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Symmetry-protected topological (SPT) order is a kind of order in zero-temperature quantum-mechanical states of matter that have a symmetry and a finite energy gap.

To derive the results in a most-invariant way, renormalization group methods are used (leading to equivalence classes corresponding to certain fixed points). The SPT order has the following defining properties:

(a) distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry.
(b) however, they all can be smoothly deformed into the same trivial product state without a phase transition, if the symmetry is broken during the deformation.

The above definition works for both bosonic systems and fermionic systems, which leads to the notions of bosonic SPT order and fermionic SPT order.

Using the notion of quantum entanglement, we can say that SPT states are short-range entangled states with a symmetry (by contrast: for long-range entanglement see topological order, which is not related to the famous EPR paradox). Since short-range entangled states have only trivial topological orders we may also refer the SPT order as Symmetry Protected "Trivial" order.

Characteristic properties

1. The boundary effective theory of a non-trivial SPT state always has pure gauge anomaly or mixed gauge-gravity anomaly for the symmetry group. As a result, the boundary of a SPT state is either gapless or degenerate, regardless how we cut the sample to form the boundary. A gapped non-degenerate boundary is impossible for a non-trivial SPT state. If the boundary is a gapped degenerate state, the degeneracy may be caused by spontaneous symmetry breaking and/or (intrinsic) topological order.
2. Monodromy defects in non-trivial 2+1D SPT states carry non-trival statistics and fractional quantum numbers of the symmetry group. Monodromy defects are created by twisting the boundary condition along a cut by a symmetry transformation. The ends of such cut are the monodromy defects. For example, 2+1D bosonic Z_n SPT states are classified by a Z_n integer m. One can show that n identical elementary monodromy defects in a Z_n SPT state labeled by m will carry a total Z_n quantum number 2m which is not a multiple of n.
3. 2+1D bosonic U(1) SPT states have a Hall conductance that is quantized as an even integer. 2+1D bosonic SO(3) SPT states have a quantized spin Hall conductance.

Relation between SPT order and (intrinsic) topological order

SPT states are short-range entangled while topologically ordered states are long-range entangled. Both intrinsic topological order, and also SPT order, can sometimes have protected gapless boundary excitations. The difference is subtle: the gapless boundary excitations in intrinsic topological order can be robust against any local perturbations, while the gapless boundary excitations in SPT order are robust only against local perturbations that do not break the symmetry. So the gapless boundary excitations in intrinsic topological order are topologically protected, while the gapless boundary excitations in SPT order are symmetry protected.

We also know that an intrinsic topological order has emergent fractional charge, emergent fractional statistics, and emergent gauge theory. In contrast, a SPT order has no emergent fractional charge/fractional statistics for finite-energy excitations, nor emergent gauge theory (due to its short-range entanglement). Note that the monodromy defects discussed above are not finite-energy excitations in the spectrum of the Hamiltonian, but defects created by modifying the Hamiltonian.

Examples

The first example of SPT order is the Haldane phase of odd-integer spin chain. It is a SPT phase protected by SO(3) spin rotation symmetry. Note that Haldane phases of even-integer-spin chain do not have SPT order. A more well known example of SPT order is the topological insulator of non-interacting fermions, a SPT phase protected by U(1) and time reversal symmetry.

On the other hand, fractional quantum Hall states are not SPT states. They are states with (intrinsic) topological order and long-range entanglements.

Group cohomology theory for SPT phases

Using the notion of quantum entanglement, one obtains the following general picture of gapped phases at zero temperature. All gapped zero-temperature phases can be divided into two classes: long-range entangled phases (ie phases with intrinsic topological order) and short-range entangled phases (ie phases with no intrinsic topological order). All short-range entangled phases can be further divided into three classes: symmetry-breaking phases, SPT phases, and their mix (symmetry breaking order and SPT order can appear together).

It is well known that symmetry-breaking orders are described by group theory. For bosonic SPT phases with pure gauge anomalous boundary, it was shown that they are classified by group cohomology theory: those (d+1)D SPT states with symmetry G are labeled by the elements in group cohomology class $H^{d+1}[G,U(1)]$. For other (d+1)D SPT states with mixed gauge-gravity anomalous boundary, they can be described by $\oplus _{k=1}^{d}H^{k}[G,iTO^{d+1-k}]$, where $iTO^{d+1}$ is the Abelian group formed by (d+1)D topologically ordered phases that have no non-trivial topological excitations (referred as iTO phases).

From the above results, many new quantum states of matter are predicted, including bosonic topological insulators (the SPT states protected by U(1) and time-reversal symmetry) and bosonic topological superconductors (the SPT states protected by time-reversal symmetry), as well as many other new SPT states protected by other symmetries.

A list of bosonic SPT states from group cohomology $H^{d+1}[G,U(1)]\oplus _{k=1}^{d}H^{k}[G,iTO^{d+1-k}]$ ($Z_{2}^{T}$ = time-reversal-symmetry group)

symmetry group	1+1D	2+1D	3+1D	4+1D	comment
$0$	$0$	$Z$	$0$	$Z_{2}$	iTO phases with no symmetry: $iTO^{d+1}$
$U(1)\rtimes Z_{2}^{T}$	$Z_{2}$	$Z_{2}$	$2Z_{2}+Z_{2}$	$Z\oplus Z_{2}+Z$	bosonic topological insulator
$Z_{2}^{T}$	$Z_{2}$	$0$	$Z_{2}+Z_{2}$	$0$	bosonic topological superconductor
$Z_{n}$	$0$	$Z_{n}$	$0$	$Z_{n}+Z_{n}$
$U(1)$	$0$	$Z$	$0$	$Z+Z$	2+1D: quantum Hall effect
$SO(3)$	$Z_{2}$	$Z$	$0$	$Z_{2}$	1+1D: odd-integer-spin chain; 2+1D: spin Hall effect
$SO(3)\times Z_{2}^{T}$	$2Z_{2}$	$Z_{2}$	$3Z_{2}+Z_{2}$	$2Z_{2}$
$Z_{2}\times Z_{2}\times Z_{2}^{T}$	$4Z_{2}$	$6Z_{2}$	$9Z_{2}+Z_{2}$	$12Z_{2}+2Z_{2}$

The phases before "+" come from $H^{d+1}[G,U(1)]$. The phases after "+" come from $\oplus _{k=1}^{d}H^{k}[G,iTO^{d+1-k}]$. Just like group theory can give us 230 crystal structures in 3+1D, group cohomology theory can give us various SPT phases in any dimensions with any on-site symmetry groups.

On the other hand, the fermionic SPT orders are described by group super-cohomology theory. So the group (super-)cohomology theory allows us to construct many SPT orders even for interacting systems, which include interacting topological insulator/superconductor.

A complete classification of 1D gapped quantum phases (with interactions)

Using the notions of quantum entanglement and SPT order, one can obtain a complete classification of all 1D gapped quantum phases.

First, it is shown that there is no (intrinsic) topological order in 1D (ie all 1D gapped states are short-range entangled). Thus, if the Hamiltonians have no symmetry, all their 1D gapped quantum states belong to one phase—the phase of trivial product states. On the other hand, if the Hamiltonians do have a symmetry, their 1D gapped quantum states are either symmetry-breaking phases, SPT phases, and their mix.

Such an understanding allows one to classify all 1D gapped quantum phases: All 1D gapped phases are classified by the following three mathematical objects: $(G_{H},G_{\Psi },H^{2}[G_{\Psi },U(1)])$ , where $G_{H}$ is the symmetry group of the Hamiltonian, $G_{\Psi }$ the symmetry group of the ground states, and $H^{2}[G_{\Psi },U(1)]$ the second group cohomology class of $G_{\Psi }$. (Note that $H^{2}[G,U(1)]$ classifies the projective representations of $G$.) If there is no symmetry breaking (ie $G_{\Psi }=G_{H}$), the 1D gapped phases are classified by the projective representations of symmetry group $G_{H}$.

topological order

In physics, topological order describes a state or phase of matter that arises system with non-local interactions, such as entanglement in quantum mechanics, and floppy modes in elastic systems . Whereas classical phases of matter such as gases and solids correspond to microscopic patterns in the spatial arrangement of particles arising from short range interactions, topological orders correspond to patterns of long-range quantum entanglement. States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition.

Technically, topological order occurs at zero temperature. Various topologically ordered states have interesting properties, such as (1) ground state degeneracy and fractional statistics or non-abelian group statistics that can be used to realize a topological quantum computer; (2) perfect conducting edge states that may have important device applications; (3) emergent gauge field and Fermi statistics that suggest a quantum information origin of elementary particles; (4) topological entanglement entropy that reveals the entanglement origin of topological order, etc. Topological order is important in the study of several physical systems such as spin liquids, and the quantum Hall effect, along with potential applications to fault-tolerant quantum computation.

Topological insulators and topological superconductors (beyond 1D) do not have topological order as defined above, their entanglements being only short-ranged, but are examples of symmetry-protected topological order.

Discovery and characterization

However, since the late 1980s, it has become gradually apparent that Landau symmetry-breaking theory may not describe all possible orders. In an attempt to explain high temperature superconductivity the chiral spin state was introduced. and the quantum Hall effect, along with potential applications to fault-tolerant quantum computation. (4) topological entanglement entropy that reveals the entanglement origin of topological order, etc. Topological order is important in the study of several physical systems such as spin liquids, and the quantum Hall effect, along with potential applications to fault-tolerant quantum computation.At first, physicists still wanted to use Landau symmetry-breaking theory to describe the chiral spin state. They identified the chiral spin state as a state that breaks the time reversal and parity symmetries, but not the spin rotation symmetry. This should be the end of the story according to Landau's symmetry breaking description of orders. However, it was quickly realized that there are many different chiral spin states that have exactly the same symmetry, so symmetry alone was not enough to characterize different chiral spin states. This means that the chiral spin states contain a new kind of order that is beyond the usual symmetry description. The proposed, new kind of order was named "topological order". describes a state or phase of matter that arises system with non-local interactions, such as entanglement in quantum mechanics, and floppy modes in elastic systems The name "topological order" is motivated by the low energy effective theory of the chiral spin states which is a topological quantum field theory (TQFT). New quantum numbers, such as ground state degeneracy (which can be defined on a closed space or an open space with gapped boundaries, including both Abelian topological orders and non-Abelian topological orders) and the non-Abelian geometric phase of degenerate ground states, were introduced to characterize and define the different topological orders in chiral spin states. More recently, it was shown that topological orders can also be characterized by topological entropy.

But experiments soon indicated that chiral spin states do not describe high-temperature superconductors, and the theory of topological order became a theory with no experimental realization. However, the similarity between chiral spin states and quantum Hall states allows one to use the theory of topological order to describe different quantum Hall states. Just like chiral spin states, different quantum Hall states all have the same symmetry and are outside the Landau symmetry-breaking description. One finds that the different orders in different quantum Hall states can indeed be described by topological orders, so the topological order does have experimental realizations.

The fractional quantum Hall (FQH) state was discovered in 1982 before the introduction of the concept of topological order in 1989. But the FQH state is not the first experimentally discovered topologically ordered state. The superconductor, discovered in 1911, is the first experimentally discovered topologically ordered state; it has Z₂ topological order.

Although topologically ordered states usually appear in strongly interacting boson/fermion systems, a simple kind of topological order can also appear in free fermion systems. This kind of topological order corresponds to integral quantum Hall state, which can be characterized by the Chern number of the filled energy band if we consider integer quantum Hall state on a lattice. Theoretical calculations have proposed that such Chern numbers can be measured for a free fermion system experimentally. It is also well known that such a Chern number can be measured (maybe indirectly) by edge states.

The most important characterization of topological orders would be the underlying fractionalized excitations (such as anyons) and their fusion statistics and braiding statistics (which can go beyond the quantum statistics of bosons or fermions). Current research works show that the loop and string like excitations exist for topological orders in the 3+1 dimensional spacetime, and their multi-loop/string-braiding statistics are the crucial signatures for identifying 3+1 dimensional topological orders. The multi-loop/string-braiding statistics of 3+1 dimensional topological orders can be captured by the link invariants of particular topological quantum field theory in 4 spacetime dimensions.

Mechanism

A large class of 2+1D topological orders is realized through a mechanism called string-net condensation. This class of topological orders can have a gapped edge and are classified by unitary fusion category (or monoidal category) theory. One finds that string-net condensation can generate infinitely many different types of topological orders, which may indicate that there are many different new types of materials remaining to be discovered.

The collective motions of condensed strings give rise to excitations above the string-net condensed states. Those excitations turn out to be gauge bosons. The ends of strings are defects which correspond to another type of excitations. Those excitations are the gauge charges and can carry Fermi or fractional statistics.

The condensations of other extended objects such as "membranes", "brane-nets", and fractals also lead to topologically ordered phases and "quantum glassiness".

Mathematical formulation

We know that group theory is the mathematical foundation of symmetry-breaking orders. What is the mathematical foundation of topological order? It was found that a subclass of 2+1D topological orders—Abelian topological orders—can be classified by a K-matrix approach. The string-net condensation suggests that tensor category (such as fusion category or monoidal category) is part of the mathematical foundation of topological order in 2+1D. The more recent researches suggest that (up to invertible topological orders that have no fractionalized excitations):

• 2+1D bosonic topological orders are classified by unitary modular tensor categories.
• 2+1D bosonic topological orders with symmetry G are classified by G-crossed tensor categories.
• 2+1D bosonic/fermionic topological orders with symmetry G are classified by unitary braided fusion categories over symmetric fusion category, that has modular extensions. The symmetric fusion category Rep(G) for bosonic systems and sRep(G) for fermionic systems.

Topological order in higher dimensions may be related to n-Category theory. Quantum operator algebra is a very important mathematical tool in studying topological orders.

Some also suggest that topological order is mathematically described by extended quantum symmetry.

periodic table of topological insulators and topological superconductors

The periodic table of topological insulators and topological superconductors, also called tenfold classification of topological insulators and superconductors, is an application of topology to condensed matter physics. It indicates the mathematical group for the topological invariant of the topological insulators and topological superconductors, given a dimension and discrete symmetry class. The ten possible discrete symmetry families are classified according to three main symmetries: particle-hole symmetry, time-reversal symmetry and chiral symmetry. The table was developed between 2008–2010 by the collaboration of Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki and Andreas W. W. Ludwig; and independently by Alexei Kitaev.

Overview

Periodic table of topological insulators and superconductors (1D up to 3D)

Symmetry class	Operation			Dimension
	$T^{2}$	$C^{2}$	$S^{2}$	1	2	3
A	X	X	X	$0$	$\mathbb {Z}$	$0$
AIII	X	X	1	$\mathbb {Z}$	$0$	$\mathbb {Z}$
AI	1	X	X	$0$	$0$	$0$
BDI	1	1	1	$\mathbb {Z}$	$0$	$0$
D	X	1	X	$\mathbb {Z} _{2}$	$\mathbb {Z}$	$0$
DIII	-1	1	1	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$\mathbb {Z}$
AII	-1	X	X	$0$	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$
CII	-1	-1	1	$2\mathbb {Z}$	$0$	$\mathbb {Z} _{2}$
C	X	-1	X	$0$	$2\mathbb {Z}$	$0$
CI	1	-1	1	$0$	$0$	$2\mathbb {Z}$

These table applies to topological insulators and topological superconductors with an energy gap, when particle-particle interactions are excluded. The table is no longer valid when interactions are included.

The topological insulators and superconductors are classified here in ten symmetry classes (A,AII,AI,BDI,D,DIII,AII,CII,C,CI) named after Altland–Zirnbauer classification, defined here by the properties of the system with respect to three operators: the time-reversal operator $T$, charge conjugation $C$ and chiral symmetry $S$. The symmetry classes are ordered according to the Bott clock (see below) so that the same values repeat in the diagonals.

An X in the table of "Symmetries" indicates that the Hamiltonian of the symmetry is broken with respect to the given operator. A value of ±1 indicates the value of the operator squared for that system.

The dimension indicates the dimensionality of the systes: 1D (chain), 2D (plane) and 3D lattices. It can be extended up to any number of positive integer dimension. Below, there can be four possible group values that are tabulated for a given class and dimension:

• A value of 0 indicates that there is no topological phase for that class and dimension.
• The group $\mathbb {Z}$ indicates that the topological invariant can take integer values (e.g. ±0,±1,±2,...).
• The group of $2\mathbb {Z}$ indicates that the topological invariant can take even values (e.g. ±0,±2,±4,...).
• The group of $\mathbb {Z} _{2}$ indicates that the topological invariant can take two values (e.g ±1).

Physical examples

The non-chiral Su–Schrieffer–Heeger model ($d=1$), can be associated with symmetry class BDI with an integer $\mathbb {Z}$ topological invariant due to gauge invariance. The problem is similar to the integer quantum Hall effect and the quantum anomalous Hall effect (both in $d=2$) which are A class, with integer $\mathbb {Z}$ Chern number.

Contrarily, the Kitaev chain ($d=1$), is an example of symmetry class D, with a $\mathbb {Z} _{2}$ binary topological invariant. Similarly, the $p_{x}+ip_{y}$ superconductors ($d=2$) are also in class D, but with a $\mathbb {Z}$ topological invariant.

The quantum spin Hall effect ($d=2$) described by Kane–Mele model is an example of AII class, with a $\mathbb {Z} _{2}$ topological invariant.

Construction

Discrete symmetry classes

There are ten discrete symmetry classes of topological insulators and superconductors, corresponding to the ten Altland–Zirnbauer classes of random matrices. They are defined by three symmetries of the Hamiltonian ${\hat {H}}=\sum _{i,j}H_{ij}c_{i}^{\dagger }c_{j}$, (where $c_{i}$, and $c_{i}^{\dagger }$, are the annihilation and creation operators of mode $i$, in some arbitrary spatial basis) : time-reversal symmetry, particle-hole (or charge conjugation) symmetry, and chiral (or sublattice) symmetry.

• Chiral symmetry is a unitary operator $S$, that acts on $c_{i}$, as a unitary rotation ($Sc_{i}S^{-1}=(U_{S})_{ij}c_{j}$,) and satisfies $S^{2}=1$. A Hamiltonian $H$ possesses chiral symmetry when $S{\hat {H}}S^{-1}=-{\hat {H}}$, for some choice of $S$ (on the level of first-quantised Hamiltonians, this means $U_{S}$ and $H$ are anticommuting matrices).

• Time-reversal symmetry (TRS) is an antiunitary operator $T$, that acts on $\alpha c_{i}$, (where $\alpha$, is an arbitrary complex coefficient, and $^{*}$, denotes complex conjugation) as $T\alpha c_{i}T^{-1}=\alpha ^{*}{(U_{T})}_{ij}c_{j}$. It can be written as $T=U_{T}{\mathcal {K}}$ where ${\mathcal {K}}$ is the complex conjugation operator and $U_{T}$ is a unitary matrix. Either $T^{2}=1$ or $T^{2}=-1$. A Hamiltonian with time reversal symmetry satisfies $T{\hat {H}}T^{-1}={\hat {H}}$, or on the level of first-quantised matrices, $U_{T}H^{*}U_{T}^{-1}=H$, for some choice of $U_{T}$.

• Charge conjugation or particle-hole symmetry (PHS) $C$ is also an antiunitary operator which acts on $\alpha c_{i}$ as $C\alpha c_{i}C^{-1}=\alpha ^{*}(U_{C}^{\dagger })_{ji}c_{j}$, and can be written as $C=U_{C}{\mathcal {K}}$ where $U_{C}$ is unitary. Again either $C^{2}=1$ or $C^{2}=-1$ depending on what $U_{C}$ is. A Hamiltonian with particle hole symmetry satisfies $C{\hat {H}}C^{-1}=-{\hat {H}}$, or on the level of first-quantised Hamiltonian matrices, $U_{C}H^{*}U_{C}^{-1}=-H$, for some choice of $U_{C}$.

In the Bloch Hamiltonian formalism for crystal structures, where the Hamiltonian $H(k)$ acts on modes of crystal momentum $k$, the chiral symmetry, TRS, and PHS conditions become

• $U_{S}H(k)U_{S}^{-1}=-H(k)$ (chiral symmetry)

• $U_{T}H(k)^{*}U_{T}^{-1}=H(-k)$ (time-reversal symmetry),

• $U_{C}H(k)^{*}U_{C}^{-1}=-H(-k)$ (particle-hole symmetry).

It is evident that if two of these three symmetries are present, then the third is also present, due to the relation $S=TC$.

The aforementioned discrete symmetries label 10 distinct discrete symmetry classes, which coincide with the Altland–Zirnbauer classes of random matrices.

Symmetry class	Time reversal symmetry	Particle hole symmetry	Chiral symmetry
A	No	No	No
AIII	No	No	Yes
AI	Yes, $T^{2}=1$	No	No
BDI	Yes, $T^{2}=1$	Yes, $C^{2}=1$	Yes
D	No	Yes, $C^{2}=1$	No
DIII	Yes, $T^{2}=-1$	Yes, $C^{2}=1$	Yes
AII	Yes, $T^{2}=-1$	No	No
CII	Yes, $T^{2}=-1$	Yes, $C^{2}=-1$	Yes
C	No	Yes, $C^{2}=-1$	No
CI	Yes, $T^{2}=1$	Yes, $C^{2}=-1$	Yes

Equivalence classes of Hamiltonians

A bulk Hamiltonian in a particular symmetry group is restricted to be a Hermitian matrix with no zero-energy eigenvalues (i.e. so that the spectrum is "gapped" and the system is a bulk insulator) satisfying the symmetry constraints of the group. In the case of $d>0$ dimensions, this Hamiltonian is a continuous function $H(k)$ of the $d$ parameters in the Bloch momentum vector ${\vec {k}}$ in the Brillouin zone; then the symmetry constraints must hold for all ${\vec {k}}$.

Given two Hamiltonians $H_{1}$ and $H_{2}$, it may be possible to continuously deform $H_{1}$ into $H_{2}$ while maintaining the symmetry constraint and gap (that is, there exists continuous function $H(t,{\vec {k}})$ such that for all $0\leq t\leq 1$ the Hamiltonian has no zero eigenvalue and symmetry condition is maintained, and $H(0,{\vec {k}})=H_{1}({\vec {k}})$ and $H(1,{\vec {k}})=H_{2}({\vec {k}})$). Then we say that $H_{1}$ and $H_{2}$ are equivalent.

However, it may also turn out that there is no such continuous deformation. in this case, physically if two materials with bulk Hamiltonians $H_{1}$ and $H_{2}$, respectively, neighbor each other with an edge between them, when one continuously moves across the edge one must encounter a zero eigenvalue (as there is no continuous transformation that avoids this). This may manifest as a gapless zero energy edge mode or an electric current that only flows along the edge.

An interesting question is to ask, given a symmetry class and a dimension of the Brillouin zone, what are all the equivalence classes of Hamiltonians. Each equivalence class can be labeled by a topological invariant; two Hamiltonians whose topological invariant are different cannot be deformed into each other and belong to different equivalence classes.

Classifying spaces of Hamiltonians

For each of the symmetry classes, the question can be simplified by deforming the Hamiltonian into a "projective" Hamiltonian, and considering the symmetric space in which such Hamiltonians live. These classifying spaces are shown for each symmetry class:

Symmetry class	Classifying space	$\pi _{0}$of Classifying space
A	$\bigcup _{n}\mathrm {U} (N)/(\mathrm {U} (N-n)\times \mathrm {U} (n))$	$\mathbb {Z}$
AIII	$\mathrm {U} (N)$	$0$
AI	$\bigcup _{n}\mathrm {O} (N)/(\mathrm {O} (N-n)\times \mathrm {O} (n))$	$\mathbb {Z}$
BDI	$\mathrm {O} (N)$	$\mathbb {Z} _{2}$
D	$\mathrm {O} (2N)/\mathrm {U} (N)$	$\mathbb {Z} _{2}$
DIII	$\mathrm {U} (N)/\mathrm {Sp} (N)$	$0$
AII	$\bigcup _{n}\mathrm {Sp} (N)/(\mathrm {Sp} (N-n)\times \mathrm {Sp} (n))$	$\mathbb {Z}$
CII	$\mathrm {Sp} (N)$	$0$
C	$\mathrm {Sp} (2N)/\mathrm {U} (N)$	$0$
CI	$\mathrm {U} (N)/\mathrm {O} (N)$	$0$

For example, a (real symmetric) Hamiltonian in symmetry class AI can have its $n$ positive eigenvalues deformed to +1 and its $N-n$ negative eigenvalues deformed to -1; the resulting such matrices are described by the union of real Grassmannians $\bigcup _{n=0}^{\infty }\mathrm {Gr} (n,N)=\bigcup _{n=0}^{\infty }\mathrm {O} (N)/\mathrm {\mathrm {O} } (n)\times \mathrm {\mathrm {O} } (N-n)$

Classification of invariants

The strong topological invariants of a many-band system in $d$ dimensions can be labeled by the elements of the $d$-th homotopy group of the symmetric space. These groups are displayed in this table, called the periodic table of topological insulators:

Symmetry class	$d=0$	$d=1$	$d=2$	$d=3$	$d=4$	$d=5$	$d=6$	$d=7$	$d=8$
A	$\mathbb {Z}$	$0$	$\mathbb {Z}$	$0$	$\mathbb {Z}$	$0$	$\mathbb {Z}$	$0$	$\mathbb {Z}$
AIII	$0$	$\mathbb {Z}$	$0$	$\mathbb {Z}$	$0$	$\mathbb {Z}$	$0$	$\mathbb {Z}$	$0$
AI	$\mathbb {Z}$	$0$	$0$	$0$	$2\mathbb {Z}$	$0$	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$\mathbb {Z}$
BDI	$\mathbb {Z} _{2}$	$\mathbb {Z}$	$0$	$0$	$0$	$2\mathbb {Z}$	$0$	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$
D	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$\mathbb {Z}$	$0$	$0$	$0$	$2\mathbb {Z}$	$0$	$\mathbb {Z} _{2}$
DIII	$0$	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$\mathbb {Z}$	$0$	$0$	$0$	$2\mathbb {Z}$	$0$
AII	$2\mathbb {Z}$	$0$	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$\mathbb {Z}$	$0$	$0$	$0$	$2\mathbb {Z}$
CII	$0$	$2\mathbb {Z}$	$0$	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$\mathbb {Z}$	$0$	$0$	$0$
C	$0$	$0$	$2\mathbb {Z}$	$0$	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$\mathbb {Z}$	$0$	$0$
CI	$0$	$0$	$0$	$2\mathbb {Z}$	$0$	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$\mathbb {Z}$	$0$

There may also exist weak topological invariants (associated to the fact that the suspension of the Brillouin zone is in fact equivalent to a $d+1$ sphere wedged with lower-dimensional spheres), which are not included in this table. Furthermore, the table assumes the limit of an infinite number of bands, i.e. involves $N\times N$ Hamiltonians for $N\to \infty$.

The table also is periodic in the sense that the group of invariants in $d$ dimensions is the same as the group of invariants in $d+8$ dimensions. In the case of no anti-unitary symmetries, the invariant groups are periodic in dimension by 2.

For nontrivial symmetry classes, the actual invariant can be defined by one of the following integrals over all or part of the Brillouin zone: the Chern number, the Wess-Zumino winding number, the Chern–Simons invariant, the Fu–Kane invariant.

Dimensional reduction and Bott clock

The periodic table also displays a peculiar property: the invariant groups in $d$ dimensions are identical to those in $d-1$ dimensions but in a different symmetry class. Among the complex symmetry classes, the invariant group for A in $d$ dimensions is the same as that for AIII in $d-1$ dimensions, and vice versa. One can also imagine arranging each of the eight real symmetry classes on the Cartesian plane such that the $x$ coordinate is $T^{2}$ if time reversal symmetry is present and $0$ if it is absent, and the $y$ coordinate is $C^{2}$ if particle hole symmetry is present and $0$ if it is absent. Then the invariant group in $d$ dimensions for a certain real symmetry class is the same as the invariant group in $d-1$ dimensions for the symmetry class directly one space clockwise. This phenomenon was termed the Bott clock by Alexei Kitaev, in reference to the Bott periodicity theorem.

Eightfold Bott clock (bold classes are chiral)

PHS TRS	-1	X	1
-1	CII	AII	DII
X	C		D
1	CI	AI	BDI

The Bott clock can be understood by considering the problem of Clifford algebra extensions. Near an interface between two inequivalent bulk materials, the Hamiltonian approaches a gap closing. To lowest order expansion in momentum slightly away from the gap closing, the Hamiltonian takes the form of a Dirac Hamiltonian $H_{\text{Dirac}}({\vec {k}})=\sum _{j=1}^{d}\Gamma _{j}v_{j}k_{j}+m\Gamma _{0}$. Here, $\Gamma _{1},\Gamma _{2},\ldots ,\Gamma _{d}$ are a representation of the Clifford Algebra $\lbrace \Gamma _{i},\Gamma _{j}\rbrace =2\delta _{ij}$, while $m\Gamma _{0}$ is an added "mass term" that and anticommutes with the rest of the Hamiltonian and vanishes at the interface (thus giving the interface a gapless edge mode at $k=0$). The $m\Gamma _{0}$ term for the Hamiltonian on one side of the interface cannot be continuously deformed into the $m\Gamma _{0}$ term for the Hamiltonian on the other side of the interface. Thus (letting $m$ be an arbitrary positive scalar) the problem of classifying topological invariants reduces to the problem of classifying all possible inequivalent choices of $\Gamma _{0}$ to extend the Clifford algebra to one higher dimension, while maintaining the symmetry constraints.

References

• Altland, Alexander; Zirnbauer, Martin R. Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures. Physical Review B. 1997, 55 (2): 1142. Bibcode:1997PhRvB..55.1142A. S2CID 96427496. arXiv:cond-mat/9602137 . doi:10.1103/PhysRevB.55.1142. 

External links

• Baez, John C. The Tenfold Way (Part 1). The n-Category Café. 2014-07-19 [2018-10-26].