text:
\section{Introduction\label{sec_intro}}

The discovery of higher order topological insulators (HOTIs) has refined the notion of the bulk-boundary correspondence \cite{benalcazar2017electric,schindler2018higher,song2017d,langbehn2017reflection,benalcazar2017quantized,geier2018second,trifunovic2019higher,schindler2018bismuth,khalaf2018symmetry,khalaf2018higher,ezawa2018minimal,ezawa2018higher,matsugatani2018connecting,imhof2018topolectrical,peterson2018quantized,serra2018observation,noh2018topological,you2018higher,queiroz2019partial,fang2020higher}.
Specifically, an order-$d$ topological insulator in $D$ dimensions exhibits gapless modes on $(D-d)$-dimensional surface, where $d=1$ corresponds to the usual bulk-boundary correspondence \cite{schnyder2008classification,schnyder2009classification,kitaev2009periodic}.

In this manuscript, we consider the case where $d=D$. An order-$D$ topological insulator in $D$ dimensions exhibits zero-dimensional corner-localized mid-gap states \cite{benalcazar2017quantized,benalcazar2017electric,benalcazar2019quantization,schindler2019fractional,watanabe2020fractional,hirayama2020higher}.
Unlike a Chern insulator \cite{panati2007triviality,brouder2007exponential}, $\mathbb{Z}_2$ topological insulator \cite{soluyanov2011wannier}, or topological crystalline insulator \cite{bradlyn2017topological,po2017symmetry}, an order-$D$ topological insulator in $D$ dimensions does not require an obstruction to the existence of symmetric, maximally localized Wannier functions.
Instead, the corner charge can result from an obstructed atomic limit (OAL) phase \cite{bradlyn2017topological}, where the bulk is a band insulator that permits maximally localized and symmetric Wannier functions, but such that the Wannier centers cannot be continuously deformed into the positions of the atoms without breaking symmetry or closing the bulk (or surface \cite{khalaf2019boundary}) band gap.
This mismatch between the bulk atoms and Wannier centers has been dubbed the filling anomaly \cite{benalcazar2019quantization}.
In a symmetric finite-sized system at charge neutrality with no polarization or surface states,
a filling anomaly results in a non-zero corner charge, quantized by crystal symmetry.

It is desirable to compute the filling anomaly and corner charge from bulk properties.
To this end, there have been two recent approaches.
The first, which applies to any lattice, regardless of symmetry, is to generalize the modern theory of polarization \cite{kingsmith1993theory,vanderbilt1993electric,resta1994macroscopic} by determining the corner charge from a bulk multipole moment \cite{PhysRevB.92.041102,wheeler2019many,kang2019many,ono2019difficulties,trifunovic2019geometric,trifunovic2020bulk,ren2020quadrupole,watanabe2020corner}.
The second, which is taken in this manuscript, is to develop a theory of symmetry indicators, i.e., formulas derived in terms of the symmetry representations of the Bloch wavefunctions at high-symmetry momenta.
Symmetry indicators have been very successful in classifying topological crystalline insulators \cite{bradlyn2017topological,vergniory2017graph,elcoro2017double,po2017symmetry,bradlyn2018band,cano2018building,khalaf2018symmetry,song2018quantitative,vergniory2019complete,zhang2019catalogue,cano2020band}
starting with the inversion eigenvalue formulas for 2D and 3D $\mathbb{Z}_2$ topological insulators \cite{fu2007topological}.

Recently, symmetry indicators have been derived for the filling anomaly and corner charge in OALs in certain 2D crystals~\cite{benalcazar2019quantization,schindler2019fractional}.
However, the results do not necessarily apply when there are multiple atoms in the unit cell.
Specifically, Refs.~\cite{benalcazar2019quantization,schindler2019fractional} limited their consideration to crystals for which there exists a symmetric finite-size termination containing an integer number of unit cells.
Such a termination does not exist for a crystal with atoms occupying multiple distinct maximal Wyckoff positions.
Instead, for such a crystal, all symmetric terminations contain a fractional number of unit cells, as shown in Fig.~\ref{fig_lattice}.

\begin{figure}[b
    \centering
    \includegraphics[width=\linewidth]{Fig1.png}
    \caption{Finite symmetric square lattices with open boundary conditions shown for (a) the simple square lattice and (b) a lattice with atoms in multiple Wyckoff positions.
    Solid blue and hollow green dots and pink squares correspond to different Wyckoff positions on the square lattice. Dashed gray lines indicate the primitive unit cell. The large black square outlines the finite-sized system (atoms on the boundary are included in the finite-sized system).
    In (b), the finite-sized system includes a fractional number of unit cells, as evidenced by the different number of solid and hollow dots, while the unit cell contains one of each.
    }
    \label{fig_lattice}
\end{figure}

In this manuscript, we develop a method to compute symmetry indicators for the filling anomaly in the general case of a crystal with atoms occupying any number of Wyckoff positions.
The method has two steps:
we first compute the filling anomaly in terms of the number of Wannier functions centered at each Wyckoff position (which has also been done recently in Ref.~\cite{watanabe2020corner}).
Second, we compute the number of Wannier centers at each Wyckoff position in terms of symmetry indicators using elementary band representations.
The second step is accomplished via an algorithm that automates the calculation of symmetry indicators, introduced in this work.
We apply our method to the square lattice with and without time reversal, inversion and their product, corresponding to the layer groups $p4/m1'$, $p4/m'$, $p4$, $p4/m$, and $p41'$; the results are in Eqs.~(\ref{eqn_eta_mult}), (\ref{eqn_eta1}), (\ref{eqn_eta2}), (\ref{eqn_eta3}), and (\ref{eqn_eta4}), respectively.
Our results provide a necessary generalization of formulas in previous work \cite{benalcazar2019quantization,schindler2019fractional}, which can give an incorrect result when there are multiple atoms in the unit cell.

In addition to diagnosing band structures in 2D, the results of our work can be used to compute 2D invariants for slices of the 3D Brillouin zone, which is crucial to diagnosing the topology of 3D semimetals \cite{wieder2020strong,lin2018topological,ghorashi2020higher} and some 3D HOTIs \cite{song2017d}.
We present an example of this in Sec.~\ref{sec_3d}.

The manuscript is organized as follows. In Sec.~\ref{sec_2d}, we review the concept of the filling anomaly and its connection to corner charge.
We derive relations between the filling anomaly and the number of Wannier centers at each Wyckoff position.
We then derive symmetry indicator formulas for the filling anomaly (and therefore corner charge) that apply to the square lattice.\par

In Sec.~\ref{sec_3d}, we build a 3D body-centered tetragonal (BCT) model. We analyze the HOTI phase and higher order semi-metal phase in this model by applying our formulas to each $k_z$ slice, which corresponds to a 2D system with two distinct atoms in the primitive unit cell.
We verify our new formulas by numerically calculating the corner/hinge states.
Our example demonstrates why the previous formulas in Refs.~\cite{benalcazar2019quantization,schindler2019fractional} do not hold for a BCT lattice. In Sec.~\ref{sec_discussion}, we summarize our results and discuss future directions.\par

\section{2D square lattice\label{sec_2d}}
We consider gapped 2D systems on the square lattice with no gapless edge modes and no bulk polarization. These systems are either (possibly obstructed) atomic limits or fragile topological phases, where all strong symmetry indicators vanish \cite{po2017symmetry}.
In addition to the $\pi/2$ rotation symmetry of the square lattice, which we denote by $C_4$,
we consider the presence of time reversal $\cal T$, inversion $\cal I$, and/or their product,  corresponding to the following 2D symmetry groups, known as layer groups: $p4/m1'$ ($C_4$, $\cal{T}$, $\cal{I}$); $p4/m'$ ($C_4$, $\cal{TI}$); $p4/m$ ($C_4$, $\cal I$); $p41'$ ($C_4$, $\cal T$); and $p4$ ($C_4$ only), where the symmetry operations in parenthesis indicate the generators, excluding translations.
The layer groups are listed in international notation, where $4$ indicates the $C_4$ rotation; $1'$ indicates $\cal{T}$, $/m$ indicates $\cal{I}$; and $m'$ indicates $\cal{IT}$ \cite{litvin}.
{The layer group $p41'$ does not have a complete symmetry indicator formula, as pointed out in Ref.~\cite{schindler2019fractional}; we derive a partial indicator in Sec.~\ref{sec_p41p}.}

In this section, we describe the method for deriving the symmetry indicator for the filling anomaly (Secs.~\ref{sec_corresp}, \ref{sec_2d_method} and \ref{sec_2d_method_symm}), which can be generalized to any crystal symmetry group in any dimension.
In Sec.~\ref{sec_2d_p4m1p}, we apply the method to the symmetry group $p4/m1'$ with spin-orbit coupling (SOC).
We first re-derive the formula for the case of only one atom in the unit cell \cite{schindler2019fractional} and then derive a new formula for the situation where atoms occupy multiple Wyckoff positions.
We generalize to the layer groups $p4/m'$, $p4$, $p4/m$, and $p41'$ with SOC in Sec.~\ref{sec_2d_gen} and summarize in Sec.~\ref{sec_summary_2D}.

\par

\subsection{\label{sec_corresp}Bulk-corner correspondence}

Topologically trivial bands have symmetric and exponentially localized Wannier functions \cite{bradlyn2017topological,po2017symmetry}.
When the Wannier centers cannot be continuously moved to coincide with the atom positions while obeying crystal symmetry, the system is in an OAL phase \cite{bradlyn2017topological}. Despite having exponentially localized Wannier centers, OALs are non-trivial in the sense that they are separated by a gap-closing phase transition from the trivial phase (where the Wannier centers coincide with the atomic positions). Canonical examples include the Su-Schrieffer-Heeger model in 1D \cite{PhysRevLett.42.1698} and the quadrupole insulator in 2D \cite{benalcazar2017quantized}.

Due to the mismatch between the atomic positions and Wannier centers, OALs can sometimes support mid-gap corner-localized states.
The connection between bulk Wannier centers and mid-gap corner charge is called the bulk-corner correspondence.
The existence of mid-gap corner states indicates that in a finite-sized system with open boundaries, the number of filled bulk valence states is different from the filling required for charge neutrality. Thus, an OAL with mid-gap corner states can either be neutral or symmetric, but not both.
This difference between the neutral and symmetric fillings is called the filling anomaly \cite{benalcazar2019quantization}.
A nontrivial filling anomaly requires not only that the number of filled states differs from the charge neutral filling, but also that the difference cannot be accounted for by adding or removing electrons to the boundary in a symmetry-preserving way.
Thus, the filling anomaly remains robust even if the mid-gap states are pushed up(down) in energy into the conduction(valence) bands by a boundary potential.

In the symmetry groups with time-reversal ($p4/m1'$ and $p41'$), the filling anomaly is defined mod $8$ because one can add or remove eight electrons to the boundary of a finite-sized system without breaking these symmetries (by adding a Kramers pair of time-reversed partners to the four corners of a square lattice).
In the systems without time-reversal ($p4/m'$, $p4/m$, or $p4$), the filling anomaly is defined mod $4$ because one can add four electrons to the boundary of a finite-sized system without breaking these symmetries, as electrons need not come in Kramers pairs.
\par

In this work, we are interested in the filling anomaly that results from purely corner charge.
Therefore, we limit ourselves to systems without gapless surface states, which excludes systems with a bulk polarization or a nontrivial topological invariant.

To compute the corner charge, we note that the symmetrically terminated square lattice with filling anomaly $\eta$ has net charge $\eta e$.
Symmetry requires the charge $\eta e$ is symmetrically sitting at the four corners, resulting in a corner charge $Q_c$ on the square lattice  \cite{benalcazar2019quantization},
\begin{equation}
\label{eqn_qc}
    Q_c=\frac{\eta }{4}e.
\end{equation}
Since, as discussed above, $\eta$ is defined either mod 8 or mod 4, it follows that $Q_c$ is defined mod $2e$ with time-reversal and mod $e$ without.
\par

\subsection{\label{sec_2d_method}Defining the filling anomaly}

\begin{figure}[b]
    \centering
    \includegraphics[width=0.85\linewidth]{Wyckoff.png}
    \caption{Maximal Wyckoff positions in the layer groups $p4/m1'$, $p4/m'$, $p4/m$, $p41'$, and $p4$. The general position $4d$ is not shown.

    }
    \label{fig_wyckoff}
\end{figure}

We now derive the filling anomaly in terms of the ion positions and Wannier centers of the occupied valence bands.
Both the ion positions and Wannier centers are described by Wyckoff positions; for a review of Wyckoff positions in the context of band theory, we refer the reader to Refs.~\cite{bradlyn2017topological,cano2018building}.
We use terminology specific to a finite square lattice of side length $L$, but the method is general.

The first step is to count the number of electrons in the charge neutral configuration.
To do this, we need to know the total number of ions in a finite-sized system with open boundary conditions.
Let $w$ be one of the four Wyckoff positions on the 2D square lattice, $1a$, $1b$, $2c$, or $4d$, shown in Fig.~\ref{fig_wyckoff}.
Define $N_w(L)$ to be the number of ions at the Wyckoff position $w$ that reside inside or on the boundary of a finite-sized square consisting of $L\times L$ unit cells.
For a periodic lattice of size $L\times L$, the number of ions is $L^2$ multiplied by the multiplicity of the Wyckoff position.
However, for open boundary conditions, this is not the case.
As shown in Fig.~\ref{fig_lattice}(b), for an $L\times L$ square with open boundary conditions:
\begin{align}
\label{eqn_N_w_ac}
    N_{a}(L)&=L^2 \nonumber\\
    N_b(L)&= (L-1)^2 \nonumber \\
    N_c(L) &= 2L(L-1).
\end{align}
The total number of electrons in the charge neutral configuration is then given by a sum over all Wyckoff positions:
\begin{equation}
\label{eqn_Nneu}
    N_{\rm neutral} = \sum_w N_w(L)a_w,
\end{equation}
where $a_w$ denotes the number of valence electrons from the ion at Wyckoff position $w$.
(More generally, if there are multiple ions not related by symmetry at the same Wyckoff position, which can happen for Wyckoff positions with a variable coordinate, such as the $4d$ position, then $a_w$ should be the sum of valence electrons from each symmetry-distinct ion in the Wyckoff position $w$.)

We now count the number of electrons required to symmetrically fill the Wannier centers of the valence bands in a finite-size system with open boundary conditions.
Each Wannier center is labelled by a Wyckoff position, $w$, and an irreducible representation (irrep), $\rho_w$, of the site symmetry group of $w$.
(The site symmetry group of $w$ is the set of symmetry operations that leave $w$ invariant; therefore, Wannier functions centered at $w$ must transform as irreps of the site symmetry group. The irreps $\rho_w$ describe the symmetry of the Wannier functions.)
Let $n_{\rho_w}$ count the number of Wannier functions centered at $w$ that transform as $\rho_w$ and are not related by symmetry; in the language of band representations \cite{bradlyn2017topological,cano2018building}, $n_{\rho_w}$ counts the number of times the band representation labelled by $\rho_w$ appears in the valence bands.
Then the number of electrons needed to symmetrically fill the Wannier functions in a finite-size $L\times L$ square with open boundary conditions is:
\begin{equation}
\label{eqn_Nsym}
    N_{\rm symmetric} = \sum_{w} N_w(L) \sum_{\rho_w} n_{\rho_w} \text{dim}(\rho_w),
\end{equation}
where $\text{dim}(\rho_w)$ is the dimension of the irrep $\rho_w$.

Eqs.~(\ref{eqn_Nneu}) and (\ref{eqn_Nsym}) define the filling anomaly:
\begin{align}
\label{eqn_eta_N_N}
    \eta &= N_{\rm neutral}-N_{\rm symmetric} \mod 4(\text{or}~8) \nonumber \\
         &= \sum_w N_w(L) (a_w-\sum_{\rho_w} n_{\rho_w}\text{dim}(\rho_w)) \mod 4(\text{or}~8),
\end{align}
where, as discussed in Sec.~\ref{sec_corresp}, mod~8 applies with time-reversal symmetry and mod~4 applies without.
Although this formula for $\eta$ includes an $L$-dependent term on the right hand side, the $L$-dependence disappears
due to our assumptions that the system has no bulk charge polarization and no bulk net charge, which would contribute terms of order $L$ and $L^2$, respectively.
Thus, $\eta$ is independent of $L$.
A similar method to compute $\eta$ was used in Refs~\cite{schindler2019fractional,benalcazar2019quantization,watanabe2020corner}.

We now specify to the square lattice.
The general Wyckoff position, denoted $4d$, has coordinates $(x,y)$, where $x,y\neq 0,\frac{1}{2}$.
Ions on the $4d$ position always come in multiples of four, even on a finite size lattice, so that $N_{d}(L)$ must be a multiple of four.
Therefore, if time-reversal symmetry is absent, the term in Eq.~(\ref{eqn_eta_N_N}) coming from the $4d$ Wyckoff position is a multiple of four and does not contribute to the filling anomaly.
If time-reversal symmetry is present, all electrons come in Kramers pairs, causing the term in Eq.~(\ref{eqn_eta_N_N}) coming from the $4d$ Wyckoff position to be a multiple of eight, which again does not contribute to the filling anomaly.
Therefore, when computing the filling anomaly, we need only concern ourselves with the maximal Wyckoff positions $1a$, $1b$, and $2c$, shown in Fig.~\ref{fig_wyckoff}.

\begin{table}[b]
\begin{tabular}{ccc}
     Layer group & $w$ & Site-symmetry group\\
     \hline
     $p4/m1'$ & $1a, 1b$ & $4/m1'$ \\
            & $2c$ & $2/m1'$ \\
    \hline
    $p4/m'$ & $1a, 1b$ & $4/m'$ \\
            & $2c$ & $2/m'$ \\
    \hline
    $p4/m$ & $1a, 1b$ &  $4/m$\\
            & $2c$ & $2/m$ \\
    \hline
    $p4$ & $1a, 1b$ &  $4$\\
            & $2c$ & $2$\\
    \hline
    $p41'$ & $1a, 1b$ &  $41'$\\
            & $2c$ & $21'$
\end{tabular}
\caption{Site-symmetry groups for the maximal Wyckoff positions, $w$, of the layer groups $p4/m1'$, $p4/m'$, $p4/m$, $p4$, and $p41'$ in international notation \cite{litvin}. The site-symmetry groups for $1a$ and $1b$ are the same groups that leave $\Gamma$ and $M$ invariant in momentum space; similarly, the site-symmetry group for $2c$ leaves $X$ invariant.}
\label{tab_ssgroups}
\end{table}

\begin{table}[b]
\begin{tabular}{c|ccc}
$4/m1'$ &$E$&$C_4$&$I$\\
\hline
$E_{\frac12 g}$&$2$&$\sqrt{2}$&$2$\\
$E_{\frac12 u}$&$2$&$\sqrt{2}$&$-2$\\
$E_{\frac32 g}$&$2$&$-\sqrt{2}$&$2$\\
$E_{\frac32 u}$&$2$&$-\sqrt{2}$&$-2$\\
\end{tabular}
\quad
\begin{tabular}{c|cc}
$2/m1'$ &$E$&$I$\\
\hline
$E_{g}$&$2$&$2$\\
$E_{u}$&$2$&$-2$\\
\end{tabular}
\quad
\begin{tabular}{c|cc}
$4/m'$ or $41'$ &$E$&$C_4$\\
\hline
$E_{\frac12}$&$2$&$\sqrt{2}$\\
$E_{\frac32}$&$2$&$-\sqrt{2}$
\end{tabular}
\caption{Characters of the irreps of the point groups $4/m1'$, $2/m1'$, $4/m'$, and $41'$ with SOC.
Characters of $2/m'$ and $21'$ are not shown because they each have only one (two-dimensional) irrep with spin-orbit coupling.
The irreps are labelled in the notation of Ref.~\cite{altmann1994point}.
In all groups, $\cal T$ or $\cal{TI}$ requires all irreps consist of two-dimensional pairs with complex conjugate eigenvalues.
The characters of $C_2 \equiv C_4^2$ and $m_z \equiv C_4^2\cal{I}$ are always zero and not listed here.
The character tables are not square because we have not included all symmetry operations and, further, have not included the irreps without SOC.}
\label{tab_irrep}
\end{table}

\begin{table}[t]
\begin{tabular}{c|ccc}
$4$&$E$&$C_4$&$C_2$\\
\hline
${}^1E_{\frac12}$&$1$&$\epsilon$&$i$\\
${}^1E_{\frac32}$&$1$&$-\epsilon^*$&$-i$\\
${}^2E_{\frac32}$&$1$&$-\epsilon$&$i$\\
${}^2E_{\frac12}$&$1$&$\epsilon^*$&$-i$\\
\end{tabular}
\qquad
\begin{tabular}{c|cc}
$2$&$E$&$C_2$\\
\hline
${}^1E_{\frac12}$&$1$&$i$\\
${}^2E_{\frac12}$&$1$&$-i$\\
\end{tabular}
\caption{Characters of irreps of the point groups $4$ and $2$ with SOC; $\epsilon=\exp(\pi i/4)$.}
\label{tab_irrep_noT}
\end{table}

\begin{table}[t]
\begin{tabular}{c|cccc}
$4/m$ &$E$&$C_4$&$C_2$&$I$\\
\hline
${}^1E_{\frac12 g}$&$1$&$\epsilon$&$i$&$1$\\
${}^1E_{\frac32 g}$&$1$&$-\epsilon^*$&$-i$&$1$\\
${}^2E_{\frac32 g}$&$1$&$-\epsilon$&$i$&$1$\\
${}^2E_{\frac12 g}$&$1$&$\epsilon^*$&$-i$&$1$\\
${}^1E_{\frac12 u}$&$1$&$\epsilon$&$i$&$-1$\\
${}^1E_{\frac32 u}$&$1$&$-\epsilon^*$&$-i$&$-1$\\
${}^2E_{\frac32 u}$&$1$&$-\epsilon$&$i$&$-1$\\
${}^2E_{\frac12 u}$&$1$&$\epsilon^*$&$-i$&$-1$\\
\end{tabular}
\quad
\begin{tabular}{c|ccc}
$2/m$&$E$&$C_2$&$I$\\
\hline
${}^1E_{\frac12 g}$&$1$&$i$&$1$\\
${}^2E_{\frac12 g}$&$1$&$-i$&$1$\\
${}^1E_{\frac12 u}$&$1$&$i$&$-1$\\
${}^2E_{\frac12 u}$&$1$&$-i$&$-1$\\
\end{tabular}
\quad
\caption{Characters of irreps of $4/m$ and $2/m$ with SOC; $\epsilon=\exp(\pi i/4)$.}
\label{tab_irrep_I_noT}
\end{table}

A second simplification for the square lattice is that $\text{dim}(\rho_w) $ is independent of $\rho_w$ and $w$ for each symmetry group we consider, as we now explain.
The maximal Wyckoff positions and their site symmetry groups are listed in Table~\ref{tab_ssgroups} for all the layer groups we consider.
In $p4/m1'$, Table~\ref{tab_ssgroups} shows the only possible site-symmetry groups are $4/m1'$ or $2/m1'$; all irreps of these groups are two-dimensional, as shown in Table~\ref{tab_irrep}.
The same is true for $p4/m'$ and $p41'$.
In $p4$, Table~\ref{tab_ssgroups} shows the only possible site-symmetry groups are $4$ or $2$; all irreps of these groups are one-dimensional, as shown in Table~\ref{tab_irrep_noT}.
The same is true for $p4/m$; its irreps are enumerated in Table~\ref{tab_irrep_I_noT}.

Thus, in all cases, $\text{dim}(\rho_w)$ is independent of both $\rho$ and $w$ and the expression for $\eta$ in Eq.~(\ref{eqn_eta_N_N}) can be simplified on the square lattice as,
\begin{equation}
\label{eqn_eta_square}
    \eta \xrightarrow[\text{lattice}]{\text{square}} \sum_{w \text{~max}} N_w(L) (a_w-n_w d)~~ \text{mod}~4(\text{or}~8),
\end{equation}
where the sum is over all maximal Wyckoff positions $w$; $\eta$ is defined mod 4 (mod 8) in the absence (presence) of time-reversal symmetry, as in Eq.~(\ref{eqn_eta_N_N});
\begin{equation}
    n_w \equiv \sum_{\rho_w} n_{\rho_w};
    \label{eqn_def_nw}
\end{equation}
and
\begin{equation}
    \label{eqn_defd}
    d=\text{dim}(\rho_w) = \begin{cases} 2 & \text{ if }p4/m1', p4/m', p41'\\
                                            1 & \text{ if }p4/m, p4 \end{cases}
\end{equation}
is the dimension of each irrep (which is independent of the choice of Wyckoff position and choice of irrep, as discussed in the previous paragraph.)

Plugging the formulas for $N_w(L)$ from Eq.~(\ref{eqn_N_w_ac}) into Eq.~(\ref{eqn_eta_square}) yields:
\begin{multline}
    \label{eqn_eta_in_nw}
    \eta \xrightarrow[\text{lattice}]{\text{square}}
    L^2 \left[ N - d(n_a + n_b +2n_c) \right] \\ - 2L\left[ a_b+a_c - d(n_b+n_c)\right]  \\+ (a_b - dn_b) ~\text{mod}~4(\text{or}~8),
\end{multline}
where $\eta$ is defined mod 4 (mod 8) in the absence (presence) of time-reversal symmetry,
\begin{equation}
    N=a_a+a_b+2a_c
    \label{eqn_defN}
\end{equation}
is the number of occupied bands, and $d$ is defined in Eq.~(\ref{eqn_defd}).

The bulk charge is determined by the number scaling with $L^2$ in Eq.~(\ref{eqn_eta_in_nw}) and must be zero in a system that is charge neutral in the bulk:
\begin{equation}
    N-d(n_a+n_b+2n_c) = 0 ~\text{mod}~4(\text{or}~8)
    \label{eqn_bulk0}
\end{equation}
The bulk polarization $p_x=p_y\equiv p$ is determined by the number scaling with $L$ in Eq.~(\ref{eqn_eta_in_nw}) and must also be zero:
\begin{equation}
    2\left[ a_b+a_c-d(n_b+n_c) \right] = 0~\text{mod}~4(\text{or}~8)
    \label{eqn_pol0}
\end{equation}
The filling anomaly $\eta$ is determined by the $L$-independent term in Eq.~(\ref{eqn_eta_in_nw}):
\begin{equation}
\label{eqn_eta_na}
    \eta = a_b-dn_b = a_a-dn_a ~\text{mod}~4(\text{or}~8),
\end{equation}
where the equality follows from Eqs.~(\ref{eqn_bulk0}) and (\ref{eqn_pol0}).
Equations (\ref{eqn_bulk0}), (\ref{eqn_pol0}) and (\ref{eqn_eta_na}) were also obtained in Ref.~\cite{watanabe2020corner}.

\subsection{\label{sec_2d_method_symm}Symmetry indicators for Wannier centers}

The formulas in the previous section derive the filling anomaly in terms of the crystal structure and Wannier centers.
We now derive $n_w$ in terms of the irreps of the little groups at high-symmetry points of the bulk band structure. This is useful because the irreps are easier to compute than the Wannier centers.
As we will see, because the irreps in momentum space do not completely determine the Wannier centers \cite{Bacry1988b,Michel1992,Bacry1993,bradlyn2017topological,cano2020band,Cano2020topology}, $n_w$ can only be determined up to some modulus from symmetry.

To this end, let $A$ be the integer ``EBR matrix'' of the symmetry group under consideration: each column of $A$ is labelled by an EBR and each row a particular irrep of the little group of a particular high-symmetry point.
The entries in the matrix indicate the number of times each momentum space irrep appears in the EBR \cite{song2020fragile,song2020twisted,cano2020band}.

A group of topologically trivial bands can be expressed as a linear combination of EBRs \cite{bradlyn2017topological} with integer coefficients $\tilde{n}_i$.
The irreps that appear at high-symmetry points in the band structure satisfy
\begin{equation}
    v=A\tilde{n},
    \label{eqn_vAn}
\end{equation}
where $v_j$ is the number of times the $j^\text{th}$ irrep appears in the band structure.
We need to invert this equation to find $n_w$ in terms of $v$, as we now explain.

Let the Smith normal form of $A$ be given by
\begin{equation}
    A = U^{-1}DV^{-1},
    \label{eqn_smithA}
\end{equation}
where $D$ is a diagonal integer matrix with diagonal entries $(d_1, \dots, d_M, 0, \dots 0)$, i.e., the first $M$ entries are positive and the remaining entries are zero, and $U,V$ are integer matrices invertible over the integers.
(Note: the stable topological crystalline insulator classification of the group is given by $\otimes_{i=1}^M \mathbb{Z}_{d_i}$, where $\mathbb{Z}_{d_i}$ is the group of integers mod $d_i$ \cite{po2017symmetry,song2018quantitative,song2020fragile,song2020twisted,cano2020band}.)

We want to express the number of EBRs corresponding to each Wyckoff position in terms of symmetry irreps.
Since we are only considering topologically trivial bands, we consider only the vectors $v$ for which there exists an integer vector $\tilde{n}$ that solves Eq.~(\ref{eqn_vAn}).
Then the Smith normal form in Eq.~(\ref{eqn_smithA}) implies $Uv = (v_1',\dots v_M', 0 \dots 0)$, where $d_i$ divides $v_i'$.
For such bands, one solution to Eq.~(\ref{eqn_vAn}) is given by $\tilde{n}=VD^pUv$, where $D^p$ is the pseudoinverse of $D$, i.e., a diagonal matrix with diagonal entries $(1/d_1, 1/d_2, \dots, 1/d_l, 0, \dots 0)$.
This solution is not generically unique:
the most general solution to Eq.~(\ref{eqn_vAn}) is $\tilde{n}=VD^pUv + V\tilde{n}_0$, where $\tilde{n}_0$ is any integer vector in the nullspace of $D$, i.e., the first $M$ entries of $\tilde{n}_0$ are zero, so that $D\tilde{n}_0=0$.
Thus, given a particular $v$, $\tilde{n}_i$ can only be determined modulo $\text{gcd}\lbrace V_{ij} | _{j>M} \rbrace$, where gcd indicates the greatest common divisor.

However, we do not need each $\tilde{n}_i$ separately: we seek $n_w$ in Eq.~(\ref{eqn_def_nw}), which is a sum of all $\tilde{n}_i$ where the EBR indicated by $i$ is induced from an irrep of the site-symmetry group of the Wyckoff position $w$; we use $i\in w$ to denote this set of EBRs.
Then, following the previous paragraph, we can express $n_w$ as
\begin{equation}
    n_w = \sum_{i\in w} \left[ VD^pUv\right]_i \mod \text{gcd}\lbrace \left( \sum_{i\in w} V_{ij} \right) | _{j > M} \rbrace.
    \label{eqn_smith_nw}
\end{equation}
We now use Eq.~(\ref{eqn_smith_nw}) to compute $n_w$ in $p4/m1'$ in terms of the symmetry indicators.
We do the same for $p4/m'$, $p4/m$, $p4$, and $p41'$ in Sec.~\ref{sec_2d_gen}.
The Smith normal form of the EBR matrix for each of these groups is computed in Appendix~\ref{app_smith}.

\subsubsection{Symmetry indicators for $n_{a,b,c}$ in $p4/m1'$}

In $p4/m1'$, there are three high-symmetry points in the Brillouin zone: $\Gamma= (0,0)$, $M=(\pi,\pi)$ and $X = (\pi,0)$. The point $(0,\pi)$ is symmetry-related to $X$, so does not add any new information.
The points $\Gamma$ and $M$ are invariant under the point group $4/m1'$, while $X$ is invariant under $2/m1'$; the irreps of both groups are listed in Table~\ref{tab_irrep}.

Define $\# K_{\frac{1}{2}u}$ ($\# K_{\frac{3}{2}u}$) to be the number of times the irrep $E_{\frac{1}{2}u}$ ($E_{\frac{3}{2}u}$) appears in the valence band spectrum at the high-symmetry point $K= \Gamma, M$ and define $\# K_{u}=\# K_{\frac12 u}+\# K_{\frac32 u}$.
Similarly, define $\# X_u $ to be the number of times the irrep $E_u$ appears in the valence band spectrum at $X$.
Then define $[K_{\rho}]$ to be the difference between the number of times the irrep indicated by $\rho$ appears at the high-symmetry point $K$ and at $\Gamma$:
\begin{equation}
    \label{eqn_defKrho}
    [K_{\rho}]=\# K_{\rho}-\# \Gamma_{\rho}.

\end{equation}
Using this notation, we find from Eq.~(\ref{eqn_smith_nw}) (details in Appendix~\ref{app_smith}):
\begin{align}
\label{eqn_na}
    n_a &= \frac{N}{2}-[X_u]-\frac32[M_{u}]+2[M_{\frac12 u}] &\mod 4,\\
\label{eqn_nb}
    n_b &= [X_u]+\frac12 [M_{u}]-2[M_{\frac12 u}]&\mod 4,\\
\label{eqn_nc}
    n_c &= \frac12 [M_{u}] &\mod 2,
\end{align}
where $N=2(n_a+n_b+2n_c)$ is the total number of filled bands, which was derived by imposing bulk charge neutrality in Eq.~(\ref{eqn_bulk0}).

\subsection{Symmetry indicators for the filling anomaly}
\label{sec_2d_p4m1p}

We are now ready to compute the filling anomaly $\eta$ in Eq.~(\ref{eqn_eta_square}) in terms of the symmetry irreps by plugging in Eqs.~(\ref{eqn_na}), (\ref{eqn_nb}), and (\ref{eqn_nc}) for the group $p4/m1'$ (the results for other groups are in Sec.~\ref{sec_2d_gen}).
In previous work \cite{benalcazar2019quantization,schindler2019fractional}, $\eta$ was computed on 2D lattices with only one maximal Wyckoff position occupied.
The main advance of this work is to compute $\eta$ for square lattices with any number of atoms in the unit cell.
In Sec.~\ref{sec_2d_1}, we compute $\eta$ for the simple square lattice with one atom in the unit cell, reproducing earlier results \cite{schindler2019fractional}.
In Sec.~\ref{sec_2d_3}, we derive $\eta$ in the general case with multiple atoms in the unit cell.

\par

\subsubsection{Simple square lattices\label{sec_2d_1}}

We now reproduce the symmetry indicator formula in Ref.~\cite{schindler2019fractional} for a square lattice with one atom in the unit cell.
Without loss of generality, we can take that atom to be in the $1a$ position.
Then the formula for the filling anomaly in Eq.~(\ref{eqn_eta_na}) (with $d=2$) simplifies to
\begin{equation}
\label{eqn_eta_nb}
    \eta = -2n_b \mod 8.
\end{equation}
Plugging the expression for $n_b$ in Eq.~(\ref{eqn_nb}) into Eq.~(\ref{eqn_eta_nb}), we obtain the symmetry indicator formula:
\begin{align}
\label{eqn_eta_simple}
    \eta = -2[X_u]-[M_{u}]+4[M_{\frac12 u}] \mod 8.
\end{align}
Noticing that $[M_u]$ must be an even number in an (obstructed) atomic limit phase because $n_c$ in Eq.~(\ref{eqn_nc}) must be an integer, we deduce that $\eta$ is in fact a $\mathbb{Z}_4$ quantity. Eq.~(\ref{eqn_eta_simple}) was introduced in Eq.~(50) in Ref.~\cite{schindler2019fractional}.
\par

\subsubsection{General case: atoms in multiple Wyckoff positions\label{sec_2d_3}}

We now consider the general case, shown in Fig.~\ref{fig_lattice}(b), where there can be atoms at any Wyckoff positions.
Thus, the number of electrons from each ion, $a_{a,b,c}$, can all be non-zero. The number of filled bands is $N=a_a+a_b+2a_c$.
Plugging Eq.~(\ref{eqn_na}) into the expression for the filling anomaly in Eq.~(\ref{eqn_eta_na}) (with $d=2$) we find the symmetry indicator formula for the filling anomaly:
\begin{align}
\label{eqn_eta_mult}
    \eta &= a_a-N+2[X_u]+3[M_{u}]-4[M_{\frac12 u}] \mod 8.
\end{align}
Since time-reversal symmetry constrains $a_a$ and $N$ to be even numbers and, as discussed below Eq.~(\ref{eqn_eta_simple}), $[M_u]$ is also even, the filling anomaly is again a $\mathbb{Z}_4$ quantity.
When $N=a_a$, which implies $a_b = a_c = 0$, Eq.~(\ref{eqn_eta_mult}) is equivalent to Eq.~(\ref{eqn_eta_simple})
(to see this, notice that the equations are mod 8, $[M_u]$ is even, and when $a_b=a_c =0$, Eq.~(\ref{eqn_pol0}) implies $n_b = n_c \mod 2$, from which Eqs.~(\ref{eqn_nb}) and (\ref{eqn_nc}) together require that $[X_u]$ is also even.)
When $a_a\neq N$, Eq.~(\ref{eqn_eta_mult}) is distinct from Eq.~(\ref{eqn_eta_simple}) and has not appeared in previous literature.

In Sec.~\ref{sec_3d}, we build an explicit body-centered tetragonal model with $C_4$, $\cal{T}$ and $\cal{I}$ symmetry.
The Hamiltonian in the $k_z = 0$ and $k_z = \pi$ planes of the model describes a square lattice with $p4/m1'$ symmetry, but
with atoms at multiple Wyckoff positions, corresponding to the projection of the 3D model onto a 2D plane.
Therefore, the 2D bulk-corner correspondence derived in this section applies to 2D planes of that model, providing a numerical check of the analytical results.
\par

\subsection{\label{sec_2d_gen}Generalization to other layer groups}

We now compute the filling anomaly in terms of the symmetry indicators for the layer groups $p4/m'$, $p4$, and $p4/m$ and explain why $p41'$ does not have an analogous formula.
The formulas for $n_{a,b,c}$ in this section are derived in Appendix~\ref{app_smith}.

\subsubsection{$p4/m'$}
\label{sec_2d_p4mp}
For the layer group $p4/m'$, the high-symmetry points $\Gamma$ and $M$ are invariant under the point group $4/m'$, while $X$ is invariant under $2/m'$.
The irreps for these groups are listed in Table~\ref{tab_irrep}.
The number of Wannier centers at each Wyckoff position, $n_a$, $n_b$ and $n_c$, are defined by Eq.~(\ref{eqn_def_nw}) with irrep dimension $d=2$ in Eq.~(\ref{eqn_defd}).
Using Eq.~(\ref{eqn_smith_nw}), we find $n_a$ and $n_b$ can be determined mod $2$:
\begin{align}
\label{eqn_na1}
    n_a &= \frac{N}{2}-[M_{\frac12 }]&\mod 2,\qquad\\
\label{eqn_nb1}
    n_b &= [M_{\frac12 }]&\mod 2,\qquad
\end{align}
where $[M_{\frac12}] = \# M_\frac12-\# \Gamma_\frac12$, where $\# K_\frac12$ indicates the number of times the irrep $E_\frac12$ appears in the valence bands at the high-symmetry point $K = \Gamma, M$.
We find $n_c=0$ mod 1, i.e., $n_c$ is not constrained by symmetry irreps.

The filling anomaly is determined by Eq.~(\ref{eqn_eta_na}) taken mod 4 with $d=2$.
Substituting in Eq.~(\ref{eqn_na1}) yields:
\begin{equation}
\label{eqn_eta1}
    \eta_1 = a_a-N+2[M_{\frac12}]\mod 4,
\end{equation}
where the subscript is to distinguish the filling anomaly in $p4/m'$ from that computed in $p4/m1'$ in Eq.~(\ref{eqn_eta_mult}).
In the case where atoms only occupy one Wyckoff position ($1a$), $a_a=N$ and this equation reduces to
\begin{equation}
\label{eqn_eta11}
    \eta_1 = 2[M_{\frac12}]\mod 4.
\end{equation}
Eq.~(\ref{eqn_eta11}) was introduced in the context of higher order Fermi arcs \cite{wieder2020strong} with $C_4$ and $M_{x,y}$ symmetries: the anti-commuting reflection symmetries there play the same role as $\cal TI$ in $p4/m'$ in generating two dimensional irreps. \par

\subsubsection{$p4$}

For the layer group $p4$, the high-symmetry points $\Gamma$ and $M$ are invariant under the point group $4$, while $X$ is invariant under $2$; the irreps for these groups are listed in Table~\ref{tab_irrep_noT}.
The number of Wannier centers at each Wyckoff position, $n_a$, $n_b$ and $n_c$, are defined by Eq.~(\ref{eqn_def_nw}) with irrep dimension $d=1$ in Eq.~(\ref{eqn_defd}).
Using Eq.~(\ref{eqn_smith_nw}), we find that $n_a$ and $n_b$ can be determined mod $4$, while $n_c$ can be determined mod $2$:
\begin{align}
    \label{eqn_na_2}
    n_a &= N-[X_2]+\frac32([M_1+M_3])+2[M_2] \mod 4\\
\label{eqn_nb_2}
    n_b &= [X_2]-\frac12 ([M_{1}]+[M_3])-2[M_2]~\mod 4\\
\label{eqn_nc_2}
    n_c &= -\frac12 ([M_1]+[M_3]) ~\mod 2
\end{align}
Here we use the notation $[M_j]=\# M_j-\# \Gamma_j$, where $j=1,2,3,4$ corresponds to the irrep with $C_4$ eigenvalue $\exp(i\frac{\pi}{2}(j-\frac12))$, and $[X_1]=\#X_1-\#\Gamma_1-\#\Gamma_3$, $[X_2]=\#X_2-\#\Gamma_2-\#\Gamma_4$, where $X_{1,2}$ corresponds to the irrep with $C_2$ eigenvalues $+i$, $-i$.
As in previous sections, $\# K_j$ indicates the number of times the irrep $j$ appears in the valence bands at the high-symmetry point $K$.

The expression for the filling anomaly is given by Eq.~(\ref{eqn_eta_na}) taken mod 4 with $d=1$:
\begin{align}
\label{eqn_eta_na2}
    \eta_2 &= a_a-n_a \mod 4 \\
         &= a_a-N+[X_2]-\frac32([M_1+M_3])-2[M_2] \mod 4,
         \label{eqn_eta2}
\end{align}
where Eq.~(\ref{eqn_eta2}) results from plugging Eq.~(\ref{eqn_na_2}) into Eq.~(\ref{eqn_eta_na2}).

\subsubsection{$p4/m$}

For the layer group $p4/m$, the high-symmetry points $\Gamma$ and $M$ are invariant under the point group $4/m$, while $X$ is invariant under $2/m$; the irreps for these groups are listed in Table~\ref{tab_irrep_I_noT}.
The number of Wannier centers at each Wyckoff position, $n_a$, $n_b$ and $n_c$, are defined by Eq.~(\ref{eqn_def_nw}) with irrep dimension $d=1$ in Eq.~(\ref{eqn_defd}).
From Eq.~(\ref{eqn_smith_nw}), we find that $n_a$ and $n_b$ can be determined mod $4$ while $n_c$ can be determined mod $2$. The formulas are:
\begin{align}
    \label{eqn_na_3}
    n_a &= N-[X_u]-\frac32[M_u]+2([M_{1u}]+[M_{2u}])\mod 4\\
\label{eqn_nb_3}
    n_b &= [X_u]+\frac12 [M_u]-2([M_{1u}]+[M_{2u}])~\mod 4\\
\label{eqn_nc_3}
    n_c &= \frac12 [M_u] ~\mod 2
\end{align}
Here we use the notation $[M_{j,\xi}]=\# M_{j,\xi}-\#\Gamma_{j,\xi}$, where the irrep of $4/m$ labelled by $j,\xi$ has
$C_4$ eigenvalue $\exp(i\frac{\pi}{2}(j-\frac12))$, $j=1,2,3,4$,
and inversion eigenvalue $+1,-1$ corresponding to $\xi = g,u$.
In addition, $[X_{1,\xi}]=\# X_{1,\xi}-\#\Gamma_{1,\xi}-\#\Gamma_{3,\xi}$ and $[X_{2,\xi}]=\# X_{2,\xi}-\#\Gamma_{2,\xi}-\#\Gamma_{4,\xi}$, where $X_{1,\xi}$, $X_{2,\xi}$ correspond to the irreps of $2/m$ with $C_2$ eigenvalues $+i$, $-i$ and inversion eigenvalues $+1, -1$ corresponding to $\xi = g,u$. We denote $[K_\xi]=\sum_j[K_{j,\xi}]$.

The filling anomaly is determined by Eq.~(\ref{eqn_eta_na}) taken mod 4 with $d=1$:
\begin{align}
\label{eqn_eta_na3}
    \eta_3 &= a_a-n_a \mod 4,\\
         &= a_a-N+[X_u]+\frac32[M_u]-2([M_{1u}]+[M_{2u}]) \mod 4,
         \label{eqn_eta3}
\end{align}
where Eq.~(\ref{eqn_eta3}) results from plugging Eq.~(\ref{eqn_na_3}) into Eq.~(\ref{eqn_eta_na3}).

\subsubsection{$p41'$}
\label{sec_p41p}

The group $p41'$ does not have a symmetry indicator formula, as we now explain.
The filling anomaly is given by Eq.~(\ref{eqn_eta_na}) taken mod 8 with $d=2$ (from Eq.~(\ref{eqn_defd})):
\begin{align}
    \label{eqn_eta_n4}
    \eta_4 &= a_a-2n_a \mod 8
\end{align}
However, $n_a$ is given by Eq.~(\ref{eqn_na1}) (as explained in Appendix~\ref{sec_Smith_p41p}, the symmetry indicator formula for $n_a$ is the same in $p41'$ as in $p4/m'$) and is only defined mod 2.
It follows from plugging Eq.~(\ref{eqn_na1}) into Eq.~(\ref{eqn_eta_n4}) that the symmetry indicator formula for $\eta_4$ is only defined mod 4, even though $\eta_4$ should be determined mod 8.
Thus, we say that the symmetry indicator formula does not exist because the symmetry indicators do not provide enough information to completely determine the filling anomaly in this group.
The mod 8 filling anomaly $\eta_4$ can only be partially determined mod 4:
\begin{equation}
    \eta_4 \mod 4= a_a-N+2[M_{\frac12}]\mod 4.
    \label{eqn_eta4}
\end{equation}

\subsection{Summary of 2D results}
\label{sec_summary_2D}
Eqs.~(\ref{eqn_eta_mult}), (\ref{eqn_eta1}), (\ref{eqn_eta2}), and (\ref{eqn_eta3}) are the symmetry indicator formulas that express the filling anomaly in OALs in terms of the symmetry invariants.
Together, these formulas encompass all square lattices with time-reversal, inversion symmetry, and/or their product,
and any number of atoms in the unit cell.\par

All the formulas derived in this section are additive, and can be applied to an insulating band structure with any number of filled valence bands, as long as it is charge neutral, polarization free, and all the strong symmetry indicators are trivial. Because of the additivity, the formulas also apply to fragile topological phases, as discussed in Ref.~\cite{benalcazar2017quantized}.

We now discuss some connections to previous work.
As we have mentioned earlier, the formulas for the filling anomaly in Refs.~\cite{benalcazar2019quantization} and \cite{schindler2019fractional} do not accommodate multiple atoms in the unit cell.
In Ref.~\cite{watanabe2020corner}, the filling anomaly $\eta$ was computed in terms of $n_w$ in the general case of having multiple atoms in the unit cell, but $\eta$ was not expressed in terms of the symmetry indicators.
Finally, the real space invariants computed in Ref.~\cite{song2020twisted} are related to the $n_w$ computed here and are computed using the EBR matrix, but there is not a one-to-one correspondence between them.
Ref.~\cite{kooi2020bulk} also discusses real space topological invariants that go beyond symmetry indicators.

In Appendix~\ref{app_diamond}, we derive the symmetry indicator formula for the filling anomaly for a finite-sized square lattice with a boundary normal to the $(1,1)$ direction. It turns out that the formulas are the same as we have derived in this section, where the boundary is normal to the $(1,0)$ direction.
The results in this section can be generalized to a finite-sized square lattice with a boundary normal to any direction by defining a square supercell with a side parallel to the boundary.
Since the supercell necessarily contains multiple atoms, the results in Sec.~\ref{sec_2d_3} apply; in order to get the correct irreps at high-symmetry points, the band structure must be computed relative to the supercell.
Non-square terminations can also obey $C_4$ symmetry (for example, an octagon); while the general logic described in this section applies, the counting in Eq.~(\ref{eqn_N_w_ac}) will be different.

Interestingly, we have found numerically that the corner states survive even if the global $C_4$ symmetry is broken, i.e., in a square lattice with an integer number of unit cells.
Although rigorously the corner states are not protected if the lattice symmetry is broken globally,
physically this makes sense because the localized states on one corner should not depend on how the lattice is terminated at other corners.
A different method to compute the presence of gapless boundary states in other systems with a fractional number of unit cells was discussed in Ref.~\cite{rhim2018unified}.

\begin{figure*}[t
    \centering
    \includegraphics[width=0.9\linewidth]{Fig2.png}
    \caption{(a) Conventional unit cell of a body-centered tetragonal lattice;
    there are two sub-layers at $z=0,~\frac12$ (solid and hollow dots).
    The eigenstates in the plane $(k_x,k_y,k_z=k_{z0})$ generically have weight on both sublattices.
    Therefore, the 2D Hamiltonian $H(k_x,k_y,k_{z0})$ describes atoms on a square lattice with two sites in the unit cell (b).

    (c) Spectrum for a rod (finite in the $x$- and $y$-directions, infinite in $z$) in the HOTI phase; gapless hinge states cross at $k_z=\pi$, forming an eight-fold degeneracy. (d) Spectrum for a rod in the DSM(i) phase, showing four-fold degenerate HOFA states between the surface Dirac point (projected at $k_z=0$) and bulk Dirac point. The HOFA regions have filling anomaly $\eta=2\mod 4$, while the crossing at $k_z=\pi$ has filling anomaly $\eta=4\mod 8$. (e) Phase diagram of our model, controlled by one parameter, $m/t$. }
    \label{fig_3d}
\end{figure*}

\section{3D model on the body-centered tetragonal lattice\label{sec_3d}}
We now apply the results derived in the previous section to classify the topology of a 3D tight-binding Hamiltonian.
We are interested in the HOTI and higher order Fermi arc (HOFA) phases.
The topology of these phases can be understood by studying 2D slices of the Brillouin zone with fixed $k_z$.
For example, the HOTI phase can be viewed as Wannier center pumping between the two time-reversal invariant planes ($k_z=0$ and $k_z=\pi$) \cite{song2017d}, while the HOFA phase requires each 2D fixed-$k_z$ slice on the Fermi arc to have a non-trivial filling anomaly \cite{wieder2020strong}.
The difference between the previous works and the present manuscript is that we consider a $C_4$ symmetric body-centered tetragonal (BCT) model, whose 2D slices in momentum space
necessarily contain atoms at different Wyckoff positions, as we will show in the next section.
Therefore, the results obtained in Sec.~\ref{sec_2d_3} and \ref{sec_2d_gen} are necessary to correctly identify the filling anomaly of 2D slices in momentum space.

\par

\subsection{Tight-binding model}
We build an explicit BCT tight-binding model in space group $87$ ($I4/m$) with a spin-$\frac32$ degree of freedom on each site.
The space group $I4/m$ is generated by body-centered lattice translations, a $C_4$ rotation about the $z$-axis, inversion, and time-reversal symmetry.
The matrix forms of these generators are given below in terms of the spin-$\frac32$ generators, $J_{x,y,z}$ (whose matrix forms are defined in Appendix~\ref{app_J}) and decomposed into Pauli matrices $\sigma_{0,x,y,z}$ and $\tau_{0,x,y,z}$:
\begin{align}
\label{eqn_C4}
    C_4 &= e^{-i\pi J_{z}/2}=\tau_z\sigma_z e^{-i\pi\sigma_z/4},\\
\label{eqn_Inv}
    {\cal I} &= \tau_0\sigma_0,\\
\label{eqn_TR}
    {\cal T} &= e^{-i\pi J_{y}}K=-i\tau_x\sigma_y K,
\end{align}
where $K$ is complex conjugation.
It will be convenient to introduce $\tau_\pm = \frac12(\tau_x\pm i\tau_y)$. \par

We define a Cartesian coordinate system by the unit vectors $e_x=(1~0~0)a$, $e_y=(0~1~0)a$, $e_z=(0~0~1)c$, where $a$ and $c$ are lattice constants.
We align the base of the conventional tetragonal unit cell diagonally with respect to the $x$ and $y$ axes (see Fig.~\ref{fig_3d}(a)).
In this basis, the primitive translation vectors of the BCT lattice are given by
$e_1 = \frac12(e_y + e_z)$, $e_2 = \frac12(e_x+e_z)$, $e_3= \frac12(e_x + e_y)$.
The conventional unit cell has two atoms, indicated by solid and hollow circles in Fig.~\ref{fig_3d}(a), which form two sublattices. The primitive unit cell has one atom.

We take the lattice constants $c>a$. Therefore, each atom has four nearest neighbors on the same sublattice and eight next-nearest neighbors on the opposite sublattice. Our model only includes hopping between each site and these 12 nearest and next-nearest neighbors, illustrated in Fig.~\ref{fig_3d}(a). One of each of these hopping terms is given below (and drawn in Fig.~\ref{fig_3d}(a)) and the others are related by symmetry:
\begin{align}
    V_{0\rightarrow e_1}&=V_1=\frac14\left( \tau_z(t\sigma_z-\gamma\sigma_y)+\beta\tau_+\sigma_0+\beta^*\tau_-\sigma_0 \right) \nonumber \\
    V_{0\rightarrow e_3}&=V_2=-\frac14\left(2t\tau_z\sigma_z+\gamma\tau_y\sigma_0 \right)
    \label{eqn_model}
\end{align}
The symmetry-related hopping terms in other directions are written explicitly in Appendix.~\ref{app_model}.
There is also an onsite mass term:
\begin{equation}
    V_{onsite} = m\tau_z\sigma_z
    \label{eqn_model_mass}
\end{equation}
The parameters $m$, $t$, and $\gamma$ are real, and $\beta$ is complex; $\beta^*$ is the complex conjugate of $\beta$.
The Hamiltonian is written in momentum space in both the primitive and conventional bases in Appendix~\ref{app_model}. \par

\subsection{Topological phases}

We now classify the topology of this model by considering the Hamiltonian in 2D slices of the 3D Brillouin zone.
Specifically, for fixed $k_{z0}$, $H(k_x,k_y,k_{z0})$ can be regarded as the Hamiltonian of a 2D system.
Since eigenstates generically have weight on both sublattices, this 2D Hamiltonian describes atoms on a square lattice with two sites in the unit cell, as shown in Fig.~\ref{fig_3d}(b). If the 2D Hamiltonian has no gapless edge states or polarization, then we can apply our results from Sec.~\ref{sec_2d_3} and \ref{sec_2d_gen} to determine the corner charge of this 2D model when the 3D system is truncated in the $x$ and $y$ directions but infinite in the $z$ direction.

We find that our model has several topological phases depending on the ratio $m/t$. The parameters $\beta$ and $\gamma$ do not change the topological phase. We list all the different phases in Fig.~\ref{fig_3d}(e) with respect to $m/t$: there is one HOTI phase and two Dirac semimetal (DSM) phases, which we now describe.

\subsubsection{$\mathbb{Z}_8$ HOTI}

\begin{table}[]
    \centering
    \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
    \hline
         $k_z$& $a_a$ & $N$ & $[X_u]$ & $[M_u]$ & $[M_{\frac12 u}]$ &$n_a$ &$n_b$ & $n_c$ &$\eta$\\
         \hline
         0& 2 & 4 & 1 & 0 & 0 & 1 & 1 & 0 &0\\
         \hline
         $\pi$& 2 & 4 & -1 & 0 & 0 & 3 & -1 & 0 & 4\\
    \hline
    \end{tabular}
    \caption{Symmetry indicators of occupied bands, $n_{a,b,c}$, and $\eta$ in the $k_z=0$ and $k_z=\pi$ planes of our BCT model in the HOTI phase. The filling anomalies $\eta$ are computed by plugging the values in the table into Eq.~(\ref{eqn_eta_mult}), while $n_{a,b,c}$ are computed from Eqs.~(\ref{eqn_na}), (\ref{eqn_nb}) and (\ref{eqn_nc}). The irreps are derived in Appendix~\ref{sec_3d_irreps}, Table~\ref{tab_calculation_HOTI}.
    Notice that $[X_u]$ need not be even because although the $(1,1)$ boundary shares the same  symmetry indicator formula as the $(1,0)$ boundary, the constraints from zero bulk charge polarization and zero bulk net charge are different, as derived in Appendix~\ref{app_diamond}.

    }
    \label{tab_BCT}
\end{table}

When $0< m/t < 2$, a $\mathbb{Z}_8$ HOTI with the non-trivial symmetry indicator $\Delta=4$ is realized.
This phase was introduced in Refs.~\cite{song2017d,khalaf2018symmetry}.
While all the $k_z$-slices are either atomic limits or fragile phases in 2D,
the 3D phase is stable topological because the 2D Wannier centers move as a function of $k_z$, which leads to helical modes on 1D hinges where the $x$- and $y$-normal surfaces meet.

Here, we derive this 3D phase on the BCT lattice by computing the filling anomaly in the $k_z=0$ and $k_z=\pi$ planes.
Since these two planes are invariant under time reversal, they are described by the layer group $p4/m1'$, for which we derived the symmetry indicator formula for the filling anomaly in Eq.~(\ref{eqn_eta_mult}) of Sec.~\ref{sec_2d_p4m1p}.

The symmetry indicators are shown in Table~\ref{tab_BCT}, from which the filling anomaly can be computed with Eq.~(\ref{eqn_eta_mult}).
We find that $\eta=4\mod 8$ in the $k_z = \pi$ plane and $\eta = 0$ in the $k_z=0$ plane.
The non-trivial value in the $k_z=\pi$ plane is responsible for the eight-fold degenerate mid-gap states we observe in Fig.~\ref{fig_3d}(c) in the $k_z=\pi$ plane for a finite-size rod geometry, which is finite in the $x$- and $y$-directions and infinite in the $z$ direction.
Since $\eta=0$ in the $k_z=0$ plane, to continuously connect the mid-gap states with the rest of the band structure requires $k_z$-dependent modes that traverse the bulk band gap; these are exactly the helical hinge modes required by the 3D HOTI phase.
Interestingly, Table~\ref{tab_BCT} shows that $n_b=-1$; thus, this slice is a fragile topological 2D insulator.
(This by itself is not enough to show that the slice is fragile because $n_{a,b}$ are only defined mod 4 (and $n_c$ mod 2); some algebra shows that there is no solution where $n_{a,b,c}>0$.)

We now reiterate the importance of our analysis in correctly describing this phase: our formula~(\ref{eqn_eta_mult}) correctly captures the filling anomaly in the $k_z=\pi$ plane (and lack of filling anomaly in the $k_z = 0$ plane), which agrees with both our numerical results and the helical edge modes predicted by the 3D formula in Ref.~\cite{song2017d}.
Earlier formulas for the filling anomaly give an incorrect result.
Specifically, Eq.~(\ref{eqn_eta_simple}) of Ref.~\cite{schindler2019fractional} yields {$\eta=6$} in the $k_z=0$ plane and {$\eta=2$} in the $k_z = \pi$ plane.
This discrepancy occurs because Ref.~\cite{schindler2019fractional} does not accommodate atoms at multiple Wyckoff positions.
While $\eta = 2$ and $\eta = 6$ are consistent with a helical mode, they do not agree with the state counting in our numerics, where there is always the same number of occupied and empty bands (and hence only consistent with $\eta = 0$ and $\eta = 4$, as we correctly obtain from Eq.~(\ref{eqn_eta_mult})).

\subsubsection{HOFA in Dirac Semimetals}

\begin{table}[]
    \centering
    \begin{tabular}{|c|c|c|c|c|c|c|c|}
    \hline
         $k_z$& $a_a$ & $N$ & $[M_{\frac12 }]$ &$n_a$ &$n_b$ & $n_c$ &$\eta$\\
         \hline
         $ 0<k_z< k_0 $& 2 & 4 & 0  & 2 & 0 & 0 &2\\
         \hline
         $k_0 < k_z < \pi $& 2 & 4 & 1 & 1 & 1 & 0 & 0\\
    \hline
    \end{tabular}
    \caption{In the two DSM phases, the bulk Dirac point is projected to $k_z=k_0$. For the two regions $ 0<k_z< k_0 $ and $k_0 < k_z < \pi$, the filling anomalies $\eta$ are computed by plugging the values in the table into Eq.~(\ref{eqn_eta1}), while $n_{a,b}$ are computed from Eqs.~(\ref{eqn_na1}) and (\ref{eqn_nb1}) ($n_c$ is always zero mod 1, as discussed below Eq.~(\ref{eqn_nb1})).
    The symmetry indicators are derived from the irreps listed in Table~\ref{tab_calculation_DSMi} of Appendix~\ref{sec_3d_irreps}.
    }
    \label{tab_BCT_HOFA}
\end{table}

When $-4 < m/t < 0$, the model is in one of two DSM phases and has two Dirac points along $\Gamma-Z$ in the bulk.
The band structure for the DSM(i) phase on a rod finite in the $x$- and $y$-directions and infinite in $z$ is shown in Fig.~\ref{fig_3d}(d).
For both of the phases, there are HOFA hinge states connecting the projection of the two bulk Dirac points. The HOFA hinge states pass through $k_z=0$, where there are also projected gapless mirror Chern surface states. The difference between the two phases is that the phase DSM(i) has filling anomaly $\eta = 4$ mod 8 at $k_z=\pi$ while the plane $k_z=\pi$ in phase DSM(ii) is trivial.
\par

The HOFA hinge states occur in the planes between the two Dirac points.
Since these planes are not time-reversal invariant, but are invariant under the product of time-reversal and inversion, they are described by the layer group $p4/m'$, for which we derived the symmetry indicator formula for the filling anomaly in Eq.~(\ref{eqn_eta1}) of Sec.~\ref{sec_2d_p4mp}.
Applying this formula, we find $\eta=2$ in the planes between the bulk Dirac points, as derived in Table~\ref{tab_BCT_HOFA}.

The HOFA are not correctly described by formulas in previous work derived by assuming atoms at only one maximal Wyckoff position: for example, Eq.~(\ref{eqn_eta11}) yields $\eta = 0$
in these planes, which would indicate a lack of hinge modes.

In the DSM(i) phase, besides the HOFA, there are two groups of corner-localized hinge states that cross at $k_z=\pi$.  These states can be pushed into the valence or conduction bands by adding some symmetry protecting potentials, however the filling anomaly at $k_z=\pi$ remains non-trivial. These states are not present in the DSM(ii) phase which has $\eta = 0$ in the $k_z = \pi$ plane.

Further, we note that the gapless surface states at $k_z=0$ are unavoidable, even if the protecting mirror symmetry is broken, because the HOFA states are four fold degenerate while the only possible degeneracy of mid gap states at a time-reversal symmetric plane is eight (corresponding to a Kramers pair of time-reversed partners at each corner.)
This discrepancy between the HOFA degeneracy and the required degeneracy at $k_z = 0$ can only be resolved by having gapless surface or bulk states projected to the point.

\section{discussion\label{sec_discussion}}

In this manuscript, we introduced a general method to derive the symmetry indicator formula for the filling anomaly that applies to crystals with any number of atoms in the unit cell.
We introduced an algorithm using the Smith normal form that makes the derivation systematic.
We applied this method to derive the filling anomaly on the 2D square lattice with time-reversal, inversion, and/or their product.
We further showed where our results go beyond earlier work that did not apply to crystals with atoms occupying multiple maximal Wyckoff positions.

We verified our results by correctly predicting the helical hinge modes and HOFAs in a 3D BCT tight-binding model, by analyzing 2D slices of the Brillouin zone.
This model served as a concrete example where previous results break down, showing the importance of our extension to crystals with atoms in multiple maximal Wyckoff positions.

Our method can be easily generalized to other $C_n$ ($n=2,3,4,6$) symmetric 2D lattices and higher dimensional lattices.
It will also be interesting to apply our results to other topological semimetals, such as those in \cite{bradlyn2016beyond} which occur on non-symmorphic lattices and therefore necessarily have multiple atoms in the unit cell.\par

\textit{Note added.} During the final stages of this work, Ref.~\cite{takahashi2021general} appeared on the ArXiv, which also computes the filling anomaly in terms of symmetry indicators for general lattices with $C_n$ symmetry.
Our results agree where they overlap.

\begin{acknowledgments}

This material is based upon work supported by the National Science Foundation under Grant No. DMR-1942447.
J.C. acknowledges the support of the Flatiron Institute, a division of the Simons Foundation.
\end{acknowledgments}


text:
Nobelist Frank Wilczek to receive 2016 Prange Prize
Nobel laureate Frank Wilczek of the Massachusetts Institute of Technology has been named the 2016 recipient of the Richard E. Prange Prize and Lectureship in Condensed Matter Theory and Related Areas. Dr. Wilczek will receive a $10,000 honorarium and deliver a public lecture entitled "Some Intersections of Art and Science" at the University of Maryland, College Park, on September 20, 2016. He will also present a Condensed Matter Theory Center/Joint Quantum Institute seminar entitled "Anyons" on Monday, September 19.
The Prange Prize, established by the UMD Department of Physics and Condensed Matter Theory Center (CMTC), honors the late Professor Richard E. Prange, whose distinguished professorial career at Maryland spanned four decades (1961-2000). The Prange Prize is made possible by a gift from Dr. Prange's wife, Dr. Madeleine Joullié, a Professor of Chemistry at the University of Pennsylvania.
As a Ph.D. student in 1973, Wilczek and his Princeton advisor David Gross discovered asymptotic freedom, which holds that the closer quarks are to each other, the weaker the interaction (color charge) between them; in extreme proximity, quarks behave almost as free particles. This insight helped lead to the Standard Model of particle physics. Wilczek and Gross shared the 2004 Nobel Prize in physics with David Politzer for this breakthrough. Wilczek has also received the Julius Edgar Lilienfeld Prize of the American Physical Society, the High Energy and Particle Physics Prize of the European Physical Society and the Lorentz Medal of the Royal Netherlands Academy of Arts and Sciences. He is the Herman Feshbach Professor of Physics at MIT, a Professor at Stockholm University, the Chief Scientist of the Wilczek Quantum Center of Zhejiang University of Technology, and a Distinguished Origins Professor at Arizona State University.
Wilczek's Prange Prize lecture will be given in Room 1412 of UMD's John S. Toll Physics Building at 4:00 p.m. on Tuesday, September 20, and is open to the public.
At the University of Chicago, Richard Prange received his Ph.D. under Nobelist Yoichiro Nambu and also worked with Murray Gell-Mann and Marvin Goldberger. At the University of Maryland, he edited a highly-respected book on the quantum Hall effect and made important theoretical contributions to the subject. His interests extended into all aspects of theoretical physics, and continued after his retirement. Dr. Prange was a member of the Maryland condensed matter theory group for more than 40 years and was an affiliate of CMTC since its inception in 2002.
"Richard enjoyed a fascinating and fulfilling career at the University of Maryland exploring condensed matter physics, and even after retirement was active in the department," said Dr. Joullié. "He spent the very last afternoon of his life in the lecture hall for a colloquium on graphene, followed by a vigorous discussion. And so I was happy to institute the Prange Prize, to generate its own robust discussions in condensed matter theory."
"The Prange Prize provides a unique opportunity to acknowledge transformative work in condensed matter theory, a field that has proven to be an inexhaustible source of insights and discoveries in both fundamental and applied physics," said Dr. Sankar Das Sarma, who holds the Richard E. Prange Chair in Physics at UMD and is also a Distinguished University Professor and Director of the CMTC.
Since its initiation in 2009, the Prange Prize has been awarded to Nobelists Philip W. Anderson, Walter Kohn, Daniel Tsui, Andre Geim, David Gross and Klaus von Klitzing.
Parking is available in the Regents Drive Garage, across the street from the Physics lecture hall; an attendant will direct visitors within the garage. A free ShuttleUM bus runs to and from the College Park Metro station at about eight-minute intervals.
Directions to the College Park campus can be found here: http://www.cvs.umd.edu/visitors/maps.html
The John S. Toll Physics building: http://maps.umd.edu/map/index.html?&LocationType=Building&LocationName=082&Welcome=False&MapView=Detailed
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College of Computer, Mathematical and Natural Sciences: http://www.cmns.umd.edu/
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Kate Sickles
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CHEMISTRY/PHYSICS/MATERIALS SCIENCES
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redpajama_set_name: RedPajamaCommonCrawl


When in Queens last weekend and while riding on an elevator at NYSUM, I was asked by a young bible student what we were doing in NYC.
To this I then got a tiny "Amen," along with the same "wide-eyed look" as before.
Let me just add to this that the power of God flows only through crucified hands and a pierced heart … not only Jesus' but ours. This is why we too must go to the Cross — to get the heart and the hands through which the love of God and the power of God can flow.


Why Are Brands Suddenly Embracing 4/20?
The nation's attitude towards marijuana is changing, and with that, we're starting to see more brands let loose and embrace cannabis culture.
Just a few years ago, we were hard-pressed to find a ton of brands acknowledging April 20, a.k.a. Weed Appreciation Day. But this year, major brands like Pepsi, Spotify, and HBO came out in droves to drum up engagement around the big day on Twitter.
Many content marketers still suffer through layers of legal and PR restrictions to get a simple white paper published. So how did these big brands get away with maybe-kinda-sorta encouraging their audiences to smoke weed?
Alec Rochford, CEO of Duby, a social networking app for cannabis enthusiasts and medical marijuana patients, has been particularly invested in this shift of attitude.
"If you think about the fact that almost 10 percent of the country consumes cannabis—depending on what industry you service—you could have a lot of cannabis consumers in [your audience]," Rochford told me.
He pointed to the gradual legalization of marijuana—especially for medical purposes—in the U.S. as a reason why we're starting to see the media present weed as more acceptable.
Major social platforms are also aiding this trend, with moderators becoming more lax about posts and pages that promote cannabis use. For example, Rochford noted that Facebook and Instagram have started to allow more cannabis businesses to create pages, whereas in the past, they just got shut down.
In fact, if it weren't for Apple's ease of attitude towards marijuana use, Rochford could have never launched Duby.
Rochford's app invites anonymous users to "light up" a new Duby in the form of a picture, video, or text post, and pass the message to a designated number of anonymous users nearby. The more influence you have (based on popularity of posts), the more users your message gets passed to. Taking a page from Tinder, Duby users can swipe left to "put out" a Duby and swipe right to keep it spreading across the map.
While Rochford doesn't plan to monetize his app just yet, he has seen a good deal of brands create accounts on the platform in just the two weeks since it launched. These brands are, of course, mostly marijuana dispensaries like Weed Maps, Stoner Daze, and Bud Puff. But now that brands like Ben & Jerry's and HBO are comfortable playing up 4/20 on social media, could they one day produce content for an app like Duby?
Down the road, Rochford says he could monetize his app by posting certain brands' ads in certain locations to target users who might be interested in their products. Given that the app is already accessible to users in Canada, the Netherlands, Jamaica, Guam, the Virgin Islands and the 23 U.S states with medical marijuana laws, there are no shortage of audiences to engage with.
Until that happens, we'll continue to marvel at how comfortable brands felt to joke about marijuana use on social media. And, oh, did I say brands? I meant police departments.


text:
The "Iraq Dinar"
Central Bank of Iraq
Central Bank sales.. High foreign remittances from the dollar in today's auction
by rocky Thu 17 Mar 2022, 6:02 am
[*][ltr][You must be registered and logged in to see this link.][/ltr]
Shafaq News/ Foreign remittances increased in the Central Bank's sales of dollars, today, Thursday, to record 166 million dollars.
Shafak News Agency correspondent stated, "The Central Bank witnessed today, during its auction for buying and selling the US dollar, an increase in its dollar sales, to reach 194 million and 394 thousand US dollars, covered by the bank at a base exchange rate of 1460 dinars per dollar, compared to yesterday, Wednesday, which sales amounted to It has 192 million and 184 thousand and 468 dollars.
Our correspondent added; Foreign remittances rose during today's auction to reach 166 million and 674 thousand dollars, compared to cash sales, which amounted to 27 million and 720 thousand dollars.
Our correspondent indicated that 27 banks have responded to requests to enhance balances abroad, and 11 banks have responded to cash requests, in addition to 97 mediating companies.
» The central bank sells more than $ 232 million in the sale of the currency today
» Auction Results posted daily by Magictime
» High central bank sales of the dollar in today's auction
» High central bank sales of the dollar at an auction Sunday
» High central bank sales of the dollar in the auction today
Parliamentary Baghdad 16 Central Bank FEDERAL airways dollar boeing Republic 9 Minister will rate region 69 2 announces Kurdistan Government budget Committee 5 President Iraq exchange
Current date/time is Wed 01 Feb 2023, 1:13 pm


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International Speaker | Variety Show Comedian | Author | Business Consultant | Team Igniter Sheli G... brings her dynamic, contagious personality, phenomenal stories that will educate and inspire, with humor and fun to keep your audience engaged.
I do such a variety of events and roles at events, both inside and outside of businesses, communities, retreats, trainings, charities and more. I use many different modalities in communicating with people, to leave them inspired, ignited, educated, laughing-feeling great, and wanting to be better in some way.
This was one of my mentors and coaches, and I traveled with his "Platinum Speakers" (top gun speakers) group for a time, until his health took him off the road. Wonderful man and speaking legend.
The Ignited Entrepreneur on Amazon, and I bring copies *as requested, to live events.
We love to educate, inspire, and have compassionate courageous dialogues on leadership, race issues and more.
Sheli G facilitating a Mastermind Retreat at the Beach, McCall Id.
Performing a hilarious dance number at a comedy variety show in 2016. Such a blast!
realizing and unlocking your strengths, using the Gallup Strengthsfinder assessment tool, and an experiential workshop. This was for a group of professional speakers.
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text:
Tag: bruce omori
July 30: Day 88 Matches 1955 Eruption Duration
Eruption continues as usual with three minor things to note. First: "fuming" on the southwestern margin of the flow near Pohoiki could mean possible breakouts, according to today's HVO Kilauea status update.
Same source also says that, rather unusually, a 4.1 earthquake at 10:02 pm was felt all the way to Hilo, far more widely than the stronger summit collapse events, possibly because it was at a depth of 7 miles.
And finally, Tuesday will mark the 89th day of the eruption, surpassing the length of the 1955 Kapoho eruption which had previously held the record for the longest LERZ eruption since westerners arrived and began keeping records.
From Other Geologists
Erik Klemetti's Rocky Planet blog in Discover magazine invites readers to "Check Out How The 2018 Eruption Has Changed At Kilauea's Summit". Although I suspect readers of this blog are well aware of pretty much everything in that post!
I tweeted him a question about "Halema'uma'u Caldera," which he seems to be using instead of "Kīlauea caldera." I know Halemaʻumaʻu crater has expanded so much it could be classified as a caldera now, but so far HVO scientists have resisted doing so, to avoid confusion with the larger, older caldera.
July 30 HCFD Overflight Photos
HCFD's July 30 album is up on Flickr. Just 12 photos plus the video clip at the top of this post. Including the clearest views we've seen of Isaac Hale Park in some time, since the laze plume wasn't in the way:
Here's a few USGS photos from today plus the LERZ map, and overflight video and photos from Mick and Bruce:
Continue reading July 30: Day 88 Matches 1955 Eruption Duration
Posted on July 31, 2018 July 31, 2018 Categories Daily Roundup, InformationTags bruce omori, fissure 8, kilauea caldera, lava flow, mick kalber, overflight, pohoiki, steam explosions, summit collapse
July 29 – HVO (and Lava Rooster's Owners) in Search of a Home
Before and after: a new aerial photo of HVO on the rim of Kīlauea Caldera shows how greatly Halemaʻumaʻu has changed. For comparison, I found this 2008 photo in the USGS archives:
BEFORE: HVO on the summit of Kilauea Caldera. September 2008. Michael Poland, USGS. (Full-sized)
And here's today's overflight photo, from farther away.
July 29, 2018. USGS: "This aerial view of Kīlauea's summit (taken early yesterday morning, looking south) shows some well-known features and some that are now more obvious as a result of ongoing collapse of Halema'uma'u and parts of the summit caldera floor. Crater Rim Drive (lower right) leads to the USGS Hawaiian Volcano Observatory and NPS Jaggar Museum (right, middle), perched on the caldera rim and overlooking the growing Halema'uma'u. Ground cracks, parallel to the crater rim, are visible on the north side of Halema'uma'u (left side of image—also see our July 28 photo). South Sulphur Bank stands out as the light-colored area on the opposite crater wall (see our July 22 photo for a closer view of this feature)." (Full-sized)
The scientists of HVO had to abandon their 100-year-old observatory on the caldera rim mid-May. During the summer session, the University of Hilo was able to provide them with plenty of space for a temporary HQ. With fall term around the corner, HVO has leased part of the U.S. Customs Building in Hilo while they seek a more permanent solution. Because if and when the summit stabilizes, the old observatory will have to be rebuilt, or at the very least, will require massive repairs.
Speaking of the summit, today's collapse event was at 12:10 pm, energy equivalent of M5.4. I captured the rockfalls 3-4 minutes before, as well as the actual summit collapse:
The clock on the Northeast Caldera Rim livestream is about a minute fast, but it's so tiny you can't see it anyway. The actual collapse starts about 1:30 into this video:
No changes on the Lower East Rift Zone eruption today.
Here's the other HVO images for July 29 (as usual, I'm mirroring them because the HVO "Photo and Video Chronology" page only shows the 20 most recent entries, making images older than two weeks somewhat inaccessible.)
Continue reading July 29 – HVO (and Lava Rooster's Owners) in Search of a Home
Posted on July 30, 2018 Categories Conference Call, InformationTags bruce omori, fissure 8, halema'uma'u, kilauea caldera, overflow, summit collapse
July 26: 2018 Kīlauea Eruption Three Months On
https://volcanoes.usgs.gov/observatories/hvo/multimedia_uploads/multimediaFile-1942.mp4
Three months ago today, shortly before 5 pm on May 3, lava began to erupt from one of several cracks that had opened in Leilani Estates in the Lower Puna district of the Big Island of Hawai'i. This followed several days of earthquakes indicating magma moving downrift from Puʻu ʻŌʻō, after its floor collapsed and its lava drained away overnight on April 30.
At first, the fissures spattered and sputtered, with most of the lava falling on both sides of the vents and building up ramparts (walls). Individual fissures erupted for several hours at a time, then died out. Some restarted, others simply steamed. A few sent out sluggish lava flows, claiming a few houses.
On May 19, the eruption began in earnest. Most of the old, stale lava stored in the rift zone since the 1955 and 1960 eruptions had been pushed out, and fresh, hot, runny lava from Puʻu ʻŌʻō began pouring out of vents, sending the first lava flows down to the sea (See the Honolulu CivilBeat livestream from that day, timestamp 6:03). Lava reached the ocean before dawn on May 20.
While Fissure 8 had originally opened on May 5, it was just one of many attempts for all that magma coming down the rift zone to find the most convenient exit. (Magma can reshape its own plumbing, just as we've seen lava do on the surface.) Fissure 8 reactivated again on May 28, and within a few days became the dominant vent for this eruption. Its lava flow reached Kapoho Bay the evening of June 3, and had covered the bay within 36 hours.
USGS overflights of Kapoho Bay, morning of June 3 and June 5.
All that magma exiting the summit caused the lava lake at the summit to drain away, then Halemaʻumaʻu fell into it and started enlarging, and eventually much of the floor of Kīlauea caldera began to subside as well. The collapses were explosive at first, then, after the lava lake's conduit had been thoroughly blocked by rubble, the collapses settled into a regular pattern.
So here we are. The LERZ eruption has added nearly 800 acres to the island, covering lower Puna with 34.0 square kilometers (13.1 square miles) of lava. We've almost come to take for granted this extraordinary eruption, which has dramatically reshaped the summit of Kīlauea and produced more lava in 3 months than Puʻu ʻŌʻō did in 35 years.
July 26, 2018. USGS: "Fissure 8 continues to erupt lava into the channel leading northeastward from the vent. This north-facing view of the cone, vent, and proximal channel was taken during HVO's overflight this morning." (Full-sized)
HVO's volcanologists have told us that eruptions like this wax and wane— Puʻu ʻŌʻō certainly did, sometimes pausing for weeks— and that part of what makes Fissure 8 extraordinary is that it's sustained such a high volume of lava effusion for so long. Today, it's finally showed signs of weakening— maybe? The USGS reported that its lava flow seemed sluggish and that lava levels are down in the lower part of the channel. The flow margin remains stalled a mere tenth of a mile from Pohoiki's boat ramp.
July 26, 2018. USGS: "At the coast, the lava flow in the Ahalanui area remains less than 0.1 miles from the Pohoiki boat ramp at Isaac Hale Park (left of center in this photo). The active ocean entry is a few hundred yards east (right) of this photograph." (Full-sized)
Today's summit collapse occurred at 12:09 pm, energy equivalent of 5.3 as usual. That's 53 hours since the previous Type A event, the longest interval so far. Clouds and fog obscured the view, but FWIW here's my video captures of the HVO tower and northeast caldera rim livestreams.
USGS: "As of 2:00 p.m. HST, July 26, 2018, the lava flow margins had not expanded since the previous map, so no red areas (indicating expansion) appear on this map." (Full-sized)
Continue reading July 26: 2018 Kīlauea Eruption Three Months On
Posted on July 27, 2018 July 27, 2018 Categories Daily Roundup, InformationTags bruce omori, fissure 8, lava flow, mick kalber, overflight, summit collapse
July 25: Lavasheds Explained (More or Less)
July 25, 2018. USGS: "Early this morning, several small lobes of lava were oozing out from crusted 'a'ā flows along the southern ocean entry." (Full-sized)
As we approach the 3-month mark— or pass it, since Puʻu ʻŌʻō's floor collapsed 3 months ago this past Monday— Kīlauea's 2018 eruption continues with no significant changes. During the USGS morning overflight, the lava's southwestern margin was holding at less than a tenth of a mile from Pohoiki boat ramp. The vigorous ocean entry continues a few hundred yards to the east. No surges in lava output have been reported at Fissure 8 since Tuesday's 6:41 am summit collapse.
July 25, 2018. USGS: " HVO's overflight early this morning revealed little advancement of the westernmost ocean entry. During the overflight, lava was approximately 145 m (475 ft) from the boat ramp at Isaac Hale Park (near the light-colored structure just left of center in this photo)." (Full-sized)
A deceptive moment of tranquility at the new, extensively remodeled summit:
July 25, 2018. USGS: "Earthquakes shook the ground at Kīlauea's summit late this afternoon, stirring up rock dust within Halema'uma'u and along the caldera walls. This view from the northeast caldera rim looks across at the Keanakāko'i and South Sulphur Banks areas." (Full-sized)
Here's HVO's Steve Brantley giving the weekly eruption update at last night's Pahoa community meeting (I'll be transcribing it tomorrow or Friday).
He explains that complicated "lavashed" map released earlier this week, which predicts where lava is most likely to go if there's a breakout along particular stretches of Fissure 8's lava channel. This is an aid to residents living in neighboring areas not affected by lava now, but who are worried about what might happen if the flow diverts.
Continue reading July 25: Lavasheds Explained (More or Less)
Posted on July 26, 2018 July 26, 2018 Categories Daily Roundup, InformationTags bruce omori, fissure 8, kilauea caldera, lava flow, pohoiki
July 24: Visible Shockwave in Steam Clouds
July 24, 2018: Two HVO geologists out standing in their field. USGS tweeted this among a batch of photos today. (Full-sized)
Today's summit collapse event at 6:41 am was upgraded to 5.6 a 5.3, as usual. The USGS apologized for calculations errors for the past 3 days
Since I was out and missed it on livestream, I'm very glad HVO captured it. Here's yet another trick from Pele's repertoire:
There's a lot to look at here. From USGS caption:
"In this video, watch as today's event unfolds from the perspective of HVO's live-stream camera. At 6:41:08 (time stamp at upper left), a small tree along the right margin of the video begins to sway. At 6:41:10, a pressure wave passes through the steam plume in the crater, and light is reflected back to the camera (highlights the passage of the expanding sound energy through the air.) At 6:41:11, a rockfall begins on the South Sulphur Banks, a distant light-colored scarp on the left."
Field crews reported no surge at Fissure 8 following today's summit collapse.
I notice the latest HVO Kilauea report says Fissure 8's cone is down to 50 m, "or 55 yards." It's definitely crumbled or eroded— that may be responsible for some of the lava boats— but I'm never sure whether they're measuring from the original ground level or the lava on which it sits. At any rate, the fountaining is lower, too, since it's not rising above the lip of the cone.
July 24, 2018. USGS: "Fissure 8 lava channel as viewed from HVO's morning overflight today. The robust volcanic gas plume in the far distance was rising from the fissure 8 vent." (Full-sized)
Down at the ocean entry, the main channel is still dumping into the sea from multiple toes near former Ahalanui.
July 24, 2018. USGS: "The Hawaii County Fire Department captured this image of Isaac Hale Park and boat ramp during their overflight of the area late this afternoon." (Full-sized)
I'm impressed HCFD was able to catch a glimpse under the thick, low laze plume, which was obstructing the view during @HotSeatHawaii's overflight, which looked about like this (in fact the USGS helicopter was down below them):
July 24, 2018. USGS: "The ocean entry has expanded to the southwest through a series of lava 'ooze-outs' from the southern flow margin that organized into an incipient channel. As of this morning, the flow margin was in or at the edge of Isaac Hale Park, approximately 175 m (575 ft) from the Pohoiki boat ramp. Unfortunately, the view was obscured by laze (the smaller plume below the larger laze plume) during the overflight." (Full-sized)
"Ooze-outs" are occurring along the west side of the active flow south Kapoho Crater, all the way down to the ocean (where it is threatening Isaac Hale).
Other news that slipped under the radar: HCFD took a swing by the summit yesterday! It was very dusty and hazy, but the new shape of Halemaʻumaʻu is becoming clear:
Look for Crater Rim Drive at lower left around 0:30, falling into the crater where it used to lead to the parking lot; HVO is really hard to glimpse at upper right.
There were several more photos, giving us a little more perspective on the whole of Kīlauea Caldera as it is now:
Continue reading July 24: Visible Shockwave in Steam Clouds
Posted on July 25, 2018 July 25, 2018 Categories Daily Roundup, InformationTags bruce omori, fissure 8, g brad lewis, lava flow, mick kalber, overflight, pohoiki, summit collapse
July 23: Pohoiki Going Tonight (I Think)
July 23, 2018. View of lava channel from F8 where it bends to south and heads for ocean (laze plume in background). USGS: "The fissure 8 channel continues to carry lava toward the coast on the west side of Kapoho Crater (vegetated cone, far left). Northwest of this cone, overflows (lower left) of the channel occurred overnight, but lava was confined to the existing flow field and did not threaten any homes or structures." (Full-sized)
The USGS has updated the energy release for yesterday evening's summit collapse to M5.5. (For more info: "Why Do Earthquake Magnitudes Sometimes Change?" from a 2016 issue of Volcano Watch by HVO.)
Fissure 8 exhibited surge behavior afterwards. Overnight, drone crews observed minor overflows just northwest of Kapoho Cone, most of which were confined to the existing flow field and/or stopped before traveling far enough to threaten structures.
July 23, 2018. USGS: "Lava continued to enter the sea near Ahalanui during HVO's early morning helicopter overflight of Kīlauea's lower East Rift Zone (view to northeast). The southern margin of the flow was still about [a quarter mile] from the boat ramp at Isaac Hale Park this morning. The jetty along the launch ramp is visible near the center of the photo." (Full-sized)
The main ocean entry and southern margin of the flow haven't moved much in the past few days. However, USGS reports "a new lobe has started from the southern lobe and is active along its southwestern margin, slowly heading towards the ocean." Today's 1 pm map clarifies the new lobe's location:
July 23, 2018, 1 pm USGS map of LERZ lava flow field. Note small overflow by Fissure 8 as well as upstream of Kapoho bend on the north side of the braided channels. (Full-sized)
Ikaika Marzo, a lava tour boat operator who stays a sensible distance from shore, reportedly saw lava is in Isaac Hale Park, within 50 m of Pohoiki boat ramp at 7:15 this evening.
No summit collapse today, unless the pattern changes; the next one ought to occur tomorrow morning.
Here's highlights of the USGS Media Conference Call from this afternoon:
Full audio file archived here.
Continue reading July 23: Pohoiki Going Tonight (I Think)
Posted on July 24, 2018 Categories Daily Roundup, InformationTags bruce omori, fissure 8, ikaika marzo, janet babb, lava flow, lidar, mick kalber, overflight, overflows, pohoiki
July 22: History Uncovered As Other Historical Places Covered
July 22, 2018. Posted by USGS on Facebook: "A telephoto view into the fissure 8 cinder cone, taken during the early morning helicopter overflight."
Status quo. No significant overflows today from Fissure 8's lava river. USGS morning overflight put the southern margin of the coastal flow field at 500 m from boat ramp at Isaac Hale Park. In other words, not much movement in that direction.
Summit collapse occurred while I was writing up this post, as expected: 8:54 pm HST, back to an energy equivalent of M5.3 on reviewing readings, they upped this one to 5.5! Let's see whether that results in an early-morning Fissure 8 surge tomorrow, er, today, Monday.
July 22, 2018. USGS: "The main ocean entry, as observed early this morning, was located a few hundred meters (yards) northeast of the southern flow margin, which remains about 500 m (0.3 mi) from the boat ramp at the Isaac Hale Park." (Full-sized)
When the National Park opens again, they're going to have a new— or rather, very old— landmark that I confess I'm rather excited about, although it's not quite as photogenic as a lava lake. Mark Twain would've seen this during his visit in 1866:
July 22, 2018. USGS: "Collapse of Kīlauea's caldera floor has exposed South Sulphur Bank, prominent in the mid-19th century but covered as lava flows filled the caldera. The flat top of the white deposit shows how high the caldera fill reached. As the caldera floor dropped in mid-June 2018, South Sulphur Bank was again exposed. The height of the bank, now more than 65 m (213 ft), increases about 2.5 m (9 ft) with each collapse event at Kīlauea's summit. On the caldera floor, white patches lie along spatter ramparts formed in 1971 and 1974." (Full-sized)
As we approach the 3-month mark, the USGS is beginning to supplement its daily reports on the eruption itself with recognition of scientists and support crew who have been working 24/7 to monitor, collect scientific data and inform civil defense and the public since this eruption began. The drone crew worked overtime last night after being grounded by weather then night before:
July 22, 2018. USGS: "The Unmanned Aircraft Systems (UAS) team frequently works into the night, flying aircraft (also referred to as drones) that hover over the active lava channel to collect data and look for changes, such as significant channel overflows. The moon (bright white area above the UAS team scientist) is partially obscured by clouds." (Full-sized)
Two maps today, one assembled from yesterday morning's overflights and one from 2 o'clock this afternoon:
Continue reading July 22: History Uncovered As Other Historical Places Covered
Posted on July 23, 2018 July 23, 2018 Categories Daily Roundup, InformationTags bruce omori, fissure 8, lava flow, laze, ocean entry
July 14: Aloha to Pohoiki
July 14, 2018. USGS: " Early morning view of fissure 8 and lava channel looking toward the east. Laze plume from the ocean entry is visible in distance (left of the fissure 8 plume). Geologists did not observe activity from any of the other fissures during this morning's overflight." (Full-sized)
The Lower East Rift Zone eruption has settled back into a routine. Unfortunately, that routine includes the far end of Fissure 8's lava flow crawling south along the coast, eating landmarks in its path. Isaac Hale Park/Pohoiki boat ramp (see flow map) is next in line. The morning USGS overflight reported that the flow front was about 1km away.
Friday evening, July 13, 2018, 6:00 pm – Kilauea's east rift zone overflight: Lava continues to pour into the sea at…
Posted by Extreme Exposure Fine Art Gallery on Saturday, July 14, 2018
That little island is still there this morning.
Fissure 8 continues to erupt into its perched channel, with its volume surging after yesterday's 7:08pm summit collapse (Mag 5.3) and returning to lower levels by morning. That pulse caused a brief overflow on the east/southeast of the channel a short way downstream from the vent, but it stayed on the apron of previous flows.
The channelized a'a flow west of Kapoho Crater continues to be the main conduit to the ocean and to the active flow expanding southwards along the shore. North of this, lava continues to "ooze out" at various points along the June-early July flow front, whose length now totals 6km (3.7 miles).
July 14, 2018. USGS: "White laze plumes mark locations where lava enters the ocean over a broad area. An open lava channel flows into the ocean at the southern-most plume (middle) near the southern flow margin. The boat ramp at Pohoiki is about 940 m (0.58 mi) farther south of the flow margin. View is toward the west-southwest." (Full-sized)
While there's no sign of activity at any fissures besides 8, the tiny island that popped up just offshore of Kapoho yesterday is still there for now.
Continue reading July 14: Aloha to Pohoiki
Posted on July 15, 2018 July 15, 2018 Categories Conference Call, InformationTags bruce omori, fissure 8, kapoho crater, lava flow, ocean entry, pohoiki