# Revealing a topological connection between stabilities of Fermi surfaces and
topological insulators/superconductors

Y. X. Zhao<EMAIL_ADDRESS>Z. D. Wang<EMAIL_ADDRESS>Department of Physics and
Center of Theoretical and Computational Physics, The University of Hong Kong,
Pokfulam Road, Hong Kong, China

###### Abstract

A topology-intrinsic connection between the stabilities of Fermi surfaces
(FSs) and topological insulators/superconductors (TIs/TSCs) is revealed. In
particular, a one-to-one relation between the topological types of FSs and
TIs/TSCs is rigorously derived; combining it with a well-established
topological theory of FSs, we produce a complete table illustrating precisely
topological types of all TIs/TSCs, while a valid part of it was postulated
before. Moreover, we propose and prove a general index theorem that relates
the topological charge of FSs on the natural boundary of a strong TI/TSC to
its bulk topological number. Implications of the general index theorem on the
boundary quasi-particles are also addressed.

###### pacs:

03.65.Vf, 71.10.-w, 73.20.-r

For the past several years, the research on topological insulators (TIs) and
superconductors (TSCs) has greatly attracted both theoretical and experimental
interests, which becomes a hot spot of contemporary physics Kane-RMP ; XLQi-
RMP . As is known, various kinds of TIs/TSCs, which possess the time-reversal
symmetry(TRS) or/and particle-hole symmetry(PHS), and chiral symmetry(CS), may
be realised whenever the symmetry constraints on the bulk topology are
enforced, while these symmetries can be preserved in the presence of weak
disorders from a viewpoint of random matrix theory Ramdom Matrix 0 ; Ramdom
Matrix I ; Ramdom Matrix II . A TI/TSC, while it is fully gapped in the bulk
spectrum, can have robust gapless modes on its boundary, being distinctly
different from those in an ordinary insulator or superconductor. According to
a conventional wisdom, these gapless modes are originated from a fact that the
boundary of a TI/TSC as a domain wall separates the system in different
topological phases, with topological order-parameters being identified as the
bulk topological numbers. As it is generally believed that there exists a
faithful bulk-boundary correspondence, the above wisdom is practically useful,
while it is likely to provide a qualitative insight on topology merely from
the bulk to boundary, rather than a complete and quantitative description of
this topological correspondence. On the other hand, stimulated by Volovik’s
pioneering work on topological Fermi surfaces (FSs) without any discrete
symmetry Volovik-Book , a topological theory of FSs has recently been
established for describing many-fermion systems with TRS or/and PHS, where a
nontrivial topological charge of an FS with a specific symmetry ensures its
topological stability against symmetry-preserving weak disorders/perturbations
and interactions FS-Classification . Therefore, it is natural and insightful
to look into directly the topological properties of boundary FSs of a TI/TSC
and to reveal quantitatively their intrinsic connection to the bulk topology,
particularly considering that the robustness of these gapless boundary modes
may have the same topological origin as that of the bulk. This new sort of
boundary-to-bulk insight on the topological essence of TIs/TSCs may further
deepen our understanding of the bulk-boundary correspondence, supplementing
the conventional one in a more efficient way because only information in the
vicinity of FSs is actually needed to disclose the topological character.

In this Letter, based on a well-established topological theory of FSs FS-
Classification , we first derive a one-to-one relation between the topological
types of FSs and TIs/TSCs, from which we establish a complete table (Tab.[1]
in a right-to-left manner) that illustrates precisely topological types of all
TIs/TSCs vs the bulk dimension and symmetry class. Intrigued by this relation,
we propose a general index theorem [Eq.(2)] as a quantitive description for
the topological boundary-bulk correspondence of TIs/TSCs and prove it in the
framework of Dirac-matrix construction. Finally, we address briefly the
predictions and restrictions on the boundary low-energy effective theories of
TIs/TSCs.

Let us begin with a brief introduction to topological FSs. An FS is actually a
region of fermionic gapless modes in the $\mathbf{k}$-space of energy
spectrum. Some FSs are stable against disorders/perturbations, while some
others are vulnerable and easy to be gapped Volovik-Book . It is found that
the topological charge of an FS is responsible for its stability Volovik-Book
; Horava ; Volovik-Vacuum ; FS-Classification . For a $(d_{s}-p)$-dimensional
FS in a $d_{s}$-dimensional $\mathbf{k}$-space, we can choose a
$p$-dimensional sphere $S^{p}$ from $(\omega,\mathbf{k})$-space to enclose it
from its transverse dimensions, where $p$ is referred to as the codimension of
the FS. Note that the spectrum is gapped on the whole $S^{p}$ since the
$S^{p}$ is constructed in the transverse dimensions of an FS. For an FS
without any discrete symmetry, its topological charge is given by the homotopy
number of the inverse Green’s function restricted on the $S^{p}$ Volovik-Book
; Horava ; Volovik-Vacuum , i.e.,
$G^{-1}(\omega,\mathbf{k})|_{S^{p}}=[i\omega-\mathcal{H}(\mathbf{k})]|_{S^{p}}$.
This idea has recently been generalized to FSs of the other classes for
characterising the corresponding types of symmetry-dependent topological
charges FS-Classification , as summarized in Tab.[1] (in a left-to-right
manner), where the $\mathbf{Z}$, $\mathbf{Z}^{(1)}_{2}$, and
$\mathbf{Z}^{(2)}_{2}$ denote the integer-valued topological charge,
$\mathbf{Z}_{2}$-valued (integers of modular $2$, i.e., $0$ or $1$ )
topological charge for the first descendant of a $\mathbf{Z}$-type, and
$\mathbf{Z}_{2}$-valued topological charge for the second descendant of a
$\mathbf{Z}$-type.

FS | AI | BDI | D | DIII | AII | CII | C | CI | $\frac{\mathrm{TI}}{\mathrm{TCS}}$
---|---|---|---|---|---|---|---|---|---
T | +1 | +1 | 0 | -1 | -1 | -1 | 0 | +1 | T
C | 0 | +1 | +1 | +1 | 0 | -1 | -1 | -1 | C
S | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | S
p\i | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | i/d
1 | 0 | 0 | $\mathbf{Z}$ | $\mathbf{Z}_{2}^{(1)}$ | $\mathbf{Z}_{2}^{(2)}$ | 0 | $2\mathbf{Z}$ | 0 | 2
2 | 0 | 0 | 0 | $\mathbf{Z}$ | $\mathbf{Z}_{2}^{(1)}$ | $\mathbf{Z}_{2}^{(2)}$ | 0 | $2\mathbf{Z}$ | 3
3 | $2\mathbf{Z}$ | 0 | 0 | 0 | $\mathbf{Z}$ | $\mathbf{Z}_{2}^{(1)}$ | $\mathbf{Z}_{2}^{(2)}$ | 0 | 4
4 | 0 | $2\mathbf{Z}$ | 0 | 0 | 0 | $\mathbf{Z}$ | $\mathbf{Z}_{2}^{(1)}$ | $\mathbf{Z}_{2}^{(2)}$ | 5
5 | $\mathbf{Z}_{2}^{(2)}$ | 0 | $2\mathbf{Z}$ | 0 | 0 | 0 | $\mathbf{Z}$ | $\mathbf{Z}_{2}^{(1)}$ | 6
6 | $\mathbf{Z}_{2}^{(1)}$ | $\mathbf{Z}_{2}^{(2)}$ | 0 | $2\mathbf{Z}$ | 0 | 0 | 0 | $\mathbf{Z}$ | 7
7 | $\mathbf{Z}$ | $\mathbf{Z}_{2}^{(1)}$ | $\mathbf{Z}_{2}^{(2)}$ | 0 | $2\mathbf{Z}$ | 0 | 0 | 0 | 8
8 | 0 | $\mathbf{Z}$ | $\mathbf{Z}_{2}^{(1)}$ | $\mathbf{Z}_{2}^{(2)}$ | 0 | $2\mathbf{Z}$ | 0 | 0 | 9
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
Table 1: Cartan Classification of systems, and topological classification of
FSs and TIs/TSCs FS-Classification . In the upper part of the table, T, C, and
S denote TRS, PHS, and CS, respectively. 0 indicates the absence of the
corresponding symmetry, $\pm$1 indicates the sign of TRS or PHS, and 1
indicates the existence of CS. In the lower part, $i$ is the index of symmetry
classes, $p$ is the codimension of an FS, and $d$ is the spatial dimension of
a TI/TSC. The elements $\mathbf{Z}$, $2\mathbf{Z}$, $\mathbf{Z}^{(1)}_{2}$,
and $\mathbf{Z}^{(2)}_{2}$ in the above periodic (eight-fold) table represent
the corresponding topological types, respectively.

We now derive the relation between the classification of FSs and that of
TIs/TSCs, as indicated in Tab.[1]. First, It is crucial to observe that all
six formulas for the topological charges of FSs can formally be used to
calculate the topological numbers of TIs/TSCs, where we make the integration
over the whole $(\omega,\mathbf{k})$-space in these formulasXLQi-RMP ;
Topological-Number-I ; Topological-Number-II , in stead of making it over
$S^{p}$ for an FS, as illustrated in Supplemental Material Supp .

Figure 1: Symmetry Identification. The left corresponds to a two-dimensional
$\mathbf{k}$-space, and the right corresponds to an $S^{2}$ enclosing a Fermi
point in a three-dimensional $\mathbf{k}$-space.

We emphasise here that the operation of TRS and PHS on the
$(\omega,\mathbf{k})$-space of a TI/TSC is essentially different from that on
an $S^{p}$ for an FS, as shown in Fig.[1]. If the symmetry operation is
regarded as an identification, the resulting topological space of $S^{p}$ for
an FS is a $p$-dimensional projective space $RP(p)$ rp , while that of
$\mathbf{k}$ space is not. This topological difference of symmetry operation
leads to that the topological types of TIs/TSCs vs. the dimension $d$ have
globally a one-dimension shift with respect to those of FSs vs. the
codimension $p$, $i.e.$,

$K_{TI}(d,i)=K_{FS}(d-1,i),$ (1)

where $K_{FS}(p,i)$ and $K_{TI}(d,i)$ denote, respectively, the topological
types of FSs (with the codimension $p$) and TIs/TSCs (with the spatial
dimension $d$) in the $i$th class. Without loss of generality, we below
consider class AII ($i=5$) and highlight the key steps in the derivation of
this relation. It is known that in the absence of any discrete symmetry, FSs
with nontrivial topological charges can exist only when $p=2n-1$ with $n$
being an integer Volovik-Book ; Horava , and TIs/TSCs can have nontrivial
topological numbers only when $d=2n$. After operations of the minus-sign TRS
on $S^{p}$ and $(\omega,\mathbf{k})$ space Supp , we obtain the following
relations for FSs and TI/TSCs:

$\begin{cases}\nu_{Z}(2n+1,5)=(-1)^{n-1}\nu_{Z}(2n+1,5)\\\
N_{Z}(2n,5)=(-1)^{n}N_{Z}(2n,5)\end{cases},$

where $\nu_{Z}(p,i)$ and $N_{Z}(d,i)$ denote respectively the topological
charge of an FS (with the codimension $p$) and the topological number of a
TI/TSC (with the dimension $d$) in the $i$th class if they are of
$\mathbf{Z}$-type. It is found for class AII that nontrivial $\mathbf{Z}$-type
(or $2\mathbf{Z}$-type) FSs can exist only when $p=4m-1$, while there are
nontrivial $\mathbf{Z}$-type (or $2\mathbf{Z}$-type) TIs/TSCs only if $d=4m$,
where $m$ is an integer. Since $\mathbf{Z}^{(1)}_{2}$ and
$\mathbf{Z}^{(2)}_{2}$, as the first and second descendants of $\mathbf{Z}$,
will follow $\mathbf{Z}$ successively when $d$ is reduced, our remaining task
is to distinguish types of $\mathbf{Z}$ and $2\mathbf{Z}$ for TIs/TSCs, which
can be done for class AII as follows: noticing that FSs are of
$\mathbf{Z}$-type when $p=8(m-1)+3$, while TIs/TSCs, whose Hamiltonians can be
constructed by Dirac matrices, possess unit topological number (i.e.,
$\mathbf{Z}$-type) for $d=8(m-1)+4$ Topological-Number-I ; Supp . Thus we have
$K_{TI}(d,5)=K_{FS}(d-1,5)$. Since similar derivations can be made for the
other seven classes (not shown here), Eq.(1) is rigorously deduced and
therefore the classification table of TIs/TSCs is established, as presented in
Tab.[1] in a right-to-left manner, which is one of our main results.

Notably, a similar classification table of TIs/TSCs was first postulated by
generalizing a Pruiskein’s theory for the integer quantum hall
effect(IQHE)(without any above-mentioned symmetry) to the other symmetry
classes IQHE ; TIClassification-I ; TIClassification-II ; TIClassification-III
. The Pruiskein’s theory is a non-linear $\sigma$ model handling disorders,
which attributes the stability of the gapless boundary modes to a topological
Wess-Zumino-Witten term on the boundary IQHE ; WZ-term . As a specific
symmetry class has a unique group for its non-linear sigma model Ramdom Matrix
0 ; Ramdom Matrix I ; Ramdom Matrix II , the classification in Tab[1] was
postulated according to whether eligible topological terms can appear on the
boundary of a system in each case Topo-Terms . While in the present work,
first, it is shown rigorously that all types of the bulk topological numbers
are just those (with one-to-one correspondence) for topological charges of FSs
identified in Ref. FS-Classification , as seen in Eq.(1); secondly, the
periodic table for TIs/TSCs is refined by classifying the two different
topological types of $\mathbf{Z}^{(1,2)}_{2}$, enabling us to choose a
corresponding formula for calculating the topological charges correctly; while
no such distinction can be seen from the existing results TIClassification-I ;
TIClassification-II ; TIClassification-III ; K-theory .

It has been known from many examples that the boundary gapless modes of a
given TI/TSC with one (or more) above symmetry are robust against
disorders/perturbations that do not break the corresponding symmetry. This
implies that the FSs corresponding to these gapless modes are actually
protected by their nontrivial topological charge with the same symmetry class
according to a theory of topologically stable FSs FS-Classification , while
such robustness can also be attributed to a nontrivial topological number of
bulk. Therefore, we attempt to find a quantitative relation between the
topological number $N$ of a bulk TI/TSC and the topological charge $\nu$ of
its boundary FSs. Let us consider a $d$-dimensional semi-infinite TI/TSC of a
symmetry class $i$ with a boundary being at $x=0$ on its left-side. Under the
natural boundary condition of the TI/TSC, i.e., no dramatic anisotropy is
induced and the symmetries in the $i$th class are preserved when the system
approaches to its boundary, FSs on the boundary will always be some Fermi
points in a topological sense. This conclusion is obviously valid for $d=1,2$.
In addition, its validity can also be extended to TIs/TSCs for $d>2$ note . As
a result, the boundary codimension $p=d-1$ . Then by appropriately choosing an
$S^{d-1}$ in the $(\omega,\mathbf{k})$ space of the boundary to enclose all
the FSs, we are able to compute the total topological charge $\nu(d-1,i)$ of
all FSs. Therefore, from Eq.(1), which enables the same type of nontrivial
topological charge of boundary FSs and bulk number to protect the FSs, we may
expect a general index theorem as a quantitive topological description of the
boundary-bulk correspondence of a TI/TSC:

$\nu(d-1,i)=N(d,i).$ (2)

Clearly, the above equation reflects essentially that the same topological
origin protects both the stability of FSs on the boundary and the gapped
spectrum of bulk against disorders/perturbations. We note that Eq.(2) accounts
for the earlier statements for classification of TIs/TSCs based on the random-
matrix theory by assuming certain boundary topological terms in a non-linear
$\sigma$ model TIClassification-I ; TIClassification-II ; TIClassification-III
, and reveals that the bulk topology can also be quantified by the topological
charge of boundary FSs. Notably, a similar index theorem for systems without
any symmetry was given in Ref.Volovik-Book , and has been generalized to other
cases with either physical arguments or strict verifications for some concrete
models Index-I ; Index-II ; Dirac-Models , though the topological charge of
FSs was not used explicitly.

To see Eq.(2) more clearly, we first present two typical examples, with the
proof to be given later. Consider a 3-dimensional TI described by the Dirac-
lattice model Topological-Number-I ; Supp , $i.e.$
$\mathcal{H}_{D}=\sum^{3}_{i=1}\sin k_{i}\Gamma^{i+2}+(3+\sum_{i=1}^{3}\cos
k_{i})\Gamma^{1}$ with $\Gamma_{i}(i=1,\cdots,5)$ being $4\times 4$ hermitian
Dirac matricesDirac-Matrix ; Supp  and $T=\Gamma^{1}\Gamma^{3}$. This model
has TRS with a minus sign and thus may have a non-trivial
$\mathbf{Z}_{2}^{(1)}$-type, as seen from Tab.[1] $(d=3)$. On the other hand,
its boundary effective model reads
$\mathcal{H}=k_{z}\sigma_{2}-k_{y}\sigma_{3}$ with $\sigma_{i}$ being Pauli
matrices Dirac-Models , which preserves the same TRS and has a Fermi point
with codimension $p=2$. One can find that $\nu(2,5)=1=N(3,5)$ Supp , verifying
Eq.(2). Similarly, another 2-dimensional model, which describes the quantum
spin Hall effect, is given by $\mathcal{H}_{spin}=\sum^{2}_{i=1}\sin
k_{i}\Gamma^{i+3}+(3+\sum_{i=1}^{2}\cos k_{i})\Gamma^{1}$. It has also TRS
with a minus sign and therefore belongs to class AII with a
$\mathbf{Z}_{2}^{(2)}$-type topological number. Its boundary effective
Hamiltonian is $\mathcal{H}=\sigma_{z}k_{z}$, which has a Fermi point ($p=1$)
with the same $\mathbf{Z}_{2}^{(2)}$-type topological charge. We can also find
that $\nu(1,5)=1=N(2,5)$ Supp .

At this stage, we turn to brief readers the proof of the general index
theorem: Eq.(2), with more details being presented in Ref.Supp . First, it is
seen from the topological nature of a TI/TSC that the total topological charge
of FSs on its boundary and the topological number of its bulk are both
invariant under continuous deformations of its Hamiltonian without closing the
bulk spectrum gap. Secondly, it is noted that Hamiltonian of each type of FS
(point) for a given codimension $p$ and class $i$ with a unit charge (or
double-unit charge for $2\mathbf{Z}$ type) can be expressed by Dirac matrices
in a unified form:

$\mathcal{H}_{FS}(\mathbf{k})=\sum_{a=1}^{p}k_{a}\Gamma_{(2n+1)}^{a+b},$ (3)

where $\Gamma^{a}_{(2n+1)}$ are $2^{n}\times 2^{n}$ Dirac matrices, and $b$ is
an integer that is specified by the corresponding topological type of the FS.
Thirdly, a modified Dirac-type model for each type of bulk TI/TSC with unit
topological number(or double-unit for $2\mathbf{Z}$ type) can also be
constructed in a unified form:

$\mathcal{H}_{TI}(\mathbf{k})=\sum_{a=1}^{d}k_{a}\Gamma^{a+b-1}_{(2n+1)}+\left(m-\epsilon
k^{2}\right)\Gamma^{\alpha}_{(2n+1)},$ (4)

where $\epsilon$ is a constant,
$\Gamma^{\alpha}=i\Gamma^{1}\Gamma^{2}\Gamma^{3}$ for $2\mathbf{Z}$ cases and
$\Gamma^{\alpha}=\Gamma^{1}$ otherwise. Note that $\epsilon$ term is necessary
for a prescription of the singularity of $\mathcal{H}_{TI}$ when $k$
approaches to infinity Volovik-Book . This model has a unified expression of
the topological number for all types of TIs/TSCs:

$N(d,i)=(1\,\,\,or\,\,\,1/2)(\mathbf{sgn}(m)+\mathbf{sgn}(\epsilon)),$ (5)

where the coefficient is $1$ for $2\mathbf{Z}$ cases, and $1/2$ otherwise.
Next, we proceed to consider an arbitrarily given TI/TSC in a certain type
with the topological number $N(d,i)$, which can be deformed continuously to a
model that is $N$-multiple (or for $2\mathbf{Z}$ cases, $(N/2)$-multiple) of
the model (4), preserving the corresponding symmetries and without closing the
bulk spectrum gap. After this deformation, both the bulk topological number
and the total topological charge of boundary FSs are preserved, as indicated
above Note-Deformation , and therefore it is sufficient to consider the unit
model (4) for proof of Eq.(2). The advantage of this deformation lies in that
the boundary low-energy effective theory of the model (4) can be
systematically derived by using a standard perturbation theory of quantum
mechanics under the open boundary conditionSupp . After some tedious
derivations detailed in Ref. Supp , the boundary effective theory of the model
(4) turns out to be

$\mathcal{H}_{eff}=\frac{1}{2}\left(\mathbf{sgn}(m)+\mathbf{sgn}(\epsilon)\right)\sum_{a=2}^{d}k_{a}\Gamma_{(2n-1)}^{a+b-3},$
(6)

for the boundary at $x=0$. Note that the symmetries of the system are now
represented in the boundary effective theory (6). By matching the symmetry
representation of the theory (6) to that of our constructed model (3), we find
that not only the topological charge of Fermi point in the boundary effective
theory (6) is equal to the topological number of the model (4), but also the
topological type of the FS is the same as that of the bulk model (4), namely,
the general index theorem is validatedSupp .

Before concluding this paper, we address briefly implications of Eq.(2) on the
form of a boundary effective theory with a given bulk topological number.
Based on the Atiyah-Bott-Singer(ABS) construction in K-theory and a fact that
an FS of multiple charge can be perturbed to a more stable state consisting of
a number of unit FSsK-theory-II ; K-theory-III ; Horava , we assert that a
typical low-energy effective theory of boundary for a $\mathbf{Z}$ type TI/TSC
is a collection of $\sum_{i=1}^{2n+1}k_{i}\Gamma_{(2n+1)}^{i}$ for non-chiral
cases or $\sum_{i=1}^{2n}k_{i}\Gamma_{(2n+1)}^{i}$ for chiral cases, with the
total topological charge being equal to the bulk topological number. On the
other hand, for $\mathbf{Z}^{(1,2)}_{2}$ type TIs/TSCs, the constraint from
ABS construction is not as strong as that of the above cases, while the
constraint on $\mathbf{Z}^{(1)}_{2}$ type is stronger than that on
$\mathbf{Z}^{(2)}_{2}$ type, since the latter needs one more extra parameter.
Nevertheless, a boundary effective theory can still be constructed by Dirac
matrices, whose expression is not uniquely determined. For instance,
$\mathcal{H}=k_{z}\sigma_{3}+\alpha k^{(2n+1)}_{z}+\beta k^{(2m+1)}_{z}$ and
$\mathcal{H}=k^{3}_{z}\sigma_{3}+\alpha k^{(2n+1)}_{z}+\beta k^{(2m+1)}_{z}$
are both eligible as the boundary effective theory of a quantum spin hall
system.

In summary, the topology-intrinsic connection between the stabilities of FSs
and TIs/TSCs has been revealed. Furthermore, we have proposed and proven a
general index theorem, providing for the first time a quantitive description
of the bulk-boundary correspondence of TIs/TCSs. It is anticipated that the
present work will also provide a deeper insight and open a wider door for
exploring exotic boundary gapless modes of TIs/TSCs such as Majorana fermions.

###### Acknowledgements.

We thank A. J. Leggett and G. E. Volovik for helpful discussions. This work
was supported by the GRF (HKU7058/11P&HKU7045/13P), the CRF (HKU8/11G) of Hong
Kong, and the URC fund of HKU.

## References

  * (1) M.Z.Hasan and C. L.Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * (2) X. L. Qi and S. C. Zhang, Rev. Mod. Phys 83, 1057 (2011).
  * (3) M. R. Zirnbauer, J. Math. Phys. 37, 4986 (1996).
  * (4) A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 (1997).
  * (5) D. Bernard and A. LeClair, J. Phys. A: Math. Gen. 35, 2555 (2002).
  * (6) G. E. Volovik, The Universe in a Helium Droplet (Clarendon, Oxford, 2003).
  * (7) Y. X. Zhao and Z. D. Wang, Phys. Rev. Lett.110, 240404 (2013).
  * (8) G. E. Volovik, arXiv:1111.4627 (2011).
  * (9) P. Hořava, Phys. Rev. Lett 95, 016405 (2005).
  * (10) X. L. Qi, T. L. Hughes, and S. C. Zhang, Phys. Rev. B 78, 195424 (2008).
  * (11) Z. Wang, X. L. Qi and S. C. Zhang, Phys. Rev. Lett. 105, 256803 (2010).
  * (12) Supplemental Material.
  * (13) M. Nakahara, Geometry, Topology, and Physics (Hilger, London, 1990).
  * (14) A. M. M. Pruisken, Nucl. Phys. B 235, 277 (1984).
  * (15) S. Ryu, A. P. Schnyder, A. Furusaki and A. W. W Ludwig, New J. Phys. 12, 065010 (2010).
  * (16) A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, AIP Conf. Proc. 1134, 10 (2009).
  * (17) A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B 78, 195125 (2008).
  * (18) E. Witten, Nucl. Phys. B 223, 422 (1983).
  * (19) Eligible topological terms in Ref.TIClassification-I ; TIClassification-II ; TIClassification-III  consisting of WZ-term and $\mathbf{Z}_{2}$ $\theta$-term have no adjustable parameters.
  * (20) J. Teo and C. L. Kane, Phys. Rev. B 82, 115120 (2010).
  * (21) To elaborate the point more clearly, let us consider a $d$-dimensional TI/TSC with the two natural boundaries at $x_{1}=\pm L$. The boundary energy-spectrum can be obtained as $E_{s}(k_{1},...,k_{d})=E_{bulk}(k_{1},...,k_{d})-E_{op}(k_{2},...,k_{d})$, where $E_{op}$ is the open-boundary energy-spectrum that is solved using the open boundary condition at $x_{1}=\pm L$ and the periodic boundary condition for the remaining $d-1$ coordinates at $\pm L$ (if $L\rightarrow\infty$, the result becomes exactly valid). Since $E_{bulk}(k_{1},...,k_{d})$ has a gap in the whole $\mathbf{k}$-space and the bulk gap must be closed at the two boundaries, the gapless points/regions at the boundary must be the intersecting points of the vector ${\bf{k}}_{1}$ and the $E_{s}=0$ equal-energy surface (the Fermi energy is set to be zero here), and they are indeed points on the $k_{2}k_{3}$-plane or some line-segments that can be continuously deformed to points for $d=3$ (the highest dimension for realistic physical systems). As for $d>3$, from the same analysis, it can be seen that they are at most some localized manifolds in the $\mathbf{k}$-space of the boundary, which are topologically equivalent to points.
  * (22) A. M. Essin and V. Gurarie, Phys. Rev. B 84, 125132 (2011).
  * (23) T. Fukui, K. Shiozaki, T. Fujiwara and S. Fujimoto, arXiv:1206.4410(2012).
  * (24) R. S. K. Mong and V. Shivamoggi, Phys. Rev. B 83,125109 (2011).
  * (25) In our notation, $\Gamma^{1}=\sigma_{1}\otimes\sigma_{1}$, $\Gamma^{2}=\sigma_{2}\otimes\sigma_{1}$, $\Gamma^{3}=\sigma_{3}\otimes\sigma_{1}$, $\Gamma^{4}=\mathbf{1}\otimes\sigma_{2}$ and $\Gamma^{5}=\mathbf{1}\otimes\sigma_{3}$, with $\sigma_{i}$ being Pauli matrices.
  * (26) When we deform the bulk Hamiltonian, the boundary effective theory may also be deformed accordingly; however, the total topological charge of FSs is invariant.
  * (27) M. Karoubi, K-Theory. An Introduction (Springer, New York, 1978).
  * (28) M. F. Atiyah, R. Bott, and A. Shapiro, Topology 3, 3 (1964), Suppl. 1.

##  I Supplemental Material

##  II SI. Derivation of Eq.(1)

In the derivation, the first key point is that for a $\mathcal{H}(\mathbf{k})$
with either TRS or PHS, the symmetries can be preserved on the $S^{p}$ if it
is chosen to be centrosymmetric with respect to the origin of
$(\omega,\mathbf{k})$-space. Secondly, It is crucial to observe that all six
formulas for the topological charges of FSs can formally be used to calculate
the topological numbers of TIs/TSCs, where we make the integration over the
whole $(\omega,\mathbf{k})$-space, in stead of making it over $S^{p}$ for an
FS. The topological difference of symmetry identification plays an essential
role in the derivation of the topological classification table of TIs/TSCs. In
practical calculations for FSs, we use the following spherical coordinates for
the chosen $S^{p-1}$,

$\begin{cases}k_{1}=\cos s_{1}\\\ k_{2}=\sin s_{1}\cos s_{2}\\\ k_{3}=\sin
s_{1}\sin s_{2}\cos s_{3}\\\ \,\,\,\,\,\,\,\,\,\,\,\,\vdots\\\ k_{p-1}=\sin
s_{1}\sin s_{2}\cdots\sin s_{p-2}\cos s_{p-1}\\\ k_{p}=\sin s_{1}\sin
s_{2}\cdots\sin s_{p-2}\sin s_{p-1}\end{cases}$ (7)

with $s_{i}\in[0,\pi]$ for $i=1,2,\cdots,p-2$ and $s_{p-1}\in[0,2\pi)$.
Without loss of generality, we have assumed that the FS is a Fermi point with
codimension $p$ in a $(p+1)$-dimensional $(\omega,\mathbf{k})$-space. In the
following, we focus on the class AII as an illustration, which has only TRS
with $\eta_{T}=-1$, while other seven classes can be treated in the same way
and results are the same. The TRS on the chosen $S^{p-1}$ with the spherical
coordinates can be represented as

$T^{\dagger}\mathcal{H}(s)T=\mathcal{H}^{T}(\pi-s_{1},\cdots,\pi-
s_{p-2},\pi+s_{p-1}).$

Note that the transformation of $s_{p-1}$ is different from those of the
others. Correspondingly, Green’s function
$G(\omega,\mathbf{k})=\left[i\omega-\mathcal{H}(\mathbf{k})\right]^{-1}$ is
transformed as

$T^{\dagger}G(\omega,s)T=G^{T}(\omega,\pi-s_{1},\cdots,\pi-
s_{p-2},\pi+s_{p-1}).$ (8)

While the Green’s function for a TI in the class AII satisfies

$T^{\dagger}G(\omega,\mathbf{k})T=G^{T}(\omega,-\mathbf{k}).$ (9)

It is seen that that all $\mathbf{k})$s of the TI reverse the signs under the
TRS transformation, which is in contrast to the situation of the chosen
$S^{p-1}$ enclosing an FS where the last coordinate $s_{p-1}$ does not reverse
its sign. We first illustrate how TRS makes the $\mathbf{Z}$-type topological
charge of an FS with codimension $p=4m+1$ vanishing. The formula for the
$\mathbf{Z}$-type topological charge with codimension $p=2n+1$ is given by

$\displaystyle\nu(2n+1,5)$ (10) $\displaystyle=$ $\displaystyle
C_{2n+1}\int_{S^{p}}d\omega
d^{2n}s\;\epsilon^{\mu_{1}\mu_{2}\cdots\mu_{2n+1}}$
$\displaystyle\mathbf{tr}\left(G\partial_{\mu_{1}}G^{-1}G\partial_{\mu_{2}}G^{-1}\cdots
G\partial_{\mu_{2n+1}}G^{-1}(\omega,s)\right),$

where $C_{2n+1}=-n!/(2n+1)!(2\pi i)^{n+1}$. Implementing the TRS
transformation, $i.e.$ Eq.(8), and the coordinate substitution
$s^{\prime}_{i}=\pi-s_{i}$ with $i=1,2,\cdots,2n-1$ and
$s^{\prime}_{2n}=\pi+s_{2n}$, we have

$\displaystyle\nu(2n+1,5)$ (11) $\displaystyle=$ $\displaystyle-
C_{2n+1}\int_{S^{p}}d\omega
d^{2n}s^{\prime}\;\epsilon^{\mu^{\prime}_{1}\mu^{\prime}_{2}\cdots\mu^{\prime}_{2n+1}}$
$\displaystyle\mathbf{tr}\left(G^{T}\partial_{\mu^{\prime}_{1}}G^{T-1}\cdots
G^{T}\partial_{\mu^{\prime}_{2n+1}}G^{T-1}(\omega,s^{\prime})\right)$
$\displaystyle=$ $\displaystyle C_{2n+1}\int_{S^{p}}d\omega
d^{2n}s\;\epsilon^{\mu_{1}\mu_{2}\cdots\mu_{2n+1}}$
$\displaystyle\mathbf{tr}\left(G\partial_{\mu_{2n+1}}G^{-1}G\partial_{\mu_{2n}}G^{-1}\cdots
G\partial_{\mu_{1}}G^{-1}(\omega,s)\right).$

Making a permutation that reverses the order of all the indices of $\epsilon$,
we obtain

$\nu(2n+1,5)=(-1)^{n-1}\nu(2n+1,5),$ (12)

where the extra $\left(-1\right)^{n}$ comes from the permutation. Now it is
clear that when $n=2m$ with $m$ being an integer, $i.e.$ $p=4m+1$, the
$\mathbf{Z}$-type topological charge of an FS is vanished for the class AII.
For the corresponding case of TI, the procedure is almost the same. First, the
integral form of topological number is the same as Eq.(10), but the
integration is made over the whole $(\omega,\mathbf{k})$-space:

$\displaystyle N(2n,5)$ (13) $\displaystyle=$ $\displaystyle C_{2n+1}\int
d\omega d^{2n}k\;\epsilon^{\mu_{1}\mu_{2}\cdots\mu_{2n+1}}$
$\displaystyle\mathbf{tr}\left(G\partial_{\mu_{1}}G^{-1}G\partial_{\mu_{2}}G^{-1}\cdots
G\partial_{\mu_{2n+1}}G^{-1}(\omega,\mathbf{k})\right).$

In a similar way to the case of FS, we make the variable substitution
$\mathbf{k^{\prime}}=-\mathbf{k}$, matrix transposition, and then a
permutation to reverse all the indices of $\epsilon$. The only difference is
that when substituting the variables, we could not obtain an extra minus sign
as that in the second equality of Eq.(11), since the number of partial
derivatives of $k_{i}$ is even. Thus we have

$N(2n,5)=\left(-1\right)^{n}N(2n,5),$ (14)

which implies the topological number of a TI with $d=4m+2$ in the class AII is
always trivial, but that with $d=4m$ can be nontrivial.

Comparing Eq.(14) with Eq.(12), we see clearly a one-dimension shift from
$K_{FS}(p,5)$ to $K_{TI}(d,5)$. From the established classification table of
FSs in class AII, we know that a sole but minor remaining uncertainty is
whether the topological types of TIs/TSCs for $d=8m+4$ are $\mathbf{Z}$ or
$2\mathbf{Z}$, which can surely be fixed by examining an appropriately chosen
case. Since the Dirac model of $d=4$ can have a unit topological charge
belonging to $\mathbf{Z}$-type, the topological charges for $d=8m+4$ is of
$\mathbf{Z}$-type, and thus $K_{FS}(d-1,5)=K_{TI}(d,5)$. The same results are
obtained for the other seven classes due to the same reason. Thus we show
rigorously that $K_{FS}(d-1,i)=K_{TI}(d,i)$, which reflects the topological
difference between the $\mathbf{k}$-space and the chosen $S^{p-1}$ to enclose
an FS when the symmetry class is concerned.

##  III SII. Topological charge of Fermi points on the boundary of TIs

###  III.1 SII.A. Topological charge of the Fermi point on the boundary of
3-dimensional TI

The boundary effective theory for a 3-dimensional TI reads

$\mathcal{H}(\mathbf{k})=k_{y}\sigma_{1}-k_{x}\sigma_{2}+\alpha(k_{x}+k_{y})\sigma_{3},$

where the term $\alpha(k_{x}+k_{y})\sigma_{3}$ merely indicates that the
Hamiltonian in the class AII allows such term, which will turn out to be
irrelevant for the topological charge of the Fermi point. This Hamiltonian has
a TRS with sign $-$, $i.e.$ FS-Classification ,

$\mathcal{H}(\mathbf{k})=\sigma_{2}\mathcal{H}^{T}(-\mathbf{k})\sigma_{2}^{-1},\quad\sigma_{2}^{T}=-\sigma_{2}.$

It belongs to the class AII. The Fermi surface is the point $\mathbf{k}=0$
with the codimension $2$. According to Ref.FS-Classification , we can
calculate its $\mathbf{Z_{2}^{(1)}}$-topological charge from Eq.(6) there. In
order to do this, we choose an $S^{2}$ in the $(\omega,\mathbf{k})$-space
enclosing the Fermi point, and make the following continuous extension of the
Green’s function restricted on the $S^{2}$:

$\displaystyle G^{-1}$ $\displaystyle=$ $\displaystyle
i\omega-\Delta\sigma_{3}\sin\theta$
$\displaystyle-\left[k\sin\phi\sigma_{1}-k\cos\phi\sigma_{2}+k(\cos\phi+\sin\phi)\sigma_{3}\right]\cos\theta,$

where $\phi\in[0,2\pi]$ parametrizes the circle in the $\mathbf{k}$-space
enclosing the Fermi surface, $\theta\in[0,\pi/2]$ is the parameter for
extension, and $\Delta$ is a positive constant. Substituting this Green’s
function into Eq.(6) in Ref.FS-Classification , we obtain

$\displaystyle\nu(2,5)$ $\displaystyle=$ $\displaystyle\frac{1}{12\pi^{2}}\int
d\omega d\phi d\theta\>\epsilon^{\mu\nu\lambda}$
$\displaystyle\,\,\,\,\,\,\times\mathbf{tr}\left(G\partial_{\mu}G^{-1}G\partial_{\nu}G^{-1}G\partial_{\lambda}G^{-1}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}d\phi\int_{0}^{\frac{\pi}{2}}d\theta\;\frac{1}{|g|^{3}}\mathbf{g}\cdot\left(\partial_{\theta}\mathbf{g}\times\partial_{\phi}\mathbf{g}\right),$

with

$\mathbf{g}=\left(k\sin\phi\cos\theta\,,-k\cos\phi\cos\theta,k(\cos\phi+\sin\phi)+\Delta\sin\theta\right).$

For simplicity, setting $\alpha=k=\Delta=1$, we obtain

$\nu[2,5]=\frac{1}{2\pi}\int_{0}^{2\pi}d\phi\int_{0}^{\frac{\pi}{2}}d\theta\,\,\,\frac{n(\theta,\phi)}{m(\theta,\phi)}=1,$

where

$n(\theta,\phi)=\cos\theta$

and

$\displaystyle m(\theta,\phi)$ $\displaystyle=$
$\displaystyle[\sin^{2}\theta+\sin(2\theta)(\cos\phi+\sin\phi)$
$\displaystyle+\cos^{2}\theta(2+\sin(2\phi))]^{\frac{3}{2}}.$

It is then found that a nontrivial $\mathbf{Z^{(1)}_{2}}$ topological charge
$\nu(2,5)=1$, which equals to $N(4,5)$ obtained before.

###  III.2 SII.B. Topological charge of the Fermi point on the boundary of
quantum spin Hall systems

For the boundary of a quantum spin Hall system, the effective boundary theory
can be written as

$\mathcal{H}(k_{x})=k_{x}\sigma_{1}+\alpha k_{x}\sigma_{2}+\beta
k_{x}\sigma_{3},$ (15)

where the last two terms preserving TRS will turn out to be irrelevant for the
topological charge. When we choose a circle in the $(\omega,k_{x})$-plane, the
Green’s function restricted on this circle is given by

$G^{-1}(\psi)=ia\sin\psi-\mathcal{H}(a\cos\psi)$

with $\psi\in[0,2\pi)$. Noting that TRS for the Green’s function is
$TG(\omega,k)T^{\dagger}=G^{T}(\omega,-k)$, this TRS is accordingly
represented as $\sigma_{2}G(\psi)\sigma_{2}=G(\pi-\psi)$ on the circle. Then
following Ref.FS-Classification , we extend the Green’s function restricted on
the circle with two additional parameters, $i.e.$,

$\displaystyle G^{-1}(\psi,\theta,\phi)$ $\displaystyle=$ $\displaystyle
ia\sin\psi-\left[\left(\mathcal{H}(a\cos\psi)\cos\theta+\sin\theta\sigma_{2}\right)\cos\phi+\sin\phi\sigma_{3}\right]$

with $\theta\in[0,\pi]$ and $\phi\in[0,\pi]$. It is straightforward to verify
that TRS is preserved for this extension:

$\sigma_{2}G\left(\psi,\theta,\phi\right)\sigma_{2}=G^{T}\left(\pi-\psi,-\theta,-\phi\right).$

The $\mathbf{Z_{2}}^{(2)}$-type topological charge is calculated as

$\displaystyle\nu(1,5)$ $\displaystyle=$ $\displaystyle\frac{1}{24\pi^{2}}\int
d\psi d\theta d\phi\epsilon^{\mu\nu\lambda}$
$\displaystyle\qquad\mathbf{tr}\left(G\partial_{\mu}G^{-1}G\partial_{\nu}G^{-1}G\partial_{\lambda}G^{-1}\right)$
$\displaystyle=$ $\displaystyle 1,$

which equals to $N(3,5)$ calculated before.

##  IV SIII. Dirac Matrix Construction of FSs and TIs/TSCs and The Proof of
General Index Theorem

###  IV.1 SIII.A. Preliminaries of Dirac matrices

The purpose of this section is to introduce the preliminaries about Dirac
matrices needed for our construction of topologically protected FSs and
TIs/TSCs. As for a pedagogical introduction to Dirac matrices, it is seen in
Ref.Kim-Dirac-Matrices . Although most results may be basis-independent, we
still introduce explicitly our convention of Dirac matrices below.

$\displaystyle\Gamma_{(2n+1)}^{a}$ $\displaystyle=$
$\displaystyle\Gamma_{(2n-1)}^{a}\otimes\sigma_{1},\quad a=1,2,3,\cdots,2n-1,$
$\displaystyle\Gamma_{(2n+1)}^{2n}$ $\displaystyle=$
$\displaystyle\mathbf{1}_{2^{n-1}}\otimes\sigma_{2},$
$\displaystyle\Gamma_{(2n+1)}^{2n+1}$ $\displaystyle=$
$\displaystyle\mathbf{1}_{2^{n-1}}\otimes\sigma_{3},$ (16)

where $\sigma_{i}$ are Pauli matrices and $\mathbf{1}_{2^{n-1}}$is the
$2^{n-1}\times 2^{n-1}$unit matrix. We input the initial condition:
$\Gamma_{(3)}^{1}=\sigma_{1},\Gamma_{(3)}^{2}=\sigma_{2}$ and
$\Gamma_{(3)}^{3}=\sigma_{3}$, to obtain Dirac matrices in all dimensions.
Notably $\Gamma$ matrices with odd superscript are purely real, while ones
with even superscript are purely imaginary. All Dirac matrices are hermitian,
and satisfy the Clifford algebra:

$\left\\{\Gamma_{(2n+1)}^{a},\>\Gamma_{(2n+1)}^{b}\right\\}=2\delta^{ab}\mathbf{1}_{2^{n}\times
2^{n}}.$ (17)

We will construct Hamiltonians by Dirac matrices and discuss their TRS and
PHS. For this purpose, we introduce the following operators.

$\displaystyle B_{(2n+1)}^{1}$ $\displaystyle:=$
$\displaystyle\Gamma_{(2n+1)}^{3}\Gamma_{(2n+1)}^{5}\cdots\Gamma_{(2n+1)}^{2n+1}$
$\displaystyle B_{(2n+1)}^{2}$ $\displaystyle:=$
$\displaystyle\Gamma_{(2n+1)}^{2}\Gamma_{(2n+1)}^{4}\cdots\Gamma_{(2n+1)}^{2n}$
$\displaystyle\tilde{B}_{(2n+1)}^{1}$ $\displaystyle:=$ $\displaystyle
B_{(2n+1)}^{1}\Gamma_{(2n+1)}^{2n+1}$ (18)
$\displaystyle\tilde{B}_{(2n+1)}^{2}$ $\displaystyle:=$ $\displaystyle
B_{(2n+1)}^{2}\Gamma_{(2n+1)}^{2n+1}$

It is direct to verify the following commutation relations below:

$\displaystyle B_{(2n+1)}^{1}\Gamma_{(2n+1)}^{a}(B_{(2n+1)}^{1})^{-1}$
$\displaystyle=$
$\displaystyle\begin{cases}(-1)^{n+1}(\Gamma_{(2n+1)}^{a})^{T},&a=2,\cdots,2n+1\\\
(-1)^{n}(\Gamma_{(2n+1)}^{1})^{T}\end{cases},$ $\displaystyle
B_{(2n+1)}^{2}\Gamma_{(2n+1)}^{a}(B_{(2n+1)}^{2})^{-1}$ $\displaystyle=$
$\displaystyle(-1)^{n}(\Gamma_{(2n+1)}^{a})^{T},\quad a=1,2,\cdots,2n+1.$

The transposition matrices of $B^{1}$, $B^{2}$, $\tilde{B}^{1}$, and
$\tilde{B}^{2}$ satisfy, respectively,

$\displaystyle B_{(2n+1)}^{1}$ $\displaystyle=$
$\displaystyle(-1)^{n(n-1)/2}\left(B_{(2n+1)}^{1}\right)^{T}$ $\displaystyle
B_{(2n+1)}^{2}$ $\displaystyle=$
$\displaystyle(-1)^{n(n+1)/2}\left(B_{(2n+1)}^{2}\right)^{T}$
$\displaystyle\tilde{B}_{(2n+1)}^{1}$ $\displaystyle=-$
$\displaystyle(-1)^{n(n+1)/2}\left(\tilde{B}_{(2n+1)}^{1}\right)^{T}$ (20)
$\displaystyle\tilde{B}_{(2n+1)}^{2}$ $\displaystyle=$
$\displaystyle(-1)^{n(n-1)/2}\left(\tilde{B}_{(2n+1)}^{2}\right)^{T}$

###  IV.2 SIII.B. Construction of all topological types of Fermi surfaces

####  IV.2.1 Fermi surfaces in non-chiral classes AI, AII, D and C

$\mathbf{Z}\,\,and\,\,\mathbf{Z}_{2}^{(1,2)}$–The simplest way to construct a
Hamiltonian with Dirac matrices is to make Weyl-type Hamiltonian, namely,

$\mathcal{H}_{(2n+1)}^{W}(\mathbf{k})=\sum_{a=1}^{2n+1}k_{a}\Gamma_{(2n+1)}^{a}.$
(21)

This Hamiltonian has either TRS or PHS depending on its spatial dimension
$2n+1$. To be explicit, the Hamiltonian satisfies

$\displaystyle
B_{(2n+1)}^{2}\mathcal{H}_{(2n+1)}^{W}(\mathbf{k})(B_{(2n+1)}^{2})^{-1}$ (22)
$\displaystyle=$
$\displaystyle(-1)^{n+1}\left(\mathcal{H}_{(2n+1)}^{W}\right)^{T}(-\mathbf{k}),$

which can be seen from Eq.(20). $B^{2}$ represents a TRS with $n=2m+1$ or a
PHS with $n=2m$, where $m$ is an integer or zero, and the corresponding
symmetry sign is given by Eq.(20). This Hamiltonian has a Fermi point at the
origin of $\mathbf{k}$-space, whose codimension is $p=2n+1$. The Fermi point
is in classes AI, AII, D, and C, respectively, for $p=8m+7$, $p=8m+3$,
$p=8m+1$, and $p=8m+5$, which precisely coincides with that of
$\mathbf{Z}$-type topological charge for the same class in the classification
table of Fermi surfaces (Tab.[I] in the main text).

Our next task is to show that this Fermi point has a unit $\mathbf{Z}$-type
topological charge. It can be enclosed by a cylinder
$A:=(-\infty,\infty)\times S^{2n}$ in $(\omega,\mathbf{k})$-space with
$S^{2n}$ centered at the origin of $\mathbf{k}$-space and $\omega$ being in
$(-\infty,\infty)$. We parametrize the Green’s function on this cylinder $A$
as

$G^{-1}|_{A}(\omega,\mathbf{k})=i\omega-k_{a}(s)\Gamma_{(2n+1)}^{a},$

where $k_{a}(s)$ are the same as those defined in Eq.(7) with $p=2n+1$. For
the Green’s function, the cylinder $A$ is essentially a topological
$(2n+1)$-dimensional sphere $S^{2n+1}$ since $G^{-1}$ tends to a constant
matrix when $\omega$ approaches to $\pm\infty$. To compute its topological
charge, we make the following integral on the cylinder with Eq.(17).

$\displaystyle\nu$ $\displaystyle=$ $\displaystyle
C_{2n+1}\int_{A}\left(G\mathbf{d}G\right)^{2n+1}$ (23) $\displaystyle=$
$\displaystyle(2n+1)C_{2n+1}\int_{-\infty}^{\infty}d\omega\int_{S^{2n}}d^{2n}s\;\epsilon^{\mu_{1}\mu_{2}\cdots\mu_{2n}}$
$\displaystyle\qquad\qquad\mathbf{tr}\left(G\partial_{\omega}G^{-1}G\partial_{\mu_{1}}G^{-1}\cdots
G\partial_{\mu_{2n}}G^{-1}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{n-1}\Omega_{2n+1}}\int_{S^{2n}}d^{2n}s\;\frac{1}{|\mathbf{n}|^{2n+1}}\epsilon^{\mu_{1}\mu_{2}\cdots\mu_{2n+1}}\epsilon^{\nu_{1}\nu_{2}\cdots\nu_{2n}}$
$\displaystyle\qquad\qquad
n_{\mu_{1}}\partial_{\nu_{1}}n_{\mu_{2}}\partial_{\nu_{2}}n_{\mu_{3}}\cdots\partial_{\nu_{2n}}n_{\mu_{2n+1}}$
$\displaystyle=$ $\displaystyle 1$

with $\Omega_{2n+1}$ being the total solid angle of $(2n+1)$-dimensional
space. Now it is clear that all Fermi points of unit topological charge with
either a TRS or PHS can be constructed by the Dirac matrices in the form of
Eq.(21). Note that we can construct an $N$-charged Fermi point for each case
simply by mutipling the unit Hamiltonian $N$ times for $\mathbf{Z}$ types, or
the double-unit models (to be constructed latter) $N/2$ times for
$2\mathbf{Z}$ types. This also holds for $\mathbf{Z}$-type and
$2\mathbf{Z}$-type TIs/TSCs with multiple topological numbers, which will not
be repeated any more.

We now turn to construct FSs protected by $\mathbf{Z}_{2}^{(1,2)}$ type
topological charge. According to our general classification table, each
$\mathbf{Z}$ type topological charge is followed by $\mathbf{Z}_{2}^{(1)}$ and
$\mathbf{Z}_{2}^{(2)}$ with decreasing codimensions, which motivates us to
construct corresponding models by lowering spatial dimensions, i.e. the number
of $k$s from the Weyl-type model (21). For $\mathbf{Z}_{2}^{(1)}$ cases, we
consider the following model

$\mathcal{H}_{W}^{(1)}=\sum_{a=1}^{2n}k_{a}\Gamma^{a+1}+\lambda\sum_{a=1}^{2n}k_{a}\Gamma^{1},$
(24)

where the last term with a coefficient $\lambda$ is added to merely indicate
that the considered model, while keeping TRS or PHS with the topological
character unchanged, may not need to have an additional CS. We may set
$\lambda=0$ without changing the topological property of the Fermi point at
$k=0$. For brevity, we have dropped the awkward subscript $(2n+1)$ in the
above expression and will do it hereafter if it can be recovered from the
context. To calculate the $\mathbf{Z}_{2}^{(1)}$ type topological charge of
the Fermi point, we first choose an $S^{2n-1}$ in $\mathbf{k}$ space, which is
parametrized by the spherical coordinates $s_{i}$ with $i=1,2,\cdots,2n-1$,
and restrict the Hamiltonian on it, i.e.
$\mathcal{H}_{W}^{(2)}|_{S^{2n-1}}=\mathcal{H}_{W}^{(2)}(s)$. According to the
formulation of $\mathbf{Z}_{2}^{(1)}$ type topological charge, we make a one-
parameter extension of $\mathcal{H}_{W}^{(2)}(s)$, which may be chosen as

$\mathcal{H}(s,\alpha)=\mathcal{H}(s)\cos\alpha+\sin\alpha\Gamma^{1},$

with $\alpha\in[-\pi/2,\pi/2]$. It is straightforward to check that this
extension satisfies all the requirementsFS-Classification . Using the extended
Green’s function $G^{-1}(\omega,s,\alpha)=i\omega-\mathcal{H}(s,\alpha)$, we
compute the topological charge as

$\nu=C_{2n+1}\int_{\tilde{A}}\mathbf{tr}\left(G\mathbf{d}G^{-1}(\omega,s,\alpha)\right)^{2n+1},$

where $\tilde{A}=A\times S^{1}$ with $A$ being the original domain
$(-\infty,\infty)\times S^{2n-1}$ in $(\omega,\mathbf{k})$ space. After
replacing the variable $\alpha$ with $\alpha^{\prime}=\alpha+\pi/2$, we are
able to treat $\alpha$ as the $2n$-th spherical coordinate of $S^{2n}$ made by
$S^{2n-1}$ and the extended dimension. Thus referring to our calculation of
$\mathbf{Z}$ type topological charge, we find that $\nu=1\mod 2$, which means
that the Fermi point at $k=0$ for model (24) is nontrivial.

With the experience of constructing nontrivial $\mathbf{Z}_{2}^{(1)}$ type
topological charge, we now give the following model as a candidate for
$\mathbf{Z}_{2}^{(2)}$:

$\displaystyle\mathcal{H}_{W}^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{a=1}^{2n-1}k_{a}\Gamma^{a+2}$ (25)
$\displaystyle+\lambda_{1}\sum_{a=1}^{2n-1}k_{a}\Gamma^{1}+\lambda_{2}\sum_{a=1}^{2n-1}k_{a}^{3}\Gamma^{2},$

where analogous to the previous case, terms with $\lambda_{1,2}$ are added to
mean that the model does not require additional symmetries, and again they may
be set as zero without changing the topological property of the Fermi point.
To see the topological property of the Fermi point, we choose an $S^{2n-2}$,
which is parametrized with a spherical coordinates $s_{i}$ with
$i=1,2,\cdots,2n-2$, in $\mathbf{k}$ space to enclose it. The Hamiltonian
restricted on the $S^{2n-2}$ is denoted by $\mathcal{H}_{W}^{(2)}(s)$, and
then according to the formulation of $\mathbf{Z}_{2}^{(2)}$ type topological
number, we make two-parameter extension of $\mathcal{H}_{W}^{(2)}(s)$, which
may be written as

$\mathcal{H}_{W}^{(2)}(s,\alpha,\beta)=\left(\mathcal{H}_{W}^{(2)}(s)\cos\alpha+\sin\alpha\Gamma^{1}\right)\cos\beta+\sin\beta\Gamma^{2},$

where $\alpha,\beta\in[-\pi/2,\pi/2]$. The above extension satisfies all the
requirements for computing the topological charge of the Fermi pointFS-
Classification . Using the extended Green’s function
$G^{-1}(\omega,s,\alpha,\beta)=i\omega-\mathcal{H}_{W}^{(2)}(s,\alpha,\beta)$,
we have the topological charge as

$\nu=C_{2n+1}\int_{\tilde{\tilde{A}}}\mathbf{tr}\left(G\mathbf{d}G^{-1}(\omega,s,\alpha,\beta)\right),$

where $\tilde{\tilde{A}}=A\times T^{2}$ with $A=[-\infty,\infty]\times
S^{2n-2}$ and $T^{2}$ being the domain of $\alpha$ and $\beta$. Through the
variable transformation: $\alpha^{\prime}=\alpha+\pi/2$ and
$\beta^{\prime}=\beta+\pi/2$, we can combine $T^{2}$ and $S^{2n-2}$ as
$S^{2n}$. Thus the Fermi point has nontrivial topological charge $\nu=1\mod
2$.

$2\mathbf{Z}\,\,type$–To construct $2\mathbf{Z}$ type Fermi surface in our
classification table, we introduce the following operator:

$\mathcal{B}^{1}=\Gamma^{2}\Gamma^{5}\Gamma^{7}\cdots\Gamma^{2n+1},$

which is a modified version of $B^{1}$ with $\Gamma^{3}$ being replaced by
$\Gamma^{2}$. $\mathcal{B}^{1}$ satisfies

$\mathcal{B}^{1}=-(-1)^{n(n-1)/2}\left(\mathcal{B}^{1}\right)^{T}$ (26)

and

$\begin{cases}\mathcal{B}^{1}\Gamma^{a}\left(\mathcal{B}^{1}\right)^{-1}=(-1)^{n+1}\left(\Gamma^{a}\right)^{T}&4\leq
a\leq 2n+1\\\
\mathcal{B}^{1}\tilde{\Gamma}\left(\mathcal{B}^{1}\right)^{-1}=\left(-1\right)^{n+1}\tilde{\Gamma}^{T}\end{cases}$
(27)

with $\tilde{\Gamma}=i\Gamma^{1}\Gamma^{2}\Gamma^{3}$. We construct the model
as

$\mathcal{H}_{W}^{2}(\mathbf{k})=\sum_{a=2}^{2n-1}k_{a}\Gamma_{(2n+1)}^{a+2}+\tilde{\Gamma}_{(2n+1)}k_{1}.$
(28)

As $\mathcal{H}_{W}^{2}$ can be diagonalized into two equivalent
$2^{n-1}\times 2^{n-1}$ blocks since
$\left[\mathcal{H}_{W}^{2},i\Gamma^{1}\Gamma^{2}\right]=0$, up to a unitary
transformation it can rewritten as

$\mathcal{H}_{W}^{2}(\mathbf{k})\sim\begin{pmatrix}\sum_{a=1}^{2n-1}k_{a}\Gamma_{(2n-1)}^{a}\\\
&\sum_{a=1}^{2n-1}k_{a}\Gamma_{(2n-1)}^{a}\end{pmatrix}.$

Since each block has unit $\mathbf{Z}$ type topological charge, the Fermi
point of $\mathcal{H}(\mathbf{k})$ has topological charge $2$. We then analyze
its TRS and PHS by using Eq.(27), which leads to

$\mathcal{B}^{1}\mathcal{H}_{W}^{2}(\mathbf{k})\left(\mathcal{B}^{1}\right)^{-1}=(-1)^{n}\mathcal{H}_{W}^{2T}(-\mathbf{k}).$

Thus based on the relation Eq.(26) and the above relation, we summarize that
the Fermi point is in class AI, AII, D, and C, respectively, for the
codimension $p=8m+3$, $p=8m+7$, $p=8m+5$, and $p=8m+1$.

####  IV.2.2 Fermi surfaces in chiral classes CI, CII, DIII and BDI

$\mathbf{Z}\,\,\,type\,\,\,and\,\,\,\mathbf{Z}_{2}^{(1,2)}\,\,\,type$-To
handle the remaining four symmetry classes with chiral symmetry(CS), we
construct the Dirac-type Hamiltonian as

$\mathcal{H}_{D}(\mathbf{k})=\sum_{a=1}^{2n}k_{a}\Gamma^{a}.$ (29)

In other words, we sum over all $\Gamma$s except the last one, which leads to
CS, $i.e.$

$\left\\{\mathcal{H}_{D},\Gamma^{2n+1}\right\\}=0.$

As is known that this Hamiltonian has a TRS(PHS) represented by $B^{2}$ just
like that of $\mathcal{H}^{W}$, using $\Gamma_{(2n+1)}^{2n+1}$ we are able to
newly construct a PHS(TRS), which is represented by
$\tilde{B}^{2}=B^{2}\Gamma^{2n+1}$ as in Eq.(18). It is verified that

$\displaystyle\tilde{B}^{2}\mathcal{H}_{D}(\mathbf{k})\left(\tilde{B}^{2}\right)^{-1}$
$\displaystyle=$
$\displaystyle-B^{2}\mathcal{H}_{D}(\mathbf{k})\left(B^{2}\right)^{-1}$
$\displaystyle=$
$\displaystyle(-1)^{n}\left(\mathcal{H}_{D}(-\mathbf{k})\right)^{T}$

Compared with Eq.(22), if $B^{2}$ refers to a TRS, then $\tilde{B}^{2}$ refers
to a PHS, and vice versa. The sign of $\tilde{B}^{2}$ as a TRS/PHS can be seen
from Eq.(20). As the codimension of the Fermi point for the Hamiltonian of
Eq.(29) is $2n$, the Fermi point is in class CI, DIII, BDI, and CII
respectively when $p=8m+6$, $p=8m+2$, $p=8m$, and $p=8m+4$, which is in
agreement with the appearance of $\mathbf{Z}$-type topological charge for the
four classes with CS in the classification table. Parallel to the case without
CS, we will show that the Fermi point of Eq.(29) has unit charge. Before doing
it, it is noted that $\mathcal{H}_{D}$ can be expressed as

$\mathcal{H}_{D}=\begin{pmatrix}0&u\\\ u^{\dagger}&0\end{pmatrix},$ (30)

where $u=-ik_{2n}\mathbf{1}_{2^{n-1}\times
2^{n-1}}+\sum_{a=1}^{2n-1}k_{a}\Gamma_{(2n-1)}^{a}$ with our convention of
Dirac matrices in Eq.(16). To calculate the topological charge, we choose an
$S^{2n-1}$ in the $\mathbf{k}$-space to enclose the Fermi point, which is
parametrized by $s_{i}$ with $i$ being from $1$ to $2n-1$. Thus we have

$\displaystyle\nu$ $\displaystyle=$
$\displaystyle\frac{C_{2n-1}}{2}\int_{S^{2n-1}}\mathbf{tr}\left[\Gamma^{2n+1}\left(\left(\mathcal{H}_{D}\right)^{-1}\mathbf{d}\mathcal{H}_{D}\right)^{2n-1}\right].$

Without loss of generality, we choose the $S^{2n-1}$ to be unit, which implies
$u^{\dagger}u=1$ and leads to

$\nu=C_{2n-1}\int_{S^{2n-1}}\mathbf{tr}\left(u\mathbf{d}u^{\dagger}\right)^{2n-1}.$

As $u^{\dagger}$ can be regarded as an inversion of Green’s function in
$(2n-1)$ spatial dimensions with $k_{2n}$ as $\omega$, referring to the case
of Eq.(23), it is seen that $\nu=1$.

Analogous to the cases without CS, we can construct $\mathbf{Z}_{2}^{(1,2)}$
type topological Fermi surfaces by lowering spatial dimension from the model
(29). The process is entirely analogous to that of non-chiral cases, so we
simply write down the corresponding models below:

$\begin{cases}\mathcal{H}_{D}^{(1)}=\sum_{a=1}^{2n-1}k_{a}\Gamma^{a+1}&for\quad\mathbf{Z}_{2}^{(1)}\\\
\mathcal{H}_{D}^{(2)}=\sum_{a=1}^{2n-2}k_{a}\Gamma^{a+2}&for\quad\mathbf{Z}_{2}^{(2)}\end{cases}.$
(31)

Note that although the above two models have additional symmetries, e.g.,
$\Gamma^{1}$ anti-commutes with $\mathcal{H}_{D}^{(1)}$, these symmetries are
not required, which means that we can add some terms to break them without
changing the corresponding topological properties, such as the $\lambda$-terms
in model (24) and model (25).

$2\mathbf{Z}\,\,\,type$-According to the corresponding model of classes AI,
AII, C and D, we can define

$\tilde{\mathcal{B}}^{1}:=\mathcal{B}^{1}\Gamma^{2n+1},$

which satisfies the relation

$\tilde{\mathcal{B}}^{1}=(-1)^{n(n+1)/2}\left(\tilde{\mathcal{B}}^{1}\right)^{T}.$
(32)

Accordingly, we introduce the following model:

$\mathcal{H}_{D}^{2}(\mathbf{k})=\sum_{a=2}^{2n-2}k_{a}\Gamma^{a+2}+k_{1}\tilde{\Gamma},$
(33)

whose Fermi point has a topological charge $2$ from the same reasoning for
model (28). It has a CS (represented by $\Gamma^{2n+1}$); and moreover, when
$\mathcal{B}^{1}$ represents a TRS(PHS), $\tilde{\mathcal{B}}^{1}$ denotes a
PHS(TRS). Considering the relation (32) and the results of nonchiral cases, it
is found that the Fermi point is in classes DIII, CI, CII, and BDI,
respectively, when $p=8m+6$, $p=8m+2$, $p=8m$, and $p=8m+4$, which is in
consistence with our classification table of Fermi surfaces.

To conclude this subsection, all types of Fermi surfaces in the classification
table have been constructed by Dirac matrices.

###  IV.3 SIII.C. Construction of all topological types of TIs/TSCs

####  IV.3.1 TIs/TSCs in non-chiral classes AI, AII, D and C

$\mathbf{Z}\,\,\,type\,\,\,and\,\,\,\mathbf{Z}_{2}^{(1,2)}\,type$– We first
wish to indicate that a part of results to be presented for the relationship
of Dirac matrices with TIs/TSCs were addressed in Ref.Ryu-TI-Classification .
As the bulk energy spectrum of a TI/TSC is fully gaped, we need one of
$\Gamma$s with a coefficient as a mass term, which is chosen to be
$\Gamma^{1}$. Thus a typical Hamiltonian reads

$\mathcal{H}(\mathbf{k})=\sum_{a=2}^{2n+1}k_{a-1}\Gamma^{a}+\left(m-\epsilon
k^{2}\right)\Gamma^{1}.$ (34)

Note that the term of $\epsilon k^{2}\Gamma^{1}$, in which a constant
$\epsilon$ can be regarded to be infinitesimally small, is included as a
prescription for the singularity of this kind of continuum model at the
infinity in the $\mathbf{k}$ spaceNote-Lattice . It will turn out that the
topological number of this Hamiltonian is either $\pm 1$ or $0$.

Let us look into the TRS/PHS of Hamiltonian in Eq.(34), which is now
represented by $B^{1}$. Using Eq.(IV.1), we obtain the relation

$B^{1}\mathcal{H}(\mathbf{k})(B^{1})^{-1}=(-1)^{n}\mathcal{H}^{T}(-\mathbf{k}).$
(35)

Thus when $n=2m$, $i.e.$ the spatial dimension $d=4m$, $B^{1}$ represents a
TRS, while it corresponds to a PHS when $n=2m+1$ or $d=4m+2$. Eq.(20) can tell
us the sign of the TRS/PHS, so that we can encapsulate that Hamiltonian of
Eq.(34) is in classes AI, AII, D, and C, respectively, for $d=8m$, $d=8m+4$,
$d=8m+2$, $d=8m+6$, which is in agreement with the distribution of
$\mathbf{Z}$-type topological numbers in the classification of TIs/TSCs. The
remaining task for these classes is to verify that Hamiltonian of Eq.(34) has
indeed unit topological number. Substituting Eq.(34) into the corresponding
formula for the integer topological number, i.e.,

$N=C_{d+1}\int_{\mathcal{M}}\mathbf{tr}\left(G\mathbf{d}G^{-1}(\omega,\mathbf{k})\right)^{d+1},$

where $\mathcal{M}$ is the whole $(\omega,\mathbf{k})$-space, we can obtain

$N=\frac{1}{2}\left(\mathbf{sgn}(m)+\mathbf{sgn}(\epsilon)\right),$ (36)

which indicates unit topological number when both $m$ and $\epsilon$ have the
same sign. Thus it is clear that the model of Eq.(34) can be regarded as a
representative for each $\mathbf{Z}$-type of the classes AI, AII, D and C.

Since the procedures to construct the $\mathbf{Z}_{2}^{(1,2)}$ type TIs/TSCs
are similar to those for constructing $\mathbf{Z}_{2}^{(1,2)}$ type Fermi
surfaces, to avoid redundant derivations, we here write down the results
directly with some necessary remarks. The corresponding models for cases of
$\mathbf{Z}_{2}^{(1)}$ and $\mathbf{Z}_{2}^{(1)}$ are, respectively,

$\begin{cases}\mathcal{H}^{(1)}=\sum_{a=1}^{2n-1}k_{a}\Gamma^{a+2}+\left(m-\epsilon
k^{2}\right)\Gamma^{1}&for\;\mathbf{Z}_{2}^{(1)}\\\
\mathcal{H}^{(2)}=\sum_{a=1}^{2n-2}k_{a}\Gamma^{a+3}+\left(m-\epsilon
k^{2}\right)\Gamma^{1}&for\;\mathbf{Z}_{2}^{(2)}\end{cases}.$

To see that nontrivial topological property of $\mathcal{H}^{(1,2)}$, we need
to make continuous extension with new parameters, which can be chosen as the
omitted momenta, compared with model (34). Concretely, for $\mathcal{H}^{(1)}$
the extension can be made by adding the term
$\alpha\Gamma^{2}-\epsilon\alpha^{2}\Gamma^{1}$ with
$\alpha\in(-\infty,\infty)$. When $\alpha=0$, we obtain the original
$\mathcal{H}^{(1)}$, and by changing $\alpha$ from $0$ to $\pm\infty$,
$\mathcal{H}^{(1)}$ is deformed smoothly without closing its gap to a trivial
model $\sim\alpha\Gamma^{2}-\epsilon\alpha^{2}\Gamma^{1}$. Since the extended
model is equivalent to the model (34) whose $\mathbf{Z}$ type topological
number is given by Eq.(36), it is concluded that the $\mathbf{Z}_{2}^{(1)}$
type topological number of $\mathcal{H}^{(1)}$ is also given by Eq.(36). The
same reasoning is applicable to $\mathcal{H}^{(2)}$ when we make two-parameter
extension, thus $\mathcal{H}^{(2)}$ has a nontrivial $\mathbf{Z}_{2}^{(2)}$
type topological number when $m$ and $\epsilon$ have the same sign. We note
again that only the symmetries in the corresponding symmetry class are
required, while other terms preserving topological properties may be added to
break unwanted additional symmetries.

$2\mathbf{Z}\,\,\,type$–Similar to the case of $2\mathbf{Z}$ type Fermi
surface, we introduce a model as

$\mathcal{H}^{2}(\mathbf{k})=\sum_{a=1}^{2n-2}k_{a}\Gamma^{a+3}+\left(m-\epsilon
k^{2}\right)\tilde{\Gamma}$

with topological number $\mathbf{sgn}(m)+\mathbf{sgn}(\epsilon)$, which can be
seen from our previous discussion of $2\mathbf{Z}$ type Fermi surfaces. Noting
that

$B^{2}\tilde{\Gamma}\left(B^{2}\right)^{-1}=\left(-1\right)^{n+1}\tilde{\Gamma}$

and relation (IV.1), we have

$B^{2}\mathcal{H}^{2}(\mathbf{k})\left(B^{2}\right)^{-1}=\left(-1\right)^{n+1}\mathcal{H}^{2T}(-\mathbf{k}).$
(37)

Combining Eq.(20) with Eq.(37), it can be found that the classes of model
belong, respectively, to AI, AII, D, and C for $d=8m+4$, $d=8m$, $d=8m+6$, and
$d=8m+2$, consistent with our classification of TIs/TSCs.

####  IV.3.2 TIs/TSCs in chiral classes CI, CII, BDI and DIII

$\mathbf{Z}\,\,\,type\,\,\,and\,\,\,\mathbf{Z}_{2}^{(1,2)}\,type$–For the
other classes with CS, the term $\Gamma^{2n+1}$ is excluded for the presence
of CS, and the corresponding model may be written as

$\mathcal{H}_{CS}(\mathbf{k})=\sum_{a=2}^{2n}k_{a-1}\Gamma^{a}+\left(m-\epsilon
k^{2}\right)\Gamma^{1}.$ (38)

Here the additional TRS/PHS is represented by
$\tilde{B}^{1}=B^{1}\Gamma^{2n+1}$ referring to Eq.(18), and we can check that

$\displaystyle\tilde{B}^{1}\mathcal{H}_{CS}(\mathbf{k})(\tilde{B}^{1})^{-1}$
$\displaystyle=$ $\displaystyle-B^{1}\mathcal{H}_{CS}(\mathbf{k})(B^{1})^{-1}$
$\displaystyle=$ $\displaystyle(-1)^{n+1}\mathcal{H}_{CS}^{T}(-\mathbf{k}).$

Compared with Eq.(35), it is seen that if $B^{1}$ denotes a TRS(PHS), then
$\tilde{B}^{1}$ corresponds to a PHS(TRS). The sign of $\tilde{B}^{1}$ is
given by Eq.(20), and thus it can be seen that the classes of model (38)
belong, respectively, to CI, DIII, BDI, and CII for $d=8m+7$, $d=8m+3$,
$d=8m+1$, and $d=8m+5$. To check the $\mathbf{Z}$-type topological number, we
use the corresponding formula

$N=\frac{C_{2n-1}}{2}\int_{M}\mathbf{tr}\left[\Gamma^{2n+1}\left(\mathcal{H}_{CS}^{-1}\mathbf{d}\mathcal{H}_{CS}\right)^{2n-1}\right],$

where $M$ denotes the whole $\mathbf{k}$ space. If we recover the $\Gamma$
matrices with their subscript to be $\Gamma_{(2n+1)}^{a}$, we recall Eq.(30)
of $\mathcal{H}_{CS}$ with

$\displaystyle u$ $\displaystyle=$ $\displaystyle-
ik_{2n-1}\mathbf{1}_{2^{n-2}}$
$\displaystyle+\sum_{a=2}^{2n-1}k_{a-1}\Gamma_{(2n-1)}^{a}+(m-\epsilon
k^{2})\Gamma_{(2n-1)}^{1}.$

Similar to the calculation of topological charge for Fermi point, we have

$N=C_{2n-1}\int_{M}\mathbf{tr}\left(u\mathbf{d}u^{\dagger}\right)^{2n-1}.$

Thus the topological number is given by

$N=\frac{1}{2}(\mathbf{sgn}(m)+\mathbf{sgn}(\epsilon)),$

which implies $N=1$ when both $m$ and $\epsilon$ are positive. Note that the
correct definition of $k^{2}$ should be $k^{2}=\sum_{a=1}^{2n-2}k_{a}^{2}$ in
order to obtain the above expression , since adding $k_{2n-1}^{2}$ in the
summation will always make the topological number vanish.

$\mathbf{Z}_{2}^{(1,2)}$ type TIs/TSCs in these classes can be obtained
directly by the method used in the previous section, thus we here present the
result:

$\begin{cases}\mathcal{H}_{CS}^{(1)}=\sum_{a=1}^{2n-2}k_{a}\Gamma^{a+2}+\left(m-\epsilon
k^{2}\right)\Gamma^{1}&for\;\mathbf{Z}_{2}^{(1)}\\\
\mathcal{H}_{CS}^{(2)}=\sum_{a=1}^{2n-3}k_{a}\Gamma^{a+3}+\left(m-\epsilon
k^{2}\right)\Gamma^{1}&for\;\mathbf{Z}_{2}^{(2)}\end{cases}$

The two models have nontrivial $\mathbf{Z}_{2}^{(1,2)}$ type topological
numbers when $m$ and $\epsilon$ have the same sign.

$2\mathbf{Z}$ type–From our experience, a desired mode in this case can be
written as

$\mathcal{H}_{CS}^{2}(\mathbf{k})=\sum_{a=1}^{2n-3}k_{a}\Gamma^{a+3}+\left(m-\epsilon
k^{2}\right)\tilde{\Gamma},$

which has topological number $\mathbf{sgn}(m)+\mathbf{sgn}(\epsilon)$. If
$B^{2}$ corresponds to a TRS(PHS), $\tilde{B}^{2}$ represents a PHS(TRS). Then
the classes of the above model belong, respectively, to DIII, CI, BDI, and CII
when $d=8m+7$, $d=8m+3$, $d=8m+5$, and $d=8m+1$, in agreement with our
classification table of TIs/TSCs.

To conclude this subsection, all types of TIs/TSCs in our classification table
have been constructed by using Dirac matrices.

###  IV.4 SIII.D. Boundary Modes of Dirac Models and General Index Theorem

Up to now, all types of TIs/TSCs have been constructed by Dirac matrices, and
all of our constructed models may be written in the unified form:

$\mathcal{H}_{G}(\mathbf{k})=\sum_{a=1}^{d}k_{a}\Gamma^{a+b-1}+\left(m-\epsilon
k^{2}\right)\Gamma^{\alpha},$ (39)

where $d$ is the bulk dimension, $b$ is the starting superscript of
$\Gamma_{(2n+1)}$ matrices, and $\Gamma^{\alpha}=\Gamma^{1}$ except in the
cases of $2\mathbf{Z}$-type where $\Gamma^{\alpha}=\tilde{\Gamma}$. The
advantage of this kind of construction in the form of (39) lies in that its
boundary low-energy effective theory can be formulated systematically through
the perturbation theory of quantum mechanics. The method used here is a
generalization of that in Ref.SQS-Book . We consider that a boundary at $x=0$
is on the left of a $d$-dimensional model of 34 and translation invariance is
still preserved along the other $d-1$ directions. To implement the
perturbation method, we will first identify the gapless subspace of the model
residing on the boundary, i.e., concentrated near $x=0$, and then compute the
transition elements in this subspace by regarding the remaining translation
invariant terms as perturbations.

A physical boundary state $\varphi(\mathbf{x})$ has zero energy and should
vanish at $x=0$ and $x\rightarrow+\infty$. If such a state exists, it
satisfies the following equation

$\left[-i\Gamma^{b}\partial_{x}+\left(m+\epsilon\partial_{x}^{2}\right)\Gamma^{1}\right]\varphi(x)=0,$

where momenta along the other directions are set to be zero since only the
ground state is relevant at present. Also note that we have set
$\Gamma^{\alpha}=\Gamma^{1}$ for explicitness, since there is no difference
essential for the cases of $\tilde{\Gamma}$. The above equation can be
rewritten as

$\left[\partial_{x}+\left(m+\epsilon\partial_{x}^{2}\right)i\Gamma^{b}\Gamma^{1}\right]\varphi(x)=0.$

Assuming that $\varphi(x)=\chi_{\eta}f(x)$, where $\eta=\pm$ and $\chi_{\pm}$
is the eigenvector of $i\Gamma^{b}\Gamma^{1}$ with $\pm$ being the
corresponding eigenvalue. Then

$\partial_{x}f(x)+\eta\left(m+\epsilon\partial_{x}^{2}\right)f(x)=0,$

with the boundary conditions:

$f(0)=0\qquad and\qquad f(x)|_{x\rightarrow\infty}=0.$

Seeking solutions with the form $f\sim e^{-\lambda x}$, we have

$\lambda^{2}-\frac{\eta}{\epsilon}\lambda+\frac{m}{\epsilon}=0.$

To satisfy the boundary condition we need the two roots $\lambda_{1,2}$ are
both positive, which requires $\eta=\mathbf{sgn}(\epsilon)$ and $m\epsilon>0$.
Thus the condition of the existence of the boundary states is

$\mathbf{sgn}(m)=\mathbf{sgn}(\epsilon),$ (40)

which is in consistence with the bulk topological number, $e.g.$, expressed in
Eq.(36). It turns out that there exist $2^{n-1}$ degenerate solutions with
degeneracy originated from the internal space:

$\varphi_{i}(x)=\chi_{\mathbf{sgn}(\epsilon)}^{i}\left(e^{-\lambda_{1}x}-e^{-\lambda_{2}x}\right),$

where $i$ labels the $2^{n-1}$ degenerate eigenvectors of
$i\Gamma^{b}\Gamma^{1}$ with eigenvalue $\mathbf{sgn}(\epsilon)$.

To obtain the low-energy effective Hamiltonian on the boundary, we consider
the remaining terms along the other directions of Eq.(34):

$\Delta\mathcal{H}=\sum_{a=2}^{d}k_{a}\Gamma^{a+b-1}-\epsilon\sum_{a=2}^{d}k_{a}^{2}\Gamma^{1}$

as perturbations. Since we are only interested in the low-energy behavior of
the boundary, it is sufficient to implement the perturbation theory of quantum
mechanics in this subspace with zero energy to obtain the low-energy effective
Hamiltonian $\mathcal{H}_{eff}$, that is

$\mathcal{H}_{eff}^{ij}=\langle\chi^{i}|\Delta\mathcal{H}|\chi^{j}\rangle.$

We first show that the quadratic terms vanish as follows. Note that

$\displaystyle\langle\chi^{i}|\Gamma^{1}|\chi^{j}\rangle$ $\displaystyle=$
$\displaystyle\langle\chi^{i}|\Gamma^{2}\Gamma^{2}\Gamma^{1}|\chi^{j}\rangle$
$\displaystyle=$
$\displaystyle-i\langle\chi^{i}|\Gamma^{2}\left(i\Gamma^{2}\Gamma^{1}\right)|\chi^{j}\rangle$
$\displaystyle=$
$\displaystyle-i\mathbf{sgn}(\epsilon)\langle\chi^{i}|\Gamma^{2}|\chi^{j}\rangle,$

and

$\displaystyle\langle\chi^{i}|\Gamma^{1}|\chi^{j}\rangle$ $\displaystyle=$
$\displaystyle\langle\chi^{i}|\Gamma^{1}\Gamma^{2}\Gamma^{2}|\chi^{j}\rangle$
$\displaystyle=$
$\displaystyle-i\langle\chi^{i}|i\Gamma^{1}\Gamma^{2}\Gamma^{2}|\chi^{j}\rangle$
$\displaystyle=$ $\displaystyle
i\langle\chi^{i}|\left(i\Gamma^{2}\Gamma^{1}\right)\Gamma^{2}|\chi^{j}\rangle$
$\displaystyle=$ $\displaystyle
i\mathbf{sgn}(\epsilon)\langle\chi^{i}|\Gamma^{2}|\chi^{j}\rangle.$

As a result,
$\langle\chi^{i}|\Gamma^{1}|\chi^{j}\rangle=-\langle\chi^{i}|\Gamma^{1}|\chi^{j}\rangle$,
and thus the quadratic terms have no contribution to $\mathcal{H}_{eff}$. For
the linear terms of $\Delta\mathcal{H}$, noting that

$\left[i\Gamma^{b}\Gamma^{1},\Gamma^{a}\right]=0,\qquad for\quad a\neq
1\;or\;b,$

all $\Gamma^{a}$s with $a\neq 1\;or\;b$ can be diagonalized into
$2^{n-1}\times 2^{n-1}$ blocks as

$\Gamma^{a}=\begin{pmatrix}\Gamma_{+}^{a}\\\ &\Gamma_{-}^{a}\end{pmatrix},$

where each block satisfies

$\left\\{\Gamma_{\eta}^{a},\Gamma_{\eta}^{b}\right\\}=2\delta^{ab}\mathbf{1}_{2^{n-1}\times
2^{n-1}},\quad a,b\neq 1\;or\;b.$

Thus either $\left\\{\Gamma_{+}^{a}\right\\}$ or
$\left\\{\Gamma_{-}^{a}\right\\}$ forms a $2^{n-1}\times 2^{n-1}$
representation of the Clifford algebra. Thus up to a unitary transformation,
we can express the effective theory as

$\mathcal{H}_{eff}=\frac{1}{2}\left(\mathbf{sgn}(m)+\mathbf{sgn}(\epsilon)\right)\sum_{a=2}^{d}k_{a}\Gamma_{(2n-1)}^{a+b-3},$
(41)

subject to the condition of Eq.(40). Note that for $2\mathbf{Z}$ cases, the
first $\Gamma$ matrix in the above expression should be $\tilde{\Gamma}$.

It is a crucial to observate that the low-energy effective theory (41), which
is obtained on the boundary of the unified model (39) of TIs/TSCs, takes also
a unified form for all types of FSs, $i.e.$, models (21), (24), (25), (28),
(29), (31), and (33). Since our final aim is to prove the general index
theorem, namely, to show that the same topological information is encoded in
both bulk and boundary, our remaining task is to look into the topological
charge of the Fermi point with the obtained effective theory. To be explicit,
we shall complete two tasks: the first one is to identify the symmetry
operators of a boundary effective theory according to those of its bulk
Hamiltonian; the second one is to check whether the topological charge of the
Fermi point on the boundary matches the bulk topological number on both
magnitude and topological type. The first one can be done by comparing the
symmetries and their signs, and thereby it turns out that, for a given
symmetry case, the corresponding symmetry operators are just those we defined
previously in the constructions of that kind of FS with unit topological
charge. This can be clearly seen from the fact that our constructions of Fermi
surfaces and TIs/TSCs matches to our classification table of both FSs and
TIs/TSCs, where the symmetry situation of an FS of codimension $d-1$ is the
same as that of a $d$-dimensional TI/TSC. For the second task, again from our
classification table of FSs and TIs/TSCs, we see that the topological charge
of the Fermi point has the same type as that of the corresponding bulk
topological number, and furthermore their numerical equality is obvious. Thus
we finally have the following general index theorem:

$\nu(i,d-1)=N(i,d),$

where $i$ is the index of the symmetry classes, $d-1$ is the codimension of
the Fermi point on the boundary, and $d$ is the spatial dimension, for a
concerned TI/TSC.

## References

  * (1) M. Nakahara, Geometry, Topology, and Physics (Hilger, London, 1990).
  * (2) Y. X. Zhao and Z. D. Wang, Phys. Rev. Lett. 110, 240404 (2013).
  * (3) J. H. Park, Lecture note on Clifford algebra.
  * (4) S. Ryu, A. P. Schnyder, A. Furusaki and A. W. W Ludwig, New J. Phys. 12, 065010 (2010).
  * (5) For lattice models, the expansion near the closest points in the spectrum is just a collection of our continuous models, which characterizes the low-energy behavior.
  * (6) S. Q. Shen, Topological Insulators: Dirac Equation in Condensed Matters(Springer, 2013).