyle\sum_{\bm{r},\bm{\delta},s,s^{\prime}}t_{\delta}\hat{c}_{r,s}^{\dagger}\sigma_{s,s^{\prime}}^{x}\hat{c}_{r+\delta,s^{\prime}}+\hat{c}_{r,s}^{\dagger}(\bm{\epsilon}\cdot\bm{\sigma})_{s,s^{\prime}}\hat{c}_{r,s^{\prime}}+h.c.$
$\displaystyle\hat{H}_{DW}$ $\displaystyle=$
$\displaystyle-\sum_{\bm{r},s,s^{\prime}}[2V\cos(\bm{q}\cdot\bm{r}+\phi_{0})\hat{c}_{r,s}^{\dagger}\sigma_{s,s^{\prime}}^{z}\hat{c}_{r,s^{\prime}}$
$\displaystyle+2\lambda\sin(\bm{q}\cdot\bm{r}+\phi_{0})\hat{c}_{r,s}^{\dagger}\sigma_{s,s^{\prime}}^{x}\hat{c}_{r,s^{\prime}}],$

where $\bm{\sigma}=(\sigma^{x},\sigma^{y},\sigma^{z})$ are the Pauli matrices,
$\bm{\epsilon}=(\epsilon_{1},0,\epsilon_{0})$ are the onsite potentials, and
$t_{\delta}=t$ for $\bm{\delta}=\hat{x},\hat{y}$ and $t_{\delta}=it^{\prime}$
for $\bm{\delta}=\hat{x}+\hat{y},\hat{x}-\hat{y}$ are the hopping parameters.
The DW term $\hat{H}_{DW}$ is similar to the spiral SDW proposed by
Overhauser[23]. The dispersion of the translation invariant Hamiltonian
$\hat{H}_{0}$:

$E_{k}^{0}=\pm
2\sqrt{[\epsilon_{1}+t(\cos{k_{x}}+\cos{k_{y}})+2t^{\prime}\sin{k_{x}}\cos{k_{y}}]^{2}+\epsilon_{0}^{2}},$
(8)

is shown in Fig. 3a, where the Fermi surface slightly above half-filling, as
shown in Fig. 3b, is rather circular and shows no obvious FSN. However, we
will show that to minimize energy, the model prefers a DW, characterized by
the term $H_{DW}$, with a preferential wave vector $|\bm{q}_{opt}|$ that shows
a continuous range.

Figure 4: The nodal lines of the 3D models in Eq. 9 in the $k_{x}-k_{y}$
planes at $k_{z}=-\frac{\pi}{3}$ for (a) $\lambda=0.8$ and (b) $\lambda=2.5$
close to a Lifshitz transition. $t=1$, $t^{\prime}=0.1$, $\epsilon_{0}=0.5$,
$\epsilon_{1}=-2.15$ as in Fig. 3, and $V=1$. The double-sided arrows denote
the spans of the nodal lines projected along different directions of the
magnetic field $\ bm{B}$. Figure 5: The DOS of the 2D system $H_{2D}$ in Eq.
III.1 with a DW wave vector $\bm{q}=(0.1,0)$ shows a peak around zero energy
theoretically attributed to the zeroth LLs of the 3D NL systems in Eq. 9 in
the presence of a magnetic field. The rest of the parameters are the same as
in Fig. 4. The integrated DOS in (b) is larger given the extent of its NLs.
Also, we see signatures of the zeroth LLs’ splittings, which remain small
compared to the gaps and keep the validity of our argument thanks to the
smallness of the magnetic field.

###  III.2 3D nodal-line system and LLs

For an incommensurate $\bm{q}$, the 2D model in Eq. III.1 is equivalent to a
3D system with a magnetic field. Without the magnetic field, the 3D system
$(\hat{a}_{k,\uparrow}^{\dagger},\hat{a}_{k,\downarrow}^{\dagger})h_{3D}(\bm{k})(\hat{a}_{k,\uparrow},\hat{a}_{k,\downarrow})^{T}$
in the momentum space:

$\begin{split}&h_{3D}(\bm{k})=[2\epsilon_{1}-2\lambda\sin k_{z}+2t(\cos
k_{x}+\cos k_{y})\\\ &+4t^{\prime}\sin k_{x}\cos
k_{y}]\sigma^{x}+[2\epsilon_{0}-2V\cos k_{z}]\sigma^{z},\\\ \end{split}$ (9)

may possess NLs where the $\sigma^{z}$ coefficient in Eq. 9 vanishes on the
$k_{z}=\pm\arccos(\epsilon_{0}/V)$ planes. We show a couple of examples in
Fig. 4.

In the presence of the magnetic field, the Dirac nodes along the nodal lines
exhibit themselves as Dirac-fermion LLs. While the $n\neq 0$ LLs depend on
Fermi-velocity details, the zeroth Landau levels remain (nearly) degenerate at
zero energy and are separated from the rest of the Landau bands with large
gaps due to the LL spacings. For example, we show in Fig. 5 the density of
states (DOS) of the models with the NLs in Fig. 4 in a magnetic field
$|\bm{q}|\ll 2\pi$, where the contributions from the zeroth LLs are clearly
visible between the red dashed lines.

###  III.3 Optimal DW wave vectors

We calculate the energy of $\hat{H}_{2D}$ in Eq. III.1 numerically via exact
diagonalization on system size $L_{x}=1000$ along $\bm{q}$ and $L_{\perp}=200$
values of $k_{\perp}$ in the perpendicular direction. We discuss our setup for
$\bm{q}$ along directions other than $\hat{x}$ and $\hat{y}$ in the Appendix
E. The results on the average energy per electron $\bar{E}$ versus the DW wave
vector $\bm{q}$ for the model parameters in Figs. 4a and 5a is shown in Fig.
6. The optimal wave vector $|\bm{q}_{opt}|\sim 0.15$ is approximately
consistent with our theoretical expectation of $|\bm{q}_{theory}|\sim 0.17$
according to Eq. 6. In addition, the DWs in the $\hat{q}=\hat{x},\hat{y}$
directions yield the lowest energy overall with negligible difference,
consistent with our analysis and the fact that the NLs projection
$|\bm{k}_{1}-\bm{k}_{2}|$ and subsequently $\bar{n}_{D}$ are largest along
these two directions. In comparison, the average electron energy without the
DW $\hat{H}_{DW}$ is $\Delta\bar{E}_{0}\sim 0.29$ above these minimums,
suggesting that the DW formation is indeed favorable energetically.

Figure 6: The (relative) average energy per electron ($E_{r}$) $\bar{E}$ for
the model in Eq. III.1 shows minimums at different optimal $|\bm{q}_{opt}|$
along different directions. The parameters are the same as in Figs. 4a and 5a:
$t=1$, $t^{\prime}=0.1$, $\epsilon_{0}=0.5$, $\epsilon_{1}=-2.15$, $V=1$,
$\lambda=0.8$, and we set $n_{e}=1.01$ slightly above half filling. The system
has a $y\rightarrow-y$ reflection symmetry thus
$\hat{q}=(\hat{x}\pm\hat{y})/\sqrt{2}$ directions are equivalent.

Also, the dependence of $\bar{E}$ with $|\bm{q}|$ is consistent with our
expectation: the optimal $|\bm{q}_{opt}|$ allows all the electrons above
charge neutrality to be accommodated by the degenerate zeroth LLs; when
$0<|\bm{q}|<|\bm{q}_{opt}|$, the filling goes to some of the higher LLs
leading to higher energy; when $|\bm{q}|>|\bm{q}_{opt}|$, on the other hand,
the electron Fermi sea suffers an energy penalty due to the higher magnetic
field. We discuss more detailed results and analysis on energetics in the
Appendix B.

Figure 7: The preferential DW wave vector $\bm{q}_{opt}$ with the minimal
energy shows good consistency between the numerical results (dashed lines) on
the 2D model in Eq. III.1 and the theory expectations (solid lines) from Eq. 6
and the geometry of the NLs versus the DW strength $\lambda$. The former is
obtained by comparing the energy for different $q$ as in Fig. 6. Different
colors denote results for $\bm{q}$ along the $\hat{x}$ and the
$(\hat{x}+\hat{y})/\sqrt{2}$ directions, and the inset shows the energy
difference $\Delta{E}=E_{\frac{1}{\sqrt{2}}(1,1)}-E_{(1,0)}$ between the two
directions at their respective $\bm{q}_{opt}$. We keep parameters $t=1$,
$t^{\prime}=0.1$, $\epsilon_{0}=0.5$, $\epsilon_{1}=-2.15$, $V=1$, and
$n_{e}=1.01$ as before while varying $\lambda$.

More generally, the value of $\bm{q}_{opt}$ versus the DW amplitude $\lambda$
as a tuning parameter is summarized in Fig. 7, and we observe a good
consistency between our analysis based on Eq. 6 and the numerical benchmark
for both $\bm{q}$ along the $(1,0)$ and the $(1,1)/\sqrt{2}$ directions.
Intuitively, when $\lambda$ is small, the nodal line is a closed contour and
grows with $\lambda$, leading to an increasing $\bar{n}_{D}$ and decreasing
$\bm{q}_{opt}$. On the contrary, as $\lambda$ increases further and beyond the
Van Hove singularities, see Fig. 4b, the NLs’ extent and thus $\bar{n}_{D}$
reverses its trend and decreases until it vanishes, giving rise to a
monotonically increasing $|\bm{q}_{opt}|$. The slight differences between
numerics and theory are likely due to the neglection of massive Dirac fermion
Landau levels (Fig. 1b) that underestimates $\bar{n}_{D}$ (overestimates
$\bm{q}_{opt}$), see more related discussions in the Appendix A. In addition,
the optimal $|\bm{q}_{opt}|$ is smaller for $\bm{q}$ along the $(1,0)$
direction until large $\lambda\sim 3.5$, suggesting that qualitatively, so
should the DW along $\hat{x}$ be the winner with lower energy, which is indeed
the case as we compare the two in the inset of Fig. 7. We also illustrate
another example with multiple nodal lines from multiple $k_{z}$ planes in the
Appendix C. Overall, we note that the DWs wave vectors change continuously
with the DW parameters, in sharp contrast to the behaviors descending from the
FSN or exciton condensation, where the DW wave vector is generally fixed by
the Fermi surface geometry.

##  IV Conclusions

In summary, we put forward a novel perspective to understand the DW tendency
in 2D systems from the energetics of a 3D NL system. Interestingly, in
scenarios where DW sees a clear FSN origin, our perspective also offers a
consistent explanation; see the Appendix D. On the other hand, our setup goes
beyond the Peierls transition as it does not require any apparent FSN to begin
with. Correspondingly, the optimal SDW wave vector depends on the geometry of
the 3D NLs and may vary continuously with respect to the DW parameters without
changes to the Fermi surface and dispersion. Also, our numerical results on
benchmark models fit with our analysis consistently. Such continuous
variations of the DW wave vectors are present in materials such as various
cuprates [27, 28, 29, 30], $2H$-Ta${\mathrm{Se}}_{2}$[33, 34] etc. While we do
not intend to relate our analysis to these materials directly, our perspective
kindles the theoretical possibility of a variable DW wave vector from more
generic origins.

On the other hand, our study points out that our current understanding of the
DW origin is still primitive and a universal understanding is not yet
available. Our current study has focused on models with rather specific
constructions. While such a setup facilitates the theoretical analysis and its
controllability, it also limits the generalization of the mechanism and its
application in practice. It will be interesting to probe the generalization of
the current mechanism beyond its model limits and its connection with FSN and
other DW mechanisms in interpolating models for further physical intuition.

Finally, we note that our theory offers a bridge between density waves and
topological nodal-line semi-metal. It is also interesting to explore the
topological property of the density wave system via the dimension-extension
perspective [48].

##  V Acknowledgement

We thank Xin-chi Zhou and Di-zhao Zhu for insightful discussions. The authors
are supported by the National Science Foundation of China(No.12174008) and the
start-up grant at Peking University. The calculations of this work is
supported by HPC facilities at Peking University.

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##  VI Appendix A: The independent-Dirac-node approximation

In the main text, we regard the Dirac nodes as independent for our
derivations. However, quantum tunneling exists between the Dirac nodes,
especially in the presence of a larger magnetic field. Here, we perform a
relatively quantitative study on the potential impact of the Dirac-node pair
separation and the magnetic field strength.

We consider the following two-bands model to account for the low-energy
scenarios at different $k_{\parallel}$ cross-sections in Fig. 1 in the main
text:

$\hat{H}_{D}=(m-k_{x}^{2})\sigma^{x}+k_{y}\sigma^{y},$ (10)

where $m$ is a parameter controlling whether the model has two massless Dirac-
fermions at $(\pm\sqrt{m},0)$ ($m>0$) or massive Dirac-fermions with a mass
$\sim|m|$ ($m<0$).

###  VI.1 1\. Dirac-node pair separation

First, we consider the $m>0$ case similar to the specific $k_{\parallel}$
cross-sections intersecting the nodal line in Fig. 1 in the main text.
Different $m$ corresponds to different Dirac-node separations, which affect
our theory’s accuracy.

We can focus on one of the Dirac node $\chi=\pm 1$ by
$k_{x}^{\prime}=k_{x}-\chi\sqrt{m},k_{y}^{\prime}=k_{y}$, and the
corresponding model becomes:

$\hat{H}_{\chi}=v_{x}k_{x}^{\prime}\sigma^{x}+v_{y}k_{y}^{\prime}\sigma^{y}-k_{x}^{\prime
2}\sigma^{x},$ (11)

where the Fermi velocities are $v_{x}=-2\chi\sqrt{m}$ and $v_{y}=1$. When we
apply a magnetic field $\bm{B}=(0,0,B)$ with the gauge $\bm{A}=(0,Bx,0)$, we
can re-express $k_{x}^{\prime}$ and $k_{y}^{\prime}$ in terms of ladder
operators $\hat{a}$ and $\hat{a}^{\dagger}$:

$\begin{split}&k_{x}^{\prime}=\sqrt{\frac{v_{y}}{|v_{x}|}}\frac{1}{\sqrt{2}l_{B}}(\hat{a}+\hat{a}^{\dagger}),\\\
&k_{y}^{\prime}+\frac{eBx}{\hbar}=\sqrt{\frac{|v_{x}|}{v_{y}}}\frac{1}{\sqrt{2}l_{B}}i(\hat{a}-\hat{a}^{\dagger}),\\\
\end{split}$ (12)

where $l_{B}=\sqrt{\frac{\hbar}{eB}}$ is the magnetic length. For instance,
the Hamiltonian for $\chi=+1$ becomes:

$\hat{H}_{+1}=-\frac{\sqrt{2|v|_{x}v_{y}}}{l_{B}}(\hat{a}\sigma^{-}+\hat{a}^{\dagger}\sigma^{+})-\frac{v_{y}|v_{x}|}{4ml_{B}^{2}}(\hat{a}+\hat{a}^{\dagger})^{2}\sigma^{x}.$
(13)

The first term on the right-hand side corresponds to the Dirac-fermion Landau
levels (LLs) of an independent Dirac node and possesses a zeroth LL at
precisely zero energy. The second term captures the tunneling from/to the
other Dirac node ($\chi=-1$) and introduces hybridization between the LLs.
When the separation surpasses the inverse magnetic length $l_{B}^{-1}\ll\Delta
k_{\perp}=2\sqrt{m}$, the influence of the second term is negligible. However,
when $l_{B}^{-1}\sim\Delta k_{\perp}$, we can no longer treat the two Dirac
nodes independently, and the second term becomes non-perturbative; their
zeroth Landau levels interact and split [46, 47]. Fig. 8 illustrates the
zeroth LL’s drift from zero energy as the magnetic field increases. A larger
separation ($m$) protects the independent-Dirac-node approximation until a
larger magnetic field $B$.

Figure 8: The zeroth LL drifts away from zero energy as the magnetic field
perpendicular to the Dirac-node separation increases, especially when the
inverse magnetic length $l_{B}^{-1}\sim\Delta k_{\perp}=2\sqrt{m}$. For
instance $\Delta k_{\perp}\approx 1.26$ for $m=0.4$, and a visible energy
drift appears at $B\sim 0.1$, $l_{B}^{-1}\approx 0.25\Delta k_{\perp}$.

For a closed nodal line, e.g., the one in Fig. 1 in the main text, there
always exist pairs of Dirac nodes with separation $\Delta k_{\perp}\sim
l_{B}^{-1}$ on cross-sections close to the turning points regardless of the
magnetic field. These Dirac nodes gap out in pairs leading to an overestimate
of the zeroth LL density and underestimate of the density wave vector. Still,
the less the magnetic field, the fewer corners we cut (metaphorically and
literally), and the more robust the approximations.

###  VI.2 2\. Massive Dirac fermions

In the main text, we focus solely on the massless Dirac fermions in our
formulations. However, when the mass is small, the massive Dirac fermion may
behave like a massless one and contribute to the zero-energy density peak,
thus the optimal wave vector.

In this subsection, we focus on the influence of massive Dirac fermions, and
consider the model in Eq. 10 with $m<0$. In the presence of a magnetic field
$\bm{B}=(0,0,B)$ with gauge $\bm{A}=(0,Bx,0)$, we can re-express the momentum
in terms of ladder operators $\hat{a}$ and $\hat{a}^{\dagger}$ as:

$\begin{split}&k_{x}=\frac{1}{\sqrt{2}l_{B}}(\hat{a}+\hat{a}^{\dagger}),\\\
&k_{y}+\frac{eBx}{\hbar}=i\frac{1}{\sqrt{2}l_{B}}(\hat{a}-\hat{a}^{\dagger}),\\\
\end{split}$ (14)

and the Hamiltonian as:

$\hat{H}_{D^{\prime}}=\\{m-[\frac{1}{\sqrt{2}l_{B}}(\hat{a}+\hat{a}^{\dagger})]^{2}\\}\sigma^{x}+i\frac{1}{\sqrt{2}l_{B}}(\hat{a}-\hat{a}^{\dagger})\sigma^{y}.$
(15)

Figure 9: Selected LLs of of Eq. 15 versus the magnetic field $B$ for massless
($m=0.01$) and massive ($m=-0.01$, $m=-0.25$) Dirac fermions dictate the
possible contributions to the DOS around zero energy.

We illustrate the low-lying LLs with selected model parameters in Fig. 9. The
results indicate that even when the Dirac nodes have just annihilated in pairs
($m<0$), the resulting LLs may still be relatively close to zero energy
depending on the mass $m$. For instance, the zeroth LLs for $m=-0.01$ in Fig.
9 differs little from that for $m=0.01$. By counting only the nodal lines, we
may underestimate the contribution from such (slightly) massive Dirac-fermion
LLs to zero-energy density peak, thus the optimal wave vector.

Still, the nodal lines offer a generally good starting point, as the zeroth
LLs for most massive fermion LLs, e.g., $m=-0.25$, and the nonzero LLs, are
too far from zero energy to contribute; see Fig. 9.

##  VII Appendix B: Energy cost of magnetic field on the Fermi sea

We calculate the energy of the Fermi sea, i.e., the average energy of the
electrons at half-filling $n_{e}=1$ for the model in Figs. 4a and 5a in the
main text in a magnetic field $B$, and summarize the results in Fig. 10. The
energy increases monotonically with respect to the amplitude of the magnetic
field (DW wave vector) and is relatively insensitive to the direction of the
magnetic field (DW wave vector). Therefore, we cannot arbitrarily increase
$|\bm{q}_{opt}|$ after reaching the complete filling of zeroth LLs due to the
associated energy cost.

Figure 10: The (relative) average energy ($E_{r}$) $E_{0}$ of the electrons in
the Fermi sea at half-filling $n_{e}=1$ depends monotonically on the largeness
of $\bm{q}$ yet little on the direction of $\bm{q}$. We have applied the same
parameter setting as in Figs. 4a and 5a in the main text.

From a phenomenological perspective, we may attribute the energy cost of the
magnetic field on the Fermi sea to the orbital diamagnetic effect of the
electrons. Otherwise, i.e., a small magnetization reduces the energy, the
Fermi sea is unstable toward spontaneous magnetization, contradictory with the
ground state precondition. Such an argument is valid within the limit of a
small magnetic field where we mainly focus.

##  VIII Appendix C: Example with multiple nodal lines

For simplicity, we focus our discussions in the main text on scenarios with a
nodal line in a single $k_{z}$-plane. It is straightforward to generalize our
conclusions to multiple nodal lines and over multiple $k_{z}$-plane, where all
contributions add to the zeroth LLs’ degeneracy and the eventual optimal wave
vector.

Here, we illustrate such an example. Let’s consider the model in Eq. 9 in the
main text:

$\begin{split}&h_{3D}(\bm{k})=[2\epsilon_{1}-2\lambda\sin k_{z}+2t(\cos
k_{x}+\cos k_{y})\\\ &+4t^{\prime}\sin k_{x}\cos
k_{y}]\sigma^{x}+[2\epsilon_{0}-2V\cos k_{z}]\sigma^{z}.\\\ \end{split}$ (16)

The model may possess two nodal lines as illustrated in Fig. 11.

Figure 11: The model in Eq. 16 possesses two nodal lines in the
$k_{z}=\frac{\pi}{3}$ (blue) and $k_{z}=-\frac{\pi}{3}$ (green) planes,
respectively, for parameters $t=1$, $t^{\prime}=0.1$, $\epsilon_{0}=0.5$,
$\epsilon_{1}=-0.5$, $V=1$, (a) $\lambda=0.8$, and (b) $\lambda=1.6$. As
$\lambda$ grows, the inner nodal line (blue) shrinks while the outer nodal
line (green) expands to the corners till it vanishes. Figure 12: The
theoretical optimal wave vector $\bm{q}_{opt}$ and the numerical results
compare relatively well and exhibit a consistent tendency over a wide range of
$\lambda$ parameter range. We carry out numerical calculations on 2D system of
size $L_{x}=1000$ and $L_{y}=200$.

According to our theoretical analysis, the optimal wave vector should follow
Eq. 6 in the main text:

$n_{e}-1=\frac{\bar{n}_{D}}{2}\frac{|\bm{q}_{opt}|}{2\pi},$ (17)

whose $\bar{n}_{D}$ incorporates both nodal lines from the two different
$k_{z}$ planes. The nodal lines possess the largest overall projection along
the $\hat{y}$ direction, predicting an optimal magnetic field (wave vector)
along the $\hat{B}=(0,1)$ ($\hat{q}=(1,0)$) direction. We compare the expected
optimal wave vector $\bm{q}_{opt}$ from Eq. 17 and the numerical results for
various values of $\lambda$ in Fig. 12, which shows overall consistency.

##  IX Appendix D: Consistency with the Fermi surface nesting

In the main text, we show that our perspective tackles density waves and their
optimal wave vectors beyond the Fermi surface nesting (FSN). An interesting
question is whether our perspective works for scenarios where FSN is still
applicable. In this subsection, we show an example where our theory is
consistent with FSN. For instance, let’s consider the quasi-1D limit of the
model in Eq. 7 in the main text:

$\begin{split}\hat{H}_{0}&=\sum_{\bm{r},\bm{\delta}}t_{\delta}\hat{c}_{r,s}^{\dagger}\sigma_{s,s^{\prime}}^{x}\hat{c}_{r+\delta,s^{\prime}}+\hat{c}_{r,s}^{\dagger}(\bm{\epsilon}\cdot\bm{\sigma})_{s,s^{\prime}}\hat{c}_{r,s^{\prime}}+\mbox{h.c.},\\\
\hat{H}_{DW}&=-\sum_{\bm{r},s,s^{\prime}}[2V\cos(\bm{q}\cdot\bm{r}+\phi_{0})\hat{c}_{r,s}^{\dagger}\sigma_{s,s^{\prime}}^{z}\hat{c}_{r,s^{\prime}}\\\
&+2\lambda\sin(\bm{q}\cdot\bm{r}+\phi_{0})\hat{c}_{r,s}^{\dagger}\sigma_{s,s^{\prime}}^{x}\hat{c}_{r,s^{\prime}}],\\\
\end{split}$ (18)

where $\bm{\sigma}=(\sigma^{x},\sigma^{y},\sigma^{z})$ are the Pauli matrices,
$\bm{\epsilon}=(\epsilon_{1},0,\epsilon_{0})$ are the onsite potentials,
$t_{\delta}=t,0$ for $\bm{\delta}=\hat{x},\hat{y}$, and
$t_{\delta}=it^{\prime}$ for $\bm{\delta}=\hat{x}+\hat{y},\hat{x}-\hat{y}$.

The Fermi surface of $\hat{H}_{0}$ in the $\bm{k}$-space:

$h_{0}(\bm{k})=[2\epsilon_{1}+2t\cos(k_{x})+4t^{\prime}\cos(k_{x})\sin(k_{y})]\sigma^{x}+2\epsilon_{0}\sigma^{z},$
(19)

exhibits nearly perfect FSN between the two pieces as illustrated in Fig. 13a,
indicating a density-wave tendency with a preferential wave vector
$\bm{q}_{DW}$:

$n_{e}\approx 1+\frac{|\bm{q}_{DW}|}{2\pi}.$ (20)

Figure 13: (a) The Fermi surface of $h_{0}(\bm{k})$, $\hat{H}_{0}$ in the
$\bm{k}$-space, shows a nearly perfect nesting wave vector $\bm{q}_{DW}$. (b)
The nodal lines of $h_{3D}(\bm{k})$ prefers the optimal magnetic field
$\bm{B}_{DW}$ along the $\hat{y}$ direction. Here, $t=1$, $t^{\prime}=0.2$,
$\epsilon_{1}=-2.15$, $\epsilon_{0}$, $V=1$, $\lambda=2$, $n_{e}=1.01$

.

On the other hand, as we map the entire 2D system $\hat{H}_{0}+\hat{H}_{DW}$
to 3D, the 3D model takes the form in the absence of a magnetic field:

$\begin{split}h_{3D}(\bm{k})&=[2\epsilon_{1}-2\lambda\sin(k_{z})+2t\cos(k_{x})+\\\
&4t^{\prime}\cos(k_{x})\sin(k_{y})]\sigma^{x}+[2\epsilon_{0}-2V\cos(k_{z})]\sigma^{z},\\\
\end{split}$ (21)

whose nodal lines are shown in Fig. 13b. The optimal magnetic field
$\bm{B}_{DW}$ favors the direction with the largest nodal-line projection,
which is the $\hat{y}$ direction in this model. Further, we use Eq. 17 (Eq. 6
in the main text) to derive the optimal density wave vector:

$\begin{split}(n_{e}-1)L_{\parallel}S_{\perp}&=\frac{\int
n(k_{\parallel})dk_{\parallel}}{2\pi/L_{\parallel}}\frac{|\bm{q}_{opt}|}{2\pi}\frac{S_{\perp}}{2}=\frac{4\pi}{2\pi/L_{\parallel}}\frac{|\bm{q}_{opt}|}{2\pi}\frac{S_{\perp}}{2}\\\
&\Rightarrow|\bm{q}_{opt}|=2\pi(n_{e}-1).\\\ \end{split}$ (22)

It is straightforward to see the consistency between $\bm{q}_{DW}$ from FSN
and $\bm{q}_{opt}$ from our novel theoretical perspective:

$|\bm{q}_{opt}|\approx|\bm{q}_{DW}|,$ (23)

and both along the $\hat{x}$ direction. Our theory is capable of incorporating
as well as going beyond the conventional FSN[31].

##  X Appendix E: Numerical method for systems with a tilted magnetic field

Generally, density waves may arise in any direction, and we need to consider
the case where $\bm{q}=(q_{x},q_{y})$ is tilted from the $\hat{x}$ or
$\hat{y}$ directions for the most energetically favorable condition. Here is
the method we used to compute the dispersion of the model in Eq. 7 of the main
text with $\bm{q}$ in any direction.

While our theoretical analysis based upon 3D NLs is a low-energy effective
theory that can apply to $\bm{q}$ along any direction, the numerical
calculations require that $\bm{q}=(q_{x},q_{y})$ points to a commensurate
direction, $\frac{q_{x}}{q_{y}}=\frac{m}{n},m,n\in\mathbb{Z}$, $gcd(m,n)=1$.
Without loss of generality, we limit ourselves to:

$\bm{q}=|q_{0}|(\frac{1}{\sqrt{1+p^{2}}},\frac{p}{\sqrt{1+p^{2}}}),$ (24)

for simplicity, so that there will not be additional sublattices. Here,
$|q_{0}|$ is the norm of the wave vector, and $p\in\mathbb{Z}$ determines the
direction of $\bm{q}$.

This wave vector in the 2D DW system corresponds to a magnetic field
$\bm{B}=|q_{0}|(\frac{p}{\sqrt{1+p^{2}}},-\frac{1}{\sqrt{1+p^{2}}},0)$ in the
3D system, whose electromagnetic vector potential takes the form of
$\bm{A}=(0,0,\bm{q}\cdot\bm{r})$. Consequently, neither $k_{x}$ nor $k_{y}$ is
good quantum number. Instead, we can take a new basis:

$\begin{split}&\hat{x}^{\prime}=(1,0)\\\
&\hat{y}^{\prime}=(-\frac{p}{\sqrt{1+p^{2}}},\frac{1}{\sqrt{1+p^{2}}}),\end{split}$
(25)

so that there remains translation symmetry along the $\hat{y}^{\prime}$
direction, and $k_{y^{\prime}}$ is a good quantum number, see Fig.14 for
illustration.

Figure 14:  For a DW vector
$\bm{q}=|q_{0}|(\frac{1}{\sqrt{1+3^{2}}},\frac{3}{\sqrt{1+3^{2}}})$ on the 2D
square lattice, we change the basis from $\hat{x},\hat{y}$ to
$\hat{x}^{\prime},\hat{y}^{\prime}$ so that we can utilize the translation
symmetry along the $\hat{y}^{\prime}$ direction.

The Hamiltonian $\hat{H}_{2D}=\hat{H}_{0}+\hat{H}_{DW}$ in the new basis takes
the same form as Eq. 7 in the main text and can be solved in a similar
fashion, with the following changes to the model settings:

$\begin{split}&\bm{\delta}=\hat{x},\hat{y}\Rightarrow\bm{\delta}=\hat{x}^{\prime},p\hat{x}^{\prime}+\hat{y}^{\prime},\\\
&\bm{\delta}=\hat{x}+\hat{y},\hat{x}-\hat{y}\Rightarrow\bm{\delta}=(p+1)\hat{x}^{\prime}+\hat{y}^{\prime},(1-p)\hat{x}^{\prime}-\hat{y}^{\prime},\\\
&\bm{q}\cdot\bm{r}\Rightarrow q_{x}x=\frac{|q_{0}|x}{\sqrt{1+p^{2}}}.\\\
\end{split}$ (26)

Finally, we apply a polynomial fit to the resulting $E_{0}(|\bm{q}|)$ to get
rid of the fluctuations in data, mainly caused by the limited system and step
sizes, for a smoother display.