We describe the homotopy classes of loops in the space of 2 × 2 simple (=non-degenerate) matrices with various symmetries. This turns out to be an elementary exercise in the homotopy of closed curves in $R3/{0}$. Since closed curves in $R3/{0}$ can be readily visualized, no advanced tools of algebraic topology are needed. The matrices represent gapped Bloch Hamiltonians in 1D with a two dimensional Hilbert space per unit cell.

## I. INTRODUCTION

We study the homotopy classes of loops in the space of simple 2 × 2 matrices that satisfy various unitary or anti-unitary symmetries. This is motivated by the program of classifying the phases of insulating quantum matter.^{1–4} The classification is enriched by allowing for phases with symmetry.^{5–8}

An insulator is characterized by a (many body) ground state separated from the excited states by a spectral gap which is bounded away from zero in the limit of large systems. The ground states of two insulators belong to the same phase if it is possible to interpolate between the two ground states without closing the gap. One can picture a phase diagram in a space of Hamiltonians with all possible parameters; the problem of finding phases can be described topologically as finding the connected components of the region corresponding to Hamiltonians with a gap. This is a challenging mathematical program, especially for interacting systems, as it concerns many-body quantum mechanics in the thermodynamic limit.^{11–18}

The phases of topological insulators are of interest for the field of condensed matter because of their mobile edge states that are impervious to disorder^{19,20} and the search for quasi-particles with exotic statistics.^{10,21,22} They are also of interest in quantum information theory because of their connections with computational complexity of many-body systems.^{23} (Besides the connected components of the space of Hamiltonians with a nonzero gap, further topological properties of this space are also interesting; for example, its fundamental group plays a role in quantized pumping^{9} and topologically protected quantum gates.^{10})

Arguably the simplest setting for studying phases of insulators is in the context of Bloch Hamiltonians for non-interacting Fermions.^{1,2} (For the study of models of free Fermions in disordered systems see e.g., Refs. 24 and 25.) Because of their discrete translation symmetry they can be studied directly in the limit of infinite systems. If the *m* lowest bands (for some *m*) are separated from the rest by a gap, an insulator with *m* Fermions per period can be formed by filling these states. We focus on the simplest case where *m* = 1 (with spinless Fermions) and where the dimension of space is one.

In one dimension, a Bloch band gives rise to a loop *P*(*k*) of orthogonal projections where *k* takes values in the Brillouin zone (BZ). The loop of band projections corresponds to a notion of a quantum state of a system with infinitely many particles.

Suppose that one restricts oneself to Bloch bands with a given symmetry, e.g., time-reversal. Two Bloch bands with projections *P*_{0}(*k*) and *P*_{1}(*k*) are said to be homotopic if there is a family of projections *P*_{t}(*k*), jointly continuous respecting the symmetry, that interpolates between *P*_{0}(*k*) and *P*_{1}(*k*). This corresponds to continuous changes between many-body quantum states of free Fermions. Describing the equivalence classes of Bloch bands with various symmetries is an interesting and non-trivial theoretical problem. (An alternative definition of a homotopy uses the space of loops which has the topology known as the compact-open topology: A homotopy is a continuous path in this space.^{26})

Our modest aim is to give an elementary invitation to this field by describing the homotopic classification of non-degenerate two-band Bloch Hamiltonians in one dimension with various symmetries. The results, summarised in Table I, go somewhat beyond the first column in the periodic table of topological insulators^{3,7,8} for the Altland–Zirnbauer symmetry classes^{5} as we allow for certain space symmetries and some results are specific to two bands (see Appendix B for a comparison with the standard classification). Our methods are elementary, as we reduce the problem to the study of the homotopy of curves in three dimensions. The elementary nature of the problem is a consequence of:

The space of one dimensional projections in $C2$ is easy to visualize. It is simply the two sphere $S2$. In contrast, the space of

*n*×*n*rank*m*projections is the Grassmannian, $Grm(Cn)$, which describes insulators of*m*occupied bands (see Sec. VI), but is more difficult to visualize.One-dimensional Bloch Hamiltonians with two gapped bands can be represented by closed curves in $R3/{0}$ hence questions regarding the homotopy of Bloch Hamiltonians reduce to questions about the homotopy of curves in three dimensional Euclidean space with the origin removed. The study of the homotopy classes of curves in two and three dimensional spaces is the most intuitive part of homotopy theory. This is in contrast with the study of the homotopy of Bloch Hamiltonians in higher dimensions which translates to the study of the homotopy of higher dimensional closed surfaces which are not easily visualized and require more advanced tools. This is true even in the case of two gapped bands, since the high homotopy groups of the 2-sphere are non-trivial.

Symmetry . | Constraint . | Homotopy classes . |
---|---|---|

None | None | {σ_{z}} |

Θ_{+} | H(k) = H*(−k) | {σ_{z}} |

Θ_{−} | H(k) = σ_{y}H*(−k)σ_{y} | Gapless |

C_{+} | H(k) = −σ_{z}H*(−k)σ_{z} | $\xb1\sigma x,\xb1R1$ |

C_{−} | H(k) = −σ_{y}H*(−k)σ_{y} | {σ_{z}} |

B | H(k) = σ_{x}H(−k)σ_{x} | {±σ_{x}, ±R_{1}} |

S | $H(k)=GkH(\u2212k)Gk*$ | {±σ_{z}} |

C_{+}◦Θ_{+} | H(k) = −σ_{z}H(k)σ_{z} | ${Rn|n\u2208Z}$ |

B◦Θ_{+} | H(k) = σ_{x}H*(k)σ_{x} | ${Rn|n\u2208Z}$ |

S◦Θ_{+} | $H(k)=GkH*(k)Gk*$ | {±σ_{z}} |

S ∧ Θ_{+} | $H(k)=GkH(\u2212k)Gk*=H*(\u2212k)$ | {±σ_{z}} |

B ∧ Θ_{+} | H(k) = σ_{x}H(−k)σ_{x} = H*(−k) | ${\xb1Rn|n\u2208Z}$ |

C_{+} ∧ Θ_{+} | H(k) = −σ_{z}H*(−k)σ_{z} = H*(−k) | ${\xb1Rn|n\u2208Z}$ |

C_{−} ∧ Θ_{+} | H(k) = −σ_{y}H*(−k)σ_{y} = H*(−k) | {σ_{z}} |

Symmetry . | Constraint . | Homotopy classes . |
---|---|---|

None | None | {σ_{z}} |

Θ_{+} | H(k) = H*(−k) | {σ_{z}} |

Θ_{−} | H(k) = σ_{y}H*(−k)σ_{y} | Gapless |

C_{+} | H(k) = −σ_{z}H*(−k)σ_{z} | $\xb1\sigma x,\xb1R1$ |

C_{−} | H(k) = −σ_{y}H*(−k)σ_{y} | {σ_{z}} |

B | H(k) = σ_{x}H(−k)σ_{x} | {±σ_{x}, ±R_{1}} |

S | $H(k)=GkH(\u2212k)Gk*$ | {±σ_{z}} |

C_{+}◦Θ_{+} | H(k) = −σ_{z}H(k)σ_{z} | ${Rn|n\u2208Z}$ |

B◦Θ_{+} | H(k) = σ_{x}H*(k)σ_{x} | ${Rn|n\u2208Z}$ |

S◦Θ_{+} | $H(k)=GkH*(k)Gk*$ | {±σ_{z}} |

S ∧ Θ_{+} | $H(k)=GkH(\u2212k)Gk*=H*(\u2212k)$ | {±σ_{z}} |

B ∧ Θ_{+} | H(k) = σ_{x}H(−k)σ_{x} = H*(−k) | ${\xb1Rn|n\u2208Z}$ |

C_{+} ∧ Θ_{+} | H(k) = −σ_{z}H*(−k)σ_{z} = H*(−k) | ${\xb1Rn|n\u2208Z}$ |

C_{−} ∧ Θ_{+} | H(k) = −σ_{y}H*(−k)σ_{y} = H*(−k) | {σ_{z}} |

## II. BLOCH HAMILTONIANS WITH TWO BANDS

*a*∈ {0, 1}.

*H*can be partially diagonalized by Fourier transform (going to momentum space). For the Fourier transform to be well defined one needs to assume locality, i.e.,

*H*(

*k*) defined by

*k*.

*H*(

*k*) is Hermitian for each

*k*since

*H*(

*k*) is not uniquely defined: There is a

*k*-dependent gauge freedom associated with the choice of unit cell as shown in Sec. II A.

### A. Gauge ambiguity of (reduced) Bloch Hamiltonians

*ℓ*= 1,

*H*transforms by

*H*(

*k*) transforms by

*G*

_{k}in

*k*-space is implemented by a diagonal,

*k*-dependent, unitary 2 × 2 matrix:

### B. Periodic matrix valued functions and loops

The space of Hermitian 2 × 2 matrices is a linear space, with trivial topology (it is a contractible space). In contrast, the space of simple (non-degenerate) 2 × 2 matrices has a non-trivial topology. Since any simple matrix can be deformed to a traceless simple matrix, while maintaining simplicity, we assume, without loss, that the (reduced) Bloch matrices we consider are traceless.

**= (**

*σ**σ*

_{x},

*σ*

_{y},

*σ*

_{z}) is a vector of Pauli matrices. The matrix is simple (non-degenerate) if

*γ*in Euclidean 3-space with the origin removed (Fig. 3):

*H*(

*k*) with

*H*(−

*k*) and therefore

**x**

_{k}with

**x**

_{−k}. In order to study deformation of

*γ*that respects the symmetry it is useful to think of the loop

*γ*as the concatenation of two curves

*γ*

_{±}related by the symmetry.

## III. SYMMETRIES

This section is a multilingual dictionary that translates several symmetries of *H* in coordinate space to symmetries of *H*(*k*) and then to symmetries of the loops *γ*. This will allow us to determine the homotopy classes for each type of symmetry in Sec. IV. (Composed symmetries, such as chiral symmetry, are relegated to Appendix A 1).

### A. Time reversal

**Θ**is an anti-unitary map. This is a consequence of the presence of

*i*in the Heisenberg equation

**A**,

**H**are operators. Since ${\Theta ,ddt}=0$ and [

**Θ**,

**H**] = 0 it follows that {

**Θ**,

*i*} = 0 and so

**Θ**is anti-linear.

**Θ**, conjugate the anticommutation relation ⟨

*g*|

*f*⟩ = {

*a*(

*g*),

*a*

^{†}(

*f*)} by

**Θ**:

**Θ**is antiunitary, it acts as complex conjugation on a scalar, so the left-hand side is equal to ⟨

*f*|

*g*⟩ which forces Θ to be antiunitary. Hence

*U*unitary and * complex conjugation.

^{2}, the only phase consistent with the anti-unitarity of Θ is ±1, i.e.,:

_{±}. Under unitary change of bases

*W*

*U*then undergoes a congruence transformation.

*U*acts on the internal coordinates $a$ in real space while leaving the spatial coordinate

*j*unchanged. It follows from [

*H*, Θ] = 0 in real space that

*H*(

*k*) is time-reversal invariant if

*U*= ±

*U*

^{t}.

#### 1. Time reversal Symmetry Θ_{+}

_{+},

*U*is a symmetric matrix. Any symmetric unitary 2 × 2 matrix

*U*is congruent to the identity. This follows from the fact that

*U*can be written as

*U*has a symmetric square root $U=U(\theta /2,\varphi ,\alpha /2)$. It follows that

*U*is congruent to the identity:

*γ*with

*γ*is symmetric under reflection in

*y*. Furthermore 3.13 implies that the end-points of γ

_{−}are in the

*x*–

*z*plane.We will thus consider the homotopies of the anchored curve γ

_{−}(see Fig. 4).

#### 2. Time reversal symmetry *Θ*_{−}

*U*up to multiplication by a scalar. Since an overall phase is for free in an anti-unitary transformation, we may choose

*U*=

*σ*

_{y}

*H*(

*k*) is

*H*(

*k*) is time reversal symmetric if

### B. Particle-hole transformation

**C**maps particle creation to particle annihilation:

*a*

^{†}(

*f*) depends linearly on

*f*while the annihilation operator is anti-linear.

**C**(the many-body operator) must commute with

*i*in order to preserve the direction of time. Thus, in order for both sides to be linearly dependent on

*f*,

*C*(the one-particle operator) must be anti-linear. Being anti-unitary,

*C*comes in two varieties

*C*

_{±}denotes the symmetry corresponding to $C2=\xb11$.

#### 1. Particle-hole symmetry

*H*there is a mirror state with an opposite energy and momentum.

*C*maps from the single particle state to its mirror image. For Bloch Hamiltonians this means

**H**commutes with

**C**. To see this write

*n*by −

*n*,

*C*and Eq. (3.20) that a Bloch Hamiltonian has Particle-Hole symmetry if

#### 2. Particle-Hole Symmetry **C**_{+}

*U*=

*σ*

_{z}, i.e.,

*γ*corresponding to

*γ*is symmetric under rotation by

*π*around the

*x*-axis and is anchored at

*k*= 0,

*π*to the punctured

*x*-axis, see Fig. 5.

The reason for choosing *U* = *σ*_{z} in the definition for Particle-Hole symmetry is to avoid a contradiction with time reversal symmetry. Had we picked the same *U* for both symmetries, the only Hamiltonian symmetric under both would have been *H* = 0.

#### 3. Particle Hole Symmetry **C**_{−}

*H*(

*k*) is

*H*(

*k*) is Particle-Hole symmetric if

*γ*corresponding to Eq. (3.29) is self-retracing

**x**

_{0}and

**x**

_{π}can then be anywhere in $R3/{0}$.

### C. Bond reflection symmetry

*B*is an involution, $B2=1$.

*k*-space is

*H*(

*k*) is bond-reflection symmetric if

*γ*corresponding to Eq. (3.35) is symmetric under a rotation by

*π*about the

*x*-axis:

*x*-axis, see Fig. 5.

### D. Site reflection symmetry

*S*is an involution that mixes different unit cells. Its action on

*H*in real space is given by

*k*-space by

*G*

_{k}is given in Eq. (2.10). Hence,

*H*(

*k*) is site-reflection symmetric if

*γ*

_{−}and

*γ*

_{+}:

*γ*is anchored at one end,

*k*= ±

*π*, to the punctured

*z*-axis

^{27}

*t*

_{1}−

*t*

_{2})

^{2}+

*v*

^{2}> 0. It has bond inversion symmetry if

*v*= 0 and site inversion symmetry if

*t*

_{1}=

*t*

_{2}, as defined by Eqs. (3.35) and (3.41).

## IV. HOMOTOPY OF LOOPS OF GAPPED 2 × 2 MATRICES

Two Bloch Hamiltonians are homotopic if one can be deformed to the other, within a given symmetry class, without closing the gap. By Sec. II B the question reduces to the homotopy of continuous closed loops $\gamma \u2208R3/{0}$ constrained by the symmetry. The symmetry constraint can sometimes complicate the question of homotopy equivalence, see Sec. IV F. Fortunately, for most of the constraints we consider, this is not the case.

When the symmetries exchange *k* and −*k*, the problem is simplified by writing *γ* as the concatenation *γ* = *γ*_{−}◦*γ*_{+} with *γ*_{±} given by Eq. (2.15). The symmetry says that *γ*_{−} determines *γ*_{+} (and vice versa), and imposes constraints on the end-points of *γ*_{±}. In some cases *γ* is also constrained to be planar. Then any homotopy of the full curve *γ* is determined by the homotopy of the arc *γ*_{−}, with the end-points required to satisfy the constraint. The homotopy equivalence of curves *γ*_{−} is readily visualized and the determination of the equivalence classes reduces to an elementary exercise. Homotopy classifications for composite and multiple symmetries are relegated to Appendix A.

### A. No symmetry

$R3/{0}$ is simply connected. This means that every (continuous) loop can be contracted to a point. Since $R3/{0}$ is connected, every point is homotopic to every other point. Hence any loop of 2 × 2 gapped Bloch matrices *H*(*k*) can be contracted to the constant *σ*_{z} (for example), which represents a periodic array of isolated cells.

### B. Time reversal symmetry Θ_{+}

By Sec. III A 1 *γ* is symmetric under reflection in the *y*-axis, and is anchored to the *x*–*z* plane for *k* = 0, *π*. A corresponding curve $\gamma \u0302$ on the unit sphere is obtained by rescaling *γ*. Its end-points are also anchored to the equator in the *x*–*z* plane, see Fig. 4.

$\gamma \u0302\u2212$ can be deformed provided it respects the anchoring to the equator. $\gamma \u0302+$ follows by the symmetry constraint. Suppose first that the anchoring points coincide. Since $S2$ is simply connected, $\gamma \u0302\u2212$ can be contracted to a point. The mirror image $\gamma \u0302+$ will contract simultaneously (Fig. 8).

*k*=

*π*can be brought to the end-point at

*k*= 0 by applying a rotation around the

*y*-axis

*R*

_{y}(

*ϕ*) (since both lie on the unit circle in the

*x*–

*z*plane). The following homotopy changes the curve continously so that the end-points at

*π*and 0 coincide at the end:

*x*–

*z*plane is connected, all points on the equator are homotopic.

It follows that the space of time reversal 1D Bloch Hamiltonians with a gap condition has a single component under homotopy, which we label by a single Pauli matrix {*σ*_{z}}. This accounts for the second line in Table I.

### C. Particle-hole symmetry

#### 1. Particle-Hole symmetry *C*_{+}

*γ*is symmetric under rotation by

*π*about the

*x*-axis, and is anchored to the (punctured)

*x*-axis for

*k*= 0,

*π*.

*γ*is homotopic to $\gamma \u0302$ on the unit sphere, anchored at the poles:

*R*≅

*R*

^{†}by a

*π*rotation about the

*x*axis.) The four elements are distinguished by the combinations of signs of the anchoring points on

*x*.

*σ*

_{x}describe a chain with disconnected cells so the ground states are pure product states

*R*

_{1}when solved on a finite chain turn out to have edge states at the ends.

#### 2. Fermionic Particle-Hole symmetry *C*_{−}

### D. Bond reflection symmetry

### E. Site reflection symmetry

*γ*of Eq. (3.42) is anchored at one point,

*k*= ±

*π*, to the punctured

*z*-axis:

*γ*

_{−}is anchored at the single point

*k*= −

*π*it can be contracted to the anchoring point by the homotopy as shown in Fig. 9,

### F. Composite symmetry *S*◦Θ_{+} : The world-line of the loop

*γ*as a world-line in the four dimensional space $S1\xd7R3$ where the first coordinate is

*k*. A case in point is the composed symmetry

*S*◦Θ

_{+}, where the Bloch Hamiltonian satisfies the constraint

*x*

_{k},

*y*

_{k}):

*z*

_{k}. The constraint implies

*λ*

_{k}is a real-valued function of

*k*.

*γ*may be viewed as the world line in the 3-dimensional manifold $MS\u25e6\Theta +$

*γ*is not a closed curve in the (

*λ*,

*z*) coordinates because the

*λ*–

*z*plane rotates relative to the

*x*–

*y*plane by

*π*as

*k*goes from −

*π*to

*π*. The end-point

**x**

_{−π}=

**x**

_{π}has different coordinates in the (

*λ*,

*z*) plane

The homotopy classes of Hamiltonians with *S*◦Θ symmetry are thus homotopy classes of curves in a punctured *λ*–*z* plane whose end-points are constrained to be mirror-images in the *z*-axis, see Fig. 10.

The end-points can be deformed to be on the +*z*-axis (since any point and its mirror image can be moved continuously to +*z*). This curve is closed so has a winding number. If one of the end-points is taken around the origin *n* times and brought back (while the other end-point mirrors the first), the winding number changes by 2*n*. Thus, there are only two classes, characterized by even and odd winding number around the origin (after the end-points are moved to the +*z*-axis). The curves with even winding number can be contracted to constants +*σ*_{z} and the curves with odd winding number can be contracted to −*σ*_{z} (by moving the two end-points half-way around a circle in opposite directions) so all Hamiltonians are homotopic to either ±*σ*_{z}. A different way of distinguishing between these two cases is to use a homotopy invariant: Consider the parity of the number of times the interior of the curve crosses the negative *z* axis. This is invariant.

## V. CHANGING UNIT CELL PERMUTES HOMOTOPY CLASSES

^{2,28}Different choices of unit cell are related by a gauge transformation, Sec. II A,

For example, in the case of bond-reflection symmetry, the classes {±*σ*_{x}} are exchanged with {±*R*_{1}} under a gauge transformation. Although this transformation exchanges these two states, they are different in a finite system. Assuming that the system is made up of some number of complete unit cells, the Hamiltonians {±*σ*_{x}} have no edge states, while {±*R*_{1}} has an edge state, as was first discovered in the Su-Schrieffer-Heeger model.^{27}

## VI. FRAGILE AND NEW PHASES

The classification of Bloch Hamiltonians with two bands in 1D can be extended to Bloch Hamiltonians with more bands. Fixing the number of bands to be *n* and the number of occupied bands to be *m* < *n*, we are interested in the homotopy of loops of *n* × *n* invertible Hermitian matrices with *m* strictly negative and *n* − *m* strictly positive eigenvalues. The homotopic classification in the general case may differ from the classification of two bands: First, distinct phases in the two bands case may dissolve into one equivalence class upon the addition of bands. Second, new phases may appear.

### A. A fragile phase

^{29}if its homotopy class changes when it is embedded into a space of a higher dimension by enlarging the projection matrix onto the band by adding extra zero blocks in the following way:

*n*×

*n*real, periodic projection

*P*(

*k*) onto a single band:

*B*◦Θ

_{+}upon choosing Θ

_{+}=

*σ*

_{x}◦*.

*P*(

*k*) is a real symmetric matrix it can be diagonalized by an orthogonal transformation and its non-trivial eigenvector

*P*(

*k*) is continuous in

*k*, we may choose $k$ so that it is a continuous family for

*k*∈ [−

*π*,

*π*]. Then $k$ traces a continuous curve on the unit sphere in $Rn$. Since

*P*(−

*π*) =

*P*(

*π*) the curve is either a closed loop or one with antipodal end-points:

*n*≥ 2.

In the case *n* = 2 the curves can be classified further according to the winding number of $k$ on the circle, which is a whole number (for the symmetric class) or a half-integer for the anti-symmetric class, see Fig. 11. This can be related to the winding number of the curve *γ* representing the Hamiltonian: Since *H* is real, *γ* is in the *x*–*z* plane. One can check that *γ* winds twice as fast as the eigenvector, so its winding number is an integer. It is odd for an anti-symmetric eigenvector and even for a symmetric one.

For *n* ≥ 3 the vector $k$ lies on $Sn\u22121$ which is simply connected, so all integer winding can be undone, reducing to a single class for the symmetric case and a single class for the anti-symmetric case.

*P*(

*k*) are labeled by:

*γ*collapse to a single equivalence class, while those with an odd winding number collapse to a second equivalence class, once the dimension of the Hamiltonian is increased above 2.

### B. New phases

*R*, which can be either bond or site reflection. The points

*k*= 0,

*π*are special as

*H*(

*k*), for

*k*= 0,

*π*are also eigenvectors of

*R*. Define the index

*N*

_{−}(

*k*) ∈ {0, …,

*m*} as the number of negative states with negative parity:

*N*

_{−}is an invariant under deformations of

*H*(

*k*) that respect the parity and do not close the gap. (The number of negative energy states with eigenvalues +1 is not a separate invariant since the total number of negative eigenvalues is fixed, see Ref. 30).

In the case of *m* occupied bands one expects to find, in general, (*m* + 1)^{2} equivalence classes labeled by [*N*_{−}(0), *N*_{−}(*π*)] and the number of invariants to increase with the number of the bands.

It is instructive to examine the results in Table I where *m* = 1, from this perspective:

In the case of bond reflection

*R*_{k}=*σ*_{x}and*N*_{−}(*k*) ∈ {0, 1}. Table I indeed gives four equivalence classes, as it should.In the case of site reflection

*R*_{k}=*G*_{k}is the identity for*k*= 0 so automatically*N*_{−}(0) = 0 and the two equivalence classes are labelled by*N*_{−}(*π*) ∈ {0, 1} in agreement with Table I.

### C. Stable phases

Since the number of invariants may either shrink or grow when the number of bands increases, it is natural to introduce a notion of “stable invariants.” For sufficiently large (invertible) matrices where both the numbers of positive and negative eigenvalues are large, the topological classification often stops changing, and are called “stable classes” in K-theory. The “fragile phases” are the phases that exist only for a sufficiently small number of bands. We shall not discuss further this issue.

## VII. CONCLUDING REMARKS

Our aim here was to give an elementary invitation to the field of topological insulators by studying the homotopic classification of 2 × 2 simple (=non-degenerate) periodic matrices in one dimension with various symmetries. Using simple intuitive facts about the homotopy of curves, we have shown how various symmetries are expressed in the homotopic classification of Bloch Hamiltonians. We have also presented an introduction to the notions of fragile and stable topological phases.

## ACKNOWLEDGMENTS

We thank Jacob Shapiro for careful reading of an earlier version of the paper and many helpful suggestions. J.E.A. also thanks Johannes Kellendonk, Oded Kenneth, Roy Meshulam, Daniel Podolsky, Immanuel Bloch, and Raffaele Resta for useful discussions. A.M.T. acknowledges the ISF for support under the Grant No. ISF 1939/18.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Joseph E. Avron**: Conceptualization (equal); Formal analysis (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Ari M. Turner**: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

### APPENDIX A: VARIOUS SYMMETRY COMPOSITIONS

#### 1. Chiral symmetry: *C*_{+}◦Θ_{+}

*composition*of Particle-Hole and time reversal symmetry. As such it is a unitary map that preserves

*k*,

_{+}=

*U*

_{t}◦ $*$ and

*C*

_{+}=

*U*

_{c}◦ $*$. Note that, in order for chiral symmetry to lead to an interesting classification, one must choose

*U*

_{t}≠

*U*

_{c}. If

*U*

_{t}=

*U*

_{c}, this would have the unfortunate consequence that $U=UcUc*=UcUc\u2020=1$ and chiral symmetry would imply

*H*(

*k*) = 0.

*H*(

*k*) with winding

*n*is homotopic to

*R*

_{n}of Eq. (4.4). It follows that chiral Bloch Hamiltonians are homotopic to

#### 2. Multiple symmetries *C*_{+} ∧ Θ_{+}

*γ*to the

*x*–

*y*plane and time reversal forces the anchoring points to lie in the

*x*–

*z*plane. It follows that the anchoring points at 0,

*π*must lie on a punctured

*x*-axis. Although

*γ*and −

*γ*have the same winding numbers and can be rotated into one another by applying a

*π*rotation about the

*z*-axis, they are homotopically distinct since such a rotation does not respect the anchoring condition. This doubles the homotopy classes

#### 3. Composite symmetry *B*◦Θ_{+}

*composition*of time reversal and bond reflection, one has

*γ*lies in the punctured

*x*–

*y*plane, and there is no constraint on the anchoring points at

*k*= 0,

*π*. The equivalence classes are labelled by their winding number and all Bloch Hamiltonians with winding

*n*are homotopic to the matrix

*R*

_{n}. This gives the ninth line in Table I:

#### 4. Multiple symmetries *B* ∧ Θ_{+}

*γ*is anchored on the punctured

*x*-axis. It follows that $\gamma \u0302$ is distinguished by its winding, and the anchoring point

**x**

_{0}= (±1, 0, 0). The equivalence classes are represented by the matrices

#### 5. Multiple symmetries *S* ∧ Θ_{+}

*x*

_{k}=

*x*

_{−k}and

*y*

_{k}= −

*y*

_{−k}; i.e.,

*λ*

_{k}=

*λ*

_{−k}. Also,

*λ*

_{±π}= 0 by Eq. (4.7).

*γ*

_{−}is anchored at −

*π*on the punctured

*z*-axis. The arc can be contracted to the point it is anchored to on the north or to the south pole while respecting the constraint, by contracting it along itself in the

*z*−

*λ*space. I.e., for

*k*< 0,

### APPENDIX B: COMPARISON WITH THE STANDARD CLASSIFICATION

The standard classification of topological insulators is concerned with the ten types of symmetry that do not have spatial symmetries other than translations. It is made from all the possible combinations of time reversal and Particle-Hole symmetry. The standard classification and the two-band classification are compared in the table below (Table II).

In the two cases marked by $*$ in first column the two band classification gives twice as many equivalence classes as in the standard classification. The extra $Z2$ is related to where the end-points of the Hamiltonian curve

*γ*_{−}are anchored rather than the topology of the whole curve, so it is usually regarded as a “weak” index.In three cases, marked by − in the second column, no comparison is made as there are no gapped 2-band Hamiltonian with Θ

_{−}symmetry.

. | Symmetry . | Two-band classification . | Standard classification . |
---|---|---|---|

A | None | 0 | 0 |

AIII | Chiral | $Z$ | $Z$ |

AI | Θ_{+} | 0 | 0 |

BDI* | Θ_{+} and C_{+} | $Z2\xd7Z$ | $Z$ |

D* | C_{+} | $Z2\xd7Z2$ | $Z2$ |

DIII^{†} | Θ_{−} and C_{+} | Gapless | $Z2$ |

AII^{†} | Θ_{−} | Gapless | 0 |

CII^{†} | Θ_{−} and C_{−} | Gapless | $Z$ |

C | C_{−} | 0 | 0 |

CI | Θ_{+} and C_{−} | 0 | 0 |

. | Symmetry . | Two-band classification . | Standard classification . |
---|---|---|---|

A | None | 0 | 0 |

AIII | Chiral | $Z$ | $Z$ |

AI | Θ_{+} | 0 | 0 |

BDI* | Θ_{+} and C_{+} | $Z2\xd7Z$ | $Z$ |

D* | C_{+} | $Z2\xd7Z2$ | $Z2$ |

DIII^{†} | Θ_{−} and C_{+} | Gapless | $Z2$ |

AII^{†} | Θ_{−} | Gapless | 0 |

CII^{†} | Θ_{−} and C_{−} | Gapless | $Z$ |

C | C_{−} | 0 | 0 |

CI | Θ_{+} and C_{−} | 0 | 0 |

For these symmetries, there are no fragile phases, which would have also caused there to be more phases in our classification than in the standard classification. Fragile phases do occur when there is reflection symmetry, see Sec. VI A.

## REFERENCES

*Physics and Mathematics of Quantum Many-Body Systems*

_{2}-index of symmetry protected topological phases with time reversal symmetry for quantum spin chains

*Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics*