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.

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 disorder19,20 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 pumping9 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 P0(k) and P1(k) are said to be homotopic if there is a family of projections Pt(k), jointly continuous respecting the symmetry, that interpolates between P0(k) and P1(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)

FIG. 1.

A periodic array with two atoms per unit cell. The red ellipse shows a unit cell.

FIG. 1.

A periodic array with two atoms per unit cell. The red ellipse shows a unit cell.

Close modal

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 insulators3,7,8 for the Altland–Zirnbauer symmetry classes5 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.

TABLE I.

In the first column Θ± stands for time-reversal with Θ±2=±1, similarly for Particle-Hole symmetry C±. B stands for bond-reflection and S for site reflection. ∧ means that both symmetries are present and ◦ that only the product is a symmetry. C+◦Θ+ is chiral symmetry. In the second column, σx, σy and σz denote the Pauli matrices, * is complex conjugation, and Gk is the diagonal unitary of Eq. (2.10). The third column gives a representative of each homotopy class. {σx}, {σy}, and {σz} represent the trivial (=null-homotopic) Bloch matrix associated with a lattice of disconnected unit cells. The choice of Pauli matrix is not always canonical (see the relevant sections). Rn is the unitary given in Eq. (4.4). The entries in the first section of the table describe generating symmetries. The entries below describe examples of composite symmetries and multiple symmetries.

SymmetryConstraintHomotopy classes
None None {σz
Θ+ H(k) = H*(−k{σz
Θ H(k) = σyH*(−k)σy Gapless 
C+ H(k) = −σzH*(−k)σz ±σx,±R1 
C H(k) = −σyH*(−k)σy {σz
B H(k) = σxH(−k)σx σx, ±R1
S H(k)=GkH(k)Gk* σz
C+◦Θ+ H(k) = −σzH(k)σz {Rn|nZ} 
B◦Θ+ H(k) = σxH*(k)σx {Rn|nZ} 
S◦Θ+ H(k)=GkH*(k)Gk* σz
S ∧ Θ+ H(k)=GkH(k)Gk*=H*(k) σz
B ∧ Θ+ H(k) = σxH(−k)σx = H*(−k{±Rn|nZ} 
C+ ∧ Θ+ H(k) = −σzH*(−k)σz = H*(−k{±Rn|nZ} 
C ∧ Θ+ H(k) = −σyH*(−k)σy = H*(−k{σz
SymmetryConstraintHomotopy classes
None None {σz
Θ+ H(k) = H*(−k{σz
Θ H(k) = σyH*(−k)σy Gapless 
C+ H(k) = −σzH*(−k)σz ±σx,±R1 
C H(k) = −σyH*(−k)σy {σz
B H(k) = σxH(−k)σx σx, ±R1
S H(k)=GkH(k)Gk* σz
C+◦Θ+ H(k) = −σzH(k)σz {Rn|nZ} 
B◦Θ+ H(k) = σxH*(k)σx {Rn|nZ} 
S◦Θ+ H(k)=GkH*(k)Gk* σz
S ∧ Θ+ H(k)=GkH(k)Gk*=H*(k) σz
B ∧ Θ+ H(k) = σxH(−k)σx = H*(−k{±Rn|nZ} 
C+ ∧ Θ+ H(k) = −σzH*(−k)σz = H*(−k{±Rn|nZ} 
C ∧ Θ+ H(k) = −σyH*(−k)σy = H*(−k{σz
Consider a one-dimensional, single particle Hamiltonian acting on the Hilbert space
2(Z)C2
(2.1)
This represents a chain of unit cells, each hosting a two dimensional Hilbert space, e.g., a spin or a pair of atoms; see Fig. 1. The basis states are j,a labeled by jZ and a ∈ {0, 1}.
FIG. 2.

The two red ellipses correspond to two different choices of unit cell. Changing the unit cell is a gauge transformation (=diagonal unitary) acting on H(k).

FIG. 2.

The two red ellipses correspond to two different choices of unit cell. Changing the unit cell is a gauge transformation (=diagonal unitary) acting on H(k).

Close modal
FIG. 3.

The one dimensional Brillouin zone is a circle parameterized by the angular variable k. The time-reversal, Particle-Hole and reflection symmetry, relate k with −k and leave invariant the two points k = 0 and k = π.

FIG. 3.

The one dimensional Brillouin zone is a circle parameterized by the angular variable k. The time-reversal, Particle-Hole and reflection symmetry, relate k with −k and leave invariant the two points k = 0 and k = π.

Close modal
A one-particle lattice Hamiltonian is a self-adjoint infinite dimensional matrix with elements
j,bHi,a=i,aHj,b*
(2.2)
A Bloch Hamiltonian is, by definition, translation invariant, i.e.,
j,bHi,a=0,bHij,a.
(2.3)
Translation invariance implies that 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.,
j0,bHj,a<
(2.4)
Locality, Eq. (2.4), implies that the (reduced) Bloch Hamiltonian H(k) defined by
Hba(k)=jj,bH0,aeikj
(2.5)
is a continuous periodic matrix-valued function of k. H(k) is Hermitian for each k since
Hba(k)=jj,bH0,aeikj=Hab*(k)
(2.6)
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.
Changing the unit cell permutes the basis vectors, i.e.,
Gj,b=jb,b,b{0,1},Z
(2.7)
See Fig. 2 for an illustration.
For the case of = 1, H transforms by
j,bGHGi,a=j+b,bHi+a,a
(2.8)
and H(k) transforms by
GkH(k)Gkb,a=Hba(k)eik(ba).
(2.9)
The action of Gk in k-space is implemented by a diagonal, k-dependent, unitary 2 × 2 matrix:
Gk=100eik
(2.10)

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.

A traceless Hermitian 2 × 2 matrix can be written as
H=xσ
(2.11)
where σ = (σx, σy, σz) is a vector of Pauli matrices. The matrix is simple (non-degenerate) if
x>0.
(2.12)
The space of simple 2 × 2 (traceless) Hermitian matrices is Euclidean 3-space with the origin removed, which is homotopic to the 2-sphere. A gapped 2 × 2 Bloch Hamiltonian is then represented by a closed curve γ in Euclidean 3-space with the origin removed (Fig. 3):
γ={xk|πkπ}R3/{0}.
(2.13)
Time-reversal, Particle-Hole, and reflection symmetries all relate H(k) with H(−k) and therefore xk with xk. 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=γγ+
(2.14)
where
γ={xk|πk0},γ+={xk|0kπ}
(2.15)
with γ± related by the symmetry.

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).

Time-reversal Θ is an anti-unitary map. This is a consequence of the presence of i in the Heisenberg equation
Ȧ=i[H,A]
(3.1)
where A, H are operators. Since {Θ,ddt}=0 and [Θ, H] = 0 it follows that {Θ, i} = 0 and so Θ is anti-linear.
Now for the problem studied in this article, where the operators act in Fock space, it is useful to represent time-reversal as an operation on single-particle states. By definition, time reversal maps particle creation to particle creation (the operation that maps creation to annihilation is Particle-Hole transformation), although a creation operator for a given state may map to a creation operator for some other state:
Θa(f)Θ=aΘ(f).
(3.2)
The bold and ordinary Θ’s refer respectively to the transformation of the creation operator and the transformation of the wave function associated to it. To derive the properties of the transformation Θ on the single particle wave functions from the properties of the many-body Θ, conjugate the anticommutation relation ⟨g|f⟩ = {a(g), a(f)} by Θ:
Θg|fΘ={Θa(g)Θ,Θa(f)Θ}={a(Θg),a(Θf)}=Θg|Θf
(3.3)
Since 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
Θ=U*
(3.4)
with U unitary and * complex conjugation.
One reasonably assumes that reversing time twice produces no observable effect, i.e., it is a pure phase. Then it follows that, since Θ commutes with Θ2, the only phase consistent with the anti-unitarity of Θ is ±1, i.e.,:
Θ2=UU*=±1U=±Ut
(3.5)
When Θ2=±1, the symmetry will be denoted Θ±. Under unitary change of bases W
ΘWΘW=WUWt*
(3.6)
U then undergoes a congruence transformation.
We assume that 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
ΘH(k)Θ1=UH*(k)U
(3.7)
Hence, H(k) is time-reversal invariant if
H(k)=UH*(k)U
(3.8)
for an appropriate unitary U = ±Ut.

1. Time reversal Symmetry Θ+

For the 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(θ,ϕ,α)=eiα[1cosθ+isinθ(σxcosϕ+σzsinϕ)]
(3.9)
U has a symmetric square root U=U(θ/2,ϕ,α/2). It follows that U is congruent to the identity:
U=U1(U)t.
(3.10)
This means that we can change to a basis in which U=1, so
Θ+=*
(3.11)
Then
H(k)=H*(k)
(3.12)
which translates to a loop γ with
zk=zk,xk=xk,yk=yk.
(3.13)
γ is symmetric under reflection in y. Furthermore 3.13 implies that the end-points of γ are in the xz plane.We will thus consider the homotopies of the anchored curve γ (see Fig. 4).
FIG. 4.

The curve γ associated with time-reversal symmetric Hamiltonians is anchored on the punctured xz plane where the red dot represents the origin. The rest of the loop is fixed by symmetry of reflection in the plane.

FIG. 4.

The curve γ associated with time-reversal symmetric Hamiltonians is anchored on the punctured xz plane where the red dot represents the origin. The rest of the loop is fixed by symmetry of reflection in the plane.

Close modal

2. Time reversal symmetry Θ

There is a unique 2 × 2 anti-symmetric unitary matrix U up to multiplication by a scalar. Since an overall phase is for free in an anti-unitary transformation, we may choose U = σy
Θ=σy*,σy=0ii0
(3.14)
whose action on H(k) is
ΘH(k)Θ1=σyH*(k)σy
(3.15)
A 2 × 2 matrix H(k) is time reversal symmetric if
xk=xk
(3.16)
Since this implies
x0=xπ=0
(3.17)
the Bloch Hamiltonian is gapless. This type of time reversal lies outside the gapped two bands framework and shall not be considered further.
Particle-hole transformation C maps particle creation to particle annihilation:
Ca(f)C=aCf
(3.18)
The particle creation operator 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
C2=±1
(3.19)
just like time-reversal. C± denotes the symmetry corresponding to C2=±1.

1. Particle-hole symmetry

Particle-hole symmetry says that for any eigenstate of the single particle Hamiltonian 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)(k)+(CHC1)(k)=0
(3.20)
In contrast, the many-particle Hamiltonian H commutes with C. To see this write
H=ϵn,kan,kan,k
(3.21)
and then, formally denoting the mirror of band n by −n,
CHC=ϵn,kan,kan,k=ϵn,kan,kan,k=ϵn,k{an,k,an,k}an,kan,k=Hϵn,k=0=H
(3.22)
It follows from Eq. (2.6), the anti-unitarity of C and Eq. (3.20) that a Bloch Hamiltonian has Particle-Hole symmetry if
H(k)=(CHC1)(k)=UH*(k)U;C=U*.
(3.23)

2. Particle-Hole Symmetry C+

By Sec. III A 1 any anti-unitary transformation whose square is 1 is equivalent to any another. Pick U = σz, i.e.,
C=σz*
(3.24)
The symmetry of γ corresponding to
H(k)=σzH*(k)σz
(3.25)
is
zk=zk,xk=xk,yk=yk
(3.26)
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 = σz 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.

FIG. 5.

The curve γ associated with Particle-Hole symmetric Hamiltonians with C2=1 is anchored on the punctured x-axis. γ+ is fixed by the symmetry as the π rotation of γ around the x-axis.

FIG. 5.

The curve γ associated with Particle-Hole symmetric Hamiltonians with C2=1 is anchored on the punctured x-axis. γ+ is fixed by the symmetry as the π rotation of γ around the x-axis.

Close modal

3. Particle Hole Symmetry C

By Sec. III A 2 we have:
C=σy*
(3.27)
whose action on H(k) is
CH(k)C1=σyH*(k)σy.
(3.28)
So a 2 × 2 matrix H(k) is Particle-Hole symmetric if
H(k)=σyH*(k)σy=H(k)
(3.29)
The loop γ corresponding to Eq. (3.29) is self-retracing
xk=xk
(3.30)
x0 and xπ can then be anywhere in R3/{0}.
Bond reflection (see Fig. 6) permutes the basis vectors
Bj,a=j,a1,B=B
(3.31)
where ⊕ is addition modulo 2. The transformation reflects unit cells, and exchanges the “atoms” in a unit cell. B is an involution, B2=1.
FIG. 6.

Bond reflection symmetry is a reflection about the green dashed line.

FIG. 6.

Bond reflection symmetry is a reflection about the green dashed line.

Close modal
The action of bond-reflection on the Hamiltonian in the coordinate basis is
j,bBHBi,a=j,b1Hi,a1
(3.32)
It follows from Eq. (2.5) that the action in k-space is
(BHB)b,a(k)=Hb1,a1(k)
(3.33)
In matrix form:
(BHB)(k)=zkxk+iykxkiykzk=σxH(k)σx
(3.34)
H(k) is bond-reflection symmetric if
H(k)=σxH(k)σx
(3.35)
The loop γ corresponding to Eq. (3.35) is symmetric under a rotation by π about the x-axis:
zk=zk,xk=xk,yk=yk
(3.36)
and is anchored to the punctured x-axis, see Fig. 5.
Reflection about a site (see Fig. 7) permutes the basis vectors by
Sj,a=ja,a,S=S
(3.37)
S is an involution that mixes different unit cells. Its action on H in real space is given by
j,bSHSi,a=jb,bHia,a
(3.38)
By Eq. (2.5) site reflection acts in k-space by
(SHS)b,a(k)=Hb,a(k)eik(ba)
(3.39)
and in matrix form
(SHS)(k)=GkH(k)Gk,
(3.40)
where Gk is given in Eq. (2.10). Hence, H(k) is site-reflection symmetric if
H(k)=GkH(k)Gk
(3.41)
The symmetry of the loop corresponding to Eq. (3.41) is a relation between γ and γ+:
zk=zk,xkyk=cosksinksinkcoskxkyk
(3.42)
γ is anchored at one end, k = ±π, to the punctured z-axis
x±π=y±π=0,|z±π|>0
(3.43)
FIG. 7.

Site reflection symmetry is a reflection about the green dashed line.

FIG. 7.

Site reflection symmetry is a reflection about the green dashed line.

Close modal

Example 3.1.
A gapped Bloch Hamiltonian that is an example for many of the phases discussed here is the Su-Schrieffer-Heeger model27 
H(k)=vt1+t2eikt1+t2eikv,v>0
(3.44)
This has a gap if (t1t2)2 + v2> 0. It has bond inversion symmetry if v = 0 and site inversion symmetry if t1 = t2, as defined by Eqs. (3.35) and (3.41).

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 γR3/{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 = γγ+ 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.

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 σz (for example), which represents a periodic array of isolated cells.

The corresponding many-body ground state is a (formal) pure product state
j0j
(4.1)
Being homotopic to a pure product state is taken as the defining property of a trivial phase. This gives the first line in Table I.

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

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

FIG. 8.

γ with particle-hole symmetry. The blue spherical curve is anchored at x = ±1. The red one is anchored at x = −1.

FIG. 8.

γ with particle-hole symmetry. The blue spherical curve is anchored at x = ±1. The red one is anchored at x = −1.

Close modal
FIG. 9.

For site inversion symmetry, the curve γ̂ is anchored at either pole. Curves anchored at the north pole are homotopic to the north pole. And curves anchored at the south pole are homotopic to the south pole. This shows that the space of loops is made of two disconnected sets.

FIG. 9.

For site inversion symmetry, the curve γ̂ is anchored at either pole. Curves anchored at the north pole are homotopic to the north pole. And curves anchored at the south pole are homotopic to the south pole. This shows that the space of loops is made of two disconnected sets.

Close modal
In the case that the anchoring points do not coincide the end-point at k = π can be brought to the end-point at k = 0 by applying a rotation around the y-axis Ry(ϕ) (since both lie on the unit circle in the xz plane). The following homotopy changes the curve continously so that the end-points at π and 0 coincide at the end:
x̂k(t)=Ry(kϕt/π)x̂K.
(4.2)
This respects the gap condition, since a rotation preserves the length of vectors. Hence γ̂ is homotopic to a point. Finally, since the equator in the xz 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 {σz}. This accounts for the second line in Table I.

1. Particle-Hole symmetry C+

By Sec. III B 2 the closed loop 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 γ̂ on the unit sphere, anchored at the poles:
x̂0,x̂π=±(1,0,0)
(4.3)
When the anchoring points coincide, γ̂ is homotopic to a point. In the case that the anchoring points are antipodal, γ̂ can be continuously shortened to a geodesic. This means that γ̂ is homotopic to a circle.
In conclusion, particle-hole symmetric Hamiltonians have four distinct equivalence classes represented by the four matrices:
{±σx,±R1},whereRn=0einkeink0
(4.4)
(RR by a π rotation about the x axis.) The four elements are distinguished by the combinations of signs of the anchoring points on x.
The Hamiltonians ±σx describe a chain with disconnected cells so the ground states are pure product states
j+j,jj,±=0±12
(4.5)
representing trivial phases, while the Hamiltonians ±R1 when solved on a finite chain turn out to have edge states at the ends.

2. Fermionic Particle-Hole symmetry C

By Sec. III B 3 the loop γ is self-retracing and is not anchored. Thus it can be contracted to a point along itself, so it is null-homotopic. This gives the fifth line in Table I.

By Sec. III C the curve γ is anchored on the punctured x-axis. This is the same scenario as in Sec. IV C 1 and hence there are four equivalence classes labeled by the four matrices
{±σx,±R1}
(4.6)
This gives the sixth line in Table I.
By Sec. III D the loop γ of Eq. (3.42) is anchored at one point, k = ±π, to the punctured z-axis:
x±π=y±π=0,|z±π|>0
(4.7)
Since 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,
xk(t)=x(1t)ktπ,t[0,1]
(4.8)
It follows that γ̂ is homotopic to either the north or south pole, and hence site symmetric Bloch Hamiltonians are homotopic to either one of two matrices
{±σz}
(4.9)
The corresponding (many-body) ground state represents a trivial phase as it is (formally) a pure product state. This gives the seventh line in Table I.
It is sometimes useful to consider the loop γ as a world-line in the four dimensional space S1×R3 where the first coordinate is k. A case in point is the composed symmetry S◦Θ+, where the Bloch Hamiltonian satisfies the constraint
H(k)=GkH*(k)Gk,Gk=100eik
(4.10)
This constrains the coordinates (xk, yk):
xkiyk=(xk+iyk)eik
(4.11)
and gives no constraint on zk. The constraint implies
(xk,yk)=λkcosk2,sink2,
(4.12)
where λk is a real-valued function of k.
The loop γ may be viewed as the world line in the 3-dimensional manifold MSΘ+
MSΘ+=k,λcosk2,λsink2,zλ,zR,πkπ
(4.13)
embedded in the 4-dimensional space S1×R3. γ is not a closed curve in the (λ, z) coordinates because the λz plane rotates relative to the xy plane by π as k goes from −π to π. The end-point xπ = xπ has different coordinates in the (λ, z) plane
λπ=λπ,zπ=zπ
(4.14)

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

FIG. 10.

γ in the λz plane. The start and end points (λ±π, z±π) are the two cyan circles that are mirror images. The red circle is the puncture in the plane. The curve can be retracted to σz by retracting the end points symmetrically relative to the z axis. The curve can not be retracted to −σz.

FIG. 10.

γ in the λz plane. The start and end points (λ±π, z±π) are the two cyan circles that are mirror images. The red circle is the puncture in the plane. The curve can be retracted to σz by retracting the end points symmetrically relative to the z axis. The curve can not be retracted to −σz.

Close modal

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 2n. 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 +σz and the curves with odd winding number can be contracted to −σz (by moving the two end-points half-way around a circle in opposite directions) so all Hamiltonians are homotopic to either ±σz. 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.

The labeling of the homotopy classes depends, in general, on the choice of unit cell.2,28 Different choices of unit cell are related by a gauge transformation, Sec. II A,
H(k)GkH(k)Gk,Z
(5.1)
Under such a transformation the equivalence classes transform as
σzσz,σxR,RnRn+.
(5.2)
This does not mean that the phases are equivalent, just that the labeling of the phase depends on how the unit cell is chosen.

For example, in the case of bond-reflection symmetry, the classes {±σx} are exchanged with {±R1} 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 {±σx} have no edge states, while {±R1} has an edge state, as was first discovered in the Su-Schrieffer-Heeger model.27 

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 nm 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 phase (with a given symmetry) is fragile29 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:
P(k)P(k)000
(6.1)
As an example of a fragile topological phase consider an n × n real, periodic projection P(k) onto a single band:
P(k)=P*(k)=P(k+2π)=P2(k),TrP(k)=1
(6.2)
The reality condition may be viewed to be a consequence of the symmetry constraint B◦Θ+ upon choosing Θ+ = σx◦*.
Since P(k) is a real symmetric matrix it can be diagonalized by an orthogonal transformation and its non-trivial eigenvector
P(k)k=k
(6.3)
can be chosen to be real vectors of unit length in Rn. The eigenvalue equation then determines k uniquely, up to a ±1 phase ambiguity. Since 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:
k=π=±k=π
(6.4)
The ± alternative divides the space of curves k on the sphere into two classes: A symmetric class where k=π=k=π and an anti-symmetric class where k=π=k=π. This holds for any 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 xz 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.

FIG. 11.

The blue solid semi-circle and the dashed semi-circle are homotopically distinct when the embedding space is the plane but are equivalent in three dimensions.

FIG. 11.

The blue solid semi-circle and the dashed semi-circle are homotopically distinct when the embedding space is the plane but are equivalent in three dimensions.

Close modal

For n ≥ 3 the vector k lies on Sn1 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.

Thus, we find that the periodic real projections P(k) are labeled by:
Z for n=2,Z2 for n3
(6.5)
Hence all real Bloch bands with an even winding number of 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.
As an example of new phases that emerge when the number of bands is increased consider the reflection symmetry R, which can be either bond or site reflection. The points k = 0, π are special as
[R,H(k)]=0,k{0,π}
(6.6)
Hence, eigenstates of 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(k)=ϵn<01ψnRψn2,H(k)ψn=ϵnψn,k{0,π}
(6.7)
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 Rk = σx and N(k) ∈ {0, 1}. Table I indeed gives four equivalence classes, as it should.

  • In the case of site reflection Rk = Gk 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.

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.

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.

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.

The authors have no conflicts to disclose.

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).

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

1. Chiral symmetry: C+◦Θ+

Chiral symmetry (aka sublattice symmetry) is the composition of Particle-Hole and time reversal symmetry. As such it is a unitary map that preserves k,
H(k)UH(k)U,U=UcUt*
(A1)
where Θ+ = Ut ◦ * and C+ = Uc ◦ *. Note that, in order for chiral symmetry to lead to an interesting classification, one must choose UtUc. If Ut = Uc, this would have the unfortunate consequence that U=UcUc*=UcUc=1 and chiral symmetry would imply H(k) = 0.
Choosing Θ+ and C+ as in Eqs. (3.11) and (3.24), H(k) has chiral symmetry if
H(k)=σzH(k)σz
(A2)
This says that the curve γ lies in a punctured xy plane, which is not simply connected and has its winding number as a topological invariant
Winding(γ)Z
(A3)
H(k) with winding n is homotopic to Rn of Eq. (4.4). It follows that chiral Bloch Hamiltonians are homotopic to
{Rn|nZ}
(A4)

2. Multiple symmetries C+ ∧ Θ+

Particle-Hole and time reversal symmetry together also imply symmetry under their composition, chiral symmetry. Chiral symmetry restricts γ to the xy plane and time reversal forces the anchoring points to lie in the xz 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
Rn±Rn.
(A5)
To summarize, the homotopy equivalence classes are given by
{±Rn|nZ}
(A6)

3. Composite symmetry B◦Θ+

For Hamiltonians with only one symmetry, the composition of time reversal and bond reflection, one has
H(k)=(BΘHΘ1B)(k)=σxH*(k)σx
(A7)
which implies
zk=0
(A8)
so γ lies in the punctured xy 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 Rn. This gives the ninth line in Table I:
{Rn|nZ}.
(A9)

4. Multiple symmetries B ∧ Θ+

Now, in addition to Eqs. (A7) and (A8) bond reflection gives the additional constraint that γ is anchored on the punctured x-axis. It follows that γ̂ is distinguished by its winding, and the anchoring point x0 = (±1, 0, 0). The equivalence classes are represented by the matrices
{±Rn|nZ}
(A10)
This gives the twelfth line in the Table I.

5. Multiple symmetries S ∧ Θ+

The conditions are that
(xk,yk)=λkcosk2,sink2,(λk,zk)=(λk,zk),λ±π=0
(A11)
The first condition follows from the site-reflection condition Eq. (3.42), as in Sec. IV F. Time-reversal implies that xk = xk and yk = −yk; i.e., λk = λk. 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 zλ space. I.e., for k < 0,
zk(t)=z(1t)ktπ,(xk,yk)=λ(1t)ktπcosk2,sink2
(A12)
There are two homotopy classes determined by the anchoring points:
{±σz}
(A13)
This gives the fourth line from the bottom of Table I.

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.

TABLE II.

Comparison between standard classification of topological insulator classification in one dimension and classification for 2 × 2 Hamiltonians, see Table I. The classifications agree except that in the standard classification a weak Z2 index is ignored in two cases (marked *), corresponding to the choice of ± in the Hamiltonians, and whenever there is Θ- symmetry (marked ), there is nothing to classify because there are no 2 × 2 Hamiltonians at all that are gapped.

SymmetryTwo-band classificationStandard classification
None 
AIII Chiral Z Z 
AI Θ+ 
BDI* Θ+ and C+ Z2×Z Z 
D* C+ Z2×Z2 Z2 
DIII Θ and C+ Gapless Z2 
AII Θ Gapless 
CII Θ and C Gapless Z 
C 
CI Θ+ and C 
SymmetryTwo-band classificationStandard classification
None 
AIII Chiral Z Z 
AI Θ+ 
BDI* Θ+ and C+ Z2×Z Z 
D* C+ Z2×Z2 Z2 
DIII Θ and C+ Gapless Z2 
AII Θ Gapless 
CII Θ and C Gapless Z 
C 
CI Θ+ and C 

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.

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