Homotopy of periodic 2 × 2 matrices

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 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 P_t(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)

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 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.

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.

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).

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.

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 $γ∈R3/{0}$ 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.

$R3/{0}$ 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.

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 $γ̂$ 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.

$γ̂−$ 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).

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.

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.

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 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 +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.

For example, in the case of bond-reflection symmetry, the classes {±σ_x} 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 {±σ_x} 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 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.

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 $Sn−1$ 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.

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.

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.

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.

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.