# Nonlocal optical conductivity of Fermi surface nesting materials

Xiamin Huang1, Xiao Jiang2, Bing Huang2 and Zhou Li1,3 1 Guangdong Aerospace
Information Research Institute, Guangzhou 510535, China  2 Beijing
Computational Science Research Center, Beijing 100193, China  3 University of
Chinese Academy of Sciences, Beijing 100039, China

###### Abstract

We investigate the nonlocal optical conductivity of Fermi surface nesting
materials which support charge density waves or spin density waves. The
nonlocal optical conductivity contains information of correlations in electron
fluids which could not be accessed by standard optical probes. Half metal
emerges from doping a charge density wave and similarly spin-valley half metal
emerges from doping a spin density wave. Based on the parabolic band
approximation, we find the Drude peak is shifted to higher frequency and
splits into two peaks in the nonlocal optical conductivity. We attribute this
to the two Fermi velocities in the half-metal or spin-valley half metal
states.

##  I Introduction

The scattering-type scanning near-field optical microscope (s-SNOM) is
frequently used to probe the response pattern of a specific material (e.g.
graphene) [1, 2, 3] driven by electromagnetic waves oscillating in both time
and space. The spatial dispersion in the response pattern is characterized by
the nonlocal optical conductivity in close connection to the complicated
structure of Fermi surface and electron correlations [4]. These short-range
effects are washed out by the standard far-field optical probe, which only
measures the optical conductivity in the long wavelength limit. The tip radius
$a$ is about 25 nm [1, 2, 3] for the atomic force microscope operating in the
tapping mode of s-SNOM. Thus the wave vector $q$ is in the order of
$1/a=0.04\mathrm{nm}^{-1}$.

The nonlocal optical conductivity is crucial for plasmons (collective
oscillations of electrons). The imaginary part of the nonlocal optical
conductivity determines the dispersion of plasmons, while the real part
determines the damping rate of plasmons. Plasmonic antennas, lenses and
resonators are successful [5] at concentrating electromagnetic energy into the
subwavelength and deep-subwavelength volume. Density-independent plasmons were
found in nodal-line semimetals [6], with possible applications in terahertz-
stable topological metamaterials.

While a weak electron-phonon interaction is one of the basic assumptions in
the BCS theory [7] for conventional superconductivity, strong electron-phonon
interaction (EPI) leads to interesting phenomena such as polarons [8, 9] and
charge density waves (CDW) [10, 11, 12, 13]. The interplay of spin degree of
freedom and strong EPI leads to spin polarons [14] and spin density waves
(SDW) [15, 16]. One prevailing understanding of charge density waves is the
Peierls instability [17] in one dimension, and the Fermi surface nesting in
any dimension. In many real materials, it is realized that the Peierls
instability fails and the CDW phase is determined from the momentum dependence
of the electron-phonon coupling matrix element [18].

Reference [19] pointed out that a spin-valley half-metal state emerges from
doping a spin density wave state. Then reference [20] clarified that weak
repulsive electron-elsectron interaction contributes to the spin density wave
ordering, following a BCS-like mean field approach. The Fermi surface in the
half metal state is fully spin polarized. The nesting vector connecting
particle and hole valleys gives inter-valley scattering. In principle, if no
electron-phonon interaction is considered, the SDW order is always energy
favored than the CDW. In the presence of electron-phonon interaction, one
needs to consider the spin-flip process when the electrons are scattered by
the lattice. The four-band Hamiltonian for the spin-valley half metal has also
been applied in twisted bilayer graphene [21, 22].

In two dimensional valleytronic materials such as $\rm{MoS_{2}}$ [23, 24] the
two nonequivalent valleys are separated in the Brillouin zone by a large
momentum so the inter-valley scattering is very small. Just like manipulating
spin has lead to spintronics [25, 26], manipulating valley index can produce
new effects including using it to carry information. As an example, in the
context of graphene, the reference [27] showed that a contrasting intrinsic
magnetic moment and Berry curvature are associated with the carrier valley
index. The inter-band optical Hall conductivity [24] is then shown to be
connected to the Berry curvature of the material, adding a dynamical factor to
the static Berry curvature.

The intra-band longitudinal optical conductivity has been less investigated.
In the spin-valley half metal (doping form SDW) or half metal (doping form
CDW), the spin and valley polarized Fermi surface and the Fermi velocities are
however closely connected to the intra-band nonlocal optical conductivity.
Recently, in transition metal dichalcogenides (such as $\rm{VSe_{2}}$,
$\rm{MoTe_{2}}$, $\rm{WTe_{2}}$ and electron-doped $\rm{MoS_{2}}$ [28, 29, 30,
31]) CDW phases were found. Half metals are predicted in two dimenional metal
selenides such as V doped $\rm{SnSe_{2}}$ and $\rm{Co_{2}Se_{3}}$ [32]. In
g-$\rm{C_{3}N_{4}}$ and g-$\rm{C_{4}N_{3}}$, the electrons of which are
entirely from s and p orbitals, metal-free half-metals are predicted [33, 34].
They may be used in bio-compatible applications, e.g. in synergistic photo-
thermal treatment to kill cancer cells [35].

In this work, we present density-functional theory(DFT) calculations of Fermi
surface nesting candidate materials, 1T-$\rm{VSe_{2}}$ and
g-$\rm{C_{4}N_{3}}$, and provide an approximate tight binding Hamiltonian for
1T-$\rm{VSe_{2}}$. For the intra-band nonlocal optical conductivity, the
anistropy of the band structure averages out, so we choose an isotropic
parabolic band Hamiltonian to capture the important physics. We then use the
Green’s function to calculate the nonlocal optical conductivity based on a
velocity-velocity correlation function approach. Both intra-band and inter-
band optical conductivity are obtained, in one, two and three dimensions. The
nonlocal ($q\neq 0$) optical conductivity becomes anistropic in two and three
dimensions, a feature not available in the ($q=0$) optical conductivity. The
imaginary part of the nonlocal optical conductivity shows two peaks in the
intra-band region for some parameters of the Fermi-surface nesting model,
suggesting two branches of plasmons in the intra-band region.

##  II Fermi surface nesting candidate materials

The numerical calculations on the mechanical and electronic properties of
$\rm{VSe_{2}}$ were performed by using a first-principles method based on the
DFT[36], as implemented in the DS-PAW which is a program under the Device
Studio platform. The DS-PAW is based on the plane wave basis and the projector
augmented wave (PAW) representation[37]. The Perdew-Burke-Ernzerhof (PBE)
exchange-correlation energy functional within the generalized gradient
approximation (GGA) were employed[38]. In the calculations, atomic positions
are fully relaxed until the force on each atom is less than $0.05eV/\AA$. The
electronic iteration convergence criterion is set to $10^{-4}eV$. The wave
functions were expanded in plane waves up to a kinetic energy cutoff of
$520eV$. The Brillourin zone integration is obtained by using a k-point
sampling mesh of $13\times 13\times 1$, generated according to the Gamma-
centered method. In the standard DFT calculations which is used in this
section, the staggered Hartree term representing the spin off-diagonal
exchange energy is not included [39]. In the future, we plan to develop
atomistic simulation for SDW and CDW, similar to DFT based on Migdal-
Eliashberg formalism [40].

(a) Crystal structure of 1T-$\rm{VSe_{2}}$.

(b) The hopping between nearest and next-nearest atoms.

Figure 1: (Color online) Red balls represent V atoms; Green balls represent Se atoms. A unit cell with one V atom and two Se atom is marked in the bottom frame. Figure 2: (Color online) The band-struture and Fermi surface of (a) spin-polarized DFT of monolayer 1T-$\rm{VSe_{2}}$, (b) non-spin-polarized DFT of monolayer 1T-$\rm{VSe_{2}}$ and (c) spin-polarized DFT of g-$\rm{C_{4}N_{3}}$. Red and blue lines in (a) and (c) represent the contributions from the spin-up and spin-down electrons, respectively, and the black dashed line in (a) represents the wannier90 fitting result. All materials are 2D. The Fermi energies of (a), (b) and (c) are -2.00 eV, -2.06 eV, -3.95 eV respectively. We then shift the whole band-structure to set the Fermi energies at 0 eV. In the right frames of (a), (b) and (c), we plot the Fermi surface (energy contours at 0 eV) in the 2D momentum Brillouin zone. We mark the nesting vector $\mathbf{Q}$ which connects different parts of the Fermi surface.  Table 1: The Se1-Se2 hopping amplitude. | Se2
---|---
$p_{y}$ | $p_{z}$ | $p_{x}$
Se1 | $p_{y}$ | 0.464 | -0.316 | 0.186
$p_{z}$ | -0.316 | 0.126 | -0.089
$p_{x}$ | 0.186 | -0.089 | 0.036
Table 2: The V-Se1 and V-Se2 hopping amplitude. | Se1 | Se2
---|---|---
$p_{y}$ | $p_{z}$ | $p_{x}$ | $p_{y}$ | $p_{z}$ | $p_{x}$
V | $d_{xy}$ | 0.515 | 0.481 | 0.283 | -0.518 | 0 | 0.568
$d_{yz}$ | 0.487 | -0.750 | -0.729 | 0 | -0.500 | 0
$d_{z^{2}}$ | 0.278 | -0.738 | 0.095 | 0.554 | 0 | 1.160
$d_{xz}$ | -0.379 | -0.139 | 0.624 | -0.761 | 0 | -0.389
$d_{x^{2}-y^{2}}$ | 0.655 | 0.464 | -0.148 | 0 | -0.696 | 0
Table 3: The Se-Se next nearest hopping amplitude. |  $p_{y}$,$p_{y}$ |  $p_{z}$,$p_{z}$ |  $p_{z}$,$p_{x}$ |  $p_{x}$,$p_{z}$ |  $p_{x}$,$p_{x}$
---|---|---|---|---|---
Se1(000)>Se1(110) | -0.169 | 0.193 | 0.344 | 0.553 | 0.654
Se2(000)>Se2(110) | -0.169 | 0.193 | 0.553 | 0.344 | 0.654

In Fig. 1(a), we give a schematic of the structure of 1T-$\rm{VSe_{2}}$ in the
trigonal phase corresponding to the space group $P-3m1$. For a single-layer
1T-$\rm{VSe_{2}}$ the primitive cell contains three atoms, and the optimized
lattice constant is $a=b=3.33\AA$, and the angle between them is
$\gamma=120^{\circ}$. In Fig. 1(b), we show a super-cell of $\rm{VSe_{2}}$,
and mark the three unit cells (000), (100), and (110). The black arrows
represent the hopping between the nearest neighbor atoms, including the
Se1-Se2, V-Se1, V-Se2 hoppings in the unit cell (000), and the V-V hopping
between the unit cell (000) and unit cell (100). The blue arrows represent the
second-nearest neighbor hopping, including Se1-Se1 and Se2-Se2 hoppings
between the unit cell (000) and unit cell (110).

In table I, II and III, we list the Wannier fitting results of the Se-Se and
V-Se hopping parameters. The V-V hopping parameters is small, however the next
nearest Se-Se hopping parameter is large. The fitted tight binding model is a
very large Hamiltonian matrix, to be manipulated numerically. From table I, II
and III, we provide an approximate tight binding model, as given in the
supplementary material. The anistropy of the tight binding model averages out
in the intra-band conductivity, so we use the isotropic parabolic band
approximation to capture important physics of the many-body calculations in
the section III.

In Fig. 2, we show the band-structure and Fermi surface of 1T-$\rm{VSe_{2}}$
and g-$\rm{C_{4}N_{3}}$. In Fig. 2(a), the spin-polarized calculations
predicts monolayer 1T-$\rm{VSe_{2}}$ is a magnetic metal, of which the
magnitude of the magnetic moment is 0.6$\mathrm{\mu B}$/cell (DFT result). The
Wannier fitting of DFT band structure is implemented within the WANNIER90
code[42]. We construct the Hamiltonian in the Wannier basis using five d
orbitals of V atom and six p orbitals of two Se atoms to generate the
localized Wannier functions. Overall, the Wannier fitted band structure agrees
well with the DFT one. Some experiment data from angle-resolved photo-emission
spectroscopy (ARPES) of $\rm{VSe_{2}}$ agrees well with the non-spin-polarized
DFT [43], while other experiment data of the Fermi surface agrees well with
the spin-polarized DFT [44], so we present both of them in Fig. 2 (a) and (b).
For the bulk $\rm{VSe_{2}}$, the band-structure is similar to 2D
$\rm{VSe_{2}}$ near the $\Gamma$, $M$, $K$ points, as given in the
supplementary material. There are more high symmetry momentum points $H$, $A$,
$L$ for the bulk $\rm{VSe_{2}}$, and the band structure around these points
are different as expected. In Fig. 2(c), the spin-polarized calculations
predicts that g-$\rm{C_{4}N_{3}}$ is a half-metal. The direct band gap for the
spin-up electrons is 2.20 $eV$ and the magnetic moment of 1 $\mathrm{\mu
B}$/cell is found for g-$\rm{C_{4}N_{3}}$. Note that in the Fermi surface of
g-$\rm{C_{4}N_{3}}$, a nesting vector $\mathbf{Q}$ clearly connects the circle
around the $\Gamma$ point and the K point. In all the calculations, the
staggered Hartree term is not included. In the following sections, we use a
parabolic band approximation to discuss the impact of the staggered Hartree
term.

##  III Hamiltonian and nonlocal optical conductivity

In below, we use the parabolic band approximation and discuss the many body
calculation for the non-local optical conductivity in the case of Fermi
surface nesting, in one, two and three dimensions. With a nesting vector
$\mathbf{Q}$ to connect (part of) the Fermi surface of a conduction band to
that of a valence band, the Hamiltonian for a charge density wave state is
given by

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}\frac{k^{2}}{2m}-\varepsilon_{0}&0&0&\Delta_{\downarrow}\\\
0&\frac{k^{2}}{2m}-\varepsilon_{0}&\Delta_{\uparrow}&0\\\
0&\Delta_{\uparrow}&-\frac{k^{2}}{2m}+\varepsilon_{0}&0\\\
\Delta_{\downarrow}&0&0&-\frac{k^{2}}{2m}+\varepsilon_{0}\end{array}\right]$
(5)

where $\Delta_{\uparrow}$ and $\Delta_{\downarrow}$ are the gap parameters for
a charge density wave insulator. In this paper we assume the gap parameters
are real numbers so the Hamiltonian [Eq. (5)] is a hermitian matrix. In the
case of complex gap parameters, two of the $\Delta_{\uparrow}$ and
$\Delta_{\downarrow}$ in Eq. (5) should be replaced by $\Delta_{\uparrow}^{*}$
and $\Delta_{\downarrow}^{*}$ to keep it a hermitian matrix. The basis states
for the Hamiltonian are
$[c^{{\dagger}}_{a,\mathbf{k},\downarrow},c^{{\dagger}}_{a,\mathbf{k},\uparrow},c^{{\dagger}}_{b,\mathbf{k}+\mathbf{Q},\uparrow},c^{{\dagger}}_{b,\mathbf{k}+\mathbf{Q},\downarrow}]$;
here $c^{{\dagger}}_{a,\mathbf{k},\sigma}$ is the creation operator for a
fermionic field with wave vector $\mathbf{k}$, spin $\sigma$ and valley $a$.
$\sigma=\pm 1$ is for the up and down direction of the spin. The spin density
wave phase is connected to the charge density wave phase by a transformation
$c^{{\dagger}}_{b\uparrow}\rightarrow c^{{\dagger}}_{b\downarrow}$ and
$c^{{\dagger}}_{b\downarrow}\rightarrow c^{{\dagger}}_{b\uparrow}$ with the
corresponding spin density wave order parameter defined as
$\Delta_{\sigma}=<c^{{\dagger}}_{a\sigma}c_{b\bar{\sigma}}>$ (here
$\bar{\sigma}=-\sigma$). Although SDW and CDW are connected mathematically, in
the experiment the detection is very different. X-Ray diffraction (XRD) and
neutron-scattering is used to detect CDW, while only neutron-scattering is
used to detect SDW. The gap parameters are generally determined by a
variational principle based on the free energy of the system. For example, in
[19] it was found
$\Delta_{\uparrow}(x)=\Delta_{0}\sqrt{1-x/(\Delta_{0}N_{F})}$,
$\Delta_{\downarrow}(x)=\Delta_{0}$ and $\mu=\Delta_{0}-x/2N_{F}$, where $x$
is the partial doping $x=-\partial\Omega/\partial\mu$, $\Omega$ is the grand
potential and $\mu$ the chemical potential. The Green’s function is defined as
the inverse of $z-\hat{H}$, where $z=i\omega_{n}$ for imaginary frequency or
$z=\omega+i\delta$ for real frequency,

$\displaystyle\hat{G}(\mathbf{k},z)$ $\displaystyle=$
$\displaystyle\frac{1}{z-\hat{H}}=\left[\begin{array}[]{cc}G_{ee}&G_{eh}\\\
G_{he}&G_{hh}\end{array}\right]$ (8)

We first define a velocity-velocity correlation function $\Pi_{xx}$ which
depends on both the wave vector and the imaginary frequency,

$\displaystyle\Pi_{xx}(\mathbf{q},i\omega_{n})=$ $\displaystyle
T\sum_{\mathbf{k},l}\mathrm{Tr}\langle\hat{v}_{x}\hat{G}(\mathbf{k}-\mathbf{q}/2,i\omega_{l})\hat{v}_{x}\hat{G}(\mathbf{k}+\mathbf{q}/2,i\omega_{l}+i\omega_{n})\rangle$
(9)

here $\omega_{n}=2n\pi T$, $\omega_{l}=(2l+1)\pi T$ are the bosonic and
fermionic Matsubara frequencies, $n$ and $l$ are integers. The nonlocal
optical conductivity in real frequency is obtained from the Kubo formula by
performing the analytical continuation $i\omega_{n}\rightarrow\omega+i\delta$,

$\sigma_{xx}(\mathbf{q},\omega)=-\frac{e^{2}}{i\omega}\Pi_{xx}(\mathbf{q},i\omega_{n}\rightarrow\omega+i\delta)$
(10)

The matrix Green’s function $\hat{G}(\mathbf{k},z)$ can be expanded in terms
of a matrix spectral density $\hat{A}(\mathbf{k},\omega)$ as

$\hat{G}(\mathbf{k},z)=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{\hat{A}(\mathbf{k},\omega)}{z-\omega}$
(11)

The matrix spectral density function is proportional to the imaginary part of
the matrix Green’s function,

$\hat{A}(\mathbf{k},\omega)=2\mathrm{Im}[\hat{G}(\mathbf{k},\omega)]$ (12)

After taking the sum over the imaginary frequencies $i\omega_{l}$, the
longitudinal conductivity $\sigma_{xx}(\omega)$ is given by

$\displaystyle\sigma_{xx}(\mathbf{q},\omega)=-\frac{e^{2}}{4\pi^{2}i\omega}\int_{-\infty}^{\infty}d\omega_{1}d\omega_{2}F(\omega)$
$\displaystyle\sum_{\mathbf{k}}\mathrm{Tr}\langle\hat{v}_{x}\hat{A}(\mathbf{k}-\mathbf{q}/2,\omega_{1})\hat{v}_{x}\hat{A}(\mathbf{k}+\mathbf{q}/2,\omega_{2})\rangle$
(13)

where
$F(\omega)=\frac{[f(\omega_{1})-f(\omega_{2})]}{\omega-\omega_{2}+\omega_{1}+i\delta}$
and $f(x)$ is the Fermi-Dirac distribution function defined as
$f(x)=1/[\exp(x/T-\mu/T)+1]$. The real part of the optical conductivity can be
obtained as

$\displaystyle\mathrm{Re}\sigma_{xx}(\mathbf{q},\omega)=\frac{e^{2}}{4\pi\omega}\int_{-\infty}^{\infty}d\omega_{1}[f(\omega_{1})-f(\omega_{1}+\omega)]\times$
$\displaystyle\sum_{\mathbf{k}}\mathrm{Tr}\langle\hat{v}_{x}\hat{A}(\mathbf{k}-\mathbf{q}/2,\omega_{1})\hat{v}_{x}\hat{A}(\mathbf{k}+\mathbf{q}/2,\omega_{1}+\omega)\rangle$
(14)

The spectral function Eq. (12) is a delta-function if no impurity or other
similar scattering are considered. Here we consider the self-energy correction
from the impurity scattering to be a pure imaginary number, $i/\tau$. The
spectral function Eq. (12) is then a broadened delta-function and the integral
over $\omega_{1}$ in Eq. (14) could not be removed. From Eq. (13) and Eq. (14)
we write the imaginary part of the optical conductivity as

$\mathrm{Im}\sigma_{xx}(\mathbf{q},\omega)=\frac{1}{\pi}\int_{-\infty}^{\infty}d\omega^{\prime}\frac{\mathrm{Re}\sigma_{xx}(\mathbf{q},\omega^{\prime})}{\omega-\omega^{\prime}}$
(15)

which recovers the Kramers-Kronig relation.

In one dimension the corresponding velocity operator can be obtained as (for
simplicity we set $\hbar=1$)

$\displaystyle\hat{v}_{x}(\mathbf{k})$
$\displaystyle=\frac{\partial\hat{H}}{\partial\mathbf{k}}=$
$\displaystyle\left[\begin{array}[]{cc}(\mathbf{k}/m)\hat{I_{2}}&0\\\
0&-(\mathbf{k}/m)\hat{I_{2}}\end{array}\right]$ (18)

where $I_{2}$ is the 2 by 2 unit matrix. For the CDW state, the spin operator
is $S_{z}=\mathrm{diag}(-1,1,1,-1)$. For the SDW state, the spin operator is
changed to $S_{z}=\mathrm{diag}(-1,1,-1,1)$, while the spin-valley operator is
$S_{z}\tau_{z}=\mathrm{diag}(-1,1,1,-1)$, so the spin operator for CDW is the
same as the spin-valley operator for SDW. For both cases the valley operator
is $\tau_{z}=\mathrm{diag}(1,1,-1,-1)$. With these operators we define the
spin conductivity (for a CDW), spin-valley conductivity (for a SDW) and valley
conductivity (for both). Details are given in the supplementary material.

Figure 3: (Color online) Real part of the optical conductivity at
$\mathbf{q}=0$ for 2D materials 1T-$\rm{VSe_{2}}$ and g-$\rm{C_{4}N_{3}}$. In
DFT calculations, the 2D material is treated as a 3D material with very large
separation between 2D layers, so the unit of the optical conductivity is 3D.
And we see an isotropic optical conductivity along x or y directions inside
the 2D plane. The optical conductivity along the z direction outside the 2D
plane is different. In the DFT software (DS-PAW) we use, the optical
conductivity is calculated from the imaginary part of the dielectric function
$\epsilon(\omega)$, given by
Re$\sigma(\omega)=\omega\epsilon_{0}$Im$\epsilon(\omega)$, $\epsilon_{0}$ is
the vacuum dielectric constant.

Figure 4: (Color online) Top panel (a): Band structure for a Fermi surface
nesting model in one dimension (1D) with the spin density wave order
parameters $\Delta_{\uparrow}=0.063E_{0}$ and $\Delta_{\downarrow}=0.2E_{0}$.
The chemical potential $\mu=0.11E_{0}$ is denoted with the red dashed line.
Bottom panel (b): schematics of the iso-energy surface of a Fermi surface
nesting model in three dimension (3D), we clearly see two Fermi-spheres
assigned with two Fermi-velocities. When the chemical potential is tuned, the
two Fermi-spheres moves close to or far away from each other, until one of
them disappears. The Fermi-velocities point outward with the same amplitude on
one Fermi-sphere, and point inward with smaller amplitude on the other,
denoted by the black arrows.

##  IV Results and discussions

In Fig. 3, we present the numerical results of the optical conductivity based
on the DFT calculations in section II. In the DFT software (DS-PAW) we use,
the real part of the optical conductivity is calculated from the imaginary
part of the dielectric function. The DFT calculation is not able to catch the
intra-band part of the nonlocal optical conductivity, usually in the frequency
region from 0 to 50 meV. Also, in the standard DFT calculation, the many body
effect from Fermi surface nesting is missing, because the staggered Hartree
term is not included. In below, we present numerical results of the nonlocal
optical conductivity based on the Hamiltonian [Eq. (5)] in section III.

The eigenvalues of the Hamiltonian [Eq. (5)] are obtained as
$E=\pm\sqrt{\Big{(}\frac{k^{2}}{2m}-\varepsilon_{0}\Big{)}^{2}+\Delta^{2}}$,
where $\Delta=\Delta_{\uparrow}$ or $\Delta_{\downarrow}$. We choose a typical
wave-vector $k_{0}=1.0$ $\mathrm{nm}^{-1}$, the corresponding energy is
$E_{0}=\hbar^{2}k_{0}^{2}/m_{e}=0.076$ $\mathrm{eV}$, where $m_{e}=9.1\times
10^{-31}$ kg is the mass of an electron. For simplicity we set the effective
mass $m=1$, the energy shift $\varepsilon_{0}=0.2E_{0}$ and define
$x^{\prime}=x/(N_{F}\Delta_{0})$ in all the numerical simulations. The
impurity-scattering self-energy correction is $1/\tau=0.005E_{0}$. Note that
the parameters of the energy shift $\varepsilon_{0}$, the gap
($\Delta_{\uparrow}$, $\Delta_{\downarrow}$) and the chemical potential $\mu$
are determined from a variational principle calculation [19], it is suggested
that the gap $\Delta_{0}$ is much smaller than $\varepsilon_{0}$, (e.g.
$\Delta_{0}$=$\varepsilon_{0}$/4). Here we choose
$\Delta_{0}=\varepsilon_{0}$. For the intra-band optical conductivity, tuning
the band gap to be much smaller will not change the key feature, in the sense
that the chemical potential still intersects with the lower conduction band
and two Fermi velocities was found.

In Fig. 4, we show a typical 1D band structure (left) or a 3D iso-energy
surface (right) for CDW/SDW phase. The doping parameter $x^{\prime}=0.9$, so
the gap parameters are $\Delta_{\uparrow}=0.063E_{0}$ and
$\Delta_{\downarrow}=0.2E_{0}$. The doping $x$ gives an asymmetry between the
gap order parameter for spin up and spin down, thus the Fermi surface is now
spin polarized typical for a half-metal. The Fermi velocities are depicted by
arrows in the right panel.

In 1D, $q$ is a number, we only have the longitudinal conductivity, in 2D and
3D, $\mathbf{q}$ is a vector, we need to consider the longitudinal one
($\mathbf{q}$ along $\hat{e}_{x}$) and the transverse one [41] ($\mathbf{q}$
perpendicular to $\hat{e}_{x}$), where $\hat{e}_{x}=(1,0)$ for 2D and
$\hat{e}_{x}=(1,0,0)$ for 3D. For the longitudinal conductivity
$\sigma^{L}_{xx}(q,\omega)$ we have the absolute value of
$\mathbf{k}+\mathbf{q}=\sqrt{(k_{x}+q)^{2}+k_{y}^{2}}=\sqrt{k^{2}+2kq\rm{cos}(\theta)+q^{2}}$.
For the transverse one $\sigma^{T}_{xx}(q,\omega)$ we have the absolute value
of
$\mathbf{k}+\mathbf{q}=\sqrt{(k_{y}+q)^{2}+k_{x}^{2}}=\sqrt{k^{2}+2kq\rm{sin}(\theta)+q^{2}}$.
Because of this difference, we see a large anistropy in the nonlocal optical
conductivity in two and three dimensions.

In 3D, we need to perform the four-fold integration (Eq. (8)) numerically,
which is very slow. For one frequency point it takes about 1 hour on 1 CPU. In
Fig. 5(a) we use 1600 points and show the real part of the nonlocal optical
conductivity in three dimensions for $q=0.2\mathrm{nm}^{-1}$. We found the
Drude peak at zero frequency is shifted to higher frequencies for the
longitudinal conductivity. In the intra-band region ($0<\omega/E_{0}<0.2$), we
found the Drude peak is separated into two-peak structure. This is due to the
two Fermi velocities in the CDW as shown in Fig. 4. If the two typical Fermi
velocities are $v_{F1}$ and $v_{F2}$, the two intra-band peaks are roughly at
$\omega_{1}=q\times v_{F1}$ and $\omega_{2}=q\times v_{F2}$. For the
transverse direction ($\mathbf{q}$ perpendicular to $\hat{e}_{x}$), the Drude
peak is not shifted. In both cases, we found two inter-band peaks at around
0.2 and 0.4, due to the fact that for a four-band system we can define two
sets of inter-band transitions. The inter-band peaks were shifted to higher
energy as the transverse mode crossovers to the longitudinal mode.

In Fig. 5(b), we investigate the 2D conductivity and found the spectral weight
is redistributed between the intra-band and inter-band peaks. The intra-band
peaks in 2D become sharper than those in 3D. In Fig. 5(c), we observe that the
1D intra-band spin conductivity is always positive, while the intra-band
valley conductivity oscillates from negative to positive. This suggests the
charge carriers at the two Fermi momenta carry the same spin and opposite
valley.

Fig. 5(d) shows the 1D imaginary part of the nonlocal optical conductivity,
which is connected to the real part through the Kramers-Kronig relation. At
$q=0$, the imaginary part does not diverge as $1/\omega$, so
$\omega\mathrm{Im}\sigma=0$. At $q=0.2$ $\mathrm{nm}^{-1}$, we see clearly the
intra-band evolves into two positive oscillations. At much larger $q=0.4$
$\mathrm{nm}^{-1}$, however we only see one positive peak in the intra-band.
This suggests that at small q, two plasmonic modes may be observed in the
Electron Energy Loss Spectroscopy. The inter-band transitions in the spin
susceptibility of the spin-valley half-metal was studied in [45]. For the
inter-band optical conductivity, it is connected to the joint density of
states for Re$\sigma(q=0,\omega)$,[46] or the joint spectral function for
Re$\sigma(q\neq 0,\omega)$.

In Fig. 6, we plot the real part of the nonlocal optical conductivity in 1D,
we see the Drude peak at $q=0$ is shifted to higher energy, the spectral
weight is transferred to the intra-band region. In the intra-band the Drude
peak splits into two peaks, due to the two different Fermi velocities. The
inter-band peaks at $\omega/E_{0}=0.4$ is not changed much by $q$. The
parameters are the same as those in Fig. 5.

In Fig. 7, we investigate the off-diagonal conductivity
$\sigma_{xy}(q,\omega)$, which is zero in the direction $\theta=0$ and
$\theta=\pi/2$. Here we set the direction $\theta=\pi/4$ (between $\mathbf{q}$
and $\hat{e}_{x}$). We show that $\sigma_{xy}$ at $\omega=0$ is negative, and
oscillates between positive and negative values at higher frequencies. The
diagonal conductivity $\sigma_{xx}$ is in the transition from the transverse
($\theta=\pi/2$) to the longitudinal ($\theta=0$) mode, we see a shifted Drude
peak and a dip in the optical conductivity around $\omega/E_{0}=0.1$, due to
the two Fermi velocities.

In Fig. 8, we observe that the $\sigma_{xx}(q,\omega)$ oscillates as the angle
changes from $\theta=0$ to $\theta=2\pi$, for four typical frequencies in the
intra-band region. For three of them $\sigma_{xx}(q,\omega)$ reaches maximum
at $\theta=0$, for one frequency at $\omega/E_{0}=0.05$, the maximum is at
$\theta=\pi/2$.

In table IV, we present the DC off-diagonal conductivity along the direction
$\theta=\pi/4$ for various wave vector q. We found the Im$\sigma_{xy}(q,0)$ is
positive at small q, firstly increases then decreases and changes sign at
larger q. In the supplementary material we present the numerical results of 1D
optical conductivity and discussed the spin and valley conductivity
respectively.

Table 4:  Im$\sigma_{xy}(\mathbf{q},\omega=0)$ along the direction $\theta=\pi/4$. | $q=0.001$ | $q=0.005$ | $q=0.01$ | $q=0.05$
---|---|---|---|---
Im$\sigma_{xy}(q,0)$ | 0.00025 | 0.0745 | -4.06 | -17.5

Note that in the Fermi surface nesting model, frequent electron-electron
collisions modify the band-structure and two Fermi velocities appear. In a
hydrodynamic approach, considering a fluid flow, the velocity changes
gradually from the center to the boundary. In that sense, the CDW/SDW state
lies somewhere between the ballistic transport picture and the fully
hydrodynamic transport picture. The intra-band optical conductivity, which
could be measured in a near-field experiment, is a good probe of these liquid-
like properties. We also observe that the valley conductivity for one intra-
band peak is negative, and for another intra-band peak is positive. This gives
a clear definition of the valley. The positive valley is associated with the
Fermi velocity $v_{F1}$ and the negative valley is associated with $v_{F2}$.
In the above discussion we found the non-local intra-band optical conductivity
is a useful tool to investigate the spin-valley physics.

##  V Conclusion

In summary, we provide DFT calculations of Fermi surface nesting materials.
Then based on the parabolic band approximation we derive the nonlocal optical
conductivity for a spin/charge density wave Hamiltonian beyond the standard
DFT. Our method is able to accurately capture the shift of the Drude peak from
zero to higher frequency in the intra-band region and splits into two peaks,
in 3D, 2D and 1D. As the direction changes from the transverse mode
($\theta=\pi/2$) to the longitudinal mode ($\theta=0$), although the
Hamiltonian is isotropic, the nonlocal optical conductivity is anistropic at
nonzero $q$. We discuss in detail the diagonal $\sigma_{xx}$ and off-diagonal
$\sigma_{xy}$ conductivities at different angles.

##  VI Methods

A first-principles method based on the density-functional theory (DFT) is
used. The Device Studio program provides a number of functions for performing
visualization, modeling and simulation. We use the projector augmented wave
method from the DS-PAW software integrated in the Device studio program. The
Wannier fitting of DFT band structure is implemented within the WANNIER90
code. Many-body calculation based on the Green’s function method is used.

##  VII COMPETING INTERESTS

The Authors declare no Competing Financial or Non-Financial Interests.

##  VIII DATA AVAILABILITY

Data available on request from the authors. The data that support the findings
of this study are available from the corresponding author, [Z.L.], upon
reasonable request.

##  IX AUTHOR CONTRIBUTIONS

Z.L. designed the project and wrote the paper, X.H. carried out the DFT
calculation, X.J. and B.H. provided the Wannier fitting, all authors
contributed to the development of the work.

###### Acknowledgements.

The authors thanks A. Rozhkov, A. Sboychakov, A. L. Rakhmanov, K. I. Kugel,
Ryusuke Matsunaga and F. Nori for useful discussions. A part of this work has
been supported by ISSP International Collaboration Program of the University
of Tokyo. Z. L. acknowledges the support of a JSPS Foreign Postdoctoral
Fellowship under Grant No. PE14052 and P16027 and the Chinese Academy of
Science funding No. E1Z1D10200 and No. E2Z2D10200. This work is supported in
part by the National Natural Science Foundation of China (Grant No. 61988102).
We gratefully acknowledge HZWTECH for providing computation facilities.

Figure 5: (Color online) (a): the real part of the nonlocal optical
conductivity for a 3D vector $\mathbf{q}$. The black solid curve is for the
longitudinal conductivity $\sigma^{L}_{xx}(q,\omega)$ ($\mathbf{q}$ along
$\hat{e}_{x}$), the Drude peak shifts to higher energy and splits into two
peaks in the intraband, due to the two different Fermi velocities. The red
dashed curve is for the transverse conductivity $\sigma^{T}_{xx}(q,\omega)$
($\mathbf{q}$ perpendicular to $\hat{e}_{x}$), the Drude peak is not shifted.
In (b), two dimensional (2D) vector $\mathbf{q}$ is considered. We observe
similar behavior as those in 3D. The second peak in the intraband is sharper
than the 3D case. (c): The real part of the optical conductivity (black solid)
for a doped CDW half-metal in 1D ($q=0.2$ $\mathrm{nm}^{-1}$) and the spin
optical conductivity (red dashed) compared with the valley optical
conductivity (green dash-dotted). It is clear that the two intra-band peaks
are associated with the same direction of spin but with opposite valley. The
two inter-band peaks are however associated with opposite spin. (d): the
imaginary part of the nonlocal optical conductivity, crucial for plasmonics,
two intra-band peaks in the imaginary part was confirmed for $q=0.2$
$\mathrm{nm}^{-1}$ (red dashed), as found in the Re$\sigma(q,\omega)$. In all
the plots, the parameters ($\Delta_{\uparrow}=0.063E_{0}$,
$\Delta_{\downarrow}=0.2E_{0}$) are the same for 1D, 2D and 3D. The chemical
potential is $\mu=0.11E_{0}$ intersecting with the lower conduction band. The
temperature $T$=0.001 K. In all the plots of the Re$\sigma(q,\omega)$, the
first two peaks (or one peak and one dip) in the region $0<\omega/E_{0}<0.2$
are considered as intra-band peaks, originating from two Fermi velocities.
Then in the region $0.2<\omega/E_{0}$, we see two inter-band peaks,
originating from the peaks in the joint density of states for
Re$\sigma(q=0,\omega)$,[46] or the joint spectral function for Re$\sigma(q\neq
0,\omega)$.  Figure 6: (Color online) Real part of the nonlocal optical
conductivity as a function of $\omega$ and $q$ in 1D. Figure 7: (Color online)
Real part of the nonlocal optical conductivity for a two dimension vector
$\mathbf{q}$. The angle $\theta$ between the wave vector $\mathbf{q}$ and
$\hat{e}_{x}$ is $\theta=\pi/4$, the wave vector is $q=0.2$ $\rm{nm}^{-1}$.The
black solid curve is for the conductivity $\sigma_{xx}$, its behavior is in
the transition from the longitudinal one $\sigma^{L}_{xx}$ to the transverse
one $\sigma^{T}_{xx}$. The red dashed curve is for the off-diagonal
conductivity $\sigma_{xy}$. Figure 8: (Color online) Real part of the nonlocal
optical conductivity for a two dimension vector $\mathbf{q}$, as a function of
the angle $\theta$ between the wave vector $\mathbf{q}$ and $\hat{e}_{x}$; the
wave vector is $q=0.2$ $\rm{nm}^{-1}$. Here $\omega^{\prime}=\omega/E_{0}$.

## References

## References

  * [1] Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene-SiO2 interface”, Nano Letters 11, 4701 (2011).
  * [2] Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging”, Nature 487, pages 82–85 (2012).
  * [3] M. B. Lundeberg, Y. Gao, R. Asgari, C. Tan, B. V. Duppen, M. Autore, P. Alonso-González, A. Woessner, K. Watanabe, T. Taniguchi, R. Hillenbrand, J. Hone, M. Polini, F. H. L. Koppens, “Tuning quantum nonlocal effects in graphene plasmonics”, Science 357, Issue 6347, pp. 187-191 (2017).
  * [4] G. Giuliani, G. Vignale, “Quantum Theory of the Electron Liquid”, Cambridge Univ. Press, 2005.
  * [5] J. A. Schuller, E. S. Barnard, W. Cai, Y. Chul Jun, J. S. White and M. l. Brongersma, “Plasmonics for extreme light concentration and manipulation”, Nature Mater. 9, 193–204 (2010).
  * [6] Jianfeng Wang, Xuelei Sui, Wenhui Duan, Feng Liu, and Bing Huang, “Density-independent plasmons for terahertz-stable topological metamaterials”, PNAS May 11, 118 (19) (2021).
  * [7] J. Bardeen, L. N. Cooper and J. R. Schrieffer, “Microscopic Theory of Superconductivity”, Physical Review 106 (1): 162-164 (1957).
  * [8] Zhou Li, Carl J. Chandler, and F. Marsiglio, “Perturbation theory of the mass enhancement for a polaron coupled to acoustic phonons”, Phys. Rev. B 83, 045104 (2011).
  * [9] Zhou Li and F. Marsiglio, “The Polaron-Like Nature of an Electron Coupled to Phonons”, Journal of superconductivity and novel magnetism 25 (5), 1313-1317 (2012).
  * [10] G. Grüner, “The dynamics of charge-density waves”, Rev. Mod. Phys. 60, 1129 (1988).
  * [11] P. Monceau, “Electronic crystals: an experimental overview”, Advances in Physics, 61, 4, pages 325-581, (2012).
  * [12] Fangdong Tang, Yafei Ren, Peipei Wang, Ruidan Zhong, John Schneeloch, Shengyuan A. Yang, Kun Yang, Patrick A. Lee, Genda Gu, Zhenhua Qiao and Liyuan Zhang, “Three-dimensional quantum Hall effect and metal-insulator transition in $\rm{ZrTe_{5}}$”, Nature 569, pages537–541 (2019).
  * [13] Peng-Lu Zhao, Hai-hou Lu and X. C. Xie, “Theory for Magnetic-Field-Driven 3D Metal-Insulator Transitions in the Quantum Limit”, Phys. Rev. Lett. 127, 046602 (2021).
  * [14] Zhou Li, L. Covaci, M. Berciu, D. Baillie, and F. Marsiglio, “Impact of spin-orbit coupling on the Holstein polaron”, Phys. Rev. B 83, 195104 (2011).
  * [15] A. W. Overhauser, “Spin Density Waves in an Electron Gas”, Phys. Rev. 128, 1437 (1962).
  * [16] Eric Fawcett, “Spin-density-wave antiferromagnetism in chromium”, Rev. Mod. Phys. 60, 209 (1988).
  * [17] R. E. Peierls, Ann. Phys. Leipzig 4, 121 (1930).
  * [18] Xuetao Zhu, Yanwei Cao, Jiandi Zhang, E. W. Plummer, and Jiandong Guo, “Classification of charge density waves based on their nature”, PNAS Feb 24, 112 (8), 2367-2371 (2015).
  * [19] A. V. Rozhkov, A. L. Rakhmanov, A. O. Sboychakov, K. I. Kugel, and Franco Nori, “Spin-Valley Half-Metal as a Prospective Material for Spin Valleytronics”, Phys. Rev. Lett. 119, 107601 (2017).
  * [20] A. V. Rozhkov, A. L. Rakhmanov, A. O. Sboychakov, K. I. Kugel, and Franco Nori, “Spin-valley half-metal in systems with Fermi surface nesting”, Phys. Rev. B 98, 155141 (2018).
  * [21] C. J. Tabert and E. J. Nicol, “Optical conductivity of twisted bilayer graphene”, Phys. Rev. B 87, 121402(R) (2013).
  * [22] A. O. Sboychakov, A. V. Rozhkov, A. L. Rakhmanov, and Franco Nori “Externally Controlled Magnetism and Band Gap in Twisted Bilayer Graphene”, Phys. Rev. Lett. 120, 266402 (2018).
  * [23] D. Xiao, G. B. Liu, W. Feng, X. Xu and W. Yao, “Coupled Spin and Valley Physics in Monolayers of $\rm{MoS_{2}}$ and Other Group-VI Dichalcogenides”, Phys. Rev. Lett. 108, 196802 (2012).
  * [24] Zhou Li and J. P. Carbotte, “Longitudinal and spin-valley Hall optical conductivity in single layer $\rm{MoS_{2}}$”, Phys. Rev. B 86, 205425 (2012).
  * [25] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelkanova and D. M. Treger, “Spintronics: A Spin-Based Electronics Vision for the Future”, Science 294, 1488, (2001).
  * [26] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano and I. Zutic, “Semiconductor spintronics”, Acta Physica Slovaca 57, No.4,5, 565-907, (2007).
  * [27] D. Xiao, W. Yao and Q. Niu, “Valley-Contrasting Physics in Graphene: Magnetic Moment and Topological Transport”, Phys. Rev. Lett. 99, 236809 (2007).
  * [28] Jiagui Feng, Deepnarayan Biswas, Akhil Rajan, Matthew D. Watson, Federico Mazzola, Oliver J. Clark, Kaycee Underwood et al., “Electronic Structure and Enhanced Charge-Density Wave Order of Monolayer VSe2”, Nano Lett., 18, 7, 4493–4499 (2018).
  * [29] P. Chen, Woei Wu Pai, Y.-H. Chan, V. Madhavan, M. Y. Chou, S.-K. Mo, A.-V. Fedorov and T.-C. Chiang, “Unique Gap Structure and Symmetry of the Charge Density Wave in Single-Layer VSe2”, Phys. Rev. Lett. 121, 196402 (2018).
  * [30] Giovanni Marini and Matteo Calandra, “Light-Tunable Charge Density Wave Orders in MoTe2 and WTe2 Single Layers”, Phys. Rev. Lett. 127, 257401 (2021).
  * [31] Mohammed K. Bin Subhan, Asif Suleman, Gareth Moore, Peter Phu, Moritz Hoesch, Hidekazu Kurebayashi, Christopher A. Howard, and Steven R. Schofield, “Charge Density Waves in Electron-Doped Molybdenum Disulfide”, Nano Letters, 21 (13), 5516-5521 (2021).
  * [32] Cheng Tang, Lei Zhang, Yalong Jiao, Chunmei Zhang, Stefano Sanvito and Aijun Du, “First-principles prediction of polar half-metallicity and out-of-plane piezoelectricity in two-dimensional quintuple layered cobalt selenide”, J. Mater. Chem. C, 9, 12046-12050 (2021).
  * [33] A. Du, S. Sanvito and S. C. Smith, “First-Principles Prediction of Metal-Free Magnetism and Intrinsic Half Metallicity in Graphitic Carbon Nitride”, Phys. Rev. Lett. 108, 197207 (2012).
  * [34] Indrani Choudhuri, Gargee Bhattacharyya, Sourabh Kumar and Biswarup Pathak, “Metal-Free Half-Metallicity in a High Energy Phase C-doped gh-$\rm{C_{3}N_{4}}$ System: A High Curie Temperature Planar System”, J. Mater. Chem. C, 4, 11530-11539 (2016).
  * [35] Jun Yang, Rui Xie, Lili Feng, Bin Liu, Ruichan Lv, Chunxia Li, Shili Gai, Fei He, Piaoping Yang, Jun Lin, “Hyperthermia and Controllable Free Radical Coenhanced Synergistic Therapy in Hypoxia Enabled by Near-Infrared-II Light Irradiation”, ACS Nano. 13(11):13144-13160 (2019).
  * [36] Kresse G, Joubert D. “From ultrasoft pseudopotentials to the projector augmented-wave method”, Phys. Rev. B, 1999, 59(3): 1758-1775.
  * [37] Blochl P E. “Projector augmented-wave method”, Phys. Rev. B, 1994, 50(24): 17953-17979.
  * [38] Perdew J P, Burke K, Ernzerhof M. “Generalized gradient approximation made simple”, Phys. Rev. Lett., 1996, 77 (18): 3865-3868.
  * [39] K. Capelle, M.F. Silva and L.N. Oliveira, “Density-functional investigation of the stability of spin density waves”, Journal of Magnetism and Magnetic Materials, 226-230, 1017-1018 (2001)
  * [40] Danhong Wu, Yiping Lin, Lingxiao Xiong, Junjie Li, Tiantian Luo, Deyi Chen, and Feipeng Zheng, “Enhanced superconductivity in bilayer $\rm{PtTe_{2}}$ by alkali-metal intercalations”, Phys. Rev. B 103, 224502 (2021).
  * [41] Pablo Rodriguez-Lopez, Wilton J. M. Kort-Kamp, Diego A. R. Dalvit and Lilia M. Woods, “Nonlocal optical response in topological phase transitions in the graphene family”, Phys. Rev. MATERIALS 2, 014003 (2018).
  * [42] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, “Maximally localized Wannier functions: Theory and applications”, Rev. Mod. Phys. 84, 1419 (2012).
  * [43] P. Chen, Woei Wu Pai, Y.-H. Chan, V. Madhavan, M. Y. Chou, S.-K. Mo, A.-V. Fedorov and T.-C. Chiang, “Unique Gap Structure and Symmetry of the Charge Density Wave in Single-Layer VSe2”, Phys. Rev. Lett., 121, 196402 (2018).
  * [44] Ganbat Duvjir, Byoung Ki Choi, Trinh Thi Ly, Nguyen Huu Lam, Kyuha Jang, Dang Duc Dung, Young Jun Chang and Jungdae Kim, “Multiple charge density wave phases of monolayer VSe2 manifested by graphene substrates”, Nanotechnology 32 364002 (2021).
  * [45] D. A. Khokhlov, A. L. Rakhmanov, A. V. Rozhkov, and A. O. Sboychakov, “Dynamical spin susceptibility of a spin-valley half-metal”, Phys. Rev. B 101, 235141 (2020).
  * [46] Zhou Li, F. Marsiglio and J. P. Carbotte, “Vanishing of interband light absorption in a persistent spin helix state”, Scientific Reports 3, 02828 (2013).

##  Appendix A Derivations of the Green’s function and the spectral function
for a four-band Hamiltonian

To obtain the Green’s function associated with the Hamiltonian (1), we firstly
use a general mathematical technique: any 4 by 4 matrix can be partitioned
into a smaller 2 by 2 block matrix. In the smaller block matrix, each element
is a 2 by 2 sub-matrix. Use $A,B,C,D$ for the four sub-matrices, we have

$\displaystyle z-\hat{H}=\left[\begin{array}[]{cc}A&C\\\
D&B\end{array}\right]$ (21)

where $z=i\omega_{n}$ for imaginary frequency or $z=\omega+i\delta$ for real
frequency. Here $A=(z-k^{2}/2m+\varepsilon_{0})\hat{I_{2}}$,
$B=(z+k^{2}/2m-\varepsilon_{0})\hat{I_{2}}$ and

$\displaystyle C=\left[\begin{array}[]{cc}0&-\Delta_{\downarrow}\\\
-\Delta_{\uparrow}&0\end{array}\right],D=\left[\begin{array}[]{cc}0&-\Delta_{\uparrow}\\\
-\Delta_{\downarrow}&0\end{array}\right]$ (26)

The Green’s function matrix is also partitioned into a smaller 2 by 2 block
matrix, with $G_{ee},G_{eh},G_{he},G_{hh}$ the four sub-matrix elements. The
equation to determine the Green’s function is written as

$\displaystyle\left[\begin{array}[]{cc}A&C\\\
D&B\end{array}\right]\left[\begin{array}[]{cc}G_{ee}&G_{eh}\\\
G_{he}&G_{hh}\end{array}\right]=\left[\begin{array}[]{cc}\hat{I_{2}}&0\\\
0&\hat{I_{2}}\end{array}\right]$ (33)

The sub-matrix elements $G_{ee}$ and $G_{he}$ satisfy the following equations,

$\displaystyle A\times G_{ee}+C\times G_{he}=\hat{I_{2}}$ (34) $\displaystyle
B\times G_{he}+D\times G_{ee}=0$ (35)

From the two equations we obtain

$\displaystyle(AB-CD)G_{ee}=B$ (36) $\displaystyle(AB-DC)G_{he}=-D$ (37)

The product of $CD$ is given by,

$\displaystyle\left[\begin{array}[]{cc}0&-\Delta_{\downarrow}\\\
-\Delta_{\uparrow}&0\end{array}\right]\left[\begin{array}[]{cc}0&-\Delta_{\uparrow}\\\
-\Delta_{\downarrow}&0\end{array}\right]=\left[\begin{array}[]{cc}\Delta_{\downarrow}^{2}&0\\\
0&\Delta_{\uparrow}^{2}\end{array}\right]$ (44)

For the other two sub-matrix elements $G_{eh}$ and $G_{hh}$, We need to solve
the two equations below

$\displaystyle A\times G_{eh}+C\times G_{hh}=0$ (45) $\displaystyle D\times
G_{eh}+B\times G_{hh}=\hat{I_{2}}$ (46)

from which we obtained

$\displaystyle(AB-CD)G_{eh}=-C$ (47) $\displaystyle(AB-DC)G_{hh}=A$ (48)

Notice the poles of the four Green’s functions are given by the equation

$(z-k^{2}/2m+\varepsilon_{0})(z+k^{2}/2m-\varepsilon_{0})-\Delta^{2}=0$ (49)

where $\Delta=\Delta_{\uparrow}$ or $\Delta_{\downarrow}$, expand this
equation in orders of $z$ we have,

$z^{2}-(k^{2}/2m-\varepsilon_{0})^{2}-\Delta^{2}=0$ (50)

From which we obtain two group of solutions for the eigenvalues corresponding
to $\Delta=\Delta_{\uparrow}$ and $\Delta_{\downarrow}$ separately,

$E_{1,2}=\pm\sqrt{(\frac{k^{2}}{2m}-\varepsilon_{0})^{2}+\Delta_{\uparrow}^{2}}$
(51)

and the other two for $\Delta_{\downarrow}$,

$E_{3,4}=\pm\sqrt{(\frac{k^{2}}{2m}-\varepsilon_{0})^{2}+\Delta_{\downarrow}^{2}}$
(52)

With the eigenvalues $E_{1,2,3,4}$ we can write the Green’s function in a
compact form

$G_{ee}=\left[\begin{array}[]{cc}\frac{(z+k^{2}/2m-\varepsilon_{0})}{(z-E_{3})(z-E_{4})}&0\\\
0&\frac{(z+k^{2}/2m-\varepsilon_{0})}{(z-E_{1})(z-E_{2})}\end{array}\right]$
(53)
$G_{he}=\left[\begin{array}[]{cc}0&\frac{\Delta_{\uparrow}}{(z-E_{1})(z-E_{2})}\\\
\frac{\Delta_{\downarrow}}{(z-E_{3})(z-E_{4})}&0\end{array}\right]$ (54)
$G_{hh}=\left[\begin{array}[]{cc}\frac{(z-k^{2}/2m+\varepsilon_{0})}{(z-E_{1})(z-E_{2})}&0\\\
0&\frac{(z-k^{2}/2m+\varepsilon_{0})}{(z-E_{3})(z-E_{4})}\end{array}\right]$
(55)
$G_{eh}=\left[\begin{array}[]{cc}0&\frac{\Delta_{\downarrow}}{(z-E_{3})(z-E_{4})}\\\
\frac{\Delta_{\uparrow}}{(z-E_{1})(z-E_{2})}&0\end{array}\right]$ (56)

To obtain the spectral function $\hat{A}(\mathbf{k},\omega)$, which is the
imaginary part of the Green’s function, we need to use the following rules to
decompose the Green’s function into a sum of single poles,

$\displaystyle\frac{(z+k^{2}/2m-\varepsilon_{0})}{(z-E_{1})(z-E_{2})}=\frac{1}{(E_{1}-E_{2})}\times$
$\displaystyle\Big{[}\frac{E_{1}+k^{2}/2m-\varepsilon_{0}}{(z-E_{1})}-\frac{E_{2}+k^{2}/2m-\varepsilon_{0}}{(z-E_{2})}\Big{]}$
(57)
$\displaystyle\frac{(z-k^{2}/2m+\varepsilon_{0})}{(z-E_{1})(z-E_{2})}=\frac{1}{(E_{1}-E_{2})}\times$
$\displaystyle\Big{[}\frac{E_{1}-k^{2}/2m+\varepsilon_{0}}{(z-E_{1})}-\frac{E_{2}-k^{2}/2m+\varepsilon_{0}}{(z-E_{2})}\Big{]}$
(58)
$\frac{\Delta_{\uparrow}}{(z-E_{1})(z-E_{2})}=\frac{\Delta_{\uparrow}}{(E_{1}-E_{2})}\Big{[}\frac{1}{(z-E_{1})}-\frac{1}{(z-E_{2})}\Big{]}$
(59)

The density of states (DOS) $N(\omega)$ can be obtained through the spectral
function:
$N(\omega)=\sum_{\mathbf{k}}\mathrm{Tr}[\hat{A}(\mathbf{k},\omega)]$.

##  Appendix B Derivation of the nonlocal optical conductivity

In order to evaluate the conductivity we need to take the trace of the matrix
product of velocity operator and Green’s function matrix, for
$\sigma_{xx}(\omega)$ we need
$\mathrm{Tr}\langle\hat{v}_{x}\widehat{G}(\mathbf{k}_{1},\omega_{1})\hat{v}_{x}\widehat{G}(\mathbf{k}_{2},\omega_{2})\rangle=(k_{x}/m)^{2}\times$

$\displaystyle\\{\mathrm{Tr}[G_{ee}(\mathbf{k}_{1},\omega_{1})G_{ee}(\mathbf{k}_{2},\omega_{2})]$
$\displaystyle-\mathrm{Tr}[G_{eh}(\mathbf{k}_{1},\omega_{1})G_{he}(\mathbf{k}_{2},\omega_{2})+G_{he}(\mathbf{k}_{1},\omega_{1})G_{eh}(\mathbf{k}_{2},\omega_{2})]$
$\displaystyle+\mathrm{Tr}[G_{hh}(\mathbf{k}_{1},\omega_{1})G_{hh}(\mathbf{k}_{2},\omega_{2})]\\}$
(60)

The spin and valley conductivity are given as
$\mathrm{Re}\sigma^{s,v}_{xx}(\mathbf{q},\omega)=\frac{e^{2}}{4\pi\omega}\int_{-\infty}^{\infty}d\omega_{1}[f(\omega_{1})-f(\omega_{1}+\omega)]\times$

$\sum_{\mathbf{k}}\mathrm{Tr}\langle
S_{z}\hat{v}_{x}\hat{A}(\mathbf{k}-\mathbf{q}/2,\omega_{1})\hat{v}_{x}\hat{A}(\mathbf{k}+\mathbf{q}/2,\omega_{1}+\omega)\rangle$

and

$\sum_{\mathbf{k}}\mathrm{Tr}\langle\tau_{z}\hat{v}_{x}\hat{A}(\mathbf{k}-\mathbf{q}/2,\omega_{1})\hat{v}_{x}\hat{A}(\mathbf{k}+\mathbf{q}/2,\omega_{1}+\omega)\rangle$

respectively.

Figure 9: (Color online) Real part of the optical conductivity for
$\Delta_{\uparrow}=0.063E_{0}$ and $\Delta_{\downarrow}=0.2E_{0}$, for wave
vectors $q=0$, $q=0.2$ $\mathrm{nm}^{-1}$ and $q=0.4$ $\mathrm{nm}^{-1}$. The
chemical potential $\mu=0.11E_{0}$. For $q=0$ (black solid), a Drude peak from
intra-band transition is found near zero frequency, at higher frequencies
$\omega=0.2E_{0}$ and $\omega=0.4E_{0}$ two peaks from inter-band transition
are found, due to the four-band feature of the Hamiltonian. For $q=0.2$
$\mathrm{nm}^{-1}$ (red dashed), the Drude peak is shifted to higher energy
and separated into two peaks in the frequency region $\omega<0.2E_{0}$. For
$q=0.4$ $\mathrm{nm}^{-1}$ (green dash-dotted), only one peak in the frequency
region $\omega<0.2E_{0}$ and the other peak shifts to the inter-band region
$\omega>0.2E_{0}$. The inter-band peaks should be compared with the peaks in
the joint density of states for Re$\sigma(q=0,\omega)$, or the joint spectral
function for Re$\sigma(q\neq 0,\omega)$.

For the charge density wave, we define the spin conductivity
$\sigma^{s}_{xx}(\omega)$, the spin operator is
$S_{z}=\mathrm{diag}(-1,1,1,-1)$; for the spin density wave, the spin operator
becomes $S_{z}=\mathrm{diag}(-1,1,-1,1)$, the spin valley operator is
$S_{z}\tau_{z}=\mathrm{diag}(-1,1,1,-1)$, the same as the spin operator for
the charge density wave. Mathematically, for the spin conductivity
$\sigma^{s}_{xx}(\omega)$ (charge density wave) and the spin valley
conductivity $\sigma^{sv}_{xx}(\omega)$ (spin density wave), we need the same
trace, $\mathrm{Tr}\langle
S_{z}\hat{v}_{x}\widehat{G}(\mathbf{k}_{1},\omega_{1})\hat{v}_{x}\widehat{G}(\mathbf{k}_{2},\omega_{2})\rangle=(k_{x}/m)^{2}\times$

$\displaystyle\\{-\mathrm{Tr}[\sigma_{z}G_{ee}(\mathbf{k}_{1},\omega_{1})G_{ee}(\mathbf{k}_{2},\omega_{2})]$
$\displaystyle-\mathrm{Tr}[\sigma_{z}(G_{he}(\mathbf{k}_{1},\omega_{1})G_{eh}(\mathbf{k}_{2},\omega_{2})-G_{eh}(\mathbf{k}_{1},\omega_{1})G_{he}(\mathbf{k}_{2},\omega_{2}))]$
$\displaystyle+\mathrm{Tr}[\sigma_{z}G_{hh}(\mathbf{k}_{1},\omega_{1})G_{hh}(\mathbf{k}_{2},\omega_{2})]\\}$
(61)

For the valley conductivity $\sigma^{v}_{xx}(\omega)$ we need
$\mathrm{Tr}\langle\tau_{z}\hat{v}_{x}\widehat{G}(\mathbf{k}_{1},\omega_{1})\hat{v}_{x}\widehat{G}(\mathbf{k}_{2},\omega_{2})\rangle=(k_{x}/m)^{2}\times$

$\displaystyle\\{\mathrm{Tr}[G_{ee}(\mathbf{k}_{1},\omega_{1})G_{ee}(\mathbf{k}_{2},\omega_{2})]$
$\displaystyle-\mathrm{Tr}[G_{eh}(\mathbf{k}_{1},\omega_{1})G_{he}(\mathbf{k}_{2},\omega_{2})-G_{he}(\mathbf{k}_{1},\omega_{1})G_{eh}(\mathbf{k}_{2},\omega_{2})]$
$\displaystyle-\mathrm{Tr}[G_{hh}(\mathbf{k}_{1},\omega_{1})G_{hh}(\mathbf{k}_{2},\omega_{2})]\\}$
(62)

In the clean limit where the impurity-scattering self-energy vanishes, the
spectral function reduces to $\delta$-function, for example,
$A_{ee}(\mathbf{k},\omega)=$

$-2\pi\left[\begin{array}[]{cc}\frac{E^{\prime}_{3}\delta(\omega-
E_{3})-E^{\prime}_{4}\delta(\omega-E_{4})}{(E_{3}-E_{4})}&0\\\
0&\frac{E^{\prime}_{1}\delta(\omega-E_{1})-E^{\prime}_{2}\delta(\omega-
E_{2})}{(E_{1}-E_{2})}\end{array}\right]$ (63)

where $E^{\prime}_{n}=E_{n}+k^{2}/2m-\varepsilon_{0}$ and $n=1,2,3,4$. The
trace
$\mathrm{Tr}[G_{ee}(\mathbf{k}_{1},\omega_{1})G_{ee}(\mathbf{k}_{2},\omega_{2})]=$

$\displaystyle
G_{ee}(\mathbf{k}_{1},\omega_{1},\downarrow)G_{ee}(\mathbf{k}_{2},\omega_{2},\downarrow)+G_{ee}(\mathbf{k}_{1},\omega_{1},\uparrow)G_{ee}(\mathbf{k}_{2},\omega_{2},\uparrow)$

where
$G_{ee}(\mathbf{k}_{1},\omega_{1},\uparrow)=\frac{(z_{1}+k_{1}^{2}/2m-\varepsilon_{0})}{(z_{1}-E_{1})(z_{1}-E_{2})}$
and
$G_{ee}(\mathbf{k}_{1},\omega_{1},\downarrow)=\frac{(z_{1}+k_{1}^{2}/2m-\varepsilon_{0})}{(z_{1}-E_{3})(z_{1}-E_{4})}$.

In 1D, the nonlocal optical conductivity contains the term
$\frac{E^{\prime}_{1}(k-q/2)E^{\prime}_{1}(k+q/2)}{E_{1}(k-q/2)E_{1}(k+q/2)}\delta(\omega_{1}-E_{1}(k-q/2))\delta(\omega_{1}+\omega-
E_{1}(k+q/2))$, the two $\delta$-functions could be used to carry out the
integral over $\omega_{1}$, and we are left with the one dimension integral
over $k$. The upper and lower limit in the integral over $k$ is set by the
Fermi-Dirac distribution function. The analytical result will be very long and
complicated because many terms from intra-band and inter-band need to be
considered. Instead we decide to carry out the integration numerically. The
self-energy from impurity scattering is nonzero in all the numerical results.

In Fig. 9, we show the real part of the optical conductivity for various wave
vectors $q$. For $q=0$, we recover the famous Drude peak for the intra-band
optical conductivity. For non-zero wave-vector $q$, we observe the shift of
the Drude peak to higher frequency. The single Drude peak is separated into
two peaks, which are associated with the two typical energies for the intra-
band transition. This is due to the fact that in one dimension, the slope near
the four Fermi momenta (the momenta at which the Fermi surface intersects with
the band) are different. The slope is related to the Fermi velocity. There are
two typical Fermi velocities $v_{F1}$ and $v_{F2}$. Thus for the same $q$ we
observe two intra-band peaks roughly at $\omega_{1}=q\times v_{F1}$ and
$\omega_{2}=q\times v_{F2}$, in the frequency region $\omega<0.2E_{0}$. The
inter-band region starts from around $\omega>0.2E_{0}$, as observed in the
optical conductivity for the case $q=0$. The peaks in the inter-band region
are determined from the peaks in the joint density of states. For the cases
$q\neq 0$, the peaks in the inter-band region are determined from the peaks in
the joint spectral functions.

##  Appendix C Band structure and a tight binding model of monolayer
$\rm{VSe_{2}}$

Figure 10: The band structure of monolayer $\rm{VSe_{2}}$ from DFT calculation with spin orbit coupling (SOC).Note that the band structure is close to the spin polarized DFT calculation as in the Fig. 2(a) of the main text. In the 2D calculation, we set the lattice constant $c=20\AA$ to ensure that there is no interaction between the layers, to achieve an ideal single-layer 2D material condition. Figure 11: (Color online) The band structure of 3D $\rm{VSe_{2}}$ from spin polarized DFT calculation. Note that the band structure around the points $\Gamma$, $M$, $K$ is close to those in the 2D monolayer $\rm{VSe_{2}}$ (Fig. 2(a) of the main text). In 3D we have more high symmetry momentum points $H$, $A$, $L$ and observe different band structure around these points as expected.  Figure 12: (Color online) A simplified tight binding model based on the Wannier fitting of the band structure of monolayer $\rm{VSe_{2}}$. Solid line represent the hopping between V and Se atoms, the hopping parameters are given in table II of the main text. Dashed line represent the hopping between $\rm{Se_{1}}$ and $\rm{Se_{2}}$ atoms, the hopping parameters are given in table I of the main text. Note that the dashed line hopping terms are not included in the approximate tight binding model $H_{t}$.  Table 5:  The V-V hopping amplitude. | V(1 0 0)
---|---
$d_{xy}$ | $d_{yz}$ | $d_{z^{2}}$ | $d_{xz}$ | $d_{x^{2}-y^{2}}$
V | $d_{xy}$ | -0.183 | 0 | -0.052 | 0.153 | 0
$d_{yz}$ | 0 | -0.052 | 0 | 0 | 0.215
$d_{z^{2}}$ | -0.052 | 0 | 0.095 | -0.085 | 0
$d_{xz}$ | 0.153 | 0 | -0.085 | -0.271 | 0
$d_{x^{2}-y^{2}}$ | 0 | 0.215 | 0 | 0 | 0.135

In this section, we present DFT calculations of the 2D and 3D $\rm{VSe_{2}}$
band structure to be compared with the results presented in the main text. In
Fig. 10, we present the DFT calculation with the spin-orbit coupling (SOC)
effect and observe the impact of SOC on 1T-$\rm{VSe_{2}}$ is minimal. In Fig.
11, we present the 3D $\rm{VSe_{2}}$ band structure.

A simplified tight binding model could be obtained by using the hopping
parameters in the table I, II and III of the main text. In Fig. 12, we use a
schematic to describe the hoppings on the lattice. The solid line represent
the hopping between V and Se atoms, the hopping parameters are given in table
II of the main text, the model is given below

$\displaystyle
H_{t}=\sum_{i,\delta,d,p}[t_{dp1}(|V_{i,d}><Se_{1,i,\delta,p}|+|Se_{1,i,\delta,p}><V_{i,d}|)+$
$\displaystyle
t_{dp2}(|V_{i,d}><Se_{2,i,-\delta,p}|+|Se_{2,i,-\delta,p}><V_{i,d}|)]$ (64)

where $i$ is the lattice site, $\delta=\delta_{1,2,3}$ are the three hopping
vectors, $d$ are the five $d$ orbitals of V atom, $p$ are the three $p$
orbitals of Se atom, $t_{dp1}$ and $t_{dp2}$ are the hopping parameters given
in table II of the main text. Here $\delta_{1}=(1.667,0.962)$,
$\delta_{2}=(0.000,-1.924)$ and $\delta_{3}=(-1.667,0.962)$ in unit of $\AA$.
The tight binding Hamiltonian is a 11 by 11 matrix. The dashed line in Fig. 12
represents the hopping between $\rm{Se_{1}}$ and $\rm{Se_{2}}$ atoms, the
hopping parameters are given in table I of the main text. The next nearest
neighbor hopping between Se atoms could be obtained from table III of the main
text. In table I of this supplementary material, we list the nearest neighbor
V-V hopping parameters for $\rm{VSe_{2}}$ from the cell (000) to (100), which
is much smaller than the other hopping parameters.