# Topological Antiferromagnetic Semimetal for Spintronics: A Case Study of a
Layered Square Net System EuZnSb2

Niraj Aryal<EMAIL_ADDRESS>Condensed Matter Physics and Materials Science
Division, Brookhaven National Laboratory, Upton, New York 11973, USA  Qiang Li
Condensed Matter Physics and Materials Science Division, Brookhaven National
Laboratory, Upton, New York 11973, USA  Department of Physics, Stony Brook
University, Stony Brook, New York 11794, USA  A. M. Tsvelik  Condensed Matter
Physics and Materials Science Division, Brookhaven National Laboratory, Upton,
New York 11973, USA  Weiguo Yin<EMAIL_ADDRESS>Condensed Matter Physics and
Materials Science Division, Brookhaven National Laboratory, Upton, New York
11973, USA

###### Abstract

We use the first principles and effective Hamiltonian methods to study the
electronic structure and magnetic properties of a recently synthesized layered
antiferromagnetic square net topological semimetal EuZnSb2 Wang  _et al._
(2020). The main message of the paper is that effects of small changes in the
band structure produced by the magnetic ordering and changes in the
orientation of the Néel vector are amplified in such transport properties as
the spin Hall conductivity. We predict that the effects of the broken symmetry
introduced by the ordering of the Néel vector, being very weak in the bulk,
are pronounced in the surface electronic dispersion, suggesting that surface
probes may be more suited to measure them. The coexistence of the magnetism
with many other competing phases make this material interesting and possibly
useful for quantum spintronics applications.

##  I Introduction

Interplay of the topological bands and competing magnetic orders could result
in novel physical properties such as large anomalous Hall effect Wang  _et
al._ (2018) and axion electrodynamics Li  _et al._ (2010). It also presents a
possibility to use magnetic reordering to manipulate electronic transport
Šmejkal  _et al._ (2018). The recent discoveries of magnetic topological
insulators Otrokov  _et al._ (2019) and magnetic Dirac Máca  _et al._ (2012);
Tang  _et al._ (2016); Niu  _et al._ (2020) and Weyl Wan  _et al._ (2011); Xu
_et al._ (2011); Yang  _et al._ (2017); Xu  _et al._ (2020) semimetals have
triggered a flurry of research activities on this topic Zou  _et al._ (2019).
In particular, antiferromagnetic (AFM) systems with broken parity
($\mathcal{P}$) and time reversal ($\mathcal{T}$) symmetries but unbroken
$\mathcal{P}\mathcal{T}$ symmetry have attracted a lot of interests recently
for novel effects such as the electrical control of AFM magnetization Wadley
_et al._ (2016); Salemi  _et al._ (2019) and the Dirac band topology Šmejkal
_et al._ (2017); Shao  _et al._ (2019).

After Young and Kane proposed the existence of topological nodal Fermions in
square net motifs Young and Kane (2015), different variants of the square net
topological materials have been studied extensively Schoop  _et al._ (2016);
Klemenz  _et al._ (2019). Recently, antiferromagnetic semimetals consisting of
strongly correlated 4f-electrons in the 111 family of type LnSbTe
[Ln=lanthanide elements] Schoop  _et al._ (2018); Hosen  _et al._ (2018) and
112 family of type LnMn(Bi,Sb)2 have been reported to host topological Fermi
surface similar to the well known ZrSiS family of materials Wang  _et al._
(2016); Kealhofer  _et al._ (2018); May  _et al._ (2014); Yi  _et al._ (2017).
It is known that when the magnetic atoms directly contribute to the formation
of the conduction bands as in such transition metal antiferromagnetic Dirac
semimetal (AFM-DSM) systems as CuMnAs, Mn3Ge etc., the magnetic ordering and
in particular, the orientation of the magnetic moments can bring subtle
changes in the electronic structure and the related transport properties Shao
_et al._ (2019, 2020). Such subtle changes in the band topology and associated
transport signatures in AFM-DSM caused by changes in the orientation of the
magnetic moments may be useful for spintronics applications such as low power
electronics and magnetic memory devices Núñez  _et al._ (2006); Shick  _et
al._ (2010).

An influence of magnetism on the electronic dispersion and transport is less
explored in the afore-mentioned $f$-electron square net systems, where it may
also lead to interesting effects. However, the conduction bands in these
systems are formed by the $p_{x}$-$p_{y}$ orbitals of the (Bi, Sb) square nets
weakly hybridized with the f-electron bands lying far away from the Fermi
level. This makes a connection between the $f$-electron magnetism and the
conduction bands as well as the associated transport anomalies less obvious.
Given the potential advantages of f-electron systems over conventional
semiconductors for spintronics applications Kontani  _et al._ (2008); Peters
and Yanase (2018) and the availability of rich material pool and magnetic
properties obtained by varying Ln elements Lei  _et al._ (2019), it is
important to study the systems where the itinerant electrons coexist with the
localized ones.

Recently, Wang et. al. have reported a discovery of a layered 4f square net
material EuZnSb2 Wang  _et al._ (2020). This material is the zintl cousin of
the more famous Mn-based 112 phases. Because of the unpaired 4f-electrons, it
orders antiferromagnetically with a Néel temperature (TN) of 20K. Density
functional theory calculations showed the presence of the extended Fermi
surface formed by the $p$-electrons. In this paper, we study in detail the
electronic structure of various AFM phases of this material. We aim to
establish if and how the orientation of the Néel vector influences the band
topology and related transport properties. We find that magnetic orderings
introduce small but non-negligible corrections in the gap size across the
Fermi surface which have consequences in the Berry curvature related transport
properties. More importantly, we find that depending on the orientation of the
Néel vector, different crystalline symmetries are broken globally. Such broken
symmetries are manifested in the electronic structure and transport
properties. Our findings suggest that systems with coexisting itinerant and
localized $f$-electrons can be useful platforms for topological spintronics
applications and more studies along this direction are necessary.

The organization of this paper is as follows. In Sec. II, we present the
details of our computational methods. In Sec. III, we present our results and
discuss them in detail. Finally, in Sec. IV, we present our conclusion and
future outlook. The derivation of the tight-binding model and the effective
Kondo exchange Hamiltonian is relegated to the Appendix.

##  II Computational Details

The density-functional-theory (DFT) calculations were done using Wien2k DFT
package Blaha  _et al._ (2020). The basis size was determined by
$R_{\textrm{mt}}K_{\textrm{max}}$ = 7 and the Brillouin zone was sampled with
a regular $18\times 18\times 3$ mesh containing 162 irreducible $k$ points to
achieve energy convergence of 1 meV. A 10,000 $k$-point mesh was used for the
Fermi surface calculations. Some of the calculations, especially in the
paramagnetic phase were verified using Quantum Espresso (QE) Giannozzi  _et
al._ (2009) package. Perdew-Burke-Ernzerhof (PBE) exchange-correlation
functional Perdew  _et al._ (1996) within the generalized gradient
approximation (GGA) were used in all the calculations. The GGA +
$U_{\textrm{eff}}$ method was used to handle the Eu $4f$ orbitals.
$U_{\textrm{eff}}$ of 6 eV was chosen in our calculations Johannes and Pickett
(2005); Kunes  _et al._ (2005); Larson and Lambrecht (2006); Zhang  _et al._
(2017); however we have also verified that the results presented here remain
robust for a large range of $U_{\textrm{eff}}$ values. The spin-orbit coupling
(SOC) was treated in the second variation method. The spin Hall conductivity
calculations were done using Wannier90 software Mostofi  _et al._ (2014); Qiao
_et al._ (2018) by taking 80$\times$80 Wannierised Hamiltonian. All Eu 4d, 4f
orbitals and Sb 5s, 5p orbitals were used in the Wannierisation procedure in
order to accurately reproduce the DFT bands in the energy window from $-1$ to
1 eV.

##  III Results and Discussion

###  III.1 Crystal Structure

Figure 1: The unit cell of EuZnSb2. Fig. (a) shows the stackings of the
layered square lattices along the c-axis and Fig. (b) shows the projection
onto the a-b plane and the denser ($\sqrt{2}\times\sqrt{2}$) square lattice of
Sb2 atoms. Eu atoms above and below Sb2 atoms occupy the interstitial site of
the Sb2 square lattice and are related by the glide (or inversion) symmetry;
Zn atoms occupy the same site as Sb2 atoms.

EuZnSb2 is a layered square net material in the space group $P4/nmm$ (no. 129)
similar to the well-known nodal-line family of materials of ZrSiS Wang  _et
al._ (2020). The crystal structure of EuZnSb2 is shown in Fig. 1. The unit
cell consists of stacking of square lattices of Eu, Zn and two types of Sb
atoms (called Sb1 and Sb2 here) along the c-direction in the arrangement of
—Zn-Sb1-Eu-Sb2-Eu-Sb1-Zn—. Sb2 and Zn atoms form a denser
($\sqrt{2}\times\sqrt{2}$) square lattice (also known as $4^{4}$ square
lattice in the crystallographic community Tremel and Hoffmann (1987)), with 2
atoms in each 2D square plane whereas Sb1 and Eu atoms form a less denser
square lattice with just 1 atom in each 2D square plane. Sb2 and Zn atoms
occupy the same site when projected on the a-b plane whereas Eu (and Sb1)
atoms above and below Sb2 atoms occupy the interstitial site of the 44 lattice
and are related by inversion (or a glide ) symmetry.

###  III.2 Non-magnetic phase and Wannier tight-binding analysis

Figure 2:  Comparison between the DFT calculated bands (gray lines) with (a) 4
and (b) 6 band TB Hamiltonian eigenvalues (red-blue dots) for the paramagnetic
phase without SOC obtained using the parameters in Table 1. In (c), we show
the 6 $p_{x}$, $p_{y}$ and $p_{z}$-like Wannier orbitals used in the
description of the 6-band TB Hamiltonian.

It was shown in Ref. Wang  _et al._ , 2020 that the lowest energy
antiferromagnetic phase of EuZnSb2 hosts broad band dispersion close to the
Fermi level. Before investigating how the band topology changes with the
change in the magnetic texture, we would like to understand the origin of the
conduction bands. For this end we first study the non-magnetic phase in the
absence of the Eu-4f electrons within the opencore approximation to simplify
the problem. Since we are interested in the electronic dispersion in the
vicinity of the Fermi level, such opencore electron description is
approximately equivalent to the application of the Hubbard $U$ within the
mean-field approximation that pushes the localized states away from the Fermi
level.

Figure 3:  Band structure from nnn 4$\times$4 Hamiltonian using the tight-
binding parameters in Table 1. In (a), sublattice and $\mathcal{R}_{4}$ is
broken explicitly in the 4$\times$4 model by introducing onsite anisotropy and
making the $\sigma$ and $\pi$ hopping different along $\hat{x}$ and $\hat{y}$
direction. In (b), the effect of SOC is included. The insets show the gap
openings in the band crossings.

The DFT calculated band dispersion for non-magnetic calculation in the absence
of SOC are shown in Fig. 2(a) (gray lines). The band diagram shows crossings
along $\Gamma$-M and $\Gamma$-X directions close to the Fermi level. Such
crossings are the consequence of the band folding due to the doubling of the
unitcell and are formed between the dispersive $p_{x}$-$p_{y}$ orbitals from
the denser square nets of Sb2 atoms. The $p_{z}$ states from Sb2 atoms have
smaller band width and are almost completely filled. Sb1 $p$ orbitals also
contribute to the states close to the Fermi level (see band character plots in
Appendix A Fig. 10).

The red and blue dots overlapped onto the gray lines in Figs. 2(a) and (b) are
the results from 4 and 6 band Wannier function description, respectively,
obtained by using the method of disentanglement Mostofi  _et al._ (2014).
Hopping parameters only up to the next-nearest neighbour (nnn) given in Table.
1 are used for this comparison. The 4-band model reproduces the crossings
between the dispersive $p_{x}$-$p_{y}$ bands along the $\Gamma$-M and
X-$\Gamma$ directions but it can not describe other features in the vicinity
of the Fermi level. It appears that 6 band tight-binding (TB) model is the
minimal model to describe the band structure in the vicinity of the Fermi
surface. Even with the nnn TB model, one can describe approximately the
generic features as seen in Fig. 2(b). In fact, by including the long range
hopping terms, we can reproduce all the features in the periphery of the Fermi
level exactly (see Fig. 12 in the Appendix A where all long range hopping
matrix elements are included in the 6-band model). The Wannier orbitals are
formed by the bond directed $p_{x}$, $p_{y}$ and $p_{z}$-like orbitals
centered on the Sb2 atoms forming the square lattice as shown in Fig. 2(c).

Qualitatively, 4-band model Hamiltonian in the basis of the $p_{x}$ and
$p_{y}$ orbitals is sufficient to understand the origin of the band dispersion
along the $\Gamma$-M and X-$\Gamma$ Klemenz  _et al._ (2020). Hence, in the
following, we only discuss the simple 4-band model to understand the effect of
$\mathcal{R}_{4}$ symmetry breaking introduced by the magnetic Eu-atoms and
the action of the spin orbit coupling. Notice that in the absence of the nnn
term, four fold degenerate bands are present right at the X high symmetry
point whereas nnn term makes the bands two fold degenerate slightly away from
the X point [see Fig. 11]. We refer the reader to Appendix A for exact
analytical expression of the eigenvalues from the 4$\times$4 Hamiltonian and
other effective Hamiltonian analysis.

Table 1:  Tight binding parameters (in eV) for the 6 band Wannier functions in the basis of the $p_{x}$, $p_{y}$ and $p_{z}$-like Wannier orbitals centered on the Sb2 atoms forming square lattice. The nearest neighbour (nn) $p_{i}$-$p_{i}$ hopping integrals are $t_{\sigma}=1.9$ and $t_{\pi}=-0.5$ for the $\sigma$\- and $\pi$-bonding-like orbital overlaps, respectively.  $\langle WFs|H|WFs\rangle$ |  Sb $p_{x}$ |  $p_{y}$ |  $p_{z}$
---|---|---|---
Sb $\epsilon-\mu$ |  0.12 |  0.12 |  -1.0
Sbnn $p_{x}$ |  tσ;tπ |  0.00 |  -0.05
Sbnn $p_{y}$ |  0.00 |  tσ;tπ |  -0.05
Sbnn $p_{z}$ |  -0.05 |  -0.05 |  -0.10
Sbnnn $p_{x}$ |  0.00 |  $\pm$0.10 |  0.00
Sbnnn $p_{y}$ |  $\pm$0.1 |  0.00 |  0.00
Sbnnn $p_{z}$ |  0.00 |  0.00 |  0.20

Gap on the Fermi surface– The Fermi surface (FS) can be gapped partially along
$\Gamma$-M by breaking the sublattice symmetry (or equivalently inversion or
glide symmetry of the 2D square plane). This can be done easily in our TB
Hamiltonian by introducing the onsite anisotropy term. However, in order to
open a gap at the X-$\Gamma$ direction, one needs to break both sublattice and
$\mathcal{R}_{4}$ symmetry; the latter can be introduced in our TB model by
making the $\sigma$ and $\pi$ hoppings asymmetric along the $\hat{x}$ and
$\hat{y}$ directions. Fig. 3(a) presents the case of opening gaps in the band
crossings by introducing such terms in our TB model.

In the presence of the SOC, the entire Fermi surface gaps out. We find the
form of SOC in our TB Hamiltonian to be
$(\lambda\sigma_{z}\tau_{y}+\delta\gamma_{z}\sigma_{z}\tau_{z})$ where
$\lambda$ and $\delta$ are constants and $\gamma$, $\sigma$, $\tau$ are the
Pauli matrices acting on the site, spin, and orbital indices, respectively.
The first term introduces coupling between the $p_{x}$ and $p_{y}$ orbitals of
same site and same spin whereas the second term introduces the on-site
asymmetry. Fig. 3(b) shows the action of SOC on the nnn tight-binding model
using $\delta=0.1$ and $\lambda=0.2$ which are extracted from the ab-initio
Wannier function analysis. In addition to gapping out the entire Fermi
surface, the introduction of SOC breaks 2 fold degeneracy at the $\Gamma$
point but preserves crossings at M and X points away from the Fermi level.
These observations are consistent with the DFT calculation.

###  III.3 Magnetic Phases and Electronic Structure

Having explored the electronic dispersion of the non-magnetic phase in detail,
we are now in the position to understand the influence of magnetic texture in
the electronic properties. Ref. Wang  _et al._ , 2020 found that two AFM
patterns (referred to as AAF and AAF3) of EuZnSb2 are competing for the ground
state, i.e., having very small energy differences well within the error bar of
the calculations. The different magnetic patterns studied in this work are
shown in Fig. 4. Throughout this manuscript, the $\hat{z}$ and $\hat{x}$
magnetic phase (alternatively called soz and sox phases in the manuscript)
imply the direction of the Néel vector ($\vec{n})$ parallel to the [001] and
[100] direction respectively. Our DFT calculations find that the energy
differences between the magnetic phases depend on the value of the Hubbard
interaction $U$. As shown in Table 2, for $U=0$, AAF3 has lower energy,
whereas as $U$ is increased, AAF becomes lower in energy. Also the magnetic
anisotropy energy is very small i.e the energy differences between the
$\hat{z}$ and $\hat{x}$ AFM pattern are almost the same. Note that the
ferromagnetic (FM) state is also closer in energy, but the nonmagnetic (NM)
phase is a very high-energy state, due to the Eu2+ $4f^{7}$ high-spin
electronic configuration yet with weak coupling between well-localized $f$
electrons. The relevant information about magnetic space groups and symmetry
relationships is listed in Table 3.

Figure 4:  (a, b) AAF and (c, d) AAF3 magnetic patterns of EuZnSb2 with magnetization along the $\hat{x}$ and $\hat{z}$ axes, respectively. Table 2: Calculated energy difference per formula unit in meV for different magnetic patterns (see text). Pattern |  GGA |  GGA+SO |  GGA+SO+U (3eV) |  GGA+SO+U (6eV)
---|---|---|---|---
AAF-x |  0 |  0 |  0 |  0
AAF-z |  - |  0.804 |  0.24 |  0.017
AAF3-x |  - |  $-2.85$ |  $-0.41$ |  1.29
AAF3-z |  $-6.46$ |  $-6.18$ |  $-0.53$ |  1.35
FM |  1.13 |  $-0.53$ |  - |  3.71
NM |  7297 |  5795 |  - |  -
Table 3: The magnetic symmetry group and symmetry operations of different magnetic patterns. In the first and last 2 rows, the operations which are different are colored in gray. Other symmetry operations can be generated by the action of $\mathcal{P}\mathcal{T}$ for the AAF phase and by the action of $\mathcal{P}$, $\mathcal{T}^{\prime}$ and $\mathcal{P}\mathcal{T}$ for the AAF3 phase, hence not mentioned here for clarity. Pattern |  MSG |  Symmetry Operations
---|---|---
AAF-z |  129.419 (P$4/n^{\prime}m^{\prime}m$) |  ${\color[rgb]{.5,.5,.5}\\{4^{+}_{001}|\frac{1}{2}00\\}}$, ${\color[rgb]{.5,.5,.5}\\{4^{-}_{001}|0\frac{1}{2}0\\}}$, ${\color[rgb]{.5,.5,.5}\\{2_{100}|\frac{1}{2}00\\}}$, $\\{2_{010}|0\frac{1}{2}0\\}$, ${\color[rgb]{.5,.5,.5}\\{2_{001}|\frac{1}{2}\frac{1}{2}0\\}}$, ${\color[rgb]{.5,.5,.5}\\{2_{110}|\frac{1}{2}\frac{1}{2}0\\}}$, ${\color[rgb]{.5,.5,.5}\\{2_{1-10}|0\\}}$, $\\{-1^{\prime}|0\\}$
AAF-x |  59.407 (P${}_{m^{\prime}mn}$) |  ${\color[rgb]{.5,.5,.5}\\{m_{100}|\frac{1}{2}00\\}}$, {2${}_{010}|0\frac{1}{2}0$}, ${\color[rgb]{.5,.5,.5}\\{m_{001}|\frac{1}{2}\frac{1}{2}0\\}}$, $\\{-1^{\prime}|0\\}$
AAF3-z |  130.432 (Pc4/ncc) |  ${\color[rgb]{.5,.5,.5}\\{4^{+}_{001}|\frac{1}{2}00\\}}$, ${\color[rgb]{.5,.5,.5}\\{4^{-}_{001}|0\frac{1}{2}0\\}}$, ${\color[rgb]{.5,.5,.5}\\{2_{110}|\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}}$, ${\color[rgb]{.5,.5,.5}\\{2_{1-10}|00\frac{1}{2}\\}}$, $\\{2_{001}|\frac{1}{2}\frac{1}{2}0\\}$, $\\{2_{100}|\frac{1}{2}0\frac{1}{2}\\}$, $\\{2_{010}|0\frac{1}{2}\frac{1}{2}\\}$, $\\{-1|0\\}$, $\\{1^{\prime}|00\frac{1}{2}\\}$
AAF3-x |  62.450 (Panma) |  $\\{2_{001}|\frac{1}{2}\frac{1}{2}{\color[rgb]{.5,.5,.5}\frac{1}{2}}\\}$, $\\{2_{100}|\frac{1}{2}0{\color[rgb]{.5,.5,.5}0}\\}$, $\\{2_{010}|0\frac{1}{2}\frac{1}{2}\\}$, $\\{-1|0\\}$, $\\{1^{\prime}|00\frac{1}{2}\\}$

####  III.3.1 AAF phase

Figure 5: Comparison of the electronic structure between the AAF-$\hat{z}$ and
AAF-$\hat{x}$ phases with the inclusion of SOC and U of 6 eV. (a) Band
structure along $\Gamma$-M-X-$\Gamma$-Z-A-R-Z high symmetry direction for the
two phases. The red-blue (green-magenta) lines are for magnetization along
$\hat{x}$ ($\hat{z}$) direction. (b, c) Fermi surface plots on the 3D
Brillouin zone. (d, e) Intensity plot of the inverse band gap value between
the valence and conduction band on the $k_{x}$-$k_{y}$ plane at $k_{z}$=0
which shows the formation of the gapped nodal line feature around the $\Gamma$
point. The rectangular boxes are drawn to highlight the fact that the
$\hat{x}$-phase has a broken $\mathcal{R}_{4}$ symmetry whereas
$\hat{z}$-phase obeys $\mathcal{R}_{4}$ perfectly. (f, g) Contour plot of the
Eu-4f orbital distribution on the valence band. The intensity range (color-
bar) is shown in a narrow to amplify the small differences between the phases.
These figures again highlight that $\mathcal{R}_{4}$ is weakly broken for the
$\hat{x}$ phase. See text for details.

The AFM arrangement of the 2 Eu atoms in the primitive unitcell located above
and below the Sb2 square lattice gives AAF phase. The two
antiferromagnetically aligned Eu atoms are no longer inversion symmetric
irrepective of any direction of the Néel vector. This is because inversion
symmetry does not operate on the spin degree of freedom. However, because of
the broken inversion ($\mathcal{P}$) and broken time reversal symmetry
($\mathcal{T}$), their product $\mathcal{P}\mathcal{T}$ is a conserved
quantity which makes the bands doubly degenerate throughout the BZ. Such
magnetic space group (MSG) falls into type-II MSG.

Unlike inversion symmetry, rotation (or screw) and mirror (or glide)
symmetries act on the spin degree of freedom. Hence, depending on the
orientation of the Néel vector, some of the symmetries could be broken. For
example, $\hat{x}$ ($\hat{z}$) direction of the Néel vector preserves (breaks)
glide symmetry $\mathcal{G}_{z}=\\{m_{001}|\frac{1}{2}\frac{1}{2}0\\}$ but
breaks (preserves) 2-fold rotation symmetry
$\mathcal{R}_{2z}=\\{2_{001}|\frac{1}{2}\frac{1}{2}0\\}$ which can be seen
from the following action of the symmetry operation on the space and spin
degree of freedom:

$\begin{gathered}(x,y,z)\xrightarrow{\mathcal{G}_{z}}(x+\frac{1}{2},y+\frac{1}{2},-z),\\\
(x,y,z)\xrightarrow{\mathcal{R}_{z}}(-x+\frac{1}{2},-y+\frac{1}{2},z),\\\
(m_{x},m_{y},m_{z})\xrightarrow{\mathcal{G}_{z}/\mathcal{R}_{z}}(-m_{x},-m_{y},m_{z})\end{gathered}$
(1)

Similarly, other symmetries absent in the AAF-$\hat{x}$ magnetic phase are
four fold rotation symmetries along the c-axis
$\\{4^{\pm}_{001}|\frac{1}{2}00\\}$, 2-fold rotation along the [110] axes
$\\{2_{110}|\frac{1}{2}\frac{1}{2}0\\},\\{2_{1-10}|0\\}$ and the product of
these symmetries with $\mathcal{P}\mathcal{T}$ symmetry. The details of the
symmetry operations present for different magnetic patterns are shown in
Table. 3.

In the following paragraphs, we will first present results from the DFT
calculations which show clear differences in the bulk and surface electronic
dispersion between the $\hat{z}$ and $\hat{x}$ phase. We will then derive
effective Kondo exchange Hamiltonian using parameters extracted from DFT
calculations to understand some of the major findings.

##### Bulk states:

The bulk band structure comparison between the AAF-$\hat{x}$ and AAF-$\hat{z}$
phases in a narrow energy window of $\pm$1 eV is shown in Fig. 5(a). Fig. 15
in Appendix C shows the electronic dispersion and density of states (DOS) in a
wider energy range. The Eu-4f states are 1 eV below the Fermi level and the
small DOS at the Fermi level mainly comes from the Sb2 $p_{x}$-$p_{y}$
orbitals.

We find that despite the absence of some symmetries in the AAF-$\hat{x}$ phase
as mentioned in the previous section, the bulk band structure for both phases
in the vicinity of the Fermi level looks almost identical except small
momentum dependent shifts. This is expected as the localized Eu-$4f$ orbitals
are pushed away from the Fermi level due to the application of the Hubbard $U$
correction. The Fermi surface plots in Fig. 5(b &c) highlight this fact which
show similar features except slight difference at the $\Gamma$-point. In fact,
the electronic dispersion from the non-magnetic phase [Fig. 2] is not very
different from the AAF phase which shows that the effect of magnetism in the
electronic dispersion is small in this system. In the absence of SOC, the
valence and conduction bands form a gapped nodal line on the $k_{x}$-$k_{y}$
plane with band crossings along the $\Gamma$-M direction. All the crossings
are gapped by the action of SOC for both phases. This is similar to the case
of nodal line semimetal ZrSiS where SOC has been found to open gap in the
Dirac crossings Schoop  _et al._ (2016).

Although there are no protected crossings for both phases, the magnitude of
the band gap is different for the two magnetic phases. In Fig. 5(d & e), we
show the inverse of the eigenvalue difference between the valence and the
conduction bands on the $k_{x}$-$k_{y}$ plane at $k_{z}=0$ to highlight the
presence of the gapped nodal line feature. In the $\hat{z}$ phase, the gap
distribution is identical in all 4-quadrants, however that is not the case in
the $\hat{x}$ phase. This is a clear signature of the broken global
$\mathcal{R}_{4}$ symmetry on the $\hat{x}$ phase only [Fig. 5(e)]. To
understand further the origin of the $\mathcal{R}_{4}$ symmetry breaking, we
also looked at the k-dependent valence band occupancy of the $4f$-electrons on
the $k_{x}$-$k_{y}$ plane [Figs. 5(f & g)]. In the vicinity of the
$\Gamma$-point, we find that the $4f$-electron contribution to the valence
band is not symmetric across the 4-quadrants unlike the $\hat{z}$ phase
indicating that $\mathcal{R}_{4}$ symmetry is weakly broken due to the
orientation of the Eu-$4f$ magnetic moments.

##### Surface states:

Figure 6: Comparison of the slab Fermi surface between the (a) $\hat{z}$ and
(b) $\hat{x}$ phase. The gray lines are the bulk-derived states and the red-
blue lines denote the hole and electron Fermi pockets obtained from the 3
layer slab calculation. Notice that the broken $\mathcal{R}_{4}$ symmetry is
amplified in the surface dispersion in panel (b).

We also performed slab calculation to see how the orientation of the Néel
vector affects the surface states. Fig. 6 shows the slab Fermi surface of the
AAF-$\hat{z}$ and $\hat{x}$ phases obtained from a 3 layer slab calculation.
The surface is terminated on the Zn and Sb1 layers. The slab calculations show
a number of additional features compared to the bulk states which are shown as
a gray background. The most notable feature which is absent in the bulk
dispersion is the closed loop (red lines) state around the $\Gamma$ point
connecting the gray ellipsoids. Remarkably, the surface states are different
for the two phases with the breaking of the $\mathcal{R}_{4}$ symmetry clearly
visible now for the $\hat{x}$ phase unlike the subtle differences we found in
in the bulk-band features. For example, along the $\Gamma$-X as well as
$\Gamma$-M directions, the surface states, especially the hole ones (red
colored lines), are related by the mirror symmetry $\mathcal{M}_{x}$ but are
asymmetric with respect to the $\mathcal{R}_{4}$ symmetry i.e. the pockets are
not identical along the 4-quadrants only for the $\hat{x}$ phase. This is
amazing given the fact that the slab is not terminated on the Eu-atoms and the
Eu-4f states have negligible contribution to the Fermi surface. This
demonstrates that the magnitude of the broken symmetry in the electronic
dispersion are more amplified on the surface compared to the bulk states.
Hence, we anticipate that surface probes could be more suitable for resolving
the broken symmetry phases.

In order to understand further how Eu-$f$ states can affect the Fermi surface
properties, we have derived an effective Kondo exchange Hamiltonian for the
$p_{x}$-$p_{y}$ electrons in the presence of the Eu-$f$ spins in Appendix B.
From such effective Hamiltonian analysis, we find that the Eu magnetic moments
can affect the dispersion of the itinerant $p_{x}$-$p_{y}$ electrons through
exchange coupling which introduces hopping between the $p$-electrons through
Eu-sites. In addition, we show that depending on the orientation of the Eu-$f$
spins, the band spectrum as well as their spin-texture will be different. This
will have consequences in the spin transport properties which will be
discussed later.

####  III.3.2 AAF3 phase

Figure 7: (a) Comparison of the electronic dispersion between the
AAF3-$\hat{x}$ (red-blue) and $\hat{z}$ (green-magenta) magnetic phases. The
inset shows that $\hat{z}$-phase has band crossings near the Fermi level
whereas they are avoided for the $\hat{x}$-phase. (b) Intensity plot of the
inverse band gap between the valence and conduction bands for the $\hat{z}$
phase showing crossings along the $\Gamma$-M directions and in the vicinity of
the X point. The small gap of $\sim$ 10 meV seen in the plot is due to the
finite k-mesh.

We also studied another competing AFM phase AAF3 in detail. AAF3 phase
requires the doubling of the unitcell along the c-axis as seen in Fig. 4 (c &
d). Unlike the AAF phase, $\mathcal{P}$ is preserved here. Moreover, despite
breaking of the $\mathcal{T}$ symmetry, $\mathcal{T}^{\prime}=\mathcal{T}\tau$
i.e. $\mathcal{T}$ followed by translation $\tau=\\{0,0,\frac{1}{2}\\}$ is the
symmetry of the system. Such MSG with non-symmorphic $\mathcal{T}$ falls into
type IV category and has been found in another AFM Dirac material EuCdAs2 Hua
_et al._ (2018). $\mathcal{P}\mathcal{T}^{\prime}$ symmetry makes the bands
doubly degenerate throughout the Brillouin zone. In addition, because of the
half translation along with $\mathcal{T}$, there are interesting symmetry
properties.

Similar to the AAF magnetic pattern, the $\hat{z}$-phase here has more
symmetry compared to the $\hat{x}$-phase due to the direction of the spins.
For example, $\mathcal{R}^{+}_{4}$ symmetry is preserved for magnetization
along $\hat{z}$ direction whereas it is not preserved for magnetization along
$\hat{x}$ or $\hat{y}$ directions. The complete list of symmetry operations
are tabulated in Table. 3.

Following closely the arguments given in Refs. Tang  _et al._ (2016); Shao
_et al._ (2019), we prove that due to the extra rotational symmetries in the
AAF3-$\hat{z}$ phase, band crossings along different high symmetry directions
are preserved whereas they are avoided (or gapped) in the AAF3-$\hat{x}$
phase. First, we find the eigenvalues of the 4-fold rotoinversion symmetry
operator $\bar{\mathcal{R}}^{+}_{4}=\\{\pm-4^{+}_{001}|\frac{1}{2},0,0\\}$:

$\begin{gathered}(x,y,z)\xrightarrow{\bar{\mathcal{R}}^{+}_{4}}(y+\frac{1}{2},-x,-z)\xrightarrow{\bar{\mathcal{R}}^{+}_{4}}(-x+\frac{1}{2},-y-\frac{1}{2},z)\\\
\xrightarrow{\bar{\mathcal{R}}^{+}_{4}}(y,x-\frac{1}{2},-z)\xrightarrow{\bar{\mathcal{R}}^{+}_{4}}(x,y,z),\\\
\end{gathered}$ (2)

Hence, $(\bar{\mathcal{R}}^{+}_{4})^{4}=-1$ and the eigenvalues are
$J_{m}=e^{i\frac{(2m+1)\pi}{4}}$, where the minus sign is from the spin
rotation and $m$=0, 1, 2, 3 such that:

$J_{0}=e^{i\pi/4}=J^{*}_{3},J_{1}=e^{i3\pi/4}=J^{*}_{2}$ (3)

Because of the $\mathcal{P}\mathcal{T}^{\prime}$ symmetry, the bands are 2
fold degenerate throughout the BZ. If $|\psi\rangle$ is the simultaneous
eigenstate of the Hamiltonian operator and the $\bar{\mathcal{R}}^{+}_{4}$, we
would like to find the $PT^{\prime}$ partner of $|\psi\rangle$. For this, we
need to find the commutation of $\bar{\mathcal{R}}^{+}_{4}$ with
$\mathcal{P}\mathcal{T}^{\prime}$.

$\begin{gathered}(x,y,z)\xrightarrow{\bar{\mathcal{R}}^{+}_{4}}(y+\frac{1}{2},-x,-z)\xrightarrow{\mathcal{P}\mathcal{T}^{\prime}}(-y-\frac{1}{2},x,z-\frac{1}{2}),\\\
(x,y,z)\xrightarrow{\mathcal{P}\mathcal{T}^{\prime}}(-x,-y,-z-\frac{1}{2})\xrightarrow{\bar{\mathcal{R}}^{+}_{4}}(-y+\frac{1}{2},x,z+\frac{1}{2}).\end{gathered}$
(4)

i.e.

$\displaystyle\bar{\mathcal{R}}^{+}_{4}PT^{\prime}$ $\displaystyle=$
$\displaystyle\tau(1,0,1)\mathcal{P}\mathcal{T}^{\prime}\bar{\mathcal{R}}^{+}_{4},$
$\displaystyle\implies\bar{\mathcal{R}}^{+}_{4}\mathcal{P}\mathcal{T}^{\prime}|\psi\rangle$
$\displaystyle=$ $\displaystyle
e^{-i(k_{x}+k_{z})}PT^{\prime}\bar{\mathcal{R}}^{+}_{4}|\psi\rangle,$
$\displaystyle\mathrm{RHS},$ $\displaystyle=$ $\displaystyle
e^{-i(k_{x}+k_{z})}\mathcal{P}\mathcal{T}^{\prime}\bar{R}^{+}_{4}|\psi\rangle,$
(5) $\displaystyle=$ $\displaystyle
e^{-i(k_{x}+k_{z})}\mathcal{P}\mathcal{T}^{\prime}J_{m}|\psi\rangle,$
$\displaystyle=$ $\displaystyle
e^{-i(k_{x}+k_{z})}J^{*}_{m}e^{(ik_{z}/2)}\mathcal{P}\mathcal{T}^{\prime}|\psi\rangle,$
$\displaystyle=$ $\displaystyle
e^{-i(k_{x})}e^{-(ik_{z}/2)}J^{*}_{m}\mathcal{P}\mathcal{T}^{\prime}|\psi\rangle.$

This implies that if $|\psi\rangle$ is an eigenstate of
$\bar{\mathcal{R}}^{+}_{4}$ operator with eigenvalue $J_{m}$, then
$\mathcal{P}\mathcal{T}^{\prime}|\psi\rangle$ is also an eigenstate of
$\bar{R}^{+}_{4}$ with eigenvalue $e^{-i(k_{x}-k_{z}/2)}J^{*}_{m}$. Now, let’s
examine the $\bar{\mathcal{R}}^{+}_{4}$ eigenvalues of $|\psi\rangle$ and its
$\mathcal{P}\mathcal{T}^{\prime}$ partner at different high symmetry points
which are invariant under the $\bar{\mathcal{R}}^{+}_{4}$ operation.

At the $\Gamma$-point where $\mathbf{k}=\mathbf{0}$, states with
$\bar{\mathcal{R}}^{+}_{4}$ eigenvalues of $(J_{0}$, $J_{3})$ and $(J_{1}$,
$J_{2})$ form degenerate pairs. Similarly, at the M high symmetry point
$(\pi,\pi,0)$, because of the extra $e^{-ik_{x}}$ factor, $(J_{0},J_{2})$ and
$(J_{1},J_{3})$ form degenerate pairs i.e. the $\bar{\mathcal{R}}^{+}_{4}$
eigenvalues of the Kramer’s pairs switches partner compared to the
$\Gamma$-point. This makes the crossing unavoidable along this line. Following
similar arguments, we find that there is an unavoidable crossing along the Z-A
line.

In Fig. 7(a), we compare the DFT calculated bands between the AAF3-$\hat{x}$
and AAF3-$\hat{z}$ phases. The differences between the two are very small but
most importantly, there are some crossings in the AAF3-$\hat{z}$ patterns
which are absent in the $\hat{x}$ pattern. For example, along the $\Gamma$-M
and Z-A directions, $\hat{z}$-pattern shows crossings whereas there is a small
gap of $\sim$10 meV in the $\hat{x}$-pattern. Similar feature is seen along
the X-$\Gamma$ and R-Z line. On the contrary, at the Z point, $\hat{x}$-phase
shows 4-fold degeneracy whereas $\hat{z}$-phase has a gap of few meVs.

In Fig. 7(b), we plot the 2D bands of the AAF3-$\hat{z}$ pattern on the
$k_{x}$-$k_{y}$ plane at $k_{z}$=0 plane. Similar to the AAF pattern, we see a
gapped nodal line feature with a small difference that point nodes survive
along $\Gamma$-M and $\Gamma$-X direction for the AAF3-$\hat{z}$ phase only.

###  III.4 Calculation of the Spin Hall Conductivity and spin current
manipulation

Figure 8: Comparison of the $\sigma_{xy}^{z}$ component of the SHC tensor as a
function of the chemical potential between the $\hat{x}$ and $\hat{z}$ phases
of the (a) AAF and (b) AAF3 magnetic patterns.

Figure 9: Comparison of the k-resolved spin berry curvature $\Omega_{xy}^{sz}$
between (a) AAF-$\hat{z}$ and (b) AAF-$\hat{x}$ patterns. Top panels: Band
structure along high symmetry directions colored by the spin Berry curvature
of individual bands $\Omega_{n,xy}^{sz}(\mathbf{k})$. The color bar is in the
log scale and sign indicates the sign of the spin Berry curvature. Bottom
panels: Sum of the $\Omega_{n,xy}^{z}(\mathbf{k})$ upto the Fermi level along
the high symmetry directions. In these plots, the values of
$\Omega_{xy}^{z}(\mathbf{k})$ below 1 Å2 are considered 0.

In the previous section, we saw that the orientation of the Néel vector gives
very small changes in the electronic structure. In this section, we show that
even though the electronic dispersion appears similar, the difference in the
spin dependent transport property is appreciable. To highlight this difference
we will study spin Hall conductivity (SHC).

The phenomenon of Spin Hall effect corresponds to generation of a purely
transverse spin current by the applied electric field Sinova  _et al._ (2015).
Out of the three mechanisms (intrinsic, skew scattering and side-jump)
contributing to the SHC, the intrinsic component of the SHC is a direct
consequence of the band topology similar to the phenomenon of anomalous Hall
conductivity when the applied electric field generates a transverse charge
current. SHC is a third rank tensor and is defined as Qiao  _et al._ (2018):

$\sigma_{ab}^{c}(\mu)=-\frac{e^{2}}{\hbar}\int_{BZ}\frac{d^{3}{\bf
k}}{(2\pi)^{3}}\sum_{n}f_{n\bf{k}}(\mu)\Omega_{n,ab}^{c}({\bf k}),$ (6)

where, $f_{nk}(\mu)$ is the $\bf k$-dependent equilibrium occupation factor of
$n^{th}$ band at the chemical potential of $\mu$. $\Omega_{n,ab}^{c}$ is the
band resolved spin Berry curvature which is, in general, a function of $\bf k$
and frequency $\omega$ and is given by:

$\displaystyle\Omega_{n,ab}^{c}(\omega,\mathbf{k})=$ (7)
$\displaystyle-2\hbar^{2}\sum_{m\neq n}\mathrm{Im}\frac{\langle
n\mathbf{k}|\\{\hat{\sigma}_{c},\hat{v}_{a}\\}|m\mathbf{k}\rangle\langle
m\mathbf{k}|\hat{v}_{b}|n\mathbf{k}\rangle}{\Delta^{2}_{nm}(k)-(\hbar\omega+i\eta)^{2}},$

where, $\hat{v}_{i}=\frac{\partial}{\partial k_{b}}$ is the velocity operator,
$\hat{\sigma}$ is the Pauli spin matrix, and $\Delta_{nm}(k)=E_{nk}-E_{mk}$.
Here, we ignore the $\omega$ dependence of the spin Berry curvature.

Table 4: The form of the spin Hall conductivity tensor for the $\hat{x}$ and $\hat{z}$ magnetic phases for both AAF and AAF3 magnetic patterns. |  $\sigma^{x}$ |  $\sigma^{y}$ |  $\sigma^{z}$
---|---|---|---
$\hat{z}$-phase |  $\left(\begin{matrix}0&0&0\\\ 0&0&-\sigma^{y}_{xz}\\\ 0&-\sigma^{y}_{zx}&0\end{matrix}\right)$ |  $\left(\begin{matrix}0&0&\sigma^{y}_{xz}\\\ 0&0&0\\\ \sigma^{y}_{zx}&0&0\end{matrix}\right)$ |  $\left(\begin{matrix}0&-\sigma^{z}_{yx}&0\\\ \sigma^{z}_{yx}&0&0\\\ 0&0&0\end{matrix}\right)$
$\hat{x}$-phase |  $\left(\begin{matrix}0&0&0\\\ 0&0&\sigma^{x}_{yz}\\\ 0&\sigma^{x}_{zy}&0\end{matrix}\right)$ |  $\left(\begin{matrix}0&0&\sigma^{y}_{xz}\\\ 0&0&0\\\ \sigma^{y}_{zx}&0&0\end{matrix}\right)$ |  $\left(\begin{matrix}0&\sigma^{z}_{xy}&0\\\ \sigma^{z}_{yx}&0&0\\\ 0&0&0\end{matrix}\right)$

Eq. (7) is deceivingly similar to the formula for the normal (charge) Berry
curvature; the only difference is that here one has to evaluate the matrix
element of the anti-commutator between the velocity operator and Pauli matrix
instead of just the velocity operator in the normal Berry curvature.

SHC, being a 3rd rank tensor, has 27 components; the magnetic symmetry
determines which of them are non-zero. From the symmetry analysis, we find
that there are only 3 (6) independent components for the $\hat{z}$ ($\hat{x}$)
phase. The exact form of the SHC tensor for each of these phases is presented
explicitly in Table 4. The reason $\hat{x}$-phase has twice the number of
independent components compared to the $\hat{z}$-phase is a consequence of the
broken tetragonal symmetry due to the broken $\mathcal{R}^{4}$ symmetry as
mentioned before. Note that due to the $\mathcal{P}\mathcal{T}$ symmetry, the
anomalous Hall conductivity is identically zero in the AFM phase.

In Fig. 8, we compare the $\sigma_{xy}^{z}$ component for the $\hat{x}$ and
$\hat{z}$ Néel vector patterns for both AAF and AAF3 magnetic phases as a
function of the chemical potential ($\mu$). Other non-zero components of SHC
$\sigma_{yz}^{x}$ and $\sigma_{zx}^{y}$ are shown in Appendix B Fig. 16.
$\sigma_{xy}^{z}$ measures the $\hat{z}$ component of the spin current along
the $\hat{x}$ direction in the presence of the external field in the $\hat{y}$
direction. We find that the differences in the SHC value between the $\hat{x}$
and $\hat{z}$ patterns are appreciable. Interestingly, the magnitude of SHC
increases by more than 2-fold for the AAF3 pattern compared to the AAF pattern
in the vicinity of the Fermi level. This is likely due to the presence of band
crossings or small band gaps in the AAF3 pattern.

Fig. 9 shows momentum resolved spin Berry curvature along the high symmetry
directions for the $\hat{z}$\- and $\hat{x}$-phases of the AAF magnetic
pattern. The intensity on the top panels shows the magnitude of the band
resolved spin Berry curvature for each k-value whereas the bottom panels show
the band summation of the k-dependent spin Berry curvature up to the occupied
states (EF). These figures highlight the fact that despite the bulk band
structure features being similar, the distribution and magnitude of the spin
Berry curvature can be different due to the small differences in the band
eigenvalues and spin texture of the bands. Our analysis from effective Kondo
exchange Hamiltonian presented in Appendix B indeed finds that the band
dispersion and their spin texture will be different depending on the
orientation of the Néel vector.

##  IV Conclusions and Outlook

In summary, we have studied the electronic structure and magnetic properties
of an antiferromagnetic square net topological semimetal EuZnSb2 by employing
the first-principles and effective Hamiltonian methods. We have found that the
effect of magnetism on the bulk low energy spectrum, especially that
introduced by the orientation of the Néel vector, is weak. Despite such weak
effects in the bulk dispersion, we find that there are consequences for the
transport properties and surface electronic dispersion. For example, our
calculations predict that the broken symmetry introduced by the direction of
the Néel vector is amplified in the surface electronic dispersion. Similarly,
the differences in the spin Hall conductivity response between different
magnetic phases is appreciable. We derived an effective Kondo exchange
Hamiltonian to understand our main findings. It will be interesting to confirm
some of the predictions made in this study by experiments like ARPES in
conjunction with the spin orbit torque experiments that can control the
orientation of the Néel vector Šmejkal  _et al._ (2017). Because of the
presence of the competing magnetic states which depend on the strength of the
interaction term in our calculations, it will also be intriguing to study the
possibilities of controlling the magnetic ground state and the band topology
by means of small external perturbations like pressure, doping, intercalation,
chemical substitution etc. Similarly, study of the surface magnetism, surface
transport properties etc. could be other directions in these investigations.

To conclude, our work provides a compelling evidence that study of the
f-electron AFM can be a promising field for band engineering and spintronics
applications. Similar investigations are necessary for other predicted
f-electron square net systems in order to make systematic comparisons and
predictions. We believe that our study will motivate future works in this
direction, especially towards prediction and search of Dirac materials showing
large electronic response to the magnetic texture, using more sophisticated
numerical techniques.

##  V Acknowledgements

This work was supported by U.S. Department of Energy (DOE) the Office of Basic
Energy Sciences, Materials Sciences and Engineering Division under Contract
No. DE-SC0012704.

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##  Appendix A Non-magnetic phase and Wannier tightbinding

###  A.1 Band characters

In Fig. 10, we show the orbital composition of the bands (or band characters)
in the vicinity of the Fermi level for the non-magnetic phase.

Figure 10: Character of the bands in the vicinity of the Fermi level for non-
magnetic phase in the absence of SOC: (a) Sb2 $p_{x}$-$p_{y}$ and $p_{z}$
character and (b) Sb1 $p_{x}$-$p_{y}$ and $p_{z}$ character .The size of the
dots is proportional to the orbital composition of the particular state.

###  A.2 Wannier bands

In Figs. 11 and 12, we compare the eigenvalues obtained from different
tightbinding Wannier Hamiltonian forms with DFT calculated bands for the non-
magnetic phase.

Figure 11: Wannier bands (red dots) from 4$\times$4 Hamiltonian formed by the
$p_{x}$-$p_{y}$ orbitals of the Sb2 square lattice by considering upto the (a)
nn and (b) nnn hopping terms given in Table 1.

Figure 12:  (a) 6$\times$6 and (b) 12$\times$12 Wannier bands (red dots)
superimposed onto the DFT bands (black lines). Long range hooping terms are
included in the Wannier Hamiltonian to reproduce the bands exactly.

###  A.3 Analytical expression of the eigenvalues for the 4$\times$4 TB
Hamiltonian

Figure 13:  (a) Fermi surface (FS) of the 4$\times 4$ tightbinding Hamiltonian
in the extended zone scheme. $t^{\prime\prime}=0$ at half-filling without SOC.
SOC opens gap for this case. (b) and (c) FS and density of states respectively
with SOC for $t^{\prime\prime}=0.04$ eV. The chemical potential is tuned to
make the system compensated which corresponds to E$\sim 0.014$eV in Fig.(c).
The red and blue colored FS pieces denote hole and electron pockets in Fig.
(b). The dashed lines in Fig. (a) and (b) show the reduced Brillouin zone.

In this Appendix, we derive the 4$\times$4 TB Hamiltonian for noninteracting
band electrons whose parameters are given in the main text. For concreteness,
we define a $\sqrt{2}\times\sqrt{2}$ square lattice containing 2 atoms in the
unit cell with lattice vectors: $\vec{a}=a_{0}(\hat{x},-\hat{y})$,
$\vec{b}=a_{0}(\hat{x},\hat{y})$, where $a_{0}$ is the distance between the
nearest neighbour atoms of the primitive square lattice unit cell.  Picture?
The tight binding Hamiltonian in $\bf{k}$-space for such 2-atom and 2-orbital
system without a spin-orbit coupling is:

$\displaystyle\hat{H_{0}}=\sum_{\bf{k}}c^{\dagger}_{\bf{k}}H_{0}({\bf{k}})c_{\bf{k}}$
(8)

where,
$c^{\dagger}_{\bf{k}}=(c^{1^{\dagger}}_{\bf{k}_{px}},c^{1^{\dagger}}_{\bf{k}_{py}},c^{2^{\dagger}}_{\bf{k}_{px}},c^{2^{\dagger}}_{\bf{k}_{py}})$
such that $c^{1^{\dagger}}_{\bf{k}_{px}}$ creates an electron at the $p_{x}$
Wannier orbital located at site 1 and so on. Similarly, the Hamiltonian matrix
in this basis is given by:

$H_{0}({\bf k})=\begin{pmatrix}t^{\prime\prime}_{\bf k}&t^{\prime}_{\bf
k}&t_{{\bf k}_{\sigma\pi}}&0\\\ t^{\prime}_{\bf k}&t^{\prime\prime}_{\bf
k}&0&t_{{\bf k}_{\pi\sigma}}\\\ t_{{\bf
k}_{\sigma\pi}}&0&t^{\prime\prime}_{\bf k}&t^{\prime}_{\bf k}\\\ 0&t_{{\bf
k}_{\pi\sigma}}&t^{\prime\prime}_{\bf k}&t"_{\bf k}\\\ \end{pmatrix},$ (9)

where we have defined the hopping matrix elements as follows: $t_{{\bf
k}_{\sigma\pi}}=-2(t_{\sigma}\cos(k_{x}a_{0})+t_{\pi}\cos(k_{y}a_{0}))$,
$t_{{\bf
k}_{\pi\sigma}}=-2(t_{\pi}\cos(k_{x}a_{0})+t_{\sigma}\cos(k_{y}a_{0}))$ and
$t^{\prime}_{\bf
k}=4t^{\prime}\sin(k_{x}a_{0})\sin(k_{y}a_{0}),~{}~{}t^{\prime\prime}_{\bf
k}=2t^{\prime\prime}(\cos(2k_{x}a_{0})+\cos(2k_{y}a_{0}))$ and
$t_{\sigma}=-1.9eV$, $t_{\pi}=0.5eV$, $t^{\prime}=0.1eV$, and
$t^{\prime\prime}\sim 0.04eV$. $t_{\sigma}$ and $t_{\pi}$ are the nearest-
neighbour $\sigma$ and $\pi$ hoppings respectively and $t^{\prime}$,
$t^{\prime\prime}$ are the next nearest and the 3rd neighbour hopping matrix
elements. Note that the $t^{\prime\prime}_{\bf k}$ term, which introduces
${\bf k}$\- dependent shift of the bands, is included here for better
comparison with the DFT results and does not change the main conclusions. The
eigenvalues of $H({\bf k})$ are:

$\displaystyle E_{0}({\bf k})=t^{\prime\prime}_{\bf k}$ (10)
$\displaystyle\pm\frac{1}{2}\Big{\\{}t_{{\bf k}_{\sigma\pi}}+t_{{\bf
k}_{\pi\sigma}}\pm[(t_{{\bf k}_{\sigma\pi}}-t_{{\bf
k}_{\pi\sigma}})^{2}+4(t^{\prime}_{\bf k})^{2}]^{1/2}\Big{\\}}$

First we discuss the condition when $t^{\prime\prime}=0$ where particle-hole
symmetry is preserved. The situation of 1/2-filling corresponds to zero
chemical potential. Then the Fermi surface is determined by the zero energy
solution of Eq. (10):

$(t_{{\bf k}_{\sigma\pi}}t_{{\bf k}_{\pi\sigma}}-t^{\prime 2}_{{\bf k}})=0.$
(11)

This condition gives a diamond-shaped Fermi surface as shown in Fig. 13(a).
When the nnn hopping strength is increased, the Fermi surface becomes more
circular and shifts away from the zone boundary (X-point).

For $t^{\prime}=0$, the eigenvalues have a trivial form given by:

$\displaystyle E_{0}({\bf k})$ $\displaystyle=$ $\displaystyle\pm t_{{\bf
k}_{\sigma\pi}}\&~{}t_{{\bf k}_{\pi\sigma}}$ (12)

When $|k_{x}|$=$|k_{y}|$=$k_{0}$, the eigenvalues are $\pm
2(t_{\sigma}+t_{\pi})cos(k_{0}a_{0})$. This gives degenerate bands along this
line as seen along the $\Gamma$-X and M-X line in Fig. 11(a).

We see that in the absence of further interactions we have a metal with a half
filled conduction band. Such situation is unstable with respect to unit cell
doubling which can gap out (at least partially) the Fermi surface. Such unit
cell doubling can occur already on the single particle level or be a
consequence of the interactions. Below we consider the former mechanism first.

The spin orbit coupling (SOC) is of the form

$H_{SOC}=\sigma^{z}(\lambda\tau_{y}\otimes
I+\delta\tau_{z}\otimes\gamma_{z}),$ (13)

where $\lambda$ and $\delta$ are constants and $\gamma$, $\sigma$, $\tau$ are
the Pauli matrices acting on the site, spin, and orbital indices,
respectively. Such form of SOC does not couple the $|\uparrow\rangle$ and
$|\downarrow\rangle$ spin sectors; hence we can still diagonalize the
Hamiltonian analytically. The eigenvalue for each spin sector in the presence
of SOC is given by:

$\displaystyle E_{\mathrm{SOC}}({\bf k})=\pm\sqrt{X(\bf
k)^{2}+\delta^{2}}+t^{\prime\prime}(\bf k),$ (14) $\displaystyle X^{2}({\bf
k})=$ $\displaystyle\frac{1}{4}\Big{\\{}t_{{\bf k}_{\sigma\pi}}+t_{{\bf
k}_{\pi\sigma}}\pm[(t_{{\bf k}_{\sigma\pi}}-t_{{\bf
k}_{\pi\sigma}})^{2}+4(\lambda^{2}+t^{\prime 2}_{\bf k}])^{1/2}\Big{\\}}^{2}.$

The estimate is $\delta\approx 0.1$eV. For $t^{\prime\prime}=0$, the FS is
fully gapped at 1/2 filling as discussed in the main text. However, for
$t^{\prime\prime}\neq 0$ case, the FS is partially gapped. In Fig. 13(b), we
show the FS in the presence of SOC for non-zero $t^{\prime\prime}$ at the
value of chemical potential when the system is fully compensated. As seen in
the figure, there are electron and hole pockets at the X point and along
$\Gamma$-M direction, respectively, similar to the FS of real system with
small DOS at the Fermi level [Fig. 13(c)].

##  Appendix B Derivation of the exchange Hamiltonian

Figure 14:  A band structure of the nnn 4$\times$4 tight binding Hamiltonian
in the presence of the effective Kondo exchange Hamiltonian for the AFM
arrangement of the Eu-$f$ spins. Fig. (a) shows a comparison with the
eigenvalues without the exchange Hamiltonian (black dots) at zero spin-orbit
coupling (SOC). Notice that the Kondo exchange splits the band degeneracies by
$\sim J_{K}<S>$ at the M, X and along X-$\Gamma$ direction. The crossing along
$\Gamma-M$ is preserved by Kondo exchange in the absence of SOC. The
dispersion is identical for any orientation of the Neel magnetization. (b)
Band dispersion in the presence of Kondo exchange with SOC when the spin
quantization axis is along the $\hat{x}$ direction. The color palette shows
the magnitude of the $\langle\hat{\sigma_{z}}\rangle$.

In order to understand how the Eu-4$f$ states may affect the band structure
and the Fermi surface properties, we have also derived an effective Kondo
exchange Hamiltonian within the second order perturbation theory in $V$. The
4$f$-shell of Eu ions is half filled. The Hund’s rule dictates that at the
ground state Eu ion has zero angular momentum and the total spin is S=7/2.
Then according to Ref. Schrieffer, 1967, the effective Kondo Hamiltonian for
an Eu ion located at the origin is the multichannel Kondo model. For a
spherically symmetric case it would have the following form:

$\displaystyle
H_{exchange}=J_{K}\sum_{m}c^{\dagger}_{m\alpha}(|k|)({\vec{\sigma}}_{\alpha\beta}\vec{S})c_{m\beta}(|k|),$
(15)

where $m$ are the electron orbital quantum numbers. In the limit when the
Hund’s coupling exceeds the Kondo temperature Schrieffer (1967), the Kondo
coupling is given by:

$J_{K}=\frac{V^{2}}{2S}\Big{[}-1/\epsilon_{f}+1/(\epsilon_{f}+U)\Big{]},$ (16)

with $V$ being the overall scale of the hybridization strength between the
Eu-$f$ and $p$ electrons and $\epsilon_{f}$ being the energy levels of the $f$
states. We estimate $J_{K}\sim 16$K.

The exchange Hamiltonian is derived under assumption that the hybridization
matrix elements are equal for all orbitals involved in the screening of the
spin. For the model on a lattice the latter assumption no longer holds. One
has to recalculate the spherical orbitals into the Wannier functions basis,
$m$-th orbital will have its own exchange $\sim|V_{m}|^{2}$ which will lead to
exchange anisotropy.

Table 5:  Hybridization matrix element (in eV) between the occupied $|Euf\sigma\rangle$ and the $|p\sigma\rangle$-states from the Sb2 square nets in the antiferromagnetic configuration without SOC obtained from the Wannier function analysis. For Eu1 atoms, the $|\uparrow\rangle$ states are occupied whereas for Eu2 atoms, $|\downarrow\rangle$ states are occupied. The matrix elements between the $|p^{2}\sigma\rangle$ and $|Eu\sigma\rangle$ states are similar with $p_{x}\leftrightarrow p_{y}$. The sign of the matrix elements are fixed by the symmetry of the corresponding hopping integral. Note that the $p$-orbitals in this table are aligned along Eu-atoms and vice-versa.  $\langle WFs|H|WFs\rangle$ |  Eu1-$f_{xz^{2}}\uparrow$ |  Eu1-$f_{yz^{2}}\uparrow$ |  Eu1-$f_{z^{3}}\uparrow$ |  Eu1-$f_{x(x^{2}-3y^{2})}\uparrow$ |  Eu1-$f_{y(3x^{2}-y^{2})}\uparrow$ |  Eu1-$f_{z(x^{2}-y^{2})}\uparrow$ |  Eu1-$f_{xyz}\uparrow$
---|---|---|---|---|---|---|---
$p_{x}^{1}\uparrow$ |  -0.13 |  0 |  -0.05 |  -0.03 |  0 |  -0.11 |  0.0
$p_{y}^{1}\uparrow$ |  0 |  0.01 |  0 |  0 |  0.07 |  0 |  0.07
$p_{z}^{1}\uparrow$ |  -0.11 |  0 |  0.06 |  -0.06 |  0 |  -0.13 |  0
$\langle WFs|H|WFs\rangle$ |  Eu2-$f_{xz^{2}}\downarrow$ |  Eu2-$f_{yz^{2}}\downarrow$ |  Eu2-$f_{z^{3}}\downarrow$ |  Eu2-$f_{x(x^{2}-3y^{2})}\downarrow$ |  Eu2-$f_{y(3x^{2}-y^{2})}\downarrow$ |  Eu2-$f_{z(x^{2}-y^{2})}\downarrow$ |  Eu2-$f_{xyz}\downarrow$
$p_{x}^{1}\downarrow$ |  0.01 |  0 |  0 |  -0.07 |  0 |  0 |  0.07
$p_{y}^{1}\downarrow$ |  0 |  -0.13 |  -0.05 |  0 |  0.03 |  0.11 |  0
$p_{z}^{1}\downarrow$ |  0 |  -0.11 |  0.06 |  0 |  0.06 |  0.13 |  0

By using the hybridization matrix elements between the localized Eu-4$f$ and
the itinerant $p_{x}$-$p_{y}$-orbitals from the ab-initio calculations as
shown in Table. 5, we obtain the following effective Kondo exchange
Hamiltonian in the basis of the $p$ orbitals $c^{\dagger}_{\bf
k\sigma}=(c^{1^{\dagger}}_{\bf{k}_{px}\sigma},c^{1^{\dagger}}_{\bf{k}_{py}\sigma},c^{2^{\dagger}}_{\bf{k}_{px}\sigma},c^{2^{\dagger}}_{\bf{k}_{py}\sigma})$:

$\displaystyle
H^{ex}_{\sigma\sigma^{\prime}}=J_{K}\sum_{k,k^{\prime},R_{1}}c^{+}_{{\bf
k}\sigma}\hat{g}_{+}(\bf k,\bf k^{\prime})({\vec{\sigma}}\vec{S}_{\bf
R_{1}})_{\sigma,\sigma^{\prime}}\mbox{e}^{-\mbox{i}({\bf k}-{\bf
k}^{\prime}){\bf R}_{1}}c_{{\bf k}^{\prime}{\sigma^{\prime}}}+$ $\displaystyle
J_{K}\sum_{k,k^{\prime},R_{2}}c^{+}_{{\bf k}{\sigma}}\hat{g}_{-}(\bf k,\bf
k^{\prime})({\vec{\sigma}}\vec{S}_{\bf
R_{2}})_{\sigma\sigma^{\prime}}\mbox{e}^{-\mbox{i}({\bf k}-{\bf
k}^{\prime}){\bf R}_{2}}c_{{\bf k}^{\prime}{\sigma^{\prime}}},$ (17)

where, $\vec{S}_{\bf R_{1}}$ and $\vec{S}_{\bf R_{2}}$ are the spins localized
on the $\bf R_{1}$ and $\bf R_{2}$ sites. $g_{+}$ and $g_{-}$ are the
effective exchange matrices given by:

$\displaystyle g_{+}(\bf k,\bf
k^{\prime})=\left(\begin{array}[]{cccccc}g_{1}&-g_{2}&g_{3}-g_{4}&-g_{3}-g_{4}\\\
-g_{2}&g_{1}&g_{3}+g_{4}&-g_{3}+g_{4}\\\ h.c.&h.c.&g_{1}&g_{2}\\\
h.c.&h.c.&g_{2}&g_{1}\end{array}\right),$ (22) $\displaystyle g_{-}(\bf k,\bf
k^{\prime})=\left(\begin{array}[]{cccccc}g_{1}&g_{2}&g_{3}-g_{4}^{*}&g_{3}+g_{4}^{*}\\\
g_{2}&g_{1}&-g_{3}-g_{4}^{*}&-g_{3}+g_{4}^{*}\\\ h.c.&h.c.&g_{1}&-g_{2}\\\
h.c.&h.c.&-g_{2}&g_{1}\end{array}\right),$ (27)

and, $g_{1}=1,~{}~{}g_{2}=1/2,~{}~{}g_{3}=As_{x}^{*}(\bf k)s_{y}^{*}(\bf
k^{\prime})$, $g_{4}=Bs_{x}^{*}(\bf k)s_{y}(\bf k^{\prime})$,
$s_{a}=1-e^{-ik_{a}}$, $A=1/16,~{}~{}B=3/16$.

The role of the Eu magnetic moments– The interaction between conduction
electrons and magnetic moments may lead to many different effects. Depending
on how strong is the coupling between these two subsystems, the net result
many be very different. Below we will consider two scenario.

In both cases we will treat the Eu spins in the mean field approximation, that
is as classical vectors $\vec{S}=\pm<S>{\bf n}$, where ${\bf n}$ is a unit
vector and $<S><7/2$.

First, we consider the case with zero SOC. Then, we will discuss the case with
strong SOC $\delta$ (we will formulate the precise criterion later). In the
latter case, the band electrons are partially gapped with small density of
states at the Fermi level and the influence of the spin order on their
spectrum is small. The nature of the spin order is determined by the
interaction between the spins through the conduction electrons.

Consider spins with opposite directions on ${\bf R}_{1}$ and ${\bf R}_{2}$
sites: $\vec{S}=\pm<S>{\bf n}$, where ${\bf n}$ is a unit vector and $<S><7/2$
is the average magnetization which has to be determined self-consistently.
Then the Kondo exchange (17) becomes an effective modification of the spin-
orbit coupling:

$\displaystyle H^{ex}_{MF}=2J_{K}<S>\sum_{k}c^{\dagger}_{\bf
k}(\vec{\sigma}\vec{n})[g_{2}\tau^{x}\otimes\gamma^{z}+\Im
mg_{4}\tau^{z}\otimes\gamma^{y}$ $\displaystyle+\Im
mg_{3}\tau^{y}\otimes\gamma_{x}-(\Re eg_{3}+\Re
eg_{4})\tau^{y}\otimes\gamma^{y}]c_{\bf k}.$ (29)

In Fig. 14(a), we show the band dispersion of the $p$-electrons in the
presence of the Kondo exchange term for zero SOC. The antiferromagnetic order
splits the band dispersion at several places highlighted in the figure insets
by magnitude of $\sim J_{K}<S>$.

In order to study the differences in the spectrum due to the spin orientation
of the Néel vector, we will need to combine (13) and (29). Close to the
X-point ($\pi/2,\pi/2)$, $t_{\sigma\pi}\sim t_{\pi\sigma}\sim 0$ i.e. there is
no coupling between the two sublattices and one can derive the effective
Hamiltonian for just one of the sublattice:

$\displaystyle H_{1}=H_{2}=\mathcal{O}(J_{K}^{2}<S>^{2})+t^{\prime\prime}_{\bf
k}$ $\displaystyle\tau^{x}(t^{\prime}_{\bf
k}-2J_{K}<S>g_{2}\sigma^{z})+(\vec{\sigma}\vec{n})(\delta\tau^{z}+\lambda\tau^{y}).$
(30)

If the Neel magnetization is along the $z$-axis, the spectrum is:

$\displaystyle E=\mathcal{O}(J_{K}^{2}<S>^{2})+t^{\prime\prime}_{\bf k}\pm$
$\displaystyle[\delta^{2}+\lambda^{2}+(t^{\prime}_{\bf{k}}-2J_{K}<S>g_{2}\sigma^{z})^{2}]^{1/2}.$
(31)

If it is perpendicular to $z$-axis then,

$\displaystyle E=\mathcal{O}(J_{K}^{2}<S>^{2})+t^{\prime\prime}_{\bf k}\pm$
$\displaystyle\Big{(}\sqrt{t^{\prime 2}_{\bf k}+\lambda^{2}+\delta^{2}}\pm
2J_{K}<S>g_{2}\Big{)}.$ (32)

With our estimates of $J_{K}$ and $\delta$, we conclude that the first term
can be neglected. Thus, we find that there is a small effect in the spectrum
due to the rotation of the Eu spins which nevertheless remains partially
gapped due to the SOC. More importantly, if ${\bf n}$ is directed along the
$\hat{z}$-axis, the $z$-projection of electron spin is a good quantum number.
For all other directions it is not. This can be seen from the spin-texture of
the bands for the $\hat{x}$-phase along the high symmetry directions in Fig.
14(b). Such differences in the spin-texture between the $\hat{z}$ and
$\hat{x}$ phases can introduce differences in the spin transport properties
like spin Hall conductivity which is proportional to the matrix element of the
spin operator.

##  Appendix C Electronic dispersion and spin Hall conductivity results for
magnetic phases

###  C.1 DFT Bands and DOS

Fig. 15 shows the DFT calculated electronic bands and DOS for AAF-$\hat{z}$
magnetic phase in the presence of SOC and U of 6 eV.

Figure 15:  DFT bands and projected density of states (PDOS) for AAF-$\hat{z}$
magnetic pattern. Positive (negative) PDOS values indicate up (down) states.
The minority spin states for AFM aligned Eu-4$f$ electrons are $\sim$10 eV
above EF, hence not shown here. The small DOS at EF comes from Sb2
$p_{x}$-$p_{y}$ orbitals.

###  C.2 Comparison with other components of the Spin hall conductivity

In the main text, we showed the comparison between the spin hall conductivity
(SHC) response as a function of the chemical potential between the $\hat{x}$
and $\hat{z}$ phase for just one component of the SHC tensor. Here, we also
show the comparison with other components of the SHC tensor.

Figure 16: Comparison of the (a) $\sigma_{xy}^{z}$, (b) $\sigma_{yz}^{x}$ and
(c) $\sigma_{zx}^{y}$component of the SHC tensor as a function of the chemical
potential between the AAF-x and z patterns.


# A tight-binding model for the excitonic band structure of a one-dimensional
molecular chain: UV-Vis spectra, Zak phase and topological properties

Wei Wu1<EMAIL_ADDRESS>Jin Zhang2 1UCL Department of Physics and Astronomy
and London Centre for Nanotechnology,
University College London, Gower Street, London WC1E 6BT  2School of Physics
and Astronomy, Yunnan University, Kunming, Yunnan Province, P. R. China

###### Abstract

Recently organic optics becomes a hot topic due to the rapid development of
organic light-emitting diodes, organic solar cells, and organic photon
detectors. The optical spectra of the molecular semiconductors are difficult
to solve an model from first-principles because (i) the very large number of
atoms in a unit cell and (ii) the accurate theoretical excited state is still
under development. Here we present a tight-binding model of an exciton band
structure in a molecular chain. We take into account the intra-molecule and
charge-transfer excitation within a molecular dimer in a unit cell, then we
apply the tight-binding model by including the coupling between two types of
excitations. We not only found that our calculations can explain a body of UV-
Vis optical spectra of transition-metal phthalocyanines, but also a one-
dimensional excitonic topological band structure if we fine-tune the couplings
in a dimerized molecular chain. We have found a large space to obtain the
topological Zak phase in the parameter space, in which there is a simple
linear relationship between the hopping integrals between cells and within
cell.

††preprint: excitonbands2019

##  I Introduction

The recent development of organic light-emitting diodes (OLED) and organic
solar cells (OSC) has stimulate an unprecedented huge amount of experimental
measurements and theoretical analysis of the optical properties of organic
materials choi2016 ; fleetham2014 . The optical properties of the organic
materials is very important because it can tell us the electronic structure,
vibrational modes, and their potential for OLED and OSC. Normally people
measure the UV-Vis optical spectra to tell the potential performance of
organic materials for OLED and OSC. The comparison between experimental
results and theoretical modelling is crucial to the understanding of the
electronic-structure origin of the optical spectra, assessment of the
potential of the organic materials for optical applications, and development
of new materials according the theoretical analysis. It is well known that the
organic materials is liable to form one-dimensional chains bogani2008 ; dd1987
, especially for the $\pi$-conjugated systems mainly due to the strong
$\pi-\pi$ interactions that can stack the molecules along a particular spatial
direction. Therefore, the main characteristics of the optical spectra of
organic chain compounds such as transition-metal phthalocyanines originates
from molecular chains serri2014 . We can then understand the optical
properties of the organic chain compounds based on the optical simulation of
molecular chains.

The optical properties of transition-metal phthalocyanines, a one-dimensional-
chain organic compound, have been studied for a long time. There are mainly
two UV-Vis absorption bands; one of them is B bands (at $\lambda\sim 350$ nm)
while the others (at $\lambda\sim 650$ nm) are called Q-band pbook . Normally
Q-band is split and very broad, which is contraversial. A popular argument is
that this is due to the Davidof splitting, however this effect is too small
comparing with the experimentally observed splitting in Q-band. Another deeper
understanding is that this is due to the coupling between the intra-molecular
and charge-transfer excitations byb2013 . This opinion is more reasonable
because (i) there indeed exist these two types of excitations and (ii) the
coupling between them should be proportional to the conduction- or valence-
bandwidth, which is close the splitting observed ($\sim 10^{-1}$ eV) wu2011 .

The common methods for analysing the optical properties of molecules or
finite-size nano-structure include configuration interaction (CI) mqc , time-
dependent Hartree-Fock (TDHF) stratmann1998 , and time-dependent density-
functional theory (TDDFT) stratmann1998 , etc. CI is normally very expensive
as the computational cost will increase with the number of atoms and basis set
factorially. People normally choose TDHF and TDDFT to compute the excited
states of the molecules or molecule clusters as they are much cheaper and the
calculation results are reasonably good as compared with CI. In practice TDDFT
is in general more accurate than TDHF although the exchange-correlation
functional is empirical through fitting to the experimental values such as
B3LYP b3lyp . For the excited state of solids, the related theoretical
approaches are still under development. For example, in CRYSTAL code ber2011 ,
people are trying to extend the idea of TDDFT to periodic sold-state
structure. To the authors’ knowledge, it is rare ber2011  to deal with the
optical properties of periodic solid from first-principles due to (i) the
large amount of atoms in the unit cell and (ii) the large number of single-
particle excited states to be taken into account in the calculation of excited
states in the solid state.

Recently the one-dimensional topological insulator su1979 ; kane2010 ;
ozawa2019  arising from the alternating coupling in the dimer chains becomes a
hot topic. The original work su1979  by Su, et. al., shows that there exists a
topological soliton owing to the alternating coupling and the interaction with
phonons. People have realized this type of topological electronic state by
using finite atom chain or arrays on the Cu surface drost2017 , in which we
can see the topological state. Similarly, people have proposed that the
similar topological state can exist in magnon zhang2013 , excitons yz2014 ,
and phonons jiang2018 . Here we propose a one-dimensional excitonic model with
alternating coupling due to the dimerization of molecular chains, which can
easily happen in the molecular crystals.

Here we propose a tight-binding method for a one-dimensional periodic
molecular chain to take into account the intra-molecule excitation, charge-
transfer excitation, and the coupling between them. We show that we can not
only qualitatively explain the optical spectra of molecule-chain compound, but
also find the topological excitonic band structure if we can fine-tune the
coupling in a dimerised molecular chain. The remaining discussion falls into
three parts. In §II, we talk about the method used. In §III, we show our
results and discuss them. In §IV, we draw some general conclusions.

##  II Methods

We mainly used the tight-bind model as shown in Fig.1 to perform the
calculations. The related Hamiltonian reads

$\displaystyle\hat{H}=\sum_{n}$ $\displaystyle
t_{2}a^{\dagger}_{\mathrm{IM_{1}},n}a_{\mathrm{CT_{1}},n}+t_{1}^{\prime}a^{\dagger}_{\mathrm{IM_{1}},n}a_{\mathrm{CT_{1}},n-1}+$
(1) $\displaystyle
t_{1}a^{\dagger}_{\mathrm{IM_{1}},n}a_{\mathrm{CT_{2}},n}+t_{2}^{\prime}a^{\dagger}_{\mathrm{IM_{1}},n}a_{\mathrm{CT_{2}},n-1}+$
$\displaystyle
t_{1}a^{\dagger}_{\mathrm{IM_{2}},n}a_{\mathrm{CT_{1}},n}+t_{2}^{\prime}a^{\dagger}_{\mathrm{IM_{2}},n}a_{\mathrm{CT_{1}},n+1}+$
$\displaystyle
t_{2}a^{\dagger}_{\mathrm{IM_{2}},n}a_{\mathrm{CT_{2}},n}+t_{1}^{\prime}a^{\dagger}_{\mathrm{IM_{2}},n}a_{\mathrm{CT_{2}},n+1}+$
$\displaystyle-\frac{d}{4}a^{\dagger}_{\mathrm{IM_{1}}}a_{\mathrm{IM_{1}}}-\frac{d}{4}a^{\dagger}_{\mathrm{IM_{2}}}a_{\mathrm{IM_{2}}}+$
$\displaystyle\frac{d}{4}a^{\dagger}_{\mathrm{CT_{1}}}a_{\mathrm{CT_{1}}}+\frac{d}{4}a^{\dagger}_{\mathrm{CT_{2}}}a_{\mathrm{CT_{2}}}$
$\displaystyle+\ h.c.$

Here $a_{s}^{\dagger}$ and $a_{s}$ are the bosonic creation and annihilation
operators for individual states $s$, respectively as we consider
singlet/triplet excitons.

Figure 1: (Colour online.) The diagram shows the excited states including
intra-molecular ($\mathrm{IM_{1}}$ and $\mathrm{IM_{2}}$) and the charge-
transfer states ($\mathrm{CT_{1}}$ and $\mathrm{CT_{2}}$). The red box
represents the unit cell. We also show the couplings between them by using
$t_{1}$, $t_{1}^{\prime}$, $t_{2}$, and $t_{2}^{\prime}$, which are between
the upper states (normally lowest-unoccupied molecular orbital (LUMO)) and the
lower states (normally highest-occupied molecular orbital (HOMO)).

.

The excited state properties, including excitation energies and oscillator
strengths, can be computed by using the time-dependent density-functional
theory module built in Gaussian 09 code gaussian09 . $t_{1}$ and $t_{2}$ are
the couplings between the single-particle LUMO (lowest-unoccupied molecular
orbital) and HOMO (highest-occupied molecular orbital) states respectively
within the unit cell, while the $t_{1}^{\prime}$ and $t_{2}^{\prime}$ are for
between cells.The couplings $t_{1}$, $t_{1}^{\prime}$, $t_{2}$, and
$t_{2}^{\prime}$ are phenomenological, which can work as fitting parameters
for the experimental results. We can also assume that there is a small energy
gap $d$ between the intra-molecular and charge-transfer excited states. We can
then transform the Hamiltonian $\hat{H}$ into the $k-$space to have
$\hat{H}_{k}$, as follows,

$\hat{H}_{k}=\begin{pmatrix}-d/2&0&t_{1}+t_{2}^{\prime}e^{-ik}&t_{1}^{\prime}e^{-ik}+t_{2}\\\
0&-d/2&t_{1}^{\prime}e^{ik}+t_{2}&t_{1}+t_{2}^{\prime}e^{ik}\\\
t_{1}+t_{2}^{\prime}e^{ik}&t_{1}^{\prime}e^{-ik}+t_{2}&d/2&0\\\
t_{1}^{\prime}e^{ik}+t_{2}&t_{1}+t_{2}^{\prime}e^{-ik}&0&d/2\\\ \end{pmatrix}$

Then we can solve the eigenvalue problems for each $k$-points (assuming the
lattice constant here is $1$) by using the standard diagonalization technique.

##  III Results and discussion

###  III.1 Mixture of the intra-molecular and charge-transfer excited states

We first solve the eigenvalue problem for a uniform molecular chain, and take
$t_{1}=t_{1}^{\prime}$ and $t_{2}=t_{2}^{\prime}$. Here we take $t_{1}$ = 0.15
eV, $t_{2}$ = 0.015 eV, the energy difference between the intra-molecular and
charge-transfer states is 0.5 eV. The oscillator strengths for the IM and CT
excited states are 0.5 and 0.02, respectively, as computed by using TDDFT in
Gaussian 09 code. According to the eigenvector obtained, we can obtain the
total oscillator strength across the first Brillouin zone from $-\pi$ to
$+\pi$ for $k$-vector (here we assume the lattice constant is 1 in our
periodic structure as show in Fig.1). As shown in Fig.2, we have computed the
oscillator strength for the above parameters (black curve), and compare this
with the calculation in which the coupling is turned off (red curve). When the
coupling is turned off, we only have the strong transition for the IM excited
state, whereas, when the coupling is turned on, we can see the lower-energy CT
excited states will strengthen and become even stronger than the IM excited
state. This is in consistent with most of the optical spectra for the
transition-metal phthalocyanines wang2010 .

Figure 2: (Colour online.) The computed absorption spectra by using the
parameters of Gaussian calculations (black curve) combined with our tight-
binding model, which is in a qualitative agreement with the previous
experimental results. Especially, we can see the longer-wave length has a
stronger absorption than the shorter one. In comparison, we have also shown
the computed optical spectra with coupling turned off (red curve).

We have computed the excitonic band structure as shown in below when taking
$t_{1}=t_{1}^{\prime}=-0.5,t_{2}=t_{2}^{\prime}=-0.1,d=-1$ (alternatively, we
use the gap between IM and CT as energy unit), in which we can see the
bandwidth is large the top and bottom bands due to $t_{1}$. We have also found
these two interesting straight bands which seem not to be coupled to any other
components. These bands are from the linear combination of the two pure IM or
CT states, therefore they are purely the IM or CT excited states. However, the
top and bottom bands contribute to the mixture between IM and CT excited
states.

Figure 3: (Colour online.) The excitonic band structure computed by using our
tight-binding model when we take the same parameters as Fig.1.

###  III.2 Analogue to the Su-Schrieffer-Heeger model: molecular dimer model
and topological phase

####  III.2.1 Topological band structure

In addition to the uniform chain calculations ($t_{1}=t_{1}^{\prime}$ and
$t_{2}=t_{2}^{\prime}$), we can also tune the couplings between the intra-
molecular and charge-transfer excited states to find the similar physics to
the Su-Schrieffer-Heeger (SSH) model su1979 . As shown in Fig.4, we have fixed
the ratio between $t_{1}^{\prime}$ and $t_{1}$ (and $t_{2}^{\prime}$ and
$t_{2}$) to be $2$, $t_{2}$ to $-3$, $d$ to $-1$ and varied $t_{1}$. We can
see the evolution of the band structure when tuning $t_{1}$ from $-8$ to $-5$
from (a)-(d) of Fig.4. When $t_{1}=-8.0$ and $t_{1}=-5.0$, the band structure
is gapped, however, when $t_{1}=-6.5$ and $t_{1}=-6.1$ we can see the band
touching. This evolution of band structure as a function of the coupling
between IM and CT excited states indicates the potential topological
properties of molecular dimer chain through excitonic aspect.

Figure 4: (Colour online.) The excitonic band structure computed by using our
tight-binding model. We can see that the evolution of the band structure as a
function of $t_{1}$. We can see the band touching when $t_{1}=-6.1$.

####  III.2.2 Zak phase

We have also computed the total Zak phase zak1989  that reads
$z=i\int_{\mathrm{FBZ}}{dk\langle\phi_{k}|\partial_{k}{\phi_{k}}\rangle}$ for
the band structure, in which FBZ represents the first Brillouin zone and
$\phi_{k}$ is the Block function. We have summed up the total Zak phase by
using $\sum_{i}|z_{i}|/\pi$ for all the bands. Here we have fixed the ratios
$t_{1}^{\prime}/t_{1}$ and $t_{2}^{\prime}/t_{2}$ to $2$ and the energy gap
$d=-1$. From the phase diagram as shown in Fig.5(a), the upper-left part is
the topological phase, whereas the lower-right part for the trivial phase. We
can also see that the phase transition happens when $t_{1}/t_{2}\simeq 2$ as
suggested by the green line. We have plotted the band structure for the
parameters at blue, purple, and red squares in the trivial, phase transition,
and toplogical phases, respectively. We can see the corresponding colour-coded
band-structure evolution in Fig.5(b). The evolution follows band-gap closing
and re-opening. As the band structure suggests, the model proposed here still
maintains the chiral symmetry similar to the SSH model. However, the band-gap
reopening happens for both the positive- and negative-energy bands separately,
which is different from the band structure arising from the SSH model. This
can be attributed to the involvement of the single-particle excited states.

Figure 5: (Colour online.) The Zak-phase digram is shown in (a) and the
corresponding colour-coded band structures are plotted in (b) to illustrate
the band-gap reopening after the phase transition. In the band structure, the
energy is in arbitrary unit.

##  IV Conclusions

We combined the first-principles TDDFT calculations for a molecular dimer with
excitonic tight-binding model to analyse the optical spectra of molecular-
chain compound. Especially we compare our theoretical modelling with the
recent optical UV-Vis measurement of transition-metal phthalocyanines crystal.
We found that our results are in a good agreement with the experimental
results. The coupling between the IM and CT excited states indeed plays an
important role to the explanation of the optical spectra of molecular chains.
Based on these uniform-chain calculations, we have tuned the coupling to
dimerize the molecules and found interesting band structures which is an
analogue to the SSH model, but in the sense of excitonic interactions within a
one-dimensional dimer chain. We found the band-structure evolution and the
band touching as we tune the interaction parameters. These calculations show
that the molecular dimer chain is a promising candidate for photonic
topological insulator. In the future, we would like to improve the model by
including the Coulomb interaction between excitons (two-electron integrals)
into the model, which may bring new physics.

###### Acknowledgements.

We thank the funding from The Department of Science and Technology of Yunnan
University (Grant No. 2016FC001) and Yunnan University (Grant No. 2016MS14).
This work is also supported by the National Science Foundation of China (Grant
No. 11664044).

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# Re-initialization-free Level Set Method via Molecular Beam Epitaxy Equation
Regularization for Image Segmentation††thanks: Submitted to the editors
DATE:18 July 2023. This work was funded by the National Natural Science
Foundation of China (12171123, U21B2075, 11971131, 12271130, 11871133), the
Natural Science Foundation of Heilongjiang Province (ZD2022A001), the
Fundamental Research Funds for the Central Universities(HIT.NSRIF202202,
2022FRFK060020, HIT.NSRIF. 2020081, 2022FRFK060014).

Fanghui Song Department of Computational Mathematics, School of Mathematics,
Harbin Institute of Technology<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Jiebao
Sun22footnotemark: 2 Shengzhu Shi Correponding author, Department of
Computational Mathematics, School of Mathematics, Harbin Institute of
Technology<EMAIL_ADDRESS>Zhichang Guo22footnotemark: 2 Dazhi
Zhang22footnotemark: 2

remarkRemark hypothesisHypothesis claimClaim LSM via MBE Regularization for
Image SegmentationF Song, J Sun, S Shi, Z Guo and D Zhang