text:
\section{Introduction}

Topological phases of free fermions protected by internal symmetries (such as time reversal) feature a gapped bulk and symmetry-protected gapless surface states
\cite{Molenkamp}.
Such bulk-boundary correspondence is an important attribute of such topological phases and it plays a key role in their classification, which was achieved in arbitrary spatial dimensions \cite{Kitaev09, Schnyder09, Ryu10}.
Crystals, however, frequently exhibit a far richer set of spatial symmetries, like lattice translations, rotations, and reflections, which can also protect new topological phases of matter \cite{Fu11,Hsieh12, Tanaka12, Robert-JanSlager, Teo, Lu14, Varjas15, Ken2015,  Wang16, Ezawa16, Varjas17, Wieder17, Benalcazar17, Song17, Schindler17, Langbehn17,  Benalcazar17Aug, Fang17}.

However, several new subtleties arise when discussing topological crystalline phases. For one, spatial symmetries are often broken at surfaces, and when that happens, the surface could be fully gapped even when the bulk is topological.
Consequentially, the diagnosis of bulk band topology is a more delicate task as, a priori, anomalous surface states in this setting are only a sufficient, but not necessary, property of a topological bulk.
Furthermore, the presence of crystalline symmetries leads to a plethora of topological distinctions even when the band insulators are individually trivial
\cite{Po17,Bradlyn17}.
For example, consider trivial (atomic) insulators, where electrons are well-localized in real space and do not support any nontrivial surface states. Suppose two atomic insulators are built from orbitals of different chemical characters, which will typically lead to different symmetry representations in the Brillouin zone (BZ). These representations cannot be modified without a bulk gap closing, and therefore the two atomic insulators are topologically distinct even though they are both derived from an atomic limit.

Since the mathematical classification of topological band structures, using K-theory, is formulated in terms of the mutual topological distinction between insulators \cite{Kitaev09, MooreFreed, Combinatorics, Ken2017}, the mentioned distinction between trivial phases is automatically incorporated. In fact, they appear on the same footing as conventional topological indices like Chern numbers. While for a physical classification one would want to discern between these different flavors of topological distinctions, they are typically intricately related.  On the one hand, the pattern of symmetry representations is oftentimes  intertwined with other topological invariants, as  epitomized by the Fu-Kane parity criterion \cite{Fu07}, which diagnoses 2D and 3D topological insulators (TIs) through their inversion eigenvalues; on the other hand, the topological distinction between atomic insulators may reside fully in their wavefunction properties is not reflected in their symmetry eigenvalues \cite{Zak2001, Bradlyn17}.

In this work, we sidestep these issues by focusing on topological crystalline phases displaying anomalous gapless surface states, which, through bulk-boundary correspondence, are obviously topological.
Unlike the  strong TI, these phases require crystalline symmetries for their protection.
Typically, their physical signatures are exposed on special surfaces where the protecting symmetries are preserved.
Such is the case for the 3D phases like weak TI \cite{Molenkamp}, the mirror Chern insulator \cite{Hsieh12, Tanaka12} and the nonsymmorphic insulators featuring so-called hourglass \cite{Wang16, Ezawa16} or wallpaper \cite{Wieder17} fermions, which are respectively protected by lattice translation, reflection and glide symmetries.
While these conventional phases feature 2D gapless surface states on appropriate surfaces, we will also allow for more delicate, 1D ``hinge'' surface states, whose existence is globally guaranteed on suitably chosen sample geometries but any given crystal facet can be fully gapped \cite{Teo, Fritz,  Zhang, Song17, Schindler17, Benalcazar17, Benalcazar17Aug, Fang17, Langbehn17}.
We will primarily focus on 3D time-reversal symmetric band structures with a bulk gap and significant spin-orbit coupling (class AII). However, our approaches can be readily extended beyond this specific setting. In particular, in symmetry classes with particle-hole symmetry, the notion of hinge states can be generalized to protected gapless points on the surface \cite{Teo}.

We will base our analysis on two general techniques, which are respectively employed to  systematically study gapless surface states and identify bulk band structures associated with them. First, we will describe a surface Dirac theory approach \cite{Molenkamp, Schnyder09, Ryu10, Witten16, Fang17}. Specifically, we will consider a surface theory with multiple Dirac cones, which may transform differently under the spatial symmetries \cite{Fang17}. Such a theory can be viewed as the surface of a ``stack of strong TI,'' which arise physically when there are multiple band inversions (assuming inversion symmetry) in the BZ. When an even number of Dirac cones are present at the same surface momentum, the Dirac cones can be locally gapped out and, naively, one expects that a trivial surface results. However, spatial symmetries can cast global constraints which obstruct the full gapping out of the Dirac cones everywhere on the surface \cite{Fang17}. When that happens, the surface remains stably gapless along certain domain walls, and hence one can infer that the bulk is topological. We will provide a comprehensive analysis on such obstructions from point-group symmetries, and catalogue all the possible patterns of such anomalous surface states in a space group.

Next, we will turn to bulk diagnostics that informs the presence of such surface states. In particular, we will focus on those which, like the Fu-Kane criterion \cite{Fu07}, expose band topology using only symmetry eigenvalues \cite{Fu07, Ari, Hughes11, Fang12}.
Such symmetry-based indicators are of great practical values, since they can be readily evaluated without computing any wavefunction overlaps and integrals. In fact, a general theory of such indicators has been developed in Ref.\ \onlinecite{Po17}, and the indicators for all 230 space groups (SGs) have been computed.

Here, we provide a precise physical meaning for all the phases captured by these indicators in class AII. In particular, we discover that the majority of them can be understood in terms of familiar indices, like the strong and weak TI $\mathbb Z_2$ indices and mirror Chern numbers. However, there exists symmetry-diagnosed band topology which persists even when all the conventional indices have been silenced. The prototypical example discussed in Ref.\ \onlinecite{Po17} is a ``doubled strong TI'' in the presence of inversion symmetry, which, as pointed out later in Ref.\ \onlinecite{Fang17}, actually features hinge-like surface states. Motivated by these developments, we relate the symmetry indicators to our surface Dirac theory and prove that, for spin-orbit coupled systems with time-reversal symmetry (TRS), any system with a nontrivial symmetry indicator is a band insulator with anomalous surface states on a suitably chosen boundary.  This is achieved by establishing a bulk-boundary correspondence between the indicators and the surface Dirac theory.
Parenthetically, we note that the symmetry indicators constitute a sufficient, but not necessary, condition for band topology, and we will discuss examples of topological phases which have gapless surface states predicted by our first approach, although the symmetry indicator is silent.

Although the techniques we adopted are tailored for weakly correlated materials admitting a band-theoretic description, many facets of our analysis have immediate bearing on the more general study of interacting topological crystalline phases \cite{Hermele17,Thorngren16, XiongMin, Huang17,XiongGlide}. A common theme is to embrace a real-space perspective, where ``topological crystals'' are built by repeating motifs that are by themselves topological phases of some lower dimension, and are potentially protected by internal symmetries or spatial symmetries that leave certain regions invariant (say on a mirror plane)  \cite{Hermele17, Huang17}. In particular, such picture allows one to readily deduce the interaction stability of the phases we describe \cite{Isobe15}, and we will briefly discuss how the Dirac surface theory analysis can be recast into such a framework.

\section{Symmetry indicators and bulk topology}
\label{Sec:Indicators}
In this section, we provide a precise physical meaning for the phases captured by the symmetry-based indicators obtained in Ref.~\onlinecite{Po17}, focusing on the physically most interesting case of strong spin-orbit coupling.

We start by considering a band insulator of spinful electrons symmetric under an SG and TRS (class AII). In this setting, we can define topological indices corresponding to weak and strong TI phase protected by TRS.  For centrosymmetric SGs, the Fu-Kane formula~\cite{Fu07} allows us to compute these indices using only the inversion eigenvalues.  Namely, they can be determined just by multiplying the parity eigenvalues of the occupied bands without performing any sort of integrals.  This is one particular instance of the relation between the symmetry representations in momentum space and band topology.

Recently, this relation was extended to all space groups (i.e., not restricted to the inversion alone) and also to a wider class of band topology~\cite{Po17}.  By comparing the symmetry representations of the bands against that of trivial (atomic) insulators, the symmetry-based indicator $X_{\rm BS}$ was computed for all 230 SGs~\cite{Po17} (and even for all 1651 magnetic SGs~\cite{WatanabePo17}). It was found to take the form
\begin{equation}
\label{XBS}
X_{{\rm BS}}=\mathbb{Z}_{n_1}\times\mathbb{Z}_{n_2}\times\cdots\times\mathbb{Z}_{n_N},
\end{equation}
where $N$ and $\{n_1, n_2,\cdots,n_N\}$ depend on the assumed SG.  The detailed nature of each nontrivial class was, nevertheless, left unclear. In the following, we identify the specific topological indices characterizing each factor in $X_{{\rm BS}}$ and clarify their meaning.

\subsection{Review of familiar indices}
Here, we review some well-known symmetry-based indicators.  Let us start with the inversion symmetry.  As mentioned above, for an SG containing the inversion symmetry $\mathcal{I}$, the inversion eigenvalues are related to the strong TI index $\nu_0\in\mathbb{Z}_2$ and the three weak TI indices $\nu_i\in\mathbb{Z}_2$ ($i=1,2,3$).  Each of the three weak TI indices gives rise to a $\mathbb{Z}_2$ factor in $X_{{\rm BS}}$ when no additional constraints are imposed.  Curiously, however, the factor in $X_{{\rm BS}}$ corresponding to the strong TI index is $\mathbb{Z}_4$, not $\mathbb{Z}_2$ \cite{Po17}.  This $\mathbb{Z}_4$ indicator will be discussed in Sec.~\ref{sec:k1}.

Next, let us discuss the mirror Chern numbers, which can be defined for every mirror symmetric plane in momentum space \cite{Hsieh12}.  On any such plane, the single-particle Hamiltonian $\mathcal{H}$ can be block-diagonalized into $\mathcal{H}_\pm$, defined respectively in the sectors corresponding to the mirror eigenvalues $M=\pm i$.  Since $\mathcal{H}_\pm$ is not necessarily time-reversal symmetric, $\mathcal{H}_{\pm}$ can possess a Chern number $C_{\pm}\in \mathbb{Z}$ which satisfies $C_+=-C_-$ ($\equiv C_M$) due to the TRS of the total Hamiltonian $\mathcal{H}$.  To be more concrete, let us assume the mirror symmetry is about the $xy$ plane.  For  primitive lattice systems, both $k_z=0$ and $k_z=\pi$ planes are mirror symmetric and can individually support $C_M^{(k_z=0)}$ and $C_M^{(k_z=\pi)}$. For body-centered systems and face-centered systems on the other hand, $k_z=0$ is the only mirror symmetric plane and there is only one mirror Chern number $C_M^{(k_z=0)}$.

When the SG is further endowed with an $n$-fold rotation symmetry $C_{n}$ ($n=2,3,4,6$) or screw symmetry ($C_n$ rotation followed by a fractional translation along the rotation axis) whose axis is orthogonal to the mirror plane, one can diagnose the mirror Chern number $C_M$ modulo $n$ by multiplying the rotation eigenvalues~\cite{Fang12}.  Naively, each $C_M$ mod $n$ produces a $\mathbb{Z}_n$ factor in $X_{{\rm BS}}$.  However, we sometimes find a ``doubled" $\mathbb{Z}_{2n}$ factor in $X_{{\rm BS}}$. The mechanism behind this enhancement is clarified in Sec.~\ref{sec:delta} and \ref{sec:k1mCh}.

Following the examination of these familiar indices, we find that $X_{{\rm BS}}$ in Eq.~\eqref{XBS} can be always factorized into ``weak factors" and a ``strong factor":
\begin{gather}
X_{{\rm BS}}=X_{\rm BS}^{(w)} \times X_{\rm BS}^{(s)}, \nonumber\\ X_{\rm BS}^{(w)}=\mathbb{Z}_{n_1}\times\cdots\times\mathbb{Z}_{n_{N-1}}, \qquad  X_{\rm BS}^{(s)}=\mathbb{Z}_{n_N}.
\label{XBS2}
\end{gather}
Every factor in $X_{\rm BS}^{(w)}$ can be completely characterized either by the weak TI indices $\nu_i$ ($i=1,2,3$) or the weak mirror Chern number $C_M^{(k_z=0)}=C_M^{(k_z=\pi)}$, both of which can be understood as arising from stacking 2D TIs.  On the other hand, for most SGs, the strong factor $X_{\rm BS}^{(s)}$ cannot be fully understood by these familiar indicators.  We will thus focus on the strong factor in the reminder of this section.

\subsection{$\mathbb{Z}_4$ index for inversion symmetry}
\label{sec:k1}
Here we show that the strong TI index $\nu_0$ can actually be promoted to a $\mathbb{Z}_4$ index in the presence of inversion symmetry. The refined index can capture topological crystalline insulators with anomalous 1D edge state, as we discuss in detail in Sec.~\ref{Sec:Surface}.

Let $n_{K}^{+}$ ($n_{K}^{-}$) be the number of occupied bands with the even (odd) parity at each time-reversal invariant momentum (TRIM) $K$.
Due to Kramers pairing, $n_{K}^{\pm}$ is even. The Fu-Kane formula~\cite{Fu07} for the strong TI index $\nu_0\in\mathbb{Z}_2$ may be expressed as
\begin{eqnarray}
(-1)^{\nu_0}&=&\prod_{K\in\text{TRIMs}} (+1)^{\frac{1}{2}n_{K}^{+}} (-1)^{\frac{1}{2}n_{K}^{-}}\notag\\
&=&(-1)^{\frac{1}{2}\sum_{K\in\text{TRIMs}}n_{K}^{-}}.
\label{defZ2}
\end{eqnarray}
We now introduce a $\mathbb{Z}$-valued index \cite{Lu14} that is simply the sum of the inversion parities of occupied bands (up to a pre-factor):
\begin{eqnarray}
\kappa_1&\equiv&\frac{1}{4}\sum_{K\in\text{TRIMs}}(n_{K}^{+}-n_{K}^{-})\in\mathbb{Z}.
\label{defk1}
\end{eqnarray}
Using the total number of occupied bands $n\equiv n_{K}^{+}+n_{K}^{-}$, one can rewrite $\kappa_1$ as $2n-\frac{1}{2}\sum_{K\in\text{TRIMs}}n_{K}^{-}$.
Comparing this with Eq.~\eqref{defZ2}, we find
\begin{equation}
(-1)^{\nu_0}=(-1)^{\kappa_1}.
\label{nu0k1}
\end{equation}
Although Eq.~\eqref{nu0k1} suggests that $\kappa_1=\nu_0$ mod 2, $\kappa_1$ contains more information than $\nu_0$ as it is ``stable" mod $4$, in the sense that any trivial insulator has $\kappa_1=4n$ ($n\in\mathbb{Z}$) and adding or subtracting trivial bands does not alter $\kappa_1$ mod $4$. Weak topological phases may realize $\kappa_1=2$ mod 4, but the most interesting case is when $\kappa_1=2$ mod 4 while $(\nu_1,\nu_2,\nu_3)$ all vanish.  One way of achieving this phase is to stack two copies of a strong TI.  The surface mode of this phase will be discussed in Sec.~\ref{Sec:Surface}.

The space group that contains only inversion $\mathcal{I}$ in addition to translations is called $P\bar{1}$ (numbered No.~2 in the International Tables~\cite{ITC}). The index $\kappa_1$ can be defined for every supergroup of $P\bar{1}$ (SGs containing $P \bar{1}$ as a subgroup), i.e., for every centrosymmetric SG.

\subsection{New $\mathbb{Z}_2$ index for four-fold rotoinversion}
\label{sec:k4}
The SGs $P\bar{4}$ (No.~81) and $I\bar{4}$ (No.~82) possess neither inversion nor mirror symmetry. Hence, the indicator $X_{\text{BS}}=\mathbb{Z}_2$ found in Ref.~\onlinecite{Po17} cannot by accounted for by the Fu-Kane parity formula \cite{Fu07} or the mirror Chern number \cite{Hsieh12}. Here, we propose a new index $\kappa_4$ in terms of the eigenvalues of four-fold rotoinversion $\bar{C}_4$ (the four-fold rotation followed by inversion) and show that the $\mathbb{Z}_2$ nontrivial phase is actually a strong TI.

To introduce the invariant, we note that there are four momenta in the BZ invariant under $\bar{C}_4$, which we will denote by $K_4$; they are $(0,0,0)$, $(\pi,\pi,0)$, $(0,0,\pi)$ and $(\pi,\pi,\pi)$ for primitive lattice systems and $(0,0,0)$, $(0,0,2\pi)$, $(\pi,\pi,-\pi)$, and $(\pi,\pi,\pi)$ for body-centered systems.  For spinful electrons, the four possible values of $\bar{C}_4$-eigenvalues are $e^{\frac{\alpha\pi}{4}i}$ ($\alpha=1,3,5,7$). Let us denote by $n_K^\alpha$ the number of occupied bands with the eigenvalue $e^{\frac{\alpha\pi}{4}i}$ at momentum $K\in K_4$.

The new $\mathbb{Z}$-valued index is (up to a pre-factor) the sum of the $\bar{C}_4$-eigenvalues of occupied bands over high-symmetry momentum in $K_4$ :
\begin{eqnarray}
\kappa_4&\equiv&\frac{1}{2\sqrt{2}}\sum_{K\in K_4}\sum_{\alpha}e^{\frac{\alpha\pi}{4}i}n_K^\alpha\in\mathbb{Z}.
\label{defk4}
\end{eqnarray}
This quantity is always an integer in the presence of TRS, and is stable modulo $2$ against the stacking of trivial bands. As we show in Appendix~\ref{appk4n0}, $\kappa_4 \mod 2$ agrees with $\nu_0$.  It is, however, still useful to distinguish $\kappa_4$ from $\nu_0$, as $\kappa_4$ conveys more information than $\nu_0$ in the presence of additional symmetries, as we will discuss in the next section.

\subsection{The combination of $\kappa_1$ and $\kappa_4$}
\label{sec:delta}
The SGs $P4/m$ (No.~83) and $I4/m$ (No.~87) contain both inversion $\mathcal{I}$ and four-fold rotation $C_4$, and the inversion center is on the rotation axis.  The product of $\mathcal{I}$ and $(C_4)^2$ is a mirror $M_z$ about the $xy$ plane and one can define the mirror Chern number $C_M^{(k_z=0)}$.  The eigenvalues of the four-fold rotation determine $C_M^{(k_z=0)}$ mod 4  \cite{Fang12}.  However, $X_{\text{BS}}$ studied in Ref.~\onlinecite{Po17} contains a $\mathbb{Z}_8$ factor, which cannot be explained by the (detected) mirror Chern number alone. Here, we argue that the combination of $\kappa_1$ and $\kappa_4$ is responsible for this enhanced factor.

Since these SGs have both inversion $\mathcal{I}$ and rotoinversion $\bar{C}_4=\mathcal{I} C_4$, $\kappa_1$ and $\kappa_4$ can be defined separately using Eqs.~\eqref{defk1} and~\eqref{defk4}.  Furthermore, the inversion center coincides with the rotoinversion center.  In this case, we find that the difference
\begin{equation}
\Delta=\kappa_1-2\kappa_4
\end{equation}
is stable modulo $8$, not only 4, against the stacking of trivial bands. Hence $\Delta$ mod 8 should be understood as a new bulk invariant, which can be reconciled with the strong factor $\mathbb{Z}_8 < X_{\rm BS}$. To see this, recall first that the definition of $\kappa_1$ and $\kappa_4$ has no ambiguity and they are perfectly well-defined as $\mathbb{Z}$-valued quantities.  Furthermore, an explicit calculation verifies that, for any trivial insulator symmetric under either $P4/m$ or $I4/m$, the two invariants $\kappa_1$ and $\kappa_4$ are not independent and $\kappa_1=2\kappa_4$ mod 8 always holds.

Even when all the mirror Chern numbers and the weak indices vanish, $\Delta$ can still be $4\neq0$ mod 8. Depending on the geometry, this phase may feature ``hinge" type 1D edge states~\cite{Song17,Fang17} which we will discuss in Sec.~\ref{Sec:Surface}.

\subsection{The combination of $\kappa_1$ and $C_M$ under six-fold rotation}
\label{sec:k1mCh}
The case of SG $P6/m$ (No.~175) is similar to $P4/m$. It has a six-fold rotation $C_6$ in addition to inversion $\mathcal{I}$.  The product of $\mathcal{I}$ and $(C_6)^3$ is the mirror $M_z$ protecting the mirror Chern number $C_M^{(k_z=0,\pi)}$.  Although the eigenvalues of the six-fold rotation can only detect mirror Chern numbers mod 6, $X_{\text{BS}}^{(s)}$ contains a $\mathbb{Z}_{12}$ factor.  In this case, the combination of $C_M^{(k_z=0)}$ mod 6 and $\kappa_1$ mod 4 can fully characterize this $\mathbb{Z}_{12}$ factor.

When the rotation $C_6$ is replaced by the screw $S_{6_3}$ (i.e., $C_6$ followed by a half translation in $z$), the resulting SG is $P6_3/m$ (No.~176). In this case, the product of $\mathcal{I}$ and $(S_{6_3})^3$ is also a mirror symmetry.  The corresponding $X_{\text{BS}}^{(s)}$ includes the same $\mathbb{Z}_{12}$ factor.

In contrast, $P\bar{6}$ (No.~174) generated by the six-fold rotoinversion $\bar{C}_6=IC_6$ does not have an inversion symmetry and $\kappa_1$ is not defined. The mirror Chern numbers $C_M^{(k_z=0,\pi)}$ protected by $M_z=(\bar{C}_6)^3$ can be diagnosed by thee-fold rotation $C_3=(\bar{C}_6)^2$ modulo 3 and this fully explains $X_{\text{BS}}^{(s)}=\mathbb{Z}_3$.

\subsection{Summary of $X_{\text{BS}}$}
The eight SGs studied above (SG 2, 81, 82, 83, 87, 174, 175, and 176) actually play the role of ``key" SGs, in the sense that they provide an anchor for understanding the  symmetry-based indicator $X_{\text{BS}}$ for all other SGs.
Suppose that we are interested in one of the 230 SGs $\mathcal{G}$. To understand the nature of a nontrivial $X_{\text{BS}}^{\mathcal{G}}$, one should first identify its maximal subgroup among the eight key SGs.  Let $\mathcal{G}_0\leq\mathcal{G}$ be the key SG. Then we know that (i) the topological indices characterizing $X_{\text{BS}}^{\mathcal{G}}$ are the same as those for $X_{\text{BS}}^{\mathcal{G}_0}$, (ii) $X_{\text{BS}}^{(s),\mathcal{G}}$ and $X_{\text{BS}}^{(s),\mathcal{G}_0}$ are the same, (iii) the weak factor $X_{\text{BS}}^{(w),\mathcal{G}}$ may be reduced from $X_{\text{BS}}^{(w),\mathcal{G}_0}$ due to the constraints imposed by additional symmetries $\mathcal{G}\setminus\mathcal{G}_0$.  In Table~\ref{tab:XBS}, we identify the key SGs for all the 230 SGs. This table reproduces $X_{\text{BS}}$ for all 230 SGs presented in Ref.~\onlinecite{Po17}, but in a way which renders their physical properties more transparent.

Physically, the key space groups play the role of the minimal SGs for which a certain non-trivial phase is possible. This means that a non-trivial phase in a SG $\mathcal{G}$ in the family of the key SG $\mathcal{G}_0$ will {\it remain} nontrivial even if some symmetries of $\mathcal{G}$ are broken so long as those of $\mathcal{G}_0$ are preserved. For instance, consider SG $P4_2/m$ (No. 84) characterized by a 4-fold screw rotation $S_{4_2}$ and the inversion $\mathcal{I}$.  The product of $\mathcal{I}$ and $(S_{4_2})^2$ gives rise to a mirror symmetry, whose mirror Chern number is diagnosed mod 4 by $X_{\rm BS}^{(s)}=\mathbb{Z}_4$.  (In this case, since the inversion center and the rotoinversion ($\mathcal{I} S_{4_2}$) center do not agree, $\kappa_1-2\kappa_2$ does not enhance $\mathbb{Z}_4$ to $\mathbb{Z}_8$.) If we break the mirror symmetry without breaking inversion, the mirror Chern number becomes undefined, but the phases corresponding to the nontrivial $X_{\rm BS}^{(s)}$ are still protected by the inversion symmetry, and are diagnosed by the inversion index $\kappa_1$.

\begin{table*}
\begin{center}
\caption{{\bf Summary of symmetry-based indicator of band topology.} Space groups are grouped into their parental key space group in the first column. The second column lists the key topological indices characterizing nontrivial classes in $X_{\text{BS}}$. The space groups are indicated by their numbers assigned in ITC, and those highlighted by an underbar are the key SGs.
\label{tab:XBS}}
\begin{tabular}{cc|ccc}\hline \hline
Key Space Group	&Key Indices	&$X_{\text{BS}}^{(w)}$ & $X_{\text{BS}}^{(s)}$	&Space Groups\\\hline\hline
\multirow{11}{*}{$P\bar{1}$} & \multirow{11}{*}{$\nu_{1,2,3}$, $\kappa_1$} & $(\mathbb{Z}_2)^3$ & $\mathbb{Z}_4$ &  \underline{2}, 10, 47. \\
&&$(\mathbb{Z}_2)^2$ & $\mathbb{Z}_4$ & 11, 12, 13, 49, 51, 65, 67, 69. \\ &\\
&&\multirow{4}{*}{$\mathbb{Z}_2$}&\multirow{4}{*}{$\mathbb{Z}_4$}
&   14, 15, 48, 50, 53, 54, 55, 57, 59, 63,\\
&&&& 64, 66, 68, 71, 72, 73, 74, 84, 85, 86, \\
&&&& 125, 129, 131, 132, 134, 147, 148, 162, \\
&&&& 164, 166, 200, 201, 204, 206, 224.\\ &\\
&& \multirow{3}{*}{0}  &\multirow{3}{*}{$\mathbb{Z}_4$}
& 52, 56, 58, 60, 61, 62, 70, 88, 126, 130, 133,\\
&&&& 135, 136, 137, 138 141, 142, 163, 165, 167,\\
&&&& 202, 203, 205, 222, 223, 227, 228, 230.\\\hline
$P\bar{4}$ & $\kappa_4$ & 0 & $\mathbb{Z}_2$ & \underline{81}, 111--118, 215, 218. \\
$I\bar{4}$ & $\kappa_4$ & 0 & $\mathbb{Z}_2$ & \underline{82}, 119--122, 216, 217, 219, 220. \\\hline
\multirow{4}{*}{$P4/m$} && $\mathbb{Z}_2\times\mathbb{Z}_4$ &$\mathbb{Z}_8$ & \underline{83}, 123.\\
& $\nu_{1,2,3}$, $C_M^{(k_z=0,\pi)}$,  &$\mathbb{Z}_2$&$\mathbb{Z}_8$ & 124. \\
& $\kappa_1-2\kappa_4$ &$\mathbb{Z}_4$&$\mathbb{Z}_8$ & 127, 221.\\
&& 0 & $\mathbb{Z}_8$ & 128.\\&\\
\multirow{2}{*}{$I4/m$} & $\nu_{1,2,3}$, $C_M^{(k_z=0)}$, & $\mathbb{Z}_2$ & $\mathbb{Z}_8$ & \underline{87}, 139, 140, 229.\\
& $\kappa_1-2\kappa_4$ & 0 & $\mathbb{Z}_8$ & 225, 226\\\hline
\multirow{2}{*}{$P\bar{6}$} &\multirow{2}{*}{$C_M^{(k_z=0,\pi)}$}& $\mathbb{Z}_3$ &$\mathbb{Z}_3$ & \underline{174}, 187, 189.\\
&& 0 & $\mathbb{Z}_3$ & 188, 190.\\\hline
\multirow{2}{*}{$P6/m$} &\multirow{2}{*}{$C_M^{(k_z=0,\pi)}$, $\kappa_1$} &$\mathbb{Z}_6$ & $\mathbb{Z}_{12}$ & \underline{175}, 191. \\
&& 0 & $\mathbb{Z}_{12}$ & 192\\ & \\
$P6_3/m$ & $C_M^{(k_z=0)}$, $\kappa_1$ & 0 & $\mathbb{Z}_{12}$ & \underline{176}, 193, 194.\\
\hline \hline
\end{tabular}
\end{center}
\end{table*}

\section{Surface states}
\label{Sec:Surface}

We have seen in the previous section that some phases diagnosed by the symmetry indicator $X_{\rm BS}$ do not fall into the standard categories of strong TI, weak TIs, and mirror Chern insulators. This raises the question on the nature of the possible surface states possessed by these phases. With this question in mind, this section will be devoted to developing a  general approach to studying anomalous surface states protected by crystalline symmetries. We will argue that our approach captures most, if not all, TCIs with anomalous surface states. In the next section, we will provide a precise relation between the surface analysis in this section and the bulk symmetry indicator discussed in the previous section.

We start by considering a sample with a specific geometry, and boundary conditions which are periodic or open along different directions. The surface is a compact manifold described by a surface Hamiltonian $h_{{\vec r},{\vec k}}$, which depends on the position on the surface ${\vec r}$ and the surface momentum ${\vec k} = i \nabla_{{\vec r}}$, a vector in the tangent space to the surface at point ${\vec r}$. The procedure of obtaining a surface Hamiltonian from a given bulk Hamiltonian is explained in detail in Appendix \ref{app:surfaceD}. To summarize, it starts by introducing some space dependent parameter in the Hamiltonian and changing it across the surface from its value inside the sample to a different value outside it. This is then followed by projecting the bulk degrees of freedom onto the space of states localized close to this surface and rewriting the Hamiltonian in terms of these surface degrees of freedom.

We call an insulator an anomalous surface topological crystalline insulator (sTCI) if its surface satisfies certain defining properties: we say a band insulator is a sTCI if there exists a geometry with some boundary conditions such that the surface Hamiltonian
\begin{enumerate}[(i)]
\item is not gapped everywhere;
\item cannot be realized in an independent lower dimensional system; and
\item can be gapped everywhere by breaking the crystalline symmetry or going through a bulk phase transition.
\end{enumerate}
This definition includes familiar phases such as weak TIs (protected by translation) and conventional TCIs, as well as the recently discovered TCI with surface hinge modes \cite{ Song17, Schindler17, Langbehn17,  Benalcazar17Aug, Fang17}.

Given a gapped bulk, the surface region where the gap vanishes is necessarily boundaryless, since otherwise the spatial spread of the surface states would diverge as we approach the boundary of the gapless region, necessitating the existence of gapless bulk states sas in Weyl semimetals \cite{Wan11, Huang15, Xu15}.
This boundaryless region could be a 2D, 1D (collection of closed curves) or a 0D region (collection of points). We will show below that the latter is impossible for an insulator in class AII, but first, let us point out that designating a certain dimensionality to a gapless region generally depends on the symmetries at play, as well as the geometry of the sample. For example, the hinge insulators protected by rotation \cite{Song17, Fang17} possess 1D helical gapless modes when placed on a sphere, but exhibit gapless 2D Dirac cones when the surface is a plane normal to the rotation axis (with periodic boundary conditions along the two in-plane directions), as we illustrate Fig.~\ref{Planes}c. The relation between 1D and 2D surface states will be explored in more detail in Sec.~\ref{Sec:Dispersion}.

To explore the types of possible surface states, we start by defining a (surface) high-symmetry point as a point left invariant by some crystalline symmetries. That is to say, the stabilizer group of this point contains elements apart from the identity. Let us first consider a generic point $\vec r$ (that is not a high-symmetry point). Since the only symmetry at $\vec r$ is time-reversal symmetry, the surface Hamiltonian can only be gapless there if it is locally identical to (i) the surface of a strong TI, or (ii) a point residing on a (movable) domain wall between two different quantum-spin Hall phases. This can be understood in terms of the classification of stable topological defects \cite{Teo10}, which implies that 1D and 2D defects are topologically stable in class AII (with a $\mathbb{Z}_2$ classification), whereas 0D defects are not. The latter generically represent vortices in superconductors and require particle-hole symmetry to be stable.

At a high-symmetry point, one extra complication arises from the fact that the crystalline symmetry acts locally as an on-site discrete unitary symmetry, which can be used to block-diagonalize the local Hamiltonian. Each block can potentially have less symmetries than the original Hamiltonian. As a result, we need to consider the additional possibility of class A (unitary-type) defects. Only 1D defects, representing domain walls between two phases with different Chern numbers, are stable in class A \cite{Teo10} and have a $\mathbb{Z}$ classification. They will only exist if the symmetry leaves a line or curve on the surface invariant, which occurs only for mirror symmetry in 3D. Such line can host an arbitrary number of 1D gapless modes due to the $\mathbb{Z}$ classification, which can be understood in terms of mirror Chern number \cite{Hsieh12}.

Our argument can now be summarized as follows: a gapless surface Hamiltonian at a certain point will be a part of (i) a single domain wall, (ii) multiple intersecting domain walls (this could only happen at a high-symmetry point since an extra symmetry is required to stabilize the crossing), or (iii) the 2D gapless surface of a strong TI. The last case does not correspond to an sTCI since it does not require a crystalline symmetry for its stability, while the former two cases correspond to sTCIs with domain walls, which are either movable or are pinned to high-symmetry points or lines. Consequently, we arrive at a unified description for the surface states of all sTCIs in terms of ``globally irremovable, locally stable topological defects.'' Our approach is reminiscent to the approach of Ref.~\onlinecite{Hermele17}, which argued that a symmetry protected topological (SPT) phase protected by a point group symmetry can be understood in terms an embedded SPT of a lower dimensionality stabilized by the point-group symmetry.

Having reduced our problem to the study of globally stable configurations of surface domain walls, the next simplification is achieved by observing that a surface domain wall in class AII can be generically constructed by first stacking together two strong TIs, and then adding a mass term which changes sign at the domain wall. This is also consistent with the analysis of Sec.~\ref{Sec:Indicators} and Refs.~\onlinecite{Fang17,Song17}, and it means that the surface theory of any sTCI can be constructed by stacking strong TIs and studying the symmetry transformation properties of the possible mass terms. Our basic building block in the following sections will be the doubled strong TI (DSTI), which is constructed by stacking two strong TIs.
In particular, we note that every even entry in the strong factor $X_{\rm BS}^{\rm (s)}$ of any space group can be viewed as the stack of two strong TIs (e.g., by rewriting $4 = 1+3$). If the order of $X_{\rm BS}^{\rm (s)}$ is even (say $X_{\rm BS}^{\rm (s)} = \mathbb Z_4$), then the DSTIs correspond precisely to the even subgroup of $X_{\rm BS}^{\rm (s)}$ (Table \ref{tab:XBS}); however, if the order is odd, then this identification does not hold, and any given entry could correspond to either a strong or a doubled strong TI. We will elaborate further on such identification ambiguity in Sec.\ \ref{Sec:BulkSurface}.

\subsection{Stacked strong TIs}
\label{Sec:DSTI}
As explained in detail in Appendix \ref{app:surfaceD}, the gapless Hamiltonian on the surface of a strong TI can be written as
\begin{equation}
\label{HTI}
h_{{\vec r},{\vec k}} = ({\vec k} \times {\vechat n}_{{\vec r}}) \cdot \mbox{\boldmath$\sigma$}.
\end{equation}
Here, ${\vechat n}_{\vec r}$ is the normal to the surface at point ${\vec r}$ and $\mbox{\boldmath$\sigma$} =(\sigma_x, \sigma_y, \sigma_z)$ is the vector of Pauli matrices representing the spin degrees of freedom. TRS is implemented as $\mathcal{T} =i \sigma_y \mathcal{K}$, where $\mathcal{K}$ denotes complex conjugation, and it protects the gap of $h_{{\vec r},{\vec k}} $.

A DSTI is constructed by stacking two strong TIs, whose surface Hamiltonian can be described by two copies of (\ref{HTI}). The only possible $\mathcal{T}$-symmetric mass term which can gap-out the surface has the form
\begin{equation}
\label{Mass}
M_{{\vec r}} = m_{{\vec r}} {\vechat n}_{{\vec r}} \cdot \mbox{\boldmath$\sigma$} \otimes \tau_y,
\end{equation}
with $\tau_{x,y,z}$ denoting the Pauli matrices in the space of the two copies.

To study the action of spatial symmetries on the DSTI model, we start by considering a generic spatial symmetry $g = \{ R_g ~|~ \vec \tau_g\}$, where $R_g \in {\rm O}(3)$ is a point-group operation and $\vec \tau_g$ denotes the (possibly fractional) translation in $g$. $R_g$ can be parameterized by three pieces of data, $\det R_g$,  $\theta_g$, and $\vechat n_g$, where $\theta_g$ and $\vechat n_g$ are respectively the rotation angle and axis, and $\det R_g = \pm 1$ indicates whether $R_g$ is a proper or improper rotation. We can construct a natural spin-1/2 representation of the point-group action of $g$, given by $v_g \equiv \exp(- i \theta_g \, \vechat n_g \cdot \vec \sigma/2)$. The action of a spatial symmetry $g$ on the surface Hamiltonian (\ref{HTI}), derived in Appendix \ref{app:surfaceD} starting from its action on the bulk Hamiltonian, is given by
\begin{equation}
g \cdot h_{{\vec r},{\vec k}} \equiv v_g h_{{\vec r},{\vec k}} v_g^\dagger =  h_{g \cdot {\vec r},R_g \vec k },
\end{equation}
which leaves the surface Hamiltonian (as a whole) invariant for any spatial symmetry $g$. Clearly, the same conclusion holds when we generalize $v_g \mapsto u_g \equiv \eta_g v_g$ for any ${\rm U}(1)$ phase factor $\eta_g$, which is further restricted by time-reversal symmetry to $\eta_g = \pm 1$.  In addition, $\eta_g $ has to respect the (projective) group structure of the point group. Specifically, for a pair of symmetry elements $g, g'$, we demand $u_{g g'} u_g^\dagger u_{g'}^\dagger = v_{g g'} v_g^\dagger v_{g'}^\dagger $, which fixes $\eta_{g g'} = \eta_g \eta_{g'}$.

Microscopically, $\eta_g$  arises naturally since the ${\rm O}(3)$ rotational symmetry is reduced to a finite, discrete point group in a crystal, and both spin and orbital angular momenta contribute to the symmetry representation.
Yet, when $u_g$ is traceless, say for $u_g = i \eta_g \sigma_z$, the sign $\eta_g$ can be removed through a basis transformation, and hence $\eta _g =\pm 1$ do not give rise to distinct representations. However, such a basis transformation will simultaneously modify the form of the surface Hamiltonian (\ref{HTI}), which in turn amounts to a redefinition of the helicity of the surface Dirac cone, as we will discuss later. In the following, we will say $u_g$ is a ``signed representation'' when we want to emphasize the importance of the sign choice $\eta_g = \pm 1$, regardless of whether this choice actually gives rise to distinct representations.

Spatial symmetries may force the mass term $m({\vec r})$ to vanish along some curve, which creates a domain wall hosting a propagating 1D helical gapless mode. To see this, consider a spatial symmetry operation $g$ which leaves the mass term (\ref{Mass}) invariant. The action of the symmetry on the mass term can be generally written as
\begin{equation}
g \cdot M_{\vec r} \equiv
u^{\tau}_g M_{\vec r} (u^{\tau}_g)^{\dagger}
= M_{g \cdot {\vec r}} .
\end{equation}
Here, $u^{\tau}_g$ is the signed representation of $g$ acting on the DSTI constructed starting from  $u_g^{(i)} = \eta_g^{(i)} v_g$ for the two strong TIs labeled by $i=1,2$ as $u^{\tau}_g = u_g^{(1)} \oplus u_g^{(2)}$.

The invariance of the mass $M_{\vec r}$ under symmetry action implies that
\begin{equation}
m_{g \cdot {\vec r}} = \pm m_{\vec r}, \quad (u_g^\tau)^{\dagger} {\vechat n}_{g \cdot {\vec r}} \cdot \mbox{\boldmath$\sigma$} \otimes \tau_y u_g^\tau = \pm  {\vechat n}_{\vec r} \cdot \mbox{\boldmath$\sigma$} \otimes \tau_y.
\end{equation}
This means that, for any signed representation $u^{\tau}_g$ of the symmetry group $\mathcal{G}$ on the DSTI, we can define its signature as a map $s: \mathcal{G} \mapsto \mathbb{Z}_2$, which assigns a $\pm$ sign to each symmetry operation $g \in \mathcal{G}$ according to whether or not the mass changes sign under the action of $g$ (in the specified signed representation):
\begin{equation}
m_{g \cdot {\vec r}} = s_g m_{\vec r}
\end{equation}
 The only condition which should be satisfied by $s_g = \pm 1$ is that it is a group homomorphism i.e.  $s_{g_1} s_{g_2} = s_{g_1 g_2}$ for $g_{1,2} \in \mathcal{G}$. The properties of the mass term on the surface are fixed by specifying the signatures of the symmetry operations, which are in turn completely fixed by the parameters $\eta_g^{(1, 2)}$ and $\det R_g$ via the relation
\begin{equation}
\label{sg}
s_g = \det R_g \, \eta_g^{(1)}  \eta_g^{(2)}.
\end{equation}
In the following, we will usually use the terminology ``a $\pm$ representation for symmetry $g$'' to indicate that the symmetry $g$ has a $\pm$ signature in this signed representation.

Let us point out that, although our DSTI model  with two flavors of Dirac fermions is not sufficient to implement all possible anomalous surface states of interest, say those with a high mirror Chern number, it is sufficient to construct the ``root states'' which generate such states upon stacking. To see this, consider a system consisting of $n$ copies of the DSTIs. In this case, there are $k_n$ independent $\mathcal{T}$-preserving mass terms $m_{i,{\vec r}}$, $i=1,\dots,k_n$. The crystalline symmetries act on the vector ${\vec m}_{\vec r} =(m_{1,{\vec r}},\dots,m_{k_n,{\vec r}})$ as orthogonal transformations leaving the length of the vector (which gives the magnitude of the gap at a given point) fixed (see appendix~\ref{app:surfaceD}). Apart from mirror symmetry, all crystalline symmetries leave at most two points on the surface invariant. As a result, they will only protect anomalous surface states if we are forced to go through a domain wall as we go from a generic point ${\vec r}$ to its image under symmetry, $g \cdot {\vec r}$. Such a domain wall will exist between the two points if and only if there is no trajectory connecting ${\vec m}$ and $O_g {\vec m}$ on the $(k_n-1)$-sphere, establishing a correspondence between the fundamental group of the $(k_n-1)$-sphere and the stable domain walls in a model with $n$ DSTIs. Since the fundamental group of the $(k-1)$-sphere is trivial for $k>1$, we deduce that we cannot build stable domain walls whenever $n>1$. For $n=1$, $k_n = 1$ and only one mass term is possible. In this case, the fact that the  0-sphere (just two points) has two disconnected components implies the possibility of having $\mathbb{Z}_2$ domain walls, thereby establishing a $\mathbb{Z}_2$ classification for sTCIs {\it not protected by mirror symmetry} in class AII.

The only exception to the previous analysis is mirror symmetry. In this case, we need to consider the mass vector in the mirror plane. We find that it has to remain invariant under the action of mirror symmetry at any point in this plane, $O_M {\vec m} = {\vec m}$, which is only possible if one of the eigenvalues of $O_M$ is $+1$. In a representation which does not satisfy this condition, e.g $O_M = -1$, the mass vector ${\vec m}$ will necessarily vanish in the mirror plane regardless of $n$. Nevertheless, the state with $n>1$ DSTIs can always be built by from the ``root'' state implemented using a single mirror-symmetric DSTI.

Notice that, up to this point, the analysis is general and would, in principle, apply to any crystalline symmetry group. For instance, weak TIs can be understood within the DSTI model by considering a surface with periodic boundary along the ``weak'' direction, and choosing a ``$-$'' representation for translation along this direction. This will enforce the existence of a domain wall for every unit lattice translation along this direction, leading to the surface states obtained by stacking 2D quantum-spin Hall systems (Fig.~\ref{WeakTI}).
Since the weak phases are relatively well-understood, we will focus in the following on sTCIs which do not correspond to weak phases. This means that, for the next two sections, we will always assume the translation symmetries take the ``$+$'' representation.

The next subsection will be devoted to crystalline point groups, whose sTCIs can be understood by considering a spherical geometry with open boundary conditions in all directions. We will describe a procedure to construct all possible sTCIs in a given symmetry group, and then use it to obtain the complete classification of sTCIs protected by point group symmetries. We will later discuss how the analysis can be extended to all space groups. Rather than exhaustively performing the same analysis for all 230 space groups, we are going to show that the classification of sTCIs in a given space group can be understood in terms of the  classification of its corresponding point group (defined by  ``modding out'' its translation subgroup).

\begin{figure}
\center
\includegraphics[width=0.35\columnwidth]{WeakTI}
\caption{Surface states for the weak TI can be built from the DSTI by choosing a ``$-$'' representation for the translation along the ``weak'' direction.}
\label{WeakTI}
\end{figure}

\subsection{Crystallographic point groups}
\label{Sec:Pointgroups}

In this section, we will first review the natural action of the different point-group symmetries on the physical degrees of freedom, and discuss what type of gapless surface states they can protect.
Afterwards, we will provide a general procedure to construct all sTCIs in a given point group and use it to obtain an exhaustive classification of sTCIs for the 32 crystallographic point groups.

\subsubsection{Crystallographic point group symmetries}

{\bf Inversion}: Inversion symmetry acts by inverting position and momentum while keeping the spin unchanged
\begin{equation}
\mathcal{I}: \quad {\vec r} \rightarrow -{\vec r}, \qquad {\vec k} \rightarrow -{\vec k}, \qquad \mbox{\boldmath$\sigma$} \rightarrow \mbox{\boldmath$\sigma$}.
\end{equation}
According to (\ref{sg}), inversion can be represented with negative signature by taking its action on the DSTI to be $\mathcal{I}^\tau = \mathcal{I} \oplus \mathcal{I} = \mathcal{I} \otimes \tau_0$ (corresponding to $\eta^{(1)} = \eta^{(2)}$), or with positive signature by acting on the DSTI as $\mathcal{I}^\tau = \mathcal{I} \oplus (-\mathcal{I}) = \mathcal{I} \otimes \tau_z$ (corresponding to $\eta^{(1)} = -\eta^{(2)}$).

In the ``$-$'' representation, the mass term changes sign between a point and its image under inversion, leading to a ``hinge'' phase with a helical gapless mode living on some inversion-symmetric curve on the sphere. The curve can be moved around but cannot be removed without breaking inversion. Such a phase is graphically illustrated in the $C_i$ entry in Fig.~\ref{HOTIPG11}. Two copies of this state can be trivialized by adding the mass terms $m^{1,2}_{\vec r} {\vechat n}_{\vec r} \cdot \mbox{\boldmath$\sigma$} \tau_{x,z} \mu_y$, which, like (\ref{Mass}), change sign under inversion, but can be chosen so that they do not both vanish at a point where the mass term $m_{\vec r}$ also vanishes, leading to a completely gapped surface. As a result, gapless modes protected by inversion will have a $\mathbb{Z}_2$ classification, as anticipated in the discussion of the previous section.

{\bf $C_n$ rotation}: An $n$-fold rotation acts by rotating the position and momentum as vectors and rotating the spin as a spinor. An $n$-fold rotation about the $z$-axis is implemented as
\begin{equation}
C_{nz}: {\vec r} \rightarrow C_{nz} {\vec r}, \qquad {\vec k} \rightarrow C_{nz} {\vec k} \qquad \mbox{\boldmath$\sigma$} \rightarrow e^{-i \frac{\pi}{n} \sigma_z} \mbox{\boldmath$\sigma$} e^{i \frac{\pi}{n} \sigma_z},
\end{equation}
A positive signature representation is obtained by taking $C_n^\tau = C_n \oplus C_n = C_n \otimes \tau_0$ (corresponding to $\eta^{(1)} = \eta^{(2)}$), whereas the choice $C_n^\tau = C_n \oplus (-C_n) = C_n \otimes \tau_z$ (corresponding to $\eta^{(1)} = -\eta^{(2)}$) leads to a negative signature \footnote{These two choices obviously do not exhaust all possible choices for the different irreps in the two copies \cite{Fang17}, but they are sufficient to construct representations with both signatures which suffices for our analysis}. Notice that, here, the ``$+$'' and ``$-$'' representations are opposite to the inversion case, as $\det \mathcal I = -1$ but $\det C_n = 1$.

We note that the ``$-$'' representation is only possible for even $n$, since it violates the condition $C_n^n = -1$ whenever $n$ is odd. Thus, $n=3$ is only consistent with a ``$+$'' representation, whereas $n=2,4,6$ can have a ``$-$'' representation which forces the mass to vanish at $n/2$ (closed) curves related by rotation and intersecting at the two points left invariant by rotation (the poles) (as shown in Fig.~\ref{HOTIPG11}, PGs $C_2$, $C_4$ and Fig.~\ref{HOTIPG22}, PG $C_6$).
 Two copies can be gapped out by adding the mass term $m_{\vec r} {\vechat n}_{\vec r} \cdot \mbox{\boldmath$\sigma$} \tau_z \otimes \mu_y$, which does not change sign under rotation and can be chosen to be positive everywhere. This means that rotations lead to $\mathbb{Z}_2$ classification as well, consistent with the discussion of the previous section.

{\bf Mirror symmetry}: Mirror symmetry acts by inverting the position and momentum components perpendicular to the mirror plane and the spin component parallel to it. For example, the action of mirror symmetry about the $xy$ plane is implemented as
\begin{equation}
M_z: \quad (x,y,z) \rightarrow (x,y,-z), \qquad  \mbox{\boldmath$\sigma$} \rightarrow \sigma_z \mbox{\boldmath$\sigma$} \sigma_z.
\end{equation}
In the mirror plane, $M_z$ flips the sign of ${\vechat n}_{\vec r} \cdot \mbox{\boldmath$\sigma$}$. Thus, in the ``$-$'' representation, implemented by taking the mirror to acts in the DSTI as $M_z^\tau = M_z \oplus M_z = M_z \otimes \tau_0$ (corresponding to $\eta^{(1)} = \eta^{(2)}$), the mass term $m_{\vec r}$ has to vanish in this plane. The ``$+$'' representation, on the other hand, is implemented in the DSTI as $M_z^\tau = M_z \oplus (-M_z) = M_z \otimes \tau_z$ (corresponding to $\eta^{(1)} = -\eta^{(2)}$) and does not impose a constraint on the mass term in the mirror plane leading to a completely gapped surface.

As we noted in the previous section, the ``$-$'' representation for the mirror implies that the mirror plane remains gapless regardless of the number of copies of the DSTI we stack together. This can be seen from the fact that, for any number of copies, any mass term will have the form $m_{\vec r} {\vechat n}_{\vec r} \cdot \mbox{\boldmath$\sigma$} \otimes \Gamma$, which will always vanish in the mirror plane. As discussed previously, this implies a $\mathbb{Z}$ classification corresponding to the mirror Chern number in real space \cite{Hsieh12}.

\subsubsection{Classification of sTCIs in the 32 crystallographic point groups}
We are now in a position to perform a systematic investigation of sTCIs in the 32 crystallographic point groups. As we argued in the beginning of this section, all these states can be built from the DSTI model considered in Sec.~\ref{Sec:DSTI} and copies of it stacked together.

In Sec.~\ref{Sec:DSTI}, we proposed that there is a one-to-one correspondence between the signatures of the different symmetries and the pattern of massless modes on the surface. One part of this correspondence is obvious since surface states corresponding to different symmetry signatures cannot be deformed into each other without changing these signatures. We conjecture that the opposite is also true: two patterns of surface modes can always be deformed into each other if they correspond to the same representation signatures for all the possible symmetries. We have checked this explicitly for several examples, where seemingly different surface state patterns corresponding to the same symmetry signatures turned out to be deformable into each other. For example, consider the point group $\bar 4$, where the only symmetry is a 4-fold rotoinversion. In this case, there seems to be two distinct surface state patterns corresponding to the ``$-$'' representation for $\bar C_4$, given by the ``equator'' state and the ``hinge'' state (cf.~Fig.~\ref{Trivial}). These two can, nevertheless, be deformed into each. The reason is that, unlike rotation, rotoinversion does not leave the poles fixed. Thus, we can move the hinges symmetrically away from the poles, brining them to the equator. This can be seen more clearly by adding the two phases and noting that a mass term can be added to gap-out the modes at their intersection points (this is possible since rotoinversion does not enforce any local constraints on the mass). The resulting surface can be deformed to a trivial one as shown in Fig.~\ref{Trivial}.
Alternatively, such correspondence can also be understood in a slightly more general language (i.e., beyond the Dirac theory analysis), as we elaborate in Sec.~\ref{Sec:Discussion}.

\begin{figure}
\center
\includegraphics[width=0.85\columnwidth]{Trivial}
\caption{In the presence of $\bar C_4$ only, the ``equator'' and the ``hinge'' state are deformable to each other since their sum can be trivialized as shown here.}
\label{Trivial}
\end{figure}

Having shown that the classification of sTCIs reduces to classifying distinct signatures of the different symmetries, we now proceed by constructing these phases explicitly by specifying the signatures for a minimal set of generators of the group (such that every group element can be written as a product of powers of generators). There are many possible choices of such generator set, but for our surface classification, we need the generator set to satisfy the extra conditions:
\begin{enumerate}
\item All independent mirror symmetries are included as generators.
\item The number of independent $C_3$ rotations included as generators is as large as possible (while respecting the condition that the generator set is minimal).
\end{enumerate}
Here, we say two symmetries $g_{1,2} \in \mathcal{G}$ are independent if they are not related by conjugation, i.e. there is no $h \in \mathcal{G}$ such that $g_1 =h^{-1} g_2 h$. For two rotations (mirrors), independence means that the rotation axes (mirror planes) are not related by any symmetry operation in the symmetry group $\mathcal{G}$. The second condition means that we should include the maximum number of independent $C_3$ rotations in the generator set as long as there is no other generator set with smaller size. For example, in PG $T$, which can be generated using either two independent $C_3$ rotations or a $C_2$ and a $C_3$, we have to choose the former since it includes more $C_3$ rotations, while in PG $C_6$, which can be generated using a single $C_6$ or a $C_3$ and a $C_6$, we have to choose the former since it contains a smaller number of generators (the latter is not really a minimal generator set). Such conditions are needed because, as explained, mirror and $C_3$ rotation symmetries behave differently compared to other symmetries in sTCI classification.

Once we have a minimal generator set satisfying these properties, we can read off the classification of sTCIs as follows:
\begin{enumerate}
\item To every mirror symmetry, we assign a factor of $\mathbb{Z}$ indicating the number of gapless modes in this mirror plane. The phase which generates this factor is obtained by choosing the ``$-$'' representation for the corresponding mirror symmetry in the DSTI model.
\item To every symmetry other than mirror and $C_3$, we assign a factor of $\mathbb{Z}_2$. Each of these $\mathbb{Z}_2$ phases is generated by choosing the ``$-$'' representation for the corresponding symmetry generator.
\end{enumerate}
Implementing these rules leads to the classification of sTCIs in all crystallographic point groups, which we tabulate under $\mathcal{S}_{\rm PG}$ in Table~\ref{Classification}. The ``hinge'' states generating the different $\mathbb{Z}$, $\mathbb{Z}_2$ factors by choosing the ``$-$'' representation in the corresponding symmetry are graphically illustrated in Figs.~\ref{HOTIPG11}-\ref{HOTIPG32}.

\begin{figure}
\center
\includegraphics[width=0.85\columnwidth]{HOTIPG1-11}
\caption{Graphical illustration of the surface states of the``hinge'' phases, which generate all sTCIs  for the first 11 crystallographic point groups, on a sphere. For each state, the operators which need to be taken in the ``$-$'' representation are shown as well as the resulting classification. Red circles indicate $\mathbb{Z}_2$ modes which would be gapped-out if two copies of the system are stacked together, whereas blue circles indicate $\mathbb{Z}$ modes protected by a mirror Chern number. Rotation axes are shown with black, blue, green, or red color for two-, four-, three- and six-fold rotations respectively.}
\label{HOTIPG11}
\end{figure}

\begin{figure}
\center
\includegraphics[width=0.85\columnwidth]{HOTIPG12-22}
\caption{Graphical illustration of the ``hinge'' surface states, which generate all sTCIs for the crystallographic PGs 12-22, on a sphere . The same conventions as in Fig.~\ref{HOTIPG11} are used.}
\label{HOTIPG22}
\end{figure}

\begin{figure}
\center
\includegraphics[width=0.85\columnwidth]{HOTIPG23-27}
\caption{Graphical illustration of the ``hinge'' surface states, which generate all sTCIs for the crystallographic PGs 23-27, on a sphere . The same conventions as in Fig.~\ref{HOTIPG11} are used.}
\label{HOTIPG27}
\end{figure}

\begin{figure}
\center
\includegraphics[width=0.85\columnwidth]{HOTIPG29-32}
\caption{Graphical illustration of the ``hinge'' surface states, which generate all sTCIs for the crystallographic PGs 29-32, on a sphere . The same conventions as in Fig.~\ref{HOTIPG11} are used.}
\label{HOTIPG32}
\end{figure}

\begin{table}
\center
\caption{{\bf Classification of point group sTCIs} $\mathcal{S}_{\rm PG}$ denotes the classification of point group sTCIs. The generators are chosen according to the criteria defined in the main text. ${\vechat n}_1$ and ${\vechat n}_2$ denotes two of the four 3-fold rotation axes in PGs 28-32. These describe systems with cubic symmetry with four 3-fold axes along the cube body-diagonal.}
\begin{tabular}{c|c|c|c}
\hline \hline
Symbol & ~~~PG~~~ & Generators & ~~~~~$\mathcal{S}_{\rm PG}$~~~~~\\
\hline
 1 & $C_1$ & 1 & 0\\
 $\bar{1}$ & $C_i$ & $\mathcal{I}$ & $\mathbb{Z}_2$\\
 2 & $C_2$ & $C_{2,z}$ & $\mathbb{Z}_2$\\
 $m$ & $C_s$ & $M_z$ & $\mathbb{Z}$\\
 $2/m$ & $C_{2h}$ & $M_z, C_{2z}$ & $\mathbb{Z} \times \mathbb{Z}_2$\\
 $222$ & $D_2$ & $C_{2z}, C_{2x}$ & $\mathbb{Z}_2^2$\\
$mm2$ & $C_{2v}$ & $M_x, M_y$ & $\mathbb{Z}^2$\\
$mmm$ & $D_{2h}$ & $M_x, M_y, M_z$ & $\mathbb{Z}^3$\\
$4$ & $C_4$ & $C_{4z}$ & $\mathbb{Z}_2$\\
 $\bar{4}$ & $S_4$ & $\bar{C}_{4z}$ & $\mathbb{Z}_2$\\
 $4/m$ & $C_{4h}$ & $C_{4z}, M_z$ & $\mathbb{Z} \times \mathbb{Z}_2$\\
$422$ & $D_4$ & $C_{4z}, C_{2x}$ & $\mathbb{Z}_2^2$\\
$4mm$ & $C_{4v}$ & $M_{x}, M_{xy}$ & $\mathbb{Z}^2$\\
$\bar{4}2m$ & $D_{2d}$ & $\bar{C}_{4z}, M_{x}$ & $\mathbb{Z} \times \mathbb{Z}_2$\\
$4/mmm$ & $D_{4h}$ & $M_{x}, M_{z}, M_{xy}$ & $\mathbb{Z}^3$\\
$3$ & $C_3$ & $C_{3z}$ & 0\\
$\bar 3$ & $S_6$ & $C_{3z}, \mathcal{I}$ & $\mathbb{Z}_2$\\
$32$ & $D_3$ & $C_{3z}, C_{2x}$ & $\mathbb{Z}_2$\\
$3m$ & $C_{3v}$ & $C_{3z}, M_{x}$ & $\mathbb{Z}$\\
$\bar 3m$ & $D_{3d}$ & $C_{3z}, M_{x}, \mathcal{I}$ & $\mathbb{Z} \times \mathbb{Z}_2$\\
$6$ & $C_6$ & $C_{6z}$ & $\mathbb{Z}_2$\\
$\bar 6$ & $C_{3h}$ & $C_{3z}, M_z$ & $\mathbb{Z}$\\
$6/m$ & $C_{6h}$ & $C_{6z}, M_z$ & $\mathbb{Z} \times \mathbb{Z}_2$\\
$622$ & $D_6$ & $C_{6z}, C_{2x}$ & $\mathbb{Z}_2^2$\\
$6mm$ & $C_{6v}$ & $M_{x}, M_{\frac{\sqrt{3}x+y}{2}}$ & $\mathbb{Z}^2$\\
$\bar 6m2$ & $D_{3h}$ & $C_{3z}, M_z, M_{x}$ & $\mathbb{Z}^2$\\
$6/mmm$ & $D_{6h}$ & $M_z, M_{x}, M_{\frac{\sqrt{3}x+y}{2}}$ & $\mathbb{Z}^3$\\
$23$ & $T$ & $C_{3,{\vechat n}_1}, C_{3,{\vechat n}_2}$ & 0\\
$m\bar 3$ & $T_h$ & $C_{3,{\vechat n}_1}, M_z$ & $\mathbb{Z}$ \\
$432$ & $O$ & $C_{3,{\vechat n}_1}, C_{3,{\vechat n}_2}, C_{4,z}$ & $\mathbb{Z}_2$\\
$\bar 43m$ & $T_d$ & $C_{3,{\vechat n}_1}, M_{{\vechat n}_1}$ &  $\mathbb{Z}$\\
$m\bar 3 m$ & $O_h$ & $C_{3,{\vechat n}_1}, M_{x},M_{xy}$ & $\mathbb{Z}^2$\\
\hline \hline
\end{tabular}
\label{Classification}
\end{table}

\subsection{Space groups}
In this section, we extend the analysis of the previous section to space groups, which requires considering symmetries which do not leave any point in space fixed (non-symmorphic symmetry). We first start by discussing the main complication arising from the inclusion of non-symmorphic symmetries, which requires a certain choice of the sample geometry and boundary conditions. We then provide a few examples of sTCIs protected by non-symmorphic symmetries. Afterwards, we will provide a complete classification of sTCIs in the 230 SGs by reducing the classification problem to the corresponding problem in point groups.

\subsubsection{Non-symmorphic symmetries}
The extension of the analysis of Sec.~\ref{Sec:Pointgroups} to include non-symmorphic symmetries is, in principle, straightforward, with the main difference being the necessity of choosing a different geometry depending on the space group. In order to preserve a non-symmorphic symmetry on the surface, we need to consider a sample with periodic boundary conditions along the directions of the fractional translation vector of the symmetry. A non-symmorphic symmetry, which can either be a screw ($n$-fold rotation followed by a fractional lattice translation along the rotation axis) or a glide (mirror reflection followed by a fractional lattice translation along a vector in the mirror plane), does not leave any point invariant. As a result, the surface states protected by non-symmorphic symmetries will have a $\mathbb{Z}_2$ classification (see the discussion at the beginning of this section).

The anomalous surface states protected by non-symmorphic symmetries can be understood by considering a cylinder geometry whose axis is parallel to the fractional translation axis, along which periodic boundary condition is taken. An $n$-fold screw in a ``$-$'' representation would give rise to a state with $n$ hinge related by $n$-fold rotation, along the sides of the cylinder. This surface state is very similar to the surface state protected by rotation, in that the hinges can be moved around freely as long as they preserve the rotational symmetry. A glide symmetry, on the other hand, gives rise to a state with two hinge modes confined to the glide plane. Despite looking similar to surface states protected by mirror symmetry, this state differs in an essential aspect, as it becomes trivial when added to itself. The reason is that, unlike mirror symmetry, a mirror Chern number cannot be defined for glide symmetry, since it does not act as an on-site symmetry in the glide plane.

As an example of surface states protected by a non-symmorphic symmetry, let us consider SG $P4_2$ (No. 77), where the only symmetry is a 4-fold screw $4_2$. On a cylinder geometry with a periodic boundary condition taken along the screw axis, a ``$-$'' representation for the screw symmetry leads to the state with four hinges shown in Fig.~\ref{CylC4}. We note that a state with domain walls at every half lattice translation along the screw axis is also consistent with the ``$-$'' representation for the screw symmetry. This state is gapless everywhere on the surface, and looks very similar to the surface of a weak TI shown in Fig.~\ref{WeakTI}. The main difference in this case is that such a state is unstable against being gapped out everywhere except for the four hinges as in Fig.~\ref{CylC4}.

Another example is SG $P6_3/m$ (No. 176), which is one of the key space groups considered in Table~\ref{tab:XBS}. This space group is generated by a 6-fold screw and a mirror perpendicular to the screw axis. Choosing a cylinder geometry with a periodic boundary condition along the screw axis, we may consider a ``$-$'' representation for either the screw or the mirror. In the former case, we get a phase with six hinges, whereas the latter is characterized by gapless modes protected by a mirror Chern number in the mirror planes. Notice that, in this case, we have several mirror planes related by screws which all have the same mirror Chern number. This phase remains non-trivial even if we break the screw symmetry , leaving behind only a single invariant mirror plane. The resulting classification in this case is $\mathbb{Z} \times \mathbb{Z}_2$.

\subsubsection{Classification of sTCIs in the 230 space groups}
We now tackle the problem of classifying all sTCI phases built from stacking strong TIs in all the 230 space groups.
From the discussion on point groups, we see that the desired classification can be reconciled with the different ways to assign the signature $\pm 1$ to a generating set of the space group. Compared to the case of point groups, a new ingredient is that a lattice translation can be assigned a $-1$ signature, which as we discussed correspond to weak TIs. As the weak TIs are relatively well-understood, we will exclude them from our discussion below, i.e., in the surface classification $\mathcal{S}_{\rm SG}$ below, we always assume the lattice translations are given $+1$ signatures.

We will now argue that the sTCIs described by our definition of $\mathcal{S}_{\rm SG}$ can be readily studied via $\mathcal{S}_{\rm PG}$. Recall that, given an SG $\mathcal{G}$, a corresponding point group $\mathcal{G}_p$ is defined by ``modding out'' the translation part of $\mathcal{G}$, i.e. $\mathcal{G}_p \equiv \mathcal{G}/T_\mathcal{G}$, where $T_\mathcal{G}$ is the translation subgroup of $\mathcal{G}$ (which is always a normal subgroup).
This procedure reduces screws/glides in $\mathcal{G}$ to rotations/mirrors in $\mathcal{G}_p$. The classification of sTCIs in any given SG $\mathcal{G}$ is the same as the classification of $\mathcal{G}_p$, except for the fact that every mirror in $\mathcal{G}_p$ which derive from a glide in $\mathcal{G}$ should be assigned a $\mathbb{Z}_2$ rather than $\mathbb{Z}$ factor.   Implementing this rule leads to Table \ref{tab:SurfaceXBS_Full} in Appendix \ref{app:XBSRep}, which provides the sTCI classification $\mathcal{S}_{\rm SG}$ for all the 230 SGs.

Several remarks about the deduction procedure from $\mathcal{S}_{\rm SG}$ to $\mathcal{S}_{\rm PG}$ are required. First, we point out that in case a mirror and a glide both exist in a given SG, they will always be independent, since a mirror plane and a glide plane can never be related by a space group symmetry. This means that the procedure described above is well-defined, and does not depend on the choice of the generator set (which satisfies the conditions of Sec.~\ref{Sec:Pointgroups}). Second, we note that, unlike 3-fold rotations, 3-fold screws do admit a ``$-$'' representation. However, the phase generated by this signed representation is the same as the weak TI phase generated by choosing the ``$-$'' for translation along the screw axis. This can be seen by observing that taking the 3-fold screw to the power of 3 yields a translation along the screw direction {\it with the same signature as the 3-fold screw}. Hence, their signatures cannot be chosen independently. Since the definition of $\mathcal{S}_{\rm SG}$ excludes weak TIs (which can be incorporated simply by including the generators of translation), such a state is excluded from $\mathcal{S}_{\rm SG}$ and the signature of the 3-screw is restricted to be $+1$, similar to the case of a $C_3$-rotation.

Finally, we comment on the meaning of adding two phases in $\mathcal{S}_{\rm SG}$, which is an abelian group representing the equivalence classes of distinct sTCI phases. The subtlety arises if there is no geometry for which both phases exhibit anomalous surface states. This will never be the case for SGs containing only symmorphic elements, but is possible in the presence of non-symmorphic symmetries. However, we have to remember elements of $\mathcal{S}_{\rm SG}$ are distinct {\it bulk} states despite being defined by their surface signature. This means that the possibility of anomalous states on any given surface is completely fixed by the bulk, as will be discussed in detail in Sec.~\ref{Sec:BulkSurface}. Moreover, it is always possible to distinguish two distinct sTCI phases by considering them on many different geometries. For instance, given two phases 1 and 2 and two geometries $G_{1}$ and $G_2$ such that 1 (2) exhibits surface states on $G_{1}$ ($G_{2}$) but not on $G_{2}$ ($G_{1}$), we can distinguish the sum of the two phases from either phase by the fact that it exhibits surfaces states on {\it both} geometries $G_1$ and $G_2$.

\begin{figure}
\center
\includegraphics[width=0.35\columnwidth]{CylC4}
\caption{Illustration for the hinge state protected by a 4-fold screw in the space group 76 on a cylinder geometry with periodic boundary conditions along the screw direction.}
\label{CylC4}
\end{figure}

\subsection{Surface dispersion at special planes}
\label{Sec:Dispersion}
Although the 1D surface modes shown in Figs.~\ref{HOTIPG11}-\ref{HOTIPG32} can be, in principle, detected by means of transport measurements, the most experimentally accessible tool to investigate such surface states is provided by ARPES, which probes the energy dispersion at a given surface. The dispersion measured by ARPES can be readily understood from the surface state pattern given in real space in Figs.~\ref{HOTIPG11}-\ref{HOTIPG32}. In brief, we consider what happens when we attach a tangent plane to the sphere at a given point.

We note that the surface modes can generally be freely moved on the surface except when they pass through a rotation-invariant point or lie in a mirror/glide plane. As a result, a tangent plane at a point that is not rotation-invariant or lies in a mirror/glide plane will generically have gapped dispersion. Therefore, the analysis of surfaces where gapless Dirac cones may exist boils down to considering the cases where the normal to the surface (i) lies in a single mirror/glide plane, (ii) lies at the intersection of multiple mirror/glide planes, or (iii) is parallel to a rotation axis. These three cases are illustrated in Fig.~\ref{Planes}.

\begin{figure}
\center
\includegraphics[width=0.7\columnwidth]{Planes}
\caption{Illustration of the surface dispersion in planes tangent to the sphere whose normal is (a) in a single mirror plane, (b) in two mirror planes, or (c) parallel to a 4-fold rotation axis.}
\label{Planes}
\end{figure}

\subsubsection{Single mirror/glide plane}
Here, we investigate the dispersion in a tangent plane whose normal lies in a single mirror plane. The glide case can be considered very similarly. The dispersion in the plane can be written as
\begin{equation}
\label{HS}
h_{\vec k} = k_y \sigma_x - k_x \sigma_y
\end{equation}
with reflection acting as $M_x \cdot h_{k_x,k_y} = \sigma_x h_{k_x,k_y} \sigma_x=   h_{-k_x,k_y} $. Reflection symmetry implies that any mass term has the form $m_{\vec k} \sigma_z  \otimes \Lambda$, which necessarily vanishes at $k_x=0$ since it satisfies $m_{-k_x,k_y} = -m_{k_x,k_y}$. It is, therefore, not possible to gap-out the surface but it is possible to move the two Dirac cones apart by adding the $\mathcal{T}$-symmetric term $m_1 \sigma_x \tau_y$, which shifts the Dirac cones to ${\vec k} = (0,\pm m_1)$.

\subsubsection{Multiple mirror/glide planes}
If the normal to the tangent plane lies in the intersection of multiple mirror planes or a glide and a mirror, all in the ``$-$'' representation, we can repeat the same argument of the previous section for each mirror/glide plane separately. The result is that the Dirac cones will have to be confined to the line invariant under mirror/glide symmetry for each mirror/glide, leading to the pair of Dirac cones being pinned at the intersection point of the mirror/glide planes.

\subsubsection{Rotation axis}
This case was considered in Ref.~\onlinecite{Fang17}. We start with the surface Hamiltonian (\ref{HS}), with 2-,4- or 6-fold rotation acting as $C_n \cdot h_{\vec k} =  e^{i \frac{\pi}{n} \sigma_z} \tau_z h_{\vec k} \tau_z e^{-i \frac{\pi}{n} \sigma_z} =   h_{C_n {\vec k}}  $. The only mass term consistent with $C_n$ and time-reversal symmetry is $m_{\vec k} \sigma_z \tau_x$ for $n=2, 6$ or $m_{\vec k} \sigma_z \tau_y$ for $n=4$, both satisfying $m_{C_n {\vec k}} = -m_{{\vec k}}$. This, in particular, implies that the mass term $m_{\vec k}$ vanishes at ${\vec k}=(0,0)$, so the Dirac cones cannot be gapped. We can, however, add the symmetry-allowed term $m_1 \tau_z$, which will move the Dirac cones away from ${\vec k}=(0,0)$, resulting in $n$ Dirac cones related by the $n$-fold rotation. To see this, let us for a moment ignore the mass term in (\ref{HS}), so that it just describes two copies of a gapless Dirac Hamiltonian. Adding the term $m_1 \tau_z$ shifts the two cones in energy by $2m_1$ so that they intersect at a ring at zero energy. The Hamiltonian of the nodal ring is given by
\begin{equation}
h_{\vec k} = (2m_1 - k_x^2 - k_y^2) \mu_z
\end{equation}
where $\mu_{x,y,z}$ denote the Pauli matrices in the space of the two bands intersecting at zero (the lower band from the Dirac cone shifted upwards and the upper band from the one shifted downwards). Projecting the mass term $m_{\vec k} \sigma_z \tau_{x,y}$ to this basis, we find that it has the form $m_{\vec k} \mu_{x,y}$, which will gap-out the nodal ring except at $n$ points where it necessarily vanishes due to $m_{C_n {\vec k}} = -m_{\vec k}$.

Note that, despite the similarity between hinge phases protected by rotation and screw symmetries, the latter are not detectable by studying the dispersion at any plane on the surface. In this sense, they are similar to hinge states protected by inversion or rotoinversion in that they can only be observed if the surface is considered as a whole.

\section{Bulk-surface correspondence}
\label{Sec:BulkSurface}
In Sec.\ \ref{Sec:Indicators}, we have discussed how the symmetry-based indicators for time-reversal symmetric systems with strong spin-orbit coupling can mostly be reconciled with either a $\mathbb Z_2$ TI index or a mirror Chern number. However, as  listed in Table~\ref{tab:XBS}, there are new invariants which indicate that certain stacks of strong TIs remain nontrivial despite the lack of a conventional index.
These unconventional phases are expected to feature hinge-like anomalous surface states, as is discussed for specific cases in Refs.\ \onlinecite{Song17, Schindler17, Langbehn17,  Benalcazar17Aug, Fang17} and analyzed throughly in Sec.\ \ref{Sec:Surface}. In this section, we tie the bulk and surface perspectives  together and describe a concrete bulk-surface correspondence.

This is achieved by making an explicit connection between the specification of bulk symmetry representations and the surface mass-term analysis. In particular, we will show that all $X_{\rm BS}$-nontrivial phases have anomalous surface states (with suitably chosen sample geometry). However, the $X_{\rm BS}$ diagnosis and our surface states classification are not generally in one-to-one correspondence; rather, for most SG, different sTCIs become indistinguishable when one focuses only on symmetry representations, and therefore the same $X_{\rm BS}$ class is identified with multiple patterns of surface states. We will discuss the general structure behind such identification ambiguity, and in Table \ref{tab:SurfaceXBS_Full} in Appendix \ref{app:XBSRep} we provide the corresponding results for all SGs.

To proceed, we make a simplifying assumption: we suppose the topological properties of the strong TI of interest can always be analyzed through a $\vec k \cdot \vec p$ (bulk) Dirac Hamiltonian about $\Gamma$ \cite{Molenkamp, Schnyder09, Ryu10, Witten16}:
\begin{equation}\begin{split}\label{eq:DiracH}
\mathcal H_{\vec k } = M \gamma_z +  \vec k \cdot \vec \sigma \, \gamma_x,
\end{split}\end{equation}
where we assume the mass term changes from $m<0$ in the bulk to $m>0$ outside the system, leading to a Dirac point localized to the surface where $m=0$ (Appendix \ref{app:surfaceD}).
Here, $\gamma$ is a set of Pauli matrices describing the orbital degrees of freedom.

We start by considering the action of a spatial symmetry $g$, defined as in Sec.~\ref{Sec:DSTI}, on the bulk Dirac Hamiltonian (\ref{eq:DiracH}), given by $g \cdot \mathcal{H}_{{\vec k}} = U_g \mathcal{H}_{{\vec k}} U^\dagger_g  =  \mathcal{H}_{R_g \vec k} $. Here, $U_g \equiv u_g \oplus (\det R_g u_g )$, with $u_g$ defined as in  Sec.~\ref{Sec:DSTI}. Next, we connect the bulk characterization with the surface theory. As detailed in Appendix \ref{app:surfaceD}, for a surface with normal $\vechat n_{\vec r}$, the surface Hamiltonian is given by $h_{\vec r, \vec k} = (\vec k \times \vechat n_{\vec r})\cdot \vec \sigma$, which transforms under symmetry as $g \cdot h_{{\vec r},{\vec k}}= u_g h_{{\vec r},{\vec k}} u_g^\dagger =  h_{g\cdot \vec r, R_g \vec k} $. Recall that, in Sec.\ \ref{Sec:Surface}, we showed that when two strong TIs are stacked together the pattern of surface modes (if any) is determined by the signature of the symmetry representation, which can be expressed in terms of $\det R_g$ and the relative sign of the symmetry representation matrices across the two copies (cf.~Eq.~\ref{sg}). This means that, by specifying the (signed) symmetry representation matrices $u_g$ of the two TIs {\it in the bulk}, we can predict the pattern of the surface modes, thereby establishing the anticipated bulk-surface correspondence.

Such correspondence, however, utilizes more information than that available from the symmetry representations alone. To make connection with the symmetry-based indicators of band topology $X_{\rm BS}$ \cite{Po17}, we have to study what information is lost when we focus only on the symmetry representations.
In other words, generally the identification of the surface states associated with a nontrivial class $[\vec b] \in X_{\rm BS}$ is not unique.
Such nonuniqueness has two origins: either the choices  $\eta_g =\pm1$ correspond to the same representation, or certain nontrivial stacking patterns of strong TIs have symmetry representations compatible with an atomic insulator, and hence evades the $X_{\rm BS}$ diagnosis.

We illustrate these ideas from two concrete examples. Consider the space group $P2$ (SG 3), which is generated by lattice translation and a $C_2$ rotation about the $z$-axis. As discussed in Ref.\ \onlinecite{Fang17}, when we stack two strong TIs with opposite helicities, described respectively by the surface Hamiltonians $h_{\pm} =\pm k_y \sigma_x  -k_x \sigma_y $ and with $C_2$ represented by $u_{\pm, C_2} = i \sigma_z$, the resulting DSTI is nontrivial and features hinge modes when subjected to the $C_2$ symmetric, spherical open boundary conditions in Sec.\ \ref{Sec:Surface}.
To reconcile with the analysis based on Eq.\ \eqref{eq:DiracH}, we can perform a basis rotation
\begin{equation}\begin{split}\label{eq:}
\left\{
\begin{array}{rl}
h_- \mapsto&  \sigma_y ( - k_y \sigma_x  - k_x \sigma_y )  \sigma_y = k_y \sigma_x  - k_x \sigma_y  \\
u_{-, C_2}  \mapsto & \sigma_y (i \sigma_z)  \sigma_y =  - i \sigma_z
\end{array}
\right.,
\end{split}\end{equation}
which implies $\eta_{\pm, C_2} = \pm 1$ in our present framework. Eq.\ \eqref{sg} then correctly predicts that the DSTI $h_{+} \oplus h_-$ features anomalous surface states. However, as $\text{tr}(\pm i \sigma_z)=0$, the distinction between $h_{+} \oplus h_-$ and the trivial phase $h_{+} \oplus h_+$ cannot be diagnosed from symmetry representations alone. This is consistent with the result $X_{\rm BS} = 0$ found in Ref.\ \onlinecite{Po17} for $P2$ (SG 3).

As a second example, we consider $P4/m$ (SG 83) generated by the lattice translations and the point-group $C_{4 h}$. In Ref.\ \onlinecite{Po17}, it was found that the strong TI generates $\mathbb Z_{8} < X_{\rm BS}$, and in the following we chart out the surface states consistent with the DSTIs, which are described by the even subgroup in $\mathbb Z_8$.
As $C_{4 h}$ is generated by the four-fold rotation $C_4$ and inversion $\mathcal I$, both of them having non-vanishing trace, the nonuniqueness in the surface signature of $X_{\rm BS}$ is more subtle.
Recall from Sec.\ \ref{Sec:Surface} that, for $C_{4 h }$, the surface classification is given by $\mathbb Z \times \mathbb Z_2$, where the $\mathbb Z$ factor is generated by a DSTI with a mirror Chern mode protected by the mirror $M_z = \mathcal I C_4^2$, and the $\mathbb Z_2$ factor generated by a DSTI with only hinge modes. Let $\mathcal H_{ \eta_{\mathcal I}, \eta_{C_4}}$ be a  strong TI described by Eq.\ \eqref{eq:DiracH}.
From Eq.\ \eqref{sg}, we see that $\mathcal H_{-,-} \oplus \mathcal H_{-,-}$ has surface states described by $(1,0) \in \mathbb Z \times \mathbb Z_2$, whereas the surface of $\mathcal H_{-,-} \oplus \mathcal H_{-,+}$ is described by $(1,1)$. Note that, here, the helicity of the mirror Chern mode is fixed by $\eta_{M_Z} = \eta_{\mathcal I} \eta_{C_4}^2 = \eta_{\mathcal I}$.

Now, consider stacking four strong TIs, $\mathcal H_K = \mathcal H_{-,-} \oplus \mathcal H_{-,-} \oplus \mathcal H_{-,-} \oplus \mathcal H_{-,+}$. The surface remains nontrivial, and is classified by $(1,0)+(1,1)=( 2, 1)\in \mathbb Z \times \mathbb Z_2$, i.e., it features two helical modes at the equator, protected by $M_z$, together with the $C_4$-hinge mode. However, the symmetry representation of $\mathcal H_K$ happens to coincide with an atomic insulator (Appendix \ref{app:XBSRep}), and therefore the nontriviality of $\mathcal H_K$ is undetected from symmetry eigenvalues alone, i.e., $\mathcal H_K$ belongs to the trivial element in $X_{\rm BS}$. In other words, $X_{\rm BS}$ could at best resolve the surface signature in $\mathbb Z \times \mathbb Z_2$ modulo the subgroup generated by $\langle (2,1) \rangle$. From the line of arguments presented in Appendix \ref{app:XBSRep}, one can further show that $\langle (2,1) \rangle$ precisely captures the nonuniqueness in identifying the surface states associated with any $X_{\rm BS}$ class. For our present problem, a band insulator described by $2$ in the strong factor $\mathbb Z_8$ can have surface states described by, for instance, either $(1,0)$ or $(1,0) - (2,1) = (-1,1)$. Now, if we can stack these two systems together, we arrive at a bulk with symmetry indicator $4 \in \mathbb Z_8$ and a surface described by $(1,0) + (-1,1) = (0,1) \in \mathbb Z \times \mathbb Z_2$, which corresponds to a system with only hinge modes but not equatorial mirror Chern modes. This can be reconciled with the more detailed analysis in Ref.\ \onlinecite{Song17} and Sec.\ \ref{Sec:Surface}.

By a similar analysis, one can map out the surface-state ambiguity of $X_{\rm BS}^{\rm (s)}$ for any space group. We perform this study for the key space groups listed in Table \ref{tab:XBS}, and the results are tabulated in Table \ref{tab:XBS_BS}. In particular, by identifying the surface signature of the minimal DSTI in any SG, we see explicitly that a $X_{\rm BS}^{\rm (s)}$-nontrivial phase is either a strong TI or a sTCI. Together with the conventional indices we identified for the weak factors, this proves that all $X_{\rm BS}$-nontrivial phases possess anomalous surface states.

\begin{table}[h]
\begin{center}
\caption{{\bf Bulk-surface correspondence for the key space groups} $X^{\rm (s)}_{\rm BS}$ denotes the symmetry-based indicator generated by the strong topological insulator; $\mathcal S$ denotes the surface classification;
$\zeta_0$ denotes the generators of the subgroup of $\mathcal S$ which is consistent with the trivial entry in  $X^{\rm (s)}_{\rm BS}$; $\zeta_2$ indicates representative surface signatures for the minimal doubled strong topological insulator, corresponding to the entry $2 \in X^{\rm (s)}_{\rm BS}$
\label{tab:XBS_BS}}
\begin{tabular}{c|c|c|c|c}\hline \hline
SG & $X_{\rm BS}^{\rm s}$ & $\mathcal S$ &  $\zeta_0 \in \mathcal S$ & $ \zeta_2 \in  \mathcal S$\\
\hline
{2}  ($P\bar{1}$) & $\mathbb Z_4$ & $\mathbb Z_2 $& $( 0)$ & $(1) $
\\
{81} ($P\bar{4}$) & $\mathbb Z_2$ & $\mathbb Z_2$ & $( 1)$ & $(0) \simeq (1)$
\\
{82}  ($I\bar{4}$) & $\mathbb Z_2$ & $\mathbb Z_2$& $( 1)$ & $(0) \simeq (1)$
\\
{83}  ($P4/m$) & $\mathbb Z_8$ & $\mathbb Z \times \mathbb Z_2 $ & $ (2,1) $ & $(1,0) \simeq (-1,1)$
\\
{87} ($I4/m$) & $\mathbb Z_8$ &  $\mathbb Z \times \mathbb Z_2 $ & $ (2,1) $ & $(1,0) \simeq (-1,1)$
\\
{174} ($P\bar 6$) & $\mathbb Z_{3}$
\footnote{
The same entry can correspond to either a strong or doubled strong topological insulator; the surface classification $\mathcal{S}$ is only applicable to the latter} & $\mathbb Z $ & $(3) $ & $(1) \simeq (-2)$
\\
{175} ($P6/m$) & $\mathbb Z_{12}$ & $\mathbb Z \times \mathbb Z_2 $ & $ (3,1) $ & $(1,0) \simeq (-2,1)$
\\
{176} ($P6_3/m$)\footnote{Nonsymmorphic; surface states on cylinder geometry periodic in the direction parallel to the screw axis} & $\mathbb Z_{12}$ & $\mathbb Z \times \mathbb Z_2 $ & $ (3,1)  $ & $(1,0) \simeq (-2,1)$
\\
\hline \hline
\end{tabular}
\end{center}
\end{table}

In closing, we make two remarks regarding the generality of the present analysis, which relies on the Dirac model Eq.\ \eqref{eq:DiracH}.
First, in the above we have restricted our attention to symmetry representations of the form $ u_g =\eta_g v_g$. This does not represent the general case, since, for instance, the valence bands at a TRIM could furnish 4D irreps \cite{Bradley}.
Alternatively, certain symmetry representations lead to surface Dirac cones dispersing as $\sim k^3$ \cite{Fang17}, and therefore also fall outside of the Dirac description.
Strictly speaking, the Dirac Hamiltonian approach does not immediately inform the surface properties of such strong TIs. However, from the analysis discussed in Appendix \ref{app:XBSRep}, we found that, for any class in the strong factor $X_{\rm BS}^{\rm s}$, one can choose a representative whose symmetry representations agree with some atomic insulator everywhere in the Brillouin zone, except for the exchange of some irreps of the form $\eta_g v_g$ at $\Gamma$, i.e., it can be viewed as a stack of strong TIs  satisfying the simplifying assumptions adopted in the present analysis. In view of this, it is likely that our results are sufficient for establishing the essential connection between the $X_{\rm BS}^{\rm (s)}$ diagnosis and the (stable) surface signatures of DSTIs, but we will leave a more elaborated justification of this completeness conjecture to future works.

Second, we note that our approach can also be adapted for the study of weak phases upon relaxing the band inversion momentum from $\Gamma$ to a  general TRIM. For instance, a weak TI can be built by superposing two strong TIs whose surface Dirac cones respectively originate from, say, $(k_x,k_y,k_z) = (0,0,0)$ and $(0,0,\pi)$, which gives rise to the symmetry representation and surface states described in Sec.\ \ref{Sec:Surface}. We anticipate that a general analysis on these weak phases can be done upon the inclusion of such additional ``valley'' index on the surface Dirac cones. However, since the surface states associated with the weak phases can be readily understood from a limit of decoupled 2D systems \cite{Molenkamp, Wang16, Ezawa16,  Varjas15, Varjas17}, we refrain from performing this additional analysis.

\section{Discussion}
\label{Sec:Discussion}

\begin{figure}
\center
\includegraphics[width=0.35\columnwidth]{Cube}
\caption{Surface ``hinge'' mode protected by inversion symmetry on a cubic geometry, assuming no other spatial symmetries are present. The mode is confined to the edges of the cube since the surface mass term cannot change on a flat face.}
\label{Cube}
\end{figure}

In closing, we make several remarks concerning the connection between our present work and other recent ideas on the study of topological phases protected by crystalline symmetries.

We first note that the ``hinge'' surface states discussed in Sec.~\ref{Sec:Surface} are expected to be robust against weak perturbations which break their protecting crystalline symmetry, as suggested by Ref.~\onlinecite{Langbehn17}. The reason is that the local stability of the hinge modes relies only on TRS. As a result, the only way to remove them, assuming TRS is preserved, is to deform them to a point, which is only possible if the symmetry-breaking perturbation is large enough (larger than the maximum value of the surface gap). We also note that, when placed on a surface with planar faces e.g. a cube, the hinge modes are expected to be localized to the edges of the sample as shown in Fig.~\ref{Cube}, since the mass term is expected to remain constant on any flat surface. We point out that, although such geometries do not fit into our description of the surface in Sec.~\ref{Sec:Surface} as a smooth manifold with a tangent space defined everywhere, this does not pose any problem since we can always round the sharp edges into a (practically indistinguishable) smooth region with very large curvature.

It is worth reiterating that our results imply that the set of sTCIs strictly contains the set of phases which are $X_{\rm BS}$-nontrivial {\it but not} strong TIs (Fig.\ \ref{fig:Venn}). That is to say, all the non-trivial $X_{\rm BS}$ phases exhibit anomalous gapless surface states, which are either of the form of a surface Dirac cone, as for strong TIs, or are described by our sTCI surface classification. The set of sTCIs also strictly contains the set of conventional TCI phases with 2D surface Dirac cones \cite{Hsieh12, Tanaka12,   Wang16, Ezawa16, Varjas17, Wieder17,  Fang17}.
sTCIs without such 2D gapless surface states then feature only 1D helical hinge modes protected by inversion, rotoinversion or screw symmetries, whose existence can only be observed by considering the surface as a whole.

\begin{figure}[h]
\begin{center}
{\includegraphics[width=0.45 \textwidth]{FigDiscussion.pdf}}
\caption{
Hierarchy of crystalline topological insulators with time-reversal symmetry and strong spin-orbit coupling (class AII) in 3D. All nontrivial entries in the symmetry indicator, $X_{\rm BS}$, are compatible with gapped topological band structures with gapless surface states on suitably chosen sample geometries.
Excluding the strong topological insulator from the $X_{\rm BS}$-nontrivial phases, we found that all the remaining phases are anomalous surface topological crystalline insulators (sTCI), defined as nontrivial insulators with anomalous surface states but can be trivialized upon the breaking of some crystalline symmetries.
There are also examples of sTCIs not diagnosed by $X_{\rm BS}$, i.e., the topological nature is not exposed by the symmetry eigenvalues in the bulk.
\label{fig:Venn}
 }
\end{center}
\end{figure}

Next, we make connections between our results and more general ideas concerning crystalline SPT classifications.
Weak TIs, which can be understood in terms of spatially stacking 2D TIs using lattice translation symmetry,  exemplify the notion of  topological crystalline insulators. Curiously, on more general grounds it has been suggested that many, if not all, crystalline symmetry-protected topological phases can be understood in terms of a similar construction, upon the replacement of lattice translation by other crystalline symmetries \cite{Varjas17, Thorngren16,Fang17,Hermele17, Huang17,XiongGlide}. Here, we remark that a similar construction is also applicable to the DSTIs. To illustrate this, consider a space group with only lattice translation and a mirror symmetry $z \leftrightarrow - z$. The $z \in  \mathbb Z$ and $z\in (1/2 + \mathbb Z)$ planes are mirror-invariant, and if we pin a quantum spin Hall insulator to each of these planes, we will arrive at a band insulator which does not immediately yield an atomic limit. However, such an insulator does not possess a weak index; rather, we should view it as interlacing two weak TIs built from stacking 2D TIs with mirror-Chern number $+1$, one from pinning them to the $z=0$ plane and the other at the $z=1/2$ planes. Computing the mirror Chern numbers, one finds that the described band insulator has mirror Chern number $2$ on the $k_z=0$ plane and $0$ on the $k_z = \pi$ plane, i.e., it can also be understood as a DSTI. Schematically, we may write ``DSTI = weak-$0$ + weak-$1/2$.''

The above picture applies to the SnTe-class of mirror-Chern TCI \cite{Isobe15, Tanaka12, Hsieh12} and is, in fact, quite general: it applies to a large class of DSTIs arising in settings where the real-space deformation of ``weak-$0$'' to ``weak-${1/2}$'' is forbidden by symmetry, as in the case when inversion \cite{Po17} and/ or nonsymmorphic \cite{Wang16, Ezawa16, Varjas17, XiongGlide} symmetries are present. In particular, we note that the glide-protected TCI showcasing the ``hourglass fermion'' surface states also falls into this category \cite{Wang16, Ezawa16, XiongGlide}.

It is also conceptually revealing to abstract our sTCI surface classification from the Dirac equation approach we have employed. Let $P$ be the group of spatial symmetries respected by the open boundary condition, e.g., in the filled-sphere geometry $P$ is the subgroup of the point-group which leaves the origin invariant.
In essence, for every spatial symmetry $g \in P$ we assign a signature $s_g = \pm 1$, which conforms to the group multiplication $s_{g g'} = s_g s_{g'}$.
Different assignment of signature can then be viewed simply as different homomorphisms from $P \rightarrow \mathbb Z_2$, which are classified by the first cohomology group of $P$ with $\mathbb Z_2$ coefficients, $H^{1}(P, \mathbb Z_2)$.

This interpretation is similar to the dimension-reduction approach advocated in Refs.\ \onlinecite{Hermele17,Huang17}, except that, due to the presence of additional internal symmetries (time-reversal for the case of DSTI), we do not fully reduce the classification to the symmetry charges at the point fixed by $P$; rather, we picture the construction of 3D phase by interlacing 2D SPTs \cite{Fang17}, and therefore the 2D classification ($\mathbb Z_2$ for class AII) enters into the coefficient of $H^1$. One can imagine a similar analysis for DSTI-like phases in a more general setting, where we replace $\mathbb Z_2$ by the appropriate lower-dimensional classification $X$ (an Abelian group), and incorporate a group action of $P$ on $X$ (orientation reversing elements in $P$ reverses the phase in $X$). However, this provides only a partial understanding on the full classification. For instance, in the presence of mirror symmetries, the DSTIs can feature $\mathbb Z$-valued mirror Chern modes (neglecting interactions) at the surface, whereas the $H^1$ description captures only the parity of the number of modes. Such discrepancy arises because the mirror symmetry is effectively reduced to an onsite unitary symmetry on the mirror-invariant plane, which enriches the classification \cite{Hermele17, Huang17}. We leave the analysis of the general structure of such classifications as an interesting open problem.

Lastly, we make a few remarks about the spin-orbit-coupling-free case (class AI), whose indicators were also obtained in Ref.~\onlinecite{Po17}.
In the absence of spin-orbit coupling, a new possibility that may be mandated by a nontrivial indicator is a gapless (semimetallic) phase (dubbed representation-enforced semimetals in Ref.\ \onlinecite{Po17}) \cite{KaneMele1, KaneMele2, Kim15, Yu15, Bian16}, and we provide an explicit example of such this scenario in the following.

An example of a semimetallic phase diagnosed by $X_{\rm BS}$ in class AI is provided by SG $P\bar 1$ (No.\ 2), where the only point-group symmetry is inversion, whose ``strong factor'' is $\mathbb{Z}_4$. The generator of such a factor has the same inversion eigenvalues at the TRIMs as the strong TI, and it can be thought of as the limit of a strong TI when the spin-orbit coupling is adiabatically switched off. The corresponding phase is a Dirac nodal ring semimetal with so-called drumhead surface states \cite{Kim15, Yu15, Bian16}. Its low-energy physics is captured by the effective spinless two-band Hamiltonian $\mathcal{H}_{\vec k} = k_z \gamma_y + (k_x^2 + k_y^2 - k_0^2) \gamma_z$, which describes a nodal ring centered at ${\vec k} = 0$ with radius $k_0$. Here, $\gamma$ denotes the Pauli matrices describing the orbital degree of freedom. $\mathcal{H}_{\vec k}$ is invariant under spinless TRS represented by charge conjugation and inversion symmetry represented by $\gamma_z$. A system constructed by stacking two copies of this Hamiltonian can be gapped out in the bulk by a mass term of the form $m_{\vec k} \gamma_x \tau_y$. Inversion, however, would require $m_{\vec k}$ to satisfy $m_{-{\vec k}} = -m_{\vec k}$, which implies that the mass has to vanish at least twice along the nodal ring, leaving behind two bulk Dirac nodes at generic, but inversion-related, momenta. This pair of Dirac nodes cannot be removed without breaking the symmetries \cite{ Ari, Hughes11} or changing the symmetry class (say by introducing spin-orbit coupling), and hence we conclude the $2 \in \mathbb{Z}_4$ entry in the $X_{\rm BS}$ diagnosis is enforced to be semimetallic by the symmetry representations.
We will leave the general question of whether any of the $X_{\rm BS}$ entries for any space group realizes a gapped topological phase in class AII (with or without surface states) for future works.

\begin{acknowledgements}
The authors would like to acknowledge stimulating discussions with P.~W.~Brouwer, M.~Geier, A.~P.~Schnyder. We thank C.~Fang for informing us of  their upcoming works (to appear), which also study the physical interpretation of phases with nontrivial symmetry indicators.  AV and HCP were supported by NSF DMR-1411343. AV acknowledges support from a Simons Investigator Award and ARO MURI program W911NF-12-1-0461.
HW acknowledges support from JSPS KAKENHI Grant Number JP17K17678.
\end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


text:
A judge ordered Tijuana's mayor not to badmouth migrants. He's appealing
A Mexican federal judge has ordered Tijuana's mayor not to spread negative messages about migrants.Ma...
Posted: Jan 6, 2019 5:08 AM
Updated: Jan 6, 2019 5:08 AM
A Mexican federal judge has ordered Tijuana's mayor not to spread negative messages about migrants.
Mayor Juan Manuel Gastelum says he's appealing the ruling, which he argues violates his right to free speech.
Government bodies and offices
Heads of government
Immigration, citizenship and displacement
International relations and national security
Gastelum made international headlines when he spoke out as thousands of migrants who'd been traveling in large groups known as caravans began arriving in his city.
He warned in November that the situation was a crisis and asked for the United Nations to intervene. And he also made more controversial comments about migrants, describing some in the group as violent and vulgar, vowing that local authorities would crack down on anyone who was caught breaking the law.
Immigrant rights advocates decried what they said was nativist rhetoric from the mayor, accusing him of fueling tensions rather than easing them. Gastelum has maintained he's simply standing up for the rights of Tijuana's residents.
Last week, a Mexican district court judge ruled that Gastelum can't make any public declarations that have "as a goal -- either implicit or explicit -- transmitting a negative message about migrant people," according to a government statement describing the ruling. The court ordered municipal authorities in Tijuana to "refrain from issuing statements contrary to the protection of and respect for migrant people."
The mayor revealed the judge's order in a Facebook post, calling it censorship.
A spokeswoman for the mayor's office told CNN Friday that the judge's order came after a civil association filed a complaint. Gastelum formally appealed the order on Wednesday, the spokeswoman said, and is awaiting an appeals court's decision.
Thursday, the mayor shared another Facebook post vowing to press forward with the appeal.
"I will appeal the order given by the federal judge that tries to censor our freedom of expression," he wrote. "Here there is a mayor that defends Tijuana and a lawyer who knows the law well. I will keep you posted."
Migrants begin arriving at new Tijuana shelter
Main migrant shelter in Tijuana closed 'due to health issues'
AMLO: entre Tapachula y Tijuana
Primera ola de migrantes de la caravana llega a Tijuana, en la frontera con Estados Unidos
Al menos 500 personas de la caravana migrante ya están en Tijuana
New shelter in Tijuana to house at least 3,000 migrants, official says
"Fuera hondureño, aquí no te queremos": Tijuana, la ciudad que no quiere a la caravana de migrantes
A sports complex is housing thousands of migrants in Tijuana. It's at least 3 times above its capacity.
México: Asesinan a dos jóvenes hondureños en Tijuana

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